Convert compensator coefficient to amplitude and phase imbalance
iqcoef2imbal to estimate the amplitude and phase imbalance for a given complex coefficient. The coefficients are an output from the
step function of the
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
C— Compensator coefficient
Coefficient used to compensate for an I/Q imbalance, specified as a complex-valued vector.
Given a scaling and rotation factor, G, compensator coefficient, C, and received signal, x, the compensated signal, y, has the form
In matrix form, this can be rewritten as
where X is a 2-by-1 vector representing the imbalanced signal [XI, XQ] and Y is a 2-by-1 vector representing the compensator output [YI, YQ].
The matrix R is expressed as
For the compensator to perfectly remove the I/Q imbalance, R = K-1 because , 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
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
where L is a constant. From this form, we can obtain Igain, Qgain, θI, and θQ. For a given phase imbalance, ΦImb, the in-phase and quadrature angles can be expressed as
Hence, cos(θQ) = sin(θI) and sin(θQ) = cos(θI) so that
The I/Q imbalance can be expressed as
The equation can be written as a quadratic equation to solve for the variable α, that is D1α2 + D2α + D3 = 0, where
When |C| ≤ 1, the quadratic equation has the following solution:
Otherwise, when |C| > 1, the solution has the following form:
Finally, the amplitude imbalance, AImb, and the phase imbalance, ΦImb, are obtained.
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, AImb = 0.