Ross Bencina wrote:
Hi Everyone,
Suppose that I generate a time series x[n] as follows:
>>>
P is a constant value between 0 and 1
At each time step n (n is an integer):
r[n] = uniform_random(0, 1)
x[n] = (r[n] <= P) ? uniform_random(-1, 1) : x[n-1]
Where "(a) ? b : c" is the C ternary operator that takes on the value b if a is
true, and
c otherwise.
<<<
What would be a good way to derive a closed-form expression for the spectrum of
x?
(Assuming that the series is infinite.)
...
Hi, from me at the moment only some generalities that appear difficult (at any level)
usually: if you want the Fourier transform of a "signal" in this case a sequence of
numbers between 0 and 1 (inclusive), the interpretations are important: is this coming
from a physical process you sampled (without much regarding Shannon), or is this some sort
of discrete Markov chain output, where you interpret the sequence of samples zeroth order
interpolated as an continuous signal that you take the FT of? In the last case, you'll get
a spectrum that shows clear multiples of the "sampling frequency" and that is highly
irregular because of the randomness, and I don't know if the FT's infinite integral of
this signal converges and is unambiguous, you might have to prove that first.
Statistically, often a problem, this sequence of numbers is like two experiments in
sequence, on depending on the other. The randomness of the P invoked choice still easily
works with the norm "big numbers" approximation, clearly, but the second one, and therefor
the result of the function prescription, has a ***dependency** which makes normal
statistical shortcuts invalid. I don't know a proper way to give a proper and correct
statistical analysis of the number sequence, and I am not even sure there is a infinite
summation based proper DC average computable. Two statistical variables with a
inter-dependency requires the use of proper summations sums or maybe Poisson sequence
analysis, I don't recall exactly, but the dependency makes it hard to do an "easy" analysis.
It could be a problem from an electronics circuit for switched supplies or something, or
maybe in a more restricted form it's an antenna signal processor step or something,
usually there are more givens in these cases, analog or digital, that you might want to
know before a proper statistical analysis can be in order, but anyhow, you could write a
simple program and do some very large sum computations, as separate experiments a number
of times with different random seeds or generators and see what happens, for instance if
simulation results soon give the impression of a fixed signal average.
T.V.
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