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From: Phil Hobbs
Subject: Re: fft in hc11 again
Date: Sat, 12 Oct 2002 20:34:47 -0400
Organization: IBM Global Services North -- Burlington, Vermont, USA
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NNTP-Posting-Date: 13 Oct 2002 00:34:59 GMT
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Chuck Simmons wrote:
> False for several reasons. There does not exist a meaningful FFT for a
> buffer length of 3998, for example. As far as I can remember, roundoff
> error is theoretically identical except that the FFT is more sensitive
> to systematic roundoff error. Roundoff error is determined by
> computational depth, number of multiply-adds per point in the final
> result. The FFT reduces computational width by reusing results at all
> depths. The depth is the same.
That isn't true. Chirp-z transforms can be used to compute DFTs
efficiently even for prime lengths, and there are fast algorithms
available for any highly-composite N.
>Data interpretation is further complicated by misconceptions. Too many
>engineers think of the DFT and the FFT as giving results that relate to
>the Fourier transform on the real line. Actually, the FFT and the DFT
>give results identical to a truncated Fourier transform on the unit
>circle. I've seen engineers chase wild geese for weeks because they
>didn't understand that fundamental fact.
(You're confusing DFTs and Z-transforms. It's the Z-transform that
happens on the unit circle--the DFT happens along the real frequency
axis. One is a conformal mapping of the other, so this is a fine
The DFT gives exact samples of the true, continuous-time Fourier
transform of a function that is band-limited and periodic with period N
samples. The error comes from failing to meet the conditions of the
sampling theorem and failing to window the data to limit the wraparound
error. It's true that people often aren't careful enough, but that
doesn't make the DFT wrong.
With care, our ordinary continuous-time notions of bandwidth and 1-Hz
SNR survive the transition into the sampled domain.
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