113_1_Week07

# 113_1_Week07 - 1 EE 113 Digital Signal Processing Week 7...

This preview shows pages 1–11. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ....
View Full Document

## This note was uploaded on 06/29/2011 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

### Page1 / 35

113_1_Week07 - 1 EE 113 Digital Signal Processing Week 7...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online