[sdiy] Power spectra

[Magnus Danielson]

From: Aaron Lanterman <lanterma at ece.gatech.edu>
Subject: [sdiy] Power spectra
Date: Mon, 11 Jul 2005 00:55:35 -0400 (EDT)
Message-ID: <Pine.GSO.4.60.0507110015290.19816 at bigzilla.ece.gatech.edu>

> 
> On Sun, 10 Jul 2005, Ian Fritz wrote:
> 
> > So from what I have been reading, popcorn noise has a 1/f^2 power spectrum 
> > above a cutoff fc.  So a first order lowpass operating on white noise might 
> > be a good way to go.
> >
> > popcorn power ~ 1/(1 + (f/fc)^2)
> 
> One thing to remember is that if you have a deterministic signal with 
> Fourier spectrum X(j w), and you filter with a filter with frequency 
> response
> H(j w), you get an output Fourier spectrum Y(j w) = H(j w) X(j w).
> 
> If instead you have a _random process_ with _power spectrum_ S_x(j w),
> and you pass it through a filter with frequency response H(j_w), you get 
> an output power spectrum of S_y(j w) = |H(j_w)|^2 S_x(j w).

You are a TEX/LATEX user, arn't you? ;O)

The differences you mention is not due to the randomness or not of the signal
but from using a amplitude response and use it either directly on the signal
response or on the power response. You can do either to both types of signals.

White noise has a flat line in its X(jw) spectra (it contains equal amount of
all frequencies, within some -3 dB points given by the system). The Sx(jw)
spectrum is only the square of the absolute and is also flat with frequency, so
we can use the same formula in either case, as long as the amplitude is scaled
properly.

> So indeed, a first-order filter would probably give you the second order 
> effect you want.

The H(jw) or H(s) properties of a -6 dB slope, that is a 1/f slope is the
amplitude responce. When you convert that into a power response you have to
square it and then you have a power response slope of 1/f².

It is the powerresponse of 1/f which is a devil, since you can not make a
theoretically perfect -3 dB/Oct response, but instead we work with
approximations for our frequency range which is sufficiently close.

Cheers,
Magnus