Chapter 8 - Power Density Spectrum

Transcription

1 EE385 Class Notes 8/8/03 John Stensby 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) ], a constant. his total average power is distributed over some range of frequencies. his distribution over frequency is described by S X (), the power density spectrum. S X () is non-negative (S X () 0) and, for real-valued X(t), even (S X () = S X (-)). Furthermore, the area under S X is proportional to the average power in X(t); that is Average Power in X(t) z - S x ( ) d. (8-) Finally, note that S X () 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 Xt () z -j t X(t)e - dt (8-) 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)] 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 > 0 denote the length of a time interval, and define the truncated process X( t) X(), t t. (8-3) 0, t Updates at 8-

2 EE385 Class Notes 8/8/03 John Stensby runcated process X can be represented as X (t) X(t)rect(t/), (8-4) where rect(t/) is the -long window depicted by Figure 8-. Signal X is absolutely integrable, that is, Fourier transform z X - () t dt. Hence, for finite, the z FX ( ) ( j t t )e X dt - (8-5) exists. For every value of, F X () is a random variable. Now, Parseval's theorem states that z z z X() t dt X() t dt FX ( ) d (8-6) Now, divide both sides of this last equation by to obtain z X () t dt F X d 4 ( ). (8-7) - - z he left-hand-side of this is the average power in the particular sample function X being integrated (It is a random variable). Average over all such sample functions to obtain rect(t/) rect( t /), t 0, t - Figure 8-: Window used in approximating the power spectum of X (t). Updates at 8-

3 EE385 Class Notes 8/8/03 John Stensby L NM z O QP L N M z O QP E X t dt E FX d () ( ), (8-8) which leads to z z - - E[ X ( t) ] dt E[ FX d 4 ( ) ]. (8-9) As, the left-hand-side of (8-9) is the formula for the average power of X(t). Hence, we can write Avg Pwr = limit E[ X (t) ] dt E[ F ( ) ]d X - limit - 4 E[ F X ( ) ] limit - = d. (8-0) he quantity S x ( ) limit L NM E[ F ( ) ] X O QP (8-) is the power density spectrum of WSS process X(t). Power density spectrum S X () is a realvalued, 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. he quantity z Sx ( )d Updates at 8-3

4 EE385 Class Notes 8/8/03 John Stensby is the power in the frequency band (, ). Finally, to obtain the power spectrum of deterministic signals, Equation (8-), without the expectation (remove the E operator ), can be applied. Example (8-): Consider the deterministic signal X(t) = Aexp[j 0 t]. his signal is not realvalued so we should not automatically expect an even power spectrum. Apply window rect(t/) to x(t) and obtain X (t) = Aexp[j 0 t]rect (8-) t he Fourier transform of X is given by FX ( ) F[ Aexp( j0t) rect( t/ )] ASa[( 0)], (8-3) where Sa(x) {sin(x)}/ x expectation is required here!). Hence, for large we have (note that nothing is random so no F ( ) ( ) A Sa [( 0)] S x, (8-4) X a result depicted by Figure 8-. he area under this graph is independent of since Sa ( ) d - (8-5) independent of. For (8-4), on either side of 0, the width between the first zero crossings (where all of the area is concentrated as approaches infinity) is on the order of /. he height is on the order of A. As a result, (8-4) approaches a delta function and Updates at 8-4

5 EE385 Class Notes 8/8/03 John Stensby S x () A 0 Width / Figure 8-: Approximation to the power spectrum of X(t). F X ( ) Sa [( 0)] x( ) limit A limit A ( 0 ), S (8-6) a result depicted by Figure 8-3. If 0 = 0, then X(t) = A, a constant DC signal. For this DCsignal case, the power spectrum is S x () = A (), as expected. Rational Power Density Spectrums In many applications S X () takes the form m m m n n n a +a a S ( ) 0 x, m n, (8-7) b +b b 0 S X () =A ( ) A 0 Figure 8-3: Power spectrum of X(t) for Example 8-. Updates at 8-5

6 EE385 Class Notes 8/8/03 John Stensby a rational function of. In (8-7), the coefficients a 0, a,, a m-, b 0, b,, b n- are realvalued. Also, only even powers of appear in the numerator and denominator since S X () is an even function of. Also, since Avg Pwr in X = z S x ( )d, (8-8) - we must have m < n (the degree of the numerator must be at least two less then the degree of the denominator for Inequality (8-8) to hold). Rational spectrums are continuous in nature. hey 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 + X AC (t), (8-9) where A is a DC constant, and zero-mean X AC has a rational spectrum, denoted here as S AC (). We compute the power spectrum of X(t) of the form (8-9). First, window X to obtain X (t) A rect(t / ) X (t)rect(t / ) (8-0) AC so that F X F( ) F ( ) F( ) F A rect(t/). (8-) F( ) F X (t)rect(t/). AC Updates at 8-6

7 EE385 Class Notes 8/8/03 John Stensby Note that F () is a deterministic function of, and F () is a random function of. A simple expansion yields X FF F ReF F F F. (8-) Use the fact that F is deterministic, and take the ensemble average of (8-) to obtain [ ] E F X F Re F E F E F. (8-3) However, note that j t EF E X AC(t)rect(t / ) E X AC(t)rect(t / )e dt F - j t EX AC(t) rect(t / )e dt - (8-4) 0 since E[X AC ] = 0. Because of (8-4), the middle term on the right-hand side of (8-3) is zero, and we have E F X F EF. (8-5) Finally, as, we have E F X limit S ( ) x A ( ) S AC( ). (8-6) Updates at 8-7

8 EE385 Class Notes 8/8/03 John Stensby hat is, the power spectrum of X is the power spectrum for the DC component A (see the sentence at the end of Example 8-) added to the rational power spectrum S AC () for the AC component. Wiener-Khinchine heorem Assume that X(t) is a wide-sense-stationary process with autocorrelation R X (). he power spectrum S X () is the Fourier transform of autocorrelation R X (). his is the famous Wiener-Khinchine heorem. 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 E F [X ] S( ) limit, (8-7) where X(t), t X (t) 0, t. (8-8) ake the inverse Fourier transform of S to obtain F limit e d - E [X j ] ( ) F S jt jt j limit E X(t )e dt X(t )e dte d (8-9) limit E[X(t )X(t )] e d dt dt j (t t ) Updates at 8-8

9 EE385 Class Notes 8/8/03 John Stensby (the fact that X is real-valued was used to obtain (8-9)). However, from Fourier transform theory we know that j (tt ) e d (tt ). (8-30) Now, use (8-30) in (8-9), and the fact that E[X(t )X(t )] = R(t -t ), to obtain F - S( ) limit R(t t ) (t t )dt dt limit R( )dt R( ) limit dt (8-3) R( ). his is the well-known, and very useful, Wiener-Khinchine heorem: the Fourier transform of the autocorrelation is the power spectrum density. Symbolically, we write R ( ) S ( ). (8-3) x A second proof of the Wiener-Khinchine heorem follows. First, note j t jt j (t t ) F [X ] X(t )e dt X(t )e dt X(t )X(t ) e dt dt (8-33) since X is assumed to be real valued. ake the expectation of this result to obtain j (tt ) -- E[ F [X ] ] R(t, t ) e dt dt. (8-34) Updates at 8-9

10 EE385 Class Notes 8/8/03 John Stensby Define = t - t and = t + t. From Example 4A- of Appendix 4A, we have j (tt ) j -- - E[ F[X ] ] R(t t )e dt dt R( )e d, (8-35) a result that leads to E[ F[X ] ] j R( )e d -. (8-36) so that E[ F[X ] ] limit limit j j SX ( ) R( )e d R( )e d - - F R( ),, (8-37) (as approaches infinity, triangle ( - /) approaches unity over all for which the integral of R() is significant). Example (8-): Power spectrum of the random telegraph signal he random telegraph signal was discussed in Chapter 7; a typical sample function is depicted by Figure 8-4. It is defined as X(t) Location of a Poisson Point Figure 8-4: A typical sample function of the Random elegraph Signal. Updates at 8-0

11 EE385 Class Notes 8/8/03 John Stensby X( 0) Xt () if number of Poisson Points in (0, t) is even = - if number of Poisson Points in (0, t) is odd, where is a random variable that takes on the two values = ± equally likely. From Chapter 7, recall that the autocorrelation of X is R X () = 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 ( ) F L N M e O QP 4 4. (8-38) he 3dB down bandwidth is. 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 P avg ( )d d ( ) S (8-39) 4 watt. his result follows immediately from the observation that [X(t)] = for all time. Example (8-3): Zero-Mean, White Noise A zero-mean, white noise process X(t) is one for which N R( ) 0 ( ), (8-40) where N 0 / is a constant. he power spectrum is N 0 / Watts/Hz. his implies that X possesses an infinite amount of power, a physical absurdity. In the mathematical literature, white noise Updates at 8-

12 EE385 Class Notes 8/8/03 John Stensby 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. hen, we can write R XY( ) E [X(t ) Y(t )][X(t) Y(t)] E X(t )X(t) E Y(t )Y(t) (8-4) R ( ) R ( ). X Y Now, take the Fourier transform of (8-4) to see that S ( ) S ( ) S ( ), (8-4) xy x y the result that power spectrums add for uncorrelated processes. his conclusion has many applications (see (8-6) for the case of a DC component added to a zero-mean process). Updates at 8-

13 EE385 Class Notes 8/8/03 John Stensby 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 ( ) h( ) h( ) R ( ) Y b g x. (8-43) ake the Fourier transform of (8-43) to obtain S Y ( ) F[ R Y] F[ h( ) h( )] SX( ). (8-44) However, F[h(t)*h(-t)] = H(j)H*(j) = H(j). (8-45) Combine this with (8-44) to obtain S Y ( ) Hj ( ) S ( ), (8-46) X 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 R X () = (N 0 /)() so that S X () = N 0 /. Let X(t) be applied to the first-order RC low-pass filter shown by Figure 8-5. Find the output power density spectrum S Y () and the first-order density function of Y(t). First, from Equation (8-46), we obtain Y N0 N0 ( ) j RC j RC (RC ) S, (8-47) Updates at 8-3

14 EE385 Class Notes 8/8/03 John Stensby S X () N 0 / + X(t) - R C Hj ( ) jrc + Y(t) - Figure 8-5: Power spectrum of white noise and a simple RC low-pass filter. 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 N0 N0 N0 AC Power in Y(t) d d 4 RC (RC ) 4RC. (8-48) Finally, output Y is Gaussian since linear filtering a Gaussian input produces a Gaussian output. As a result, we can write f (y) Y y exp Y Y, (8-49) N 0 / S Y () RC Figure 8-6: Power Spectrum of RC low-pass filter output. Updates at 8-4

15 EE385 Class Notes 8/8/03 John Stensby + X(t) F V(j) - Vj ( ) j / Hj ( ) Xj ( ) j / j where Figure 8-7: RC low-pass filter and transfer function. Y = N 0 /(4RC). Note that S Y (), 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 watts/hz. Find the average power absorbed by the resistor in the circuit depicted by Figure 8-7. he spectral density of the power absorbed by the resistor is given by S R ( ) H(j ), (8-50) an even-symmetry, rational function with denominator degree two more than numerator degree. Hence, the total power absorbed by the Ohm resistor is given by P avg R ( )d d / watt S - - (8-5) 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 Updates at 8-5

16 EE385 Class Notes 8/8/03 John Stensby H(0) -B N B N -axis Figure 8-8: Amplitude response of an ideal low-pass filter function of a low-pass system/filter. Let X(t) be white noise with a power spectrum of N 0 / watts/hz. he average power output is z N0 Pavg H( j) d. (8-5) - Now, consider an ideal low-pass filter that has a gain equal to H(0) and a one-sided bandwidth of B N Hz (see Figure 8-8). Apply the white noise X(t) to this ideal filter. he output power is z BN N0 Pavg H( 0) d N H 0 BN B 0 ( ). (8-53) - N Again, consider H(j). he 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 (8-5) to obtain z N0 N0 H( 0) BN H( j) d. (8-54) - his yields B N z Hj d 4 H 0 ( ) ( ) - (8-55) Updates at 8-6

17 EE385 Class Notes 8/8/03 John Stensby R + + X(t) C Y(t) Hj ( ) jrc - - Figure 8-9: RC low-pass filter and transfer function. 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 BN d d Hz 4-4 RC - R C 4RC (8-56) Example (8-7): Find the noise equivalent bandwidth of an n th -order Butterworth low-pass filter. By definition Hj ( ), n ( / ) c Magnitude H() n = n = 4 n = / c Figure 8-0: Magnitude response of an n th -order Butterworth filter with 3db cutoff frequency c. he horizontal axis is / c. he filter approaches an ideal low-pass filter as order n becomes large. Updates at 8-7

18 EE385 Class Notes 8/8/03 John Stensby for positive integer n. he quantity c is the 3dB cut off frequency (see Figure 8-0 for magnitude response). he noise equivalent bandwidth is z z BN d c d. 4 - n n 4 ( / ) - c his last integral appears in most integral tables. Using the tabulated result, the noise equivalent bandwidth B N is B N F H G c 4n sin( / n) I KJ, n =,, 3,, (8-57) Hz. As n, the Butterworth filter approaches the ideal LPF. he limit of (8-57) is F H limit B c N limit n n 4nG J ( ) ( ) c 3 5 n 3! n 5! n I K (8-58) as expected. Updates at 8-8