6.011: Signals, Systems & Inference Lec 3 Energy spectral - PowerPoint PPT Presentation

6.011: Signals, Systems & Inference Lec 3 Energy spectral density 1

Inner (dot) product of signals X < x , v > = x [ k ] v [ k ] for real signals x [ · ] , v [ · ] k ← [ n ] ← X More generally, p [ n ] = x [ k ] v [ k − n ] = x ∗ v − k ← ← [ ` ] = v [ − ` ] where − v 2

Transform of reversed signal X V ( e j Ω ) = v [ n ] e − j Ω n − DTFT of v [ n ] = n ← v [ − n ] e − j Ω n = − ← [ n ] = X X v [ k ] e j Ω k DTFT of v n k − = V ∗ ( e j Ω ) = V ( e − j Ω ) 3

Transform of inner product ← ← [ n ] = V ( e − j Ω ) so − DTFT of v ← ← [ n ] X p [ n ] = x [ k ] v [ k � n ] = x ⇤ v � k l P ( e j Ω ) = X ( e j Ω ) V ( e − j Ω ) − 4

Zero lag (n=0): Parseval, Rayleigh, Plancherel 1 Z π X P ( e j Ω ) d Ω p [0] = x [ k ] v [ k ] = 2 π − π k so 1 Z π X j Ω ) V ( e − j Ω ) d Ω x [ k ] v [ k ] = X ( e 2 π − π k 5

Energy Setting v [ k ] = x [ k ] , 1 Z π X | x [ k ] | 2 = 2 π X ( e j Ω ) X ( e − j Ω ) d Ω E x = − π k − 1 Z π | X ( e j Ω ) | 2 d Ω = 2 π π − 6

Energy spectral density (ESD) X ¯ r [ n ] = x [ k ] x [ k � n ] = R xx [ n ] | {z } k deterministic autocorrelation l transform pair � � 2 ¯ � X ( e j Ω ) � xx ( e j Ω ) = S � � | {z } energy spectral density 7

Cross (energy) spectral density ← ¯ ← X y [ k ] x [ k � n ] = y ⇤ x [ n ] R yx [ n ] = � k l − ¯ yx ( e j Ω ) = Y ( e j Ω ) X ( e – j Ω ) S So if y [ n ] = h ⇤ x [ n ] then − ¯ ¯ yx ( e j Ω ) = H ( e j Ω ) X ( e j Ω ) X ( e – j Ω ) = H ( e j Ω ) S xx ( e j Ω ) S 8

Similarly … ... if y [ n ] = h ∗ x [ n ] then ¯ yy ( e j Ω ) = Y ( e j Ω ) Y ( e − j Ω ) S = H ( e j Ω ) X ( e j Ω ) X ( e − j Ω ) H ( e − j Ω ) 2 � � ¯ j Ω ) j Ω ) = � H ( e � S xx ( e � � 9

Energy of x[.] in a specified band x [ n ] y [ n ] H ( e j Æ ) H ( e j Æ ) ¢ ¢ 1 -Æ 0 Æ 0 Æ 10

Noise-free signal 11

Magnitude spectrum of noise-free signal 12

ESD of noise-free signal 13

+ unit-intensity white Gaussian noise 14

Magnitude spectrum of noisy signal 15

ESD of noisy signal 16

Heart rate variability ECG signal 1.2 1 ECG amplitude (mV) 0.8 0.6 0.4 (a) 0.2 0 − 0.2 − 0.4 − 0.6 − 0.8 54 56 58 60 62 64 66 Time (sec) Instantaneous HR signal 1.3 x ( t ) (beats/sec) 1.25 (b) 1.2 1.15 1.1 45 50 55 60 65 70 75 80 Time (sec) Power spectrum 90 80 70 60 xx ( e ) 50 j Æ (c) 40 D 30 20 17 10 0 − 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1 Frequency (Hz)

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