Systems, Sampling and Quantization
13
conformal mapping
band of interest of the discrete-time system. However, if the frequency response
of the continuous-time system is sufficiently close to zero in high frequency, the
aliasing can be neglected and the resulting discrete-time system turns out to be
a good approximation of the continuous-time template.
Often, the continuous-time impulse response is derived from a decomposition
of the transfer function of a system into simple fractions. Namely, the transfer
function of a continuous-time system can be decomposed
into a sum of terms
such as
H
s
(s) =
a
s - s
a
,
(32)
which are given by impulse responses such as
h
s
(t) = ae
s
a
t
1(t) ,
(33)
where 1(t) is the ideal step function, or Heaviside function, which is zero for
negative (anticausal) time instants. Sampling the (33) we produce the discrete-
time response
h(n) = T a e
s
a
T n
1(n) ,
(34)
whose transfer function in z is
H(z) =
T a
1 - e
s
a
T
z
-1
.
(35)
By comparing (35) and (32) it is clear what is the kind of operation that we
should apply to the s-domain transfer function in order to obtain the z-domain
transfer function relative to the impulse response sampled with period T .
It is important to recognize that the impulse-response method preserves the
stability of the system, since each pole of the left s hemiplane is matched with a
pole that stays within the unit circle of the z plane, and vice versa. However, this
kind of transformation can not be considered a conformal mapping, since not
all the points of the s plane are coupled to points of the z plane by a relation
z = e
sT
. An important feature of the impulse-invariance method is that, being
based on sampling, it is a linear transformation that preserves the shape of the
frequency response of the continuous-time system, at least where aliasing can
be neglected.
It is clear that the method of the impulse invariance can be used when the
continuous-time reference model is a lowpass or a bandpass filter (see sec. 2 for
a treatment of filters). If the template is an high-pass filtering block the method
is not applicable because of aliasing.
1.5.2
Bilinear Transformation
An alternative approach to using the impulse invariance to discretize continuous
systems is given by the bilinear transformation, a conformal map that creates
8
This holds for simple distinct poles. The reader might try to extend the decomposition to
the case of coincident double poles.
9
To be convinced of that, consider a second order continuous-time transfer function with
simple poles and a zero and convert it with the method of the impulse invariance. Verify that
the zero does not follow the same transformation that the poles are subject to.