On 11/27/05, dasimov@earthlink.net <dasimov@earthlink.net> wrote:
I hope some math-funner can help me with a DFTF question:
The DTFT of a complex doubly-infinite sequence x := {x[n] in C | -oo < n < oo} is given by
X(w) = sum over all integers n of x[n]*exp(i n w).
A textbook I'm teaching from states that if x[n] is defined as the "unit step sequence":
x[n] = 0, n < 0; x[n] = 1, n >= 0
by definition, X[w] =sum_n=0^oo exp((i n w), and that by some mystical procedure,
(***) X[w] = 1/(1 - exp(iw)) + pi * sum over all integers n of delta(w + 2*pi*n)
(where delta = Kronecker delta function).
I don't know how to interpret this, since of course in the normal sense X(w) doesn't converge for any w.
But in some generalized function sense, can someone please explain equation (***), and if possible at least hint at a rigorous foundation (just in case it has one).
Thanks,
Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's a long time since I did any of this engineering stuff, and my library is currently stuffed into crates; but I'll stick my neck out and try to help. The "delta" is not a Kronecker delta, but an infinite spike whose integral is nonzero (here pi): this is an example of a "distribution", a generalisation of the classical notion of "function" which permits such limiting cases to be discussed compactly. The spike series on the RHS takes care of the divergent to oo cases when w is a multiple of 2 pi; the function part takes care of the other w [when the DFT sum does converge!]. Having said this, I'm still puzzled about what's supposed to happen to the function on the RHS when it evaluates to 1/0 --- hard to tell at this distance, but perhaps the author considers it a spike with weight zero ... Maybe google on "distribution+function+fast/discrete Fourier transform" could turn up some rapid assistance; or a quick look at Wikipedia? Fred Lunnon