Lecture18 - Power Spectrum For a deterministic signal x(t),...

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1 Lecture 18 Spectral Analysis Power Spectrum ± For a deterministic signal x(t), the spectrum is a FT ± The energy is then and by Parseval () ( ) , jt Xx t e d t ω +∞ −∞ = 2 |() | X 22 1 2 | ( )| . x td t X d E π ωω + ∞+ ∞− = = ∫∫ Power Spectrum ± The expression represents the energy in the band 2 |( ) | X Δ (, ) t 0 X t 0 2 | X ) Energy in Power Distribution ± For stochastic processes, a direct application of FT yields a sequence of random variables for every ± For a stochastic process represents the ensemble average power (instantaneous energy) at instant t ± To obtain a spectral distribution of power versus frequency, it is common to adopt a finite interval ( T , T ) . 2 E{| X(t) | } ( ) T j t T T XX t e d t =
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2 Power Distribution ± A time average leads to a power distribution based on ( T , T ) of a specific realization ² An ensemble average power is 2 2 |( ) | 1 () 22 T jt T T X Xte d t TT ω = 12 2 * 1 2 1 2 ) | 1 {() () } 1 (, ) 2 T XX jtt T X P E E X t X t e dt dt R t t e dt dt T −− ⎧⎫ == ⎨⎬ ⎩⎭ = ∫∫ Power Distribution ± For wide sense stationary (w.s.s) processes, it is possible to further simplify by using ± To yield ± Letting 1 2 ( ) XX XX R tt R t t =− 1 2 1 ( ) . 2 TX X PR t t e d t d t T T →∞ 2 2 2 || 2 2 1 ( 2 || ) 2 ( 1 ) 0 X XX T j T T j T T e T d T Re d ωτ τ ττ Power Spectral Density ± To yield a PSD of a WSS process ± We thus have ± The autocorrelation function and the power spectrum of a wss process form a Fourier transform pair, known as the Wiener- Khinchin Theorem () l im () 0 XX T XX j T SP R e d ωω +∞ −∞ →∞ FT ( ) 0 . XX XX RS τω ←⎯→ Power spectral Density ± The inverse works out as ± In particular we have Justifying that is a PSD 1 2 ( ) XX XX j R Se d π +∞ −∞ = 2 1 2 ( 0 ) { | ( ) |} , XX XX S d R E X t P the total power. +∞ −∞ = XX S
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3 Power Spectral Density ± The non-negativity of autocorrelation leads to non negativity of PSD ω 0 represents the power in the band (, ) ωω + Δ () XX S Δ XX S ** 11 2 1 1 2 1 2 ( ) 0 . ij XX XX i XX nn jtt i j n jt i i aa R t t aa S e d Sa e d π +∞ −∞ == +∞ = −∞ −= =≥ ∑∑ ( ) ( ) 0.
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This note was uploaded on 09/24/2009 for the course ECE 514 taught by Professor Krim during the Fall '08 term at N.C. State.

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Lecture18 - Power Spectrum For a deterministic signal x(t),...

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