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A random process is defined by X(t) = 10cos(100t+ θ) where θ is uniformly distributed in (0,2π). Verify whether X(t) WSS random Process and correlation ergodic.

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

Marks: 10M

Year: Dec 2014

1 Answer
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To prove X(t) is WSS random process

i.e. E(X(t))=μ

& E(X(t)×X(tτ))=R(τ)

Given: X(t) = 10 cos (100t+6)

∴ E(X(t))=10E(cos⁡(100t+θ))

=10E(cos100t.cosθ-sin100t.sinθ)

E(X(t)) =10[cos100t×E(cosθ)-sin100t×E(sinθ)]--- (1)

Given θ is uniformly distributed ∴ f_θ (θ)=\frac{1}{2π}

∴ E(cosθ)=∫_0^{2π}cosθ.f_θ (θ)dθ

=∫_0^2πcosθ.\frac1{2π} dθ

E(cosθ)=0 -------(2)

Similarly

E(sinθ)=∫_{0}^{2π}sinθ.f_θ (θ)dθ

=∫_0^{2π}sinθ.\frac1{2π} dθ

E(sinθ)=0---(3)

From eqn(1), eqn(2) and eqn(3) we get

∴ E(X(t))=0

R(τ)=E(X(t)×X(t+τ)) \\ =E(10 cos⁡(100t+θ).10cos⁡(100t+100τ+θ))\\ =\frac{100}2 E[2 cos⁡(100t+θ).cos⁡(100t+100τ+θ)]\\ =50E[cos⁡(200t+100τ+θ)+cos⁡(100τ))] =50[∫_0^{2π}cos⁡(200t+100τ+θ).\frac1{2π} dθ+ ∫_0^{2π}cos⁡(100τ).\frac1{2π} dθ]

R(τ)=50cos⁡(100τ)

Correlation Ergodic

A stationary random process is said to be correlation ergodic if the Time average tends to the ensemble average .i.e.,

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(\bar{R_T }) =\frac1{2T} ∫_{-T}^T X(t) X (t+τ) dt

(\bar{R_T} ) =\frac1{2T} ∫_{-T}^T100 cos⁡(100t+θ)cos⁡(100t+100τ+θ) dt

(\bar{R_T} ) =\frac{25}T ∫_{-T}^Tcos⁡(200t+100τ+θ)+cos⁡(100τ) dt

(\bar{R_T}) =\frac{25}T ∫_{-T}^Tcos⁡(200t+100τ+θ)dt+\frac{25}T ∫_{-T}^Tcos⁡(100τ) dt

(\bar{R_T} ) =\frac{25}T ∫_{-T}^Tcos⁡(200t+100τ+θ)dt+50cos⁡(100τ) dt

Now \lim_{T→∞}⁡R_T=50cos⁡{100τ}=R(τ)

∴ {X(t)} is correlation ergodic.

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