113_1_Week07_print

113_1_Week07_print - 1 EE 113: Digital Signal Processing...

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1 EE 113: Digital Signal Processing Week 7 Frequency response (Ch 13) All pass and minimum phase systems (part of Ch 14)
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2 Sinusoids as Eigenfunctions ± IR h [ n ] completely describes LTI system: ± Complex sinusoid input i.e. ± Output is sinusoid scaled by FR at ω 0 = hk [ ] xn k [ ] k x [ n ] h [ n ] y [ n ] = x [ n ] h [ n ] [ ] = e j 0 n yn [] = [ ] e j 0 n k ( ) k = e j 0 k e j 0 n k = He j 0 ( ) [ ] = j 0 ( ) e j 0 n +θ ω 0 ( ) ( ) H ( e j ) = | H ( e j ) | e j θ ( )
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3 System Response from H ( e j ω ) ± If x [ n ] is a complex sinusoid at 0 then the output of a system with IR h [ n ] is the same sinusoid scaled by | H ( e j ) | and phase-shifted by arg{ H ( e j )} = θ ( ) where H ( e j ) = DTFT{ h [ n ]} ± | H ( e j ) | magnitude response gain ± arg{ H ( e j )} phase resp. phase shift
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4 Why study DTFT? Frequency Response (FR) ± Knowing the scaling for every sinusoid fully describes the system behavior frequency response describes how a system affects each pure frequency
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5 Magnitude and Phase of the DTFT ± Examples on the board!! The DTFT of a sequence x(n) is a complex function of ω
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6 Example 2
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7 Example 3
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8 Real sequences
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9 ± In practice signals are real e.g. ± Real h [ n ] H ( e -j ω )= H * ( e j ) = | H ( e j ) | e -j θ ( ) xn [] = A cos 0 n ( ) = A 2 e j 0 n () + e j 0 n = A 2 e j φ e j 0 n + A 2 e j e j 0 n Real Sinusoids yn = A 2 e j He j 0 ( ) e j 0 n + A 2 e j j 0 ( ) e j 0 n = AH e j 0 ( ) cos 0 n +φ + θω 0 ( ) ( ) - ω 0 ω 0 X ( e j )
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10 Real Sinusoids ± A real sinusoid of frequency ω 0 passed through an LTI system with a real impulse response h [ n ] has its gain modified by | H ( e j 0 ) | and its phase shifted by θ ( 0 ) . A cos( 0 n + φ ) h [ n ] | H(e j 0 ) | A cos( 0 n + + ( 0 ))
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11 Transient / Steady State ± Most signals start at a finite time e.g. What is the effect? ± xn [] = e j ω 0 n μ n [ ] yn = hn = hk e j 0 n k ( ) k =−∞ n = e j 0 n k ( ) k =−∞ e j 0 n k ( ) k = n = He j 0 () e j 0 n e j 0 k k = n ( ) e j 0 n Steady state - same as with pure sine input Transient response
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12 -40 -30 -20 -10 0 10 20 30 40 time / n Total output Steady State Transient Transient / Steady State ± ±
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This note was uploaded on 06/01/2010 for the course EE EE113 taught by Professor Mihaela during the Spring '10 term at UCLA.

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113_1_Week07_print - 1 EE 113: Digital Signal Processing...

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