Re: Phase of FFT compared to phase of Sinusoid

in article 1143513341.346923.144510@xxxxxxxxxxxxxxxxxxxxxxxxxxxx, Ron N. at
rhnlogic@xxxxxxxxx wrote on 03/27/2006 21:35:

robert bristow-johnson wrote:
so what's the "time width" of a single sample, x[n], of a sampled signal?
T = 1/Fs? zero? or undefined?

None of the above.

no, the answer *is* above and the answer is "undefined".

x[n] is a sequence of numbers. an ordered set. there is no time width
assigned to it. sorry Rick for anthropomorphizing, but x[n] and X[k] know
nothing about time or frequency. they're just sequences of numbers with a
particular relationship to each other. they don't know any T or Fs, but
they *do* know about N, the size of the DFT. x[n] and X[k] don't have to be
attached to any particular physical process (but they may, if you want), so
there are no physical parameters like T or Fs or V_ref as a property of the
sequences of numbers, x[n] and X[k]. but there *is* N. N is the period of
x[n] and X[k].

In the mathematical sense, x[n] is the limit of a sequence of ever
smaller windows, all of non-zero width. (e.g. for every episilon
there is a non-zero delta, etc.)

x[n] (and X[k]) is just a sequence of numbers (that are periodic with period
N). there is no *necessary* concept of them being drawn from any physical
context or source. what you're saying sounds like it is relating the values
of x[n] to the string of hypothetical dirac impulses from the ideal sampling
process. i would say those impulses have a width of "zero" which is what
Stan was alluding to. if you want to construct such a (continuous-time)
physical signal to attach x[n] and X[k] to, it might be:


x(t) = T * SUM{ x[n] * delta(t - n*T) }
n

and the spectrum or Fourier Transform of x(t) would be:


X(f) = 1/N * SUM{ X[k] * delta(f - k/(N*T)) }
k

note the convention where parenths are for continuous-domain functions and
brackets are for discrete-domain functions. delta(.) is the Dirac impulse
function and T is the sampling time (N*T is the period in the time domain).
you can see that the periodicity remains but has scalers:


x(t + N*T) = x(t) for all t


X(f + 1/T) = X(f) for all f


this is now a case where an infinite and periodic sequence of equally spaced
impulse functions in one domain is mapped to an infinite and periodic
sequence of equally spaced impulse functions in the reciprocal domain. this
special case is very closely related to the DFT, but it is not the DFT, it
is simply a Fourier Transform (and inverse) of a contrived continuous-time
function to a continuous-frequency function that has a strict corresponding
analog in the DFT (and inverse). this case has T (and 1/T) as a parameter
as well as N. but the corresponding DFT has only N.

if we were talking about something like this, then what Stan said is sorta
true: "If you assume a periodic input, the bins are interpreted as having
infinitesimal band width", but i wouldn't call those "bins". X[k] is a
"bin" in the DFT or FFT but it's "X[k]/N * delta(f - k/(N*T))" which is an
infinitesimally thin frequency component in the spectrum of x(t) just as
"x[n] * delta(t - n*T)" is an infinitely thin function of time. but x[n] or
X[k] aren't infinitely thin anything, nor some "limit of a sequence of ever
smaller windows, all of non-zero width". they're just numbers. they could
come from anywhere or any process and someone stuffs them into a DFT.

sorry to be anal, but about this stuff, i feel compelled to be, lest
anything i write be misunderstood or misrepresented.

In the physical sense, there is a width that must be above some
multiple of Planck's constant,

you mean the Planck Time? dunno what either have to do with this.

and usually much wider; but it's all an approximation
anyway as there a no perfectly bandlimited signals in the real world,
AFAIK.

i wasn't talking about the real world or anything physical anyway. i was
talking about a mathematical issue regarding the DFT which maps a sequence
of numbers to another sequence of numbers.


(p. 532): "In recasting Eqs. (8.11) and (8.12) {Discrete Fourier Series and
inverse} in the form of Eqs. (8.61) and (8.62) {DFT and inverse} we have not
eliminated the inherent periodicity. ... The inherent periodicity is
always present. Sometimes it causes us difficulty and sometimes we can
exploit it, but to totally ignore it is to invite trouble."

There are artifacts due to any single finite DFT aperature that can
be interpreted as due to periodicity. However that interpretation
fails when considering a sequence of overlapped DFT results as
a consistant single set of data.

listen, when computing the DFT or inverse on a finite set of data, you are
fitting a collection of basis functions that are all periodic with period N.
there is no time that this periodicity is not there. whether that
periodicity is responsible for some artifact or not is an issue of whether
or not there is some operation that causes shifting in one domain or the
other. if there is, there is an artifact (you have to use modulo(N)
addressing or recognize the periodic extension). if there is no shifting
(like there is just amplitude scaling), there is no artifact that the
periodicity is responsible for.

Is there anything in O&S regarding phase-vocoder, overlap-add, or other
sequential DFT usages? (my copy isn't handy right now...)

there certainly is discussion of overlap-add and overlap-save to do "fast
convolution" in O&S and that periodic property of the DFT is certainly an
issue there. i think, in the case of fast convolution, the inherent
periodicity is considered to be a cause of difficulty and not a property to
exploit. about the only time this inherent periodic extension done by the
DFT is something to exploit is in computing the coefficients of a Fourier
series of a periodic sequence, which is fundamentally what the DFS or DFT
(they're the same thing) does. i used that fact a lot in the Wavetable
Synthesis 101 paper i did a decade ago (it's at musicdsp.org, if you want
it). otherwise, this inherent periodic extension that the DFT does is
usually some kind of headache that you have to worry about (whether it's
aliasing of spectral leakage or the overlap-add/overlap-save mess or the
modulo addressing when shifting or convoluting or something else).

i don't think O&S have a phase-vocoder in it.


--

r b-j rbj@xxxxxxxxxxxxxxxxxxxx

"Imagination is more important than knowledge."


.

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