Random variables, Random processes

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Random variables, Random processes

• Analog and Digital Communications
• Autumn 2005-2006

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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

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Random Variables

• Probability mass function (discrete RV)
• Probability density function (cont. RV)
• Joint pdf of independent RVs
• Mean
• Variance
• Characteristic function
• (IFT of pdf)

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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

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Random Processes

• Mean
• Autocorrelation
• Autocovariance

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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