Let x[n] = anu[n] h[n] = bnu[n] What is the expression for

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Let x[n] = anu[n] h[n] = bnu[n] What is the expression for y[n], for a discrete-time system?

1. _∞
  ∑ a^k u[k]b^{n – k} u[n – k]
  ^{k = – ∞}
2. _∞
  ∑ aⁿ u[k]b^{n – k} u[n – k]
  ^{k = – ∞}
3. _∞
  ∑ a^k u[n – k]bⁿ u[k]
  ^{k = – ∞}
4. _∞
  ∑ a^k u[n – k]bⁿ u[k]
  ^{k = – ∞}

Correct Option: A

For any arbitrary signal x(n) the response of the system is given by

	_∞
y(n) =	∑	x(k).x(n – k)
	^{k = – ∞}

Here, x(n) = aⁿ.u(n)
x(n) = bⁿ u(n)

	_∞
So, y(n) =	∑	a^k u(k).b^{n – k} u[n – k]
	^{k = – ∞}

Hence, alternative (A) is the correct answer.