Although not posting here for some years, I have nevertheless been
a regular reader.

For some reason, two of the kindergarten class from uk.radio.amateur,
reay and cole, in order to draw attention to themsleves, decided to
annoy the denizens of this NG by raking over the embers from 10 years ago,
so perhaps it is appropriate to lay the ghost of Big K, for those
interested in the fundamental basis of sampling.

So...

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

(1/T)sum (0, inf)(d(t-T) * f(t-T) )

... 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 (0, inf)(d(t-T) * f(t-T) )

... which lacks the essential divisor of T.

(In the recent diatribe from reay, he tried to claim that this division factor of T is mentioned in all the texts which is simply untrue, so I suspect that in his haste to want to blurt out an infantile insult that reay is confusing the descriptions of sampling with the derivation of the Fourier Series)

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 (0, inf)( f(t-T) * ( U(t-2T) - U(t-T)) )

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

So, as the Diracian Delta is a doddle to deal with, 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 T volts high and 1/T seconds long, with T tending towards zero, which in out asciification comes out as ..

T * ( U(t-2T) - U9t-T) )

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

sum (0, inf)( f(t-T) * ( U(t-2T) - U(t-T)) )

... can also be represented as ..

(1/T)sum (0, inf)(d(t-T) * f(t-T) )

... 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 Bristow's article about sampling and reconstruction (in one of Bristow's rare manifestation as as a grown up?) that he had to re-introduce the factor of T out-of-thin-air for reconstruction, so I'd like to suggest from my analusis 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

-----ooooo-----

(Cross-posted to uk.radio.amateur for the benefit of cole who has demanded such a explanation, although I doubt he will understand any of it, and will respond, if he ever dares to show his face, with abuse and bluster)

For some reason, two of the kindergarten class from uk.radio.amateur,

So...

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

(1/T)sum (0, inf)(d(t-T) * f(t-T) )

... 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 (0, inf)(d(t-T) * f(t-T) )

... which lacks the essential divisor of T.

(In the recent diatribe from reay, he tried to claim that this division factor of T is mentioned in all the texts which is simply untrue, so I suspect that in his haste to want to blurt out an infantile insult that reay is confusing the descriptions of sampling with the derivation of the Fourier Series)

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 (0, inf)( f(t-T) * ( U(t-2T) - U(t-T)) )

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

So, as the Diracian Delta is a doddle to deal with, 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 T volts high and 1/T seconds long, with T tending towards zero, which in out asciification comes out as ..

T * ( U(t-2T) - U9t-T) )

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

sum (0, inf)( f(t-T) * ( U(t-2T) - U(t-T)) )

... can also be represented as ..

(1/T)sum (0, inf)(d(t-T) * f(t-T) )

... 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 Bristow's article about sampling and reconstruction (in one of Bristow's rare manifestation as as a grown up?) that he had to re-introduce the factor of T out-of-thin-air for reconstruction, so I'd like to suggest from my analusis 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

-----ooooo-----

(Cross-posted to uk.radio.amateur for the benefit of cole who has demanded such a explanation, although I doubt he will understand any of it, and will respond, if he ever dares to show his face, with abuse and bluster)