diff --git a/content/GA_2_3/CSIRO_Alt_seas_inc.txt b/content/GA_2_3/CSIRO_Alt_seas_inc.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9163035c7c484890620f3144d5748f5219d000e0
--- /dev/null
+++ b/content/GA_2_3/CSIRO_Alt_seas_inc.txt
@@ -0,0 +1,331 @@
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diff --git a/content/GA_2_3/P7.ipynb b/content/GA_2_3/P7.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..b3ec5ec1534654e8148c9a3573bb9ac9d20131fa
--- /dev/null
+++ b/content/GA_2_3/P7.ipynb
@@ -0,0 +1,899 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Project 7: Signal Processing\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\"\\>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\"\\>\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): week 2.3. Due: Friday December 1, 2023.*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Analyzing cantilever-beam accelerations and global Mean Sea-Level measurements\n",
+ "\n",
+ "This project covers the week on Signal Processing (week 2.3).\n",
+ "\n",
+ "As a warming up you will create and analyze some elementary signals yourself, and next, you will carry out frequency domain analyses on two given data-sets, namely acceleration measurements of a Cantilever-Beam experiment, and (in optional Task 10) Global Mean Sea-Level measurements.\n",
+ "\n",
+ "Most of the Tasks in this notebook consist of both coding, producing a plot, and answering (open) questions. Typically, as you work your way through the Tasks, you can often re-use code, or part of it, from earlier Tasks and assignments."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "id": "0491cc69"
+ },
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\"> <p>In many of the code blocks below, template code to create figures is provided. Note that there is a lot of code missing, and one line of <code>YOUR_CODE_HERE</code> does not imply that only one line of code is missing!</p></div>"
+ ]
+ },
+ {
+ "attachments": {
+ "verticalcantileverbeam.png": {
+ "image/png": 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IBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYuKwrpzAgGLisK6cwIBi4rCunMCAYu68871yOIGeEoz7tX/7riWj6Ftnwebfk82vJ5tOXzaMvnsU3LRebfAIt+pFUH3+/odQ/8sOan9eo/X2TVwfc7et0DP6yP6Vl9+I+XWXXw/Y5e98CP6tWntei9ZtXB9zt63QM/rNfpWS1Muebg+x297oEf1vyxsehprTr4fkeve+CHNT+tfuuJ58fGx6Knterg+x297oEf1fwNYNGXuVUH3+/odQ/8sPqt52nMHxuLntaqg+939LoHfljz01r0sbHq4Psdve6BH9b8sbHoaa06uN96rqvfep5Gv/U8jfljY9HTWnXw/Y5e98APa35aiz42Vh18v6PXPfDDmj82Fj2tVQf3W8+VzU/Lf7zMqoPvd/S6B35U/dbzNOaPjUVPa9XB9zt63QM/rKZ8GvPTWvQNYNXB9zt63QM/rH7reRb91vM0mvJpzB8bi57WqoPvd/S6B35Y89Pqt5548zeARU9r1cH3O3rdAz+q/mXl0+i3nqcxf2wselqrDr7f0ese+GHNT2vRx8aqg+939LoHfljzx8aip7Xq4H7rua5+63ka/dbzNOaPjUVPa9XB9zt63QM/rPlpLfrYWHXw/Y5e98APa/7YWPS0Vh3cbz3X1W89T6Pfep7G/LGx6GmtOvh+R6974Ic1P61FHxurDr7f0ese+GH5tJZ9A1h18P2OXvfAD2vVN4B1Xx/udnS/9Rzpt56H4u/QZU9r1cH3O3rdAz+u6ffowqe16uD7Hb3ugR/Y+MQWfwNYdfD9jl73wA/sfc03gFUH3+/odQ9ctaU/f/4PDvMXtjrnOloAAAAASUVORK5CYII="
+ }
+ },
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Data Acquisition System\n",
+ "\n",
+ "A data acquisition (DAQ) system usually consists of four components:\n",
+ "\n",
+ "<img src=\"https://gitlab.com/JelleKnibbe/public-files/-/raw/main/SP_Practical/DAQ.png\" style=\"margin:auto\"/>\n",
+ "\n",
+ "<ol>\n",
+ " <li>A sensor transforms a physical signal into a small voltage. The transformation is often frequency dependent or nonlinear, and can show drift or bias.</li>\n",
+ " <li>Signal conditioning electronics (often included in the sensor housing) filter unwanted frequencies, such as the 50 Hz line frequency caused by power plant generators, using low-pass or notch filters. In addition the sensor voltage may be amplified.</li>\n",
+ " <li>An analog-to-digital (A/D) converter samples the analog voltage with a certain rate, such as 100 Hz. The A/D converter can measure analog voltages within a certain range, the measurement range. The measured voltage is quantized with a certain resolution, such as 14-bit, which means that the full measurement range is divided into $2^{14}=16,384$ discrete intervals. Together, the resolution and measurement range determine the precision. Usually the sample rate, the measurement range, and the sample resolution are configurable. Sampling and quantizing together turns an analog signal into a digital signal.</li>\n",
+ " <li>A digital computer reads the sampled data from the A/D converter at specific times, and allows for further processing, analysis and storage of the (digital) signal.</li>\n",
+ "</ol>\n",
+ "\n",
+ "The result is that an analog, physical signal, is turned into a series of numbers (samples of the signal, in the time domain), ready for processing and analysis in a digital computer.\n",
+ "\n",
+ "\n",
+ "\n",
+ "One of the data sets that you will analyze during this MUDE Q2 project was acquired by an accelerometer in a smartphone, on top of a vertical cantilever-beam. The accelerometer measures the side-ward accelerations, expressed in m/s<sup>2</sup>, and sampled at 50 Hz. Detailed information about the experiment and the sensor can be found in \"Experimental evaluation of smartphone accelerometer and low-cost dual frequency GNSS sensors for deformation monitoring\", by Alexandru Lapadat, Christian Tiberius and Peter Teunissen, Sensors 2021, 21, 7946, https://doi.org/10.3390/s21237946.\n",
+ "\n",
+ "A quick impression of the test setup can be gained by watching the short video [Cantilever Beam Experiment](https://youtu.be/o4moRwvlBLU?si=aKelBMWm3HB2Of26) (1 minute).\n",
+ "\n",
+ "A theoretical description of the motion of the smartphone accelerometer fixed to the cantilever beam is presented in Appendix A. Pulling the beam at the tip and releasing it, results in a (nearly) horizontal side-ward motion of the smartphone, and the (horizontal) position can be described by a *damped harmonic* as a function of time. Consequently, also the first and second derivative with respect to time, the velocity and acceleration as a function of time as the smartphone will measure it, are harmonics.\n",
+ "\n",
+ "### Basic sinusoid signal\n",
+ "\n",
+ "You will start with first creating (and analyzing) a few simple signals yourself. In the first few Tasks of this project, we will take the damping ratio zero and use a fairly short measurement time (duration), and hence the acceleration measured by the smartphone is a plain (undamped) sinusoid as shown in Appendix A (a stationary signal). In the following Tasks we also add a phase offset $\\varphi$ to the plain sinusoid:\n",
+ "\n",
+ "$$\n",
+ "x(t) = A \\sin(2 \\pi f_c t + \\varphi)\n",
+ "$$\n",
+ "\n",
+ "The result is taken from the last equation in Appendix A. We consider here acceleration, though for convenience, we omit the dots on top of the $x(t)$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "%matplotlib inline"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 1:</b>\n",
+ "\n",
+ "- Create a time-array starting at $t=0$ s, ending at $t=5.0$ s, with a sampling rate of 100 Hz. Hint: what is the number of samples $N$? And what should then be the last value in the time-array? Note that, for example, with $N=10$ samples at a sampling rate of $f_s = 100$ Hz, we have signal samples at times $t=0.00, 0.01, 0.02, ... , 0.08, 0.09$ seconds, hence $T_{meas}= N \\Delta t =0.1$ seconds (the sample-and-hold convention).\n",
+ "- Create a sinusoidal signal $x(t) = A \\sin(2 \\pi f_c t + \\varphi)$, with amplitude $A=1.0$ Volt, carrier frequency $f_c=1.0$ Hz, and initial phase $\\varphi = 5$ degrees, to be converted into radians.\n",
+ "- Make a plot of the signal against time. Note that this is strictly a *sampled signal* $x_n$ rather than $x(t)$, but since we use a rather high sampling rate, the signal shown is close to continuous in time.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "id": "0491cc69"
+ },
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\"> <p>We will give you the answers in this code cell for free!</p></div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "T_meas = 5\n",
+ "f_s = 100 #sampling rate [Hz]\n",
+ "\n",
+ "t_vec = np.arange(0, T_meas, 1 / f_s) # ends at 4.99, length 500 according to the sample-and-hold convention\n",
+ "\n",
+ "A = 1 \n",
+ "f_c = 1 \n",
+ "phi = 5 * np.pi / 180\n",
+ "x = A * np.sin(2 * np.pi * f_c * t_vec + phi)\n",
+ "\n",
+ "plt.plot(t_vec, x, color='b', label='signal')\n",
+ "plt.xlabel('$t \\: [s]$')\n",
+ "plt.ylabel('$x(t) \\: [V]$')\n",
+ "plt.legend(loc='upper right')\n",
+ "plt.title(fr'Sinusoidal signal with $A$={A} V, $f_c$={f_c} Hz and initial phase $\\phi$={phi:.3f} °')\n",
+ "plt.grid()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### The Discrete Fourier Transform (chapter 5)\n",
+ "\n",
+ "To discover which frequencies are contained in the signal $x(t)$, we can transform it from the time domain to the frequency domain using the Fourier transform:\n",
+ "\n",
+ "$$\n",
+ "X(f) = \\int_{-\\infty}^{\\infty} x(t) e^{-j 2\\pi f t} dt\n",
+ "$$\n",
+ "\n",
+ "Which can then be expressed in terms of magnitude and phase:\n",
+ "\n",
+ "$$\n",
+ "X(f) = |X(f)|e^{j\\theta(f)}\n",
+ "$$\n",
+ "\n",
+ "Where $|X(f)|$ plotted against frequency $f$ is called the *magnitude spectrum* (and practically often referred to as amplitude spectrum).\n",
+ "\n",
+ "The set and number of sinusoids required to approximate or re-create a given signal $x(t)$ depends on the shape of that signal. Of course, when the signal is a pure sinusoid, we only need one term. For an example, consider a 2 Hz sinusoidal signal and its magnitude spectrum below:\n",
+ "\n",
+ "<img src=\"https://gitlab.com/JelleKnibbe/public-files/-/raw/main/SP_Practical/SignalSpectrumExample.png\" style=\"margin:auto\" width=800/>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "You can see a sharp spike in the magnitude plot (which has two logarithmic axes, and we only plot the spectrum for positive frequencies $f>0$). A continuous time sinusoid with amplitude $A$ will produce a magnitude of $\\frac{A T_{meas}}{2}$ in the amplitude spectrum (with $T_{meas}$ the *measurement time* or duration), rather than $\\frac{A}{2}$ which you would expect (in a double-sided spectrum). This is because of the limited time duration of the input signal, which actually implies multiplication of $x(t)$ by a *rectangular time window* $w(t)=\\Pi(\\frac{t}{T_{meas}})$. Since the sinusoid has an amplitude of $A=1$ V, and was measured for $T_{meas}=5$ seconds, the magnitude becomes 2.5. The magnitude at all other frequencies is very small, approximately $10^{-15}$, or approximately zero.\n",
+ "\n",
+ "The Fourier transform is a continuous-time operation, mapping the continuous time-domain to the continuous frequency domain. Usually, we don't have continuous-time signals, but discrete time signals $x_n$ sampled at a certain sampling rate. The discrete Fourier Transform (DFT) is the discrete-time equivalent of the continuous time-Fourier transform:\n",
+ "\n",
+ "$$\n",
+ "X_k = \\sum_{n=0}^{N-1}x_ne^{-j2\\pi kn/N}, k=0,1,...,N-1\n",
+ "$$\n",
+ "\n",
+ "Where $X_k$ is the sequence of frequency domain samples. NumPy contains a function to perform the DFT using a fast numerical algorithm, the Fast Fourier Transform:\n",
+ "\n",
+ "<code>X_discr = np.fft.fft(x)</code>\n",
+ "\n",
+ "For more information on the function see [here](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft.html).\n",
+ "\n",
+ "The discrete-time signal $x_n$ and the frequency domain samples $X_k$ returned by Python are of equal length ($N$ samples in, $N$ samples out). When you compare the equations of the DFT with the continuous Fourier Transform, you will see that in the continuous equation we integrate over time $t$, while in the DFT *no information* about the time or sample rate is required (there is no $\\Delta t$, as noted at the end of Chapter 5). This results in a *scale factor* in the spectrum when the DFT of signals with different sample rates are compared. To solve this, we have to multiply the DFT as calculated by NumPy by the sample time interval $\\Delta t = 1/f_s$ to restore the time dimension and obtain a *discrete approximation of the continuous Fourier Transform*. To account for the measurement duration as described above, we have to divide by $T_{meas} = N \\Delta t$. So, in total, we have to divide the NumPy result by the number of samples $N$:\n",
+ "\n",
+ "<code>X_cont = np.fft.fft(x) / N</code>\n",
+ "\n",
+ "The DFT only contains information (the complex value of $X_k$) at discrete frequencies, known as the *analysis frequencies*, which are integer multiples of the frequency resolution $f_0$. If we have a sampled signal $x_n$ consisting of $N$ samples, sampled at $f_s$ Hz with $f_s=1/ \\Delta t$, the frequency resolution is:\n",
+ "\n",
+ "$$\n",
+ "f_0 = \\frac{1}{T_{meas}} = \\frac{1}{N \\Delta t} = \\frac{f_s}{N}\n",
+ "$$\n",
+ "\n",
+ "The first value of the complex vector $X_n$ returned by NumPy is the mean of the time-domain signal corresponding to $f=0$ Hz. So, the frequency vector corresponding to the discrete Fourier transform starts at zero and has $N$ elements, the analysis frequencies:\n",
+ "\n",
+ "<code>f_vec = np.arange(0, f_s, f_0)</code>\n",
+ "\n",
+ "With task 2 you visualize the entire output as you get it from the `np.fft.fft`, hence for frequencies $[0,f_s)$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 2:</b> \n",
+ "\n",
+ "- Calculate the Fast Fourier Transform of the signal you generated in Task 1. Remember: the signal has 500 samples and thus does *not* end at exactly 5 second.\n",
+ "- Create the frequency vector $f$ with the analysis frequencies.\n",
+ "- Plot the modulus of the Fourier Transform against the frequency $f$ using a log scale on both axes. Use plot markers to see at which frequencies the Fourier transform was calculated.\n",
+ "\n",
+ "On top of that, answer to the following questions:\n",
+ "<ol>\n",
+ " <li>Describe the amplitude spectrum.</li>\n",
+ " <li>What is the magnitude of $X(f)$ at $f$ = 1 Hz?</li>\n",
+ " <li>Do you notice anything peculiar about the amplitude spectrum?</li>\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE\n",
+ "\n",
+ "# Write your own code or use the template below to create your figure(s)\n",
+ "\n",
+ "# f, axes = plt.subplots(1,1,figsize=(10,5))\n",
+ "\n",
+ "# axes[0].plot(YOUR_CODE_HERE, YOUR_CODE_HERE, 'x', color='b', label='Fourier transform')\n",
+ "# axes[0].loglog()\n",
+ "# axes[0].set_xlabel('$f \\: \\: [Hz]$')\n",
+ "# axes[0].set_ylabel('$|X(f)| \\: [V]$')\n",
+ "# axes[0].grid()\n",
+ "# axes[0].set_title('Log/Log')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Symmetry\n",
+ "\n",
+ "\n",
+ "Due to symmetry properties of the DFT, the following holds for a real signal $x(t)$:\n",
+ "\n",
+ "$$\n",
+ "|X_k| = |X_{-k}|\n",
+ "$$\n",
+ "\n",
+ "Which means that the modulus of the Fourier coefficients is symmetric about $f=0$. The NumPy <code>fft</code> command appends the Fourier coefficients for $k<0$ to the right side of the Fourier coefficients for $k \\ge 0$. This is called the *two-sided* spectrum.\n",
+ "\n",
+ "Since we are working with real signals, the symmetry property is valid, and we can just ignore the coefficients for $k<0$. This means that we consider the following range in the frequency domain:\n",
+ "\n",
+ "$[0, \\frac{f_s}{2}]$ for even values of $N$\n",
+ "\n",
+ "$[0, \\frac{f_s}{2})$ for odd values of $N$\n",
+ "\n",
+ "In other words, we are only considering frequencies up to half of $f_s$. This is easily implemented in Python using the floor division operator <code>//</code>:\n",
+ "\n",
+ "<code>X_cont = X_cont[:N//2]</code>\n",
+ "\\\n",
+ "<code>f_vec = f_vec[:N//2]</code>\n",
+ "\n",
+ "If you're unfamiliar with this, consider that <code>A//B</code> returns the number of times <code>B</code> 'fits into' <code>A</code>. In principle, this will always return an integer, so it can be easily used for indexing an array. In this specifc example, <code>N//2</code> will return $\\frac{N}{2}$ for even $N$ and $\\frac{N-1}{2}$ for odd N, which is exactly what we want. See the example below."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "N = 100\n",
+ "print(f'{N} floor divided by 2: {N//2}')\n",
+ "print(f'{N+1} floor divided by 2: {(N+1)//2}')\n",
+ "print(f'{N-1} floor divided by 2: {(N-1)//2}')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "\n",
+ "<p>\n",
+ "<b>Task 3:</b> \n",
+ "\n",
+ "- Repeat Task 2 with different measurement times $T_{meas}$ for the signal. Use measurement times such that the oscillation fits *exactly* 1 time, 5 times and 20 times.\n",
+ "- Plot the amplitude spectrum for all three measurement times, **only for positive frequencies**, in separate graphs (log-log scale) with the same domains and answer to the following questions:\n",
+ "<ol>\n",
+ " <li>What is the effect of changing $T_{meas}$ on the frequency range in the amplitude spectrum? Does the highest analysis frequency change?</li>\n",
+ " <li>Does the frequency resolution change?</li>\n",
+ " <li>Does the magnitude of at the peaks change?</li>\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE\n",
+ "\n",
+ "# Write your own code or use the template below to create your figure(s)\n",
+ "\n",
+ "plt.figure(figsize=(12,4))\n",
+ "for i, T_meas in enumerate(YOUR_CODE_HERE):\n",
+ " YOUR_CODE_HERE\n",
+ "\n",
+ " plt.subplot(1, 3, i+1)\n",
+ " \n",
+ " YOUR_CODE_HERE\n",
+ " \n",
+ " plt.grid()\n",
+ " plt.tight_layout()\n",
+ "plt.legend()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Aliasing (chapter 4)\n",
+ "\n",
+ "So far we have been analyzing a signal with a fairly low frequency, just $1$ or $2$ Hz. Now suppose, due to some unexpected source, our signal contains a second sinusoid with a much higher frequency. This is often caused by the line frequency ($50$ Hz) due to power plant generators providing the $220$ V current in regular buildings, or by electrical interference. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "\n",
+ "<p>\n",
+ "<b>Task 4:</b> \n",
+ "\n",
+ "- Create a time-array starting $t=0$, with a sampling rate of 100 Hz, and ending such that a sinusoid with a frequency of 1 Hz fits exactly 5 times in the measurement time.\n",
+ "\n",
+ "- Re-create the signal $x(t) = A \\sin(2 \\pi f_c t + \\varphi)$ from the first Task, with amplitude $A=1.0$ V, carrier frequency $f_c=1.0$ Hz, and initial phase $\\varphi = 5$ degrees, to be converted into radians.\n",
+ "\n",
+ "- Add to this signal a second sinusoid with a frequency of $80$ Hz and amplitude of $0.1$ V (and zero initial phase).\n",
+ "\n",
+ "- Create a plot of the signal against time, and of the amplitude spectrum as in the previous Tasks. \n",
+ "\n",
+ "Then answer to the following questions:\n",
+ "<ol>\n",
+ " <li>What do you see in the frequency plot? Are there peaks? How many? Where?</li>\n",
+ " <li>Does this match what you see in the time plot?</li>\n",
+ " <li>Does changing the measurement time (duration) help?</li>\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE\n",
+ "\n",
+ "# Write your own code or use the template below to create your figure(s)\n",
+ "\n",
+ "# plt.figure(figsize=(12,6))\n",
+ "# plt.subplot(211)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.grid()\n",
+ "# plt.title('Time signal')\n",
+ "# plt.legend()\n",
+ "\n",
+ "# plt.subplot(212)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.loglog()\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.grid()\n",
+ "# plt.tight_layout()\n",
+ "# plt.title('Amplitude spectrum')\n",
+ "# plt.legend()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The effect seen in Task 4 is called *aliasing*, and the cause is that if your sample rate is too low for the signals you're measuring, you will not capture their oscillation period sufficiently - with at least two samples per cycle. Instead, the signal will appear at a much lower frequency, as you can see in the figure below. Here $f_c = 5$ Hz, and the signal was sampled with $f_c = 7$ Hz, and based on the discrete time samples we incorrectly conclude that there is a frequency component at $2$ Hz (at the end of Chapter 4).\n",
+ "\n",
+ "<img src=\"https://gitlab.com/JelleKnibbe/public-files/-/raw/main/SP_Practical/NewSlide38.png\" style=\"margin:auto\" width=800/>\n",
+ "<img src=\"https://gitlab.com/JelleKnibbe/public-files/-/raw/main/SP_Practical/NewSlide51.png\" style=\"margin:auto\" width=800/>\n",
+ "\n",
+ "So, how fast do we need to sample to capture an $80$ Hz signal?"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 5:</b> \n",
+ "\n",
+ "Repeat Task 4 with different sample rates: $110, 150, 160,$ and $200$ Hz. Plot the signal and amplitude spectrum for each one (you might want to use a loop). \n",
+ "\n",
+ "Then answer the following questions:\n",
+ "<ol>\n",
+ " <li>At what frequency does the (aliased) 80 Hz signal appear in the spectrum, for the above values of $f_s$?</li>\n",
+ " <li>Can you figure out the relationship between the sample rate and the frequency of the aliased signal?</li>\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE\n",
+ "\n",
+ "# Write your own code or use the template below to create your figure(s)\n",
+ "\n",
+ "# for YOUR_CODE_HERE:\n",
+ " \n",
+ "# YOUR_CODE_HERE\n",
+ " \n",
+ "# plt.figure(figsize=(12,6))\n",
+ "# plt.suptitle(f'$f_s = {YOUR_CODE_HERE}$ Hz')\n",
+ "# plt.subplot(211)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.grid()\n",
+ "# plt.legend()\n",
+ " \n",
+ "# plt.subplot(212)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.loglog()\n",
+ "# plt.grid()\n",
+ "# plt.tight_layout()\n",
+ "# plt.legend()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "As you probably figured out, the sample rate needs to be faster than twice the largest frequency in the signal. In other words, you can measure signals with a frequency up to half the sample rate without aliasing becoming a problem. The frequency above which aliasing occurs (half the sampling rate) is called the *Nyquist frequency* (Chapter 4)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Damped vibrations\n",
+ "\n",
+ "We now know how to create an amplitude/magnitude spectrum. We've seen that we get aliasing if our signal has components at frequencies higher than the *Nyquist frequency* (and leakage if the measurement time is too short). Now, let's consider the damped vibration aceleration signal that an accelerometer would measure (see *Appendix A*). In this case the damping ratio is not equal to zero, unlike in Tasks 1 to 5."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 6:</b> \n",
+ "\n",
+ "- Create a time-array starting at $t=0$ s, ending at $t = 50$ s, with a sampling rate of $100$ Hz.\n",
+ "- Create a damped harmonic acceleration signal $x(t) = \\frac{x(0)}{\\sqrt{1-\\zeta^2}} e^{-\\zeta \\omega_0 t} \\sin(\\omega_d t)$ with $\\zeta = 0.05$, $\\omega_0 = 10 \\pi$ rad/s (corresponding to $5$ Hz), $\\omega_d = \\omega_0 \\sqrt{1-\\zeta^2} = 9.987 \\pi$ rad/s, and initial displacement $x(0)=1$ (for convenience the initial phase of the signal is kept to zero).\n",
+ "- Create a plot of the signal against time, and of the amplitude spectrum as in the previous Tasks.\n",
+ "\n",
+ "Then answer the following questions:\n",
+ "<ol>\n",
+ " <li>Do you see any changes in the time plot, compared to the earlier plot? Describe them!</li>\n",
+ " <li>What is the dominant frequency of the signal now?</li>\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE\n",
+ "\n",
+ "# # Write your own code or use the template below to create your figure(s)\n",
+ "\n",
+ "# plt.figure(figsize=(12,6))\n",
+ "# plt.subplot(211)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.grid()\n",
+ "# plt.legend()\n",
+ "\n",
+ "# plt.subplot(212)\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.loglog()\n",
+ "# YOUR_CODE_HERE\n",
+ "# plt.grid()\n",
+ "# plt.tight_layout()\n",
+ "# plt.legend()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Cantilever-beam acceleration measurements\n",
+ "\n",
+ "Now you're ready to start analyzing the cantilever-beam acceleration measurements.\n",
+ "\n",
+ "Read in the data-file: *cantileverbeam_acc50Hz.csv*.\n",
+ "\n",
+ "This dataset contains 5 minutes of measurements with a sampling rate of $50$ Hz (a total of $N=15001$ samples). The first column in the file contains the UTC time of day (in seconds), the second column contains the measured acceleration (in m/s<sup>2</sup>)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Detrending\n",
+ "\n",
+ "In this project the signal of interest is the damped, harmonic motion of the cantilever-beam. In addition the measurements may contain (unwanted) effects of the sensor (think of an offset, due to imperfect manufacturing and/or prior calibration, or a drift over time, for instance due to temperature changes in the electronics during the experiment). By detrending we mean to remove such effects, prior to our spectral analysis, so that we can start with a nice, zero mean signal.\n",
+ "\n",
+ "There are built-in functions in Python to remove such an offset and/or trend (and you're free to use them). But, earlier in the MUDE, in week 1.3, you learned about least-squares parameter estimation, and that's what you can apply to do the detrending (and actually built-in functions just apply the very same principle). Next week, on Time Series Analysis, you will actually be covering (again) the estimation of a trend in a time series of measurements.\n",
+ "As a re-cap for week 1.3: when a series of observations $y_1, …, y_m$ (in our case with $m=N$) is supposed or expected to exhibit a functional linear trend (a straight line in terms of a graph), this can be modelled as\n",
+ "\n",
+ "$$\n",
+ "\\mathbb{E} = \\begin{pmatrix} \\begin{bmatrix} Y_{1} \\\\ Y_{2} \\\\ \\vdots \\\\ Y_{m} \\end{bmatrix} \\end{pmatrix} \n",
+ " = \n",
+ " \\begin{bmatrix} 1 & t_1 - t_1 \\\\ 1 & t_2 - t_1 \\\\ \\vdots & \\vdots \\\\ 1 & t_m - t_1 \\end{bmatrix} \n",
+ " \\begin{pmatrix} x_{1} \\\\ x_{2} \\end{pmatrix}\n",
+ "$$\n",
+ "\n",
+ "with $x_1$ the offset at time $t_1$ (rather than $t=0$), and $x_2$ the slope of the line. The two unknown parameters in this vector $x$ can be estimated through (unweighted) least-squares, $\\hat{x}=(A^T A)^{-1} A^T y$, and next the residuals are obtained as $\\hat{\\epsilon}=y-\\hat{y}=y-A\\hat{x}$. The residuals are the 'left-over part' of the observations, once the (estimated) trend has been taken out; these residuals are of interest for further spectral analysis!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 7:</b> \n",
+ "\n",
+ "- For the cantilever-beam acceleration measurements, perform a least-squares estimation according to the above model, and report the estimated offset and slope of the trend.\n",
+ "- The detrended acceleration measurements, hence the elements of vector $\\hat{\\epsilon}$ are the input to your spectral analysis, so from here on, we denote them by $x(t)$ (in continuous time), and by $x_n$ (in discrete time) with $n=0,…,N-1$.\n",
+ "- Make a plot of the input signal as a function of time, hence of the detrended accelerations.\n",
+ "\n",
+ "Report the estimated offset and slope of the trend (i.e. numerical values).\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "df = pd.read_csv('cantileverbeam_acc50Hz.csv', header=0)\n",
+ "\n",
+ "t = np.array(df['time']) #\n",
+ "dat = np.array(df['acceleration']) #\n",
+ "\n",
+ "N = len(t)\n",
+ "\n",
+ "# observation record length (as N*dt, according to sample-and-hold convention)\n",
+ "T = (t[N-1] - t[0])*N/(N - 1)\n",
+ "dt = T/N\n",
+ "plt.figure()\n",
+ "plt.plot(t, dat, color='b', label='acceleration signal')\n",
+ "plt.xlabel('time [s]')\n",
+ "plt.ylabel('acceleration [m/s2]')\n",
+ "plt.title('Vertical cantilever beam acceleration')\n",
+ "plt.legend()\n",
+ "\n",
+ "YOUR_CODE_HERE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 8:</b> \n",
+ "\n",
+ "Previously, in Task 3, you computed and plotted the magnitude spectrum $|X_k|$ of a signal for positive frequencies $f>0$ (putting a lot of attention to correctly labelling the horizontal frequency axis of the plot!), and now, with the material of Chapter 6 on spectral estimation, you will estimate the power spectral density $S$ of the signal through the periodogram, which is just: $S(k\\Delta f)=\\frac{|X_k|^2}{T}$ (in [W/Hz] when $x_n$ is a voltage signal; and $|X_k|$ being the result straight from the <code>np.fft</code>, multiplied by sampling interval $\\Delta t$), for frequency $k\\Delta f$, with frequency resolution $\\Delta f=\\frac{1}{T}$, and $k=0,…,N-1$ (hence, pretty much the same procedure as with the magnitude spectrum, though just taking the square of the modulus, and dividing by $T$).\n",
+ "\n",
+ "Compute and plot the periodogram (if you prefer, feel free to use a linear scaling of the axes here, rather a log-log). Please, pay attention to correctly labelling the axes, and stating dimensions of the quantities along the axes!\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 9:</b> \n",
+ "\n",
+ "Answer the following questions:\n",
+ "<ol>\n",
+ " <li>Report the damped natural frequency (in Hertz) of this one-degree-of-freedom (1DOF) mechanical system. Does it match the motion of the beam shown in the cantilever-beam video?</li>\n",
+ " <li>The acceleration was measured at quite a high sample rate of $50$ Hz. What is the minimum sampling frequency to correctly identify the damped natural frequency in the periodogram?</li>\n",
+ "</ol>\n",
+ "<p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Task 10: Global Mean Sea-Level (GMSL) data (optional)\n",
+ "\n",
+ "Repeat the steps with Tasks 7-9, but now with the Global Mean Sea Level data set. Data is from the Australia's National Science Agency (CSIRO): [GMSL data](https://www.cmar.csiro.au/sealevel/sl_hist_last_decades.html). \n",
+ "\n",
+ "These data result from nearly 3 decades of satellite altimetry (with satellite missions such as TOPEX/Poseidon and the Jason-series). The first column contains the time tag or epoch (in decimal years), the second column is the global mean sea level (in mm). There is one measurement per month (monthly average, so that for instance tide-effects are averaged out, and the measurement typically refers to the middle of the month, hence 1993.042 is mid January in 1993). The single monthly measurement is the global mean sea level, so, the average of the entire world.\n",
+ "\n",
+ "The sampling frequency $f_s = 12$ per year ($\\Delta t = 1/12 \\sim 0.083$ year), and there are $N=331$ measurements in total."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "data = pd.read_csv('CSIRO_Alt_seas_inc.txt', names=['month', 'sl'])\n",
+ "data.head()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# create time-array, time relative to t0 [yr] (epoch-time of the first observation;\n",
+ "# t0=1993.042 refers to mid January 1993)\n",
+ "t = data.iloc[:, 0] - data.iloc[0, 0]\n",
+ "\n",
+ "# number of observations\n",
+ "N = len(t)\n",
+ "\n",
+ "# observation record length (as N * dt, according to sample-and-hold convention)\n",
+ "T = (t[N - 1] - t[0]) * N / (N - 1)\n",
+ "\n",
+ "# Delta t [yr]; dt = T/N = (N*dt)/N\n",
+ "dt = T / N\n",
+ "\n",
+ "# observed sea-level height\n",
+ "y = data.iloc[:,1]\n",
+ "\n",
+ "# plot observed time-series, as it is, versus epoch-time in [year]\n",
+ "plt.plot(data.iloc[:,0],y, color='b', label='sea level')\n",
+ "plt.xlabel('time [yr]')\n",
+ "plt.ylabel('sea-level height [mm]')\n",
+ "plt.title('Global Mean Sea-Level (GMSL) rise')\n",
+ "plt.legend()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>Task 10 (Optional!):</b> \n",
+ "\n",
+ "Detrend the data. \n",
+ "\n",
+ "Estimate and plot power spectral density (PSD), hence the periodogram, for the (detrended) global mean sea-level data. \n",
+ "\n",
+ "Identify the largest peak in the spectrum, what is the frequency, and can you come up with a physical explanation of this behaviour?\n",
+ "</ol>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%;\">\n",
+ "<p>\n",
+ "<b>End of task.</b>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**You may write your answer in this Markdown cell, but remember that it needs to be included in your `Report.md`.** "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#C8FFFF; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Appendix A: Cantilever-beam dynamics:</b> \n",
+ "\n",
+ "The dynamics of the smartphone suspended on a cantilever beam can be considered as the mass-spring-damper system shown below:\n",
+ "\n",
+ "<img src=\"https://gitlab.com/JelleKnibbe/public-files/-/raw/main/SP_Practical/mass_spring_damper.png\" style=\"margin:auto\" width=200/>\n",
+ "\n",
+ "The equation of motion of the deflection $x$ of this mass-spring-damper system (a damped harmonic oscillator) can be described by the following second order differential equation:\n",
+ "\n",
+ "$$\n",
+ "\\ddot{x}(t) + \\frac{c}{m} \\dot{x}(t) + \\frac{k}{m} x(t) =0\n",
+ "$$\n",
+ "\n",
+ "Where $\\ddot{x}(t)$, $\\dot{x}(t)$ and $x(t)$ are the acceleration, velocity and displacement as a function of time of the oscillating mass $m$ [kg] respectively. (with a unit transfer function, $x(t)$ describes the motion of the proof mass inside the smartphone accelerometer). For simplicity, we consider here the homogeneous equation, corresponding to free motion. The other parameters are the damping coefficient $c$ [kg/s] and spring constant $k$ [N/m]. The damping ratio can be obtained from the system parameters: $\\zeta = \\frac{c}{2 \\sqrt{mk}}$, which is dimensionless. The undamped natural frequency is $\\omega_0 = \\sqrt{\\frac{k}{m}}$ [rad/s]. The differential equation becomes:\n",
+ "\n",
+ "$$\n",
+ "\\ddot{x}(t) + 2 \\zeta \\omega_0 \\dot{x}(t) + \\omega_0^2 x(t) =0\n",
+ "$$\n",
+ "\n",
+ "For the under-damped case ($0 \\le \\zeta \\le 1)$ of our smartphone and assuming an initial zero tip velocity $\\dot{x}(t=0)=0$ (release from stand-still), the solution for the position as a function of time is given by:\n",
+ "\n",
+ "$$\n",
+ "x(t)=e^{-\\zeta \\omega_0 t} \\frac{x(0)}{\\sqrt{1-\\zeta^2}}\\sin(\\omega_d t + \\varphi)\n",
+ "$$\n",
+ "\n",
+ "Where $x(0)$ is the initial position $x(t=0)$, $\\omega_d$ is the damped natural frequency $\\omega_d = \\omega_0 \\sqrt{1-\\zeta^2}$, and the phase shift $\\varphi = \\arctan \\left( \\frac{\\sqrt{1-\\zeta^2}}{\\zeta} \\right)$. The sinusoid term represents the harmonic motion, and the exponential term represents the damping of that motion over time. Next, the velocity of the smartphone's oscillation can be derived as:\n",
+ "\n",
+ "$\\dot{x}(t) = e^{-\\zeta \\omega_0 t} \\frac{x(0)}{\\sqrt{1-\\zeta^2}}\\sin(\\omega_d t)$.\n",
+ "\n",
+ "The acceleration of the smartphone (which is what is being measured) is found as:\n",
+ "\n",
+ "$\\ddot{x}(t) = e^{-\\zeta \\omega_0 t} \\frac{x(0)}{\\sqrt{1-\\zeta^2}} \\sin (\\omega_d t - \\phi)$.\n",
+ "\n",
+ "Note that $\\omega_0$ and $\\omega_d$ are the angular frequencies expressed in radians per second. $\\ddot{x}(t)$ is a damped harmonic signal where the rate of damping is determined by the damping ratio $\\zeta$.\n",
+ "\n",
+ "In order to get a pure harmonic signal (as used in Tasks 1 to 5), set the damping ratio $\\zeta=0$, and optionally set the phase-shift $\\phi$ to zero as well, then the tip acceleration is given by the following simple sinusoidal expression:\n",
+ "\n",
+ "$$\n",
+ "\\ddot{x}(t)=x(0) \\sin(\\omega_0 t).\n",
+ "$$\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**End of notebook.**\n",
+ "<h2 style=\"height: 60px\">\n",
+ "</h2>\n",
+ "<h3 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; bottom: 60px; right: 50px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.4"
+ },
+ "widgets": {
+ "application/vnd.jupyter.widget-state+json": {
+ "state": {},
+ "version_major": 2,
+ "version_minor": 0
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/content/GA_2_3/P7_solution.pdf b/content/GA_2_3/P7_solution.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..d282036b53b58b10f4f11017c293f0e7cded846f
Binary files /dev/null and b/content/GA_2_3/P7_solution.pdf differ
diff --git a/content/GA_2_3/README.md b/content/GA_2_3/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..a563f132b4193780027514ffa76a015fce7013dc
--- /dev/null
+++ b/content/GA_2_3/README.md
@@ -0,0 +1,48 @@
+# Project 7: Signal Processing
+
+*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/).*
+
+
+## Project 7 Info
+
+The focus of this assignment is on signal processing, and specifically on spectral analysis (i.e. analysis of signals in the frequency domain).
+
+**Your primary objective is to complete all tasks in the notebook `P7.ipynb` and summarize your answers in `Report.md`.**
+
+### Overview of material
+
+- this `README.md` with instructions
+- `Report.md`, primary notebook in which you are supposed to copy your plots and include your answers to the questions (typically a short, one-line answer each time; though a simple 'yes' or 'no' is not sufficient, please include a short reasoning, justification or argumentation).
+- `P7.ipynb`, the secondary Jupyter notebook with description and tasks, to be used for actual coding
+- `cantileverbeam_acc50Hz.csv`, data file with acceleration measurements from the cantilever beam (tasks 7-9)
+- `CSIRO_Alt_seas_inc.txt`, data file with Global Mean Sea Level measurements (task 10; optional)
+- [Cantilever Beam Experiment Video](https://youtu.be/o4moRwvlBLU?si=aKelBMWm3HB2Of26): a short supplementary one minute movie illustrating the cantilever-beam experiment.
+
+We will use `pandas` to import the dataset, so make sure this is included in your `mude` environment before starting Jupyter Lab.
+
+## Grading
+
+This project will be graded with the following weights for each assessment category:
+
+* 0.40 for **Interpretation** and 0.40 for **Application**
+* 0.10 for **Documentation** and 0.10 for **Programming**.
+
+All 10 tasks are weighted equally (1 point each) and the maximum grade is 10 (points). Task 10 is optional; each group will receive 1 point by default for this one.
+
+### Submission and deadline
+
+- Submit your answers, together with any relevant plots, in the Markdown file `Report.md`. This is the primary document that will be used to determine your grade; however, the auxiliary files (e.g., `*.ipynb` or `*.py` files) may be checked in case something is not clear.
+- The deadline is to submit your work by making commits to your Group's GitLab repository by Friday at 12:30h.
+
+### Repository, Formatting and Static Check
+
+There is no static check for this project. Be sure to leave the outputs from your code cells in your `*.ipynb` file so that they are readable.
+
+You are always expected to provide well-formatted figures and Markdown text in your `Report.md` file, as well as logically organize any auxiliary files you may use (e.g., try to put your figures in a sub-directory, if there are a lot of them). If you run out of time it is OK if your `*ipynb` files do not run.
+
+**Importing figures into a Markdown file:** in Project 5 and 6 we noticed a lot of figures not rendering properly in the Markdown file (i.e., broken image links); this was typically due to spaces in the file name. To avoid rendering issues in your Markdown reports, we recommend *not* including spaces in the filename as a general rule (e.g., `my_image.png` instead of `my image.png`). If you need to include an image in Markdown with a space in it, replace the space with `%20`. For example, to show file `my image.png` you should use: ``
+
+**End of file.**
+
+<span style="font-size: 75%">
+© Copyright 2023 <a rel="MUDE Team" href="https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595">MUDE Teaching Team</a>, TU Delft. This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0 License</a>.
diff --git a/content/GA_2_3/Report.md b/content/GA_2_3/Report.md
new file mode 100644
index 0000000000000000000000000000000000000000..ce5b335d10de24a62535c595798f8b9500d8e52c
--- /dev/null
+++ b/content/GA_2_3/Report.md
@@ -0,0 +1,70 @@
+# Project 7 Report: Signal Processing
+
+*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/).*
+
+**YOUR GROUP NAME HERE**
+
+
+## Primary Task
+
+**Complete the notebook `P7.ipynb`, copy the plots and write your answers to the questions below.** Typically a short, one-line answer is sufficient; though a simple 'yes' or 'no' is _not_ sufficient, please include a short reasoning, justification or argumentation.
+
+_You will be graded on the plots and answers provided in this file. You can delete the instructions and any other unnecessary text prior to submission._
+
+## Report Instructions
+
+Remember to use Markdown features to clearly indicate your answers for each question below.
+
+**Importing figures into a Markdown file:** in Project 5 and 6 we noticed a lot of figures not rendering properly in the Markdown file (i.e., broken image links); this was typically due to spaces in the file name. To avoid rendering issues in your Markdown reports, we recommend *not* including spaces in the filename as a general rule (e.g., `my_image.png` instead of `my image.png`). If you need to include an image in Markdown with a space in it, replace the space with `%20`. For example, to show file `my image.png` you should use: ``
+
+## Answers to Questions
+
+### Task 1
+
+_Your answer here._
+
+### Task 2
+
+_Your answer here._
+
+### Task 3
+
+_Your answer here._
+
+### Task 4
+
+_Your answer here._
+
+### Task 5
+
+_Your answer here._
+
+### Task 6
+
+_Your answer here._
+
+### Task 7
+
+_Your answer here._
+
+### Task 8
+
+_Your answer here._
+
+### Task 9
+
+_Your answer here._
+
+### Task 10 (optional)
+
+_Your answer here._
+
+
+## General Comments on the Assignment [optional]
+
+_Use this space to let us know if you encountered any issues completing this assignment (but please keep it short!). For example, if you encountered an error that could not be fixed in your Python code, or perhaps there was a problem submitting something via GitLab. You can also let us know if the instructions were unclear. You can delete this section if you don't use it._
+
+**End of file.**
+
+<span style="font-size: 75%">
+© Copyright 2023 <a rel="MUDE Team" href="https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595">MUDE Teaching Team</a>, TU Delft. This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0 License</a>.
diff --git a/content/GA_2_3/cantileverbeam_acc50Hz.csv b/content/GA_2_3/cantileverbeam_acc50Hz.csv
new file mode 100644
index 0000000000000000000000000000000000000000..771c7fd6a28f4e69e63bfa67631ad18bc5a0ea29
--- /dev/null
+++ b/content/GA_2_3/cantileverbeam_acc50Hz.csv
@@ -0,0 +1,15002 @@
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diff --git a/content/Week_2_3/Lecture_Slides.pdf b/content/Week_2_3/Lecture_Slides.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..cc5cb4496da8d829f715e9a833c8ae463d069116
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diff --git a/content/Week_2_3/PA11_iter_remoto.ipynb b/content/Week_2_3/PA11_iter_remoto.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..7c6d5f2086e0698d4cfdb2da0b51af6bed7ad7d5
--- /dev/null
+++ b/content/Week_2_3/PA11_iter_remoto.ipynb
@@ -0,0 +1,803 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "c96d6259-08d6-4289-aea2-589d67cdb5ee",
+ "metadata": {},
+ "source": [
+ "# Programming Assignment 11: iTer-remoto\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\" />\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\" />\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): Week 2.3. Due: complete this PA prior to class on Friday, Dec 1, 2023 (but better if you do it by Wed, Nov 29!).*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7a28e541-d2d0-48a9-abf6-7b73075c8fd3",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Overview of Assignment\n",
+ "\n",
+ "This assignment will address a few simple, but useful, topics related to indexing and iterating in Python, illustrated with an annotated stem figure. The content of this PA is simple, but will be used directly in the Wednesday workshop this week.\n",
+ "\n",
+ "Specifically, we will look at:\n",
+ "- iterables and iterable objects\n",
+ "- iterables `range`, `enumerate` and `zip`\n",
+ "- the modulo operator `%`\n",
+ "- plot type `stem`\n",
+ "\n",
+ "## Assignment Criteria\n",
+ "\n",
+ "**You will pass this assignment as long as your respository fulfills the following criteria:** \n",
+ "\n",
+ "- You have completed this notebook and it runs without errors"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "75681df5-7a73-469a-ad83-ebd05451e0b7",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b0584fa2-7f4b-4566-9217-e6ca58d0e191",
+ "metadata": {},
+ "source": [
+ "## Introduction\n",
+ "\n",
+ "Suppose we are writing code to summarize some simple data, perhaps a group of teams that each have a score for a particular game."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4d6bb586-3ade-473e-8dd7-19dd350fa3f6",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "07ed2fba-5f83-47d8-af7e-b3a83bebbbbf",
+ "metadata": {},
+ "source": [
+ "Suppose we want to list the names of each team. One typical approach is to loop over each item in the list:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "878ff8c2-fe3c-421a-b1d8-400450fc6bf5",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "N_teams = len(team)\n",
+ "print('The team names are:')\n",
+ "for i in range(N_teams):\n",
+ " print(f'{team[i]}')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "172fea58-dd93-4af0-a09c-f4129338ab54",
+ "metadata": {},
+ "source": [
+ "However, this is not the most efficient way to do this, as we need to define two extra variables: `N_teams` and the iteration index `i`. There is a better way! \n",
+ "\n",
+ "## Iterable Objects\n",
+ "\n",
+ "Python supports what is referred to as _iterable_ objects: generally, this means objects that can be easily used by Python's built-in and efficient iteration schemes. Examples are: list, tuple, dictionaries, and strings. You may be interested in reading about all of [the technical details here](https://docs.python.org/3/glossary.html#term-iterator); however, for this assignment the explanations here and examples below are sufficient. It is important to recognize this type of object exists in Python, so we can take advantage of it in our code, or so you can recognize the algorithms in other authors code.\n",
+ "\n",
+ "Let's see an iterable object in action. First, an easy way to test if an object is iterable is to use it as an argument of the `iter()` method:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fc48118b-014b-4044-95fc-787e110de380",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "iter(team)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "111ef874-1b80-45fc-bb2c-ce9f7982681b",
+ "metadata": {},
+ "source": [
+ "Other objects can also be made into an iterator, like numpy arrays:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ee686157-6923-4d2e-b74f-35ed1049c70f",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "iter(np.array([1, 5, 67]))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e4302934-4828-4dcd-bc3b-eafbc59ad199",
+ "metadata": {},
+ "source": [
+ "The cell above should return `iterator`, which indicates the object (argument) is iterable. An integer, however, is _not_ iterable. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "41269037-161e-4c50-9dd7-5346bfc41b7a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1:</b> \n",
+ " \n",
+ "Try running the code below to confirm that an integer is not iterable (an error will occur), then fix it by converting the argument to an iterable object. Experiment by turning the integer into a list, string and np.array. \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5b65450f-dd18-4456-ac6b-d4856ae47b23",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "iter(5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "47cf00b4-6946-44e6-bbeb-4176eb1e5674",
+ "metadata": {},
+ "source": [
+ "Great, you can make an iterator! But what do you _do_ with it? \n",
+ "\n",
+ "_Most of the time we simply use it in a `for` loop, but it is worthwhile to understand this object a bit more deeply._\n",
+ "\n",
+ "One simple way to understand an iterator is to imagine that it is a way of (efficiently!) turning your iterable object into something that can go sequentially through all its values. To do this it has two \"abilities\":\n",
+ "\n",
+ "1. the iterator knows its current value, and \n",
+ "2. the iterator knows its next value\n",
+ "\n",
+ "Let's try using the iterator by running the following cells:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f98b14ac-aa6b-4342-ba0f-faef9bc3f2b4",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "letters = 'abcde'\n",
+ "letter_iterator = iter(letters)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a48aadf-4e61-4c01-a23c-f0c28f7615e9",
+ "metadata": {},
+ "source": [
+ "Nothing new yet, but now watch what happens when we use the Python built-in command `next()`\n",
+ "\n",
+ "_Run the cell multiple times._"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a925cca3-ad71-4744-a6c8-7d578c7b69f0",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "next(letter_iterator)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fbc2e084-0032-42ec-9d21-9cec68d4e6ec",
+ "metadata": {},
+ "source": [
+ "There are two key things to note here:\n",
+ "\n",
+ "1. The iterator \"knows\" how to go through all of its values, and\n",
+ "2. Calls to `next` returns to value itself, _not the index!_\n",
+ "\n",
+ "This is a very useful feature, as we can see in this simple example below:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "59642a7a-df92-44e0-984d-f054f1e5162d",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "for i in letters:\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5629acaa-81b5-43e3-bd27-7713ad5b2ae2",
+ "metadata": {},
+ "source": [
+ "Obviously this is a simple `for` loop, but hopefully the explanation above indicates what is actually happening \"under the hood\" of the `for` loop:\n",
+ " \n",
+ "1. identifies `letters` as an iterable object\n",
+ "2. converts it into an iterator, and then it\n",
+ "3. finds the next value in the sequence and assigns it to variable `i`\n",
+ "4. stop once there is no next value\n",
+ " \n",
+ "This is a bit of a simplification of what happens in reality, but it gets the idea across.\n",
+ "\n",
+ "Now let's take a look at something you have probably used a lot: the `range()` method:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "16660b30-08ba-43dd-8ddd-d34f5fa8f094",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "print(type(range(5)))\n",
+ "print(range(5))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0531da9d-bec3-4d57-a653-0b1e58f54523",
+ "metadata": {},
+ "source": [
+ "Although it looks like it produces a tuple, `range` is a special iterable object that simply counts. As we can see it works the same as our `for` loop above:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "04019b3e-9853-449c-a230-4c9033f55311",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "for i in range(5):\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09db8005-1a9b-45b0-aac8-9821ea087fd9",
+ "metadata": {},
+ "source": [
+ "Hopefully this explains why `range` is such a useful feature for accessing sequential indexed elements of arrays.\n",
+ "\n",
+ "It turns out there are two more built-in Python methods that produce useful iterable objects: `enumerate` and `zip`. Let's take a look ath their doc strings."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0eb9605e-b4bf-4dc1-9a38-b64e78ff8563",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2:</b> \n",
+ " \n",
+ "Read the docstrings below for the two methods. Confirm that they produce an iterable, but compare them to observe how the input and output of each is different. Can you imagine why one could be more useful than another in certain situations?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "045f749d-1d12-4f70-954e-301ddcd9be32",
+ "metadata": {},
+ "source": [
+ "\n",
+ "`enumerate`:\n",
+ "\n",
+ "```Python\n",
+ "Init signature: enumerate(iterable, start=0)\n",
+ "Docstring: \n",
+ "Return an enumerate object.\n",
+ "\n",
+ " iterable\n",
+ " an object supporting iteration\n",
+ "\n",
+ "The enumerate object yields pairs containing a count (from start, which\n",
+ "defaults to zero) and a value yielded by the iterable argument.\n",
+ "\n",
+ "enumerate is useful for obtaining an indexed list:\n",
+ " (0, seq[0]), (1, seq[1]), (2, seq[2]), ...\n",
+ "```\n",
+ "\n",
+ "And `zip`:\n",
+ "\n",
+ "```Python\n",
+ "Init signature: zip(self, /, *args, **kwargs)\n",
+ "Docstring: \n",
+ "zip(*iterables, strict=False) --> Yield tuples until an input is exhausted.\n",
+ "\n",
+ " >>> list(zip('abcdefg', range(3), range(4)))\n",
+ " [('a', 0, 0), ('b', 1, 1), ('c', 2, 2)]\n",
+ "\n",
+ "The zip object yields n-length tuples, where n is the number of iterables\n",
+ "passed as positional arguments to zip(). The i-th element in every tuple\n",
+ "comes from the i-th iterable argument to zip(). This continues until the\n",
+ "shortest argument is exhausted.\n",
+ "\n",
+ "If strict is true and one of the arguments is exhausted before the others,\n",
+ "raise a ValueError.\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f7b34d40-e7c3-477c-9a78-68d26ea2dd81",
+ "metadata": {},
+ "source": [
+ "<br><br><br><br><br><br><br><br><br><br><br>\n",
+ "\n",
+ "The main takeaways, should be as follows:\n",
+ "\n",
+ "1. Yes, they both produce an iterator\n",
+ "2. `enumerate` takes one iterable object and returns indices along with the value\n",
+ "3. `zip` takes two iterable objects and returns their values\n",
+ "\n",
+ "Let's try them out by running the next cell. Does it behave as expected?"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "bdea9148-e2b0-4107-92b5-61a83de02efc",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "thing_1 = 'roberts_string'\n",
+ "thing_2 = [2, 3, 36, 3., 1., 's', 7, '3']\n",
+ "\n",
+ "test_1 = enumerate(thing_1)\n",
+ "print(f'We created: {test_1}')\n",
+ "print(next(test_1), next(test_1), next(test_1))\n",
+ "\n",
+ "test_2 = zip(thing_1, thing_2)\n",
+ "print(f'We created: {test_2}')\n",
+ "print(next(test_2), next(test_2), next(test_2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d72f609b-0167-40f4-8e8f-c4476acf6417",
+ "metadata": {},
+ "source": [
+ "Can you see the difference?\n",
+ "\n",
+ "Looking at them in a for loop will also illustrate what's going on:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f3687fdf-c949-44f7-b5df-961e63703d59",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print('First, enumerate:')\n",
+ "for i, j in enumerate(thing_1):\n",
+ " print(i, j)\n",
+ "print('\\nThen, zip:') \n",
+ "for i, j in zip(thing_1, thing_2):\n",
+ " print(i, j)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3eedb9f0-29f9-4fea-b9ab-742f40ea8813",
+ "metadata": {},
+ "source": [
+ "Now let's return to our teams from the beginning of this assignment: let's apply our knowledge of `enumerate` and `zip` to see if we can print out the points per team in an efficient way."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf6e780a-d153-4e68-b7f5-c4b83ffd564f",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 3:</b> \n",
+ " \n",
+ "Use <code>enumerate</code> to print out the summary of points per team according to the print statement.\n",
+ "<br><br>\n",
+ "<em>Hint: you only need to use one of the lists as an argument.</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "73ea50cb-5c87-4f84-b618-4dd9ae38cc0c",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]\n",
+ "\n",
+ "for YOUR_CODE_WITH_enumerate_HERE:\n",
+ " print(f'Team {} has {} points.')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ac1ef287-e43a-4c78-9536-bdba4828906f",
+ "metadata": {},
+ "source": [
+ "You may have noticed that `enumerate` is a bit awkward for this case, since we still need to define an unnecessary iteration index to access the team name. Let's see if `zip` makes things easier:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0bdfde09-88cf-4358-9af0-edc9c75acabf",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 4:</b> \n",
+ " \n",
+ "Use <code>zip</code> to print out the summary of points per team according to the print statement.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6bcfc8a9-57a1-41db-8449-fe689def8b5a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]\n",
+ "\n",
+ "for YOUR_CODE_WITH_zip_HERE:\n",
+ " print(f'Team {} has {} points.')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "edd3cf5a-4ad3-47ed-be1d-6aa8dc1cbd88",
+ "metadata": {},
+ "source": [
+ "That's really compact!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf60de78-247e-458e-9235-25ca4e2e4eb0",
+ "metadata": {},
+ "source": [
+ "## Modulo Operator `%`\n",
+ "\n",
+ "The modulo operator `%` is common in programming languanges, but not so much in our typical engineering problems. It turns out it can be very useful in iteration loops. It's actually quite simple: _the expression_ `a % b` _returns the remainder of `a/b`._ That's it!\n",
+ "\n",
+ "Take a look at the folloiwing examples that illustrate the operator:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3db0d267-1440-4846-aca1-5a1fcb2d6c0f",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "print(6 % 5)\n",
+ "print(5 % 6)\n",
+ "print(1 % 10)\n",
+ "print(5 % 5)\n",
+ "print(0 % 5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "123bf24c-913e-43dd-ae19-630711f2c8e6",
+ "metadata": {},
+ "source": [
+ "You can't divide by zero, just as in normal division.\n",
+ "\n",
+ "Below is an illustration for how the modulo is useful for doing things in a `for` loop that doesn't need to be done on every iteration."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "55f4af15-fc83-4233-ad80-93487141ca11",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "for i in range(100):\n",
+ " if i%25 == 0:\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d2fe6e7b-b3ef-47f4-8798-44e21e9a9545",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 5:</b> \n",
+ " \n",
+ "Experiment with the two Python cells above and make sure you understand what the modulo operator is and how it works.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dfef5493-eae2-43e2-88fe-9d0328c9c0b1",
+ "metadata": {},
+ "source": [
+ "We won't do anything else with `%` now, but will apply this in the last task of the assignment."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9df653a5-7e35-402c-a89c-4c1ef10802c0",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Stem Plot Type\n",
+ "\n",
+ "You have probably used the matplotlib plot types `plot`, `hist` and `scatter` frequently; another type is `stem`, which is useful for indicating the magnitude of various points along a number line. As with a scatter plot, the data does not need to be ordered, and missing values are easy to handle."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "76776113-9f57-473e-b39c-dffea2f7ad5b",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 6:</b> \n",
+ " \n",
+ "Run the cell below and play with the values to make sure you are comfortable with the stem plot type. Do you see how it can handle two values with the same index?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "835f184e-005a-4d2e-a673-0257a10660ad",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "value = [2, 7, 5, 1, 8]\n",
+ "index = [0, 2, 2, 6, 4]\n",
+ "plt.plot(index, value, 'o')\n",
+ "plt.stem(index, value);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "793f0d58-c8ba-492f-a450-29654d0a754b",
+ "metadata": {},
+ "source": [
+ "## Bringing it all together: Earthquake analysis\n",
+ "\n",
+ "To put the ideas above into practice, consider the following data, in the form of three lists, that describes the largest earthquakes in the world since 2025, that are also located on the ring of fire (Source: [Wikipedia](https://en.wikipedia.org/wiki/Lists_of_earthquakes))."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0228a5a7-49a8-475a-a63a-bd07f8128e01",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "year = [2003, 2011, 2013,\n",
+ " 2006, 2022,\n",
+ " 2017, 2019, 2001, 2010, 2015,\n",
+ " 2014, 2022, 2016,\n",
+ " 2000, 2007, 2012, 2005, 2004]\n",
+ "\n",
+ "magn = [8.3, 9.1, 8.3,\n",
+ " 8.3, 7.6,\n",
+ " 8.2, 8.0, 8.4, 8.8, 8.3,\n",
+ " 8.2, 7.6, 7.9,\n",
+ " 8.0, 8.4, 8.6, 8.6, 9.1]\n",
+ "\n",
+ "site = ['Japan, HokkaidÅ',\n",
+ " 'Japan, Honshu',\n",
+ " 'Russia, Sea of Okhotsk',\n",
+ " 'Russia, Kuril Islands',\n",
+ " 'Mexico, Michoacán',\n",
+ " 'Mexico, Chiapas',\n",
+ " 'Peru, Loreto',\n",
+ " 'Peru, Arequipa',\n",
+ " 'Chile, Concepción',\n",
+ " 'Chile, Coquimbo',\n",
+ " 'Chile, Iquique',\n",
+ " 'Papua New Guinea, Morobe',\n",
+ " 'Papua New Guinea, New Ireland',\n",
+ " 'Papua New Guinea',\n",
+ " 'Indonesia, Sumatra',\n",
+ " 'Indonesia, Indian Ocean',x\n",
+ " 'Indonesia, Simeulue',\n",
+ " 'Indonesia, Sumatra']"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7addd27e-f9c8-4966-83ce-df66a516c2ab",
+ "metadata": {},
+ "source": [
+ "We can visualize the data in the following figure. Can you find an easy way to recreate this plot?\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8aa46742-7241-4788-a344-2a8d878c2746",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 7:</b> \n",
+ " \n",
+ "Using the tools described in this assignment, recreate the plot above. Note that to avoid cluttering the figure, the detailed earthquake information is only plotted for years that are a multiple of 5 (i.e., 2000, 2005, etc.).\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "33264465-fd6d-487a-9663-9b25e71dc5dc",
+ "metadata": {},
+ "source": [
+ "**Hints:** create a for loop using 3 iterables (`x`, `y` and `z`) based on _one_ of these iterable tools: `range`, `enumerate` or `zip`. Use the modulo operator `%` to choose the 5-year multiples. An outline of the solution in pseudo-code is also provided here, each of which should be one line in your code (except item 5):\n",
+ "```\n",
+ "1 for each iterable item\n",
+ "2 if the year is a multiple of 5\n",
+ "3 create the label (string)\n",
+ "4 annotate the plot (i.e., annotate(...))\n",
+ "5 include any additional formatting (more than one line)\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "de8d3186-1025-46f5-874b-16ef0a79c280",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "80580ab9-4d79-46b1-ae6e-775af04d43ad",
+ "metadata": {},
+ "source": [
+ "**End of notebook.**\n",
+ "<h2 style=\"height: 60px\">\n",
+ "</h2>\n",
+ "<h3 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; bottom: 60px; right: 50px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.4"
+ },
+ "widgets": {
+ "application/vnd.jupyter.widget-state+json": {
+ "state": {},
+ "version_major": 2,
+ "version_minor": 0
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/content/Week_2_3/PA11_solution.ipynb b/content/Week_2_3/PA11_solution.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..9a5994cfa71c00bab8cf4f03a8eccc73fb7b6920
--- /dev/null
+++ b/content/Week_2_3/PA11_solution.ipynb
@@ -0,0 +1,1028 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "c96d6259-08d6-4289-aea2-589d67cdb5ee",
+ "metadata": {},
+ "source": [
+ "# Programming Assignment 11: iTer-remoto\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\" />\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\" />\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): Week 2.2. Due: complete this PA prior to class on Friday, Dec 1, 2023 (but better if you do it by Wed, Nov 29!).*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7a28e541-d2d0-48a9-abf6-7b73075c8fd3",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Overview of Assignment\n",
+ "\n",
+ "This assignment will address a few simple, but useful, topics related to indexing and iterating in Python, illustrated with an annotated stem figure. The content of this PA is simple, but will be used directly in the Wednesday workshop this week.\n",
+ "\n",
+ "Specifically, we will look at:\n",
+ "- iterables and iterable objects\n",
+ "- iterables `range`, `enumerate` and `zip`\n",
+ "- the modulo operator `%`\n",
+ "- plot type `stem`\n",
+ "\n",
+ "## Assignment Criteria\n",
+ "\n",
+ "**You will pass this assignment as long as your respository fulfills the following criteria:** \n",
+ "\n",
+ "- You have completed this notebook and it runs without errors"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 101,
+ "id": "75681df5-7a73-469a-ad83-ebd05451e0b7",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b0584fa2-7f4b-4566-9217-e6ca58d0e191",
+ "metadata": {},
+ "source": [
+ "## Introduction\n",
+ "\n",
+ "Suppose we are writing code to summarize some simple data, perhaps a group of teams that each have a score for a particular game."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 208,
+ "id": "4d6bb586-3ade-473e-8dd7-19dd350fa3f6",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "07ed2fba-5f83-47d8-af7e-b3a83bebbbbf",
+ "metadata": {},
+ "source": [
+ "Suppose we want to list the names of each team. One typical approach is to loop over each item in the list:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 209,
+ "id": "878ff8c2-fe3c-421a-b1d8-400450fc6bf5",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "The team names are:\n",
+ "green\n",
+ "red\n",
+ "blue\n"
+ ]
+ }
+ ],
+ "source": [
+ "N_teams = len(team)\n",
+ "print('The team names are:')\n",
+ "for i in range(N_teams):\n",
+ " print(f'{team[i]}')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "172fea58-dd93-4af0-a09c-f4129338ab54",
+ "metadata": {},
+ "source": [
+ "However, this is not the most efficient way to do this, as we need to define two extra variables: `N_teams` and the iteration index `i`. There is a better way! \n",
+ "\n",
+ "## Iterable Objects\n",
+ "\n",
+ "Python supports what is referred to as _iterable_ objects: generally, this means objects that can be easily used by Python's built-in and efficient iteration schemes. Examples are: list, tuple, dictionaries, and strings. You may be interested in reading about all of [the technical details here](https://docs.python.org/3/glossary.html#term-iterator); however, for this assignment the explanations here and examples below are sufficient. It is important to recognize this type of object exists in Python, so we can take advantage of it in our code, or so you can recognize the algorithms in other authors code.\n",
+ "\n",
+ "Let's see an iterable object in action. First, an easy way to test if an object is iterable is to use it as an argument of the `iter()` method:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 170,
+ "id": "fc48118b-014b-4044-95fc-787e110de380",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "<list_iterator at 0x18960118820>"
+ ]
+ },
+ "execution_count": 170,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "iter(team)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "111ef874-1b80-45fc-bb2c-ce9f7982681b",
+ "metadata": {},
+ "source": [
+ "Other objects can also be made into an iterator, like numpy arrays:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 169,
+ "id": "ee686157-6923-4d2e-b74f-35ed1049c70f",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "<iterator at 0x189601199c0>"
+ ]
+ },
+ "execution_count": 169,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "iter(np.array([1, 5, 67]))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e4302934-4828-4dcd-bc3b-eafbc59ad199",
+ "metadata": {},
+ "source": [
+ "The cell above should return `iterator`, which indicates the object (argument) is iterable. An integer, however, is _not_ iterable. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "41269037-161e-4c50-9dd7-5346bfc41b7a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1:</b> \n",
+ " \n",
+ "Try running the code below to confirm that an integer is not iterable (an error will occur), then fix it by converting the argument to an iterable object. Experiment by turning the integer into a list, string and np.array. \n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 175,
+ "id": "5b65450f-dd18-4456-ac6b-d4856ae47b23",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "<iterator at 0x1896011b1c0>"
+ ]
+ },
+ "execution_count": 175,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# iter(5)\n",
+ "iter([5])\n",
+ "iter('5')\n",
+ "iter(np.array([5]))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "47cf00b4-6946-44e6-bbeb-4176eb1e5674",
+ "metadata": {},
+ "source": [
+ "Great, you can make an iterator! But what do you _do_ with it? \n",
+ "\n",
+ "_Most of the time we simply use it in a `for` loop, but it is worthwhile to understand this object a bit more deeply._\n",
+ "\n",
+ "One simple way to understand an iterator is to imagine that it is a way of (efficiently!) turning your iterable object into something that can go sequentially through all its values. To do this it has two \"abilities\":\n",
+ "\n",
+ "1. the iterator knows its current value, and \n",
+ "2. the iterator knows its next value\n",
+ "\n",
+ "Let's try using the iterator by running the following cells:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 187,
+ "id": "f98b14ac-aa6b-4342-ba0f-faef9bc3f2b4",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "letters = 'abcde'\n",
+ "letter_iterator = iter(letters)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a48aadf-4e61-4c01-a23c-f0c28f7615e9",
+ "metadata": {},
+ "source": [
+ "Nothing new yet, but now watch what happens when we use the Python built-in command `next()`\n",
+ "\n",
+ "_Run the cell multiple times._"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 188,
+ "id": "a925cca3-ad71-4744-a6c8-7d578c7b69f0",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "'a'"
+ ]
+ },
+ "execution_count": 188,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "next(letter_iterator)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fbc2e084-0032-42ec-9d21-9cec68d4e6ec",
+ "metadata": {},
+ "source": [
+ "There are two key things to note here:\n",
+ "\n",
+ "1. The iterator \"knows\" how to go through all of its values, and\n",
+ "2. Calls to `next` returns to value itself, _not the index!_\n",
+ "\n",
+ "This is a very useful feature, as we can see in this simple example below:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 189,
+ "id": "59642a7a-df92-44e0-984d-f054f1e5162d",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "a\n",
+ "b\n",
+ "c\n",
+ "d\n",
+ "e\n"
+ ]
+ }
+ ],
+ "source": [
+ "for i in letters:\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5629acaa-81b5-43e3-bd27-7713ad5b2ae2",
+ "metadata": {},
+ "source": [
+ "Obviously this is a simple `for` loop, but hopefully the explanation above indicates what is actually happening \"under the hood\" of the `for` loop:\n",
+ " \n",
+ "1. identifies `letters` as an iterable object\n",
+ "2. converts it into an iterator, and then it\n",
+ "3. finds the next value in the sequence and assigns it to variable `i`\n",
+ "4. stop once there is no next value\n",
+ " \n",
+ "This is a bit of a simplification of what happens in reality, but it gets the idea across.\n",
+ "\n",
+ "Now let's take a look at something you have probably used a lot: the `range()` method:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 191,
+ "id": "16660b30-08ba-43dd-8ddd-d34f5fa8f094",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "<class 'range'>\n",
+ "range(0, 5)\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(type(range(5)))\n",
+ "print(range(5))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0531da9d-bec3-4d57-a653-0b1e58f54523",
+ "metadata": {},
+ "source": [
+ "Although it looks like it produces a tuple, `range` is a special iterable object that simply counts. As we can see it works the same as our `for` loop above:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 192,
+ "id": "04019b3e-9853-449c-a230-4c9033f55311",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "0\n",
+ "1\n",
+ "2\n",
+ "3\n",
+ "4\n"
+ ]
+ }
+ ],
+ "source": [
+ "for i in range(5):\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09db8005-1a9b-45b0-aac8-9821ea087fd9",
+ "metadata": {},
+ "source": [
+ "Hopefully this explains why `range` is such a useful feature for accessing sequential indexed elements of arrays.\n",
+ "\n",
+ "It turns out there are two more built-in Python methods that produce useful iterable objects: `enumerate` and `zip`. Let's take a look ath their doc strings."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0eb9605e-b4bf-4dc1-9a38-b64e78ff8563",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 2:</b> \n",
+ " \n",
+ "Read the docstrings below for the two methods. Confirm that they produce an iterable, but compare them to observe how the input and output of each is different. Can you imagine why one could be more useful than another in certain situations?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "045f749d-1d12-4f70-954e-301ddcd9be32",
+ "metadata": {},
+ "source": [
+ "\n",
+ "`enumerate`:\n",
+ "\n",
+ "```Python\n",
+ "Init signature: enumerate(iterable, start=0)\n",
+ "Docstring: \n",
+ "Return an enumerate object.\n",
+ "\n",
+ " iterable\n",
+ " an object supporting iteration\n",
+ "\n",
+ "The enumerate object yields pairs containing a count (from start, which\n",
+ "defaults to zero) and a value yielded by the iterable argument.\n",
+ "\n",
+ "enumerate is useful for obtaining an indexed list:\n",
+ " (0, seq[0]), (1, seq[1]), (2, seq[2]), ...\n",
+ "```\n",
+ "\n",
+ "And `zip`:\n",
+ "\n",
+ "```Python\n",
+ "Init signature: zip(self, /, *args, **kwargs)\n",
+ "Docstring: \n",
+ "zip(*iterables, strict=False) --> Yield tuples until an input is exhausted.\n",
+ "\n",
+ " >>> list(zip('abcdefg', range(3), range(4)))\n",
+ " [('a', 0, 0), ('b', 1, 1), ('c', 2, 2)]\n",
+ "\n",
+ "The zip object yields n-length tuples, where n is the number of iterables\n",
+ "passed as positional arguments to zip(). The i-th element in every tuple\n",
+ "comes from the i-th iterable argument to zip(). This continues until the\n",
+ "shortest argument is exhausted.\n",
+ "\n",
+ "If strict is true and one of the arguments is exhausted before the others,\n",
+ "raise a ValueError.\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f7b34d40-e7c3-477c-9a78-68d26ea2dd81",
+ "metadata": {},
+ "source": [
+ "<br><br><br><br><br><br><br><br><br><br><br>\n",
+ "\n",
+ "The main takeaways, should be as follows:\n",
+ "\n",
+ "1. Yes, they both produce an iterator\n",
+ "2. `enumerate` takes one iterable object and returns indices along with the value\n",
+ "3. `zip` takes two iterable objects and returns their values\n",
+ "\n",
+ "Let's try them out by running the next cell. Does it behave as expected?"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 198,
+ "id": "bdea9148-e2b0-4107-92b5-61a83de02efc",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "We created: <enumerate object at 0x000001896037AED0>\n",
+ "(0, 'r') (1, 'o') (2, 'b')\n",
+ "We created: <zip object at 0x0000018960392680>\n",
+ "('r', 2) ('o', 3) ('b', 36)\n"
+ ]
+ }
+ ],
+ "source": [
+ "thing_1 = 'roberts_string'\n",
+ "thing_2 = [2, 3, 36, 3., 1., 's', 7, '3']\n",
+ "\n",
+ "test_1 = enumerate(thing_1)\n",
+ "print(f'We created: {test_1}')\n",
+ "print(next(test_1), next(test_1), next(test_1))\n",
+ "\n",
+ "test_2 = zip(thing_1, thing_2)\n",
+ "print(f'We created: {test_2}')\n",
+ "print(next(test_2), next(test_2), next(test_2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d72f609b-0167-40f4-8e8f-c4476acf6417",
+ "metadata": {},
+ "source": [
+ "Can you see the difference?\n",
+ "\n",
+ "Looking at them in a for loop will also illustrate what's going on:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f3687fdf-c949-44f7-b5df-961e63703d59",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "First, enumerate:\n",
+ "0 r\n",
+ "1 o\n",
+ "2 b\n",
+ "3 e\n",
+ "4 r\n",
+ "5 t\n",
+ "6 s\n",
+ "7 _\n",
+ "8 s\n",
+ "9 t\n",
+ "10 r\n",
+ "11 i\n",
+ "12 n\n",
+ "13 g\n",
+ "\n",
+ "Then, zip:\n",
+ "r 2\n",
+ "o 3\n",
+ "b 36\n",
+ "e 3.0\n",
+ "r 1.0\n",
+ "t s\n",
+ "s 7\n",
+ "_ 3\n"
+ ]
+ }
+ ],
+ "source": [
+ "print('First, enumerate:')\n",
+ "for i, j in enumerate(thing_1):\n",
+ " print(i, j)\n",
+ "print('\\nThen, zip:') \n",
+ "for i, j in zip(thing_1, thing_2):\n",
+ " print(i, j)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3eedb9f0-29f9-4fea-b9ab-742f40ea8813",
+ "metadata": {},
+ "source": [
+ "Now let's return to our teams from the beginning of this assignment: let's apply our knowledge of `enumerate` and `zip` to see if we can print out the points per team in an efficient way."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf6e780a-d153-4e68-b7f5-c4b83ffd564f",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 3:</b> \n",
+ " \n",
+ "Use <code>enumerate</code> to print out the summary of points per team according to the print statement.\n",
+ "<br><br>\n",
+ "<em>Hint: you only need to use one of the lists as an argument.</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "73ea50cb-5c87-4f84-b618-4dd9ae38cc0c",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Team green has 5 points.\n",
+ "Team red has 9 points.\n",
+ "Team blue has 7 points.\n"
+ ]
+ }
+ ],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]\n",
+ "\n",
+ "# for YOUR_CODE_WITH_enumerate_HERE:\n",
+ "# print(f'Team {} has {} points.')\n",
+ "\n",
+ "for i, j in enumerate(score):\n",
+ " print(f'Team {team[i]} has {j} points.')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ac1ef287-e43a-4c78-9536-bdba4828906f",
+ "metadata": {},
+ "source": [
+ "You may have noticed that `enumerate` is a bit awkward for this case, since we still need to define an unnecessary iteration index to access the team name. Let's see if `zip` makes things easier:"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0bdfde09-88cf-4358-9af0-edc9c75acabf",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 4:</b> \n",
+ " \n",
+ "Use <code>zip</code> to print out the summary of points per team according to the print statement.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 203,
+ "id": "6bcfc8a9-57a1-41db-8449-fe689def8b5a",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Team green has 5 points.\n",
+ "Team red has 9 points.\n",
+ "Team blue has 7 points.\n"
+ ]
+ }
+ ],
+ "source": [
+ "team = ['green', 'red', 'blue']\n",
+ "score = [5, 9, 7]\n",
+ "\n",
+ "# for YOUR_CODE_WITH_zip_HERE:\n",
+ "# print(f'Team {} has {} points.')\n",
+ "\n",
+ "for i, j in zip(team, score):\n",
+ " print(f'Team {i} has {j} points.')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "edd3cf5a-4ad3-47ed-be1d-6aa8dc1cbd88",
+ "metadata": {},
+ "source": [
+ "That's really compact!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf60de78-247e-458e-9235-25ca4e2e4eb0",
+ "metadata": {},
+ "source": [
+ "## Modulo Operator `%`\n",
+ "\n",
+ "The modulo operator `%` is common in programming languanges, but not so much in our typical engineering problems. It turns out it can be very useful in iteration loops. It's actually quite simple: _the expression_ `a % b` _returns the remainder of `a/b`._ That's it!\n",
+ "\n",
+ "Take a look at the folloiwing examples that illustrate the operator:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 205,
+ "id": "3db0d267-1440-4846-aca1-5a1fcb2d6c0f",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "1\n",
+ "5\n",
+ "1\n",
+ "0\n",
+ "0\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(6 % 5)\n",
+ "print(5 % 6)\n",
+ "print(1 % 10)\n",
+ "print(5 % 5)\n",
+ "print(0 % 5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "123bf24c-913e-43dd-ae19-630711f2c8e6",
+ "metadata": {},
+ "source": [
+ "You can't divide by zero, just as in normal division.\n",
+ "\n",
+ "Below is an illustration for how the modulo is useful for doing things in a `for` loop that doesn't need to be done on every iteration."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 206,
+ "id": "55f4af15-fc83-4233-ad80-93487141ca11",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "0\n",
+ "25\n",
+ "50\n",
+ "75\n"
+ ]
+ }
+ ],
+ "source": [
+ "for i in range(100):\n",
+ " if i%25 == 0:\n",
+ " print(i)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c2277b2c-19ed-41ca-912c-c62ed8b836a8",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 5:</b> \n",
+ " \n",
+ "Experiment with the two Python cells above and make sure you understand what the modulo operator is and how it works.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "688ce8a1-c990-4f9c-b750-43c6b235a8e4",
+ "metadata": {},
+ "source": [
+ "We won't do anything else with `%` now, but will apply this in the last task of the assignment."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9df653a5-7e35-402c-a89c-4c1ef10802c0",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Stem Plot Type\n",
+ "\n",
+ "You have probably used the matplotlib plot types `plot`, `hist` and `scatter` frequently; another type is `stem`, which is useful for indicating the magnitude of various points along a number line. As with a scatter plot, the data does not need to be ordered, and missing values are easy to handle."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f0ec055f-0e8a-469d-8463-afba0379d839",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 6:</b> \n",
+ " \n",
+ "Run the cell below and play with the values to make sure you are comfortable with the stem plot type. Do you see how it can handle two values with the same index?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 100,
+ "id": "835f184e-005a-4d2e-a673-0257a10660ad",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "value = [2, 7, 5, 1, 8]\n",
+ "index = [0, 2, 2, 6, 4]\n",
+ "plt.plot(index, value, 'o')\n",
+ "plt.stem(index, value);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "793f0d58-c8ba-492f-a450-29654d0a754b",
+ "metadata": {},
+ "source": [
+ "## Bringing it all together: Earthquake analysis\n",
+ "\n",
+ "To put the ideas above into practice, consider the following data, in the form of three lists, that describes the largest earthquakes in the world since 2025, that are also located on the ring of fire (Source: [Wikipedia](https://en.wikipedia.org/wiki/Lists_of_earthquakes))."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 61,
+ "id": "0228a5a7-49a8-475a-a63a-bd07f8128e01",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "year = [2003, 2011, 2013,\n",
+ " 2006, 2022,\n",
+ " 2017, 2019, 2001, 2010, 2015,\n",
+ " 2014, 2022, 2016,\n",
+ " 2000, 2007, 2012, 2005, 2004]\n",
+ "\n",
+ "magn = [8.3, 9.1, 8.3,\n",
+ " 8.3, 7.6,\n",
+ " 8.2, 8.0, 8.4, 8.8, 8.3,\n",
+ " 8.2, 7.6, 7.9,\n",
+ " 8.0, 8.4, 8.6, 8.6, 9.1]\n",
+ "\n",
+ "site = ['Japan, HokkaidÅ',\n",
+ " 'Japan, Honshu',\n",
+ " 'Russia, Sea of Okhotsk',\n",
+ " 'Russia, Kuril Islands',\n",
+ " 'Mexico, Michoacán',\n",
+ " 'Mexico, Chiapas',\n",
+ " 'Peru, Loreto',\n",
+ " 'Peru, Arequipa',\n",
+ " 'Chile, Concepción',\n",
+ " 'Chile, Coquimbo',\n",
+ " 'Chile, Iquique',\n",
+ " 'Papua New Guinea, Morobe',\n",
+ " 'Papua New Guinea, New Ireland',\n",
+ " 'Papua New Guinea',\n",
+ " 'Indonesia, Sumatra',\n",
+ " 'Indonesia, Indian Ocean',x\n",
+ " 'Indonesia, Simeulue',\n",
+ " 'Indonesia, Sumatra']"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7addd27e-f9c8-4966-83ce-df66a516c2ab",
+ "metadata": {},
+ "source": [
+ "We can visualize the data in the following figure. Can you find an easy way to recreate this plot?\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8aa46742-7241-4788-a344-2a8d878c2746",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px\">\n",
+ "<p>\n",
+ "<b>Task 1:</b> \n",
+ " \n",
+ "Using the tools described in this assignment, recreate the plot above. Note that to avoid cluttering the figure, the detailed earthquake information is only plotted for years that are a multiple of 5 (i.e., 2000, 2005, etc.).\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f380a6f2-3de3-4e58-8495-cc4db2d5e232",
+ "metadata": {},
+ "source": [
+ "**Hints:** create a for loop using 3 iterables (`x`, `y` and `z`) based on _one_ of these iterable tools: `range`, `enumerate` or `zip`. Use the modulo operator `%` to choose the 5-year multiples. An outline of the solution in pseudo-code is also provided here, each of which should be one line in your code (except item 5):\n",
+ "```\n",
+ "1 for each iterable item\n",
+ "2 if the year is a multiple of 5\n",
+ "3 create the label (string)\n",
+ "4 annotate the plot (i.e., annotate(...))\n",
+ "5 include any additional formatting (more than one line)\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 99,
+ "id": "de8d3186-1025-46f5-874b-16ef0a79c280",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plt.plot(year, magn, 'o')\n",
+ "plt.stem(year, magn)\n",
+ "\n",
+ "\n",
+ "for x, y, z in zip(year, magn, site):\n",
+ " if x%5 == 0:\n",
+ " label = f\"M_w = {y} in {x} at {z}\"\n",
+ " plt.annotate(label, \n",
+ " (x,y), # these are the coordinates to position the label\n",
+ " textcoords=\"offset points\", \n",
+ " xytext=(15,50),\n",
+ " fontsize=8,\n",
+ " ha='left',\n",
+ " rotation=35,\n",
+ " arrowprops=dict(facecolor='black', shrink=0.1, width=0.5, headwidth=5)) \n",
+ "plt.xlim(1999, 2025)\n",
+ "plt.ylim(5,15)\n",
+ "plt.xlabel('Year of Earthquake')\n",
+ "plt.ylabel('Magnitude of Earthquake')\n",
+ "plt.title('Biggest earthquakes each year since 2000 on Ring of Fire');\n",
+ "# plt.savefig('earthquakes.svg')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "80580ab9-4d79-46b1-ae6e-775af04d43ad",
+ "metadata": {},
+ "source": [
+ "**End of notebook.**\n",
+ "<h2 style=\"height: 60px\">\n",
+ "</h2>\n",
+ "<h3 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; bottom: 60px; right: 50px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.4"
+ },
+ "widgets": {
+ "application/vnd.jupyter.widget-state+json": {
+ "state": {},
+ "version_major": 2,
+ "version_minor": 0
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/content/Week_2_3/WS11_DFT_you_try_meow.ipynb b/content/Week_2_3/WS11_DFT_you_try_meow.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..46437f7a9d8910be29a8c46e413fa656661c61e3
--- /dev/null
+++ b/content/Week_2_3/WS11_DFT_you_try_meow.ipynb
@@ -0,0 +1,469 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "4af6b45f-f7aa-46a8-9d79-c1579ae0a799",
+ "metadata": {},
+ "source": [
+ "# WS11: Discrete Fourier Transform (DFT): You Try Meow (Miauw)\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\" />\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\" />\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): Week 2.3, Signal Processing. For: November 29, 2023*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eff9791a",
+ "metadata": {},
+ "source": [
+ "The goal of this workshop to work with the _Discrete Fourier Transform_ (DFT), implemented in Python as the _Fast Fourier Transform_ (FFT) through `np.fft.fft`, and to understand and interpret its output.\n",
+ "\n",
+ "The notebook consists of two parts:\n",
+ "- The first part (Task 0) is a demonstration of the use of the DFT (_you read and execute the code cells_),\n",
+ "- The second part is a simple exercise with the DFT (_you write the code_).\n",
+ "\n",
+ "To start off, let's do a quick quiz question: _what is the primary purpose of the DFT?_\n",
+ "\n",
+ "_Run the code cell below to find the answer._"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "36a05bbd-69a1-4805-a4ad-e84de89de560",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "%%html\n",
+ "<iframe src=\"https://tudelft.h5p.com/content/1292126914399042257/embed\" aria-label=\"Meow\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" allow=\"autoplay *; geolocation *; microphone *; camera *; midi *; encrypted-media *\"></iframe><script src=\"https://tudelft.h5p.com/js/h5p-resizer.js\" charset=\"UTF-8\"></script>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "03f3b3c6-d3f6-4121-9f39-c30c8bf70ea5",
+ "metadata": {},
+ "source": [
+ "That's right! We convert our signal into the frequency domain.\n",
+ "\n",
+ "And if you would like an additional explanation of the key frequencies, you can find it [here](https://medium.com/@kovalenko.alx/fun-with-fourier-591662576a77)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bfca25a5-75ce-4837-9387-01f95be10bd0",
+ "metadata": {
+ "id": "0491cc69"
+ },
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\"> <p>Note the use of <code>zip</code>, <code>stem</code>, <code>annotate</code> and the modulo operator\n",
+ "<code>%</code>. Refer to PA11 if you do not understand these tools. Furthermore, note that the term _modulus_ is also used here (and in the textbook), which is another term for _absolute value._</p></div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "da13fbf3",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "from matplotlib import pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "13234855",
+ "metadata": {},
+ "source": [
+ "## Task 0: Demonstration of DFT using pulse function\n",
+ "\n",
+ "In the first part of this notebook, we use $x(t)=\\Pi(\\frac{t}{4})$, and its Fourier transform $X(f)=4 \\,\\textrm{sinc}(4f)$, as an example (see the first worked example in Chapter 3 on the Fourier transform). The pulse lasts for 4 seconds in the time domain; for convenience, below it is not centered at $t$ = 0, but shifted (delayed) to the right.\n",
+ "\n",
+ "The signal $x(t)$ clearly is non-periodic and is an energy signal; apart from a short time span of 'activity' it is zero elsewhere.\n",
+ "\n",
+ "We create a pulse function $x(t)$ in discrete time $x_n$ by numpy. The total signal duration is $T$ = 20 seconds (observation or record length). The sampling interval is $\\Delta t$ = 1 second. There are $N$ = 20 samples, and each sample represents 1 second, hence $N \\Delta t = T$.\n",
+ "\n",
+ "Note that the time array starts at $t$ = 0 s, and hence, the last sample is at $t$ = 19 s (and not at $t$ = 20 s, as then we would have 21 samples).\n",
+ "\n",
+ "### Task 0.1: Visualize the Signal"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b33e1be2",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "t = np.arange(0,20,1)\n",
+ "xt = np.concatenate((np.zeros(8), np.ones(4), np.zeros(8)))\n",
+ "\n",
+ "plt.plot(t, xt,'o')\n",
+ "plt.stem(t[8:12], xt[8:12]) \n",
+ "plt.xticks(ticks=np.arange(0,21,5), labels=np.arange(0,21,5))\n",
+ "plt.xlabel('time [s]')\n",
+ "plt.ylabel('xn');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a338eede",
+ "metadata": {},
+ "source": [
+ "### Task 0.2: Evaluate (and visualize) the DFT\n",
+ "\n",
+ "We use `numpy.fft.fft` to compute the one-dimensional Discrete Fourier Transform (DFT) of $x_n$, which takes a signal as argument and returns an array of coefficients (refer to the [documentation](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft.html) as needed).\n",
+ "\n",
+ "The DFT converts $N$ samples of the time domain signal $x_n$, into $N$ samples of the frequency domain. In this case, it produces $X_k$ with $k = 0, 1, 2, \\ldots, N-1$, with $N$ = 20, which are complex numbers. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "26d0f558-caa7-41f2-9477-60d74940fc10",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.2:</b> \n",
+ "Read the code cell below before executing it and identify the following:\n",
+ "<ol>\n",
+ " <li>Where is the DFT computed, and what is the output?</li>\n",
+ " <li>Why is the modulus (absolute value) used on the DFT output?</li>\n",
+ " <li>What are the values are used for the x and y axes of the plot?</li>\n",
+ " <li>How is frequency information found and added to the plot (mathematically)?</li>\n",
+ "</ol>\n",
+ "Once you understand the figure, continue reading to understand <em>what do these 20 complex numbers mean, and how should we interpret them?</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3acb465b",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "index_fft = np.arange(0,20,1)\n",
+ "plt.plot(index_fft, abs_fft, 'o')\n",
+ "\n",
+ "freq = np.arange(0, 1, 0.05)\n",
+ "for x,y in zip(index_fft, abs_fft):\n",
+ " if x%5 == 0 or x==19:\n",
+ " label = f\"f={freq[x]:.2f} Hz\"\n",
+ " plt.annotate(label, \n",
+ " (x,y),\n",
+ " textcoords=\"offset points\", \n",
+ " xytext=(0,10),\n",
+ " fontsize=10,\n",
+ " ha='center') \n",
+ "plt.ylim(0,5)\n",
+ "plt.xlim(-2,21)\n",
+ "plt.xlabel('fft-index')\n",
+ "plt.ylabel('$|X_k|$')\n",
+ "plt.stem(index_fft, abs_fft);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09c72cdd",
+ "metadata": {},
+ "source": [
+ "The frequency resolution $\\Delta f$ equals one-over-the-measurement-duration, hence $\\Delta f = 1/T$. With that knowledge, we can reconstruct the frequencies expressed in Hertz.\n",
+ "\n",
+ "The spectrum of the sampled signal is periodic in the sampling frequency $f_s$ which equals 1 Hz ($\\Delta t$ = 1 s). Therefore, it is computed just for one period $[0,f_s)$. The last value, with index 19, represents the component with frequency $f$ = 0.95 Hz. A spectrum is commonly presented and interpreted as double-sided, so in the above graph we can interpret the spectral components with indices 10 to 19 corresponding to negative frequencies. \n",
+ "\n",
+ "### Task 0.3: Identify negative frequencies"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "df1f483b",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "plt.stem(index_fft, abs_fft)\n",
+ "plt.plot(index_fft, abs_fft, 'o')\n",
+ "\n",
+ "freq = np.concatenate((np.arange(0, 0.5, 0.05), np.arange(-0.5, 0, 0.05)))\n",
+ "for x,y in zip(index_fft, abs_fft):\n",
+ " if x%5 == 0 or x==19:\n",
+ " label = f\"f={freq[x]:.2f} Hz\"\n",
+ " plt.annotate(label, \n",
+ " (x,y),\n",
+ " textcoords=\"offset points\", \n",
+ " xytext=(0,10),\n",
+ " fontsize=10,\n",
+ " ha='center') \n",
+ "plt.ylim(0,5)\n",
+ "plt.xlim(-2,21)\n",
+ "plt.xlabel('fft-index')\n",
+ "plt.ylabel('$|X_k|$');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d87a9a6",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "Now we can interpret the DFT of the $x(t)=\\Pi(\\frac{t}{4})$. The sampling interval is $\\Delta t$ = 1 second, and the obsesrvation length is T=20 seconds, and we have N=20 samples. \n",
+ "- We can recognize a bit of a sinc function.\n",
+ "- The DFT is computed from 0 Hz, for positive frequencies up to $\\frac{fs}{2} = 0.5$ Hz, after which the negative frequencies follow from -0.5 to -0.05 Hz.\n",
+ "- $X(f)=4 \\textrm{sinc}(4f)$ has its first null at $f=0.25$Hz.\n",
+ "\n",
+ "### Task 0.4: Create symmetric plot\n",
+ "\n",
+ "For convenient visualization, we may want to explicitly shift the negative frequencies to the left-hand side to create a symmetric plot. We can use `numpy.fft.fftshift` to do that. In other words, the zero-frequency component appears in the center of the spectrum. Now, it nicely shows a symmetric sprectrum. It well resembles the sinc-function (taking into account that we plot the modulus/absolute value). The output of the DFT still consists of $N$ = 20 elements. To enable this, we set up a new frequency array (in Hz) on the interval $[-fs/2,fs/2)$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e1eef6b6-1e90-43db-b059-6521b2a840d3",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.4:</b> \n",
+ "Read the code cell below before executing it and identify how the plot is modified based on the (new) specification of frequency. Note that it is more than just the <code>freq</code> variable!\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a8aac894",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "abs_fft_shift = np.abs(np.fft.fftshift(np.fft.fft(xt)))\n",
+ "freq = np.arange(-0.5, 0.5, 0.05)\n",
+ "plt.stem(freq, abs_fft_shift)\n",
+ "plt.plot(freq, abs_fft_shift, 'o')\n",
+ "plt.ylabel('|Xk|')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "569ca2d3",
+ "metadata": {},
+ "source": [
+ "### Task 0.5: Showing spectrum only for positive frequencies\n",
+ "\n",
+ "In practice, because of the symmetry, one typically plots only the right-hand side of the (double-sided) spectrum, hence only the part for positive frequencies $f \\geq 0$. This is simply a matter of preference, and a way to save some space."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a26e361d-c5fa-47a2-b32c-416cc3d10659",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.5:</b> \n",
+ "Can you identify what has changed (in the code and visually), compared to the previous plot?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b9608638",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "N=len(xt)\n",
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "freq = np.arange(0.0, 1.0, 0.05)\n",
+ "plt.plot(freq[:int(N/2)], abs_fft[:int(N/2)], 'o')\n",
+ "plt.stem(freq[:int(N/2)], abs_fft[:int(N/2)])\n",
+ "plt.ylabel('$|X_k|$')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83c4ca67-9234-4741-a56f-294a12bcd98a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.6:</b> \n",
+ "Confirm that you understand how we have arrived at the plot above, which illustrates the magnitude (amplitude) spectrum for frequencies $f \\in [0,f_s/2)$, rather than $[0,f_s)$.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9f5558b4",
+ "metadata": {},
+ "source": [
+ "## Task 1: Application of DFT using simple cosine"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "46dd5c0c",
+ "metadata": {},
+ "source": [
+ "It is always a good idea, in spectral analysis, to run a test with a very simple, basic signal. In this way you can test and verify your coding and interpretation of the results. \n",
+ "\n",
+ "Our basic signal is just a plain cosine. We take the amplitude equal to one, and zero initial phase, so the signal reads $x(t) = \\cos(2 \\pi f_c t)$, with $f_c$ = 3 Hz in this exercise. With such a simple signal, we know in advance how the spectrum should look like. Namely just a spike at $f$ = 3 Hz, and also one at $f$ = -3 Hz, as we're, for mathematical convenience, working with double sided spectra. The spectrum should be zero at all other frequencies.\n",
+ "\n",
+ "As a side note: the cosine is strictly a periodic function, not a-periodic (as above); still, the Fourier transform of the cosine is defined as two Dirac delta pulses or peaks (at 3 Hz and -3 Hz). You may want to check out the second worked example in Chapter 3 on the Fourier transform: Fourier transform in the limit."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "87de3cc5",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 1:</b> \n",
+ " \n",
+ "Create a sampled (discrete time) cosine signal by sampling at $f_s$ = 10 Hz, for a duration of $T$ = 2 seconds (make sure you use exactly $N$ = 20 samples). Plot the sampled signal, compute its DFT and plot its magnitude spectrum $|X_k|$ with proper labeling of the axes (just like we did in the first part of this notebook). Include a plot of the spectrum of the sampled cosine signal using only the positive frequencies (up to $f_s/2$, as in the last plot of the previous task).\n",
+ "\n",
+ "<em>Note: you are expected to produce three separate plots.</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "28ad034e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE_PLOT_1"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "af87d120",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE_PLOT_2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6fd9014c",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "YOUR_CODE_HERE_PLOT_3"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a4a306d-a372-460e-b312-e8fa138a9797",
+ "metadata": {},
+ "source": [
+ "**End of notebook.**\n",
+ "<h2 style=\"height: 60px\">\n",
+ "</h2>\n",
+ "<h3 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; bottom: 60px; right: 50px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
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+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
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+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
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+ },
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diff --git a/content/Week_2_3/WS11_solution.ipynb b/content/Week_2_3/WS11_solution.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..e96f8fdd6a2fc8e153e51ed8adec4174b32b0662
--- /dev/null
+++ b/content/Week_2_3/WS11_solution.ipynb
@@ -0,0 +1,568 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "4af6b45f-f7aa-46a8-9d79-c1579ae0a799",
+ "metadata": {},
+ "source": [
+ "# WS11: Discrete Fourier Transform (DFT): You Try Meow (Miauw)\n",
+ "\n",
+ "<h1 style=\"position: absolute; display: flex; flex-grow: 0; flex-shrink: 0; flex-direction: row-reverse; top: 60px;right: 30px; margin: 0; border: 0\">\n",
+ " <style>\n",
+ " .markdown {width:100%; position: relative}\n",
+ " article { position: relative }\n",
+ " </style>\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\" style=\"width:100px\" />\n",
+ " <img src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\" style=\"width:100px\" />\n",
+ "</h1>\n",
+ "<h2 style=\"height: 10px\">\n",
+ "</h2>\n",
+ "\n",
+ "*[CEGM1000 MUDE](http://mude.citg.tudelft.nl/): Week 2.3, Signal Processing. For: November 29, 2023*"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eff9791a",
+ "metadata": {},
+ "source": [
+ "The goal of this workshop to work with the _Discrete Fourier Transform_ (DFT), implemented in Python as the _Fast Fourier Transform_ (FFT) through `np.fft.fft`, and to understand and interpret its output.\n",
+ "\n",
+ "The notebook consists of two parts:\n",
+ "- The first part (Task 0) is a demonstration of the use of the DFT (_you read and execute the code cells_),\n",
+ "- The second part is a simple exercise with the DFT (_you write the code_).\n",
+ "\n",
+ "To start off, let's do a quick quiz question: _what is the primary purpose of the DFT?_\n",
+ "\n",
+ "_Find the answer [here](https://tudelft.h5p.com/content/1292126914399042257/embed)._"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "03f3b3c6-d3f6-4121-9f39-c30c8bf70ea5",
+ "metadata": {},
+ "source": [
+ "That's right! We convert our signal into the frequency domain. And if you would like an additional explanation of the key frequencies, you can find it [here](https://medium.com/@kovalenko.alx/fun-with-fourier-591662576a77)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bfca25a5-75ce-4837-9387-01f95be10bd0",
+ "metadata": {
+ "id": "0491cc69"
+ },
+ "source": [
+ "<div style=\"background-color:#facb8e; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\"> <p>Note the use of <code>zip</code>, <code>stem</code>, <code>annotate</code> and the modulo operator\n",
+ "<code>%</code>. Refer to PA11 if you do not understand these tools. Furthermore, note that the term _modulus_ is also used here (and in the textbook), which is another term for _absolute value._</p></div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "da13fbf3",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "from matplotlib import pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "13234855",
+ "metadata": {},
+ "source": [
+ "## Task 0: Demonstration of DFT using pulse function\n",
+ "\n",
+ "In the first part of this notebook, we use $x(t)=\\Pi(\\frac{t}{4})$, and its Fourier transform $X(f)=4 \\,\\textrm{sinc}(4f)$, as an example (see the first worked example in Chapter 3 on the Fourier transform). The pulse lasts for 4 seconds in the time domain; for convenience, below it is not centered at $t$ = 0, but shifted (delayed) to the right.\n",
+ "\n",
+ "The signal $x(t)$ clearly is non-periodic and is an energy signal; apart from a short time span of 'activity' it is zero elsewhere.\n",
+ "\n",
+ "We create a pulse function $x(t)$ in discrete time $x_n$ by numpy. The total signal duration is $T$ = 20 seconds (observation or record length). The sampling interval is $\\Delta t$ = 1 second. There are $N$ = 20 samples, and each sample represents 1 second, hence $N \\Delta t = T$.\n",
+ "\n",
+ "Note that the time array starts at $t$ = 0 s, and hence, the last sample is at $t$ = 19 s (and not at $t$ = 20 s, as then we would have 21 samples).\n",
+ "\n",
+ "### Task 0.1: Visualize the Signal"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "b33e1be2",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "t = np.arange(0,20,1)\n",
+ "xt = np.concatenate((np.zeros(8), np.ones(4), np.zeros(8)))\n",
+ "\n",
+ "plt.plot(t, xt,'o')\n",
+ "plt.stem(t[8:12], xt[8:12]) \n",
+ "plt.xticks(ticks=np.arange(0,21,5), labels=np.arange(0,21,5))\n",
+ "plt.xlabel('time [s]')\n",
+ "plt.ylabel('xn');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a338eede",
+ "metadata": {},
+ "source": [
+ "### Task 0.2: Evaluate (and visualize) the DFT\n",
+ "\n",
+ "We use `numpy.fft.fft` to compute the one-dimensional Discrete Fourier Transform (DFT) of $x_n$, which takes a signal as argument and returns an array of coefficients (refer to the [documentation](https://numpy.org/doc/stable/reference/generated/numpy.fft.fft.html) as needed).\n",
+ "\n",
+ "The DFT converts $N$ samples of the time domain signal $x_n$, into $N$ samples of the frequency domain. In this case, it produces $X_k$ with $k = 0, 1, 2, \\ldots, N-1$, with $N$ = 20, which are complex numbers. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "26d0f558-caa7-41f2-9477-60d74940fc10",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.2:</b> \n",
+ "Read the code cell below before executing it and identify the following:\n",
+ "<ol>\n",
+ " <li>Where is the DFT computed, and what is the output?</li>\n",
+ " <li>Why is the modulus (absolute value) used on the DFT output?</li>\n",
+ " <li>What are the values are used for the x and y axes of the plot?</li>\n",
+ " <li>How is frequency information found and added to the plot (mathematically)?</li>\n",
+ "</ol>\n",
+ "Once you understand the figure, continue reading to understand <em>what do these 20 complex numbers mean, and how should we interpret them?</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "3acb465b",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "index_fft = np.arange(0,20,1)\n",
+ "plt.plot(index_fft, abs_fft, 'o')\n",
+ "\n",
+ "freq = np.arange(0, 1, 0.05)\n",
+ "for x,y in zip(index_fft, abs_fft):\n",
+ " if x%5 == 0 or x==19:\n",
+ " label = f\"f={freq[x]:.2f} Hz\"\n",
+ " plt.annotate(label, \n",
+ " (x,y),\n",
+ " textcoords=\"offset points\", \n",
+ " xytext=(0,10),\n",
+ " fontsize=10,\n",
+ " ha='center') \n",
+ "plt.ylim(0,5)\n",
+ "plt.xlim(-2,21)\n",
+ "plt.xlabel('fft-index')\n",
+ "plt.ylabel('$|X_k|$')\n",
+ "plt.stem(index_fft, abs_fft);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09c72cdd",
+ "metadata": {},
+ "source": [
+ "The frequency resolution $\\Delta f$ equals one-over-the-measurement-duration, hence $\\Delta f = 1/T$. With that knowledge, we can reconstruct the frequencies expressed in Hertz.\n",
+ "\n",
+ "The spectrum of the sampled signal is periodic in the sampling frequency $f_s$ which equals 1 Hz ($\\Delta t$ = 1 s). Therefore, it is computed just for one period $[0,f_s)$. The last value, with index 19, represents the component with frequency $f$ = 0.95 Hz. A spectrum is commonly presented and interpreted as double-sided, so in the above graph we can interpret the spectral components with indices 10 to 19 corresponding to negative frequencies. \n",
+ "\n",
+ "### Task 0.3: Identify negative frequencies"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "df1f483b",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "plt.stem(index_fft, abs_fft)\n",
+ "plt.plot(index_fft, abs_fft, 'o')\n",
+ "\n",
+ "freq = np.concatenate((np.arange(0, 0.5, 0.05), np.arange(-0.5, 0, 0.05)))\n",
+ "for x,y in zip(index_fft, abs_fft):\n",
+ " if x%5 == 0 or x==19:\n",
+ " label = f\"f={freq[x]:.2f} Hz\"\n",
+ " plt.annotate(label, \n",
+ " (x,y),\n",
+ " textcoords=\"offset points\", \n",
+ " xytext=(0,10),\n",
+ " fontsize=10,\n",
+ " ha='center') \n",
+ "plt.ylim(0,5)\n",
+ "plt.xlim(-2,21)\n",
+ "plt.xlabel('fft-index')\n",
+ "plt.ylabel('$|X_k|$');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d87a9a6",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "Now we can interpret the DFT of the $x(t)=\\Pi(\\frac{t}{4})$. The sampling interval is $\\Delta t$ = 1 second, and the obsesrvation length is T=20 seconds, and we have N=20 samples. \n",
+ "- We can recognize a bit of a sinc function.\n",
+ "- The DFT is computed from 0 Hz, for positive frequencies up to $\\frac{fs}{2} = 0.5$ Hz, after which the negative frequencies follow from -0.5 to -0.05 Hz.\n",
+ "- $X(f)=4 \\textrm{sinc}(4f)$ has its first null at $f=0.25$Hz.\n",
+ "\n",
+ "### Task 0.4: Create symmetric plot\n",
+ "\n",
+ "For convenient visualization, we may want to explicitly shift the negative frequencies to the left-hand side to create a symmetric plot. We can use `numpy.fft.fftshift` to do that. In other words, the zero-frequency component appears in the center of the spectrum. Now, it nicely shows a symmetric sprectrum. It well resembles the sinc-function (taking into account that we plot the modulus/absolute value). The output of the DFT still consists of $N$ = 20 elements. To enable this, we set up a new frequency array (in Hz) on the interval $[-fs/2,fs/2)$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e1eef6b6-1e90-43db-b059-6521b2a840d3",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.4:</b> \n",
+ "Read the code cell below before executing it and identify how the plot is modified based on the (new) specification of frequency. Note that it is more than just the <code>freq</code> variable!\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "a8aac894",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "abs_fft_shift = np.abs(np.fft.fftshift(np.fft.fft(xt)))\n",
+ "freq = np.arange(-0.5, 0.5, 0.05)\n",
+ "plt.stem(freq, abs_fft_shift)\n",
+ "plt.plot(freq, abs_fft_shift, 'o')\n",
+ "plt.ylabel('|Xk|')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "569ca2d3",
+ "metadata": {},
+ "source": [
+ "### Task 0.5: Showing spectrum only for positive frequencies\n",
+ "\n",
+ "In practice, because of the symmetry, one typically plots only the right-hand side of the (double-sided) spectrum, hence only the part for positive frequencies $f \\geq 0$. This is simply a matter of preference, and a way to save some space."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a26e361d-c5fa-47a2-b32c-416cc3d10659",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.5:</b> \n",
+ "Can you identify what has changed (in the code and visually), compared to the previous plot?\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "b9608638",
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "N=len(xt)\n",
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "freq = np.arange(0.0, 1.0, 0.05)\n",
+ "plt.plot(freq[:int(N/2)], abs_fft[:int(N/2)], 'o')\n",
+ "plt.stem(freq[:int(N/2)], abs_fft[:int(N/2)])\n",
+ "plt.ylabel('$|X_k|$')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83c4ca67-9234-4741-a56f-294a12bcd98a",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 0.6:</b> \n",
+ "Confirm that you understand how we have arrived at the plot above, which illustrates the magnitude (amplitude) spectrum for frequencies $f \\in [0,f_s/2)$, rather than $[0,f_s)$.\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9f5558b4",
+ "metadata": {},
+ "source": [
+ "## Task 1: Application of DFT using simple cosine"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "46dd5c0c",
+ "metadata": {},
+ "source": [
+ "It is always a good idea, in spectral analysis, to run a test with a very simple, basic signal. In this way you can test and verify your coding and interpretation of the results. \n",
+ "\n",
+ "Our basic signal is just a plain cosine. We take the amplitude equal to one, and zero initial phase, so the signal reads $x(t) = \\cos(2 \\pi f_c t)$, with $f_c$ = 3 Hz in this exercise. With such a simple signal, we know in advance how the spectrum should look like. Namely just a spike at $f$ = 3 Hz, and also one at $f$ = -3 Hz, as we're, for mathematical convenience, working with double sided spectra. The spectrum should be zero at all other frequencies.\n",
+ "\n",
+ "As a side note: the cosine is strictly a periodic function, not a-periodic (as above); still, the Fourier transform of the cosine is defined as two Dirac delta pulses or peaks (at 3 Hz and -3 Hz). You may want to check out the second worked example in Chapter 3 on the Fourier transform: Fourier transform in the limit."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "87de3cc5",
+ "metadata": {},
+ "source": [
+ "<div style=\"background-color:#AABAB2; color: black; vertical-align: middle; padding:15px; margin: 10px; border-radius: 10px; width: 95%\">\n",
+ "<p>\n",
+ "<b>Task 1:</b> \n",
+ " \n",
+ "Create a sampled (discrete time) cosine signal by sampling at $f_s$ = 10 Hz, for a duration of $T$ = 2 seconds (make sure you use exactly $N$ = 20 samples). Plot the sampled signal, compute its DFT and plot its magnitude spectrum $|X_k|$ with proper labeling of the axes (just like we did in the first part of this notebook). Include a plot of the spectrum of the sampled cosine signal using only the positive frequencies (up to $f_s/2$, as in the last plot of the previous task).\n",
+ "\n",
+ "<em>Note: you are expected to produce three separate plots.</em>\n",
+ "</p>\n",
+ "</div>"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "28ad034e",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "### SOLUTION\n",
+ "\n",
+ "fc=3\n",
+ "fs=10\n",
+ "dt=1/fs\n",
+ "T=2\n",
+ "N=T*fs\n",
+ "t=np.arange(0,T,dt)\n",
+ "xt=np.cos(2*np.pi*fc*t)\n",
+ "\n",
+ "plt.plot(t,xt)\n",
+ "plt.xlabel('time [s]')\n",
+ "plt.ylabel('xn');"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "id": "af87d120",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "### SOLUTION\n",
+ "\n",
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "freq=np.arange(0,fs,1/T)\n",
+ "plt.stem(freq, abs_fft)\n",
+ "plt.plot(freq, abs_fft, 'o')\n",
+ "plt.ylabel('|Xk|')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "id": "6fd9014c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "### SOLUTION\n",
+ "\n",
+ "abs_fft = np.abs(np.fft.fft(xt))\n",
+ "freq=np.arange(0,fs,1/T)\n",
+ "plt.stem(freq[:int(N/2)], abs_fft[:int(N/2)])\n",
+ "plt.plot(freq[:int(N/2)], abs_fft[:int(N/2)], 'o')\n",
+ "plt.ylabel('|Xk|')\n",
+ "plt.xlabel('frequency [Hz]');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a4a306d-a372-460e-b312-e8fa138a9797",
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+ "source": [
+ "**End of notebook.**\n",
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+ " article { position: relative }\n",
+ " </style>\n",
+ " <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">\n",
+ " <img alt=\"Creative Commons License\" style=\"border-width:; width:88px; height:auto; padding-top:10px\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" />\n",
+ " </a>\n",
+ " <a rel=\"TU Delft\" href=\"https://www.tudelft.nl/en/ceg\">\n",
+ " <img alt=\"TU Delft\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/tu-logo/TU_P1_full-color.png\"/>\n",
+ " </a>\n",
+ " <a rel=\"MUDE\" href=\"http://mude.citg.tudelft.nl/\">\n",
+ " <img alt=\"MUDE\" style=\"border-width:0; width:100px; height:auto; padding-bottom:0px\" src=\"https://gitlab.tudelft.nl/mude/public/-/raw/main/mude-logo/MUDE_Logo-small.png\"/>\n",
+ " </a>\n",
+ " \n",
+ "</h3>\n",
+ "<span style=\"font-size: 75%\">\n",
+ "© Copyright 2023 <a rel=\"MUDE Team\" href=\"https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=65595\">MUDE Teaching Team</a> TU Delft. This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>."
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.4"
+ },
+ "latex_envs": {
+ "LaTeX_envs_menu_present": true,
+ "autoclose": false,
+ "autocomplete": true,
+ "bibliofile": "biblio.bib",
+ "cite_by": "apalike",
+ "current_citInitial": 1,
+ "eqLabelWithNumbers": true,
+ "eqNumInitial": 1,
+ "hotkeys": {
+ "equation": "Ctrl-E",
+ "itemize": "Ctrl-I"
+ },
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+ "user_envs_cfg": false
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+ "widgets": {
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+ }
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+}
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