solns1_600(2)

solns1_600(2) - ECE-600 Introduction to Digital Signal...

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Unformatted text preview: ECE-600 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 time-invariant. (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....
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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.

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solns1_600(2) - ECE-600 Introduction to Digital Signal...

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