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Describes signals that cannot be precisely characterized.

Integration

Z t a b X t

Linear processing

Y t h t X

Differentiation

X t t X t

    Properties

  • Z t a b X t t a b X t
  • Z 2 t 2 a b X t 2 t 1 a b X t 1 t 2 a b t 1 a b R X t 2 t 1

Y t h t X h t X
If X t is wide sense stationary and the linear system is time invariant
Y t h t X X t h t Y
R Y X t 2 t 1 Y t 2 X t 1 h t 2 X X t 1 h t 2 R X t 1
R Y X t 2 t 1 h t 2 t 1 R X h R X t 2 t 1
where t 1 .
R Y t 2 t 1 Y t 2 Y t 1 Y t 2 h t 1 X h t 1 R Y X t 2 h t 1 R Y X t 2
R Y t 2 t 1 h t 2 t 1 R Y X R Y t 2 t 1 h ~ R Y X t 2 t 1
where t 2 and h ~ h for all . Y t is WSS if X t is WSS and the linear system is time-invariant.

X t is a wide sense stationary process with X 0 , and R X N 0 2 .Consider the random process going through a filter with impulse response h t a t u t .The output process is denoted by Y t . Y t 0 for all t .

R Y N 0 2 h h N 0 2 a 2 a
X t is called a white process. Y t is a Markov process.

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Power Spectral Density
The power spectral density function of a wide sense stationary (WSS) process X t is defined to be the Fourier transform of the autocorrelation functionof X t .
S X f R X 2 f
if X t is WSS with autocorrelation function R X .

    Properties

  • S X f S X f since R X is even and real.
  • Var X t R X 0 f S X f
  • S X f is real and nonnegative S X f 0 for all f .

If Y t h t X then

S Y f R Y h h ~ R X H f H ~ f S X f H f 2 S X f
since H ~ f t h ~ t 2 f t H f

X t is a white process and h t a t u t .

H f 1 a 2 f
S Y f N 0 2 a 2 4 2 f 2

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Questions & Answers

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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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