Page 217 - Mechanical Engineers' Handbook (Volume 2)

206 Signal Processing

4 STABILITY AND PHASE ANALYSIS
4.1 Stability Analysis
Consider a transfer function

1
H(s) (62)
s p
where the pole p is a complex number + j . The inverse Laplace transform of this function
t
is e [cos( t)+ j sin( t)]. This function is bounded as t → if and only if 0. From
this, we can determine the stability of a function by inspecting the real components of all
poles of a given transfer function. The procedure for a rational function (a ratio of polyno-
mials) would be to factor the polynomials in the denominator and inspect to ensure that the
real components to all of the poles are less than zero. Suppose
a as as m
H(s) 0 1 m
b bs bs 2
n
0
1
(s
)(s
) (s
)
0 1 m (63)
(s p )(s p ) (s p )
0
n
1
In a similar way, by inspection of the S-to-z transformation z e , we see that the entire
s
left half of the plane in the S-domain maps inside the unit circle in the z-domain. For this
reason, we analyze the stability of systems in the z-domain by inspecting the poles of the
transfer function. The system is stable if the norm of all poles is less than 1.
4.2 Phase Analysis
While processing the data in real time, our filters must act on the signal history. For this
reason, there will always be some delay in the output of our process. Worse, certain filters
will delay some frequency components by more or less than other frequency components.
This results in a phase distortion of the filter. For a certain class of FIR filters, it is possible
to design filters that shift each frequency component by a time delay in proportion to the
frequency. In this way, all frequency components are shifted by an equal time delay. Though
it is possible to design certain non-real-time, noncausal IIR filters that are phase shift dis-
tortionless, in general, IIR filters will produce some phase shift distortion. We can determine
the actual phase shift for each frequency component by computing

Table 3 Comparison of FIR and IIR Characteristics

FIR IIR
Run time efficiency Less efficient; requires high-order Higher efficiency; usually possible to
filter. achieve a desired design
specification in fewer
computations
Stability Always stable Stable if all poles are inside the unit
circle
Phase shift distortion Can be designed to be phase Generally distorts phase
shift distortionless
Ease of design Simpler design process, usually Design is more complex, involving
involving Fourier transforms special functions or solving
or solving linear systems nonlinear systems