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Following peer review - the derivation of the representation of sampling by Diracian Delta functions

Started by gareth February 26, 2015
Revised after peer review, approached with a little less haste


So...

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

(1/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 divisor 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 ..

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

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

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

... can also be represented as ..


(1/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!)




"gareth" <no.spam@thank.you.invalid> wrote:
> Revised after peer review, approached with a little less haste > > > So... > > Sampling with a period of T is given by (after asciification) as .. > > (1/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 divisor 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 .. > > T * ( U(t-(n+1)T) - U(t-nT) ) > > ... and therefore our sampling mechanism is strongly related to > the Diracian Delta except for the multiplication factor of T > and thus ... > > sum (n : 0, inf)( f(nT) * ( U(t-(n+1)T) - U(t-nT)) ) > > ... can also be represented as .. > > > (1/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!)
Where's your references? What a sloppy approach to presenting a paper. Why am I not surprised? -- STC // M0TEY // twitter.com/ukradioamateur
"Stephen Thomas Cole" <usenet@stephenthomascole.com> wrote in message 
news:mco3nj$vpd$1@dont-email.me...
> > Where's your references? What a sloppy approach to presenting a paper. Why > am I not surprised?
Your malicious cross-post to uk.radio.amateur was noted and removed. It remains that 100% of your postings consist of rather silly and infantile personal remarks. As amateur radio is a technical pursuit, and you yourself elected to cross post to comp.dsp which is even more technical, why not make your first technical post ever, since you first arrived to pollute Usenet just over 2 years ago with your tirades of abuse? I think that you lack any technical acumen and your bluster is your attempt to cover up.
On 2/26/15 4:28 PM, gareth wrote:
> Revised after peer review, approached with a little less haste > > > So... > > Sampling with a period of T is given by (after asciification) as .. > > (1/T)sum (n : 0, inf)(d(t-nT) * f(nT) ) >
sure you want 1/T here? it *compounds* the problem rather than correct it. (i think we all understand that "d(t)" is the Dirac impulse function.)
> > 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,
nope. it's the textbooks' (except for Ken Pohlmann's "Principles of Digital Audio") treatment of reconstruction that has to have a brick-wall reconstruction filter with passband gain of T (whereas "my" brick-wall reconstruction filter has a passband gain of 1 thus is dimensionally tractable). and the reason they need "T" in the reconstruction is because they don't put the factor in the sampling function (the "Dirac comb").
> 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!) >
-- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."