# Specify a random process such that $R_Y[0]=3+u$, $R_Y[1]=-2+u$, and $R_Y[k]=u$ otherwise

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.

• 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 $k \to \infty$? Commented Jan 14, 2022 at 16:16
• @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. Commented Jan 14, 2022 at 21:50

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.