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there is a problem that I should specify a real-valued random process $Y[n]$ such that the autocorrelation function $R_Y[k]$ satisfies $$R_Y[0]=3+u,\ R_Y[1]=R_Y[-1]=-2+u,\ \text{and}\ R_Y[k]=u, |k|>1.$$

First subproblem is to find a feasible set of u with |u|>0. I found region $-\frac{1}{2}\leq u$.

The main problem is to specify a random process $Y[n]$.

Considering that $R_Y[k]=u$ for $|k|>1$, I thought $Y[n]$ is likely to have a DC value. Besides, the autocorrelation function has unique values for k=0,1, I set $$Y[n]=a\,w[n]+b\,w[n-1]+c$$ for constant $a,b,c$ and $w[n]$ is white Gaussian noise with zero mean and variance 1, and has autocorrelation $R_w[k]=\delta[k]$. But the final results becomes $a,b$ are imaginary numbers, which cause complex-valued $Y[n]$.

Is there any way I can get a random process? The problems say a) specify a random process with the given autocorrelation function (that is, specify the stochastic generation mechanism for the process). b) Is there a unique random process with the given autocorrelation function? If the answer is no, identify possible sources of difference between the various random processes that all have this given autocorrelation function.

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  • $\begingroup$ Homework? Does the book or your course notes have any mention of processes with autocorrelation $R[k] = u$? How about processes with any autocorrelation function that does not go to zero as $k \to \infty$? $\endgroup$
    – TimWescott
    Jan 14, 2022 at 16:16
  • $\begingroup$ @TimWescott This is from previous exam, but unluckily I do not have a solution.. I've never seen non-diminishing autocorrelation function like this. I do not know what is the referred textbook, sorry. $\endgroup$
    – Junho
    Jan 14, 2022 at 21:50

1 Answer 1

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If I read your problem correctly, $R_Y[k]$ is given by

$$R_Y[k]=3\delta[k]-2\big(\delta[k-1]+\delta[k+1]\big)+u\tag{1}$$

The power spectrum is the DTFT of $R_Y[k]$, which is given by

$$S_Y(\omega)=3-4\cos(\omega)+2\pi u\,\delta(\omega),\qquad -\pi\le\omega<\pi\tag{2}$$

From $(2)$ it is obvious that $S_Y(\omega)\ge 0$ cannot be satisfied, regardless of the value of $u$. Hence, $R_Y[k]$ is not a valid autocorrelation function, and, consequently, there is no solution to your problem.

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  • $\begingroup$ Thank you for the answer. Yes, your statement totally make sense. I might need to answer for the problem that such process cannot exist. $\endgroup$
    – Junho
    Jan 14, 2022 at 21:46

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