4 Adaptive Autoregressive Spectrum Analysis

The power spectrum of a discrete-time stochastic process can be estimated by assuming that the process can be modeled using a linear process as shown in Fig. 2.12. In this figure, the process $x(n)$ is modeled as the output of the linear filter $H(\omega)$, the input of which is a white noise $u(n)$. In this case, the power spectrum of $x(n)$ is given by

$$S_x(\omega) = \sigma_{u}^{2} \vert H(\omega)\vert^{2},$$ (24)

where $\sigma_{u}^{2}$ is the variance of the white noise $u(n)$ which has a flat spectrum. From eq. (2.24) it is clear that the power spectrum of $x(n)$ can be obtained by estimating the transfer function $H(\omega)$. This can be done by using an adaptive prediction structure. A case of practical interest is when the filter $H(\omega)$ can be modeled as an all-pole autoregressive model with transfer function given by

$$H(\omega) = \frac{1}{1 - \Sigma_{k=1}^{M} a_k e^{-j\omega k}}.$$ (25)

Figure 2.12: Autoregressive process modeling.

In this case, the AR parameters, $\{a_1, a_2, \cdots a_M \}$ can be estimated using an adaptive transversal prediction error filter as shown in Fig. 2.13. The power spectrum function of $x(n)$ can then be calculated from

$$S_x(\omega,n) = \frac{\sigma_{u}^{2}}{\vert 1 - \Sigma_{k=1}^{M} a_k(n) e^{-j \omega k} \vert^2}.$$ (26)

In eq. (2.26), the power spectrum $S_x(\omega,n)$ as well as the autoregressive parameters $a_k; k=1,2,\cdots M$ are considered time varying. This provides a practical procedure for measuring the instantaneous frequency contents of the process $x(n)$ from the autoregressive parameters.

Figure 2.13: Block diagram of the adaptive transversal forward prediction error filter.