Justin Salamon wrote:
... That said, whilst zero padding will give you an interpolated spectrum in the frequency domain, you may still miss the "true location" of your peaks, ...
I think there's a difference between using an FFT on a sampled signal to have an idea of what frequencies might be there, or to try to decode information (like in digital communication systems), or have a specific purpose with it (like in Musical DSP).
To make an Fast Fourier Transform approximate a "full" Fourier Transform requires two main issues to be resolved: the "binning" that is inherent in the FFT, and sampling aspect, that matters as soon as frequencies aren't far away from the Niquist rate, or when you want more accuracy than the basic accuracy you get from run of the mill FFTs in things like Software Radio and Audio Filtering.
For the theoretically inclined: approximating a full Fourier Transform requires time interpolation of the samples to a (possibly much higher) sampling frequency, and on top of that a very long FFT, and proper analysis of the results of the FFT. Usually FFT results differ if you take a different length for the FFT, at least all kinds of errors because of waves not "fitting" in the FFT (time/sample) interval will make the signal to noise ratio not as high as you'd want. The full FT can distinguish between waves with only a very little frequency difference, which is usually a problem with FFTs and all kinds of averaged FFTs.
It's a pretty sport to write down some formulas and do computations, but I've noticed that all too often, real accuracy can easily get lost at the theoretical level, before that stage.
T.V. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
