solns1_600

# solns1_600 - ECE-600 Introduction to Digital Signal...

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Autumn 2011 Homework #1 Sep. 28, 2011 HOMEWORK SOLUTIONS #1 1. Here, the input/output relationship of H is y [ n ] = ∑ 2 m =0 (3 − m ) x [ n − m ] = 3 x [ n ]+2 x [ n − 1]+ x [ n − 2]. (a) For H to be linear, we need that H{ αx [ n ] + βw [ n ] } = α H{ x [ n ] } + β H{ w [ n ] } for all α and β and x [ n ] and w [ n ] . From the I/O relationship, we know that H{ αx [ n ] + βw [ n ] } = 2 summationdisplay m =0 (3 − m )( αx [ n − m ] + βw [ n − m ]) = α 2 summationdisplay m =0 (3 − m ) x [ n − m ] + β 2 summationdisplay m =0 (3 − m ) w [ n − m ] = α H{ x [ n ] } + β H{ w [ n ] } . Thus, the system is linear. (b) Time invariance means that, if H{ x [ n ] } = y [ n ], then H{ x [ n − d ] } = y [ n − d ] for any integer shift d . Investigating whether this is the case here, we first use H{ x [ n ] } = ∑ 2 m =0 (3 − m ) x [ n − m ] to evaluate H{ x [ n − d ] } = 3 x [ n − d ] + 2 x [ n − d − 1] + x [ n − d − 2] and then we use y [ n ] = 3 x [ n ] + 2 x [ n − 1] + x [ n − 2] evaluate y [ n − d ] = 3 x [ n − d ] + 2 x [ n − d − 1] + x [ n − d − 2] . Since these two quantities are equal, the system is indeed 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 only on the current input x [ d ] and the past inputs x [ d − 1] and x [ d − 2], and thus the system is causal. (d) Stability means that a bounded input guarantees a bounded output. To examine whether the system is stable, we can assume a generic 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 ] | = | 3 x [ n ] + 2 x [ n − 1] + x [ n − 2] | ≤ 3 | x [ n ] | + 2 | x [ n − 1] | + | x [ n...
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solns1_600 - ECE-600 Introduction to Digital Signal...

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