Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 54

This can be approximated using the standard trapezoidal rule

y ( nT ) − y [( n − 1) T ]= T

2 ( x ( nT )+ x [( n − 1) T ])

(2.142)

This analog difference equation can now be digitized to hold for n discrete
samples:

y ( n ) − y ( n − 1) = T

2 [ x ( n )+ x ( n − 1)]

(2.143)

which can be transformed to

1+ z − 1
1

H ( z )= Y ( z )

X ( z ) = T

(2.144)

z − 1

2 −

Comparing Equations 2.140 and 2.144 gives us the approximation

2
T

1 z − 1

1+ z − 1

−

s = (2.145)

Alternatively, rearranging Equation 2.145 yields the relationship

z = 1+( T/ 2) s
1

(2.146)

− ( T/ 2) s

To design a digital filter from the analog version, we therefore have to replace
the s of the analog Laplace transfer function using Equation 2.145 to obtain the
digital transfer function in z -transform. Suppose s = a + bj , it can be seen that
if

=1 if a = 0. This is inter-
preted as previously where analog poles are mapped to the digital poles within,
outside, and on the unit circle. The bilinear transformation is preferable to the
previous method because it does not suffer from anti-aliasing effects, though it
sometimes results in oscillatory behavior in the resulting digital design. This
method is frequently used to derive higher-order filters.

| z | > 1if a> 0 and | z | < 1if a< 0. Furthermore, | z |

2.6.3 Finite Impulse Response Filters
In FIR filters, the output is determined using only the current and previous
input samples. The output then has the following form:

M

b j
a 0 x ( n

y ( n )= − j )

(2.147)

j =0

As with IIR digital filters, several configurations of FIR filters are possible as
seen in the following sections.