solns5_600(2)

# solns5_600(2) - 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 Autumn 2010 Homework #5 Nov. 5, 2010 HOMEWORK SOLUTIONS #5 1. (a) To resample a discrete-time signal from a sampling rate of 990 MHz to an effective rate of 999 MHz, we change the sampling rate by the ratio 999000 990000 = 999 990 = 111 110 , noticing that { 111 , 110 } are co-prime (i.e., we cannot simplify their ratio any further). Re- sampling then consists of the following three steps: i. Upsample by 111. ii. Lowpass filter the result with cutoff frequency min { π 111 , π 110 } = π 111 (radians/sample) and DC gain 111. iii. Downsample by 110. 2. (a) We start with the definition of the DTFT: ˜ X ( e jω ) = ∞ summationdisplay n = −∞ ˜ x [ n ] e − jωn (1) = ∞ summationdisplay l = −∞ N − 1 summationdisplay m =0 ˜ x [ m + lN ] e − jω ( m + lN ) via n = m + lN (2) = ∞ summationdisplay l = −∞ e − jωlN N − 1 summationdisplay m =0 x [ m ] e − jωm (3) where in the last equation we used the facts that x [ m ] = ˜ x [ m ] = ˜ x [ m + lN ] for any m = , 1 ,...,N − 1 and any l ∈ Z . Next, we recall from Homework 2 that ∞ summationdisplay l = −∞ e − jωlN = 2 π ∞ summationdisplay k = −∞ δ ( ωN − k 2 π ) (4) = 2 π N ∞ summationdisplay k = −∞ δ ( ω − k 2 π N ) (5) where the last equation used the Dirac scaling rule. Pluggin (5) into (3), we get ˜ X ( e jω ) = 2 π N ∞ summationdisplay k = −∞ δ ( ω − k 2 π N ) N − 1 summationdisplay m =0 x [ m ] e − jωm (6) which shows that the only values of ω yielding a non-zero DTFT are ω = k 2 π N for integers k . We can use this fact to rewrite the previous expression as ˜ X ( e jω ) = 2 π N ∞ summationdisplay k = −∞ δ ( ω − k 2 π N ) N − 1 summationdisplay m =0 x [ m ] e − j 2 π N km . (7) P. Schniter, 2010P....
View Full Document

## This note was uploaded on 02/14/2011 for the course ECE 600 taught by Professor Clymer,b during the Fall '08 term at Ohio State.

### Page1 / 4

solns5_600(2) - 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