1. Sep 20, 2005CS477: Analog and Digital Communications1 Random variables, Random processes Analog and Digital Communications Autumn 2005-2006

2. Sep 20, 2005CS477: Analog and Digital Communications2 Random Variables Outcomes and sample space Random Variables Mapping outcomes to: Discrete numbers  Discrete RVs Real line  Continuous RVs Cumulative distribution function One variable Joint cdf

3. Sep 20, 2005CS477: Analog and Digital Communications3 Random Variables Probability mass function (discrete RV) Probability density function (cont. RV) Joint pdf of independent RVs Mean Variance Characteristic function (IFT of pdf)

4. Sep 20, 2005CS477: Analog and Digital Communications4 Random Processes Mapping of an outcome (of an experiment) to a range set R where R is a set of continuous functions Denoted as or simply For a particular outcome is a deterministic function For or simply is a random variable

5. Sep 20, 2005CS477: Analog and Digital Communications5 Random Processes Mean Autocorrelation Autocovariance

6. Sep 20, 2005CS477: Analog and Digital Communications6 Random Processes Cross-correlation (Processes are orthogonal if ) Cross-covariance

7. Sep 20, 2005CS477: Analog and Digital Communications7 Example

8. Sep 20, 2005CS477: Analog and Digital Communications8 Example Mean is constant and autocorrelation is dependent on

9. Sep 20, 2005CS477: Analog and Digital Communications9 Example

10. Sep 20, 2005CS477: Analog and Digital Communications10 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!)

11. Sep 20, 2005CS477: Analog and Digital Communications11 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