Random Processes and PSD

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Random Processes and PSD

• Analog and Digital Communications
• Autumn 2005-2006

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Random Processes

• Cross-correlation
• (Processes are orthogonal if )
• Cross-covariance

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Example
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Example
Mean is constant and autocorrelation is dependent
on
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Example
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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!)

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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

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PSD Example
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Deterministic Signals and PSD
For energy signals, multiply above expression
with time
ACF is a more generic function than average power
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Deterministic Signals
Cross-correlation function
Cross power spectral density
(Also applicable to jointly stationary random
signals)
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LTI Systems Revisited
For random and deterministic signals
(Prove it at home!)
(For real channels!)
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Example Random Signal
Consider White Noise input to an LTI filter