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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 ω n ω n − k ⇒ s Output is sinusoid scaled by FR at ω y n [ ] = h k [ ] e j ω n − k ( ) k ∑ = h k [ ] e − j ω k ⋅ e j ω n k ∑ = H e j ω ( ) ⋅ e j ω n +θ ω ( ) ( ) ⇒ y n [ ] = H e j ω ( ) ⋅ x n [ ] H ( e j ω ) = | H ( e j ω ) | e j θ ( ω ) 3 System Response from H ( e j ω ) s If x [ n ] is a complex sinusoid at ω then the output of a system with IR h [ n ] is the same sinusoid scaled by | H ( e j ω ) | and phase shifted by arg H j ω )} = θ ω and phase-shifted by arg{ H ( e )} = θ ( ω ) where H ( e j ω ) = DTFT{ h [ n ]} s | H ( e j ω ) | “ magnitude response ” → gain s arg{ H ( e j ω )} “ 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 ω 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 ω n +φ ( ) = A 2 e j ω n +φ ( ) + e − j ω n +φ ( ) ( ) Real Sinusoids X ( e j ω ) s Real h [ n ] → H ( e-j ω ) = H * ( e j ω ) = | H ( e j ω ) | e-j θ ( ω ) = A 2 e j φ e j ω n + A 2 e − j φ e − j ω n ⇒ y n [ ] = A 2 e j φ H e j ω ( ) e j ω n + A 2 e − j φ H e − j ω ( ) e − j ω n ⇒ y n [ ] = A H e j ω ( ) cos ω n +φ +θ ω ( ) ( ) ω- ω ω 10 Real Sinusoids s A real sinusoid of frequency ω A cos( ω n + φ ) h [ n ] | H(e j ω ) | A cos( ω n + φ + θ ( ω )) passed through an LTI system with a real impulse response h [ n ] has its gain modified by | H ( e j ω ) | and its phase shifted by θ ( ω ) ....

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