(New page: == Test Correction of # 3 == An LTI system has unit impulse response <math> h[n] = u[-n] </math> Compute the system's response to the input <math> x[n] = 2^{n}u[-n] </math>) |
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== Test Correction of # 3 == | == Test Correction of # 3 == | ||
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Compute the system's response to the input <math> x[n] = 2^{n}u[-n] </math> | Compute the system's response to the input <math> x[n] = 2^{n}u[-n] </math> | ||
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| + | (Simplify answer until all summation signs disappear.) | ||
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| + | h[n] = u[-n] | ||
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| + | x[n] = <math>2^{n}</math>u[-n] | ||
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| + | y[n] = x[n] * h[n] | ||
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| + | = <math> \sum^{\infty}_{k = -\infty} 2^{k}u[-k]u[-n--k]</math> | ||
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| + | since -k > 0 and k < 0 the summation parameters change | ||
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| + | = <math> \sum^{0}_{k = -\infty} 2^{k}u[-n+k]</math> | ||
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| + | other step function changes parameters again | ||
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| + | n = k | ||
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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> since n <= 0; </math> | ||
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| + | This is then a property that can be exchanged to get rid of the summation signs using a general solution. | ||
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| + | { = <math> \frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}} </math> For all k < n and n <= 0 | ||
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| + | { = 0 , n > 0 | ||
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| + | The transform of this is then= | ||
| + | <math> (2-2^{n})u[-n]</math> | ||
Latest revision as of 09:57, 12 October 2008
Test Correction of # 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 answer until all summation signs disappear.)
h[n] = u[-n]
x[n] = $ 2^{n} $u[-n]
y[n] = x[n] * h[n]
= $ \sum^{\infty}_{k = -\infty} 2^{k}u[-k]u[-n--k] $
since -k > 0 and k < 0 the summation parameters change
= $ \sum^{0}_{k = -\infty} 2^{k}u[-n+k] $
other step function changes parameters again
n = k
= $ \sum^{0}_{k = n} 2^{k} $
= $ \sum^{-n}_{0} \frac{1}{2}^{k} $ since n <= 0; </math>
This is then a property that can be exchanged to get rid of the summation signs using a general solution.
{ = $ \frac{1-\frac{1}{2}^{-n+1}}{1-\frac{1}{2}} $ For all k < n and n <= 0
{ = 0 , n > 0
The transform of this is then= $ (2-2^{n})u[-n] $
