Signals and Signal Spaces (779448), страница 3

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A wide-sense stationarycontinuous-time noise process x ( t ) is said to be white if its power spectraldensity is a constant:(1.97)S z z ( W ) = CJ 2 .191.3. Random SignalsThe autocorrelation function of the process is a Dirac impulse withweight2:r z z ( 7 )= uz d(7).(1.98)Since the power of such a process is infinite it is not realizable. However,the white noise process is a convenient model process which is often used fordescribing properties of real-world systems.Continuous-Time Gaussian White Noise Process. We consider a realvalued wide-sense stationary stochastic process ~ ( tand) try to represent iton the interval [-a, a] via a series expansion4 with an arbitrary real-valuedorthonormal basis cpi(t)for L2 (-a, a).

The basis satisfiesIf the coefficients of the series expansion given byai =1;cpi(t)X ( t ) dtare Gaussian random variables withE { a ? } = cT2viwe call x ( t ) a Gaussian white noise process.Bandlimited White Noise Process. A bandlimited white noise process isa whitenoise process whose power spectral density is constant within a certainfrequency band and zero outside this band. See Figure 1.2 for an illustration.t-%laxumax0Figure 1.2. Bandlimited white noise process.Discrete-Time White Noise Process.

A discrete-time white noise processhas the power spectral densitySZZ(&) = cTz4Series expansions are discussed in detail in Chapter 3.(1.99)20Chapter 1 . Signals and Signal Spacesand the autocorrelationsequenceTZdrn)1.3.3=2fJdmo.(1.100)Transmission of Stochastic Processes throughLinear SystemsContinuous-Time Processes. We assume a linear time-invariant systemwith the impulse response h(t),which is excited by a stationary process ~ ( t ) .The cross correlation function between the input process ~ ( tand) the outputprocess y ( t )is given byLcm-=E { ~ * (xt()~ + T - - X ) h(X)dX}TZZ(T)(1.101)* h(.).The cross power spectral density is obtained by taking the Fourier transform of (1.101):SZY(W) = S Z Z ( W ) H ( w ) .(1.102)Calculating the autocorrelation function of the output signal is done asfollows:= / / E { x * ( ~ - Q ! z) ( t + ~ - P ) } h*(a)h(P)dadP(1.103)+= /rZZ(.

- X) /h*(a)h(a X) dadX211.3. Random SignalsThus, we obtain the following relationship:Taking the Fourier transform of (1.104), we obtain the power spectraldensity of the output signal:= Szz(w) I H ( w ) I 2 .Sy,(w)(1.105)We observe that the phase of H ( w ) has no influence on Syy(w).Consequently,only the magnitude frequency response of H ( w ) canbedeterminedfromS Z Z ( W ) and S y y ( 4 .Discrete-Time Processes.

Theresults for continuous-time signals andsystems can be directly applied to the discrete-time case, where a systemwith impulse response h(n) is excited by a process z ( n ) ,yielding the outputprocess y(n). The cross correlation sequence between input and output is%y(m)= r z z ( m ) * h(m).(1.106)The cross power spectral density becomesH(ej").Szy(ejw)= Szz(ej")(1.107)For the autocorrelation sequence and thepower spectral density at the outputwe get(1.108)s,,(ej")= szz(eju) IH(eju)l"(1.109)As before, the phase of H(ej'") has no influence on S,,(ej").Here we ceasediscussion of the transmission of stochastic processesthrough linear systems, but we will returntothistopic in Section 5 ofChapter 2, where we will study the representation of stationary bandpassprocesses by means of their complex envelope..

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