Question

The auto correlation of x(t) = e^-atu(t) is ________________

(a) \(\frac{e^{-at}}{a^2} \)

(b) \(\frac{e^{-at}}{2a} \)

(c) \(\frac{e^{-aλ}}{a^2} \)

(d) \(\frac{e^{-aλ}}{2a} \)

The question was asked in exam.

The query is from Concept of Convolution topic in chapter Signal Transmission Through Linear Systems of Signals and Systems

Answer 1

Correct option is (d) \(\frac{e^{-aλ}}{2a} \)

The best explanation: R (λ) = \(\int_{-∞}^∞ x(t)  x(t±λ)  \,dt\)

= \(\int_{-∞}^∞ e^{-at} u(t)  e^{-a(t-λ)}  u(t-λ) \,dt\)

= \(\int_λ^∞ e^{-2at}  e^{aλ} \,dt\)

= \(\frac{e^{aλ}}{-2a}\)[0-e^-2aλ]

= \(\frac{e^{-aλ}}{2a} \).