How do I freqency shift a signal by a fractional amount using IFFT and FFT?

Let me preface this with: I am an extreme novice when it comes to FFTs.

I have a signal vector x of length 1200. To shift the frequency of the signal I am doing the following:

    y=IFFT(x);

    SHIFT=1;
    t=[1:1200]*2*pi/1200;
    for j=1:1200
    mult(j)=exp(-i*SHIFT*t(j));
    z=mult.*y;
    end

   w=real(fft(z));

When SHIFT is an integer I get what I am looking for in w, which is the signal x shifted by the value of SHIFT. However, when SHIFT is fractional the vector w looks like some combination between a shifted x and its derivative.

My question is: what do I do to shift x by a fractional amount using FFTs and IFFTs?

Thanks,

Victoria

Answer 1

[Matt J]

You would do something like this

    N=length(y);
    mult = exp(-1i*2*pi/N*(0:N-1)*SHIFT);

    w=ifft((fft(y).*mult ),'symmetric')

[Matt J]

This is equivalent to shifting a sinc-interpolated version of y. Unless you really can't live with linear interpolation, though, directly shifting/resampling in the original domain, without FFTs is going to be more efficient. Compare:

    SHIFT=0.3;
    y=rand(1,1e7);
    N=length(y);
    ax=(0:N-1);

    tic
         mult = exp(-1i*2*pi/N*(0:N-1)*SHIFT);

         w=ifft((fft(y).*mult ),'symmetric');
    toc;
    %Elapsed time is 1.242511 seconds.

    tic;
      w= interp1(y,ax+SHIFT,'linear');
    toc
    %Elapsed time is 0.512226 seconds.

    F=griddedInterpolant(y,'linear');
    tic
      w=F(ax+SHIFT);
    toc
    %Elapsed time is 0.312686 seconds.

[Matt J]

Victoria commented:

Thanks. That did the trick.

[Matt J]

You're welcome. If this resolves your question, the normal thing for you to do now would be to Accept-click my Answer.

[rashi]

but the output for w,w1 and w2 are different, will those be not same?

[Matt J]

No, the first version does not use linear interpolation and so will be different from the others.