Question

In Matlab, I frequently compute power spectra using Welch’s method (pwelch), which I then display on a log-log plot. The frequencies estimated by pwelch are equally spaced, yet logarithmically spaced points would be more appropriate for the log-log plot. In particular, when saving the plot to a PDF file, this results in a huge file size because of the excess of points at high frequency.

What is an effective scheme to resample (rebin) the spectrum, from linearly spaced frequencies to log-spaced frequencies? Or, what is a way to include high-resolution spectra in PDF files without generating excessively large files sizes?

The obvious thing to do is to simply use interp1:

  rate = 16384; %# sample rate (samples/sec)  
  nfft = 16384; %# number of points in the fft

  [Pxx, f] =  pwelch(detrend(data), hanning(nfft), nfft/2, nfft, rate);

  f2 = logspace(log10(f(2)), log10(f(end)), 300);
  Pxx2 = interp1(f, Pxx, f2);

  loglog(f2, sqrt(Pxx2)); 

However, this is undesirable because it does not conserve power in the spectrum. For example, if there is a big spectral line between two of the new frequency bins, it will simply be excluded from the resulting log-sampled spectrum.

To fix this, we can instead interpolate the integral of the power spectrum:

  df = f(2) - f(1);
  intPxx = cumsum(Pxx) * df;                     % integrate
  intPxx2 = interp1(f, intPxx, f2);              % interpolate
  Pxx2 = diff([0 intPxx2]) ./ diff([0 F]);       % difference

This is cute and mostly works, but the bin centers aren’t quite right, and it doesn’t intelligently handle the low-frequency region, where the frequency grid may become more finely sampled.

Other ideas:

• write a function that determines the new frequency binning and then uses accumarray to do the rebinning.
• Apply a smoothing filter to the spectrum before doing interpolation. Problem: the smoothing kernel size would have to be adaptive to the desired logarithmic smoothing.
• The pwelch function accepts a frequency-vector argument f, in which case it computes the PSD at the desired frequencies using the Goetzel algorithm. Maybe just calling pwelch with a log-spaced frequency vector in the first place would be adequate. (Is this more or less efficient?)
• For the PDF file-size problem: include a bitmap image of the spectrum (seems kludgy–I want nice vector graphics!);
• or perhaps display a region (polygon/confidence interval) instead of simply a segmented line to indicate the spectrum.

Answer 1

Solution found: https://dsp.stackexchange.com/a/2098/64

Briefly, one solution to this problem is to perform Welch’s method with a frequency-dependent transform length. The above link is to a dsp.SE answer containing a paper citation and sample implementation. A disadvantage of this technique is that you can’t use the FFT, but because the number of DFT points being computed is greatly reduced, this is not a severe problem.