Is there relationship between autocorrelations Rx(2) and Rx(1)?

Question

There is a random process, $X:\mathbb{Z}\rightarrow\mathbb{R}$ in discrete-time domain, which is wide-sense stationary with zero mean and autocorrelation function $R_X(\tau)$.

a) What range of values can $R_X(1)$ take? For this, using the relationship $|R_X(\tau)|\leq R_X(0)$, I derived $-1\leq a \leq 1$.

b) If $R_X(0)=1$ and $R_X(1)=a$, show that $2a^2-1\leq R_x(2) \leq 1$.

I can understand its upper bound, but I do not know how I can derive the lower bound $2a^2-1$.

Could anyone help with this?

Answer 1

Some definitions: $$R_{X}(0) = 1$$ $$R_{X}(1) = a$$ $$R_{X}(2) = b$$

I will use the the condition that the autocorrelation be positive semi-definite, and the above definitions for all parts of the problem.

The derivation below is an application of Sylvester's criterion.

Lag 0 $$\begin{vmatrix} 1 \end{vmatrix} \geq 0\hspace{1cm} \checkmark$$

Lag 1 $$\begin{vmatrix} 1 & a\\ a & 1 \end{vmatrix} \geq 0$$ $$1 - a^{2} \geq 0$$ $$a^{2} \leq 1$$ $$-1 \leq a \leq 1\hspace{1cm} \checkmark$$

Lag 2 $$\begin{vmatrix} 1 & a & b \\ a & 1 & a \\ b & a & 1 \end{vmatrix} \geq 0$$ $$1 + a^{2}b + a^{2}b - b^{2} - a^{2} - a^{2} \geq 0$$ $$-b^{2} + 2a^{2}b - 2a^{2} + 1 \geq 0$$ $$b^{2} - 2a^{2}b + \left(2a^{2} - 1\right) \leq 0$$

Solving for the equality first:

$$b = \frac{2a^{2} \pm \sqrt{4a^{4} - 4\left(2a^{2} - 1\right)}}{2} $$ $$b = a^{2} \pm \sqrt{\left(a^{2}-1\right)^{2}}$$ $$b = a^{2} \pm \left(a^{2}-1\right)$$ $$b \in \{2a^{2} - 1, 1\}$$

The solution is valid between these bounds and invalid outside so:

$$2a^{2} - 1 \leq b \leq 1$$

Using our original definition:

$$2a^{2} - 1 \leq R_{X}(2) \leq 1\hspace{1cm} \checkmark$$

Answer 2

Let me give you a counterexample showing that the given limits for $R_X[2]$ in terms of $R_X[0]$ and $R_X[1]$ are not sufficient for guaranteeing the non-negativity of the corresponding power spectrum.

Let's fix $R_X[2]$ at its given lower limit:

$$R_X[2]=2a^2-1$$

Furthermore, let's assume that $R_X[k]=0$ for $|k|>2$. The DC value of the corresponding power spectrum $S_X(\omega)$ is just the sum over all values $R_X[k]$, and is given by

$$S_X(0)=1+2a+2(2a^2-1)=4a^2+2a-1$$

It's straightforward to show that for

$$a\in\left[\frac{-1-\sqrt{5}}{4},\frac{-1+\sqrt{5}}{4}\right]$$

the value of $S_X(0)$ becomes negative, which shouldn't be the case for a power spectrum. Hence, it appears that the given lower bound for $R_X[2]$ is not correct.

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EDIT: As shown in GrapefruitIsAwesome's answer, it appears that the given limits on $R_X[2]$ have been derived from the requirement that the autocorrelation matrix be positive semi-definite. However, for the sequence $R_X[k]$ to be a valid autocorrelation sequence, it is necessary that all Toeplitz matrices with first row $R_X[0],R_X[1],\ldots,R_X[n]$ are positive semi-definite for $n=1,2,\ldots,\infty$.

In the example above I've assumed $R_X[k]=0$ for $|k|>2$, in which case the given limits on $R_X[2]$ are not sufficient, but only necessary. The given limits just guarantee that there exist valid autocorrelation sequences with the first three elements as given. However, not all sequences with those first three elements are valid autocorrelation sequences.