Re: DFT or DFS: Are they the same thing?

On 17 Aug, 10:25, "Phil O. Sopher" <inva...@xxxxxxxxxxxxxxx> wrote:
"glen herrmannsfeldt" <g...@xxxxxxxxxxxxxxxx> wrote in message

news:h5i9nr$c7r$3@xxxxxxxxxxxxxxxxxxx

robert bristow-johnson <r...@xxxxxxxxxxxxxxxxxxxx> wrote:
< or are these several variants of the Fourier Transform merely
< degenerate or special cases of one unified Fourier Transform
< definition.
I think there is some advantage to the discrete transforms
over continuous integrals of delta functions.  It is easier
to think about, which is often reason enough.
Also, some people don't like delta functions.

The proto-Bible (or proto-Koran, etc,  depending upon into which version
of make-believe you were brain-washed when young) of DSP,
Oppenheim & Schafer, 1975,
....
It is unfortunate IMHO that a throw-away remark targetted at the
mathematically
juvenile should have been seized upon and promulgated by later authors when
a simpler and more rigorous presentation has been available from the very
early days via O&S.

I don't see what sampling/reconstruction has to do
with the relation between the DFT and the DTFT?

Oppeheim & Schafer didn't have the best starting point for
their treatment of the DFT. The paper

Cooley, Tukey & Lewis: "The Finite Fourier Transform"
IEEE Trans. Audio & Electronics, Vol 17 No 2, 1969,

seems to be where the mess starts.

In the introduction the authors start out right.
They point out that one usually wants to discuss
functions on a continuous domain, but have to use
discrete sequences of finite length for numerical
computations:

"Given that we have to operate on such sequences,
say, X(j) for j = 0,1,...,N-1, it is natural to
develop a spectral theory for them, i.e., to define
a particular orthogonal transformation which takes
the sequence X(j) into another sequence, say A(n),
of the same length as X(j) and which describes
the frequency structure of X(j)."

The next paragraph shows how the authors totally
underestimate the topic, as well as their readers:

"There are few specific and extensive expositions
of the [DFT] in th eliterature and in particular
in books on Fourier analysis. This is probably
because the theory is relatively simple, and
because it can be subsumed under the theory of
Fourier analysis on locally compact Abelian groups."

So the authors consider the theory of the DFT to
be trivial, if the readers know Abelian groups.
They may be right, I don't know. I don't know
Abelian groups.

At last the damage is done:

"[I]t will be shown below that both computationally
and mathemathically the defining relationship of the
[DFT] automatically forces the assumption that the
sequence X(j) repeats itself periodically on all
integers outside the of the range j=0,1,...,N-1.
Any other assumption, say, that the X(j) has the
value zero for j outside the range 0,1,...,N-1,
produces a different theory, i.e., the theory
of Fourier series."

The blunder, of course, is that the authors don't
see or understand the difference between the statements
"X(j) is infinitely long and periodic with period N"
and "The DFT behaves as if the input sequence is
infinitely long and periodic with period N."

Again, I don't know what an "Abelian group" is.
Maybe this periodic *behaviour* of the DFT is a
consequence of it satisfying the axioms of an
Abelian group.

As for O&S' 1975 book, it is clear from the phrase
on page 87,

"There are several points of view that can be taken
toward the derivation and interpretation of the DFT
representation of a finite-duration sequence"

that Oppenheim & Schafer are fully aware of the blunder
made by Cooley & al being so unequivocal about the
interpretation of the DFT.

Rune
.

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