Random Variables and Stochastic Processes 0903720 Lecture22 Dr

Random Variables and Stochastic Processes – 0903720 Lecture#22 Dr. Ghazi Al Sukkar Email: ghazi. alsukkar@ju. edu. jo Office Hours: Refer to the website Course Website: http: //www 2. ju. edu. jo/sites/academic/ghazi. alsukkar 1

Chapter 9 Power Spectrum § Definition of Power Spectral Density § Properties of PSD § Baseband Bandpass Processes § Cross-PSD § Properties of Cross-PSD § PSD and Linear Time Invariant Systems (LTI) § Multi-terminal Systems and PSD § PSD for DT systems 2

Power Spectrum For a deterministic signal x(t), the spectrum is well defined. If represents its Fourier transform, i. e. , then represents its energy spectrum. This follows from Parseval’s theorem since the signal energy is given by Thus represents the signal energy in the band Energy in 3

• Taking the Fourier transform of the random process des not work. 4

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• represents the power in the band 9

Properties of PSD • 10

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Baseband Bandpass Processes • 12

Cross-Power Spectrum • 13

Properties of CPSD • 14

Power Spectra and Linear Systems • X(t) h(t) Y(t) 15

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Example: Let represent a “smoothing” operation using a moving window on the input process X(t). Find the spectrum of the output Y(t) in term of that of X(t). Sol. : If we define an LTI system with impulse response h(t) as in the Figure, then in term of h(t), so that Here 19

so that 20

Multi-terminal Systems and PSD • MIMO 24

PSD for Discrete-time Processes •