OK, I‘m stumped. I’ve been trying to calculate the power spectrum of a
particular random waveform, and I’m not making much progress. Help!
Let w(t) be a random signal with a white spectrum. If we integrate this
signal we obtain a random walk, r(t). That is, r(t) = Int[0..t] w(y) dy,
for t > 0. As well known, the expectation of the amplitude of r(t)
increases with the square root of time, i.e., t^(0.5).
I want to force this amplitude to decrease over time, and do so by
multiplying the waveform by a decaying power law. That is, s(t) = u(t)
r(t) t^alpha, where u(t) is the unit step, and 0 > alpha > -1.5.
(Nasty
things happen in the math for alpha <= -1.5, and I’m not concerned with
this region). The question is, what is the power spectrum of s(t), as a
function of alpha?
I’ve done extensive computer simulations of the problem and know that
the general form is: P(w) = w^beta, where P(w) is the power spectrum of
s(t). For alpha = 0, beta = -1 (i.e., the power spectrum of the
unmodified
random walk).
However, for -0.5 < alpha < -1.5, beta reaches an asymptotic value of -(2
alpha + 3). This is not as strange as it seems, it is simply the power
spectrum of the envelope of the waveform. That is, the power spectrum of
the random signal: u(t) r(t) t^(alpha) is the same as the power spectrum
of the deterministic waveform: u(t) t^(alpha + 0.5). The factor of 0.5
in
this is the natural expansion of the random walk is being overcome.
In short, if the amplitude of the random walk is allowed to grow in time,
its power spectrum remains that of a random walk. On the other hand,
forcing the amplitude to decrease over time produces a power spectrum that
corresponds to the envelope of the decrease.
Now, how do I show this analytically? I keep hitting dead ends when I go
through the techniques I know. Thanks in advance!
Steve
|
