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# sm8_36

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```Problem 8.36 The power spectral density of a random process X(t) is shown in Fig. 8.27.
(a) Determine and sketch the autocorrelation function RX(τ) of the X(t).
(b) What is the dc power contained in X(t)?
(c) What is the ac power contained in X(t)?
(d) What sampling rates will give uncorrelated samples of X(t)? Are the samples statistically

independent?
Solution
(a) Using the results of Problem 8.35, and the linear properties of the Fourier transform

( ) ( )ττ 0221 sinc1 fR +=
(b) The dc power is given by power centered on the origin

1

)(lim

)(limpower

0

0

=
=

=

∫
∫

−→

−→
ε

εε

ε

εδ

δ dff

dffSdc X

(c) The ac power is the total power minus the dc power

2
1

1)0(
power)0(

=
−=
−=

X

X

R
dcRpowerac

(d) The correlation function RX(τ) is zero if samples are spaced at multiples of 1/f0.

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page…8-46```