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==Linearity and Time Invariance== | ==Linearity and Time Invariance== | ||
| + | '''6.a)''' | ||
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| + | '''the system is defined as''' | ||
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| + | <math>X_k[n] = \delta[n - k] \to sys \to Y_k[n] = (k + 1)^2 \delta[n - (k + 1)]</math> | ||
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| + | '''let us check for time invariance''' | ||
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| + | '''System followed by time delay''' | ||
| + | |||
| + | '''now,let us apply a time-delay of <math>t_0</math> to the system.''' | ||
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| + | |||
| + | <math>\delta[n - k] \to sys \to (k + 1)^2 \delta[n - (k + 1)] \to timedelay \to (k + 1)^2 \delta[n - t_0 -(k + 1)] = (k + 1)^2 \delta[n -(k + 1 +t_0)] </math> | ||
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| + | '''Time-delay followed by system:''' | ||
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| + | <math>\delta[n - k] \to timedelay \to \delta[n-(k + t_0)] \to sys \to (k + t_0 + 1)^2 \delta[n - (k + t_0 + 1)]</math> | ||
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| + | '''For the system to be time invariant both the outputs should be same but they are not. so the system is not time variant it is rather a time variant system as the output varies with time.''' | ||
Revision as of 09:19, 12 September 2008
Linearity and Time Invariance
6.a)
the system is defined as
$ X_k[n] = \delta[n - k] \to sys \to Y_k[n] = (k + 1)^2 \delta[n - (k + 1)] $
let us check for time invariance
System followed by time delay
now,let us apply a time-delay of $ t_0 $ to the system.
$ \delta[n - k] \to sys \to (k + 1)^2 \delta[n - (k + 1)] \to timedelay \to (k + 1)^2 \delta[n - t_0 -(k + 1)] = (k + 1)^2 \delta[n -(k + 1 +t_0)] $
Time-delay followed by system:
$ \delta[n - k] \to timedelay \to \delta[n-(k + t_0)] \to sys \to (k + t_0 + 1)^2 \delta[n - (k + t_0 + 1)] $
For the system to be time invariant both the outputs should be same but they are not. so the system is not time variant it is rather a time variant system as the output varies with time.
