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From: sjreeves@eng.auburn.edu (Stan Reeves)
Subject: Re: DFT of non-equispace samples?
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Date: Tue, 2 May 1995 14:06:02 GMT
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In article <3mrtg2$k76@csgrad.cs.vt.edu>
liu@csgrad.cs.vt.edu (Xiangdong Liu) writes:
> 
> I have the problem of interpreting the DFT of non-equispace sampled
> functions or signals.  I would appreciate any comments or references.
> As an example, let f(t) be a continuous function.  Sample the function
> to get f_i = f(t_i) where  
> 
>   t_i = 3k         i = 3k
>         3k + p     i = 3k + 1
>         3k + q     i = 3k + 2 
> 
> and where 0 < p < q < 3, but p <> 1, q <> 2.
> 
> Now I have a sequence of numbers {f_i}, i= 0, 1, 2, ...., N-1.
> 
> I can compute the discrete Fourier transform of {f_k}.  How is this
> computed DFT related to the DFT of equispaced samples?  How is it 
> related to the true Fourier transform of f(t)?
> 
> It seems easy to deal with the above kind of periodic non-equispace
> sampling using continuous FT if i runs from -infinity to +infinity.
> The samples can be thought of as the sum of three sets of equal spaced 
> samples where each set has a simple FT representation in terms of
> that of f(t).  But when i is finite, the closed form DFT of the
> non-equispaced samples seems to be so complicated that I don't know
> what the computed DFT complex numbers really mean.

Check out the following references:



"Symmetry Stabilization for Fast Discrete Monomial Transforms and
Polynomial Evaluation", Moore, Healy, and Rockmore, Linear Algebra
and its Applications - Special Issue on Computational Linear Algebra
in Algebraic and Related Problems, 192:249-299, October, 1993.

J.D.\ Markel, ``FFT Pruning,'' {\it IEEE Trans.\ Audio and
Electroacoustics,} vol.\ AU-19, No.\ 4, Dec.\ 1971.

Also, Burrus and the Rice crowd did a lot on this subject.  I don't have
any specific references, but you may be able to do a literature search to
dig them up.

Here are a couple more:

B.M. Kim and L.P. Heck, ``Automatic Design of Parallel
        Implementations of DSP Algorithms,'' {\em Proc. of 
        IEEE Intern. Conf. on Acoustics, Speech, and Signal
        Processing}, Albuquerque, New Mexico, April 1990.

L.P. Heck, D.A. Schwartz, R.M. Mersereau, and
        J.H. McClellan, ``Symbolic Simplification of Digital Signal
        Processing Software,'' {\em Proc. of IEEE
        Intern. Symp. on Circuits and Systems}, Portland,
        Oregon, May 1989.


--
Stan Reeves
Auburn University, Department of Electrical Engineering, Auburn, AL  36849
INTERNET: sjreeves@eng.auburn.edu
WWW: http://www.eng.auburn.edu/~sjreeves
