IIR filter design - Continuous and Discrete Time Signals and Systems - page 716

Fig. 16.1. Impulse invariance
transformation from the s-plane
(a) to the z-plane (b).

Im{s}

Im{z}

p

T

Re{s}

Re{z}

(1, 0)

p

−

T

(a) (b)

and the z-transform of Eq. (16.4) given by

H ( z ) = ∞ h ( nT ) z − n , z-transform (16.6)

k =−∞

we note that the two expressions are equal provided

z = e Ts . (16.7)

In terms of real and imaginary components of s = σ + j ω , Eq. (16.7) can be
expressed as follows:

z = e σ T e j ω T . (16.8)

Equation (16.7) provides a mapping between the DT variable z and the CT
variable s . The mapping, commonly referred to as the impulse invariance trans-
formation, is illustrated in Fig. 16.1, where we observe that the s-plane region

Re s =σ< 0 and Im s =ω <π/ T ,

shown as the shaded region, in Fig. 16.1(a) maps into the interior of the unit
circle z < 1 shown in Fig 16.1(b). Equations (16.7) and (16.8) can also be
used to derive the following observations.

Right-half s-plane Re s > 0 Taking the absolute value of Eq. (16.8) yields

z = e σ T e j ω T = e σ T .

(16.9)

In the right-half s-plane, Re { s } = σ> 0 , resulting in z > 1 . Therefore, the
right-half s-plane is mapped to the exterior of the unit circle.

Origin s = 0 Substituting s = 0 into Eq. (16.7) yields z = 1 . The origin s = 0
in the s-plane is therefore mapped to the coordinate (1, 0) in the z-plane.

Imaginary axis Re s = 0 Taking the absolute value of Eq. (16.8) and substi-
tuting Re { s =σ = 0 yields z = 1. The imaginary axis Re { s }= 0 is therefore
mapped on to the unit circle z = 1 .