sinc() question

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0307010915.48add7d5@posting.google.com...
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:<k4EKa.1728$Jk5.809467@feed2.centurytel.net>...
> > "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> > news:f56893ae.0306260414.2a72e7c2@posting.google.com...
> > > yates@ieee.org (Randy Yates) wrote in message
> >  news:<567ce618.0306250841.698282f8@posting.google.com>...
> > >
> > > > Then the inverse FT lies?
> > >
> > > Well, that's an interesting question... the truth is, I have never
> > > seen a good treatment of the Inverse Discrete Fourier Transform.
> > >
> > > I believe the continuous case was treated by Papoulis in his
> > > nowhere-to-be-found book, but the *discrete* inverse transform
> > > has eluded me (does anybody know of such a treatment?).
> > >
> > > As far as I can remember, we always computed forward transforms
> > > and implemented the inverse transform by "recognizing X(f) as the
> > > transformed x(t)" by means of table lookup. I wouldn't be surprised
> > > if answering your questions would require the full math-chinery, and
> > > very careful evaluation and analysis of some more or less obscure
> > > boundary condition.
> > >
> > > Rune
> >
> > Rune,
> >
> > I think the reason for this is Duality.  If you know one, you know the
> > other.  If you Google on [duality "discrete fourier transform"] you may
find
> > something that makes sense for you.  Or, did you mean something else?
>
> Duality is no problem. My problem is the technicalities when manipulating
> Fourier integrals of complex-valued functions. This thread has made it
> embarrasingly clear to me that I have relied solely on the Plug'n Pray
> method for evaluating the inverse Fourier transform, and that I'm in way
> over my head once the functions in question (as well as the questions!)
> depart from the "everybody knows" answers to "standard" problems.

Rune,

I once took a brief course on DSP that was reasonably technical but not a
math class by any means - it was very practical in approach.  They showed us
how to do graphical manipulations that I use to this day.  Here's how it
works:

Draw the time and frequency domain "pictures" of a waveform of interest.
Sometimes these are simply "cartoons" that show a random looking waveform
and a spectrum of a handy shape and bandlimits.  These can be continuous or
sampled, infinite or periodic or time-limited (and the equivalents in the
frequency domain).

Perform a convolution on one and the corresponding multiplication on the
other (or vice versa) according to what you're trying to study.  Just assume
that all of the functions are real - that's good enough.  So, you may be
dealing with envelopes - that's OK.  I'm sure there are pathological cases
where the cartoons don't work but that's never been a great problem for me.

Continue to perform these operations until you reach the point you want.
The ability to envison one domain or another side-by-side is very helpful.
Of course, this is only good for general cases - not pathological ones or
near pathological ones or "special" ones.

Maybe this is what you meant by plug 'n pray..... but it works for me in a
whole lot of situations.

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:4aEKa.1729$Jk5.807866@feed2.centurytel.net...
>
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3EFAF198.43303370@ieee.org...

(snip)

> > Sort of apropos: a "mentee"'s homework was to write two-dimentional DFT,
> > then test the code with an image that is a uniform gray. His code
> > produced a transform that was zero everywhere. Clearly, it needs to be
> > zero a.e., but the "DC" term needs a value, else all shades of gray,
> > when DFTed then IDFTed would return black. That argument was enough to
> > convince him that he had a bug. But the DFT is zero everywhere with a
> > single exception. Hmmm.

> Do you mean that he/she had a bug or that the 2DFT was truly zero
> everywhere?  It shouldn't be, should it?  The gray scale value should
return
> a spike - which I take to be your "single exception"?

How about if it was a Sine transform?

-- glen

"Randy Yates" <yates@ieee.org> wrote in message
news:3EF90221.7789F30@ieee.org...
> robert bristow-johnson wrote:
> >
> > In article 3EF83ECD.72947E50@ieee.org, Randy Yates at yates@ieee.org
wrote
> > on 06/24/2003 08:03:
> >
> > > What's the difference in the impulse response between a filter which
> > > has the frequency-domain response
> > >
> > > |H(w)| = 1, |f| < Fs/2, 0 otherwise, angle(H(w)) = 0
> > >
> > > and
> > >
> > > |H(w)| = 1, |f| <= Fs/2, 0 otherwise, angle(H(w)) = 0
> > >
> > > ???
> >
> > i think there would be no difference since the inverse FT integral of
the
> > difference function of the two would be zero.

> So does it pass Fs/2 or not???

If I remember, a related discussion to Nyquist came up some time ago.  Does
Nyquist require a < or <=?

All practical solutions require a finite number of sample points over a
finite range of time, which may not be an integral multiple of Fs.  (It it
obvious that the length of a concert will always be an integer multiple of
1/44100 s long?  I believe that, using previously mentioned theorems, the
probability is zero.)   As long as the length (or period for periodic
signals) is not a mulitple of 1/Fs, it doesn't matter.

-- glen

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<0OmMa.2108$Jk5.1127722@feed2.centurytel.net>...
> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> news:f56893ae.0307010915.48add7d5@posting.google.com...
> > This thread has made it
> > embarrasingly clear to me that I have relied solely on the Plug'n Pray
> > method for evaluating the inverse Fourier transform, and that I'm in way
> > over my head once the functions in question (as well as the questions!)
> > depart from the "everybody knows" answers to "standard" problems.
> 
> Rune,
> 
> I once took a brief course on DSP that was reasonably technical but not a
> math class by any means - it was very practical in approach.  They showed us
> how to do graphical manipulations that I use to this day.  Here's how it
> works:
[snipped]
> Maybe this is what you meant by plug 'n pray..... but it works for me in a
> whole lot of situations.

More or less, yes. "Plug'n Pray" means the intuitive approach that has 
developed after a few years work based on using very basic ideas and 
principles and the tables of tranform pairs.

Every once in a while I come across questions like those of this thread, 
where the basic ideas are taken to some limit, or some not quite 
"standard" questions are asked, etc. In those cases I try to follow the 
advice of Morse and Ingard in the introduction to "Theoretical Acoustics", 
i.e. to use some sort of intuitive understanding of the problem and 
formulate it in terms of maths, then put any "intuitive concerns" aside 
and let the math-chinery grind, and only at the very end try to interpret 
what comes out of it. 

Of course, that's not how things actually happen. More often than not,
I know (or at least suspect) in advance where I want to end up, and 
how to get there. The maths serves more as a check if it is really 
possible to get there from whatever starting point I have. 

Anyway, I'm a bit too aware now that I am not really able to use the 
mathematical tools for handeling complex-valued FTs that fall outside 
the tabulated "standards". Probably a good starting point for learning 
how to use them...

Rune

Glen Herrmannsfeldt wrote:
> 
  ...
> (It it
> obvious that the length of a concert will always be an integer multiple of
> 1/44100 s long?  I believe that, using previously mentioned theorems, the
> probability is zero.)   As long as the length (or period for periodic
> signals) is not a mulitple of 1/Fs, it doesn't matter.
> 
> -- glen

It is always possible to pad out the length of a concert with enough
silence to make its functional length an integer multiple of 1/44100s od
a second. Not many would complain that the composer's intent had been
desecrated. 

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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