ch09-p372536.tex 29/3/2007 14: 40 Page 170
170 9 Quantization and Two’s Complement Arithmetic in Digital Filters
1
0.8
0.6
0.4
0.2
0
x
1
(k)
0.2
0.4
0.6
0.8
1
(a)
0 50 100
Clock cycle k
150 200
1
0.8
0.6
0.4
0.2
0
x
2
(k)
0.2
0.4
0.6
0.8
1
(b)
0 50 100
Clock cycle k
150 200
1
0.8
0.6
0.4
0.2
0
s
1
(k)
0.2
0.4
0.6
0.8
1
(c)
0 50 100
Clock cycle k
150 200
Figure 9.18 State and symbolic responses of a high bit second order digital ﬁlter associated with
two’s complement arithmetic when a
=0, b =−2andx(0) =[0.2137 0.0280]
T
. (a) State variable
x
1
(k); (b) State variable x
2
(k); (c) Symbolic sequence s(k).
Solution:
The state and symbolic responses of the system is shown in Figure 9.18. As can
be seen from Figure 9.18, even though the system matrix is unstable, both the
state and symbolic responses become zero after certain number of iterations.
SUMMARY
In this chapter we have looked at the nonlinear behaviors of digital ﬁlters asso-
ciated with both quantization and two’s complement arithmetic. A ﬁnite state
machine may exhibit a near chaotic behavior. Evenfor the same ﬁlter parameters
and initial conditions, its corresponding inﬁnite state machine exhibits linear or
limit cycle behaviors. Also, for some ﬁlter parameters in the extended bound-
aries of the stability triangle, the state vector of a high bit digital ﬁlter associated
with two’s complement arithmetic will converge to a quasi periodic orbit after
a number of iterations no matter what the initial conditions are. Hence, a new
trajectory pattern, which looks like a rotated letter ‘X’, is exhibited on the phase
plane. The center of the rotated letter is located at the origin, and the slopes
of the ‘straight lines’ of the rotated letter are equal to the values of the pole
locations. Moreover, when all the ﬁlter parameters are even numbers, no matter
what initial conditions, order and the stability of the system matrix of the system

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