Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 175

FIGURE 6.8
Mapping between s-plane and z-plane.
If a > 0, this means jzj ¼ e aT < 1. Then the left-hand side half plane (LHHP) of the s-plane
is mapped to the inside of the unit circle of the z-plane. When a ¼ 0, this causes jzj ¼ e aT ¼ 1.
Thus the ju axis of the s-plane is mapped on the unit circle of the z-plane, as shown in Figure 6.8 .
Obviously, the right-hand half plane (RHHP) of the s-plane is mapped to the outside of the unit
cycle in the z-plane. A stable system means that for a given bounded input, the system output must
be bounded. Similar to the analog system, the digital system requires that all poles plotted on the
z-plane must be inside the unit circle. We summarize the rules for determining the stability of
a DSP system as follows:
1. If the outmost pole(s) of the z-transfer function HðzÞ describing the DSP system is (are) inside the
unit circle on the z-plane pole-zero plot, then the system is stable.
2. If the outmost pole(s) of the z-transfer function HðzÞ is (are) outside the unit circle on the z-plane
pole-zero plot, the system is unstable.
3. If the outmost pole(s) is (are) first-order pole(s) of the z-transfer function HðzÞ and on the unit circle
on the z-plane pole-zero plot, then the system is marginally stable.
4. If the outmost pole(s) is (are) multiple-order pole(s) of the z-transfer function HðzÞ and on the unit
circle on the z-plane pole-zero plot, then the system is unstable.
5. The zeros do not affect the system stability.
Notice that the following facts apply to a stable system (bounded-in/bounded-out [BIBO] stability
discussed in Chapter 3):
1. If the input to the system is bounded, then the output of the system will also be bounded, or the
impulse response of the system will go to zero in a finite number of steps.
2. An unstable system is one where the output of the system will grow without bound due to any
bounded input, initial condition, or noise, or the impulse response will grow without bound.
3. The impulse response of a marginally stable system stays at a constant level or oscillates between
two finite values.
Examples illustrating these rules are shown in Figure 6.9 .