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Spectral Properties of Random Signals

Module by: Nick Kingsbury

Summary: 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 a function of time shift ττ. In particular, for τ=0 τ 0
r X X 0=limT→∞12T∫-TTX2tdt=mean  power  of  X(t) r X X 0 T 1 2 T t T T X t 2 mean  power  of  X(t) (1)
Note that if T→∞ T , for all ττ
r X X τ= r X X -τ≤ r X X 0 r X X τ r X X τ r X X 0 (2)
As ττ becomes large, Xt X t and Xt+τ X t τ will usually become decorrelated and, as long as XX is zero mean, r X X r X X will tend to zero.
Hence the ACF will have its maximum at τ=0 τ 0 and decay symmetrically to zero (or to μ2 μ 2 , if μ≠0 μ 0 ) as |τ| τ increases.
The width of the ACF (to say its half-power points) tells us how slowly XX is fluctuating or how band-limited it is. subfigure 1.2 shows how the ACF of a rapidly fluctuating (wide-band) random signal, as in subfigure 1.1 upper plot, decays quickly to zero as |τ| τ increases, whereas, for a slowly fluctuating signal, as in subfigure 1.1 lower plot, the ACF decays much more slowly.
Subfigure 1.1
Subfigure 1.2
Subfigure 1.3
Figure 1: Illustration of the different properties of wide band (upper) and narrow band (lower) random signals: (a) the signal waveforms with unit variance; (b) their autocorrelation functions (ACFs); and (c) their power spectral densities (PSDs). In (b) and (c), the thin fluctuating curves shows the actual values measured from 4000 samples of the random waveforms while the thick smooth curves show the limits of the ACF and PSD as the lengths of the 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=FTut=∫utⅇ-ⅈωtdt ℱ u ω FT u t t u t ω t (3)
The Power Spectral Density (PSD) of a random process XX is defined to be the Fourier Transform of its ACF:
SXω=FT r X X τ=∫ r X X τⅇ-ⅈωτdτ SX ω FT r X X τ τ r X X τ ω τ (4)
r X X τ=FT-1SXω=12π∫SXωⅇⅈωτdω r X X τ FT SX ω 1 2 ω SX ω ω τ (5)
N.B. Xt X t must be at least Wide Sense Stationary (WSS).
From Equation 1 and Equation 5 we see that the mean signal power is given by:
r X X 0=12π∫SXωdω=∫SX2πfdf r X X 0 1 2 ω SX ω f SX 2 f (6)
Hence SX SX has units of power per Hertz. Note that we must integrate over all frequencies, both positive and negative, to get the correct total power.
subfigure 1.3 shows how the PSDs of the signals relate to the ACFs in subfigure 1.2.
Properties of PSDs for real-valued Xt X t :
  1. SXω=SX-ω SX ω SX ω
  2. SXω SX ω is Real-valued
  3. SXω≥0 SX ω 0
Properties 1 and 2 are because ACFs are real and symmetric about τ=0 τ 0 ; and 3 is because SX SX represents power density.

Linear system (filter) with WSS input

Figure 2: Block diagram of a linear system with a random input signal, Xt X t .
Let the linear system with input Xt X t and output Yt Y t have an impulse response ht h t , so
Yt=ht*Xt=∫hαXt-αdα Y t h t X t α h α X t α (7)
Then the ACF of YY is
r Y Y t1t2=EYt1Yt2=E∫hα1Xt1-α1dα1∫hα2Xt2-α2dα2=E∫∫hα1hα2Xt1-α1Xt2-α2dα1dα2=∫∫hα1hα2EXt1- α 1 Xt2-α2dα1dα2=∫∫hα1hα2 r X X t1-α1t2-α2dα1dα2 r Y Y t1 t2 Y t1 Y t2 α1 h α1 X t1 α1 α2 h α2 X t2 α2 α2 α1 h α1 h α2 X t1 α1 X t2 α2 α2 α1 h α1 h α2 X t1 α 1 X t2 α2 α2 α1 h α1 h α2 r X X t1 α1 t2 α2 (8)
If XX is WSS then
r Y Y τ=EYtYt+τ=∫∫hα1hα2 r X X τ+α1-α2dα1dα2= r X X τ*h-τ*hτ r Y Y τ Y t Y t τ α2 α1 h α1 h α2 r X X τ α1 α2 r X X τ h τ h τ (9)
Taking Fourier transforms:
SYω=FT r Y Y τ=∫∫∫hα1hα2 r X X τ+ α 1 - α 2 dα1dα2ⅇ-ⅈωτdτ=∫∫hα1hα2∫ r X X τ+ α 1 - α 2 ⅇ-ⅈωτdτdα1dα2=∫∫hα1hα2∫ r X X λⅇ-ⅈωλ- α 1 + α 2 dλdα1dα2=∫hα1ⅇⅈωα1dα1∫hα2ⅇ-ⅈωα2dα2∫ r X X λⅇ-ⅈωλdλ=ℋω¯ℋωSXω SY ω FT r Y Y τ τ α2 α1 h α1 h α2 r X X τ α 1 α 2 ω τ α2 α1 h α1 h α2 τ r X X τ α 1 α 2 ω τ α2 α1 h α1 h α2 λ r X X λ ω λ α 1 α 2 α1 h α1 ω α1 α2 h α2 ω α2 λ r X X λ ω λ ℋ ω ℋ ω SX ω (10)
where ℋω=FTht ℋ ω FT h t . i.e:
SYω=|ℋω|2SXω SY ω ℋ ω 2 SX ω (11)
Hence the PSD of YY = the PSD of XX ×× the power gain |ℋ|2 ℋ 2 of the system at frequency ωω.
Thus if a large and important system is subject to random perturbations (e.g. a power plant subject to random load fluctuations), we may measure r X X τ r X X τ and r Y Y τ r Y Y τ , transform these to SXω SX ω and SYω SY ω , and hence obtain
|ℋω|=SYωSXω ℋ ω SY ω SX ω (12)
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 ℋω ℋ ω .
However, if instead we measure the Cross-Correlation Function (CCF) between XX and YY, we get:
r X Y t1t2=EXt1Yt2=EXt1∫hα2Xt2-α2dα2=E∫hα2Xt1Xt2-α2dα2=∫hα2EXt1Xt2- α 2 dα2=∫hα2 r X X t1t2-α2dα2 r X Y t1 t2 X t1 Y t2 X t1 α2 h α2 X t2 α2 α2 h α2 X t1 X t2 α2 α2 h α2 X t1 X t2 α 2 α2 h α2 r X X t1 t2 α2 (13)
If Xt X t , and hence Yt Y t , are WSS:
r X Y τ=EXtYt+τ=∫hα r X X τ-αdα=hτ* r X X τ r X Y τ X t Y t τ α h α r X X τ α h τ r X X τ (14)
and taking Fourier transforms:
S X Y ω=FT r X Y τ=ℋωSXω S X Y ω FT r X Y τ ℋ ω SX ω (15)
where S X Y ω S X Y ω is known as the Cross Spectral Density between XX and YY. Therefore,
ℋω= S X Y ωSXω ℋ ω S X Y ω SX ω (16)
Hence we obtain the amplitude and phase of ℋω 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 r X Y τ r Y X τ and that (unlike r X X r X X and r Y Y r Y Y ) these need not be symmetric about τ=0 τ 0 . Hence the cross spectral density S X Y ω S X Y ω need not be purely real (unlike SXω SX ω ), and the phase of S X Y ω S X Y ω gives the phase of ℋω ℋ ω .

Physical Interpretation of Power Spectral Density

Figure 3: Narrowband filter frequency response and PSD of filter input and output.
Let us pass Xt X t through a narrow-band filter of bandwidth δω=2πδf δ ω 2 δ f , as shown in Figure 3:
ℋω=1ifω0<|ω|≤ω0+δω0otherwise ℋ ω 1 ω0 ω ω0 δ ω 0 (17)
Find average power at the filter output (shaded area in Figure 3, divided by 2π 2 ):
P0= r Y Y 0=12π∫-∞∞SYωdω=12π∫-∞∞SXω|ℋω|2dω=12π∫-ω0+δω0-ω0SXωdω+∫ω0ω0+δω0SXωdω≈2SXω012πδω0 P0 r Y Y 0 1 2 ω SY ω 1 2 ω SX ω ℋ ω 2 1 2 ω ω0 δ ω0 ω0 SX ω ω ω0 ω0 δ ω0 SX ω 2 SX ω0 1 2 δ ω0 (18)
since SX-ω=SXω SX ω SX ω . Expressed in terms of f0=ω02π f0 ω0 2 :
P0≈2SX2πf0δf P0 2 SX 2 f0 δ f (19)
with the factor of 2 appearing because our filter responds to both negative and positive frequency components of XX.
Hence SX SX is indeed a Power Spectral Density with units V2Hz V 2 Hz (assuming unit impedance).

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