# A system has impulse response h[n] = cos (n) u[n]. The system is:

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A system has impulse response h[n] = cos (n) u[n]. The system is:
1. Causal and stable
2. Non-causal and stable
3. Non-causal and not stable
4. Causal and not stable

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Correct Answer - Option 4 : Causal and not stable

Concept:

Condition for Causality:

A system with impulse response h[n] is stable if it satisfies:

h[n] = 0  ;  n < 0

Condition for stability:

A system is said to be stable if the impulse response is absolutely integrable, i.e.

$\mathop \sum \limits_{n = - \infty }^\infty \left| {h\left[ n \right]} \right| < \infty$

Application:

Given h[n] = cos (n) u[n]

Multiplication of u[n] ensures that h[n] = 0 for n < 0

Hence given system is causal.

$\mathop \sum \limits_{n = - \infty }^\infty \left| {h\left[ n \right]} \right| = \mathop \sum \limits_{h = - \infty }^\infty \left| {\cos \left[ n \right]u\left[ n \right]} \right|$

$= \mathop \sum \limits_{h = 0}^\infty \left| {\cos \left[ n \right]} \right|$

Since -1 ≤ cos [n] ≤ 1

|cos [n]| < 1

$\mathop \sum \limits_{n = - \infty }^\infty \left| {h\left[ n \right]} \right| = \mathop \sum \limits_{h = 0}^\infty \left| {\cos \left[ n \right]} \right|$

For every n, the value of cos [n] is finite but the summation is going for n → ∞, which causes the

$\mathop \sum \limits_{h = - \infty }^\infty \left| {h\left[ n \right]} \right| \to \infty$

So, the given system is not stable.