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<math>X_k[n] = \delta[n - k] \rightarrow system \rightarrow Y_k[n] = (k + 1)^2 \delta[n - (k + 1)]</math> | <math>X_k[n] = \delta[n - k] \rightarrow system \rightarrow Y_k[n] = (k + 1)^2 \delta[n - (k + 1)]</math> | ||
| − | Let us apply a time-delay of <math>n_0</math> to the system. | + | Let us apply a time-delay of <math>n_0</math> to the system. |
| − | <math>\delta[n - k] \rightarrow system \rightarrow (k + 1)^2 \delta[n - (k + 1)] \rightarrow time-delay \rightarrow (k + 1)^2 \delta[n - n_0 -(k + 1)] = (k + 1)^2 \delta[n -(k + 1 | + | System followed by time-delay: |
| + | |||
| + | <math>\delta[n - k] \rightarrow system \rightarrow (k + 1)^2 \delta[n - (k + 1)] \rightarrow time-delay \rightarrow (k + 1)^2 \delta[n - n_0 -(k + 1)] = (k + 1)^2 \delta[n -(k + 1 +n_0)] </math> | ||
| + | |||
| + | |||
| + | Time-delay followed by system: | ||
| + | |||
| + | <math>\delta[n - k] \rightarrow time-delay \rightarrow \delta[n-(n_0 + k)] \rightarrow system \rightarrow (k + 1)^2 \delta[n - (k + 1)]</math> | ||
Revision as of 16:43, 11 September 2008
6 a) The system cannot be time-invariant.
$ X_k[n] = \delta[n - k] \rightarrow system \rightarrow Y_k[n] = (k + 1)^2 \delta[n - (k + 1)] $
Let us apply a time-delay of $ n_0 $ to the system.
System followed by time-delay:
$ \delta[n - k] \rightarrow system \rightarrow (k + 1)^2 \delta[n - (k + 1)] \rightarrow time-delay \rightarrow (k + 1)^2 \delta[n - n_0 -(k + 1)] = (k + 1)^2 \delta[n -(k + 1 +n_0)] $
Time-delay followed by system:
$ \delta[n - k] \rightarrow time-delay \rightarrow \delta[n-(n_0 + k)] \rightarrow system \rightarrow (k + 1)^2 \delta[n - (k + 1)] $
