This module introduces spectral properties of random signals, such as relation of power spectral density to ACF, linear system (filter) with WSS input, and physical interpretation of power spectral density.
Relation of power spectral density to acf
The autocorrelation function (ACF) of an ergodic random signal
tells us how correlated the signal is with itself as afunction of time shift
. In particular, for
Note that if
, for all
As
becomes large,
and
will usually become decorrelated and, as long as
is zero mean,
will tend to zero.
Hence the ACF will have its
maximum at
and decay symmetrically to zero (or to
, if
) as
increases.
The width of the ACF (to say its half-power points) tells us
how slowly
is fluctuating or
how band-limited it is.
shows how the ACF of a rapidly fluctuating (wide-band) random
signal, as in
upper
plot, decays quickly to zero as
increases, whereas, for a slowly fluctuating
signal, as in
lower
plot, the ACF decays much more slowly.
The ACF measures an entirely different aspect of
randomness from amplitude distributions such as pdf and cdf.
As with deterministic signals, we may formalize our ideas of
rates of fluctuation by transforming to the
Frequency
(Spectral) Domain using the
Fourier
Transform :
The
Power Spectral Density (PSD) of a random
process
is defined to be the
Fourier Transform of its ACF:
N.B.
must be
at least Wide Sense
Stationary (WSS).
From
and
we see that the mean signal power
is given by:
Hence
has units of power per Hertz. Note that we must
integrate over
all frequencies, both
positive and negative, to get the correct total power.
shows how the PSDs of
the signals relate to the ACFs in
.
Properties of PSDs for real-valued
:
is Real-valued
Properties 1 and 2 are because ACFs are real and symmetric
about
; and 3 is because
represents
power density.
Linear system (filter) with wss input
Let the linear system with input
and output
have an impulse response
, so
Then the ACF of
is
If
is WSS then
Taking Fourier transforms:
where
. i.e:
Hence the PSD of
= the PSD of
the power gain
of the system at frequency
.
Thus if a large and important system is subject to random
perturbations (e.g. a power plant subject to random loadfluctuations), we may measure
and
, transform these to
and
, and hence obtain
Hence we may measure the system frequency response
without taking the plant off line. But this does not give any information about the
phase of
.
However, if instead we measure the
Cross-Correlation
Function (CCF) between
and
, we get:
If
, and hence
, are WSS:
and taking Fourier transforms:
where
is known as the
Cross Spectral Density between
and
. Therefore,
Hence we obtain the
amplitude and phase of
. As before, this is achieved without taking the
plant off line.
Note that for WSS processes,
and that (unlike
and
) these need not be symmetric about
. Hence the cross spectral density
need not be purely real (unlike
), and the phase of
gives the phase of
.
Physical interpretation of power spectral density
Let us pass
through a narrow-band filter of bandwidth
, as shown in
:
Find average power at the filter output (shaded area in
, divided by
):
since
. Expressed in terms of
:
with the factor of 2 appearing because our filter responds to
both negative and positive frequency components of
.
Hence
is indeed a
Power Spectral Density with
units
(assuming unit impedance).