Theory for convolution

Here's a mind-challenge with no solution as far as I know. Maybe someone he=
re will have an idea.=20
Let's say I have a continuous discrete-time signal that is 5 seconds long w=
ith the Dirac impulses occurring once per second starting at 0. Now lets sa=
y that I multiply this signal by a rectangular pulse that starts at t=3D1/2=
 and ends at t=3D4.5, so the first and last sample are 0. There are 2 ways =
to compute the continuous Fourier transform of this , one is just to make a=
 new time signal with first and last sample=3D0 and take the FT, and the ot=
her is to convolve the original spectrum (including the first and last impu=
lses ) with the FT of the pulse (which will be a Sinc function with some ph=
ase delay). The second method is where things get weird, because in the tim=
e domain  the pulse can start anywhere between 0+ epsilon and 1-epsilon, an=
d end anywhere between 4+epsilon and 5-epsilon, and the time domain wavefor=
m will be the same (0 for the 1st and last impulse ), so obviously the FT r=
esult will be identical, but if you force yourself to use the frequency-dom=
ain convolution method, the width of the sinc signal will vary continually =
as you change the pulse width, and yet somehow you must get an identical co=
nvolution result for that entire range of sinc widths (since the time domai=
n signal doesn't change).=20
Can anyone explain this without resorting to the time domain ?

Bob

On Monday, July 17, 2017 at 7:03:47 AM UTC-7, radam...@gmail.com wrote:
... 
> Let's say I have a continuous discrete-time signal ...
> Bob

Let's not. You either have continuous time signals or discrete time signals. Convolution theorems are defined for each case, not for crossed cases.

You also don't get to assign a 'length' to a signal without accounting for the windowing that requires.

And, a discrete signal of length 5 with an initial sample at time 0 has samples at 0, 1, 2, 3, and 4.

Dale B. Dalrymple

I'm saying that I have a signal which is some finite number of Dirac impuls=
es on an integer time grid. Such a signal has a continuous Fourier transfor=
m. This is well-known and different from a discrete-time signal that is sim=
ply a sequence of unit-less numbers. I may have used the wrong terminology =
for this signal but do you get the point?

Bob

On Monday, July 17, 2017 at 4:33:05 PM UTC-4, radam...@gmail.com wrote:
> I'm saying that I have a signal which is some finite number of Dirac impu=
lses on an integer time grid. Such a signal has a continuous Fourier transf=
orm. This is well-known and different from a discrete-time signal that is s=
imply a sequence of unit-less numbers. I may have used the wrong terminolog=
y for this signal but do you get the point?


i might call that an "ideally-uniform-sampled signal". i think i know what =
you mean.  but i am having trouble understanding what the question is.  the=
 best that i can understand your question, we have:

          5
x_s(t) =3D SUM x[n]  delta(t-nT)
         n=3D0

and you're multiplying  x_s(t) times this rect window:

w(t) =3D rect( (t - 2.5*T)/(4T) )

where rect(x) is the standard unit-wide rect centered at x=3D0.

so the multiplied product is


                5
x_s(t) w(t) =3D  SUM{ x[n]  delta(t-nT)} rect( (t - 2.5*T)/(4T) )
               n=3D0



                4
x_s(t) w(t) =3D  SUM{ x[n]  delta(t-nT)}=20
               n=3D1


now what is the problem you're having, Bob?

r b-j

Robert

Thanks. I'm not really having any problem, this is a purely theoretic quest=
ion.=20

The point I'm trying to make is that the width of the multiplying rect puls=
e can vary continuously over some range and as long as it passes the same s=
et of impulses, the FT of the result willl not change. But if I use the con=
volution -in-the-frequency domain rule, I convolve with a sinc function, an=
d depending on the width of the rect pulse (again, varied continuously over=
 some range that passes the same set of impulses), I will have Sinc functio=
ns of varying "spread". So how do I explain using the convolution rule only=
 that I get a FT result that does not vary as the rect width varies?

Bob

On Monday, July 17, 2017 at 9:27:46 PM UTC-4, radam...@gmail.com wrote:
> Robert
>=20
> Thanks. I'm not really having any problem, this is a purely theoretic que=
stion.=20
>=20
> The point I'm trying to make is that the width of the multiplying rect pu=
lse can vary continuously over some range and as long as it passes the same=
 set of impulses, the FT of the result willl not change. But if I use the c=
onvolution -in-the-frequency domain rule, I convolve with a sinc function, =
and depending on the width of the rect pulse (again, varied continuously ov=
er some range that passes the same set of impulses), I will have Sinc funct=
ions of varying "spread". So how do I explain using the convolution rule on=
ly that I get a FT result that does not vary as the rect width varies?

it's a good question and similar to the reconstruction formula when the ban=
dwidth is *known* to be less than the Nyquist frequency.

Bob, do you do Stack Exchange yet?  just like how markets change when somet=
hing like Amazon comes on the scene, comp.dsp is becoming like Borders book=
store.  i might ask this question, (or the one that is more interesting to =
me about reconstruction) on the dsp.se site.

r b-j

On Monday, July 17, 2017 at 9:27:46 PM UTC-4, radam...@gmail.com wrote:
\>=20
> Thanks. I'm not really having any problem, this is a purely theoretic que=
stion.=20
>=20
> The point I'm trying to make is that the width of the multiplying rect pu=
lse can vary continuously over some range and as long as it passes the same=
 set of impulses, the FT of the result willl not change. But if I use the c=
onvolution -in-the-frequency domain rule, I convolve with a sinc function, =
and depending on the width of the rect pulse (again, varied continuously ov=
er some range that passes the same set of impulses), I will have Sinc funct=
ions of varying "spread". So how do I explain using the convolution rule on=
ly that I get a FT result that does not vary as the rect width varies?
>=20

so Bob, i spent about 80 minutes spiffying this up with LaTeX and the quest=
ion (with me swapping time and frequency domains w.r.t. your question) is p=
resented as in the DSP Stack Exchange:

https://dsp.stackexchange.com/questions/42495/implication-of-sampling-and-r=
econstruction-theorem=20

wanna look at that and make sure i don't misrepresent the question?

Thanks!

Looks good to me, I think you're right that the dual problem occurs in reconstruction. Thanks for putting in the effort. 

Bob

On Tuesday, July 18, 2017 at 6:46:56 AM UTC-4, radam...@gmail.com wrote:
> Thanks!
> 
> Looks good to me, I think you're right that the dual problem occurs in reconstruction. Thanks for putting in the effort. 
> 

i really stripped the question down and asked it on the math Stack Exchange.

https://math.stackexchange.com/questions/2362378/another-mathy-question-from-the-signal-processing-community

it has an answer right away.  the math guys have a notation that replaces the rect() or the unit step that i have to get used to.

so i still haven't groked the whole thing yet.

L8r,

r b-j

Very interesting , I'll see if I can wade through it. 

Thanks again !

Bob