On the root distribution of general polynomials with respect to the unit circle

SIGNAL

PROCESSING Signal Processing 53 (1996) 75-82

On the root distribution of general polynomials with respect to the unit circle Messaoud Benidir* Lahoratoire des Siynaux et Systtimes, Supelec, University

of‘ Paris-&d, Plateau de Moulon, 91192

Received 14 January 1994; revised 15 September

Gf~sur- Yvette, France

1995

Abstract A modification and an extension of Schur-Cohn’s algorithm and Jury’s table are presented. The new versions of the classical algorithms allow us to associate a sequence of coefficients k, to every polynomial P. We establish that the number of zeros of P outside the unit circle equals the number of k, satisfying lkjl > 1. A simple expression for the number of zeros of P on the unit circle is also established. An extension of the modified algorithm which introduces an arbitrary parameter allows us to study the critical situations: 1 - 8 d Ik,I d 1 + 8, where E is an arbitrary small positive number. Zusammenfassung Es wird eine Modifikation und eine Erweiterung des Schur-Cohn-Algorithmus und der Jury-Tabelle vorgestellt. Die neuen Versionen der klassischen Algorithmen erlauben uns, zu jedem Polynom P eine Sequenz von Koeffizienten k, zuzuordnen. Wir begriinden, daB die Zahl der Nullstellen von P aufierhalb des Einheitskreises gleich der Anzahl der k, ist, mit Ik, I > 1. Eine einfache Beziehung fIir die Zahl der Nullstellen von P auf dem Einheitskreis wird ebenfalls gegeben. Eine Erweiterung des modifizierten Algorithmus fihrt einen Entscheidungsparameter ein, der es erlaubt, die kritische Situation: 1 - E < (k, I d 1 + E zu studieren, wobei 8 ein beliebig kleiner positiver Entscheidungsparameter ist. RCsumL: Nous prCsentons une modification et une extension de l’algorithme de Schur-Cohn et de la table de Jury. Ces nouvelles versions de I’algorithme classique permettent d’associer une suite de coefficients k, B tout polyname P. Nous etablissons que le nombre de zCros de P, sit&s B 1’extCrieur du cercle unit&, est &gal au nombre de coefficients k, vCrifiant lk, I > 1. Nous donnons Cgalemcnt une expression simple du nombre de zCros situ&s sur le cercle unit& L’extension proposke introduit un paramitre arbitraire qui permet d’Ctudier les situations critiques: 1 - E< lk, / < 1 + c, oh E est un parambtre positif arbitrairement petit. Keywords:

Polynomial;

Linear system; Stability test; Schur-Cohn’s

* Tel.: 33 169 85 1717; fax: 33 169 41 3060; e-mail: benidir@

algorithm; Jury’s table

Iss.supelec.fr.

0165-1684/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved PIISOl65-1684(96)00077-l

76

M. Benidir / Signal Processing 53 (1996) 7542

1. Introduction

2. Classical Schur-Cohn algorithm and Jury table

One of the main problems in the study of linear systems is the determination of their stability. For linear discrete and causal systems the stability condition can be deduced from the study of the zero distribution of a polynomial with respect to the unit circle (UC). This problem has been extensively studied in [l-l 1, 13-151. There are several procedures to solve this problem: Schur-Cohn’s algorithm, Jury’s table, Bistritz’s table, etc. Some extensions and comparisons between these methods are proposed in [3,6, 151. All these procedures introduce two kinds of singularities and the main method to solve these singularities consists of using the derivative and adding a root inside the UC [5, lo]. Another recent contribution to this problem based on the LDL* factorization of Bezoutians is proposed in [ 111. In this paper, we restrict our study to the well-known procedures as Schur-Cohn’s algorithm and Jury’s table. In fact, in the literature, we encounter different versions of these procedures which are the basis of any recent method for studying the zero location of polynomials with respect to the UC. In the following, we have referred to as Schur-Cohn’s algorithm and Jury’s table not exactly the original works of the authors but forms actually well-known to be equivalent to the original results. These procedures consist of associating a sequence of coefficients kj to every polynomial P. To obtain the number of zeroes of P outside of unit circle, we have to compare lkjl to 1. Our purpose is first to introduce a simple modification into the classical Schur-Cohn algorithm which allows us to obtain a new version referred to as the modified Schur-Cohn (MSC) algorithm [l]. This version gives the zero distribution with respect to the UC for every polynomial. In particular, we obtain a simple expression for the number of zeros outside the UC in terms of the kj satisfying lkj ( > 1. Then we introduce a class of algorithms, depending on an arbitrary parameter and referred to as extended Schur-Cohn (ESC) algorithm, which allows us to obtain a new way to solve the singular cases and to study the critical situations corresponding to lkjl = 1. The ESC algorithm is a polynomial version of a similar procedure proposed in [ 151 and based on rational functions. Several examples are given to illustrate the results of the paper and complete proofs are given in Appendix A as they are of restricted interest.

2.1. Notations Let P be a polynomial of degree n with complex coefficients ci, written in the form P(z) = cgz” + c1zn-’ + . . + c,.

(1)

We can associate to P its reciprocal defined by P*(z)

!? zv(z-l)

= 20 + ClZ’ + . . . + c,z”

(2)

where CJ denotes the complex conjugate of cj. AS P is of degree n, we have P*(O) # 0 and we can define in a unique way a complex number k, an integer d 2 1 and a polynomial Q given by k e -P(O)/P*(O), Q(0) # 0

or

zdQ(z) e P(z) + k P*(z), Q = 0.

(3)

The integer d denotes the multiplicity of the root 0 of polynomial P(z) + kP*(z). Denoting the degree of R by d”(R) and, by convention, d”(O) = --co, polynomials P and Q satisfy the following equivalences that will be used in this paper: d”(P) = d’(P*)

iff k # 0,

(4)

d”(Q) = d”(P) - d iff Ikl # 1,

(5)

d’(Q)
(6)

- 2d iff Ikl = 1.

For example, one has P(z) = z3 + 22 - k, P*(z)

5 z~(z-~ + 22-l - k) = 1 + 2z2 - kz3,

(7)

and this shows that d’(P) = d”(P*) iff k # 0 and the proof of (4) is obvious in the general case. The properties (5) and (6) can easily be established in the same way. In the following, n,,(R), n;(R) and n,(R) denote the number of zeros of a polynomial R, respectively outside, inside and on the UC. The algorithms which are proposed in this paper are based on the construction of polynomial Q from P via (3) and the main results are the relations between no(P), ni(P), nc(P) and

no(Q), nice>, at(Q).

M. Benidir/

2.2. Schur-Cohn’s

(SC)

Signal

Processmg

53 (1996)

1kj j > 1 or in terms of the sign changes in the sequence:

algorithm

Sg(a,,-1), . . . , Mao).

Sg(a,), One version of this algorithm is a recursive computational scheme which allows us to compute, from a given polynomial P, = P of degree n, a sequence of n coefficients kj as follows [2, IO].

Algorithm Forj=n,n-

l,...

kj 5 -P,(O)/Pjf(O); ~4-1 (z) g i;(z)

+ k,PjE(z).

(8)

in the recursion (8), the subscripts j and j - I denote the degrees of the corresponding polynomials. The algorithm ends with either j = 1 or with Pj-1 such that d”(P,_ r ) < j - 1. The first situation defines the regulur case and the second one corresponds to the singular case. 2.3. Schur-Cohn’s

stability

test

A form well-known as Schur-Cohn’s stability test is given by the following equivalence [2, lo]: n;(P) = n iff l&ii < 1, for j = n,n - 1, . . ., 1. 2.4. Jury’s stability

71

75-82

(9)

test

Let ajZJ denote the highest degree term of the polynomial Pj, j = n, n - 1,. . . , 1. Assuming the coefficient a,, to be real, then all the coefficients aj are real and we have the following equivalence:

2.6. Number

of’ unstable zeros in the singular case

In this case, it is well-known [2] that n(P) # n and different methods are proposed in the references [l-11, 13-151 to determine n,(P). In the rest of the paper, we introduce a modification in the Schur-Cohn algorithm and an extension of this algorithm. This yields a very simple expression of n,(P) valid in both regular and singular cases.

3. Reformulation of Cohn’s algorithm In this section, we propose a new version of the SC algorithm which allows us to obtain a sequence of kj which makes the expression for n,(P) in terms of the kJ very simple. Denoting by P’ the derivative of P, we can associate to any polynomial P of degree n, a sequence of polynomials P,, Pn-l,. . ,P,, m>,O, d”(e)
Algorithm 1. Initial condition: P, = P. 2. For j = n, n - 1,. ., compute 4-i

as follows:

kj ’ -P,(O)/Py(O), m(P) = n iff Sg(a,)

= Sg(a,-1 ) =

= Sg(ao), ZdiQ(Z)

’

(10) where Sg(a) denotes the sign of a. Indeed, the identification of the two member sides of (8) yields Qj_1 = Uj(1 - //Cj1*). According to (1 l), the equivalence ( 10) is obvious.

Pj(Z)

+ kjP]*(Z),

4?(o)

# 0 or Q = 0,

(kjcj(<

1,

(12)

take&i

fi Q

(13)

take <,-I

6 Q*

(14)

(11) between

(9) and

2.5. Number of unstable zeros in the regular case The number n,(P) of the zeros of P outside the UC can be expressed [2, IO] in terms of the ki satisfying

> 1, =I,

andQ=O,

=I,

andQ#O,

takeP,_,

“P,’

replace Pj by PJ e Q + (c-‘Pj

(15)

+ kjcPJ*), (16)

M. Benidir / Signal Processing 53 (1996)

78

where c is an arbitrary real parameter satisfying 0 < c < 1 and then go back to (12). If the last case appears, we replace the coefficient kj associated with Pj by ij computed from jj which satisfies jij\ # 1, and we continue the algorithm. The algorithm ends with the construction of P, such that d’(P,) = 0. At each step of the algorithm, we can replace Pj by NjPj where Nj denotes an arbitrary non-zero real or complex number. If both m = 0 and all the kj satisfy lkil < 1 for 1 <
(17)

and the main property of the sequence stated as follows.

if lkl < 1,

(21)

n,(P) = n,(Q*) + d”(P) - d”(Q*)

if /k( > 1. (22)

Lemma 2 (Singular case). Assume that the toe@cient k appearing in (3) satisfies (k( = 1. Then we have the two following cases. If Q = 0, then n,(P) = rz,(P’) + u,

(23)

n,(P) = %(P’),

(24)

where u denotes the number of distinct zeros of P appearing on the UC. If Q # 0, then n,(P) = M,(p),

(25)

n,(P) = n,(P),

(26)

where P(z) = Q(z) + {c-‘P(z)

+ kcP*(z)},

)I.

(27)

is a polynomial of the same degree as P for which

+ ...

(18)

(ii) Zf the MSC algorithm does not introduce any

step j for which Pj-1 = PI!, we have n,(P) = 0 and if jo is the largest index such that J&l = PjO,we have n,(P) = do(&) - 2n,(&).

n,(P) = no(Q)

O
n,(P) = Cd”@, ) - do@, -1 >I

+ [do(&) - d”(&

and

S(P) can be

Theorem 1. (i) If aN the kj uppearing in S(P) satisfy [kit< 1, then we have n,(P) = 0 and if there are h indices jl > j2 > . . >jhsuchthat(kjJ) > 1, Idi
+[d”(I$) - do(+)]

7542

lmu~*(o)I# 1.

Example 1 (Regular case). Let P(z) = z3 + z* - 22 - 2.

(28)

Then

(19) P*(z)=-2z3-2z2+z+1,

k=2,

(29)

Comment 1. In the general case, we have m = 0 and d”(q) = j for 0 1. Theorem 1 is established hereunder as a direct consequence of the two following lemmas for which proofs are proposed in Appendix A.

Lemma 1 (Regular case). Assume that the coeficient k appearing iit (3) sutisfies (k( # 1. Then we have n,(P) = n&Z?) = n,(Q*)

(20)

P(z) + 2P*(z) = -3z*(z + 1) b zdQ(z), d=2,

Q(Z)=-3(z+l).

We obtain from (20), n,(P)=n,(Q*)=n,(z As Ik/ = 2 > 1, (22) gives

+ I)= 1.

n,(P) = n,(z + 1) + d”(P) - d”(Q*) = 2. This result can be directly deduced from the factorized form P(z) = (22 - 2)(z + 1).

Example 2 (Singular case). Let P(z) = z2 + 22 - 1.

(30)

M. Benidir 1 Signul Processing

53 (1996)

75-82

19

4. A modified Cohn’s algorithm: an extended recursion

We have P*(Z) = 1 + 22 - z2,

k = 1,

4.1. The extended recursion P(z) + kP*(z)

= 42 b zdQ@), In this section, we propose a class of algorithms that extends the MSC algorithm. The principle of the SC or MSC algorithms is based on the computation of a polynomial Q from P via (3). Replacing relation (3) by

d = 1, Q(z) = 4 # 0. Thus, for c = i, the polynomial

(Z - o)~Q(z)

P(z)= Q(Z) + [c-‘P(z) + kcP*(z)]

= P(z) + kP*(z),

Q(a) # 0 or Q = 0. = 3/2z2 + 5z + ; is such that IP(O)/P*(O)l n,(P)

:= n,(P),

n,(P)

=

i

#

1 and satisfies

= n,(P).

Proof of Theorem 1. (i) Let us consider the four cases appearing

in the construction

(33)

where (T is an arbitrary real number and k a complex number depending on P and cr, we obtain a class of recursions referred to as the ESC algorithm. The validity of (33) is based on the following results established in Appendix A. Note that, for G = 0, relation (33) leads to (3).

of Pj-1 from Pj.

(1) In the case defined by ( 13), +I

= Q and (21)

give n,(q) = n,(+i ). In the case defined by (14), 4-l = Q* and (22) (2) give n,(e) = n,(&i) + {d”(P) - d”(P,_I)}. In the case defined by (15), p/ satisfies P,+kjP,* = (3) 0, q-, = p/ and (24) give n,(P) = n,(P_t ). (4) In the case defined by (16), we replace 4 by P and relation (26) yields n,(Pj) = n,(i)). We continue the computation of Pj-1 from the polynomial P. Iterating for j = n, n - 1, . . . , m, we obtain (18). (ii) According to the definition of j,, the steps j = n, n - 1,. . , jo do not introduce the case (15). We therefore deduce from Lemmas 1 and 2 that

Lemma 3. Let P be a non-self-reciprocal polynomial, i.e. P is not proportional to its reciprocul P*. Then, there always exist a real 0 < 0 < 1, a complex number k, lkl # 1, and a polynomial Q satkfying (33). The number k is given by k = -P(o-)/P*(o). 4.2. The ESC algorithm 1. Initial condition: P, = P. 2. For j = n,n - l,..., choose a real 0 satisfying 0 6 CT< 1 and compute P,- 1 as follows: k, = -P(a)/P;(,), (Z

(31) In addition, as PjO satisfies Pj(, + kj,PjT, = 0, i.e. P;(]is self-reciprocal. The number n,(Pj,) therefore equals the number n;(Pjo) of its zeros inside the UC and we get n,&) + 2n,(PJ,) = d”(c,). Thus we obtain (32) Relation (19) is a direct consequence (31). 0

(34)

-

CT)~‘Q(Z)

=

e(Z)

+

kjPT(z),

(35)

Q(o) # 0 or Q = 0

lkjl < 1, > 1,

take&l

a Q

(36)

take pi-1 fi Q*

=I,

andQ=O,

=l,

andQ#O,

take&i

(37) :P,!

(38)

of (32) and change the value of r~ to obtain Ik, 1 # 1

(39)

M. Benidir / Signal Processing 53 (1996)

80

and then go back to (35). The algorithm ends with the first polynomial P, such that d’(P,) = 0. The ESC algorithm allows us to compute from P a sequence W)

= {k,, k?7-I,...,

km}.

(40)

The sequence (40) depends on the different values of G introduced at each step of the algorithm. The value G = 0 corresponds to the minimum of computations.

Theorem 2. Ifthe sequence in (40) is constructed from a real sequence of satisjying 0
7542

5. Conclusion The modified version of the Schur-Cohn algorithm introduced in this paper allows us to determine the zero location with respect to the unit circle for every polynomial. The modification of the classical algorithm does not introduce any supplementary computations. Extensions are given to show that the Schur-Cohn algorithm can be considered as an important case of a general class of algorithms. This class allows us to study the critical situation corresponding to lkjl Fz 1.

Appendix A The proofs given in this appendix are based on the following well-known theorem.

1, we consider the four possible situations appearing in the algorithm. The proof can easily be established in the same way as in the case of Theorem 1 by using Lemma 4 below instead of Lemma 1. 0

Theorem 3 (Rouche’s theorem). polynomials such that

Lemma 4. Let a be real such that 0
Thus

0 < (R(z) - S(z)1 < [S(z)/

Let R and S be two

for IzI = 1.

(42)

lkl = IP(a)/P*(a)l # 1. If the polynomial Q is deduced from k and P via (33), then the equalities in (20)-(22) still hold.

n,(R) = n,(S) = 0,

(43)

n(R) = n(S).

(44)

4.3. Application to the critical case

Proof of Lemma 4. As Ikl # 1, relation (5) gives

The decision we have to take depends on the three situations: lkj\< 1, lkjl = 1 or lkjl > 1. The difficulty appears when )kj 1m 1. In this case, the two following situations are possible. 1. lkjl = 1 and Q = 0. This case can be tested by a simple examination of the coefficients of the polynomial Pj which have to be self-reciprocal. 2. lkjl # 1 but lkjl M 1 and Q # 0. In this case, kj is given by (34), which can be written as

U(z - a)dQo = U(Po + koP,*)

with (kol = (k(. (45)

kj=-fi?? u=l

d”(Q) = d’(P) - d. Let us denote by zero(m) a zero of P with multiplicity m and belonging to the UC. If a is zero(m) of P, (6)-l = a is zero(m) of P* by virtue of definition (2). Then a is zero(m’) of Q and ml 2 m. Let U be the factor of P given by the product of all the zeros of P appearing on the UC, i.e. n,(U) = nc(P). It is clear that U is self-reciprocal [2] and relation (3) can be factorized as

1 - aZ,’

(41)

where z,,, u = 1, . . , j, denote the zeros of Pj. According to the properties of the bilinear transform, we can choose a value of a for which kj does not satisfy lkjl G 1 and there is no difficulty to decide if /kil < 1 or lkjl > 1.

In addition, if Qo(a) = 0 for Ia/ = 1, we obtain /PO(a)1= JkoJ[P,*(a)1 and this contradicts the two conditions /kol # 1 and IPo(a)I = (P:(a)1 # 0. Thus n,(Qo) = 0 and we deduce nc(Q) = n,(U) = n,(P). Finally, as nc(Q) = n,(Q*), we obtain (20). To establish (21) and (22), we distinguish two cases, n,(P) = 0 and n,(P) # 0.

M. Benidir J Siynal

Processing

Case 1. n,(P) = 0. Jkj < 1: If k = 0, we have (z - G)~Q(z) = P(z) which easily gives (21). We can therefore assume k # 0. Definition (2) implies IP(z)I = IP*(z)l

for jz/ = 1

(46)

and (3 ) gives

for Jz/ = 1.

We deduce ni{(z - cr)“Q} = ni(p) = n,(Q) + d from Rouche’s theorem. Thus d”(P) - n,,(P) = d”(Q) no(Q) + d by virtue of (20). As (k( # 1, (5) gives d”(P) = d”(Q) + d. Finally, we obtain (21). lkl > 1: As jkl # 1, (5) gives d”(Q) = d”(P) - d and relation (33) can be written as = P*(z) + kP(z);

(49)

or, equivalently, h( 1 - z~)~Q*(z)

k = -k - Q(0)/co(c-’

- c)

to choose a real c,

(52)

satisfies $1 # 1. To prove (25) let CIbe a zero of p such that 1x1= 1. Then (27) and (3) give + kP*(a)]

+ [c-‘P(x)

+ kc?‘*(u)] = 0,

= P(z) + U*(z)

with Ih( 2 (k-‘( = (kl-’ < 1.

(53) or

(48)

(1 - ~a)~Q*(z)

possible

for Iz/ = 1. (47)

# 0 for IzI = 1 and 0 < (kl < 1, we get

0 < I(2 - G)~Q(z) - P(z)1 < IP(

81

75-82

It is therefore always Obc < 1, such that

r-“[P(x)

l(z - G)~Q(z) - P(z)1 = \kjlP(z)l As IP(

53 (1996)

(c+ + c-‘llf(cc)l

= lkllb

+ cllP*(cx)(.

As Ikl = 1 and (P*(a) = IP(a

we obtain

{ix-” + c-11 - (x-d + ci}lP(x)l

= 0

(54)

(55)

and therefore P(x) = 0. Conversely, P(x) = 0 yields p(a) = 0. To complete the proof of (25 ), it is sufficient to show that if a is a zero(m) of P it is also a zero(&) of P and m’>m. If x is zero(m) of P, (i)-’ = a is zero(m) of P* by virtue of definition (2 ). According to (3) a is zero(m’) of Q and m’>m. Finally, H is zero(m”) of P by virtue of (27) and m” 13m’ > m. Now let us consider the Proof of (26). It is easy to see that the reciprocal polynomial of p can be written in the form

(50)

As in the previous case, we obtain ni{h( 1 -~cr)~Q*} = tii(Q*) = ni(k’). This can equivalently be written as d”(Q*)-nc(Q*)-n,(Q*) = d”(P)-n,(P)-n,(P). According to n,(Q*) = n,(P) = 0, this gives (22). Case 2. n,(p) # 0. Polynomial P can be factorized in the form (45). Deleting factor U, we return to the previous case n,(Po) = 0. 0 Proof of Lemma 1. This lemma is the particular case of Lemma 4 obtained for r~ = 0. 0

I;* =z-“(P*

+ iP)

+ (c-lP*

+ ckP) :z R + hR*, (56)

where R(z) e (z-d + c-‘)P*,

h e ck.

As lhl = (ckl = lc( < 1, according to Roucht’s rem, we obtain ni(P*)

(57) theo-

= n,(R).

(58)

?Zi(P*) = ni(P*)

(59)

Thus Proof of Lemma 2. The proofs of (23) and (24) are given in [2; 10, Chapter lo]. If IZdenotes the degree of P, f’ is the sum of a polynomial Q of degree n - 2d and the polynomial c-‘P + kcP* of degree n. The polynomial P is of degree n and can be written as P(z)

= [cO(c-’ - c)]z”+

. + [Q(O) + c,(c-’

- c)]. (51)

because 0 < c < 1. Finally, conditions P(0) # 0 and B(O) # 0 yield ni(P) = n,(p*) and n,(P*) = n,(P). We obtain (26). q Proof of Lemma 3. Assume that for all real G, 0 < 0 < 1, the numbers k defined in (34) satisfies

M. Benidir / Signal Processing 53 (1996) 75-32

82

jkj = 1. In this case, we have IP( ___ P*(e)12

P(e) WJ) = a”&+)a”JJ(a-l)

= l

forOdcr
(60)

or G(o)

P(a) 5 o”P(a-1)

for060<

1.

d+(d) -

&)

=O (61)

However, either the function G(t) equals zero for all complex numbers z or the number of its zeros is countable [ 121. Thus the equality in (6 1) holds for all complex values of z and we therefore have P(z) = az”P(z-‘) and z”P(z-‘) = d(z), where c( is a complex number. This contradicts hypothesis P # ctP*. Finally, there always exists a real 6, 0 < CJ < 1, for which the complex number k in (34) satisfies (k[ # 1. For k given by (34), the polynomial P(z) + kP*(z) has (Tas zero and can, therefore, be written as in (33). This shows the existence of the polynomial Q. q

[ 11 M. Benidir, “Modified Schur-Cohn’s algorithm and Jury’s test in the complex case”, in: Fundamentals of Discrete-Time Systems, TSI Press, 1993, pp. 195-202. [2] M. Benidir and B. Picinbono, “Extensions of the stability criterion for ARMA filters”, IEEE Trans. Acoust. Speech Signal Process., Vol. ASSP-35, April 1987, pp. 425-431.

[3] M. Benidir and B. Picinbono, “Comparison between some stability criteria for discrete-time filters”, IEEE Trans. Acousi. Speech Signal Process., Vol. 36, No. 7, July 1988, pp. 993-1001. [4] Y. Bistritz, “A circular stability test for general polynomials”, Systems Control Lett., No. 7, 1986, pp. 89-97. [5] A. Cohn, “ijber die Anzahl der Wurzeln algebraischen Gleichung in einem Kreise”, Math., No. 2, 1922, pp. 110-148. [6] P. Delsarte, Y.V. Genin and Y. Kamp, “Pseudo-1ossless functions with application to the problem of locating the zeros of a polynomial”, IEEE Trans. Circuits Systems, Vol. CAS-32, April 1985, pp. 373-381. [7] EL Jury, “A simplified stability criterion for linear discrete systems”, Proc. IRE, 1962, pp. 1493-1500. [8] E.1. Jury, “Theory and application of inners”, Proc. IEEE, No. 63, 1975, pp. 1044-1069. [9] E.I. Jury, “A note on the modified stability table for linear discrete time systems”, IEEE Trans. Circuits Systems, Vol. 38, No. 2, February 1991, pp. 221-223. [lo] M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Amer. Math. SOL, New York, 1949. [l l] D. Pal and T. Kailath, “Displacement structure approach to singular root distribution problems: the unit circle case”, IEEE Trans. Automat. Coritrol, Vol. 39, No. 1, January 1994, pp. 238-245. [ 121 W. Rudin, Analyse Reel/e et Complexe, Masson, Paris, 1975. [13] C.W. Schelin, “Counting zeros of real polynomials within the unit disk”, SIAM J. Numer. ANUS, Vol. 20, No. 5, October 1983, pp. 1023-1031. [14] J. Schur, “ijber Potemrichen, die im innern des Einheitskreise beschrankt sind” (On series which are bounded in the unit circle), J. Reine Angew. Math., Vol. 147, No. 4, 1917, pp. 205-232. [15] P.P. Vaidyanathan and S.K. Mitra, “A unified structural interpretation of some well-known stability-test procedures for linear systems”, Proc. IEEE, Vol. 75, No. 4, April 1987, pp. 418-497.