The sequence is time limited and the pulse width is constant, suppose 'T'
The Amplitudes of the pulse are either 0 or 1 and this is controlled by a Pseudo Random Binary Sequence.
So the occurrences of 1s and 0s is random and unknown
Something like the following needs careful analysis of interchange of limits and sense of convergence, but lets just go ahead and do it anyway.
Suppose we have a waveform W(t) \in L_2(R) and let
W(t) = int_{-\infty}^{\infty} w(\omega) exp(i\omega t) d\omega
be its Fourier representation i.e.
w(\omega) = (1/2\pi) \int_{-\infty}^\infty W(t) exp(-i\omega t) dt
Then your random sequence is upto a shift of zero and scale (0,1 --> -1, 1)
S(t) = \sum_{n = -\infty}^\infty X_n W(t - n)
for a sequence of X_n of independent identically distributed -1,1 Bernouilly stochasts that take on the value 1 with probability 1/2. Then the Fourier transform should be
S(\omega) = (sum_{n=-\infty}^\infty X_n exp(i n\omega) ) w(\omega)
But sum_{n=-\infty}^\infty X_n exp(i n\omega) is exactly the "derivative" of what people call the Brownian bridge.
See also the https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics