17 years ago
Sum d(t - n*T) is 2*Pi/T Sum d(s/j - 2*Pi/T)
I unsuccessfully tried to find the steps between the function and its
transform via two paths.
First, since the Dirac's comb is an infinite sequence of Dirac's
impulses, some advanced before t=0, and others delayed after t=0, the
delay theorem comes to the mind. Moreover, the Laplace Transform of
the Dirac's impulse being 1, it could not be simpler!
... + exp(-2*s*T) + exp(-2*s*T) + exp(0*s*T) + exp(1*s*T) +
exp(2*s*T) + ...
Oops! This series does not converge in the normal sense. It should
converge in the sense of distributions, but I do not know how to
handle it. :-(
Second, I start from the definition of the Laplace Transform.
Integral Sum d(t - n*T) exp(-s*t) dt
I change the order of the sum and the integral.
Sum Integral d(t - n*T) exp(-s*t) dt
Since the Dirac's impulse differs from 0 only when t=n*T and has an
«area» of 1, I can drop the integration.
Sum exp(-s*t) dt
Alas! I fall in the same dead-end as I fell above!
Could you help me find the way?
Thank you very much in advance.