[sdiy] Dirac deltas

[Ian Fritz]

At 02:50 PM 7/9/05, Aaron Lanterman wrote:

>A dirac delta x(t) = delta(t) has a flat spectrum X(j omega) = 1. Its 
>integral, a unit step function x(t) = u(t), would have a spectrum of X(j 
>omega) = 1/(j omega). (Uhm, sort of: u(t) doesn't have a finite absolute 
>integral, so it doesn't technically have a Fourier transform per se - 
>notice the spectrum blows up as omega -> infinity. But it works operationally).

Isn't this one of these deals where it depends on how you take the 
limit?  A square wave has well-defined Fourier coefficients Cn that (of 
course) don't depend on the frequency of the wave.  So to produce a single 
step by taking the limit of a long period square wave you would just keep 
using the same coefficients. Of course,  the fundamental frequency would be 
going to zero, so the coefficients corresponding to audio frequencies would 
have progressively larger n and would become small.  (Cn ~ 1/n, n -> infin.)

But to get back to popcorn noise, I think we shouldn't be interested in 
single isolated steps, but rather a collection of steps with some 
statistical time distribution.  So the correct comparison with square 
waves, if one is needed, would be with square waves having periods 
comparable to the time between steps.

   Ian