Basic Query about Sampling.

[Alex Miua]

Hello people,

I am a novice in DSP so please bear with me. Suppose I want to sample
a continous signal at regular intervals of period T, then will I be
using the dirac delta impulse train or the kronecker delta impulse
train ?

I think it should be the kronecker delta impulse train because it has
a height of one whereas the Dirac delta has a height of infinity but
when I read oppenheims book section four point two says its the dirac
delta function

any one understand it better than i do ?

thank you
alex

[Randy Yates]

Alex Miua <[email protected]> writes:

> Hello people,
>
> I am a novice in DSP so please bear with me. Suppose I want to sample
> a continous signal at regular intervals of period T, then will I be
> using the dirac delta impulse train or the kronecker delta impulse
> train ?
>
> I think it should be the kronecker delta impulse train because it has
> a height of one whereas the Dirac delta has a height of infinity but
> when I read oppenheims book section four point two says its the dirac
> delta function
>
> any one understand it better than i do ?
>
> thank you
> alex

Hi Alex,

It is the dirac delta impulse train. Many folks have a problem with the
mathematical integrity of this construction since theoretically you are
not supposed to use a dirac delta function outside of an integral, but
if you go on to look at the resulting signal in the frequency domain by
taking the Fourier transform of the impulse train / signal product, then
you resolve that problem and, via the sifting property of the dirac
delta function, you see exactly the effect that is sought after, namely
that only the values of the input signal at n*T are used in the
transform.

In general you don't use the kronecker delta unless you're already
working with discrete-time signals.
--
% Randy Yates % "Bird, on the wing,
%% Fuquay-Varina, NC % goes floating by
%%% 919-577-9882 % but there's a teardrop in his eye..."
%%%% <[email protected]> % 'One Summer Dream', *Face The Music*, ELO
http://www.digitalsignallabs.com

[Tim Wescott]

On Sat, 22 Mar 2008 02:48:07 -0700, Alex Miua wrote:

> Hello people,
>
> I am a novice in DSP so please bear with me. Suppose I want to sample a
> continous signal at regular intervals of period T, then will I be using
> the dirac delta impulse train or the kronecker delta impulse train ?
>
> I think it should be the kronecker delta impulse train because it has a
> height of one whereas the Dirac delta has a height of infinity but when
> I read oppenheims book section four point two says its the dirac delta
> function
>
> any one understand it better than i do ?
>
> thank you
> alex

If you are sampling a signal in the real world then you can't use either
the Dirac delta or the Kroneker delta -- these are both pure mathematical
constructs that can't exist in a finite-amplitude, continuous-time
world. In the real world you usually use a sample-and-hold circuit,
which these days is usually buried in the monolithic ADC chip that you
also usually use.

If you're modeling the sampling of the signal in mathemagic land (which
is what you mean, I think) then you want to use the Dirac delta function
(although for subsampling an already sampled signal I suppose it'd be
appropriate to model the process using the Kroneker).

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

[Fred Marshall]

"Tim Wescott" <[email protected]> wrote in message
news:[email protected]...
> On Sat, 22 Mar 2008 02:48:07 -0700, Alex Miua wrote:
>
>> Hello people,
>>
>> I am a novice in DSP so please bear with me. Suppose I want to sample a
>> continous signal at regular intervals of period T, then will I be using
>> the dirac delta impulse train or the kronecker delta impulse train ?
>>
>> I think it should be the kronecker delta impulse train because it has a
>> height of one whereas the Dirac delta has a height of infinity but when
>> I read oppenheims book section four point two says its the dirac delta
>> function
>>
>> any one understand it better than i do ?
>>
>> thank you
>> alex
>
> If you are sampling a signal in the real world then you can't use either
> the Dirac delta or the Kroneker delta -- these are both pure mathematical
> constructs that can't exist in a finite-amplitude, continuous-time
> world. In the real world you usually use a sample-and-hold circuit,
> which these days is usually buried in the monolithic ADC chip that you
> also usually use.
>
> If you're modeling the sampling of the signal in mathemagic land (which
> is what you mean, I think) then you want to use the Dirac delta function
> (although for subsampling an already sampled signal I suppose it'd be
> appropriate to model the process using the Kroneker).
>
> --
> Tim Wescott
> Control systems and communications consulting
> http://www.wescottdesign.com
>

I had to refresh my memory and look up the Kronecker. The conclusion I
reached, perhaps incorrectly, that the Kronecker delta is the same as what
we often refer to as the "unit sample". It seems to me that a unit sample
is implementable and a train of them is as well. How does this apply?
Well, subsampling or decimation would be multiplication by a train of unit
samples at a higher integer multiple interval of the original. Isn't that
right?

Fred