Hi,
I understand that you want to reconstruct the input signal (x) from the channel output (y) in a multipath frequency selective channel scenario using the path gains (h). For this, equalization techniques need to be employed. In MATLAB, you can design a equalizer based on the channel's impulse response using the comm.LinearEqualizer, comm.DecisionFeedbackEqualizer, comm.MLSEEqualizer system objects, etc.
Please follow the example MATLAB code below to understand equalization using the communication toolbox of MATLAB:
Simulate the Multipath Channel: Simulate the multipath channel using the appropriate model (e.g., Rayleigh or Rician fading). This usually involves defining the number of paths and their respective delays and average path gains.
% Example of creating a Rayleigh fading channel object
fs = 1000; % Sampling frequency in Hz
channel = comm.RayleighChannel('SampleRate',fs,'PathDelays',[0 1e-4 3e-4],'AveragePathGains',[0 -3 -6],'NormalizePathGains',true);
Pass the Signal Through the Channel: Pass your input signal (x) through the channel to obtain the channel output (y) and the path gains (h).
% Assuming x is your input signal
[y, h] = channel(x);
Equalization: To reconstruct (x) from (y), you can use various equalization techniques. For frequency selective channels, typical equalizers include Linear Equalizers (LE), Decision Feedback Equalizers (DFE), etc.
Linear Equalizers attempt to invert the channel's effect linearly.
% Example of using a linear equalizer
eq = comm.LinearEqualizer('Algorithm','LMS','NumTaps',5,'StepSize',0.01);
[x_est, err] = eq(y, x); % Assuming x is known at the receiver for training
Decision Feedback Equalizer (DFE) uses past decision outputs to cancel intersymbol interference (ISI), which is not possible in LE.
% Example of using a DFE
dfe = comm.DecisionFeedbackEqualizer('Algorithm','LMS','NumForwardTaps',5,'NumFeedbackTaps',4,'StepSize',0.01);
[x_est, err] = dfe(y, x); % Assuming x is known at the receiver for training
MLSE Equalizer equalizes modulated signals using maximum likelihood sequence estimation.
mlse = comm.MLSEEqualizer(TracebackDepth=10, Channel=chCoeffs, Constellation=pskmod(0:3,4,pi/4));
If you're using OFDM, equalization is typically performed in the frequency domain, which simplifies the process especially for channels with long impulse responses.
% Assuming Y is the received OFDM symbol in frequency domain and H is the channel frequency response
X_est = Y ./ H; % Simple division for equalization
Then the signal can be demodulated to obtain the final result.
Please follow the below-mentioned MATLAB R2024a documentation links to understand more on comm.RayleighChannel, comm.LinearEqualizer, comm.DecisionFeedbackEqualizer, comm.MLSEEqualizer system objects respectively:
- https://www.mathworks.com/help/comm/ref/comm.rayleighchannel-system-object.html
- https://www.mathworks.com/help/comm/ref/comm.linearequalizer-system-object.html
- https://www.mathworks.com/help/comm/ref/comm.decisionfeedbackequalizer-system-object.html
- https://www.mathworks.com/help/comm/ref/comm.mlseequalizer-system-object.html
Hope this helps!
