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
Unformatted text preview: ECE600 Introduction to Digital Signal Processing Autumn 2010 Homework #1 Sept. 29, 2010 HOMEWORK SOLUTIONS #1 1. Here, the input/output relationship of H is y [ n ] = m = 2 x [ n m ] = x [ n ] + x [ n + 1] + x [ n + 2]. (a) For H to be linear, we need that H{ x [ n ] + w [ n ] } = H{ x [ n ] } + H{ w [ n ] } . From the I/O relationship, we know that H{ x [ n ] + w [ n ] } = summationdisplay m = 2 ( x [ n m ] + w [ n m ]) = summationdisplay m = 2 x [ n m ] + summationdisplay m = 2 w [ n m ] = H{ x [ n ] } + H{ w [ n ] } . Thus, the system is linear. (b) Time invariance means that H{ x [ n d ] } = y [ n d ] for any shift d . Substituting n n d in H{ x [ n ] } , we get H{ x [ n d ] } = x [ n d ] + x [ n d + 1] + x [ n d + 2] and substituting n n d in the expression for y [ n ], we get y [ n d ] = x [ n d ] + x [ n d + 1] + x [ n d + 2] . Since the latter two quantities are equal, the system is timeinvariant. (c) Causality means that, for any time d , the output y [ d ] does not depend on the future inputs { x [ n ] } n>d . From the I/O relationship, we see that y [ d ] depends directly on x [ d + 1] and x [ d + 2], and thus the system is not causal. (d) Stability means that a bounded input guarantees a bounded output. To examine whether the system is stable, we can assume a bounded input and check to see whether the output is bounded. Say that the input is bounded. This means that there exists finite M x such that  x [ n ]  < M x for all n . Then, from the I/O relationship,  y [ n ]  =  x [ n ] + x [ n + 1] + x [ n + 2]   x [ n ]  +  x [ n + 1]  +  x [ n + 2]  < 3 M x which implies that the output is also bounded. Hence, the system is stable....
View
Full
Document
This note was uploaded on 03/05/2011 for the course ECE 600 taught by Professor Clymer,b during the Fall '08 term at Ohio State.
 Fall '08
 Clymer,B
 Digital Signal Processing, Signal Processing

Click to edit the document details