solns2_600

# solns2_600 - ECE-600 Introduction to Digital Signal...

This preview shows pages 1–2. Sign up to view the full content.

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: ECE-600 Introduction to Digital Signal Processing Winter 2011 Homework #2 Jan. 21, 2011 HOMEWORK SOLUTIONS #2 1. (a) DTFT modulation property: F DTFT { x [ n ] e jω n } = ∞ summationdisplay n =-∞ x [ n ] e jω n e- jωn = ∞ summationdisplay n =-∞ x [ n ] e- j ( ω- ω ) n = X ( e j ( ω- ω ) ) via the DTFT definition (b) DTFT shift property: F DTFT { x [ n − d ] } = ∞ summationdisplay n =-∞ x [ n − d ] e- jωn = ∞ summationdisplay m =-∞ x [ m ] e- jω ( m + d ) via substitution m = n − d (limits don’t change) = e- jωd ∞ summationdisplay m =-∞ x [ m ] e- jωm = e- jωd X ( e jω ) via the DTFT definition (c) DTFT property of real-valued X ( e jω ): x * [ − n ] = parenleftbigg 1 2 π integraldisplay π- π X ( e jω ) e jω (- n ) dω parenrightbigg * using the IDTFT definition = 1 2 π integraldisplay π- π X * ( e jω ) e jωn dω distributing the conjugation & ( e- jωn ) * = e jωn = 1 2 π integraldisplay π- π X ( e jω ) e jωn dω since real-valued X ( e jω ) implies X * ( e jω ) = X ( e jω ) = x [ n ] using the IDTFT definition 2. (a) The Fourier series says that we can write any T-period signal p ( t ) as p ( t ) = ∞ summationdisplay k =-∞ P [ k ] e jk 2 π T t with P [ k ] = 1 T integraldisplay T/ 2- T/...
View Full Document

## This note was uploaded on 02/21/2011 for the course ECE 600 taught by Professor Clymer,b during the Spring '08 term at Ohio State.

### Page1 / 3

solns2_600 - ECE-600 Introduction to Digital Signal...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online