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autocorrelation function of the equation

If the spectral density of a WSS process is given by

S(ω) =b(a|ω|)a(ω)       |ω|≤ α

=0             |ω|>α

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

Marks: 5M

Year: May 2015

1 Answer
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The autocorrelation function R(τ) is given by

R(τ)=F(1){S(ω)}

=12πS(ω)eiτωdω

=12πaab(a|ω|)aeiτωdω

=12π×2baa0(aω)eiτωdω

=bπaa0(aω)cosτωdω

=bπa[(aω)[sinτωτcosτωτ2]a0

=b(πaτ2)(1cosaτ)

=b(πaτ2)×2sin2aτ2

=abπ×2sin((aτ/2)aτ)2

R(τ)=ab2π×sin((aτ/2)(aτ/2))2

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