Path: news.mathworks.com!newsfeed-00.mathworks.com!newsfeed2.dallas1.level3.net!news.level3.com!postnews.google.com!k30g2000yqf.googlegroups.com!not-for-mail
From: Greg <heath@alumni.brown.edu>
Newsgroups: comp.soft-sys.matlab
Subject: Re: FFT and DFT in matlab
Date: Tue, 11 Aug 2009 11:10:19 -0700 (PDT)
Organization: http://groups.google.com
Lines: 49
Message-ID: <863221bc-5347-4d03-b46b-1855f757921e@k30g2000yqf.googlegroups.com>
References: <32904364.31308.1240942509387.JavaMail.jakarta@nitrogen.mathforum.org> 
	<h5c7nj$6ba$1@fred.mathworks.com> <5d217906-c5db-4cc7-97bf-ef26c62e04a2@t13g2000yqt.googlegroups.com>
NNTP-Posting-Host: 69.141.163.135
Mime-Version: 1.0
Content-Type: text/plain; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
X-Trace: posting.google.com 1250014219 18018 127.0.0.1 (11 Aug 2009 18:10:19 GMT)
X-Complaints-To: groups-abuse@google.com
NNTP-Posting-Date: Tue, 11 Aug 2009 18:10:19 +0000 (UTC)
Complaints-To: groups-abuse@google.com
Injection-Info: k30g2000yqf.googlegroups.com; posting-host=69.141.163.135; 
	posting-account=mUealwkAAACvQrLWvunjg50tRAnsNtJR
User-Agent: G2/1.0
X-HTTP-UserAgent: Mozilla/4.0 (compatible; MSIE 7.0; Windows NT 5.1; .NET CLR 
	2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe)
Xref: news.mathworks.com comp.soft-sys.matlab:562467


On Aug 5, 6:39 pm, "Steven G. Johnson" <stev...@alum.mit.edu> wrote:
> On Aug 5, 11:16 am, "Haroun Youssef" <aga...@yahoo.com> wrote:
>
> > Matlab is usingFFTnot DFT, so it is considering that the samples are unifrom.
>
> AnFFTis an algorithm for computing the DFT (the name of the abstract
> mathematical transformation, not a specific method to compute this
> transformation).
>
> It would be more accurate to say that a transform with non-equispaced
> data is not a DFT.

Given the amount of literature on NDFTs (NUDFTs, etc) this
appears to be a personal view point.

Speaking of viewpoints, I think there are basically 2:

1. Approximations for the analysis of continuous signals
   and systems.
2. Analysis of discrete signals and systems.

Taking the former viewpoint, a DFT should be just what the
initials imply: A specific sum-based approximation to the
continuous CFT integral for functions sampled over a finite
interval T = t2-t1. There should be no constraint that
samples are uniformly spaced.

The primary constraint should be that the discrete time
transform converge to the continuous time transfornm as
N --> inf but sum(dti,i = 1:N) = T.

A secondary constraint should be that the DFT reduce to
the well known formula when t1 = 0, dti = dt = T/N =
constant and tN = (N-1)*dt = T-dt.

However, the best rectangular approximation to the integral
approximates the area from t1 to t2 with endpoint rectangles
of width dt/2. Whereas the traditional uniformly spaced DFT
best approximates the area from t1-dt/2 to tN+dt/2 with the
time sample centered at the middle of a rectangle.

So ... when applying a NDFT formula to uniformly spaced
points take the endpoint factors of 2 into account when
comparing with the traditional result.

Hope this helps.

Greg