Random Processes and PSD

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# Random Processes and PSD - PowerPoint PPT Presentation

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

Analog and Digital Communications

Autumn 2005-2006

CS477: Analog and Digital Communications

Random Processes
• Cross-correlation

(Processes are orthogonal if )

• Cross-covariance

CS477: Analog and Digital Communications

Example

CS477: Analog and Digital Communications

Example

Mean is constant and autocorrelation is dependent on

CS477: Analog and Digital Communications

Example

CS477: Analog and Digital Communications

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)
• 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

CS477: Analog and Digital Communications

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

Cross-correlation function

Cross power spectral density

(Also applicable to jointly stationary random signals)

CS477: Analog and Digital Communications

LTI Systems Revisited

For random and deterministic signals:

(Prove it at home!)

(For real channels!)

CS477: Analog and Digital Communications

LTI

Example: Random Signal

Consider White Noise input to an LTI filter

CS477: Analog and Digital Communications