iqcoef2imbal

Convert compensator coefficient to amplitude and phase imbalance

Syntax

[A,P] =
iqcoef2imbal(C)

Description

example

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

Examples

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Estimate I/Q Imbalance from Compensator Coefficient

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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

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`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

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`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

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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

$y = G [x + C conj (x)] .$

In matrix form, this can be rewritten as

$Y = R X,$

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

$R = [\begin{matrix} 1 + Re {C} & Im {C} \\ Im {C} & 1 - Re {C} \end{matrix}]$

For the compensator to perfectly remove the I/Q imbalance, R = K^-1 because $X = K S$ , 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

$M = [\begin{matrix} 1 & - α \\ α & 1 \end{matrix}]$

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

$K M = L [\begin{matrix} I_{g a i n} \cos (θ_{I}) & Q_{g a i n} \cos (θ_{Q}) \\ I_{g a i n} \sin (θ_{I}) & Q_{g a i n} \sin (θ_{Q}) \end{matrix}]$

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

$\begin{matrix} θ_{I} = - (π / 2) (Φ_{I m b} / 180) \\ θ_{Q} = π / 2 + (π / 2) (Φ_{I m b} / 180) \end{matrix}$

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

$L [\begin{matrix} I_{g a i n} \cos (θ_{I}) & Q_{g a i n} \cos (θ_{Q}) \\ I_{g a i n} \sin (θ_{I}) & Q_{g a i n} \sin (θ_{Q}) \end{matrix}] = L [\begin{matrix} I_{g a i n} \cos (θ_{I}) & Q_{g a i n} \sin (θ_{I}) \\ I_{g a i n} \sin (θ_{I}) & Q_{g a i n} \cos (θ_{I}) \end{matrix}]$

The I/Q imbalance can be expressed as

$\begin{matrix} K M = [\begin{matrix} K_{11} + α K_{12} & - α K_{11} + K_{12} \\ K_{21} + α K_{22} & - α K_{21} + K_{22} \end{matrix}] \\ = L [\begin{matrix} I_{g a i n} \cos (θ_{I}) & Q_{g a i n} \sin (θ_{I}) \\ I_{g a i n} \sin (θ_{I}) & Q_{g a i n} \cos (θ_{I}) \end{matrix}] \end{matrix}$

Therefore,

$(K_{21} + α K_{22}) / (K_{11} + α K_{12}) = (- α K_{11} + K_{12}) / (- α K_{21} + K_{22}) = \sin (θ_{I}) / \cos (θ_{I})$

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

$\begin{matrix} D_{1} = - K_{11} K_{12} + K_{22} K_{21} \\ D_{2} = K_{12}^{2} + K_{21}^{2} - K_{11}^{2} - K_{22}^{2} \\ D_{3} = K_{11} K_{12} - K_{21} K_{22} \end{matrix}$

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

$α = \frac{- D_{2} - \sqrt{D^{2} - 4 D_{1} D_{3}}}{2 D_{1}}$

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

$α = \frac{- D_{2} + \sqrt{D^{2} - 4 D_{1} D_{3}}}{2 D_{1}}$

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

$\begin{matrix} K^{'} = K [\begin{matrix} 1 & - α \\ α & 1 \end{matrix}] \\ A_{I m b} = 20 \log_{10} ({K^{'}}_{11} / {K^{'}}_{22}) \\ Φ_{I m b} = - 2 \tan^{- 1} ({K^{'}}_{21} / {K^{'}}_{11}) (180 / π) \end{matrix}$

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.

Documentation

iqcoef2imbal

Syntax

Description

Examples

Estimate I/Q Imbalance from Compensator Coefficient

Input Arguments

`C` — Compensator coefficient
complex-valued scalar or vector

Output Arguments

`A` — Amplitude imbalance
real-valued vector

`P` — Phase imbalance
real-valued vector

More About

I/Q Imbalance Compensation

Extended Capabilities

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

See Also

Functions

Objects

Communications Toolbox Documentation

Support

Documentation

iqcoef2imbal

Syntax

Description

Examples

Estimate I/Q Imbalance from Compensator Coefficient

Input Arguments

C — Compensator coefficient complex-valued scalar or vector

Output Arguments

A — Amplitude imbalance real-valued vector

P — Phase imbalance real-valued vector

More About

I/Q Imbalance Compensation

Extended Capabilities

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

See Also

Functions

Objects

Communications Toolbox Documentation

Support

`C` — Compensator coefficient
complex-valued scalar or vector

`A` — Amplitude imbalance
real-valued vector

`P` — Phase imbalance
real-valued vector

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