lecture3-f08

lecture3-f08 - Administrative EE264 Digital Signal...

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EE264 Digital Signal Processing Lecture 3 Discrete-Time Random Signals II September 29, 2008 Ronald W. Schafer Department of Electrical Engingeering Stanford University Stanford University, EE264 Administrative • Problem Set #1 is posted. It is due on Weds., Oct. 1. • Lecture slides will be posted by 9 am on class days. https://ccnet.stanford.edu/ee264 will take you to the course website. https://myvideosu.stanford.edu will take you to where you can find the videos of the lectures. Stanford University, EE264 Overview of Lecture • Review: Discrete-time random signals as inputs to LTI systems • The power spectrum • Properties of the autocorrelation and power spectrum • LTI systems and the power spectrum • “White noise” • Demos and Examples. Stanford University, EE264 Linear System with a Random Input LTI System h [ n ], H ( e j ω ) x [ n ] y [ n ] c hh [ m ] = h [ m + k ] k =−∞ h [ k ] = h [ m ]* h [ m ] φ yy [ m ] = xx [ m A ] A c hh [ A ] = xx [ m ] c hh [ m ] xx [ m ] yy [ m ] = xx [ m ] c hh [ m ] Autocorrelation of output Deterministic autocorrelation yy [ m ] = Ey [ n + m ] y [ n ] {} = Ex [ n + m r ] h [ r ] r x [ n k ] h [ k ] k
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Stanford University, EE264 Derivation of Convolution Theorem φ yy [ m ] = Ey [ n + m ] y [ n ] {} = Ex [ n + m r ] h [ r ] r =−∞ x [ n k ] h [ k ] k = h [ r ] r h [ k ] k [ n + m r ] x [ n k ] { } = h [ r ] r h [ k ] k xx [ m r + k ] Let A = r k or r = A + k = h [ A + k ] h [ k ] k A xx [ m A ] = c hh [ A ] A xx [ m A ] = xx [ m ] c hh [ m ] c hh [ A ] ± ² ³ ³ ´ ³ ³ Stanford University, EE264 Properties of the Autocorrelation • Definition: • Average power: • Symmetry: • Shape: xx [ m ] = [ n + m ] x [ n ] { } xx [ m ] = xx [ m ] xx [ m ] = xx [ m ] if x is real xx [ m ] xx [0] lim m →∞ xx [ m ] = m x 2 { } ] 0 [ power average 2 xx x E = = Stanford University, EE264 Power Density Spectrum • The DTFT of a random signal does not exist theoretically.
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lecture3-f08 - Administrative EE264 Digital Signal...

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