Читать книгу Artificial Intelligence and Quantum Computing for Advanced Wireless Networks - Savo G. Glisic - Страница 46

Linear filters: As already indicated so far in this chapter, linear filters have been exploited for the structures of predictors. In general, there are two families of filters: those without feedback, whose output depends only upon current and past input values; and those with feedback, whose output depends upon both input values and past outputs. Such filters are best described by a constant coefficient difference equation, as

(3.48)

where y(k) is the output, e(k) is the input, a_i, i = 1, 2, … , p, are the AR feedback coefficients and b_j, j = 0, 1, … , q, are the moving average (MA) feedforward coefficients. Such a filter is termed an autoregressive moving average (ARMA (p, q)) filter, where p is the order of the autoregressive, or feedback, part of the structure, and q is the order of the MA, or feedforward, element of the structure. Due to the feedback present within this filter, the impulse response – that is, the values of (k), k ≥ 0, when e(k) is a discrete time impulse – is infinite in duration, and therefore such a filter is referred to as an infinite impulse response (IIR) filter.

The general form of Eq. (3.48) is simplified by removing the feedback terms as

(3.49)

Such a filter is called MA (q) and has an FIR that is identical to the parameters b_j, j = 0, 1, … , q. In digital signal processing, therefore, such a filter is called an FIR filter. Similarly, Eq. (3.48) is simplified to yield an autoregressive (AR(p)) filter

(3.50)

which is also an IIR filter. The filter described by Eq. (3.50) is the basis for modeling the speech generating process. The presence of feedback within the AR(p) and ARMA (p, q) filters implies that selection of the a_i, i = 1, 2, … , p, coefficients must be such that the filters are bounded input bounded output (BIBO) stable. The most straightforward way to test stability is to exploit the ‐domain representation of the transfer function of the filter represented by (3.48):

(3.51)

To guarantee stability, the p roots of the denominator polynomial of (z), that is, the values of z for which D(z) = 0, the poles of the transfer function, must lie within the unit circle in the z‐plane, ∣z ∣ < 1.

Nonlinear predictors: If a measurement is assumed to be generated by an ARMA (p, q) model, the optimal conditional mean predictor of the discrete time random signal {y(k)}

(3.52)

is given by

(3.53)

where the residuals ê j = 1, 2, … , q. The feedback present within Eq. (3.53), which is due to the residuals ê (k − j), results from the presence of the MA (q) part of the model for y(k) in Eq. (3.48). No information is available about e(k), and therefore it cannot form part of the prediction. On this basis, the simplest form of nonlinear autoregressive moving average (NARMA (p, q)) model takes the form

(3.54)

where Θ(·) is an unknown differentiable zero‐memory nonlinear function. Notice e(k) is not included within Θ(·) as it is unobservable. The term NARMA (p, q) is adopted to define Eq. (3.54), since except for the (k), the output of an ARMA (p, q) model is simply passed through the zero‐memory nonlinearity Θ(·).

The corresponding NARMA (p, q) predictor is given by

(3.55)

where the residuals ê j = 1, 2, … , q. Equivalently, the simplest form of nonlinear autoregressive (NAR(p)) model is described by

(3.56)

and its associated predictor is

(3.57)

The two predictors are shown together in Figure 3.10, where it is clearly indicated which parts are included in a particular scheme. In other words, feedback is included within the NARMA (p, q) predictor, whereas the NAR(p) predictor is an entirely feedforward structure. In control applications, most generally, NARMA (p, q) models also include also external (exogeneous) inputs, (k − s), s = 1, 2, … , r, giving

Figure 3.10 Nonlinear AR/ARMA predictors.

(3.58)

and referred to as a NARMA with exogenous inputs model, NARMAX (p, q, r), with associated predictor

(3.59)

which again exploits feedback.