iqcoef2imbal

Convert compensator coefficient to amplitude and phase imbalance

Syntax

[A,P] = iqcoef2imbal(C)

Description

[A,P] = iqcoef2imbal(C) converts compensator coefficient C to its equivalent amplitude and phase imbalance.

Examples

Estimate I/Q Imbalance from Compensator Coefficient

Use iqcoef2imbal to estimate the amplitude and phase imbalance for a given complex coefficient. The coefficients are an output from the step function of the IQImbalanceCompensator.

Create a raised cosine transmit filter to generate a 64-QAM signal.

M = 64;
txFilt = comm.RaisedCosineTransmitFilter;

Modulate and filter random 64-ary symbols.

data = randi([0 M-1],100000,1);
dataMod = qammod(data,M);
txSig = step(txFilt,dataMod);

Specify amplitude and phase imbalance.

ampImb = 2; % dB 
phImb = 15; % degrees

Apply the specified I/Q imbalance.

gainI = 10.^(0.5*ampImb/20);
gainQ = 10.^(-0.5*ampImb/20);
imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180);
imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180));
rxSig = imbI + imbQ;

Normalize the power of the received signal

rxSig = rxSig/std(rxSig);

Remove the I/Q imbalance using the comm.IQImbalanceCompensator System object. Set the compensator object such that the complex coefficients are made available as an output argument.

hIQComp = comm.IQImbalanceCompensator('CoefficientOutputPort',true);
[compSig,coef] = step(hIQComp,rxSig);

Estimate the imbalance from the last value of the compensator coefficient.

[ampImbEst,phImbEst] = iqcoef2imbal(coef(end));

Compare the estimated imbalance values with the specified ones. Notice that there is good agreement.

[ampImb phImb; ampImbEst phImbEst]

ans = 2×2

    2.0000   15.0000
    2.0178   14.5740

Input Arguments

C — Compensator coefficient

complex-valued scalar or vector

Coefficient used to compensate for an I/Q imbalance, specified as a complex-valued vector.

Example: 0.4+0.6i

Example: [0.1+0.2i; 0.3+0.5i]

Data Types: single | double

Output Arguments

A — Amplitude imbalance

real-valued vector

Amplitude imbalance in dB, returned as a real-valued vector with the same dimensions as C.

P — Phase imbalance

real-valued vector

Phase imbalance in degrees, returned as a real-valued vector with the same dimensions as C.

More About

I/Q Imbalance Compensation

The function iqcoef2imbal is a supporting function for the comm.IQImbalanceCompensator System object™.

Given a scaling and rotation factor, G, compensator coefficient, C, and received signal, x, the compensated signal, y, has the form

$$y=G\left[x+C\mathrm{conj}(x)\right].$$

In matrix form, this can be rewritten as

$$Y=RX,$$

where X is a 2-by-1 vector representing the imbalanced signal [X_I, X_Q] and Y is a 2-by-1 vector representing the compensator output [Y_I, Y_Q].

The matrix R is expressed as

$$R=\left[\begin{array}{cc}1+\mathrm{Re}\left\{C\right\}& \mathrm{Im}\left\{C\right\}\\ \mathrm{Im}\left\{C\right\}& 1-\mathrm{Re}\left\{C\right\}\end{array}\right]$$

For the compensator to perfectly remove the I/Q imbalance, R = K^-1 because $$X=KS$$, where K is a 2-by-2 matrix whose values are determined by the amplitude and phase imbalance and S is the ideal signal. Define a matrix M with the form

$$M=\left[\begin{array}{cc}1& -\alpha \\ \alpha & 1\end{array}\right]$$

Both M and M^-1 can be thought of as scaling and rotation matrices that correspond to the factor G. Because K = R^-1, the product M^-1 R K M is the identity matrix, where M^-1 R represents the compensator output and K M represents the I/Q imbalance. The coefficient α is chosen such that

$$KM=L\left[\begin{array}{cc}{I}_{gain}\cos ({\theta }_I)& Q_{gain}\cos ({\theta }_Q)\\ I_{gain}\sin ({\theta }_I)& Q_{gain}\sin ({\theta }_Q)\end{array}\right]$$

where L is a constant. From this form, we can obtain I_gain, Q_gain, θ_I, and θ_Q. For a given phase imbalance, Φ_Imb, the in-phase and quadrature angles can be expressed as

$$\begin{array}{c}{\theta }_I=-\left(\pi /2\right)\left({\Phi }_{Imb}/180\right)\\ {\theta }_Q=\pi /2+\left(\pi /2\right)\left({\Phi }_{Imb}/180\right)\end{array}$$

Hence, cos(θ_Q) = sin(θ_I) and sin(θ_Q) = cos(θ_I) so that

$$L\left[\begin{array}{cc}{I}_{gain}\cos ({\theta }_I)& Q_{gain}\cos ({\theta }_Q)\\ I_{gain}\sin ({\theta }_I)& Q_{gain}\sin ({\theta }_Q)\end{array}\right]=L\left[\begin{array}{cc}{I}_{gain}\cos ({\theta }_I)& Q_{gain}\sin ({\theta }_I)\\ I_{gain}\sin ({\theta }_I)& Q_{gain}\cos ({\theta }_I)\end{array}\right]$$

The I/Q imbalance can be expressed as

$$\begin{array}{c}KM=\left[\begin{array}{cc}{K}_{11}+\alpha K_{12}& -\alpha K_{11}+K_{12}\\ K_{21}+\alpha K_{22}& -\alpha K_{21}+K_{22}\end{array}\right]\\ =L\left[\begin{array}{cc}{I}_{gain}\cos ({\theta }_I)& Q_{gain}\sin ({\theta }_I)\\ I_{gain}\sin ({\theta }_I)& Q_{gain}\cos ({\theta }_I)\end{array}\right]\end{array}$$

Therefore,

$$\left(K_{21}+\alpha K_{22}\right)/\left(K_{11}+\alpha K_{12}\right)=\left(-\alpha K_{11}+K_{12}\right)/\left(-\alpha K_{21}+K_{22}\right)=\sin ({\theta }_I)/\cos ({\theta }_I)$$

The equation can be written as a quadratic equation to solve for the variable α, that is D₁α² + D₂α + D₃ = 0, where

$$\begin{array}{c}{D}_1=-K_{11}{K}_{12}+K_{22}{K}_{21}\\ D_2=K_{12}^2+K_{21}^2-K_{11}^2-K_{22}^2\\ D_3=K_{11}{K}_{12}-K_{21}{K}_{22}\end{array}$$

When |C| ≤ 1, the quadratic equation has the following solution:

$$\alpha =\frac{-D_2-\sqrt{D^2-4D_1{D}_3}}{2D_1}$$

Otherwise, when |C| > 1, the solution has the following form:

$$\alpha =\frac{-D_2+\sqrt{D^2-4D_1{D}_3}}{2D_1}$$

Finally, the amplitude imbalance, A_Imb, and the phase imbalance, Φ_Imb, are obtained.

$$\begin{array}{c}{K}^{\prime }=K\left[\begin{array}{cc}1& -\alpha \\ \alpha & 1\end{array}\right]\\ A_{Imb}=20{\log }_{10}\left({K^{\prime }}_{11}/{K^{\prime }}_{22}\right)\\ {\Phi }_{Imb}=-2{\tan }^{-1}\left({K^{\prime }}_{21}/{K^{\prime }}_{11}\right)\left(180/\pi \right)\end{array}$$

Note

• If C is real and |C| ≤ 1, the phase imbalance is 0 and the amplitude imbalance is 20log₁₀((1–C)/(1+C))

• If C is real and |C| > 1, the phase imbalance is 180° and the amplitude imbalance is 20log₁₀((C+1)/(C−1)).

• If C is imaginary, A_Imb = 0.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.