Conceptual question on FFT and chirp signal

Question

If I take the FFT of a sinusoid I will get a plot whit all the energy of the signal concentrated at the sinusoid frequency. But what happens if I have a signal in which the frequency keeps changing?(like the chirp: linear and exponential)

Is it possible to compute the FFT of a chirp signal? And what should I expect as result?

I tried to do it in Matlab creating a chirp using the matlab function and then passing it to the FFT function but I do not know if this makes any sense and how to 'read ' the result I get.

Could you also suggest me some readings (papers, books) on the topic?

Answer 1

FFT is an fast algorithm to compute DFT

So it works on finite length of samples. This fact has some side effects on the spectrum that it will generate for signal, but generally it consist only signal frequencies that it includes on its time window of length nfft.

When u sweep frequency in your sinusoidal, in fact you are doing some sort of frequency modulation. when you calculate FFT of this signal. You would see power in all frequencies that sinusoidal has been modulated to in the window that you calculate your FFT.

Consider following example I wrote in Matlab:

fs = 48e3;
t = 0:1/fs:20e-3;
x = sin(2.*pi.*(10e3 - (2e3 .*(abs(t-10e-3)/10e-3))).*t);

This signal would sweep frequency from 8K to 10K in [0 , 10ms] and from 10K to 8K in [10ms, 20ms] and has been sampled by 48KHz frequency. To calculate its FFT:

L = length(x);
xf = fft(x)/L;
xf_s = fftshift(xf);
f = Fs/2 * linspace(-1,1-2/L,L);

Then you can plot its DFT spectrum representation using

plot(f,abs(xf_s));

and:

Just like most of frequency modulated signals. Note that here we calculated FFT for full length of signal, if we calculate it for only some portion of signal we would have frequencies power for that portion.

Answer 2

Your question is deeper than you think. For signals whose spectra is time varying,like your chirp signal, you lose this spectral variation information in simple Fourier transforms. You must consult time-frequency analysis to retain the full information of both time variation and spectral variations.

Consider using "specgram()" function of matlab which computes and plots time-frequency spectograms of signals.

Consider the following code from matlab:

  t=[0 0.5 1.0 1.5 2.0]; % time breakpoints
  f=[0 200 100 150 300];  % instantaneous frequency breakpoints
  p=polyfit(t,f,4);      % fit 4th order polynomial over time
  t=0:0.001:2;           % 2 secs @ 1kHz sample rate
  y=chirp(t,p);
  figure,specgram(y,128,1E3,128,120); % 2D time-frequency display

On the contrary a single FFT of the complete signal, which includes all the frequencies, nevertheless, loses their time variation information.

Answer 3

If you were talking about a Laplace (or Fourier) transform then any function with time varying coefficients has a valid transform. But you have to do the analog/continuous calculation. I would imagine that the same holds true for z transforms except for aliasing; which is not trivial. My experience is that this problem,chirp, is a standard example in Wavelet analysis.

Some internet references:

In some detail: http://perso.ens-lyon.fr/patrick.flandrin/SPIE01_PF.pdf

Practical application?: http://www.phys.ufl.edu/ireu/IREU2014/pdf_reports/Eve_Report.pdf

If you become interested is line of analysis try: @book{meyer1992wavelets, title={Wavelets and applications}, author={Meyer, Yves}, year={1992}, publisher={Paris [etc.]: Masson; Berlin [etc.]: Springer-Verlag} }

Quite dated but understandable since they were still in the justification phase. Section 1a. A comparison with short term Fourier analysis is 1b. Except amazon lists \$83 new; looking at my copy I paid $7.50. I really don't understand technical book pricing.