Re: Appendix A: Types of Fourier Transforms
- From: robert bristow-johnson <rbj@xxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 24 Jan 2011 19:33:46 -0800 (PST)
On Jan 24, 8:08 pm, dbd <d...@xxxxxxxx> wrote:
On Jan 23, 9:07 pm, robert bristow-johnson <r...@xxxxxxxxxxxxxxxxxxxx>
that is where the window is applied. whether it's a rectangular
window or you apply something else to it like a Hamming or Kaiser.
and you *model* what it does by multiplying the samples that are
outside this window by zero. those N samples are appended by an
infinite string of zeros on both sides and, if we could get a
hypothetical machine to do a DTFT, that is the information it would
since a finite number of samples are non-zero, and if the input is
assumed sampled from a bandlimited source, you could attach a sinc()
function to each of those N samples, CFT that, and you have the DTFT
for -pi < omega < +pi. uniformly sample those frequency-domain points
(with a spacing of 2*pi/N) and you have the DFT. funny thing is, when
you sample in one domain, it causes periodic repeating and overlapping
in the reciprocal domain. you cannot escape the periodic extension no
matter how you want to view the DFT.
You cannot use sinc interpolation to recover arbitrary bandlimited
signals from only N samples.
the sinc functions attached to the samples outside of the window have
those sample values (what are zero) multiplying them. doesn't matter,
forget about attaching sinc functions to the samples, just leave them
attached to the dirac deltas (all by N have zero coefficients) and
apply a continuous Fourier Transform to that. then you get the DTFT
for omega not just between -pi and +pi, but it will repeat (with
period 2*pi) forever. you still sample that with N equally spaced
samples (with spacing 2*pi/N) and you get the DFT.
Stochastic, non-stationary, and periodic (but not periodic in the
N samples) signals cannot be recovered from only the N samples.
the DFT doesn't know or care where those N samples came from. it's
just presented with N samples. whether they come from stochastic or
deterministic sources, stationary or not, periodic or not, they're
just N samples.
but the DFT still fits N periodic basis functions (with coefficients
X[k]) to those N samples. any rectangular window embedded to those
basis functions serve no utility. if you attach the rectangular
window functions to the basis functions, then, if any shifting is
needed (by multiplying by e^(-i*2*pi*k*m/N) or by a transfer function,
H[k]), all that rectangular window does for you is force you to use
modulo-N arithmetic in the indices.
As has been said all along, your process can only
work for the signal space of linear combinations of signals periodic
in N samples (and within the bandlimit). You have only a special case
for your argument.
no, what i had said is that whether or not you are thinking the N
samples were extracted from a periodic signal, it doesn't matter. the
DFT treats those N samples as defining a single period of a periodic
signal. the DFT fits periodic basis functions to those N samples.
Not everyone is willing to live only in that corner.
you have no choice. (but it's not a corner, it's the whole space of N
arbitrary sample values.)
In dynamic signal analyzers (DSA), data outside the current window
will be used for weighted overlapped spectral averaging (WOSA or
Welch's method) and/or generating other lines in the waterfall/
spectrogram displays. These are common basic digital signal processing
applications of DFTs. These are common systems where data sets are
much larger than interesting transform sizes and all the data gets
processed, sometimes within multiple windows. These are applications
where the processing is selected because it is known (and verifiable)
that there are signal components that cannot be treated correctly as
periodically extended. Because of this, stochastic signals and non-
stationary signals can be either enhanced or reduced by the selection
of appropriate post-DFT algorithms. Applications outside of DSAs
include filter banks, frequency domain adaptive filters,
demultiplexers and intercept receivers.
blah blah blah blah blah....
the above says absolutely nothing regarding the issue at hand. it's
fluff. not even that.
You are correct that this is not responsive to your issue. As it is a
response to a question by Jerry about a snippet quoted from one of my
you weren't responding to his question either. just a form of techno-
the issue (using the briefest language i can think of) is whether or
not the DFT is something different than the DFS. whether or not the
DFT necessarily periodically extends the data passed to it. whether
or not there is any utility in defining the DFT basis functions
differently from those of the DFS, whether or not there is any utility
to windowing the DFT basis functions outside of the interval 0 <= n <
i am keeping the issue focused on that point.
your interpretation of it as a response to you is disingenuous
whatever you say.
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