Random processes and psd
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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

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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