(New page: == IS THIS SYSTEM TIME INVARIANT? == '''<math>X_k[n] = d[n-k] \Rightarrow Y_k[n]=(k+1)2 d[n-(k+1)]</math>''' '''TEST''' '''<math>d[n-k] \Rightarrow (k+1)2 d[n-(k+1)] \rightarrow [time ...) |
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'''TEST''' | '''TEST''' | ||
| − | '''<math>d[n-k] \Rightarrow (k+1)2 d[n-(k+1)] \rightarrow [time delay] \rightarrow (k+1)^2 d[n -(k+1) - t_o]</math>''' | + | '''<math>d[n-k] \Rightarrow (k+1)2 d[n-(k+1)] \rightarrow [time delay] \rightarrow Z(t) = (k+1)^2 d[n -(k+1) - t_o]</math>''' |
| − | '''<math>d[n-k] \rightarrow [time delay] \rightarrow d[(n-k) - t_o] \Rightarrow (k+1)^2 d[n -(k+1) - t_o]</math>''' | + | '''<math>d[n-k] \rightarrow [time delay] \rightarrow d[(n-k) - t_o] \Rightarrow W(t) = (k+1)^2 d[n -(k+1) - t_o]</math>''' |
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| + | since '''<math>Z(t) = W(t)</math>''' the system is time invariant. | ||
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| + | '''PART B''' | ||
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| + | Assuming that this system is linear, what input '''<math>X[n]</math>''' would yield the output '''<math>Y[n]=u[n-1]</math>'''? | ||
| + | |||
| + | The input would have to be '''<math>u[n]</math>''' | ||
Latest revision as of 17:45, 12 September 2008
IS THIS SYSTEM TIME INVARIANT?
$ X_k[n] = d[n-k] \Rightarrow Y_k[n]=(k+1)2 d[n-(k+1)] $
TEST
$ d[n-k] \Rightarrow (k+1)2 d[n-(k+1)] \rightarrow [time delay] \rightarrow Z(t) = (k+1)^2 d[n -(k+1) - t_o] $
$ d[n-k] \rightarrow [time delay] \rightarrow d[(n-k) - t_o] \Rightarrow W(t) = (k+1)^2 d[n -(k+1) - t_o] $
since $ Z(t) = W(t) $ the system is time invariant.
PART B
Assuming that this system is linear, what input $ X[n] $ would yield the output $ Y[n]=u[n-1] $?
The input would have to be $ u[n] $
