Maths: the Laplace Transform of the Dirac's comb?

-- snip --

Whenever I get wound up in the infinities of the dirac delta function I start thinking of taking the limit of a pulse of unit area as it's width goes to zero. Usually I use a rectangular pulse.

For the following proof you'll have to accept that the Fourier transform of a repeating waveform is a set of dirac deltas weighted by the Fourier series for the waveform. You can prove this using the similar techniques.

I'm going to prove this for the Fourier transform, and leave it to you to clean it up and extend it to the Laplace domain. I come out different by a factor of 2pi, I suspect because we define the transforms differently.

So say that h(t, a) defines a pulse such that

{ 0 t < -a/2 h(t, a) = { 1/a -a/2 0, and you'll get the correct expression.

-- snip --

Thank you, Mr. Wescott.

The two words «repeating waveform» woke me up! Of course, the Dirac's comb is a sequence of delta functions which fire (pardon me for the violent word ;-) ) every nT unit of time where T is the period. Being a periodic function, it is legitimate to calculate its Fourier coefficients. Let us assume that our Dirac's comb is made up of unit impulses and let us calculate its Fourier coefficients over the interval [0..T[. In this interval, the delta function fires only once at exactly t = 0.

T a[n] = (1/T) Integral[ d(t-n/T) exp(-j*n*2*pi*t/T) dt 0

At t = 0, exp(-j*n*2*pi*t/T evaluates to 1 for all values of n, the delta function integrates to 1, so a[n] = 1/T for all n!

So I am allowed to write: +inf +inf Sum d(t - n*T) = 1/T Sum exp(j*n*2*pi*t/T) -inf -inf

Amazing, is it not? The Dirac's comb is an infinite sum of phasors whose angular speed (in radians) is n*2*pi/T and modulus 1/T.

The plot of the Fourier coefficients versus frequency is repetitive to say the least: a dot of ordinate 1/T at every abscissa n/T. From this plot, we can go intuitively to the Fourier transform. No doubt, this will not satisfy the mathematicians in this group; so let us call them to fill the void. The latter being the «density» of the frequency component in the signal, we can say that there is an infinite density of the frequency component at every frequency n/T. Let us use the delta function to express it.

Most of the time, I prefer to express the spectrum with respect to the natural frequency (2*pi*f - is this the correct word for French word «pulsation»?). Making two substitutions in the above expression does it: replace f by w, and replace T by T/2*pi.

P.S.: In French, we use the word «transformation» to describe the Transform mathematical process and the word «transformée» to describe the result. Is there such a distinction in English? The Dirac impulse and the complex exponential are so useful that I am surprised that their «Intellectual Property» is not yet protected by a rain of USPO patents.