Big K - laying the ghost to rest

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)
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Come along now, Gareth. This screed simply cannot be peer reviewed as you've rather evasively refused to make any references whatsoever. I can't waste my time with this hogwash, try harder man.
--
STC // M0TEY // twitter.com/ukradioamateur

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I see he has wasted another day bouncing off the walls of his self constructed cell.
As you say, his post is less than convincing, I gave up on it due to the 'revisionist' nature, those familiar with his crack pot theories will know what I mean.
His habit of claiming that he understands things 'to his satisfaction' explains a lot, after all, success relies on satisfying others.
Anyway, I've had a busy and productive day, time for some dinner, a chat with my XYL, then turn on the soldering iron.
I've check in later to see how the ball game is going, I can see this one bouncing for sometime yet ;-)
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The term e^(-jwt) isn't some magical time machine relating to "minus time", e^(-jwt) is simply another way of writing 1/(e^jwt) which is a value that decreases as t increasing.
-----ooooo-----
... but humility would be preferable to your blustering with personal remarks as youdo above.
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I remember trying to tell you exactly the same thing back in 2003 (Blimey, was it really 12 years ago?) and you said that because there was an "imaginary" j component (j=SQRT(-1)) the normal rules of algebra didn't apply! Would you care to enlarge upon your statement?
--
;-)
.
73 de Frank Turner-Smith G3VKI - mine's a pint.
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He also claimed you couldn't divide complex numbers which, if true, would invalidate countless standard techniques etc.
I think that was the time I went through some complex number theory and Matt commented it was the fist time he'd understood some of it. It may have been another time, you know how repetitive he is and how often I tried to help him. He always responded with abuse.
He eventually claimed that the standard technique used to divide complex numbers was flawed as it didn't actually divide complex numbers due to the use of the conjugate of the denominator. Such a claim is tantamount to claiming that 4/8 is not equal to 1/2, which I would expect a primary school pupil to know was nonsense.
As usual, the above can all be verified in the archives.
It is almost impossible to believe he could lack such basic skills. Were it simply his behaviour on the newsgroups we could perhaps dismiss it all as just obtuseness but when you look at his employment problems.... After all, can you imagine even a test technician with such gaps?
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Put your money where your mouth is and evaluate the following quotient, other than by multiplying top and bottom by the complex conjugate which is a method of evaluating quotients but is not actually division ...
(17 + 23i) / (3 + 7i)

See above, your coment is nonsense.

Once again, it is you who tries to stir things up with abuse despite your trying to lay that at others' doors.
But again, brian, why are you digging up discussions, arguments even, from 10 or more years ago? Is your life today so shallow and worthless? Are you bouncing off the walls in your downmarket house in a back lane?
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< snip >

Zoopla reckons Brian's "down-market house in a back lane" is worth several times more than your starter semi in a cul-de-sac. HTH
--
;-)
.
73 de Frank Turner-Smith G3VKI - mine's a pint.
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On Fri, 27 Feb 2015 10:07:38 +0000, FranK Turner-Smith G3VKI wrote:

<grumpy rant> Who really cares, I mean who does? If Batman lives in a more expensive area than Robin it matters nought. Seems to me that too many people know the price of everything and the value on zilch. Just saying.
The abusive back and forth between Gareth and his detractors is tedious and immature. Grown men, my watzit!
had enough :-(
</grumpy rant>
Now I'm off to the sunny garden to assemble a potting table.
Charlie.
--
Hello Wisconsin!

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Oh well, so long as I've got the biggest willy I don't care.
--
;-)
.
73 de Frank Turner-Smith G3VKI - mine's a pint.
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On Fri, 27 Feb 2015 12:05:51 +0000, FranK Turner-Smith G3VKI wrote:

I've just had enough of Gareth, STC, BR and those who can't get by with dragging up each other's failing on a weekly basis. None of those three appear to have the self-awareness or ability to change their behaviour or let bygones be bygones. They have far too much emotional investment in each other to walk away. The holes in their lives would be too great as far as I can see. Life's too short to spend time waiting for them to get better.
As for your willy, what ever the size, keep it to yourself :-)
Charlie.
--
Hello Wisconsin!

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You misjudge me because I _NEVER_ originate such matters.
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On Fri, 27 Feb 2015 14:01:42 +0000, gareth wrote:

Irregular verbs, innit; I'm responding, he's originating, you're a gormless arsehole.
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He's Gormless Amateur Radio Enthusiast, Total Headcase!
--
STC // M0TEY // twitter.com/ukradioamateur

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_S_T_U_P_I_D_ boy, Cole.
It is interesting the malicious way that you deal with challenges that have arisen from the things that you yourself have posted.
Your credibility has just gone down to zero.

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Oh dear, I failed to notice that in your malicious intent to divert attention away from your own mental state, that you had cross-posted to uk.rec.models.rail, a completely unrelated NG, but it does show evidence that you are stalking me.
Shame on you.
Now, you complained that there were no refences to accompany my dissertation, so I have provided you with a number of them, but now you seem to be squirming and running away with your tail between your legs, shouting rather silly and infantile remarks as you go.
Any credibility that you may have had, if any, has just disappeared.
Why are you even posting to technical NG, for you seem to lack any technical acumen whatsoever and 100% of your posts contain rather silly and infantile remarks that only serve to show you up for the fool that you undoubtedly are.
<CAPTAIN MAINWARING> _S_T_U_P_I_D_ boy. </CAPTAIN MAINWARING>


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