# Random Processes and PSD - PowerPoint PPT Presentation

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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

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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