Power spectrum of a modified random walk

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&rsquo;m not concerned with
this region).  The question is, what is the power spectrum of s(t), as a
function of alpha?

I&rsquo;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 

>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!

Hi Steve,

I tried to have a crack at it, by first finding the autocorrelation
function of r(t)  as E{ r(t) r*(t-tau) }.  It turns out it depends both on
t and tau, so your random walk is not a stationary process.  So it is not
possible to speak of a power spectrum that is independent of time! (Maybe
this is the problem:  you are looking for something that does not exist.)
If I remember correctly, PSD of an energy signal is defined only if it is
stationary.

I am not sure if this will work, but you might try finding the
time-dependent spectrum as the Fourier transform of the autocorrelation
with respect to the lag variable (tau above). 

Emre

Above I meant PSD of a "power" signal, instead of an "energy" signal.  (In
your case w(t) is a power signal; it has infinite energy.)

Emre

>I am not sure if this will work, but you might try finding the
>time-dependent spectrum as the Fourier transform of the autocorrelation
>with respect to the lag variable (tau above). 

The following paper gives some background and references on non-stationary
spectral analysis:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.9899

Hi Emre,
It hadn't occured to me that the signal would be nonstationary.  This
explains why I am having so much trouble, and gives me some ideas to look
at. Thanks!
Steve   

>Hi Emre,
>It hadn't occured to me that the signal would be nonstationary.  This
>explains why I am having so much trouble, and gives me some ideas to
look
>at. Thanks!
>Steve   

Glad I could help.  :-)

Emre