1. Sep 22, 2005CS477: Analog and Digital Communications1 Random Processes and PSD Analog and Digital Communications Autumn 2005-2006

2. Sep 22, 2005CS477: Analog and Digital Communications2 Random Processes Cross-correlation (Processes are orthogonal if ) Cross-covariance

3. Sep 22, 2005CS477: Analog and Digital Communications3 Example

4. Sep 22, 2005CS477: Analog and Digital Communications4 Example Mean is constant and autocorrelation is dependent on

5. Sep 22, 2005CS477: Analog and Digital Communications5 Example

6. Sep 22, 2005CS477: Analog and Digital Communications6 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!)

7. Sep 22, 2005CS477: Analog and Digital Communications7 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

8. Sep 22, 2005CS477: Analog and Digital Communications8 PSD: Example

9. Sep 22, 2005CS477: Analog and Digital Communications9 Deterministic Signals and PSD For energy signals, multiply above expression with time ACF is a more generic function than average power

10. Sep 22, 2005CS477: Analog and Digital Communications10 Deterministic Signals Cross-correlation function Cross power spectral density (Also applicable to jointly stationary random signals)

11. Sep 22, 2005CS477: Analog and Digital Communications11 LTI Systems Revisited For random and deterministic signals: (Prove it at home!) (For real channels!)

12. Sep 22, 2005CS477: Analog and Digital Communications12 Example: Random Signal Consider White Noise input to an LTI filter LTI