The causal systems is/are— (i) y(n) = x2(n) u(n) (ii)

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The causal systems is/are—
(i) y(n) = x²(n) u(n)
(ii) y(n) = x(|n|)
(iii) y(n) = x(n) – x(n² – n)
_N
(iv) y(n) = ∑ x(n – k)
^{k = 1}

1. (i), (ii) and (iii)
2. (i) and (iv)
3. (iii) and (iv)
4. (i) and (iv)

Correct Option: D

Given
(i) y(n) = x²(n) u(n)
This system is causal for any value of n.
(ii) y(n) = x(|n|) If we put any negative value of n, say n = – 2, we get
y(– 2) = x(|– 2|) = x(2)
i.e., output depends on the future value of input.
Hence, this system is non-causal.
(iii) y(n) = x(n) – x(n² – n)
This system is causal for n = 1, 2 and non-causal for values other than 0, 1 and 2.
Let n = 3, we get
y(3) = x(3) – x(3² – 3)
y(3) = x(3) – x(6)
which is non-causal

	_N
(iv) y(n) =	∑	x(n – k)
	^{k = 1}

Since, the limit of k is positive, therefore, this system is causal for any value of k.
Hence, alternative (D) is the correct choice.