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OFDM with FFT Based Oversampling

This example modifies an OFDM+CP signal to efficiently output an oversampled waveform from the OFDM modulator. Configure the simple case with the sample rate related to subcarrier spacing and FFT length.

k = 4;       % Number of bits per symbol 
M = 2^k;     % Modulation order
nFFT = 128;  % Number of FFT bins
cplen = 8;   % CP length
txsymbols = randi([0 M-1],nFFT,1);
txgrid = qammod(txsymbols,M,UnitAveragePower=true);
txout = ifft(txgrid,nFFT);
txout = txout(:); % Vectorize matrix if processing multiple symbols
txcp = txout(nFFT-cplen+1:nFFT);
txout = [txcp; txout];

scs = 20e3;        % Subcarrier spacing in Hz
Fs = scs * nFFT/2; % Sampling rate (1.28e6 Hz)
Ts = 1 / Fs;       % Sample duration in seconds  

Tend = Ts * (length(txout)-1);
subplot(211)
hold off
plot(0:Ts:Tend,real(txout),"*")
title("Real component of transmitter output")
subplot(212)
hold off
plot(0:Ts:Tend,imag(txout),"*")
title("Imaginary component of transmitter output")

Define an FFT length longer than nFFT to cause oversampling in time domain. To aid comparison later, insert zeros into the middle of txgrid to maintain correspondence between bin centers for the original and upsampled signals. A control here allows you to adjust the integer oversampling rate used by the OFDM modulator output and demodulator input.

upFactor = 3;
nFFTUp  = upFactor * nFFT;
fftgrid = [txgrid(1:nFFT/2); ...
    zeros((upFactor-1)*nFFT,1); ...
    txgrid((nFFT/2+1):nFFT)];
% Each column of fftgrid is one OFDM symbol
txout = upFactor * ifft(fftgrid,nFFTUp);
% Vectorize the matrix to process multiple OFDM symbols
txout = txout(:);
cplenUp = cplen * upFactor;
txcp = txout(nFFTUp-cplenUp+1:nFFTUp);
txout = [txcp; txout];
Ts = 1 / (upFactor*Fs);
Tend = Ts * (length(txout)-1);
subplot(211)
hold on
plot(0:Ts:Tend,real(txout))
legend ("Original","Upsampled","Location","southeast")
subplot(212)
hold on
plot(0:Ts:Tend,imag(txout))
legend ("Original","Upsampled","Location","southeast")

Figure contains 2 axes objects. Axes object 1 with title Real component of transmitter output contains 2 objects of type line. These objects represent Original, Upsampled. Axes object 2 with title Imaginary component of transmitter output contains 2 objects of type line. These objects represent Original, Upsampled.

Filter the transmission through a channel that adds noise, frequency dependency, and delay to the received signal.

hchan = [0.4 1 0.4].';
rxin = awgn(txout,40);       % Add noise   
rxin = conv(rxin,hchan);     % Add frequency dependency
channelDelay = dsp.Delay(1); % Could use fractional delay
rxin = channelDelay(rxin);   % Add delay

Add a random offset less than the CP length. An offset setting of zero models perfect synchronization between transmitted and received signals. Any timing offset less than the CP length can be compensated by equalization via an additional linear phase. To directly compare signals at different rates, prior to the FFT processing, normalize the synchronized signal by the upsampling factor.

offset = (randi(cplenUp) - 1); % random offset less than length of CP
% Remove CP and synchronize the received signal
rxsync = rxin(cplenUp+1+channelDelay.Length-offset:end);

rxgrid = fft(rxsync(1:nFFTUp),nFFTUp)/upFactor;

Practical systems require estimation of the channel as part of the signal recovery process. The combination of OFDM and a CP simplifies equalization to a complex scalar for each frequency bin. As long as the latency falls within the length of the CP, synchronization is accomplished by the channel estimator. A control here allows you to experiment by disabling the equalization at the receiver front end.

useEqualizer = true;
if useEqualizer
    hfchan = fft(hchan,nFFTUp);
    % Linear phase term related to timing offset
    offsetf = exp(-1i * 2*pi*offset * (0:nFFTUp-1).'/nFFTUp);
    rxgrideq = rxgrid ./ (hfchan .* offsetf);
else % Without equalization errors occur
    rxgrideq = rxgrid;
end
rxgridNoZeroPad = [rxgrideq(1:nFFT/2); ...
    rxgrideq((1+(upFactor-0.5)*nFFT):end)];
rxsymbols = qamdemod(rxgridNoZeroPad,M,UnitAveragePower=true);
if max(txsymbols - rxsymbols) < 1e-8
    disp("Oversampled receiver output matches transmitter input.");
else
    disp("Received symbols do not match transmitted symbols.")
end
Oversampled receiver output matches transmitter input.

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