Linearity Property : A system is said to be linear if superposition theorem applies to that system. If it
does not satisfy the superposition theorem, then it is said to be a nonlinear system.
Shift invariant Property: A system is time invariant if the time shift in the input signal results in
corresponding time shift in the output. A system which does not satisfy the
above condition is time variant system.
check whether the corresponding systems given are linear and time invariant or not.
a. Y[n] = x2[n-2]
b. Y[n] = 4x[n] + 6
a. Y[n] = x2[n-2]
Linearity:
Y1[n] = x12[n-2]
Y2[n] = x22[n-2]
Y3[n] = a1x12[n-2]+a2 x22[n-2]
Y3 ‘[n] = T[a1x1[n]+a2 x2[n]]
Y3 ‘[n] = [a1x1[n-2]+a2 x2 [n-2]]2
Y3 ‘[n] = a12x12[n-2]+a22x22[n-2]+2 a1x1[n-2].a2 x2 [n-2]
Y3[n] ≠Y3 ‘[n] It is a non linear system.
Shift invariant:
Y[n,k] = x2[(n-2)-k]
Y[n-k] = x2[n-k-2]
Y[n,k] = Y[n-k] it is shift invariant system.
b. Y[n] = 4x[n] + 6
Linearity:
Y1[n] = 4x1[n] + 6 Y2[n] = 4x2[n] + 6 Y3[n]
= a1 Y1[n]+a2 Y2[n]
= a1 (4x1[n] + 6)+a2 (4x2[n] + 6)
Y3 ‘[n] = T[a1x1[n]+a2 x2[n]]
= 4[a1x1[n]+a2 x2[n]] + 6
Y3[n] ≠ Y3 ‘[n] . It is Non linear system.
Shift invariant :
Y[n,k] = 4x[n-k] + 6
Y[n-k] = 4x[n-k] + 6
Y[n,k] = Y[n-k] it is shift invariant system.
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