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Circuits Syst Signal Process (2018) 37:4336–4362 4341
(−2
ρ
,
ρ
2
)
(2
ρ
,
ρ
2
)
(0, −
ρ
2
)
a
n1
2h
2h
Stability triangle
a
n2
Trust quadrate
(a
n1
( ), a
n2
( ))
Fig. 1 The stability triangle for the filter 1/ A
n
(z, a
n
) and the trust quadrate for δa
n
at a
n
()
E(ω, a, b) ≡ H (e
jω
, a, b) − D(ω) =
B(e
jω
, b)
Q+sign(J )
q=1
˜
A
q
e
jω
,
˜
a
q
− D(ω). (3b)
For the filter H (z, a, b), a well-known necessary and sufficient stability condition
is that all poles of the filter or all zeros of its denominator lies inside the unit circle.
With the representation (2c) of the filter denominator, the stability domain S of the
filter H (z, a, b) can be described by S =
˜
S
1
×
˜
S
2
×···×
˜
S
Q+sign(J )
, where
˜
S
q
={
˜
a
q
|
all zeros of
˜
A
q
(z,
˜
a
q
) lie inside the unit circle} and the symbol “×” is the Cartesian
product operator. If the filter H (z, a, b) is required to have certain robustness on its
stability, say, all poles are required to lie inside the circle of radius ρ<1.0, the stability
domain becomes
S(ρ) =
˜
S
1
(ρ) ×
˜
S
2
(ρ) ×···×
˜
S
Q+sign(J )
(ρ) with (4a)
˜
S
q
(ρ) =
S
(q−1)P+1
(ρ) × S
(q−1)P+2
(ρ) ×···×S
qP
(ρ), q ≤ Q,
S
QP+1
(ρ) × S
QP+2
(ρ) ×···×S
QP+J
(ρ), q = Q + 1,
(4b)
and S
n
(ρ) ={a
n
∈ R
2
|both zeros z
kn
of A
n
(z, a
n
) satisfy |z
kn
| <ρ,k = 1, 2} or
S
n
(ρ) =
a
n
∈ R
2
|
ρa
n1
|
− a
n2
≤ ρ
2
, a
n2
≤ ρ
2
, (4c)
where S
n
(ρ) in (4c) is the stability triangle of the filter 1/ A
n
(z, a
n
) with pole radius
ρ<1.0[15,18,19], as shown in Fig. 1.
Remark 1 We have implicitly assumed that the IIR digital filter will be implemented
with infinite word-length arithmetic. It is pointed out that, however, if the filter is
implemented with finite word-length arithmetic, its stability issue will be much more
involved since the filter may become a nonlinear one, and some recent results can be
found in [2,3,21] for this case.