Chapter 8 - Power Density Spectrum

Chapter 8 - Power Density Spectrum

Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t)2_{], a constant.}

This total average power is distributed over some range of frequencies. This distribution over frequency is described by SX(), the power density spectrum. SX() is non-negative

(SX()  0) and, for real-valued X(t), even (SX() = SX(-)). Furthermore, the area under SX is

proportional to the average power in X(t); that is

Average Power in X(t) d  

z

1 2 - Sx( ) . (8-1)

Finally, note that SX() has units of watts/Hz.

Let X(t) be a WSS random processes. We seek to define the power density spectrum of X(t). First, note that

F X t( ) 

z

_-_X(t)e-j tdt (8-2)

does not exits, in general. Random process X(t) is not absolutely integrable, and F[X(t)] does not converge for most practical applications. Hence, we cannot use _{F[X(t)]}2 as our definition of power spectrum (however, _F



X(t)



exists as a generalized random function that can be used to develop a theory of power spectral densities).

We seek an alternate route to the power spectrum. Let T > 0 denote the length of a time interval, and define the truncated process

X t X t t T t T T( ) ( ), ,    0  . (8-3)

Truncated process XT can be represented as

T

X (t) X(t)rect(t / 2T) , (8-4)

where rect(t/2T) is the 2T-long window depicted by Figure 8-1. Signal XT is absolutely integrable, that is, _-XT( )t dt



z

 . Hence, for finite T, the

Fourier transform F_XT( )  ( )et j t dt  

z

X_T - (8-5)

exists. For every value of , F_XT() is a random variable. Now, Parseval's theorem states that

X_T X_T -T ( )t dt - ( )t dt - F ( ) d T XT

z



z



z

    2 2 1 2 2   . (8-6)

Now, divide both sides of this last equation by 2T to obtain

1 2 1 4 2 2 T T t dt T F d T XT X_T - ( ) - ( )

z



z

     . (8-7)

The left-hand-side of this is the average power in the particular sample function XT being integrated (It is a random variable). Average over all such sample functions to obtain

T -T rect(t/2T) 1 rect t T) t t ( / 2 1 0     , T , T

Figure 8-1: Window used in approximating the power spectum of XT(t).

E T T t dt E T F d T XT 1 2 1 4 2 2 X_T - ( ) - ( )

z

z

L

NM

O

QP



L

_NM

O

_QP

     , (8-8) which leads to 1 2 1 4 2 2 T TE t dt T E F d T XT [X_T( ) ] [ ( ) ] -

-z



z

     . (8-9)

As T  , the left-hand-side of (8-9) is the formula for the average power of X(t). Hence, we can write T _XT XT T 2 2 -T -T T 2 - T 1 1

Avg Pwr = _limit E[ X (t) ] dt _limit E[ F ( ) ]d

2T 4 T E[ F ( ) ] 1 = dlimit 2 2T                   _ _







. (8-10) The quantity S_x( ) 

L

[ ( ) ]

N

MM

O

_Q

PP

 T E F T XT limit 2 2 (8-11)

is the power density spectrum of WSS process X(t). Power density spectrum SX() is a real-valued, nonnegative function. If X(t) is real-valued the power spectrum is an even function of . It has units of watts/Hz, and it tells where in the frequency range the power lies. The quantity 1 2 1 2      S_x( ) d

z

is the power in the frequency band (1, 2). Finally, to obtain the power spectrum of deterministic signals, Equation (8-11), without the expectation (remove the “E operator”), can be applied.

Example (8-1): Consider the deterministic signal X(t) = Aexp[j0t]. This signal is not real-valued so we should not automatically expect an even power spectrum. Apply window rect(t/2T) to x(t) and obtain

X (t) = A exp[j_T ₀t]rect _2Tt (8-12)

The Fourier transform of XT is given by

F_XT( ) _F[ exp(A j₀t rect t) ( /2T)]2AT Sa[(  ₀)T], (8-13)

where Sa(x) {sin(x)}/ x . Hence, for large T we have (note that nothing is random so no expectation is required here!)

XT 2 2 2 ₀ F ( ) _{T Sa [( )T]} ( ) 2 A 2T  _{    }     _{ }    x S , (8-14)

a result depicted by Figure 8-2. The area under this graph is independent of T since

2 -T Sa ( -T) d 1    _{ } 



(8-15)

independent of T. For (8-14), on either side of 0, the width between the first zero crossings (where all of the area is concentrated as T approaches infinity) is on the order of 2/T. The height is on the order of 2A2T. As a result, (8-14) approaches a delta function and

XT 2 2 2 0 2 0 T _T F ( ) _{T Sa [( )T]} limit ( ) 2 A limit 2 A ( ), 2T    _{  }            x S (8-16)

a result depicted by Figure 8-3. If 0 = 0, then X(t) = A, a constant DC signal. For this

DC-signal case, the power spectrum is Sx() = 2A2(), as expected. Rational Power Density Spectrums

In many applications SX() takes the form

2m 2m 2 2 2m 2 2 2n 2n 2 2 2n 2 2 0 0 a +a a ( ) , m n b +b b                     x _ S , (8-17) 2A2 ₀  S_X() =2A2(_)

Figure 8-3: Power spectrum of X(t) for Example 8-1.

Width  2/T



₀

2A2_T S_x()

a rational function of . In (8-17), the coefficients a0, a2, …, a2m-2, b0, b2, …, b2n-2 are

real-valued. Also, only even powers of  appear in the numerator and denominator since SX() is an

even function of . Also, since

Avg Pwr in X = 1

2 -Sx   

z

( )d  , (8-18)

we must have m < n (the degree of the numerator must be at least two less then the degree of the denominator for Inequality (8-18) to hold).

Rational spectrums are continuous in nature. They contain no delta function(s), an observation that implies that X has no DC or sinusoidal component(s). However, in many applications, one encounters processes that have a DC component and/or an AC component, in addition to a random component with a rational spectrum. For the case of a DC component, we can write

X(t) = A + XAC(t), (8-19)

where A is a DC constant, and zero-mean XAC has a rational spectrum, denoted here as SAC().

We compute the power spectrum of X(t) of the form (8-19). First, window X to obtain

T AC X (t) Arect(t / 2T) X (t)rect(t / 2T) (8-20) so that

 









T AC 1 2 1 2 X F ( ) F ( ) F ( ) rect(t / 2T) F ( ) X (t)rect(t / 2T) . A         F F F . (8-21)

Note that F1() is a deterministic function of , and F2() is a random function of . A simple expansion yields

 

T 2 2 2 2 1 2 1 1 2 2 X  F F  F 2 Re F F_  _ F F . (8-22)

Use the fact that F1 is deterministic, and take the ensemble average of (8-22) to obtain

 

T 2 2 2 1 1 2 2 E_ X _ F 2 Re F E F_ 

[ ]

__{ }E F _   F  . (8-23)

However, note that

 









AC AC AC j t 2 -j t

-E F E X (t)rect(t / 2T) E X (t)rect(t / 2T)e dt

E X (t) rect(t / 2T)e dt 0  _{ }   _{ }       _{ } _{ }    





F (8-24)

since E[XAC] = 0. Because of (8-24), the middle term on the right-hand side of (8-23) is zero,

and we have

 

T 2 ₂ 2 ₂ 1 E X _F E F 2T 2T 2T   _{ }     F  _{ }  _{. (8-25)} Finally, as T  , we have

 

T AC 2 2 T E X limit ( ) 2 ( ) ( ) 2T A                x F S S . (8-26)

That is, the power spectrum of X is the power spectrum for the DC component A (see the sentence at the end of Example 8-1) added to the rational power spectrum SAC() for the AC

component.

Wiener-Khinchine Theorem

Assume that X(t) is a wide-sense-stationary process with autocorrelation RX(). The

power spectrum SX() is the Fourier transform of autocorrelation RX(). This is the famous Wiener-Khinchine Theorem.

Proof:

Recall that the power spectrum of real-valued (the proof can be generalized to include complex-valued random processes) random process X(t) is

2 T T E [X ] ( ) limit 2T          S F , (8-27) where T X(t), t T X (t) 0, t T        . (8-28)

Take the inverse Fourier transform of S to obtain





1 2 1 2 2 -1 _T j T T _{j t} T _{j t j} 1 1 2 2 T T T T T _{j (t t )} 1 2 2 1 T T T 1 _limit E [X ] _{e d} ( ) 2 _2T 1 1 limit E X(t )e dt X(t )e dt e d 2 2T 1 1 limit E[X(t )X(t )] e d dt dt 2T 2  _    _{   }      _{  }              _  _  _{ }    _{ }      _ _ 









 



S F F (8-29)

(the fact that X is real-valued was used to obtain (8-29)). However, from Fourier transform theory we know that

1 2 j (t t ) 1 2 1 e d (t t ) 2  _{  }        



. (8-30)

Now, use (8-30) in (8-29), and the fact that E[X(t1)X(t2)] = R(t2-t1), to obtain





T T -1 2 1 1 2 2 1 T T T T T 1 1 T T T T 1 limit R(t t ) (t t )dt dt ( ) 2T 1 1 limit R( )dt R( ) limit dt 2T 2T R( ).                    _{  }    

 





F S (8-31)

This is the well-known, and very useful, Wiener-Khinchine Theorem: the Fourier transform of the autocorrelation is the power spectrum density. Symbolically, we write

x

R ( )  S( ). (8-32)

A second proof of the Wiener-Khinchine Theorem follows. First, note

1 2 1 2 T T T T j (t t ) 2 j t j t T _-T 1 1 _-T 2 2 _{-T -T} 1 2 1 2 [X ] 



X(t )e  dt



X(t )e  dt 

 

X(t )X(t ) e   dt dt F (8-33)

since X is assumed to be real valued. Take the expectation of this result to obtain

1 2

T T j (t t )

2

T _{-T -T} 1 2 1 2

Define  = t1 - t2 and  = t1 + t2. From Example 4A-2 of Appendix 4A, we have





1 2 T T j (t t ) 2T 2 j T _{-T -T} 1 2 1 2 _-2T E[ [X ] ]_F 

 

R(t t ) e   dt dt 



2T  R( )e  d , (8-35)

a result that leads to

2 2T T j -2T E[ [X ] ] 1 R( )e d 2T 2T       _  _    



F _{. (8-36)} so that





X 2 2T T j j -2T -T T E[ [X ] ] limit limit ( ) 1 R( )e d R( )e d 2T 2T R( ) ,               _  _         





S F F , (8-37)

(as T approaches infinity, triangle (1 - /2T) approaches unity over all  for which the integral of R() is significant).

Example (8-2): Power spectrum of the random telegraph signal

The random telegraph signal was discussed in Chapter 7; a typical sample function is depicted by Figure 8-4. It is defined as



 









Location of a Poisson Point X(t)

X X t ( ) ( ) 0     

if number of Poisson Points in (0, t) is = - if number of Poisson Points in (0, t) is

even odd ,

where  is a random variable that takes on the two values  = ±1 equally likely. From Chapter 7, recall that the autocorrelation of X is RX() = e, where  is the average point density (also, in X(t),  is the average number of zero crossings per unit length). By the Wiener-Khinchine theorem, the power spectrum is

S_x( ) e      

L

_NM

O

_QP

   F 2 ₂4 ₂ 4 . (8-38)

The 3dB down bandwidth is 2. Large values for "average-toggle-density"  make waveform X(t) toggle faster; they also make the bandwidth larger, as shown by (8-38). For the random telegraph signal, the average power is

avg 1 1 ₂2 ₂ 1 P ( )d d ( ) 1 2 ₄              



S 



_{  }  (8-39)

watt. This result follows immediately from the observation that [X(t)]2= 1 for all time.

Example (8-3): Zero-Mean, White Noise

A zero-mean, white noise process X(t) is one for which

0 N

R( ) ( )

2

    , (8-40)

where N0/2 is a constant. The power spectrum is N0/2 Watts/Hz. This implies that X possesses an infinite amount of power, a physical absurdity. In the mathematical literature, white noise

processes are called generalized random processes (the rational being somewhat similar to that used when delta functions are called generalized functions). Intentionally, we have not stated how X(t) is distributed (do not assume that X(t) is Gaussian unless this is explicitly stated). In the name assigned to X(t), the adjective “white” is include to draw a parallel to white light, light containing all frequencies.

White noise X(t) exists only as a mathematical abstraction. However, it is a very useful abstraction. For example, suppose we have a finite bandwidth system driven by a wide-band noise process with spectrum that is flat over the system bandwidth (noise bandwidth >> system bandwidth). Under these conditions, the analysis could be simplified by assuming that input X(t) is white noise.

Addition of Power Spectrums for Uncorrelated Processes

Suppose WSS, zero-mean processes X(t) and Y(t) are uncorrelated so that E[X(t+)Y(t)] = E[X(t+)]E[Y(t)] = 0 for all t and . Then, we can write







 



X Y X Y R ( ) E [X(t ) Y(t )][X(t) Y(t)] E X(t )X(t) E Y(t )Y(t) R ( ) R ( ) .                    (8-41)

Now, take the Fourier transform of (8-41) to see that

x y ( )  x( )  y( )

S S S , (8-42)

the result that power spectrums add for uncorrelated processes. This conclusion has many applications (see (8-26) for the case of a DC component added to a zero-mean process).

Input-Output of Power Spectrums

Let X(t) be a W.S.S process. Y(t) = L[ · ] denotes a linear, time-invariant system. From Chapter 7, recall the formula

R_Y( ) 

b

h( ) (   h )

g

R_x( ) . (8-43)

Take the Fourier transform of (8-43) to obtain

S_Y( ) F[RY]F[ ( ) (h   h )] ( )S_X  . (8-44)

However,

F[h(t)*h(-t)] = H(j)H*(j) = H(j). (8-45)

Combine this with (8-44) to obtain

SY( )  H j( ) SX( )

2 _{, (8-46)}

an important result for computing the output spectral density.

Example (8-4): Let X(t) be modeled as zero-mean, white Gaussian noise. We assume RX() = (N0/2)() so that SX() = N0/2. Let X(t) be applied to the first-order RC low-pass filter shown by Figure 8-5. Find the output power density spectrum SY() and the first-order density function of Y(t). First, from Equation (8-46), we obtain

Y 0 0 ₂ N N 1 1 1 ( ) 1 j RC 1 j RC 2 _{2 1 (RC )}     _{  }      _{ }    S , (8-47)

a result that is depicted by Figure 8-6. Output Y(t) has a mean of zero (why?) and a variance equal to the AC power. Hence, the variance of Y(t) is





Y 2 0 0 0 2 2 N N N 1 1 1 AC Power in Y(t) d d 2 2 _{1 (RC )} 4 RC ₁ 4RC            



_{ } 



_{ } . (8-48)

Finally, output Y is Gaussian since linear filtering a Gaussian input produces a Gaussian output. As a result, we can write

Y Y Y 2 2 1 y f (y) exp 2 2        _  _, (8-49) -2 -1 0 1 2RC N₀/2 SY()

Figure 8-6: Power Spectrum of RC low-pass filter output. N₀/2 S_X()  C R + -X(t) Y(t) + -H j j RC ( )    1 1

where  = N_Y2 0/(4RC).

Note that SY(), given by (8-47), is an even-symmetry, rational function of  with a denominator degree is two more than the numerator degree (which is a requirement for a finite power output process). Generally speaking, we should expect an even-symmetry rational output spectrum from a lumped-parameter, time-invariant system (an RLC circuit/filter, for example) that is driven by noise that has a flat spectrum over the system bandwidth.

Example (8-5): Let X(t) be a white Gaussian noise ideal current source with a double sided spectral density of 1 watts/Hz. Find the average power absorbed by the resistor in the circuit depicted by Figure 8-7. The spectral density of the power absorbed by the resistor is given by

R 2 ₂ 1 ( ) H( j ) 1 1        S , (8-50)

an even-symmetry, rational function with denominator degree two more than numerator degree. Hence, the total power absorbed by the 1 Ohm resistor is given by

R avg 1 _- 1 _- 1 ₂ P ( ) d d 1/ 2 watt 2 2 ₁           



S 



_{ } (8-51)

Noise Equivalent Bandwidth of a Low-pass System/Filter

We seek to quantify the idea of system/filter bandwidth. Let H(j) be the transfer

1 1F X(t) H j V j X j j j j ( ) ( ) ( ) / /             1 1 1 V(j) +

function of a low-pass system/filter. Let X(t) be white noise with a power spectrum of N0/2 watts/Hz. The average power output is

P_avg  N H j d  

z

1 2 0 2 2  - ( ) . (8-52)

Now, consider an ideal low-pass filter that has a gain equal to H(0) and a one-sided bandwidth of BN Hz (see Figure 8-8). Apply the white noise X(t) to this ideal filter. The output power is

P_avg N H d N H B B B N N N  1

z

 2 0 0 0 2 2 0 2 2 2     - ( ) ( ) . (8-53)

Again, consider H(j). The noise equivalent bandwidth of H(j) is defined to be the one-sided bandwidth (in Hz) of an ideal filter (with gain H(0)) that passes as much power as H(j) does when both filters are supplied with the same input white noise. Hence, equate (8-53) and (8-52) to obtain N H₀ 0 2B_N 1 N₂ H j 2d 2 0 ( )  ( )  

z

 -  . (8-54) This yields B H H j d N  _ 

z

1 4 0 2 2  ( ) - ( )  (8-55) 2B_N -2B_N H(0) -axis

as the noise equivalent bandwidth of filter H(j).

Example (8-6): Find the noise equivalent bandwidth of the single pole RC low-pass filter depicted by Figure 8-9. Direct application of Formula (8-55) yields

N _{- 2 2 2 - 2} 1 1 1 1 1 B d d Hz 4 _{1 R C} 4 RC ₁ 4RC          



_{ } 



_{ } (8-56)

Example (8-7): Find the noise equivalent bandwidth of an nth-order Butterworth low-pass filter. By definition H j c n ( ) ( / )    2 2 1 1   , 0.0 0.5 1.0 1.5 2.0 2.5 3.0/_c 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ma gn it ude n = 2 n = 4 n = 6 H()

Figure 8-10: Magnitude response of an nth_{-order Butterworth filter with 3db cutoff}

frequency c. The horizontal axis is /c. The filter approaches an ideal low-pass filter

as order n becomes large.

C R + -X(t) Y(t) + -H j j RC ( )    1 1

for positive integer n. The quantity c is the 3dB cut off frequency (see Figure 8-10 for

magnitude response). The noise equivalent bandwidth is

B_N d d c n c n        

z

z

1 4 1 1 4 1 1 2 2  _{ }    _  ( / ) - - .

This last integral appears in most integral tables. Using the tabulated result, the noise equivalent bandwidth BN is B n n N 

F

HG

c

I

KJ

1 4 2   sin( / ) , n = 1, 2, 3, , (8-57)

Hz. As n  , the Butterworth filter approaches the ideal LPF. The limit of (8-57) is

n n c n n n c B n N      

F

H

GG

I

_K

JJ

 limit limit 1 4 2 2 3 21 3 5 21 5       !( ) !( )  (8-58) as expected.