Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 53

Poles on the right-handside are mapped outside the circle, which is the region
of instability although poles lying on the imaginary axis are transformed to
points on the unit circle itself. In summary, the impulse-invariant transfor-
mation causes areas of width 2 π/T to be mapped to the unit circle in the
z -plane.
Writing the z -transform of the digital transfer function as a Laplace trans-
form, we obtain

H A s + j 2 πp
T

∞

H ( z )= 1
T

−∞

H A j 2 πp + ω
T

∞

1
T

= (2.136)

−∞

where z =e sT , ω =Ω T , and Ω is the analog frequency whereas ω the digital
frequency. If the analog filter is bandlimited, the digital filter function takes
the form

T H A j ω

H ( z )= 1 (2.137)

T

where |ω|≤π is the cutoff frequency. At this point we have

H A j ω
T

= 0

(2.138)

for

≥

ω
T

π
T .

2.6.2.2 Bilinear Transformation
Another method for deriving digital filters from analog counterparts is the
bilinear transformation. This method begins with the assumption that the
derivative of the analog signal is obtainable, that is, we have

d y ( t )
d t

= x ( t ) (2.139)

The Laplace transform of this gives us
sY ( s )= X ( s )

Y ( s )
X ( s )

1
s

⇒ = (2.140)

Integrating both sides of Equation 2.140 gives us the difference equation as
follows:

nT

d y ( t )
d t

= y ( nT ) − y [( n − 1) T ]

( n− 1) T

nT

= x ( t ) (2.141)

( n − 1) T