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Integration

Z ω t a b X t ω

Linear processing

Y t τ h t τ X τ

Differentiation

X t t X t

    Properties

  1. Z t a b X t ω t a b μ X t
  2. 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

μ Y t τ h t τ X τ τ h t τ μ X τ
If X t is wide sense stationary and the linear system is time invariant
μ Y t τ h t τ μ X μ X t h t μ Y
R Y X t 2 t 1 Y t 2 X t 1 τ h t 2 τ X τ X t 1 τ h t 2 τ R X τ t 1
R Y X t 2 t 1 τ h t 2 t 1 τ R X τ h R X t 2 t 1
where τ τ t 1 .
R Y t 2 t 1 Y t 2 Y t 1 Y t 2 τ h t 1 τ X τ τ h t 1 τ R Y X t 2 τ τ h t 1 τ R Y X t 2 τ
R Y t 2 t 1 τ h τ t 2 t 1 R Y X τ R Y t 2 t 1 h ~ R Y X t 2 t 1
where τ t 2 τ and h ~ τ h τ for all τ . 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 μ X 0 , and R X τ N 0 2 δ τ .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 . μ Y t 0 for all t .

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

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 τ R X τ 2 f τ
if X t is WSS with autocorrelation function R X τ .

    Properties

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

If Y t τ h t τ X τ then

S Y f R Y τ h h ~ R X τ 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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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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