Signals/IQ-Modulation.md

 ---
 title: I/Q Modulation
 ---
+In many cases, qubit resonance frequencies are much higher than the maximum output frequency of the AWG. I/Q modulation is a method to up-convert the output signal of the AWG to a much higher frequency. The frequency of a microwave (MW) source is added to the output of the AWG in a process called I/Q mixing.
 
-TODO:
-* formulas I/Q modulation
-* offset I and Q
-* gain, phase corrections
\ No newline at end of file
+The AWG generates an in-phase and a quadrature (I/Q) signal that are mixed with the I/Q output of a MW source. See [In-phase and quadrature components](https://en.wikipedia.org/wiki/In-phase_and_quadrature_components).
+
+![IQ_mixing](uploads/133d6d3789174f646cb2e129a05fe85c/IQ_mixing.png){width=354 height=207}
+
+The output of this mixer is:
+
+$`y(t) = I(t) cos(\omega_c t) - Q(t) sin(\omega_c t)`$
+
+IQ modulation can be used to create a single frequency output signal which is higher of lower in frequency
+than the MW source. For this the IQ modulation signals should be sinusoids with a 90 degrees shift.
+
+$`I(t) = A(t) cos(\omega_m t + \phi(t))`$
+
+$`Q(t) = A(t) sin(\omega_m t + \phi(t))`$
+
+$`y(t) = A(t) [cos(\omega_m t + \phi(t)) cos(\omega_c t) - sin(\omega_m t + \phi(t)) sin(\omega_c t)]`$
+
+$`y(t) = A(t) cos((\omega_c + \omega_m) t + \phi(t))`$
+
+IQ modulation allows frequency multiplexing, because it is a linear operation. 
+The I and Q components can contain multiple frequencies which will all be shifted with $`f_c`$.
+IQ modulation can also be used for fast chirps (frequency sweeps), because phase 
+and frequency of the I and Q components can be swiftly changed by the AWG.
+
+# I/Q modulation in pulse_lib
+
+Users of pulse_lib don't have to care about the calculation of I and Q outputs.
+They can specify the desired MW pulses after IQ modulation and pulse_lib calculates the I and Q output signals for the AWG.
+
+The user has to specify the I and Q output channels of the AWG as an I/Q pair and pass the frequency of the vector signal generator, the LO frequency.
+For coherent qubit control every qubit needs a channel with the resonant frequency of the qubit.
+
+Example defining a qubit control channel 'q1' on I/Q output pair 'IQ1':
+```Python
+pulse.define_iq_output("IQ1", awg.name, i_ch_num, q_ch_num, marker_name="M_IQ")
+pulse.set_iq_lo("IQ1", mw_source.frequency)
+
+pulse.define_qubit_channel("q1", "IQ1", 12.34e9)
+```
+
+# I/Q modulation errors
+
+The I/Q mixing is not a perfect process and additional frequencies will be generated.
+
+Ideally, the I/Q mixer creates single side-band signal.
+
+![ideal_modulation](uploads/8b5b39f3a3fc54cf3b6156e2efeedcbd/ideal_modulation.png){width=900 height=212}
+
+*Ideal I/Q modulation of single sideband*
+
+In practice the signals and electrical components are not perfect. Small differences between amplifiers,
+filters and path length result in small errors in the I/Q output.
+
+The I and Q output of the AWG can have a voltage offset, a (frequency dependent) gain difference and
+a (frequency dependent) phase difference. The output signal after modulation will contain the carrier frequency
+and the mirror side-band.
+
+![real_modulation](uploads/704a0eac1aa034f74edb2c69dba114f5/real_modulation.png){width=900 height=216}
+
+*IQ modulation in practice containing the carrier frequency and a mirrored side-band.*
+
+## Offset error
+
+A voltage offset in the AWG output results in the output of the carrier frequency.
+
+$`y(t) = [a+cos(\omega_m t)] cos(\omega_c t) - [b + sin(\omega_m t)] sin(\omega_c t)]`$
+
+$`y(t) = cos((\omega_c + \omega_m) t) + a \cdot cos(\omega_c t) - b \cdot sin(\omega_c t)`$
+
+![IQ_offset_error](uploads/a1a291c09c6fe90fb333b7e50f8e8052/IQ_offset_error.png){width=413 height=309}
+
+*Remainder of carrier frequency due to offset error.*
+
+
+## Gain error
+
+A difference in gain between I and Q components add the mirrored side-band frequency,
+$`f_c - f_m`, to the output.$
+
+$`y(t) = (1 + a) cos(\omega_m t) cos(\omega_c t) - sin(\omega_m t) sin(\omega_c t)`$
+
+$`y(t) = (1+\frac{a}{2}) cos((\omega_c + \omega_m) t) + \frac{a}{2} cos((\omega_c - \omega_m) t)`$
+
+![IQ_gain_error](uploads/7859cbec3d28450170e098c3dbba9110/IQ_gain_error.png){width=409 height=311}
+
+*Mirrored side-band due to gain error*
+
+## Phase error
+
+When the phase difference between the I and Q components is not exactly 90 degrees then the mirrored
+side-band frequency is output as well. The resulting output is similar to the output with a gain error.
+
+$`y(t) = cos(\omega_m t + \frac{a}{2}) cos(\omega_c t) - sin(\omega_m t - \frac{a}{2}) sin(\omega_c t)`$
+
+$`y(t) = cos(\frac{a}{2}) cos((\omega_c + \omega_m) t) - sin(\frac{a}{2}) sin((\omega_c - \omega_m) t)`$
+
+
+# IQ corrections in pulse_lib
+
+A vector signal generator will have options to correct the offset, phase and gain error of the IQ input, but only with a frequency independent correction.
+Pulse_lib can also correct for these errors, where gain and phase corrections are frequency dependent.
+
+Example:
+  Add gain and phase offset to qubit channels.
+
+  ```Python
+  pulse.set_qubit_correction_phase("q1", phase_correction)
+
+  pulse.set_qubit_correction_gain("q1", correction_I, correction_Q)
+  ```