% the fftbased spectra
NN=2048; th=linspace(0,2*pi,NN);
Y =abs(fft(y,NN))/sqrt(N);
Y = Y.^2;
plot(th,Y);
FFTbased methodsWe begin with two classical formulae. The correlogram is defined as where denotes an estimate of the covariance lag , whereas the periodogram The two coincide when the correlation lag is estimated via where . The periodogram can be efficiently computed using the fast Fourier transform (FFT). There is a variety of methods, such as Welch and BlackmanTukey methods, designed to improve the performance using lag window functions either in the time domain or in the correlation domain. In situations when the data length is short, to get a smooth spectrum, we may increase the data length by padding zeros to the sequence. Using the above example, we pad up by adding zeros. The corresponding FFTbased spectrum is shown in the following figures. Rudimentary code is displayed as a demonstration. A file to reproduce the following results can be downloaded. Alternative matlab buildin routines for periodograms are periodogram, pwelch, etc. % the fftbased spectra
NN=2048; th=linspace(0,2*pi,NN);
Y =abs(fft(y,NN))/sqrt(N);
Y = Y.^2;
plot(th,Y);
