System represented by equations— (i) y[n] = anx[n] (ii)

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System represented by equations—
(i) y[n] = anx[n]
(ii) y[n] = ax[n – 1] + bx[n – 3]

1. (i) is time-variant, (ii) is time-invariant
2. (i) is time-invariant, (ii) is time-invariant
3. Both (i) and (ii) are time-variant
4. Both (i) and (ii) are time-invariant

Correct Option: A

(i) y[n] = an x[n]
F[x(n)] = an x[n]
The response to a delayed excitation.
F[x(n – n₀)] = an[x(n – n₀)] …(A)
The delayed response is
y[(n – n₀)] = a[(n – n₀) x(n – n₀] …(B)
Since F[x(n – n₀)] ≠ y(n – n₀)
Hence, the system is not time invariant i.e., the system is time dependent.
(ii) y[n] = ax[n – 1] + bx[n – 2]
F[x(n)] = ax[n – 1] + bx[n – 2]
The response to a delayed excitation
F[x(n – n₀)] = ax[n – n₀ – 1] + bx[n – n₀ – 2]
The delayed response is
y(n – n₀) = ax[n – n₀ – 1] + bx[n – n₀ – 2]
Since, F[n – n₀] = y[n – n₀]
Therefore, the given system is time invariant.
Hence, alternative (A) is the correct choice.