{Xn}^=∞ n=-∞ is an independent and identically distributed (i.i.d) random process with Xn equally likely to be +1 or –1. {Yn}^n=∞ n=-∞ is another random process obtained as Yn = Xn + 0.5 X n-1. The autocorrelation function of {Yn}^n=∞ n=-∞ , denoted by RY[k], is

Question

[latex]\left\{X_{n}\right\}_{n=-\infty}^{n=\infty}[/latex] is an independent and identically distributed (i.i.d) random process with [latex]X_{n}[/latex] equally likely to be +1 or [latex]-1 \cdot\left\{Y_{n}\right\}_{n=-\infty}^{n=\infty}[/latex] is another random process obtained as [latex]Y_{n}=X_{n}+0.5 X_{n-1}[/latex]. The autocorrelation function of [latex]\left\{Y_{n}\right\}_{n=-\infty}^{n=\infty}[/latex] , denoted by [latex]R_{Y}[k][/latex], is

Accepted Answer

[latex]\begin{gathered}R_{y}(k)=R_{y}(n, n+k) \=E[Y[n] \cdot Y[n+k)] \Y[n]=X[n]+0.5 x[n-1] \R_{y}(k)=E[(X[n]+0.5 X[n-1]) \cdot(x[n+k] \+0.5 x[n+k-1])] \=E[x[n] x[n+k]+0.5 x[n-1] x[n+k] \+0.5 x[n] x[n+k-1]+0.25 x[n-1] x[n+k-1]\end{gathered}[/latex]

[latex]\begin{aligned}&=R_{x}(k)+0.5 R_{x}[k+1]+0.5 R_{x}(k-1)+0.25 R_{x}(k) \&R_{y}(k)=1.25 R_{x}(k)+0.5 R_{x}(k+1)+0.5 R_{x}(k-1) \&R_{x}(k)=E[x[n] x[n+k]]\end{aligned}[/latex]

for [latex]k=0 E\left[x^{2}[n] ight]=I^{2} \frac{1}{2}+(-1)^{2} \frac{1}{2}=1[/latex]

For k ≠ 0

[latex]\begin{aligned}&R_{x}(k)=E[x[n]] \cdot E[X(n+k)]=0 \&\operatorname{So} R_{y}(k)=1.25 R_{x}(k)+0.5 R_{x}(k-1)+0.5 R_{x}(k+1) \end{aligned}[/latex] .

[latex]\begin{array}{l} ext { For } R_{y}(0)=1.25 R_{x}(0)+0.5 earrow R_{x}^{\circ}(-1)+0.5 earrow R_{x}^{\circ}(+1) \\quad=1.25 \\begin{aligned}R_{y}(1) &=1.25 earrow \hat{R}_{x}^{\circ}(1)+0.5 R_{x}(0)+0.5 earrow R_{x}^{\circ}(+2) \&=0.5\end{aligned} \\begin{aligned}R_{y}(-1) &=1.25 earrow \vec{R}_{x}^{\circ}(-1)+0.5 earrow R_{x}^{\circ}(-2)+0.5 R_{x}(0) \ &=0.5\end{aligned} \ R_{y}(2)=1.25 earrow R_{x}^{\circ}(2)=0.5 earrow \hat{R}_{x}^{\circ}(1)+0.5 earrow R_{x}^{\circ}(3) \end{array}[/latex]

So all values of [latex]R_{y}(k)[/latex] are zero except at k =0, 1 and –1.
So

Hence, the correct option is (B).

Question 2.1.3: {Xn}^=∞ n=-∞ is an independent and identically distributed (...

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