CTFT of a discrete signal

Randy Yates wrote:
> Tim Wescott <tim@seemywebsite.com> writes:
>> As a model, the Delta function is a great thing.  If I do make an
>> issue of anything it is all the places where I was taught the model as
>> reality, then tripped up once I got into the real world.
> 
> Well, yeah. Do you think anyone out of college really expects to see a
> Dirac generator sitting on the bench? :)

You mean they've been dropped from this year's Agilent catalogue? :-)

Steve

On 14 Aug, 20:54, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> > Depends on what you want. If you want to troll and start a war
> > on comp.dsp you are almost certain to succeed. If you want to
> > get on to a side track whic will cause you huge greaf and provide
> > no understanding about either DSP or continuous-time signal
> > processing, you are well on your way.
>
> This is not at all my goal, I am just trying to get a clearer view on both
> transforms...or like some others have replied in some way unify both.
> Maybe I am making it difficult for myself, that can be true....so be it...

Just be aware that this is an area that is by no means easy.
As I understand it, Dirac's Delta is some sort of simplification
that can be used instead of Lebesgue's integral. The only
thing I know about Lebesgue's integral is that it can integrate
weird functions and is taught only to graduate maths students.

As long as you aware of this fact, you might just get through
in one piece.

> Also like others have mentioned, often in text books discrete signals are
> obtained by multiplication with a dirac comb. So if it possible to go from
> continuous to discrete in that way, is it such a stupid question trying to
> go the other way?

The stupidity here lies with the textbook authors. Except for
two notable exceptions (Oppenheim and Schafer's 1975 book
and Rick Lyon's "Understanding DSP") they all look the same,
address the issues in more or less the same order, in more or
less the same way. Which means they all make the same
mistakes. As far as I am concerned, the relationships
between the different forms of the Fourier transforms are
not covered remotely near sufficiently in standard texts.
I suppose authors find the stuff too basic / fundamental
to really look into, and it is usually included as a matter
of form, before they start doing the stuff that interest them.

Just to illustrate the oversimplification by authors -- have
you ever seen a derivation of the convolution sum formula in
the textbooks? Me neither. But that derivation is first of all
very simple to do, second it illustrates how one thinks in
terms of mathematical concepts. As far as I am concerned,
it is a grave mistake on the authors part not to include it.

Only after I had observed some of Mr Beans threads here on
comp.dsp, did I notice the potential for confusion over Dirac's
Delta. I have, in fact, seen one, maybe two, texts on DSP
which attempt to address these questions. There is one
book by two French authors, from Springer. I don't remember
the title of the book, but it was published soe time
around 1997-97.

Rune

>
> You're on the right track. When you model the discrete signal x(nT) as
> a series of scaled diracs, then plug that into the CTFT, the integral
> from the CTFT turns them into a sum of numbers ala DTFT. Remember the
> sifting property of the Dirac? \int_{-infty}^{+\infty} x(t)\delta(t - tau) 
> dt = x(tau).
> -- 

first of all, thanks to all the people that have responded, all your inputs 
were  very useful to me...also rbj I liked a lot the link to your 
mathematical description of sampling & reconstruction.

Below I tried to do continue with what Randy introduced above. Are there any 
flaws in this?

----------------

my discrete signal, only k= 0 : N-1 values are non-zero (spaced T)

     s[k]

discrete signal represented as series of scaled dirac's as if it were a 
'continuous signal'

     s(t) = SUM_{k=0}^_{k=N-1} s[k] * d(t-k*T)

     where d(t) is the dirac delta

FT of s(t):

     S(f) = INT_{-inf}_{+inf} SUM_{k=0}^_{k=N-1} s[k] * d(t-k*T) 
exp(-i*2*pi*f*t) dt

     S(f) = SUM_{k=0}^_{k=N-1} s[k] INT_{-inf}_{+inf} d(t-k*T) 
exp(-i*2*pi*f*t) dt

     S(f) = SUM_{k=0}^_{k=N-1} s[k] exp(-i*2*pi*f*k*T)

     S(f) = SUM_{k=-inf}^_{k=+inf} s[k] exp(-i*2*pi*f*k*T)


replacing 2*pi*f*T with the normalized radial frequency w (radians per 
sample) this becomes:

     SUM_{k=-inf}^_{k=+inf} s[k] exp(-i*k*w)

which is if I am not mistaken the DTFT. 

On 15 Aug, 11:55, "NewLine" <qw897jhk_remove_this_and_t...@skynet.be>
wrote:
> > You're on the right track. When you model the discrete signal x(nT) as
> > a series of scaled diracs, then plug that into the CTFT, the integral
> > from the CTFT turns them into a sum of numbers ala DTFT. Remember the
> > sifting property of the Dirac? \int_{-infty}^{+\infty} x(t)\delta(t - tau)
> > dt = x(tau).
> > --
>
> first of all, thanks to all the people that have responded, all your inputs
> were  very useful to me...also rbj I liked a lot the link to your
> mathematical description of sampling & reconstruction.
>
> Below I tried to do continue with what Randy introduced above. Are there any
> flaws in this?

None other than that there is no justification for bringing in
Dirac's delta.

To see why Dirac's delta came into being, consider
the spectrum of the power signal

x[n] = exp(jw'n)      n = ...,-2,-1,0,1,2,...  [1]

The required computations go like (view with fixed-width
font)

               N     1
X(w) = lim    sum  -----  exp(jw'n)exp(-jwn)   [2.a]
      N->inf  n=-N 2N-1

              N     1
     = lim   sum  -----  1                    [2.b]
      N->inf n=-N  2N-1

       /          2N-1
       |   lim   ------ = 1      w = w'
     = |   N->inf 2N-1
       |
       \                  0      w =/= w'

So X(w) == 0 "almost everywhere ," i.e. everywhere
except for one point, at w = w', where it equals 1.
It's a half-weird function if you are only used to
think in terms of continuous function, but for
somebody with a college maths course under the
belt, it is nothing extraordinary about X(w).

Now, how do you get something like Parseval's
relation, which relates the integral of squared
x[n] in time domain with the integral of squared
X(w) in frequency domain, to work with an X(w)
as above?

Using the old workhorse, the Riemann integral, fails:

  inf                 -e               e
integral X(w) = lim  integral 0 dw + integral 1 dw
 -inf           e->0 -inf             -e

                    inf
                + integral 0 dw  = 0.              [3]
                    e

Riemann's integral is stertched beyond its limits
by an integrand which is 0 "almost everywhere."
But the would of maths be a lot nicer place if
X(w) could be integrated, right?

This was the problem Lebesgue solved. He developed
a way of integrating functions which are 0 "almost
everywhere" so that

  inf
integral_Lebesgue X(w) = 1.
 -inf

with X(w) above.

The one problem with Lebesque's integral is that
it is far from easy to use (well, that's what
my maths teachers told me; I have no personal
experience with it) so Dirac set about to come
up with a way to integrate functions which are
0 "almost everywhere" without having to resort
to the Lebesgue integral.

The result is Dirac's delta function. The function

   inf
X = integral d(t - t0) dt
   -inf

is basically nothing more than a tag, a note to
the analyst, to say that "add 1 to the integral
X when you  pass t = t0."

That's all there is to it. Once you get a clear
picture of why the Dirac delta is useful, you
also see why it is plain wrong to use it
outside the integral sign.

Rune

"NewLine" <qw897jhk_remove_this_and_this@skynet.be> writes:

>>
>> You're on the right track. When you model the discrete signal x(nT) as
>> a series of scaled diracs, then plug that into the CTFT, the integral
>> from the CTFT turns them into a sum of numbers ala DTFT. Remember the
>> sifting property of the Dirac? \int_{-infty}^{+\infty} x(t)\delta(t - tau) 
>> dt = x(tau).
>> -- 
>
> first of all, thanks to all the people that have responded, all your inputs 
> were  very useful to me...also rbj I liked a lot the link to your 
> mathematical description of sampling & reconstruction.
>
> Below I tried to do continue with what Randy introduced above. Are there any 
> flaws in this?
>
> ----------------
>
> my discrete signal, only k= 0 : N-1 values are non-zero (spaced T)
>
>      s[k]
>
> discrete signal represented as series of scaled dirac's as if it were a 
> 'continuous signal'
>
>      s(t) = SUM_{k=0}^_{k=N-1} s[k] * d(t-k*T)
>
>      where d(t) is the dirac delta

You've prematurely represented the signal as a sequence s[k]. Keep the
original signal intact. By the way, I don't see the original signal
here, so let's call it s_a(t) (for "analog"), and let s(t) be the
"discretized" version, i.e.,

  s(t) = s_a(t) * sum_{n=-\infty}^{+\infty} d(t-n*T).

Then when you take the FT of s(t) you use the sifting property
of the Dirac delta function:

  S(f) = \int_{-infty}^{+\infty} (s_a(t) * sum_{n=-\infty}^{+\infty} d(t-n*T)) * e^{-j\omega t} dt.
       = sum_{n=-\infty}^{+\infty} \int_{-infty}^{+\infty} s_a(t) * d(t-n*T) e^{-j\omega t} dt.
       = sum_{n=-\infty}^{+\infty} s_a(n*T) * e^{-j\omega n*T}

and continue from there.
-- 
%  Randy Yates                  % "Watching all the days go by...    
%% Fuquay-Varina, NC            %  Who are you and who am I?"
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://home.earthlink.net/~yatescr

robert bristow-johnson wrote:

(snip, I wrote)

>>Among the transform pairs there is

>>discrete <--> periodic

>>Applying that twice, you get

>>discrete periodic <--> discrete periodic

>>which is the DTFT.

> Glen, i think that's the DFT  (discrete periodic <--> discrete
> periodic).

> the DTFT is
> 
>    aperiodic, discrete time <--> periodic, continuous frequency

I think you are right.  I was trying to complement the CTFT in
the original post, and somehow DTFT seemed more natural.

The other that always confuses me is, why isn't that a Fourier series?

continuous time, periodic <--> aperiodic discrete frequency

should be a Fourier series, so why not the other way around?

-- glen

Randy Yates wrote:

(snip)

> Well, yeah. Do you think anyone out of college really expects to see a
> Dirac generator sitting on the bench? :)

I do remember generators with a capacitor (and charging circuit)
and reed relay to generate very short pulses.  It seems too simple,
but it works pretty well.  (As long as they aren't too close together.)

It is pretty easy to make pulses shorter than you can measure.

-- glen

Randy Yates wrote:

(snip)

> I'm not sure I respect your attempt to make this such an issue. First
> of all, you act like you're the only mathematician around here and the

I, at least, never thought that.

> rest of us need you to enlighten us. Some of us have had some analysis
> as well. I think we all must admit that there is always more to be
> learned.

When I learned about delta functions, I was told that mathematicians
didn't like them, and often didn't believe in them.  There was a story
about some mathematician who rewrote a quantum mechanics book to remove
delta functions from all the explanations.

-- glen

On Aug 15, 4:26 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> robert bristow-johnson wrote:
>
> (snip, I wrote)
>
> >>Among the transform pairs there is
> >>discrete <--> periodic
> >>Applying that twice, you get
> >>discrete periodic <--> discrete periodic
> >>which is the DTFT.
> > Glen, i think that's the DFT
> > (discrete periodic <--> discrete periodic).
> > the DTFT is
>
> >    aperiodic, discrete time <--> periodic, continuous frequency
>
> I think you are right.  I was trying to complement the CTFT in
> the original post, and somehow DTFT seemed more natural.
>
> The other that always confuses me is, why isn't that a Fourier series?

why do you say that it isn't?  the discrete samples in the time domain
are the fourier coefficients of the periodic spectrum in the freqency
domain.  if you normalize out the 2*pi factor (by expressing frequency
in normalized cycles/sample rather than radians/sample), the period of
that periodic spectrum is 1, and the spacing between your discrete
coefs (which are the samples) is 1.

> continuous time, periodic <--> aperiodic discrete frequency
>
> should be a Fourier series, so why not the other way around?

it *is* a Fourier series.

r b-j

On Aug 15, 4:33 pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
>
> When I learned about delta functions, I was told that mathematicians
> didn't like them, and often didn't believe in them.

what they don't believe is that the dirac delta "function" is a true
function as the mathematicians understand a function of a real
variable.  from the POV ofLebesgue integration (which is more general
than the Riemann integration we learned in calculus), if two functions
agree "almost everywhere" (which is "everywhere but for a countable
number of discrete points"), then the integrals of the two functions
over the same region (or limits of integration) must agree.  well, we
electrical engineers like to think of the dirac impulse as have zero
value everywhere but at t=0, and its integral (from -eps to +eps) is
1. but the dirac impulse "function" agrees with the zero everywhere
function at every point except t=0, and we know that the integral of 0
is 0.  so this understanding of the dirac impulse violates that
property of Lebesgue integration of two virtually identical functions.

>  There was a story
> about some mathematician who rewrote a quantum mechanics book to remove
> delta functions from all the explanations.

yeah, some math guys really don't like it, and like our Neanderthal
engineering treatment of the dirac impulse even less.  i'm completely
fine with the Neanderthal engineering usage of the dirac delta.


r b-j