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The derivation of the representation of sampling by the use of the delta function

Started by gareth February 26, 2015
Further to Robert's peer review, and the then swapping around
of the multiply / divide words. hopefully correct at last ...

-----ooooo-----

Revised after peer review, approached with a little less haste


So...

Sampling with a period of  T is given by (after asciification) as ..

(T)sum (n : 0, inf)(d(t-nT) * f(nT) )

... with * representing multiplication and not convolution as we
are still in the time domain.

However, (and this is where my protest came in having
previously fully revised Fourier, Laplace, Butterworth, Tchebyschev,
Elliptical, and PID, etc,   to degree standard thus giving me a full
understanding of the
Diracian Delta and its characteristics), all the texts that I encountered,
and, indeed, much of the Interweb give it as ...

sum (n : 0, inf)(d(t-nT) * f(nT) )

 ... which lacks the essential multiplier of  T.




What is the justification for this derivation?

It is because the real representation of sampling is not done with
Diracian Delta Funcions, but with Unit Steps, as follows ...

sum (n : 0, inf)( f(nT)  * ( U(t-(n+1)T)  - U(t-nT))  )

... but this is very messy to deal with analytically.

So, as the Diracian Delta is a easier to deal with mathematically, having
a frequency spectrum of unity (ie, every possibly cosine in
phase at t = 0), is there some way that the sampling expression
could be re-represented with Diracian Deltas?

The answer is a resounding, "Yes!"!

Consider the definition of the Diracian Delta, as it is presented to
electronics engineers (in my case, the second year at Essex Uni 1970 - 1971)
which is a pulse of unity area 1/T volts high and T seconds long, with
T tending towards zero, which in out asciification comes out as ..

(1/T) * (  U(t-(n+1)T) - U(t-nT)  )

... and therefore our sampling mechanism is strongly related to
the Diracian Delta except for the division factor of  T
and thus ...

sum (n : 0, inf)( f(nT)  * ( U(t-(n+1)T)  - U(t-nT))  )

... can also be represented as ..


(T)sum (n : 0, inf)(d(t-nT) * f(nT) )

... with T (or even 1/T) being the missing factor which
I had dubbed Big K.

Now, having resolved this issue, and not having any further direct use
for DSP, I retired from my studies knowing that my fundamental mathematical
understanding was on such a strong footing that I could easily move on from
there
should the need arose.

However, ISTR that in Robert Bristow-Johnson's article about sampling and 
reconstruction
that he had to
re-introduce
the factor of T out-of-thin-air for reconstruction, so I'd like to suggest
from
my analysis above that it is not necessary to bring in the deus-ex-machina
of T at the
end because it should always have been there from the beginning?

EOE (Hopefull none this time!)