[RP]Lecture Note XIV - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XIV-1 C1002900 RP Lecture Handout VIV: Power Spectrum and Spectral Representation Reading: Chapter 9.3 and 11.4 We are often interested in how power in a stochastic process is distributed as a func- tion of frequency. A natural measure of this distribution is obtained by the power spectral density (PSD) of the process, which is also frequently referred to as simply the power spectrum. 14.1. Power Spectral Density From an engineering perspective, a natural way to define the PSD of a process is in terms of the power at the output of a narrow bandpass filter centered at the frequency of interest whose input is  Xt (what a spectrum analyzer would measure) . Definition 14.1: Let ht be the impulse response of an ideal bandpass filter whose frequency response is , if o t h e r w i s e Hj     0 22 0 (XIV.1) and let denote the output of this filter when its input is the WSS . Then, the PSD of at frequency 0 is   lim XX Sj X t  2 0 0 . (XIV.2) Two alternative interpretations of the PSD are important consequences of this definition.
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XIV-2 1) First interpretation: Theorem 14.1 (Wiener-Khinchine): Let  XX Sj be the PSD of a WSS Xt in the sense of definition 14.1. Then,      j XX XX XX R R e d      (XIV.3) i.e., the autocorrelation function and PSD are Fourier transform pairs. Proof: Let XX Tj and XX  be the Fourier transform of XX R j and R j , respectively. Then, it suffices to show that XX is equal to the PSD at frequency . Note first that      Rh h R   . (XIV.4) Hence,   TjH j T j 2 . (XIV.5) Now,   XX R H jTj d   2 2 0 1 2 . (XIV.6) Since lim Hj   2 0 0 2 . (XIV.7) We obtain   lim . XX XX XX X t d   2 0 0 0 0 (XIV.8) Hence,
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) XIV-3     . XX XX XX Sj Tj R   (XIV.9)      XX XX j XX XX j XX XX RS j R e d R Se d        1 2 Since XX XX Rj R , XX is real.
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[RP]Lecture Note XIV - Kyung Hee University Department of...

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