Sampling frequency offset vs carrier frequency offset effects

Question

I am reading about the sampling frequency offset (SFO) of OFDM system. According to my understanding and all sources I read, it can be defined in similar way to the carrier frequency effect (CFO). I mean the effect is added into the signal in time domain as following:

$x[n] = x[n] e^{2\pi j \frac{\Delta f_s}{f_s}n}$

where $n$ is $0,1,2,... N-1$ and $N$ the length of the signal $x$, and $\varepsilon = \frac{\Delta f_s}{f_s}$ is the presented effect of the sampling frequency offset. Here is a source explaining it with a clear way : https://ieeexplore.ieee.org/document/957773

I have also read those sources describing that effect, Offset in Carrier and Timing

and

What is the difference between (Sample Timing Offset), (Carrier Frequency offset), and (Sampling Frequency Offset)?

I agree with the above description, but what I am looking for is related how to add the effect of sampling frequency offset into the signal, is my way above right? if so, how can I convert the added percentage $\varepsilon$ into ppm? for example if I added $\varepsilon = 0.00001$, what is the equivalent value in ppm?

On the other hand, if the way of adding that effect is not right, could please guide me how to add it with clear steps? However, that was described here How to Emulate Sample Frequency Offset for OFDM in Matlab? but I couldn't understand how to do it in matlab.

Update

Here is the the code where I tried to add the SFO using the Eq. explained here How to Emulate Sample Frequency Offset for OFDM in Matlab? :

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,[]));
    for sy = 1 : n_sym
        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)))/200000;   %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


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); 

And, here is the figure with the effect of 5 ppm.

It's ok now, But the question now why do I find a rotation also when I add 1 ppm ? However, that normally should be more similar into the ideal constellation.

Answer 1

If $x[n]$ is your OFDM signal in the time domain, and you compute $x[n] e^{j 2\pi \frac{\Delta f_s}{f_s} n}$ then you are simulating a carrier frequency offset of magnitude $\Delta f_s$ given a carrier $f_s$. For an OFDM signal with only one active carrier (SC-FDM), that might be a good enough approximation to carrier frequency offset as well.

If I were you and wanted to learn more about the fundamentals of OFDM, and the effect of sampling frequency offset (SFO) on the OFDM signal. I would write a function that computes the OFDM signal naively and allows me to vary the sampling period.

If you recall, an OFDM signal transforms a stream of symbols $s[n]$ into $N$ parallel streams of symbols $s_k[n] = s[n N + k]$. Let's ignore the computationally efficient transmitter and just try to replicate the signal. For a basic OFDM system (by basic I mean that I will not apply a window or a filter to the system to improve its spectral properties, but you could do that on your own once you understand the basic approach I'm putting forth here), you can form the $N^{\mbox{th}}$ OFDM symbol by multiplying each subcarrier by the complex gain $s_k[n]$ and summing the result. This is done for a length of time equal to your symbol period $T$ plus your cyclic prefix length $T_{CP}$. We can compute the OFDM symbol from time $t=n (T + T_{CP})$ until time $t=(n + 1) (T + T_{CP})$ at any given time instant. Let's define a new variable $\tau=0$ when $t = n (T + T_{CP})$. Now we can compute the OFDM symbol for any instant $0 \leq \tau \leq T + T_{CP}$ as follows $$ y_n(\tau) = \sum_{k=0}^{N-1} s_k[n] e^{j 2\pi \frac{k}{f_s} \tau}. $$ and every value outside the range $0 \leq \tau \leq T + T_{CP}$ is understood to be zero.

This can be used to write a formula for the entire sequence: $$ y(t) = \sum_{n=0}^{M/N-1} y_n(t - n (T + T_{CP})). $$

This formula assumes that the original sequence $s[n]$ was of length $M$. If it wasn't, just zero pad it and the equation still holds. Now, we can sample this at any rate we want by simply computing the samples at $t = m T'_s$, $$ y(m T'_s) = \sum_{n=0}^{M/N-1} y_n(m T'_s - n (T + T_{CP})). $$

So, now we can define a function that computes $y_n(\tau)$ and $y(t)$ and use them to compute your OFDM transmit signal sampled at an arbitrary rate $T'_s$. It is late where I am, so I am going to head to bed. Maybe you can see how to translate this into Matlab already? If so, that is great. If not, I will see what I can do to provide you with a simple example later.