Signals and Systems - Electrical Engineering

690 C H A P T E R 11: Introduction to the Design of Discrete Filters

FIGURE 11.25

Block diagrams of different components

used to realize discrete filters: (a) delay,

(b) constant multiplier, and (c) adder.

z−^1

x[n] x[n−1]

(a)

y[n]

x[n] x[n]+y[n]

+

(c)

x[n] α x[n]

(b)

α

In choosing a structure over another to realize a filter, two factors to consider are:

n Computational complexity, which relates to the number of operations (mainly multiplications

and additions), but more importantly to the number of delays used. The aim is to reduce to a

minimum the number of delays in the structure.

n Quantization effectsor the representation of filter parameters using finite-length registers. The aim

is to minimize quantization effects on parameters and on operations.

We will consider here the computational complexity of the structures seeking to obtain mini-

mal realizations—that is, to optimize the number of delays used. The quantization effects are not

considered.

11.6.1 Realization of IIR Filters

The structures commonly used to realize IIR filters are:

n Direct forms I and II

n Cascade

n Parallel

The direct forms represent the difference equation resulting from the transfer function of the IIR filter

while attempting to minimize the number of delays. The cascade and parallel structures are based on

the product or sum of first- and second-order filters to express the filter transfer function, which are

in turn implemented using a direct form.

Direct Form Realizations

Given the transfer function of an IIR filter

H(z)=

Y(z)

X(z)

=

∑M− 1

k= 0 bkz

−k

1 +

∑N− 1

k= 1 akz

−k

(11.62)

whereY(z)andX(z)are the Z-transforms of the outputy[n] and the inputx[n], the input–output

relationship is given by the difference equation

y[n]=−

N∑− 1

k= 1

aky[n−k]+

M∑− 1

k= 0

bkx[n−k] (11.63)

The direct forms attempt to realize this equation with no more thanN−1 delays.