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.

Answer 2

[Duncan Carlsmith]

analytic_segment = hilbert(segment); % Complex analytic signal

shifted_segment = real(analytic_segment .* exp(1j*2*pi*f_shift*t));

[Paul]

Hi Duncan,

Can you illustrate this approach with a concrete example?

[Duncan Carlsmith]

%% Frequency-Shift Demo with Synthetic Audio
% This script creates a 1-s synthetic audio signal consisting of four
% sinusoids (1, 2, 3, 4 kHz) of decreasing amplitude plus Gaussian noise,
% shifts it up by 500 Hz, and compares the spectra and waveforms.

% -------------------------------------------------------------------------
% Parameters
% -------------------------------------------------------------------------
fs       = 44100;    % Sampling rate (Hz)
duration = 1.0;      % Signal length (s)
f_shift  = 500;      % Desired frequency up-shift (Hz)

% -------------------------------------------------------------------------
% Construct synthetic signal  (column vector)
% -------------------------------------------------------------------------
t      = (0:1/fs:duration-1/fs).';          % Time axis
amp    = [1.0 0.8 0.6 0.4];                 % Amplitudes
freqs  = [1000 2000 3000 4000];             % Frequencies (Hz)

segment = zeros(size(t));
for k = 1:numel(freqs)
    segment = segment + amp(k)*sin(2*pi*freqs(k)*t);
end
segment = segment + 0.2*randn(size(t));     % Additive white Gaussian noise

% -------------------------------------------------------------------------
% Play original audio
% -------------------------------------------------------------------------
soundsc(segment,fs);
pause(2);                                   % Brief pause

% -------------------------------------------------------------------------
% Frequency-shift via analytic signal
% -------------------------------------------------------------------------
analytic_segment = hilbert(segment);        % Analytic (complex) signal
shifted_segment  = real(analytic_segment ...
    .* exp(1j*2*pi*f_shift*t));

% -------------------------------------------------------------------------
% Play shifted audio
% -------------------------------------------------------------------------
soundsc(shifted_segment,fs);
pause(2);

% -------------------------------------------------------------------------
% Power spectra (overlaid)
% -------------------------------------------------------------------------
% 1 Hz frequency resolution for clear peak separation
[P_orig,F]  = pspectrum(segment,        fs,'FrequencyResolution',3);
[P_shift,~] = pspectrum(shifted_segment,fs,'FrequencyResolution',3);

figure;
plot(F,10*log10(P_orig),'b','LineWidth',1.2,'DisplayName','Original');
hold on;
plot(F,10*log10(P_shift),'r','LineWidth',1.2,'DisplayName','Shifted');
xlim([0 6000]);
xlabel('Frequency (Hz)');
ylabel('Power/Frequency (dB/Hz)');
title('Power Spectrum: Original vs. Frequency-Shifted');
legend('Location','best');
grid on;
ylim([-60,0])










% -------------------------------------------------------------------------
% Time-domain waveforms (overlaid)
% -------------------------------------------------------------------------
figure;
plot(t,segment,        'b','DisplayName','Original'); hold on;
plot(t,shifted_segment,'r','DisplayName','Shifted');
xlabel('Time (s)');
ylabel('Amplitude');
title('Waveforms: Original vs. Frequency-Shifted');
legend('Location','best');
grid on;
% Uncomment the next line to zoom into the first 10 ms:
xlim([0 0.01]);