Mixer imbalance numerical demonstration
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Disclaimer:
This article is a personal research note and a record of my own understanding. It may contain simplifications, incomplete arguments, or unintentional mistakes. It is shared for interest and discussion only, not as a formal reference.
note
For an ideal IQ Mixer, the generated RF signal is ($\omega_C$ is the carrier / signal source LO frequency)
$ RF = I(t) \cos ( \omega_{C} t) + Q(t) \sin ( \omega_{C} t) $
In physical world, the IQ Mixer has
$ RF = I’‘(t) \cos ( \omega_{C} t) + Q’‘(t) \sin ( \omega_{C} t) $
where we denote the IQ mismatch as
$ I’’ = (1+\delta_A) I’ \cos(\delta_\phi) + Q’ \sin(\delta_\phi) $
$ Q’’ = Q’ $
which is one of the denotation as an example, and you may seek more supporting mathematical information for LLM-AI.
The LO leakage contributed by the DC-part can be considered as
$ I’ = I + I_0 $
$ Q’ = Q + Q_0 $
where $I_0, Q_0$ are constants。
In experiment, the spectrum analyzer was used to acquire RF spectral data, while the supplied I and Q values were adjusted to minimize leakage from the LO and image components. The resulting parameters correspond to the required mixer correction parameters.
RF spectral data:
spectrum analyzer->Computer->Python scriptsI, Q Parameters:
Python scripts-> ChangeNumpy array-> SendAWG
Just point-by-point mapping from mathematical understanding to your own scripts
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# I' = I + I_offset
I_offset = 0.001 # for example
I_array_change_dc = I_array + I_offset
# fake_function
some_awg_sendTo_some_port(array=I_array_change_dc, *args, **kwargs)
python demo
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import numpy as np
from matplotlib import pyplot as plt
import scipy
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def mixer_iq2rf(t, i, q, fc=5e9, dA=-0.0, dphi=0.0, i0=0.0, q0=-0.0):
wc = 2*np.pi*fc
# i2 = i + i0
# q2 = q + q0
i3 = (1+dA)*i*np.cos(dphi) + q*np.sin(dphi) + i0
q3 = q + q0
res = i3*np.cos(wc*t) + q3*np.sin(wc*t)
return res
def get_fft_spectrum(t, rf_raw):
N = len(rf_raw)
fs = 1.0 / (t[1] - t[0]) # 采样率
rf_fft = np.fft.rfft(rf_raw-np.mean(rf_raw),norm='ortho')
freqs = np.fft.rfftfreq(N, d=1.0/fs)
return freqs, np.abs(rf_fft)
def plot_spectrum(freqs, power_fft):
fig = plt.figure(figsize=(5,2), dpi=150)
plt.plot(freqs/1e9, power_fft)
plt.ylim(-130,50)
plt.grid(linestyle='dotted')
plt.xlabel("freq (GHz)")
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f_sb = 200e6 # sideband
fc = 5e9
qpara = {"fc": fc, "f_sb": f_sb}
t = np.arange(0, 20_000, 0.05)*1e-9 # ns
i = np.sin(f_sb*2*np.pi*t)
q = np.cos(f_sb*2*np.pi*t)
show plot
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rf_raw = mixer_iq2rf(t, i, q, dA=0.0, dphi=0.0, i0=0.0, q0=0.0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft)
plt.xlim(4.7,5.3)
plt.title(r"Ideal $\delta_A, \delta_\phi, I_0, Q_0 =0$")
You will see a peak of leakage in the frequency of $f_c$, which is induced by the DC-part i0, q0, and usually called as zero (LO) leakage
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rf_raw = mixer_iq2rf(t, i, q, dA=0.0, dphi=0.0, i0=0.01, q0=-0.01)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft)
plt.xlim(4.7,5.3)
plt.title(r"Non-zero $I_0, Q_0$")
You will see a peak of leakage in the frequency of $f_c-f_{sb}$, which is induced by the mixer imbalance parameter dA, dphi, and usually called as mirror leakage
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rf_raw = mixer_iq2rf(t, i, q, dA=0.01, dphi=-0.02, i0=0.0, q0=0.0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft)
plt.xlim(4.7,5.3)
plt.title(r"Non-zero $\delta_A, \delta_\phi$")
You will see a peak of leakage in both frequencies of $f_c$ and $f_c-f_{sb}$
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rf_raw = mixer_iq2rf(t, i, q, dA=1e-4, dphi=-2e-4, i0=5e-3, q0=-1e-3)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft)
plt.xlim(4.7,5.3)
plt.title(r"More physical case")
correct demo
Assume artificial parameters to numerically emulate mixer imbalance and DC offset
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dA=1e-4
dphi=-2e-4
i0=5e-3
q0=-1e-3
rf_raw = mixer_iq2rf(t, i, q, dA=dA, dphi=dphi, i0=i0, q0=q0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft)
plt.xlim(4.7,5.3)
plt.title(r"Before correct")
Search paramters to correct LO leakage
In a manner analogous to a real experiment, the correction parameters are numerically searched as the following demonstration
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def binary_search_2d(f, x_bounds, y_bounds, tol=1e-10, max_iter=100):
"""
Two-dimensional alternating-direction binary search（2D）
"""
xl, xu = x_bounds
yl, yu = y_bounds
for it in range(max_iter):
# ---- x ----
xm = 0.5 * (xl + xu)
yc = 0.5 * (yl + yu)
f_left = f(xl, yc)
f_right = f(xu, yc)
if f_left < f_right:
xu = xm
else:
xl = xm
# ---- y ----
ym = 0.5 * (yl + yu)
xc = 0.5 * (xl + xu)
f_down = f(xc, yl)
f_up = f(xc, yu)
if f_down < f_up:
yu = ym
else:
yl = ym
# ---- Convergence criterion ----
if (xu - xl) < tol and (yu - yl) < tol:
break
x_opt = 0.5 * (xl + xu)
y_opt = 0.5 * (yl + yu)
return x_opt, y_opt, f(x_opt, y_opt)
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def correct_zero(corr_i0, corr_q0):
i2 = i-corr_i0
q2 = q-corr_q0
rf_raw = mixer_iq2rf(t, i2, q2, dA=dA, dphi=dphi, i0=i0, q0=q0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
# power_fft2 = 10*np.log10(rf_fft)
fc = qpara['fc']
idxs = np.where(np.round(freqs/1e9,3)==fc/1e9)[0]
p_max = np.max(rf_fft[idxs])
print(f"corr_i0, corr_q0 = ({corr_i0, corr_q0})")
return p_max
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res_opt = binary_search_2d(
correct_zero,
x_bounds=(-0.5, 0.5),
y_bounds=(-0.5, 0.5),
tol=1e-14,
)
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corr_i0, corr_q0 = res_opt[:2]
print(corr_i0, corr_q0)
i2 = i.copy()-corr_i0
q2 = q.copy()-corr_q0
rf_raw = mixer_iq2rf(t, i2, q2, dA=dA, dphi=dphi, i0=i0, q0=q0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft2 = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft2)
plt.xlim(4.7,5.3)
plt.title(r"After correct LO")
Search paramter to correct mirror leakage
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def correct_mirror(dA, dphi):
i3 = (i-q*np.sin(dphi))/(np.cos(dphi)*(1+dA)) - corr_i0
q3 = q.copy() - corr_q0
rf_raw = mixer_iq2rf(t, i3, q3, dA=dA, dphi=dphi, i0=i0, q0=q0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
# power_fft2 = 10*np.log10(rf_fft)
f_mirror = qpara['fc'] - qpara['f_sb']
idxs = np.where(np.round(freqs/1e9,3)==f_mirror/1e9)[0]
p_max = np.max(rf_fft[idxs])
print(f"dA, dphi = ({dA, dphi}), p_max = {p_max}")
return p_max
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res_mirror = binary_search_2d(
correct_mirror,
x_bounds=(-0.5, 0.5),
y_bounds=(-0.5, 0.5),
tol=1e-14,
)
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corr_dA, corr_dphi = res_mirror[:2]
# dA=1e-4
# dphi=-2e-4
print(corr_dA, corr_dphi)
i3 = (i-q*np.sin(dphi))/(np.cos(dphi)*(1+dA)) - corr_i0
q3 = q.copy() - corr_q0
rf_raw = mixer_iq2rf(t, i3, q3, dA=dA, dphi=dphi, i0=i0, q0=q0)
freqs, rf_fft = get_fft_spectrum(t, rf_raw)
power_fft2 = 10*np.log10(rf_fft)
plot_spectrum(freqs, power_fft2)
plt.xlim(4.7,5.3)
plt.title(r"After correct LO and mirror")