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solns1_600

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

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Winter 2011 Homework #1 Jan. 12, 2011 HOMEWORK SOLUTIONS #1 1. Here, the input/output relationship of H is y [ n ] = ∑ 2 m =0 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 ] } 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 ( αx [ n- m ] + βw [ n- m ]) = α 2 summationdisplay m =0 x [ n- m ] + β 2 summationdisplay m =0 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 . 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 these 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 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....
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solns1_600 - ECE-600 Introduction to Digital Signal...

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