IIR filter design - Continuous and Discrete Time Signals and Systems - page 715

(a) finite-difference discretization of differential equations;
(b) mapping poles and zeros from the s-plane to the z-plane;
(c) impulse invariance method;
(d) bilinear transformation.

The finite-difference discretization of differential equations is a straightforward
method to derive difference equation representations for digital filters. First,
the s-transfer functions, obtained by using the CT filter design techniques, are
used to calculate the input-output relationship of the equivalent CT filter. These
relationships are generally in the form of linear, constant-coefficient differential
equations, and are discretized to obtain difference equations that represent the
input-output relationships of the designed DT filters.
In the second method, referred to as the matched z-transform technique, the
s-plane poles and zeros of a designed CT filter are mapped to the z-plane. The
s-plane poles and zeros are then used to derive the transfer function H ( z ) of the
digital IIR filter.
The impulse invariance method samples the impulse response h ( t )ofan
LTIC system to derive the impulse response h [ k ] of the corresponding LTID
system. Finally, the bilinear transformation provides a one-to-one, non-linear
mapping from the s-plane to the z-plane. The impulse invariance and bilin-
ear transformations are the focus of this chapter. In Section 16.2, we cover
the impulse invariance method followed by the bilinear transformation, in
Section 16.3.

16.2 Impulse invar iance

To derive the impulse invariance transformation, we approximate the impulse
response h ( t ) of a CT filter with its sampled representation,

∞ ∞

h ( t ) ≈ h ( t ) δ ( t − nT ) = h ( nT ) δ ( t − nT ) , (16.3)

n =−∞ n =−∞

obtained by sampling h ( t ) with an impulse train δ ( t - nT ). Clearly, the

approximation in Eq. (16.3) improves as the sampling interval T → 0. The DT
impulse response h [ k ] of the equivalent IIR filter is obtained from the samples
h ( kT ) and is given by

∞

h [ k ] = h ( kT ) = h ( nT ) δ ( k − n ) . (16.4)

n =−∞

Comparing the expressions for the Laplace transform of Eq. (16.3) given by

∞

h ( nT )e − nTs

Laplace transform H ( s ) = (16.5)

n =−∞