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**Unformatted text preview: **1 EE 113: Digital Signal Processing Week 7 Frequency response (Ch 13 Frequency response (Ch 13) All pass and minimum phase systems (part of Ch 14) 2 Sinusoids as Eigenfunctions s IR h [ n ] completely describes LTI system: s Complex sinusoid input i.e. = h k [ ] x n k [ ] k x [ n ] h [ n ] y [ n ] = x [ n ] h [ n ] x n [ ] = e j n n k s Output is sinusoid scaled by FR at y n [ ] = h k [ ] e j n k ( ) k = h k [ ] e j k e j n k = H e j ( ) e j n + ( ) ( ) y n [ ] = H e j ( ) x n [ ] H ( e j ) = | H ( e j ) | e j ( ) 3 System Response from H ( e j ) s If x [ n ] is a complex sinusoid at 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 j )} = and phase-shifted by arg{ H ( e )} = ( ) where H ( e j ) = DTFT{ h [ n ]} s | H ( e j ) | magnitude response gain s arg{ H ( e j )} phase resp. phase shift 4 Why study DTFT? Frequency Response (FR) s Knowing the scaling for every sinusoid fully describes the system behavior frequency response describes how a system affects each pure frequency 5 Magnitude and Phase of the DTFT The DTFT of a sequence x(n) is a complex function of s Examples on the board!! 6 Example 2 7 Example 3 8 Real sequences 9 s In practice signals are real e.g. x n [ ] = A cos n + ( ) = A 2 e j n + ( ) + e j n + ( ) ( ) Real Sinusoids X ( e j ) s Real h [ n ] H ( e-j ) = H * ( e j ) = | H ( e j ) | e-j ( ) = A 2 e j e j n + A 2 e j e j n y n [ ] = A 2 e j H e j ( ) e j n + A 2 e j H e j ( ) e j n y n [ ] = A H e j ( ) cos n + + ( ) ( ) - 10 Real Sinusoids s A real sinusoid of frequency A cos( n + ) h [ n ] | H(e j ) | A cos( n + + ( )) passed through an LTI system with a real impulse response h [ n ] has its gain modified by | H ( e j ) | and its phase shifted by ( ) ....

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