AR models and PARCOR coefficients

Let $\{s(0),s(1),\cdots,s(t-1)\}$ be a sequence of signals. Then forward autoregressive (AR) model of order m is given by

$$s(l)=\sum_{i=1}^{m}a^m(i)s(l-i) + \epsilon_f^m(l),$$ (1)

where $\{a^m(i)\}_{i=1}^{m}$ and $\epsilon_f^m(l)$ are forward AR coefficients and forward prediction error, respectively. Similarly backward AR model of order m is given by

$$s(l-m-1)=\sum_{i=1}^{m}b^m(i)s(l-i)+\epsilon_b^m(l),$$ (2)

where $\{b^m(i)\}_{i=1}^{m}$ and $\epsilon_b^m(l)$ are backward AR coefficients and backward prediction error, respectively. The AR coefficients $\{a^m(i)\}_{i=1}^{m}$ or $\{b^m(i)\}_{i=1}^{m}$ can be determined so that the mean squares error is minimized.

PARtial autoCORelation (PARCOR) coefficients are often used in speech signal processing and are often more useful than AR coefficients $\{a^m(i)\}_{i=1}^{m}$ or $\{b^m(i)\}_{i=1}^{m}$. PARCOR coefficient P^m of order m is defined as a correlation coefficient between forward and backward prediction errors in the autoregressive model of order m - 1. Namely,

$$P^m = \frac{\sum_{i=m}^{t-1}\epsilon_f^{m-1}(i)\epsilon_b^{m-... ...{t-1} \{ \epsilon_f^{m-1}(i)\}^2 \{ \epsilon_b^{m-1}(i)\}^2}}.$$ (3)

It is well known that the PARCOR coefficients P^m is the same as the AR coefficient a^m(m) or b^m(m) of AR models of order m. These coefficients are calculated from autocorrelations of the sequence of signals.

There is a fast recursive algorithm for calculating AR and PARCOR coefficients. The algorithm can compute all AR and PARCOR coefficients of orders 1 through m with O(m²) [11].

For online computation of PARCOR coefficients, we can use the recursive formula with forgetting factor $0 < \alpha \le 1$ to estimate the autocorrelations r(l) as

$$r(l)^{(t+1)} \leftarrow (1 - \alpha) r(l)^{(t)} + \alpha s(t+1) s(t+1).$$ (4)

From the estimated autocorrelations, we can calculate the PARCOR coefficients with O(m²) computation.