etc etc etc :-( Big K's ghost, hopefully the finalisedversion!

This has been edited to resolve that the Unit Steps were the
wrong way around ...

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

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

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

... 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!)

On 27.02.2015 11:19, gareth wrote:

> EOE (Hopefull none this time!)

And YET another brand new posting of you, keeping the "ghost" alive by
constantly reiterating it.

Good job, buddy. Just a few more of these and everyone (including people
who didn't even know you before) will think you're a tool.

Cheers,
Johannes

-- 
>> Wo hattest Du das Beben nochmal GENAU vorhergesagt?
> Zumindest nicht �ffentlich!
Ah, der neueste und bis heute genialste Streich unsere gro�en
Kosmologen: Die Geheim-Vorhersage.
 - Karl Kaos &#4294967295;ber R&#4294967295;diger Thomas in dsa <hidbv3$om2$1@speranza.aioe.org>

"Johannes Bauer" <dfnsonfsduifb@gmx.de> wrote in message 
news:mcpi22$dc2$1@news.albasani.net...
> On 27.02.2015 11:19, gareth wrote:
>> EOE (Hopefull none this time!)
> And YET another brand new posting of you, keeping the "ghost" alive by
> constantly reiterating it.
> Good job, buddy. Just a few more of these and everyone (including people
> who didn't even know you before) will think you're a tool.

Do you think, perhaps, that you've started something based on a pointless 
triviality?

I cannot see why any normal rational person would wish to pursue your
current line of posts.

On 27.02.2015 11:58, gareth wrote:

> I cannot see why any normal rational person would wish to pursue your
> current line of posts.

I did post because I thought that you might want to be interested in the
perspective that someone has of you who didn't know you before this.
Obviously, you're ignorant to any well-meant advice.

You've since *started* four different threads in which you reiterate the
same thing over and over. I hope for you that you'll eventually realize
the pattern which causes others to see you as a loony. Hint, hint: could
have to do with multiple posts of identical content.

Cheers,
Johannes

-- 
>> Wo hattest Du das Beben nochmal GENAU vorhergesagt?
> Zumindest nicht �ffentlich!
Ah, der neueste und bis heute genialste Streich unsere gro�en
Kosmologen: Die Geheim-Vorhersage.
 - Karl Kaos &#4294967295;ber R&#4294967295;diger Thomas in dsa <hidbv3$om2$1@speranza.aioe.org>

"Johannes Bauer" <dfnsonfsduifb@gmx.de> wrote in message 
news:mcpn7h$na1$1@news.albasani.net...
> I did post because I thought that you might want to be interested in the
> perspective that someone has of you who didn't know you before this.
> Obviously, you're ignorant to any well-meant advice.
> You've since *started* four different threads in which you reiterate the
> same thing over and over. I hope for you that you'll eventually realize
> the pattern which causes others to see you as a loony. Hint, hint: could
> have to do with multiple posts of identical content.

I cannot see why any normal rational person would wish to pursue your
current line of posts.

You are welcome to comment upon the technical matters raised
but there is really no call for resorting to repeated abusive posts
such as you do.

It does seem very hypocritical for you to complain about me
doing something many times when you seem intent on
representing yourself many times as a childish abuser.