FFT based channel estimation

MATLAB code for channel impulse response estimation using FFT-based channel estimation method

% The code is developed by SalimWireless.com

clc;
clear all;
close all;

% Parameters
N = 128;                  % Signal length
epsilon = 1e-3;           % Regularization term
useWhiteNoise = true;     % Set false to use delta function probe
SNR_dB = 100;              % Signal-to-Noise Ratio in dB

% True channel (FIR with 3 taps)
true_h = [0.8, 0.5+0.3j, 0.2, 0.1];
L = length(true_h);

% Generate probe signal
if useWhiteNoise
    x = randn(1, N);  % White noise
else
    x = zeros(1, N);  % Delta function
    x(1) = 1;
end

% Pass through the channel (linear convolution)
y_clean = conv(x, true_h, 'same');

% Add AWGN
signal_power = mean(abs(y_clean).^2);
noise_power = signal_power / (10^(SNR_dB/10));
noise = sqrt(noise_power/2) * (randn(1, N) + 1j * randn(1, N));
y_noisy = y_clean + noise;

% FFTs
Xf = fft(x);
Yf = fft(y_noisy);

% Estimate channel frequency response with regularization
Hf_est = Yf ./ (Xf + epsilon);

% IFFT to estimate impulse response
h_est = ifft(Hf_est);

% Plot results
figure;
subplot(3,1,1);
stem(0:L-1, real(true_h), 'bo', 'LineWidth', 1.5); hold on;
stem(0:L-1, real(h_est(1:L)), 'rx');
title('True vs Estimated Impulse Response (Real Part)');
legend('True', 'Estimated');

subplot(3,1,2);
plot(abs(Hf_est));
title('|Estimated H(f)|');

subplot(3,1,3);
plot(angle(Hf_est));
title('∠Estimated H(f)');

% Diagnostics
disp('Maximum imaginary part of estimated impulse response (should be small):');
disp(max(abs(imag(h_est))));

disp('First few samples of estimated impulse response:');
disp(h_est(1:10));
web('https://www.salimwireless.com/search?q=channel%20Impuse%20response', '-browser');

Output

More precisely, the aforementioned method is a basic channel estimation simulation where:

You know the input (x - white noise or delta function).
You measure the output (y_noisy - affected by the channel and noise).
You estimate the channel (h_est) by:
- Taking the FFT of both input and output.
- Dividing in the frequency domain (H(f) = Y(f) / (X(f) + ε) with a small regularization term).
- Taking the IFFT of H(f) to get the estimated impulse response in time domain.

In a real wireless system:

The channel is unknown.
You send pilots (known symbols) and apply the same principle (but with more practical techniques like Least Squares (LS), Minimum Mean Square Error (MMSE), etc.).

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