High error when adding small value of SFO into the OFDM signal

Question

I am trying to emulate the SFO effect for OFDM system, I did it using linear interpolation as shown HERE . That can be explained as if I have the time domain signal $s_n$ representing the output of ifft (let’s ignore the noise and the guard interval), so the SFO can be inserted using linear interpolation. Below is the code I made for that, and the resulted rotated signal without compensating the SFO.

clc; clear all; close all;
n_sym = 10;  % The number of symbols
N = 1024;    % The symbole length 
mod = 4;     %the modulation order 
len= n_sym* log2(mod)*N;  %Lenght of whole data (N * Number of symbol* M)


%This part will generate binary vector as per length entered by user
data=floor(rand(1,log2(mod)*len)+0.5);
%Mapping of binary data
mapper_out = qammod(bi2de(reshape(data,[],log2(mod))), mod,'UnitAveragePower', true);
% Take the iFFt operation after S/P operation 
sn = ifft(reshape(mapper_out,N,[]));

%===== Here using interpolation to add 1 ppm SFO =====
for sy = 1 : size(sn,2)
    S_y = sn(:,sy);       %Taking every symbol separatly
    for nn = 1 : length(S_y)-1 
        X_te(nn) = S_y(nn) + (nn*(S_y(nn+1) - S_y(nn))/1e6);   %Doing linear interpolation with 5ppm
    end 
        X_te2(:,sy) = [S_y(1); X_te.'];                 %[x[1];  x[nn]]
end 
  out = fft(X_te2);  out = out(:);  %P/S

  %===Calculate the BER 
data_rec = qamdemod(out, mod); 
b_rec = reshape(de2bi(data_rec,log2(mod)),[],1).';
[BER_1 Ratio_1] = biterr(b_rec(1:68),data(1:68))
%========== 

figure;plot(real(out),imag(out),'b+');title('constellation with and without  Sampling Frequency Offset'); hold on; 
plot(real(mapper_out),imag(mapper_out),'r+','LineWidth',3); 

The problem is that, when adding an SFO effect of 1 ppm, the signal is completely rotated as shown in the above figure, and the BER cannot be recovered at all as shown in the above code. However, when having 1 ppm SFO, it’s expected to have little bit similar constellation to the ideal signal, and the BER performance is supposed to be almost 0.

Update

Normally, the added SFO effect into the signal $x[n]$ as following:

$y\left [ n \right ] = x\left [ n \right ] + n \times \frac{x\left [ n + 1 \right ] - x\left [ n \right ]}{10^6}$

The dominator is $10^6$ means that 1 ppm is added. Assuming the length of the symbols is $N = 1024$ as I insert the SFO for each symbol alone, so the maximum added SFO will be $1024$ multiplied by the difference between $x[N]$ and $x[N-1]$.

On the other hand, if we can normalize the added SFO by $N$ to have the maximum added SFO 1ppm multiplied by the difference between $x[N]$ and $x[N-1]$, the above equation will be:

$y\left [ n \right ] = x\left [ n \right ] + n \times \frac{x\left [ n + 1 \right ] - x\left [ n \right ]}{N \times 10^6}$

Unfortunately, the rotation is still very big and the signal is completely deteriorated.

Answer 1

The sampling frequency offset will induce a time offset on every symbol that is increasing from symbol to symbol. This results in the rotation as observed.

Frequency is the derivative of phase with respect to time, so a constant frequency offset would result in a linearly increasing phase versus time.

The referenced link provides more detailed formulas as to the actual offsets but the following graphic should provide more intuitive insight into both Sampling Frequency Offset (SFO) and Sampling Time Offset (STO) on one OFDM subcarrier, where below we see the sampling clock versus the ideal sampling location on the time domain.

What else is interesting which is a direct result of time - frequency duality is if we change the above axis to frequency instead of time. With this we can observe the constellations over frequency at one time instead of above with is at one frequency over time. In this case the results will swap in that if we observed the constellation from sub-carrier to sub-carrier (over frequency) at any given time, the SFT will result in a fixed offset such as the bottom plot and the STO will result in a rotation. This may be the source of confusion for the OP.