I have a great confusion with the autocorrelation function of an AR process. Its derivation usually follows in this way (Haykin, 2007):

The difference equation for an AR(M) process, $u(n)$, is

\begin{equation}\label{AR_diff_eq} \sum_{k=0}^M a_ku(n-k) = v(n) \tag{1} \end{equation}

with $a_0 =1$, and $v(n)$ is white noise that satisfies $E[v(n)v(k)] = \sigma_v^2\delta(n-k)$.

Assuming that $u(n)$ is asymptotically stationary, i.e., the roots $\{p_k\}$of the characteristic equation $\sum_{k=0}^M a_kz^{-k} = 0$ are inside the unit circle, if we multiply at both sides of \eqref{AR_diff_eq} by $u(n-l)$ and then we take the expected value, we get

\begin{equation} \sum_{k=0}^M a_kE[u(n-k)u(n-l)] = E[v(n)u(n-l)] \end{equation}

\begin{equation}\label{AR_ACF} \sum_{k=0}^M a_kr(l-k) = E[v(n)u(n-l)] \tag{2} \end{equation}

where $r(k)$ is the autocorrelation function of $u(n)$.

We note that for $l>0$, $E[v(n)u(n-l)] = 0$, since $u(n-l)$ only depends on samples of the white noise up to time $n-l$, and $v(n)$ is decorrelated with those samples. Therefore, \eqref{AR_ACF} becomes the following difference equation:

\begin{equation}\label{ACF_diff_eq} \sum_{k=0}^M a_kr(l-k) = 0, \qquad l>0 \tag{3} \end{equation}

and its solution has the form

\begin{equation}\label{ACF} r(m) = \sum_{k=1}^M C_kp_k^m \tag{4} \end{equation}

where $\{C_k\}$ are constants.

My confusion is with \eqref{ACF}, because it is an expression for the autocorrelation function that is not an even function. Moreover, in the derivation of the Yule-Walker equations, the difference equation in \eqref{ACF_diff_eq} is used, and is also assumed that $r(-m) = r^*(m)$, where the superscript $^*$ represents the complex conjugate operation.

In the the Wikipedia article for an AR process, the autocorrelation function it is presented as an even function, but there is a lack of reference for its derivation.

Any hint to get out of my confusion would be greatly appreciated. Best.

Edition (05/13/16)

I took another look to this question and I realized that in the derivation is assumed that the process is (asymptotically) stationary, and therefore $r(-m) = r(m)$. Using that we could write \eqref{ACF} as

\begin{equation} r(m) = \sum_{k=1}^M C_kp_k^{\lvert m \rvert} \tag{5} \end{equation}

Looking retrospectively, I guess that my source of confusion was that \eqref{ACF} was presented without making any note about the valid values for $m$ and that when the property $r(-m) = r(m)$ is used to derive the Yule-Walker equations, it was not established a clear link between that and \eqref{ACF}.


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