Path: utzoo!utgpu!jarvis.csri.toronto.edu!mailrus!cs.utexas.edu!uunet!zephyr.ens.tek.com!tekcrl!tekfdi!videovax!bart
From: bart@videovax.tv.tek.com (Bart Massey)
Newsgroups: comp.dsp
Subject: Re: FFTs of Low Frequency Signals (really: decimation)
Message-ID: <5624@videovax.tv.tek.com>
Date: 12 Nov 89 00:04:54 GMT
References: <5619@videovax.tv.tek.com> <10208@cadnetix.COM> <2586@irit.oakhill.UUCP> <5305@orca.WV.TEK.COM>
Reply-To: bart@videovax.tv.tek.com (Bart Massey)
Organization: Tektronix TV Measurement Systems, Beaverton OR
Lines: 36

In article <5305@orca.WV.TEK.COM> mhorne%ka7axd.wv.tek.com@relay.cs.net writes:
> 
> You *can* get better resolution by doing 1) zero padding of the data, or
> 2) sampling longer at F and then decimating to reduce the data set, then
> performing the DFT on the reduced data.  Both of these methods effectively
> increase your sample duration, thereby increasing your frequency domain
> resolution.


No!  (2) will increase your resolution, (1) will not!

Certainly, zero-padding is not equivalent to increasing the sample duration
-- if you take a longer sequence of samples, you don't expect to obtain the
original sequence with a bunch of zeros appended.  As it turns out,
zero-padding the DFT input is equivalent to applying sin(x)/x interpolation
to the DFT output.  Zero-padding thus can't be adding resolution, since you
can obtain the same effect by applying a post-transformation to the output
of the original DFT on the original data.  What the zero-padding *does* do is
increase the *accuracy* of the *representation* of the data!  In other
words, in a DFT there is usually better resolution than the obvious
representation will indicate.

Given a fixed record length, the DFT can be shown to be at least as good as
any other estimator of the spectral composition of a signal, given that you
have no special a priori knowledge about the source of that signal.  There
are really only two choices for increasing your resolution; you can take
into account a better model of the signal source, as e.g. ARMA techniques
do, or you can obtain more data.

But, if you just want to better use the resolution you have, input
zero-padding, a cheap form of sin(x)/x interpolation for DFT output, is a
useful technique.

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