hello
if I have a system with QPSK modulation and channel response in MATLAB. I need to do equalization (MMSE) to correct the BER. anyone has any reference on how to implement equalizer for QPSK system
i used the MATLAB function but I faced problems in the demodulation process due to size mismatch
The MMSE (Minimum Mean Square Error) equalizer is a popular technique used to combat the effects of channel distortion in QPSK (Quadrature Phase Shift Keying) systems. Here are the steps to implement the MMSE equalizer for a QPSK system:
1. Channel estimation: The first step is to estimate the channel by sending a known training sequence through the channel and measuring the received signal at the receiver. This can be done using techniques such as LS (Least Squares) or LMMSE (Linear Minimum Mean Square Error) estimation.
2. Calculate the channel matrix: Once the channel is estimated, the next step is to calculate the channel matrix. This can be done by using the coefficients of the estimated channel impulse response to construct a matrix H.
3. Calculate the noise variance: The next step is to estimate the noise variance of the received signal. This can be done by measuring the power of the received signal in the absence of the transmitted signal.
4. Calculate the MMSE equalizer coefficients: Using the estimated channel matrix and noise variance, the MMSE equalizer coefficients can be calculated using the following equation:
W = (H^H * H + σ^2 * I)^-1 * H^H
Where, H^H is the conjugate transpose of H, σ^2 is the noise variance, and I is the identity matrix.
5. Apply the MMSE equalizer: The final step is to apply the MMSE equalizer to the received signal. This can be done by multiplying the received signal by the MMSE equalizer coefficients:
y = W * r
Where, r is the received signal and y is the equalized signal.
By following these steps, the MMSE equalizer can be implemented for a QPSK system. This can help to improve the performance of the system by reducing the effects of channel distortion and improving the BER (Bit Error Rate) performance.
hello Ahmed Al-Abrt
thank you for answering
i do all these steps but i don't how to correctly equalize the signal and demodulate it in MATLAB
i create the equalizer
i have channel matrix
i convolve it with the signal and noise
but it give me different number of points so the demodulated signal different from the original so i can't calculate the error rate
Hello Ahmed Al-Abrt
Let's try a slighty different implentation of the MMSE equalizer. Suppose that the transmitted (Tx) signal at the emitter is x[n] and that the received signal (Rx) y[n] is obtain by linear convolution between the channel impulse response h[h] and the transmiited signal x[n], i.e. y[n] = h(n]*x[n], where * stands for the linear convolution operator.
Try this...this should work but I cannot verify it because I so not have matlab available at this moment :)
----------------Estimation of the L taps of a FIR MMSE equalizer wo---------
1- estimate the (2L+1) auto-correlation coefficients of the received signal :
>> L = 20 % number of taps in the equalizer
>> ryy = xcorr(y,y,L) ;
2 - compute the (LxL) autoccorelation matrix of the received signal
>> Ryy = toeplitz(ryy(L+1:end));
3- estimate the (2*L+1) inter-correlation coefficients between received and transmitted signals
>> r_yx = xcorr(y,x,L) ;
4- estimate the L coefficients of the linear MMSE equalizer
>> wo = inv(Ryy)*conj(r_yx(L+1:-1:2)) ;
5- apply the equalizer to the received signal
>> z[n] = filter(wo,1,y);
6 -take decisions in the Q-PSK alphabet and then compute the BER but you need to synchronize the emitted and recieved bistreams to do that !
This corresponds to the theory (see slide n°17 of https://www.eecis.udel.edu/~arce/files/Courses/StatSP/Slides_Set_8.pdf).
I hope this will help,
Hello Mariam Qutiaba
Let's try a slighty different implentation of the MMSE equalizer. Suppose that the transmitted (Tx) signal at the emitter is x[n] and that the received signal (Rx) y[n] is obtain by linear convolution between the channel impulse response h[h] and the transmiited signal x[n], i.e. y[n] = h(n]*x[n], where * stands for the linear convolution operator.
Try this...this should work but I cannot check it because I do not have matlab available at this moment :)
----------------Estimation of the L taps of a FIR MMSE equalizer wo---------
1- estimate the (2L+1) auto-correlation coefficients of the received signal :
>> L = 20 % number of taps in the equalizer
>> ryy = xcorr(y,y,L) ;
2 - compute the (LxL) autoccorelation matrix of the received signal
>> Ryy = toeplitz(ryy(L+1:end));
3- estimate the (2*L+1) inter-correlation coefficients between received and transmitted signals
>> r_yx = xcorr(y,x,L) ;
4- estimate the L coefficients of the linear MMSE equalizer
>> wo = inv(Ryy)*conj(r_yx(L+1:-1:2)) ;
5- apply the equalizer to the received signal
>> z[n] = filter(wo,1,y);
6 -take decisions in the Q-PSK alphabet and then compute the BER but you need to synchronize the emitted and recieved bistreams to do that !
This corresponds to the theory (see slide n°17 of https://www.eecis.udel.edu/~arce/files/Courses/StatSP/Slides_Set_8.pdf).
I hope this will help,
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