If the spectral density of a WSS process is given by

S(ω) = b(a-|ω| )/a |ω|≤ a

=0 |ω|>a

Find the autocorrelation function of the equation.

**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

The autocorrelation function R(τ) is given by

R(τ)=$F^{-1} {S(ω) }$

=1 ($∫_{-∞}^∞ S(ω) {e^{iτω}} dω$)/2π

=1/2π ($∫_{-a}^a b(a-|ω| )/a {e^{iτω}} dω$)/2π

=2b ($∫_0^a (a-ω) {e^{iτω}} dω$)/2πa

=b/πa ($∫_0^a (a-ω) cosτω dω$)/2πa

=b (a-ω) { $[sin τω/τ-cosτω/{τ^2}]_0^a$}/πa

=b(1-cosaτ)/(πa$τ^2$ )

=(2b$sin^2$⁡aτ/2) /(πa$τ^2$ )

=ab/π×2sin⁡$(aτ/2)/aτ)^2$

R(τ)=absin⁡(aτ/2)/$(aτ/2)^2$2π