The post on IQ imbalance in transmitter, briefly discussed the effect of amplitude and phase imbalance and also showed that IQ imbalance results in spectrum at the image frequency. In this article, we will quantify the power of the image with respect to the desired tone (also known as** IMage Rejection Ratio IMRR**) for different values of gain and phase imbalance.

## System Model

Consider an IQ modulator having gain of and on each arm and phase imbalance of as shown in figure below.

**Figure : IQ modulator with gain and phase imbalance**

The output signal is,

.

Considering an ideal IQ demodulator multiplying the received signal with and respectively,

.

.

Ignoring the common term and writing the base band equivalent form,

.

**This is the model for transmit IQ imbalance. **

## Image Rejection Ratio (IMRR) with transmit IQ imbalance

By sending a complex sinusoidal , and by taking ratio of the power of the signal at the image frequency and desired frequency , the image rejection ratio can be computed.

Let and correspondingly, and .

**Finding the component**

To find the component, multiply the received signal with and integrate over period .

The power of the component is,

**Finding the component**

To find the component, multiply the received signal with and integrate over period .

**The Image Rejection Ratio (IMRR) is**

.

Substituting and with variable and , the equation simplifies to,

.

**A useful approximation to IMRR**

When there is no phase imbalance i.e , the equation reduces to,

.

When there is no gain imbalance i.e , the equation reduces to,

.

As these two are independent, they can be added to give an approximate value of Image Rejection Ratio.

**Summarizing, the Image Rejection Ratio for a given value of gain imbalance and phase imbalance is,**

.

**Simulation Results**

Simple Matlab/Octave code plotting the simulated and theoretical values of Image Rejection for different values of gain and phase imbalance.

clear; close all N = 64; fm = 2; gammadB_v = [-3:.1:3]; phiDeg_v = [-6:.2:6]; [tt gammadB_zeroIdx ] = min(abs((gammadB_v-0))); [tt phiDeg_zeroIdx ] = min(abs((phiDeg_v-0))); for (ii = 1:length(gammadB_v)) for (jj = 1:length(phiDeg_v)) gammadB = gammadB_v(ii); phiDeg = phiDeg_v(jj); gammaLin = 10^(gammadB/20); phiRad = phiDeg*pi/180; epsilonLin = gammaLin -1 ; % transmitted signal xt = exp(j*2*pi*fm*[0:N-1]/N); % received signal with IQ imbalance xht_re = gammaLin*cos(phiRad/2)*real(xt) + sin(phiRad/2)*imag(xt); xht_im = gammaLin*sin(phiRad/2)*real(xt) + cos(phiRad/2)*imag(xt); xht = xht_re + j*xht_im; % taking ifft() to find the +fm and -fm components yF = fft(xht,N); y_pfm = yF(fm+1); y_nfm = yF(N-fm+1); est_imrr_lin = (abs(y_nfm)./abs(y_pfm))^2; theory_imrr_lin = (gammaLin^2 + 1 - 2*gammaLin*cos(phiRad))./(gammaLin^2 + 1 + 2*gammaLin*cos(phiRad)); approx_imrr_lin = (epsilonLin^2 + phiRad^2)/4; est_imrr_dB(ii,jj) = 10*log10(est_imrr_lin); theory_imrr_dB(ii,jj) = 10*log10(theory_imrr_lin); approx_imrr_dB(ii,jj) = 10*log10(approx_imrr_lin); end end figure plot(gammadB_v,theory_imrr_dB(:,phiDeg_zeroIdx),'bs-'); hold on plot(gammadB_v,est_imrr_dB(:,phiDeg_zeroIdx),'md-'); plot(gammadB_v,approx_imrr_dB(:,phiDeg_zeroIdx),'gx-'); xlabel('gain imbalance, dB'); ylabel('image rejection, dB'); grid on; legend('theory','estimated','approx') title('Image Rejection Ratio with gain imbalance alone'); axis([-3 3 -50 -10]); figure plot(phiDeg_v,theory_imrr_dB(gammadB_zeroIdx,:),'bs-'); hold on plot(phiDeg_v,est_imrr_dB(gammadB_zeroIdx,:),'md-'); plot(phiDeg_v,approx_imrr_dB(gammadB_zeroIdx,:),'gx-'); xlabel('phase imbalance, degree'); ylabel('image rejection, dB'); grid on; legend('theory','estimated','approx') title('Image Rejection Ratio with phase imbalance alone'); axis([-6 6 -50 -20]);

**Figure : Image Rejection Ratio (IMRR) with gain imbalance alone**

**Figure : Image Rejection Ratio (IMRR) with phase imbalance alone**

**Observations**

1) The approximate expression holds good for reasonable values of gain and phase imbalance.

2) As a rule of thumb, the following numbers are useful :

** - For 1 degree of phase imbalance, the Image Rejection Ratio (IMRR) is around -41dB**

** - For 1dB of gain imbalance, the Image Rejection Ratio (IMRR) is around -25dB**

## References

**[1] Cavers, J.K.; Liao, M.W.; , “Adaptive compensation for imbalance and offset losses in direct conversion transceivers,” Vehicular Technology, IEEE Transactions on , vol.42, no.4, pp.581-588, Nov 1993 doi: 10.1109/25.260752**

[2] Table of trignometric identities http://www.sosmath.com/trig/Trig5/trig5/trig5.html

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