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The spectral density of random process, where
Px(τ) = e– 2a|τ| is—
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2a a2 + ω2 -
4a 4a2 + ω2 -
a a2 + ω2 - None of these
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Correct Option: B
We know that spectral density is nothing but a Fourier transform of auto correlation function. i.e.
| Sxx(ω) = |  | ∞ | Rxx(τ) e–jωτ.dτ | -∞ |
| = | | ∞ | e–2a(τ) e–jωτ.dτ | -∞ |
| = | | 0 | e2a(τ) e–jωτ.dτ + | | 0 | e–2a(τ) e–jωτ.dτ | -∞ | -∞ |
| = |  |  | 0 | + | | | -∞ | ||
| 2a - jω | -∞ | - (2a + jω) | 0 |
| = | + | 2a – jω | 2a + jω |
| = | 4a2 + ω2 |
Hence, alternative (B) is the correct choice.
