13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if.

Transcription

1 For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if jt X ( ) = xte ( ) dt, (3-) then X ( ) represents its energy spectrum. his follows from Parseval s theorem since the signal energy is given by 3. Power Spectrum x ( tdt ) = π X ( ) d = E. hus X ( ) represents the signal energy in the band (see Fig 3.). X() t X ( ) X ( ) (3-) (, + ) Energy in (, + ) 0 t Fig

2 However for stochastic processes, a direct application of (3-) generates a sequence of random variables for every. Moreover, for a stochastic process, E{ X(t) } represents the ensemble average power (instantaneous energy) at the instant t. o obtain the spectral distribution of power versus frequency for stochastic processes, it is best to avoid infinite intervals to begin with, and start with a finite interval (, ) in (3-). Formally, partial Fourier transform of a process X(t) based on (, ) is given by jt X ( ) = Xte ( ) dt (3-3) so that X ( ) jt = X () te dt (3-4) represents the power distribution associated with that realization based on (, ). Notice that (3-4) represents a random variable for every, and its ensemble average gives, the average power distribution based on (, ). hus

3 X ( ) * j ( t t) P ( ) = E { ( ) ( )} = E X t X t e dtdt j ( t t) = R ( t, t) e dt dt (3-5) represents the power distribution of X(t) based on (, ). For wide sense stationary (w.s.s) processes, it is possible to further simplify (3-5). hus if X(t) is assumed to be w.s.s, then R ( t, t) = R ( t t) and (3-5) simplifies to P R t t e dt dt j t t ( ( ) = ( ) ). Let τ = t t and proceeding as in (4-4), we get jτ P ( ) = R ( τ) e ( τ ) dτ jτ τ τ = R ( ) e ( ) dτ 0 to be the power distribution of the w.s.s. process X(t) based on (, ). Finally letting in (3-6), we obtain (3-6) 3

4 jτ S ( ) = lim P( ) = R ( τ) e dτ 0 to be the power spectral density of the w.s.s process X(t). Notice that R F ( ) S ( ) 0. (3-7) (3-8) i.e., the autocorrelation function and the power spectrum of a w.s.s Process form a Fourier transform pair, a relation known as the Wiener-Khinchin heorem. From (3-8), the inverse formula gives π jτ R ( τ) = S ( ) e d (3-9) and in particular for τ = 0, we get π S ( ) d = R (0) = E{ X ( t) } = P, the total power. (3-0) From (3-0), the area under S ( ) represents the total power of the process X(t), and hence S ( ) truly represents the power spectrum. (Fig 3.). 4

5 S ( ) S ( ) represents the power in the band (, + ) Fig 3. he nonnegative-definiteness property of the autocorrelation function in (4-8) translates into the nonnegative property for its Fourier transform (power spectrum), since from (4-8) and (3-9) n n n n * * i j i j = i j i= j= i= j= 0 π j ( t t ) i j a ar ( t t) aa S ( ) e d n jti = π S ( ) ae i= i d 0. (3-) From (3-), it follows that + R ( τ) nonnegative - definite S ( ) 0. (3-) 5

6 If X(t) is a real w.s.s process, then R ( τ) = R ( τ) so that jτ S ( ) = R ( τ) e dτ = 0 R ( τ)cosτdτ = R ( τ)cos τdτ = S ( ) 0 so that the power spectrum is an even function, (in addition to being real and nonnegative). (3-3) 6

7 Power Spectra and Linear Systems If a w.s.s process X(t) with autocorrelation function R ( τ) S ( τ) 0 is X(t) h(t) applied to a linear system with impulse response h(t), then the cross correlation Fig 3.3 function R ( τ ) XY and the output autocorrelation function R ( τ ) YY given by (4-40)-(4-4). From there are Y(t) * * R ( τ) = R ( τ) h ( τ), R ( τ) = R ( τ) h ( τ) h( τ). (3-4) XY YY But if f() t F( ), g() t G( ) (3-5) hen f() t g() t F( ) G( ) (3-6) since jt F{ f() t g()} t = f() t g() t e dt 7

8 { τ τ τ} jt F{ f( t) g( t)}= f( ) g( t ) d e dt jτ j ( t τ ) = f( τ) e dτ g( t τ) e d( t τ) = F( ) G( ). Using (3-5)-(3-7) in (3-4) we get S F R h S H * * ( ) = { ( ) ( τ)} = ( ) ( ) XY (3-7) (3-3) since where ( ) jt * * jτ * h ( τ) e dτ = h( t) e dt = H ( ), jt H ( ) = hte ( ) dt (3-9) represents the transfer function of the system, and S ( ) = F{ R ( τ)} = S ( ) H( ) YY YY XY = S ( ) H( ). (3-0) 8

9 From (3-3), the cross spectrum need not be real or nonnegative; However the output power spectrum is real and nonnegative and is related to the input spectrum and the system transfer function as in (3-0). Eq. (3-0) can be used for system identification as well. W.S.S White Noise Process: If W(t) is a w.s.s white noise process, then from (4-43) R ( τ) = qδ( τ) S ( ) = q. (3-) WW WW hus the spectrum of a white noise process is flat, thus justifying its name. Notice that a white noise process is unrealizable since its total power is indeterminate. From (3-0), if the input to an unknown system in Fig 3.3 is a white noise process, then the output spectrum is given by SYY ( ) = q H( ) (3-) Notice that the output spectrum captures the system transfer function characteristics entirely, and for rational systems Eq (3-) may be used to determine the pole/zero locations of the underlying system. 9

10 Example 3.: A w.s.s white noise process W(t) is passed through a low pass filter (LPF) with bandwidth B/. Find the autocorrelation function of the output process. Solution: Let X(t) represent the output of the LPF. hen from (3-) q, B / S ( ) = q H( ). = (3-3) 0, > B / Inverse transform of S ( ) gives the output autocorrelation function to be B/ jτ B/ jτ R ( τ) = ( ) S e d = q e d B/ B/ sin( Bτ / ) = qb = qb sinc( Bτ / ) (3-4) ( Bτ /) H ( ) R qb ( τ ) B / B / τ (a) LPF Fig. 3.4 (b) 0

11 Eq (3-3) represents colored noise spectrum and (3-4) its autocorrelation function (see Fig 3.4). Example 3.: Let Y ( t) t+ = X( τ ) d τ (3-5) represent a smoothing operation using a moving window on the input process X(t). Find the spectrum of the output Y(t) in term of that of X(t). Solution: If we define an LI system with impulse response h(t) as in Fig 3.5, then in term of h(t), Eq (3-5) reduces to so that Here t Y ( t) = h( t τ) X( τ) dτ = h( t) X( t) S S H YY ( ) = ( ) ( ). + jt ht () / t Fig 3.5 H ( ) = e dt = sinc( ) (3-8) (3-6) (3-7)

12 so that S ( ) = S ( )sinc ( ). (3-9) YY S ( ) sinc ( ) S YY ( ) π Fig 3.6 Notice that the effect of the smoothing operation in (3-5) is to suppress the high frequency components in the input (beyond π / ), and the equivalent linear system acts as a low-pass filter (continuoustime moving average) with bandwidth π / in this case.

13 Discrete ime Processes For discrete-time w.s.s stochastic processes X(n) with autocorrelation sequence { r } (proceeding as above) or formally k, defining a continuous time process X ( t) = X( n) δ ( t n), we get n the corresponding autocorrelation function to be R ( τ) = rδ( τ k). Its Fourier transform is given by k = k S j ( ) = r e 0, k = and it defines the power spectrum of the discrete-time process X(n). From (3-30), so that S ( ) is a periodic function with period π B =. S ( ) = S ( + π / ) k (3-3) (3-30) (3-3) 3

14 his gives the inverse relation B jk rk = S ( ) e d B B (3-33) and B r0 = E{ X( n) } = S ( ) d B B (3-34) represents the total power of the discrete-time process X(n). he input-output relations for discrete-time system h(n) in (4-65)-(4-67) translate into and where S S H e XY * j ( ) = ( ) ( ) represents the discrete-time system transfer function. j S ( ) = S ( ) H( e ) YY j He ( ) = hn ( ) e n= jn (3-35) (3-36) (3-37) 4