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Random Processes and PSD. Analog and Digital Communications Autumn 2005-2006. Random Processes. Cross-correlation (Processes are orthogonal if ) Cross-covariance. Example. Example. Mean is constant and autocorrelation is dependent on. Example. Stationary and WSS RP.

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random processes and psd

Random Processes and PSD

Analog and Digital Communications

Autumn 2005-2006

CS477: Analog and Digital Communications

random processes
Random Processes
  • Cross-correlation

(Processes are orthogonal if )

  • Cross-covariance

CS477: Analog and Digital Communications

example
Example

CS477: Analog and Digital Communications

example1
Example

Mean is constant and autocorrelation is dependent on

CS477: Analog and Digital Communications

example2
Example

CS477: Analog and Digital Communications

stationary and wss rp
Stationary and WSS RP
  • Stationary Random Process (RP)
  • Wide sense stationary (WSS) RP
    • Mean constant in time
    • Autocorrelation depends only on
    • Stationary  WSS (Converse not true!)

CS477: Analog and Digital Communications

power spectral density psd
Power Spectral Density (PSD)
  • Defined for WSS processes
  • Provides power distribution as a function of frequency
  • Wiener-Khinchine theorem
    • PSD is Fourier transform of ACF

CS477: Analog and Digital Communications

psd example
PSD: Example

CS477: Analog and Digital Communications

deterministic signals and psd
Deterministic Signals and PSD

For energy signals, multiply above expression with time

ACF is a more generic function

than average power

CS477: Analog and Digital Communications

deterministic signals
Deterministic Signals

Cross-correlation function

Cross power spectral density

(Also applicable to jointly stationary random signals)

CS477: Analog and Digital Communications

lti systems revisited
LTI Systems Revisited

For random and deterministic signals:

(Prove it at home!)

(For real channels!)

CS477: Analog and Digital Communications

example random signal

LTI

Example: Random Signal

Consider White Noise input to an LTI filter

CS477: Analog and Digital Communications

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