(New page: == Linearity and Time Variance == From the given equation: <math>\,\!Y_k[n]=(k+1)^2\delta [n-(k+1)]</math> , It is clear that this is a non-time-invariant, or more simply, a time varia...) |
|||
| Line 7: | Line 7: | ||
It is clear that this is a non-time-invariant, or more simply, a time variant function because the amplitude changing <math>(k+1)^2</math> term. | It is clear that this is a non-time-invariant, or more simply, a time variant function because the amplitude changing <math>(k+1)^2</math> term. | ||
| − | The inputs to this system are all shifted <math>\delta</math> functions, this means that if the <math>X[n]</math> inputs were changed to unit step functions <math>u[n]</math> , then the output will be a time shifted step function. | + | The inputs to this system are all shifted <math>\delta</math> functions, this means that if the <math>X[n]</math> inputs were changed to unit step functions <math>u[n]</math> , then the output will be a time shifted step function. For this specific example we want an output of <math>Y[n]=u[n-1]</math> , so we can notice that when : |
| + | |||
| + | <math>\,\!X[n]=\delta [n-1]</math> yields <math>\,\!Y_0[n]=\delta [n-1]</math> , then | ||
| + | |||
| + | |||
| + | <math>\,\!X[n]=u[n-1]</math> will yield <math>\,\!Y_0[n]=u[n-1]</math> | ||
Revision as of 15:22, 11 September 2008
Linearity and Time Variance
From the given equation:
$ \,\!Y_k[n]=(k+1)^2\delta [n-(k+1)] $ ,
It is clear that this is a non-time-invariant, or more simply, a time variant function because the amplitude changing $ (k+1)^2 $ term.
The inputs to this system are all shifted $ \delta $ functions, this means that if the $ X[n] $ inputs were changed to unit step functions $ u[n] $ , then the output will be a time shifted step function. For this specific example we want an output of $ Y[n]=u[n-1] $ , so we can notice that when :
$ \,\!X[n]=\delta [n-1] $ yields $ \,\!Y_0[n]=\delta [n-1] $ , then
$ \,\!X[n]=u[n-1] $ will yield $ \,\!Y_0[n]=u[n-1] $
