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Unformatted text preview: Random Processes Chapter 11 t 1 t 2 v(t, E 1 ) v(t, E 2 ) v(t, E 3 ) v(t, E 4 ) t t t t v 1 = v(t 1 ) = v(t 1 ,E i ) for all i v 2 = v(t 2 ) = v(t 2 ,E i ) for all i Randomnoise source and some sample functions of the random noise process Noise Source v(t) In order to do the noise analysis of a given communication system, we assume that such a noise source was present at the input to the receiver. s(t) n(t) r(t) How does this concept help us? A random variable (RV) maps events into constants A random process maps events into time functions Sample function amplitudes at some instant t = t 1 are the values taken by the RV v(t 1 ) in various trials Specifying a random process ( ) cos( ) c x t A t ω = +Θ Θ is URV in (0, 2 ) π Ref: B.P. Lathi, Modern Analog & Digital Communication Systems, Oxford, 3/E A collection of all sample functions is called an ensemble An ensemble has complete information about the random process When we are dealing with random process or processes, we do not know which sample function will occur in a given trial Therefore we need to average over the entire ensemble Statistical Averages Mean : ( ) ( ) [ ( )] ( ) X X t m t E X t xp x dx ∞∞ = = ∫ Mean of a random process is deterministic function of time and at every instant t is equal to the mean of the RV X(t ). Autocorrelation of a random process measures the extent to which the process changes with time Completely describes the random process’s PSD or power content Autocorrelation: 1 2 1 2 1 2 1 2 ( ) ( )...
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 Spring '10
 Ahsan
 Probability theory, Stochastic process, Autocorrelation, Stationary process

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