Re: A FAQ for Xmas

Pierian Spring wrote:
(snip)


2. Where have your Diracian impulses come from?

The Diracian has some interesting properties - zero width,
area of unity, infinite sum of all possible cosines, a height
which is not discussed but which appears to be greater in magnitude
than any voltage appearing in your circuits. In the
systems that you deal with, what experimental evidence do you
have that there are pulses with the attributes of Diracians upon
which to base your theory?


A simple example:
Take a LP filter consisting of one resistor R and one capacitor C.

If we apply a short pulse of voltage E to the input, the output rises by delta_v = (1/RC)E.delta_t

This is a good approximation if the pulse width (delta_t) is much shorter than the time constant (1/RC) and becomes a better approximation as delta_t approaches 0.

Having delta_t equal to 0 brings up mathematical difficulties, so we recognise that the important thing here is not E or delta_t alone, but the product E.delta_t, which is of course an impulse. If the product is 1.0 then it is called a unit impulse.

As long as we know the value of the impulse, and as long as delta_t is short, the precise value of E and delta_t do not matter because the response of the system will be the same.

You ask: "In the systems that you deal with, what experimental evidence do you have that there are pulses with the attributes of Diracians upon which to base your theory?"

The experimental evidence is that when the systems that I deal with are tested with an impulse, the response depends on the product E.delta_t. As long as delta_t is short enough, the response is independent of its precise length. In fact, the voltage output control of the function generator or its pulse-width control can be, within limits, used interchangeably as an 'impulse' control and this affects only the amplitude of the output of the system and not any other aspect of the system response.

I acknowledge that I am not the first in this newsgroup to point out all this, admittedly with arguments that are far more elegant than mine, but thought it may be of interest expressed in this particular way.

Regards,
John

.

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