3.1 Introduction

A digital filter can be designed in a variety of ways. A practical digital filter implementation is very dependent on the filter structure whether the filter is nonrecursive or recursive. We define the nonrecursive and recursive structure of a filter at the outset.

The output sequence y(n) of a nonrecursive filter is a function only of the past inputs and the present input x(n).

The transfer function of nonrecursive filter is given by, in the z domain,

The structure of a nonrecursive filter of order N is shown in Figure 3.1.

Figure 3.1: Nonrecursive filter of order N

The output sequence y(n) of a recursive filter is a function of the past output, the past inputs and the present input x(n).

The transfer function of a recursive filter in the z domain is,

(we usually normalize a ₀=1)

The structure of a recursive filter is shown in Figure 3.2.

Figure 3.2: Recursive filter of order N

The transfer function (3.4) may be implemented as shown in Figure 3.3. We call the structure a canonical form. It has the least number of delay elements but twice the number of summers.

Figure 3.3: Recursive filter, canonical form

The transfer function can be interpreted as a cascade of several transfer functions where H _i(z) is either a first-order section or a second-order section.

Another variation in implementation is a partial expansion of the transfer function (3.4) (see Figure 3.4). The structure corresponding to the partial expansion is a parallel...