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== Question 3. == | == Question 3. == | ||
* An LTI system has unit impulse response h[n] =u[-n]. Compute the system's response to the input <math>x[n] = 2^{n}u[-n].</math> Simplify your answer until all <math> \sum</math> signs disappear.) | * An LTI system has unit impulse response h[n] =u[-n]. Compute the system's response to the input <math>x[n] = 2^{n}u[-n].</math> Simplify your answer until all <math> \sum</math> signs disappear.) | ||
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| + | == Answer == | ||
| + | <math> y[n] = x[n] * h[n] , where * is convolution/,</math> | ||
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| + | <math> = \sum^{\infty}_{k=-\infty} 2^{k}u[-k]u[-n+k]</math> | ||
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| + | <math> = \sum^{0}_{k=-\infty} 2^{k}u[-n+k]</math> | ||
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| + | <math> = \sum^{0}_{k=n} 2^{k}</math> | ||
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| + | <math> = \sum^{-n}_{0} \frac{1}{2}^{k},</math> for n=<0 <br> | ||
| + | <math> = 0 ,</math>for n>0<br> | ||
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| + | <math> =\frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}}</math>for n=<0 <br> | ||
| + | <math> = 0, </math>for n>0<br> | ||
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| + | <math> =(2-2^{n})u[-n]</math><br> | ||
Latest revision as of 13:31, 10 October 2008
Question 3.
- An LTI system has unit impulse response h[n] =u[-n]. Compute the system's response to the input $ x[n] = 2^{n}u[-n]. $ Simplify your answer until all $ \sum $ signs disappear.)
Answer
$ y[n] = x[n] * h[n] , where * is convolution/, $
$ = \sum^{\infty}_{k=-\infty} 2^{k}u[-k]u[-n+k] $
$ = \sum^{0}_{k=-\infty} 2^{k}u[-n+k] $
$ = \sum^{0}_{k=n} 2^{k} $
$ = \sum^{-n}_{0} \frac{1}{2}^{k}, $ for n=<0
$ = 0 , $for n>0
$ =\frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}} $for n=<0
$ = 0, $for n>0
$ =(2-2^{n})u[-n] $
