Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 50

x ( n ) b 0 y ( n )

+

z − 1

z − 1

b 1

− a 0

z − 1

z − 1

b 2 − a 1

z − 1

z − 1

a N

−

b M

FIGURE 2.14
A direct IIR filter configuration.

Applying the z -transform and rearranging gives the filter transfer function

j =0 b j x ( n

M

− j )

H ( z )= Y ( z )
X ( z )

=

k =1

N

1+ a k y ( n − k )

b 0 + b 1 z − 1 + + b M z −M

···

= (2.130)

1+ a 1 z − 1 + ··· + a N z −N

In the time domain this may be written as the difference equation

y ( n )= − a 1 y ( n − 1) − a 2 y ( n − 2) −···− a M y ( n − M )+ b 0 x ( n )

+ b 1 x ( n − 1) + b N x ( n − M )

2.6.1.2 Canonical IIR Filter

This filter can be implemented as a system of difference equations where the
internal state of the system, s i can be estimated for a given input x ( n ) from
previous internal states. This may be written as

s ( n )= x ( n ) − a 1 s ( n − 1) − a 2 s ( n − 2) −···− a N s ( n − N )

The output in this case is the summation of the history of internal state values
written as

y ( n )= b 0 s ( n )+ b 1 s ( n − 1) + ··· + b N s ( n − N )