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Random Process

# Random Process - Random(Stochastic Process Random...

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1 Random (Stochastic) Process Random signals (voice, T.V., Digital computer data, electrical noise) have two properties: Function of time (defined in some interval) Random (impossible to describe exactly the waveforms) An indexed set of functions of some parameter ( t,d..) that has some statistical properties

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2 Note that to qualify as a R.P., x could be a function of any other variable (distance for example) and a function of more than one variable. Outcome of the First trial of The experiment Outcome of the 2 nd trial of The experiment ( sample function ) Outcome of the n th trial of The experiment x 1 (t ) x 2 (t ) x n (t ) x 1 (t k ) x 2 (t k ) x n (t k ) 0 s 1 s 2 s n Sample space S ( ensemble )
3 Example: Consider the voltage waveforms emitted from a noise source One possible waveform is x(t,E i ) is a sample function of the sample space All possible sample functions is called the ensemble and defines the random process x(t) that describes the noise source A random process maps events into functions of the parameter ‘t’ . (R.V. maps events into constants)

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4 But the noise voltage is function of time. Thus the R.V. ‘x’ is a function of time and can be expressed as x (t) A R.V that is a function of time is called a random process (or stochastic process) To continue with the example of the random process x(t), we need to record noise voltage for each ‘t’ in order to specify the random process. We obtain many waveforms, ( called sample function ). Sample function amplitudes at some instant t=t 1 are x(t 1 ) in various trials The collection of all possible waveforms is called ensemble (sample space) of the random process
5 How to specify a random process ?

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