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# Chapter3 - PEM3066 Random Processes Queueing Theory Chapter...

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PEM3066: Random Processes & Queueing Theory Chapter 3: Spectral Analysis Page 1 of 18 Chapter 3: Spectral Analysis Objective: To analyse the response of systems due to random processes in the frequency domain. Syllabus: Power density spectrum or power spectral density (psd). Noise modelling. Linear system response to random signal. Narrowband, bandlimited and bandpass processes. Contents: 3.1 Introduction 3.2. Power Spectral Density (PSD) Function 3.2.1 Definition 3.2.2 Properties of PSD 3.2.3 Cross-Power Spectral Density 3.3. Noise Modelling 3.3.1 Coloured noise or band-limited white noise 3.3.2 White Gaussian Noise 3.4 System Response to Random Signal 3.4.1 Review of Basic Terminology in Linear Systems Theory 3.4.2 Fixed Linear Systems with WSS Random Processes as Inputs 3.4.2.1 Output Mean 3.4.2.2 Output PSD & autocorrelation function 3.4.2.3 System Transfer Function 3.4.3 Fixed Linear Systems with AWGN as Input (Taub pp320) 3.4.4 Special System 3.4.4.1 Hilbert Transform 3.5 Summary Appendix: Fourier Transform Pair

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PEM3066: Random Processes & Queueing Theory Chapter 3: Spectral Analysis Page 2 of 18 3.1 Introduction i) Very often we are interested to feed random processes to a system that “transform” them into another time functions. ii) For example, FM signal from the broadcast station is the input to a communication channel. The system (the channel in this case) transforms the typical sample function by adding noise into it, attenuating it, etc. Figure 3.1: Typical Channel Model iii) The processing problem can be stated as in the figure above. Given a random process X t ( ) , in continuous or discrete time, and a system ( 29 g , we are interested in characterising the stochastic behaviour of the output process Y t ( ) . iv) Recall that questions relating the input to the output of a deterministic linear system with deterministic inputs can be easily answered in the frequency domain. We may also use this convenient domain in the analysis of problem related to random processes. v) Frequency domain representation of a random process must be defined. 3.2. Power Spectral Density (PSD) Function 3.2.1 Definition i) In the deterministic case, the Fourier transform of a signal ) ( t x represents the frequencies present in the signal. Example : ( 29 ( 29 ( 29 { } [ ] [ ] ) ( ) ( 2 1 ) ( ) ( 2 1 ) ( ) ( 2 sin 2 cos ) ( 1 1 0 0 2 1 f f f f j f f f f dt e t x t x F f X t f t f t x ft j o + - - + + + - = = = + = - - δ δ δ δ π π π ii) For a fixed ζ S , the sample path ( 29 X t , ζ is a deterministic time signal. In order to determine the frequency domain representation for a random process, can we define { } ) , ( ) , ( ζ ζ t x F f X = ? g( X (t)) X (t) Y (t) N (t)
PEM3066: Random Processes & Queueing Theory Chapter 3: Spectral Analysis Page 3 of 18 iii) Problem with the definition above : We must know the dependence of the sample functions on time It does not take into account the relative probabilities among the sample functions.

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Chapter3 - PEM3066 Random Processes Queueing Theory Chapter...

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