# iqimbal2coef

Convert I/Q imbalance to compensator coefficient

## Description

example

C = iqimbal2coef(A,P) converts an I/Q amplitude and phase imbalance to its equivalent compensator coefficient.

## Examples

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Generate coefficients for the I/Q imbalance compensator System object™ using iqimbal2coef. The compensator corrects for an I/Q imbalance using the generated coefficients.

Create a raised cosine transmit filter System object.

txRCosFilt = comm.RaisedCosineTransmitFilter;

Modulate and filter random 64-ary symbols.

M= 64;
data = randi([0 M-1],100000,1);
dataMod = qammod(data,M);
txSig = txRCosFilt(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 by creating and applying a comm.IQImbalanceCompensator object. Set the compensator such that the complex coefficients are made available as an output argument.

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

Compare the final compensator coefficient to the coefficient generated by the iqimbal2coef function. Observe that there is good agreement.

idealcoef = iqimbal2coef(ampImb,phImb);
[coef(end); idealcoef]
ans = 2×1 complex

-0.1137 + 0.1296i
-0.1126 + 0.1334i

## Input Arguments

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Amplitude imbalance in dB, specified as a real-valued row or column vector.

Example: 3

Example: [0; 5]

Data Types: double

Phase imbalance in degrees, specified as a real-valued row or column vector.

Example: 10

Example: [15; 45]

Data Types: double

## Output Arguments

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Coefficient that perfectly compensates for the I/Q imbalance, returned as a complex-valued vector having the same dimensions as A and P.

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### I/Q Imbalance Compensation

The function iqimbal2coef is a supporting function for the comm.IQImbalanceCompensator System object™.

Define S and X as 2-by-1 vectors representing the I and Q components of the ideal and I/Q imbalanced signals, respectively.

$X=K\cdot S$

where K is a 2-by-2 matrix whose values are determined by the amplitude imbalance, A, and phase imbalance, P. A is expressed in dB and P is expressed in degrees.

The imbalance can be expressed as:

$\begin{array}{l}{I}_{gain}={10}^{0.5A/20}\\ {Q}_{gain}={10}^{-0.5A/20}\\ {\theta }_{i}=-\left(\frac{P}{2}\right)\left(\frac{\pi }{180}\right)\\ {\theta }_{q}=\frac{\pi }{2}+\left(\frac{P}{2}\right)\left(\frac{\pi }{180}\right)\end{array}$

Then K has the form:

$K=\left[\begin{array}{cc}{I}_{gain}\mathrm{cos}\left({\theta }_{i}\right)& {Q}_{gain}\mathrm{cos}\left({\theta }_{q}\right)\\ {I}_{gain}\mathrm{sin}\left({\theta }_{i}\right)& {Q}_{gain}\mathrm{sin}\left({\theta }_{q}\right)\end{array}\right]$

The vector Y is defined as the I/Q imbalance compensator output.

$Y=R\cdot X$

For the compensator to perfectly remove the I/Q imbalance, R must be the matrix inversion of K, namely:

$R={K}^{-1}$

Using complex notation, the vector Y can be rewritten as:

$\begin{array}{c}y={w}_{1}x+{w}_{2}\mathrm{conj}\left(x\right)\\ ={w}_{1}\left(x+\left({w}_{2}}{{w}_{1}}\right)\mathrm{conj}\left(x\right)\right)\end{array}$

where,

$\begin{array}{l}\mathrm{Re}\left\{{w}_{1}\right\}=\left({R}_{11}+{R}_{22}\right)/2\\ \mathrm{Im}\left\{{w}_{1}\right\}=\left({R}_{21}-{R}_{12}\right)/2\\ \mathrm{Re}\left\{{w}_{2}\right\}=\left({R}_{11}-{R}_{22}\right)/2\\ \mathrm{Im}\left\{{w}_{2}\right\}=\left({R}_{21}+{R}_{12}\right)/2\end{array}$

The output of the function is w2/w1. To exactly obtain the original signal, the compensator output needs to be scaled and rotated by the complex number w1.

Note

There are cases for which the output of iqimbal2coef is unreliable.

• If the phase imbalance is ±90°, the in-phase and quadrature components will become co-linear; consequently, the I/Q imbalance cannot be compensated.

• If the amplitude imbalance is 0 dB and the phase imbalance is 180°, w1 = 0 and w2 = 1i; therefore, the compensator takes the form of y = 1i*conj(x).

## Version History

Introduced in R2014b