On Feb 14, 5:39 am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 14 Feb, 09:57, "Nasser M. Abbasi" <n...@12000.org> wrote:
>
>
>
> > "Rune Allnor" <all...@tele.ntnu.no> wrote in message
>
> >news:a117e885-bc9a-4bdc-91a9-6efdf4a8a813@j31g2000yqa.googlegroups.com...
>
> > > On 14 Feb, 07:48, "Nasser M. Abbasi" <n...@12000.org> wrote:
>
> > >> So, since this formula assumed the signal being sampled is periodic,
> > >> (that
> > >> is after all how this formula same about),
>
> > > Could you justify both those claims, please?
> > > That the signal is periodic and that this property
> > > is somehow necessary or essential for the analysis?
>
> > > Rune
>
> > hi Rune,
>
> > May be what I said was not too clear. What I meant is this:
>
> > The derivation of the sinc interpolation formula starts by taking the
> > Fourier transform X(w) of x(t), then it finds the relation of X(w) to the
> > discrete Fourier transform of x[n], which is obtained from x(t) by sampling.
> > right? This is how it is derived in my textbook (DSP by oppenheim, 1975,
> > page 29).
>
> Sure.
>
> > But when we are given some x(t) of limited time frame, say t=0 to t=t0, and
> > we want the Fourier transform of x(t), then we assume this signal is
> > periodic of period 0..t0 and then let the period go to infinity to obtain
> > the Fourier transform of x(t) from the fourier coefficients. Right?
>
> Wrong.
>
> What one needs to understand is that there are several variations
> of the Fourier transform around, as well as some more "Fourier-
> like" transforms. The four basic forms of the FT are classified
> in terms of whether the signal domain is
>
> 1) Countinous-time (CT) or discrete-time (DT)
> 2) The signal supposrt is of finite (F) or infinite extent (I).
>
> From these classes alone, you have four transforms:
>
> 1) CT-I - Fourier transform
> 2) CT-F - Fourier series
> 3) DT-I - Discrete-Time Fourier Transform, DTFT
> 4) DT-F - Discrete Fourier Transform, DFT.
>
> For all these variants the requirement is that the sum or
> integral of the squared time signal over the signal support
> is finite (i.e. finite energy signals) for the FT to exist:
>
> sum |x[n]|^2 < infinite
>
> for the DT variants and
>
> integral |x(t)|^2 dt < infinite
>
> for the CT variants.
>
> One can also modify the FT to allow for finite *power* signals,
> (view with fixed-width font)
>
> 1 N
> lim ---- sum |x[n]|^2 < inf
> N->inf 2N-1 n=-N
>
i think you meant 2N+1 in the denom, but the condition is sufficient
nonetheless.
> (and a similar relation for the CT-I variant), i.e. the infinite-
> duration signals have finite power.
>
> The problem is that these different variants were not recognized
> at first, so some early authors, who wanted to compute the DT-I
> coefficients by means of the FFT, which computes DT-F coefficients,
> started messing around with sloppy remarks about the DFT being
> governed by assumptions about signal periodicity.
now, i dunno what Nasser meant about his assumption of signal
periodicity, but even with Fred's and Dale's support, i (as well as
early authors like O&S) continue to recognize the utter non-difference
in the equations regarding the Discrete Fourier Series (DFS) and the
Discrete Fourier Transform (DFT). the DFT maps a discrete and
periodic sequence in one domain (call it the "time domain") to another
discrete and periodic sequence in the reciprocal domain (we'll call it
the "frequency domain") having the same integer period. that's what
the DFT is, whether or not any human beings make any assumptions about
periodicity of the signal they are considering.
we're not assuming this. we're recognizing it.
> Since this blunder somehow lipped through peer reviews
it's no blunder. in fact, it is *missing* this fact and assuming
otherwise (that perhaps the DFT assumes something else, like zero
extension) that is the blunder.
> - no one
> seem to have been aware of the relevance and importance of the
> details at the time - and got published in one of the leading
> journals, the myths and superstitions about periodic properties
> of data have remained to this day.
we know from the continuous Fourier Transform that periodically
extending a function in one domain (by summing together shifted
copies) causes sampling of the transform of that function in the other
domain. likewise sampling (with impulses) in one domain causes
periodic extension in the other. something that is both sampled (made
discrete) *and* periodic in one domain will have its transform also
discrete and periodic in the reciprocal domain.
> > This is how my textbook finds the Fourier transform
> > of a signal that is defined over some limited time duration.
listen, when you simply look at a finite snippet of signal (which is
the same as the signal of possible infinite extent being multiplied by
a window function), you have no idea what it was outside of that
finite domain. but as soon as you pass it to the DFT, you've
periodically extended, that is how the DFT is going to deal with it,
as if you've passed to it exactly one period of a discrete and
periodic signal. whether it was periodic or not before applying the
windowing operation does not matter, the DFT (and its associated
operations and theorems) will "assume" so and will treat it as such.
>
> > I mean x(t) might not be periodic of course, ie. it can be aperiodic. but we
> > have limited time view of it.
>
> Sure. Which mean we do not know anything about the properties
> of the function outside the time gate.
>
> Now, a complicating factor is that the finite-domain *basis*
> *functions* are periodic, so for the purposes of arithmetics,
> it makes a lot of sense to review the signal as periodic outside
> the observation interval.
>
> This is, however, a consequence of the choice of basis functions
> and *not* a property of the signal as such.
i agree with that Rune. but when you pass it to the DFT, the DFT uses
periodic basis functions and views that finite snippet of data as
periodic to fit to it.
> > So, what I meant to say is that x(t) is *assumed* to be periodic over the
> > length of the time record. Again, this is just to be able to talk about the
> > Fourier transform for it.
the *Discrete* Fourier transform for it.
> > Maybe you do not think this is the correct way to look at it.
>
> It depends on the context. Circular convolution is a fact.
> Unless you take certain precautions, the result of a convolution
> implemented in spectrum domain as multiplications of DFTs of
> signals *behaves* *as* *if* the signals were periodic. This
> does *not* mean that the signals actually *are* periodic,
it does means, as far as the DFT (and its properties) are concerned,
the signal *it* is looking at is one period of a periodic sequence.
> but is merely a computational artifact caused by the periodic
> properties of the [sinusoidal] basis functions.
Nasser, i have been involved in "periodic" fights here at comp.dsp
about this inherent property of the DFT that periodically extends the
finite set of data passed to it (because the basis functions of the
DFT are all periodic sharing a common period).
r b-j