A linear system is characterized by, H(jω) = e

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A linear system is characterized by, H(jω) = e^{– bω3}. The system is physically—

1. unrealizable
2. realizable
3. depends on the value of b
4. None of these

Correct Option: A

Given that H(jω) = e^{– bω³}
According to Paley-Wiener criterion, the given system will be physically realizable, if condition given below is satisfied.

= ∫^∞_{– ∞}	\|In H(jω)\|	· dω < ∞
	1 + ω²

= ∫^∞_{– ∞}	\|In e^{– bω³} \|	dω
	1 + ω²

= ∫^∞_{– ∞}	bω³	dω
	1 + ω²

= b∫^∞_{– ∞}	ω²	dω
	1 + ω²

= b∫^∞_{– ∞}	t	, (put ω² = t, 2ω dω = dt)
	1 + t

= b		∫^∞_-∞		T + 1	–	1		dt
				T + 1		1 + t

= b		∫^∞_-∞ dt - ∫^∞_-∞	1	dt
			1 + t

∵		ω²	is even function
		1 + ω²

= 2[ω – tan– 1ω]^∞₀

= 2		∞ –	π	– 0 + 0
			2

= 2· ∞
= ∞