4.4 Spectral properties of random signals

Random processes Page 1 / 1

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 $τ$ . In particular, for $τ 0$

r_{X X} 0 T

∞ 1 2 T t T T X t 2 mean power of X(t)

Note that if

T

∞ , for all

τ

r_{X X} τ r_{X X} τ r_{X X} 0

τ

becomes large,

X t

and

X t τ

will usually become decorrelated and, as long as

X

is zero mean,

r_{X X}

will tend to zero.

Hence the ACF will have its maximum at $τ 0$ and decay symmetrically to zero (or to $μ 2$ , if $μ 0$ ) as $τ$ increases.

The width of the ACF (to say its half-power points) tells us how slowly $X$ 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 $τ$ increases, whereas, for a slowly fluctuating signal, as in lower plot, the ACF decays much more slowly.

Illustration of the different properties of wide band (upper) and narrow band (lower) random signals: (a) thesignal waveforms with unit variance; (b) their autocorrelation functions (ACFs); and (c) their powerspectral densities (PSDs). In (b) and (c), the thin fluctuating curves shows the actual values measured from4000 samples of the random waveforms while the thick smooth curves show the limits of the ACF and PSD as the lengths ofthe waveforms tend to infinity.

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 :

ℱ_{u} ω FT u t t u t ω t

The Power Spectral Density (PSD) of a random process

X

is defined to be the Fourier Transform of its ACF:

S_{X} ω FT r_{X X} τ τ r_{X X} τ ω τ

r_{X X} τ FT S_{X} ω 12 ω S_{X} ω ω τ

N.B.

X t

must be at least Wide Sense Stationary (WSS).

From and we see that the mean signal power is given by:

r_{X X} 0 12 ω S_{X} ω f S_{X} 2 f

Hence

S_{X}

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 $X t$ :

$S_{X} ω S_{X} ω$
$S_{X} ω$ is Real-valued
$S_{X} ω 0$

Properties 1 and 2 are because ACFs are real and symmetric about

τ 0

; and 3 is because

S_{X}

represents power density.

Linear system (filter) with wss input

Block diagram of a linear system with a random input signal,

X t

Let the linear system with input $X t$ and output $Y t$ have an impulse response $h t$ , so

Y t h t X t α h α X t α

Then the ACF of

Y

r_{Y Y} t_{1} t_{2} Y t_{1} Y t_{2} α_{1} h α_{1} X t_{1} α_{1} α_{2} h α_{2} X t_{2} α_{2} α_{2} α_{1} h α_{1} h α_{2} X t_{1} α_{1} X t_{2} α_{2} α_{2} α_{1} h α_{1} h α_{2} X t_{1} α_{1} X t_{2} α_{2} α_{2} α_{1} h α_{1} h α_{2} r_{X X} t_{1} α_{1} t_{2} α_{2}

X

is WSS then

r_{Y Y} τ Y t Y t τ α_{2} α_{1} h α_{1} h α_{2} r_{X X} τ α_{1} α_{2} r_{X X} τ h τ h τ

Taking Fourier transforms:

S_{Y} ω FT r_{Y Y} τ τ α_{2} α_{1} h α_{1} h α_{2} r_{X X} τ α_{1} α_{2} ω τ α_{2} α_{1} h α_{1} h α_{2} τ r_{X X} τ α_{1} α_{2} ω τ α_{2} α_{1} h α_{1} h α_{2} λ r_{X X} λ ω λ α_{1} α_{2} α_{1} h α_{1} ω α_{1} α_{2} h α_{2} ω α_{2} λ r_{X X} λ ω λ ℋ ω ℋ ω S_{X} ω

where

ℋ ω FT h t

. i.e:

S_{Y} ω ℋ ω 2 S_{X} ω

Hence the PSD of

Y

= the PSD of

X

\times

the power gain

ℋ 2

of the system at frequency

ω

Thus if a large and important system is subject to random perturbations (e.g. a power plant subject to random loadfluctuations), we may measure $r_{X X} τ$ and $r_{Y Y} τ$ , transform these to $S_{X} ω$ and $S_{Y} ω$ , and hence obtain

ℋ ω S_{Y} ω S_{X} ω

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

ℋ ω

However, if instead we measure the Cross-Correlation Function (CCF) between $X$ and $Y$ , we get:

r_{X Y} t_{1} t_{2} X t_{1} Y t_{2} X t_{1} α_{2} h α_{2} X t_{2} α_{2} α_{2} h α_{2} X t_{1} X t_{2} α_{2} α_{2} h α_{2} X t_{1} X t_{2} α_{2} α_{2} h α_{2} r_{X X} t_{1} t_{2} α_{2}

X t

, and hence

Y t

, are WSS:

r_{X Y} τ X t Y t τ α h α r_{X X} τ α h τ r_{X X} τ

and taking Fourier transforms:

S_{X Y} ω FT r_{X Y} τ ℋ ω S_{X} ω

where

S_{X Y} ω

is known as the Cross Spectral Density between

X

and

Y

. Therefore,

ℋ ω S_{X Y} ω S_{X} ω

Hence we obtain the amplitude and phase of

ℋ ω

. As before, this is achieved without taking the plant off line.

Note that for WSS processes, $r_{X Y} τ r_{Y X} τ$ and that (unlike $r_{X X}$ and $r_{Y Y}$ ) these need not be symmetric about $τ 0$ . Hence the cross spectral density $S_{X Y} ω$ need not be purely real (unlike $S_{X} ω$ ), and the phase of $S_{X Y} ω$ gives the phase of $ℋ ω$ .

Physical interpretation of power spectral density

Narrowband filter frequency response and PSD of filter input and output.

Let us pass $X t$ through a narrow-band filter of bandwidth $δ ω 2 δ f$ , as shown in :

ℋ ω 1 ω_{0} ω ω_{0} δ ω 0

Find average power at the filter output (shaded area in , divided by

2

P_{0} r_{Y Y} 0 12 ω

∞ ∞ S Y ω 1 2 ω ∞ ∞ S X ω ℋ ω 2 1 2 ω ω 0 δ ω 0 ω 0 S X ω ω ω 0 ω 0 δ ω 0 S X ω 2 S X ω 0 1 2 δ ω 0

since

S_{X} ω S_{X} ω

. Expressed in terms of

f_{0} ω_{0} 2

P_{0} 2 S_{X} 2 f_{0} δ f

with the factor of 2 appearing because our filter responds to both negative and positive frequency components of

X

Hence $S_{X}$ is indeed a Power Spectral Density with units $V 2 Hz$ (assuming unit impedance).

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Source: OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3

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