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

Question

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.

Answer 1

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.