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Using the system then time delay method: | Using the system then time delay method: | ||
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dirac[n-2] -> SYSTEM -> 9 dirac[n-3] -> Time Delay -> 9 dirac[n-5] | dirac[n-2] -> SYSTEM -> 9 dirac[n-3] -> Time Delay -> 9 dirac[n-5] | ||
Using the time delay then system method: | Using the time delay then system method: | ||
| + | |||
dirac[n-2] -> Time Delay -> dirac[n-4] -> SYSTEM -> 25 dirac[n-6] | dirac[n-2] -> Time Delay -> dirac[n-4] -> SYSTEM -> 25 dirac[n-6] | ||
Because the two outputs are not the same, the system is not time-invariant. | Because the two outputs are not the same, the system is not time-invariant. | ||
Revision as of 11:18, 12 September 2008
Linearity and Time Invariance
Given system:
Input Output
X0[n]=δ[n] -> Y0[n]=δ[n-1]
X1[n]=δ[n-1] -> Y1[n]=4δ[n-2]
X2[n]=δ[n-2] -> Y2[n]=9 δ[n-3]
X3[n]=δ[n-3] -> Y3[n]=16 δ[n-4]
... -> ...
Xk[n]=δ[n-k] -> Yk[n]=(k+1)2 δ[n-(k+1)] -> For any non-negative integer k
Time Invariant System?
Suppose the system is defined as the third line where input is $ X_2[n]= dirac[n-2] $ and output: $ Y_2[n]=9 dirac[n-3] $ with a time delay of 2 seconds.
Using the system then time delay method:
dirac[n-2] -> SYSTEM -> 9 dirac[n-3] -> Time Delay -> 9 dirac[n-5]
Using the time delay then system method:
dirac[n-2] -> Time Delay -> dirac[n-4] -> SYSTEM -> 25 dirac[n-6]
Because the two outputs are not the same, the system is not time-invariant.
