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solns1_600(2)

# solns1_600(2) - ECE-600 Homework#1 Introduction to Digital...

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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 ] = 0 m = - 2 x [ n m ] = x [ n ] + 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 ] } . From the I/O relationship, we know that H{ αx [ n ] + βw [ n ] } = 0 summationdisplay m = - 2 ( αx [ n m ] + βw [ n m ]) = α 0 summationdisplay m = - 2 x [ n m ] + β 0 summationdisplay m = - 2 w [ n m ] = α H{ x [ n ] } + β 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. (e) The impulse response h [ n ] is defined as h [ n ] = H{ δ [ n ] } , where δ [ n ] denotes the Kronecker delta. Then, the I/O relationship implies h [ n ] = δ [ n ] + δ [ n + 1] + δ [ n + 2] = braceleftBigg 1 n ∈ { 0 , 1 , 2 } 0 else

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solns1_600(2) - ECE-600 Homework#1 Introduction to Digital...

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