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From: bart@videovax.tv.tek.com (Bart Massey)
Newsgroups: comp.dsp
Subject: Re: FFTs of Low Frequency Signals
Message-ID: <5619@videovax.tv.tek.com>
Date: 9 Nov 89 22:57:07 GMT
References: <10208@cadnetix.COM> <2586@irit.oakhill.UUCP>
Reply-To: bart@videovax.tv.tek.com (Bart Massey)
Distribution: usa
Organization: Tektronix TV Measurement Systems, Beaverton OR
Lines: 46

In article <2586@irit.oakhill.UUCP> diaz@irit.UUCP (Rafael Diaz) writes:
> In article <10208@cadnetix.COM> jensen@cadnetix.COM () writes:
> >What I am wondering about is the behavior of the transform when the
> >signals being examined have significant energies in the lower frequency
> >ranges.
> 
> From all spectral estimation techniques the FFT/DFT based has the LARGEST
> variance, therefore, using it for low frequency high resolution detection
> is the worst.
> 
> What you need is a good ARMA based technique, namely BURG and and LS. 

I don't think your choice of estimation technique really has very much to
with the frequency band of interest.  It's true that AR and ARMA models are
much better estimators of the characteristics of a single sinusoid than the
DFT, but that's sort of independent of frequency.  As other posters have
pointed out, the problem with really low frequency signals is that the
corresponding record lengths are very large, and most algorithms for signal
estimation have costs which grow as the square of the record length or so.

The typical technique for coping the problem is, I believe, to "make the
signals higher-frequency", through the magic of decimation.  Assume that all
the signals of interest are in the range 0-1KHz, and you're sampling at 16KHz
(32000 samples/sec).  If necessary, run a low-pass filter over the input, to
remove signals in the range 1-16KHz.  Note that the record length of this
filter will itself be very large, but at least it's a linear process, so
*relatively* cheap.  Now, just take every 16th sample!

What've you just done?  You've mapped the sample frequency domain 0-1KHz
onto the range 0-16KHz!  Now, if you do e.g. a 16-point FFT on your new
record, your bin centers will be at 500Hz, 1500Hz, .. , 15500Hz in the new
range, which corresponds to (500/16)Hz .. (15500/16)Hz, i.e. 31.25Hz ..
968.75Hz in the original domain.  Note that even with the FFT cost of O(N) =
N lg N, you have saved quite a bit over doing the FFT directly on the
original data.  In particular, for the FFT on the original data, the cost
was roughly 256 * 8 = 2048 units, whereas the low-pass filter plus the short
FFT costs roughly 256 + 16 * 4 = 320 units -- a significant savings, even
though you get the same amount of info about the band in question either way!

Note that using ARMA techniques on the decimated data provides an even bigger
efficiency win, since these techniques are typically O(N) >= N^2 !  In short
(no pun intended :-), decimate your data -- IMHO you'll be glad you did!

					Bart Massey
					..tektronix!videovax.tv.tek.com!bart
					..tektronix!reed.bitnet!bart