User:Cxw/How to use the DFT, with Matlab examples

DFT quick reference

What is it?	DFT element	matlab example and comments
How often do you want to sample?	sampling frequency $f_{s}$	fsample=100; sample at 100Hz psample=1/fsample; sample period
For how long do you want to sample?	time range $t=0\ldots t_{\rm {max}}$	tmax=10; run from 0 to 10 sec. inclusive
How many samples does that give you?	$N$	nsamples=tmax/psample+1; +1 because we include t=0 and t=tmax. nsamples=2^ceil(log2(nsamples)); round up to a power of 2 in length so FFT will work. times=(0:psample:(nsamples-1)*psample)'; Create a column vector of sample times.
How far apart are each of the frequency-domain result points?	$\Delta f={f_{s} \over N}$	delf=fsample/nsamples; frequencies=(0:delf:(nsamples-1)*delf)'; Column vector of result frequencies
What signal do you want to sample?	input $x(t)$	x=sin(2*pi*10*times)+sin(2*pi*3*times); Make a 10Hz sine wave plus a 3Hz sine wave
What are the results?	Fourier transform $X(f)=\sum _{n=0}^{N-1}x(n)e^{-j2\pi nf/N}$	fft_x=fft(x, length(x));
What frequencies does the signal have?	$X_{m}(f)=|X(f)|$	fft_x_mag=abs(fft_x);
What phase relationships?	$X_{\phi }(f)=\arctan \left({\operatorname {Im} (X(f)) \over \operatorname {Re} (X(f))}\right)$	fft_x_phase=unwrap(angle(fft_x));
How do you view the results?		plot(frequencies, fft_x_mag); Or, to match the amplitude of the magnitude peak to the amplitude of the sine wave, plot(frequencies, (2/nsamples)*fft_x_mag);
What about the power spectrum?	$X_{p}(f)=|X(f)|^{2}$	fft_x_power=fft_x_mag.^2; plot(frequencies, fft_x_power);

References

Lyons, Richard G. Understanding digital signal processing. Upper Saddle River: Prentice Hall PTR, 2001. ISBN 0-201-63467-8. Chapter 3 discusses the DFT.