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MATLAB Code for Zak Transform (Time-Frequency Transform)

 

MATLAB Code for Zak Transform (Time-Frequency Transform) 

%% The code is developed by www.salimwireless.com
clc; clear; close all;
%% Step 1: Parameters
N = 16; % Total number of symbols
M = 4; % Lattice rows (time slots)
SNR_dB = 20; % Noise level
modOrder = 4; % QPSK
%% Step 2: Generate random QPSK symbols
data = randi([0 modOrder-1], N, 1);
txSymbols = pskmod(data, modOrder, pi/4);
disp('Transmitted symbols:');
disp(txSymbols);
%% Step 3: Discrete Zak Transform (map to 2D lattice)
lattice = zeros(M, N/M);
for n = 0:N-1
m = mod(n, M) + 1; % row (time slot)
k = floor(n / M) + 1; % column (frequency slot)
lattice(m,k) = lattice(m,k) + txSymbols(n+1)*exp(-1j*2*pi*(k-1)*n/N);
end
% Optional: Apply simple phase rotation per column to illustrate Zak Transform
%for k = 1:N/M
% lattice(:,k) = lattice(:,k) .* exp(-1j*2*pi*(k-1)*(0:M-1)'/N);
%end
disp('Zak lattice:');
disp(lattice);
%% Step 4: Serialize lattice for transmission
txSerial = lattice(:); % Column-wise flattening
disp('Serial transmission stream:');
disp(txSerial);
%% Step 5: Channel (flat fading + noise)
h = (randn(size(txSerial)) + 1j*randn(size(txSerial)))/sqrt(2);
rxSerial = txSerial .* h; % Apply channel
rxSerial = awgn(rxSerial, SNR_dB, 'measured'); % Add noise
disp('Received serial stream:');
disp(rxSerial);
%% Step 6: Reshape back into lattice at receiver
rxLattice = reshape(rxSerial, M, N/M);
disp('Received lattice (reshaped):');
disp(rxLattice);
%% Step 7: Equalization (remove channel effect)
hLattice = reshape(h, M, N/M);
eqLattice = rxLattice ./ hLattice;
disp('Equalized lattice:');
disp(eqLattice);
%% Step 8: Inverse Zak Transform (flatten back to 1D symbols)
rxSymbols = zeros(N,1);
for m = 1:M
for k = 1:N/M
n = (k-1)*M + (m-1); % original index
rxSymbols(n+1) = eqLattice(m,k) * exp(1j*2*pi*(k-1)*n/N);
end
end
disp('Recovered 1D symbols:');
disp(rxSymbols);
%% Step 9: Demap symbols
rxData = pskdemod(rxSymbols, modOrder, pi/4);
disp('Recovered data bits:');
disp(rxData);
%% Step 10: Compute Symbol Error
numErrors = sum(data ~= rxData);
fprintf('Symbol errors: %d out of %d\n', numErrors, N);
web('https://www.salimwireless.com/search?q=otfs%20ofdm', '-browser'); 
 

 Output

Transmitted symbols:
   0.7071 - 0.7071i
   0.7071 - 0.7071i
  -0.7071 + 0.7071i
  -0.7071 - 0.7071i
   0.7071 + 0.7071i
  -0.7071 - 0.7071i
  -0.7071 + 0.7071i
  -0.7071 + 0.7071i
  -0.7071 + 0.7071i
  -0.7071 - 0.7071i
   0.7071 + 0.7071i
   0.7071 + 0.7071i
  -0.7071 + 0.7071i
  -0.7071 + 0.7071i
   0.7071 - 0.7071i
   0.7071 - 0.7071i

Zak lattice:
   0.7071 - 0.7071i   0.7071 - 0.7071i  -0.7071 + 0.7071i   0.7071 + 0.7071i
   0.7071 - 0.7071i  -0.3827 + 0.9239i  -1.0000 - 0.0000i   0.9239 - 0.3827i
  -0.7071 + 0.7071i   1.0000 + 0.0000i   0.7071 - 0.7071i   0.0000 + 1.0000i
  -0.7071 - 0.7071i   0.9239 - 0.3827i   0.0000 - 1.0000i   0.9239 + 0.3827i

Serial transmission stream:
   0.7071 - 0.7071i
   0.7071 - 0.7071i
  -0.7071 + 0.7071i
  -0.7071 - 0.7071i
   0.7071 - 0.7071i
  -0.3827 + 0.9239i
   1.0000 + 0.0000i
   0.9239 - 0.3827i
  -0.7071 + 0.7071i
  -1.0000 - 0.0000i
   0.7071 - 0.7071i
   0.0000 - 1.0000i
   0.7071 + 0.7071i
   0.9239 - 0.3827i
   0.0000 + 1.0000i
   0.9239 + 0.3827i

Received serial stream:
   0.8979 - 0.2337i
   0.6302 + 0.0081i
   0.5156 - 0.1635i
  -0.1600 + 0.7016i
  -0.5541 + 0.6423i
  -0.2204 + 1.0218i
  -0.4239 - 0.2612i
  -0.8138 + 1.0080i
   1.5777 - 0.8193i
   0.3525 + 0.1960i
   1.2537 - 0.5949i
   0.6953 - 0.4729i
   1.1553 - 1.2822i
  -0.2305 + 0.5419i
   0.4042 - 0.7063i
  -0.7030 + 1.3476i

Received lattice (reshaped):
   0.8979 - 0.2337i  -0.5541 + 0.6423i   1.5777 - 0.8193i   1.1553 - 1.2822i
   0.6302 + 0.0081i  -0.2204 + 1.0218i   0.3525 + 0.1960i  -0.2305 + 0.5419i
   0.5156 - 0.1635i  -0.4239 - 0.2612i   1.2537 - 0.5949i   0.4042 - 0.7063i
  -0.1600 + 0.7016i  -0.8138 + 1.0080i   0.6953 - 0.4729i  -0.7030 + 1.3476i

Equalized lattice:
   0.6493 - 0.5983i   0.6756 - 0.6211i  -0.7041 + 0.7505i   0.6701 + 0.7524i
   0.7856 - 0.8601i  -0.3434 + 1.0541i  -0.6994 + 0.1083i   0.7405 - 0.4090i
  -0.5754 + 0.6370i   0.8883 - 0.0777i   0.6756 - 0.6978i   0.0747 + 1.0447i
  -0.6840 - 0.6918i   0.8995 - 0.4555i  -0.0477 - 0.8712i   0.9284 + 0.4356i

Recovered 1D symbols:
   0.6493 - 0.5983i
   0.7856 - 0.8601i
  -0.5754 + 0.6370i
  -0.6840 - 0.6918i
   0.6211 + 0.6756i
  -0.8425 - 0.7206i
  -0.5732 + 0.6831i
  -0.6567 + 0.7651i
  -0.7041 + 0.7505i
  -0.5711 - 0.4179i
   0.6978 + 0.6756i
   0.6498 + 0.5823i
  -0.7524 + 0.6701i
  -0.5276 + 0.6612i
   0.6859 - 0.7915i
   0.7577 - 0.6910i

Recovered data bits:
     2
     2
     1
     3
     0
     3
     1
     1
     1
     3
     0
     0
     1
     1
     2
     2

Symbol errors: 0 out of 16 

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