Determining the autocorrelation sequence from an AR model

I have the following equation:

$$x(n)=\frac{14}{24}x(n-1)+\frac{9}{24}x(n-2)-\frac{1}{24}x(n-3)+w(n)$$ where, $w(n)$ is a stationary white noise process with variance $\sigma^2_w$

Now, I want to determine the auto correlation sequence $\gamma_{xx}(m)$, for $m = 0,1,\ldots,5$

Now, I know that the autocorrelation function for AR process which is given below:

$$\gamma_{xx}(m)= \begin{cases}\displaystyle-\sum_{k=1}^p a_k\gamma_{xx}(m-k), & m > 0\\ \displaystyle -\sum_{k=1}^p a_k\gamma_{xx}(m-k) + \sigma_w^2, & m = 0\\ \gamma_{xx}^*(-m), & m < 0\\ \end{cases}$$

Here, I already found the coefficients of ${a_k}$ which are $a_1=-\frac{14}{24}, a_2=-\frac{9}{24}, a_3=\frac{1}{24}$.

Now, I don't get, how can I calculate $\gamma_{xx}(0)$ to $\gamma_{xx}(5)$ recursively?

Please someone tell me how can I solve $\gamma_{xx}(0)$ and $\gamma_{xx}(1)$? Then I can do the the rest.

Looking at your definition of $\gamma_{xx}(m)$, it seems that you have 4 equations with 4 unknown (for $m=0,\ldots,3$):
Solving the above system of equations is straight forward, and after that you can easily obtain $\gamma_{xx}(m)$ for $m=4,5$
• Hi: I'm not sure where you got the formula for $\gamma_{xx}(0)$ but the zero lag autocorrelation for any stochastic process is 1.0. – mark leeds Jan 23 '18 at 16:10