If the spectral density of a WSS process is given by

S(ω) $=\frac{b(a-|ω|)}{a(ω)}$       |ω|≤ $\alpha$

=0             |ω|>$\alpha$

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 5M

Year: May 2015

The autocorrelation function R$(τ)$ is given by

$$R(τ)=F^{(-1)} \{S(ω)\}$$

$$=\frac1{2π} ∫_{-∞}^∞S(ω) e^{iτω} dω$$

$$ =\frac1{2π} ∫_{-a}^a\frac{b(a-|ω|)}{a} e^{iτω} dω$$

$$=\frac{1}{2π}×\frac{2b}a ∫_0^a(a-ω)e^{iτω} dω$$

$$=\frac{b}{πa} ∫_0^a(a-ω)cosτωdω$$

$$=\frac{b}{πa} \left[(a-ω) [sin⁡\frac{τω}τ-\frac{cosτω}{τ^2} \right]_0^a$$

$$=\frac{b}{(πaτ^2 )}(1-cosaτ)$$

$$=\frac{b}{(πaτ^2 )}×2 sin^2⁡\frac{aτ}2$$

$$ =\frac{ab}π×2sin⁡\left(\frac{(aτ/2)}{aτ}\right)^2$$

$$R(τ)=\frac{ab}{2π}×sin⁡\left(\frac{(aτ/2)}{(aτ/2)}\right)^2$$