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Random Processes and PSDPowerPoint Presentation

Random Processes and PSD

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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

Example: Random Signal

Consider White Noise input to an LTI filter

CS477: Analog and Digital Communications

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