CIRCUITS SYSTEMS SIGNAL PROCESS VOL. 10, NO. 2, 1991

ON LIMIT CYCLES IN N-ORDER DIRECT-FORM DIGITAL FILTERS*

Tamal Bose I and David P. Brown 2

Abstract. The relationship between the elements in the vector of any limit cycle due to rounding in an n-order direct-form digital filter is established. Some bounds on the elements in such vectors are also determined. Sufficient conditions for the accessibility of period-r limit cycles due to rounding in n-order digital filters are presented.

1. Introduction

Digital filters are often used as a part of digital instruments for processing measured data. When digital filters are implemented, they are subjected to finite wordlength effects of the hardware. One of the undesirable effects of finite register length is the occurrence of periodic oscillations called limit cycles. Limit cycles are a potential problem not only in the implementation of digital filters but also in any instrument where some type of arithmetic is performed. Therefore it is important to study the conditions under which limit cycles exist and obtain constraints for suppressing them. It is also desirable to find bounds on the limit cycles, since they give a good estimate of the error in measurement (or filtering).

Limit cycles due to adder overflow and magnitude truncation quantiza- tion in digital filters have been thoroughly investigated [1]-[6], and several designs constraints have been obtained in order to suppress them. However, if rounding is employed, limit cycles can still occur. Limit cycles due to rounding in digital filters have also been investigated to some extent [7]-[9], but the analyses were mostly restricted to second-order systems.

In this paper n-order digital filters are considered for zero-input limit cycles due to rounding. For n-order direct-form digital filters the relationship between the vectors comprising a period-r limit cycle due to rounding is

* Received November 13, 1989. 1 Electrical Engineering Department, The Citadel, Charleston, South Carolina 29409, USA. 2 Department of Electrical Engineering, Southern Illinois University, Carbondale, Illinois

62901-6603, USA.

154 Bose AND BROWN

established (Theorem 1). Some bounds on the elements of such vectors are presented (Theorems 2 and 3). In addition, conditions are presented for the accessibility (Theorem 4) of limit cycles of period-r in direct-form digital filters.

2. Structure of limit cycles

The n-order direct-form digital filter is shown in Figure 1, where [-R l denotes the rounding operat ion, u(n) is the input, and y(n) is the output. With zero input, the state equation of the filter is

x(k + 1) -- lAx(k)] R, k = 0, 1 . . . . (1)

where x e R", A e R n • ", [. ]R is the rounding operator and

0 1 0 ... 0 ] /

0 0 1 ... O[

A-- / 0 0 0 .-- 1 |

1 al a2 " " an J

The filter state vector is x(k)= [xi(k)], and al, a2 , . . . , am are the filter coefficients, a l r 0. Without loss of generality, rounding to the nearest integer is assumed. Also, it is assumed that the linear system (without quantization) is stable, that is, all the eigenvalues of A are inside the unit circle.

In the following the relationship between the elements of the vectors comprising a period-r limit cycle is presented.

u(n)

Cn

<t a2

Figure 1. Direct-form digital filter of order n.

<1 al

l > C2

>

y(n)

L I M I T C Y C L E S I N N - O R D E R D I R E C T F O R M D I G I T A L F I L T E R S 1 5 5

Theorem 1. I f the system described by (1) sustains a period-r limit cycle, then the vectors Ua, U2, . . . , U,, Ui ~ Uj, comprising the limit cycle satisfy

u~ -. u . ] , U ~ = [ u 2 u3 " " u . u . + l ] , . . . ,

u 1 ... u ,_l] for r > n ,

(i) U] = [ul

and

(ii) U ] = [ u 1 u 2 ..

U ~ = [ u ~ u3 . .

�9 " " U j + 1 ] ,

Utr = [ U r U 1 . . .

for r <_ n,

where all u i are integers

U r U 1 U 2 �9 �9 " U r �9 " �9 U 1 " �9 �9 U j ] ,

U r /11 U 2 U 3 � 9 U r U 1 � 9 a 2

. . . ~

U r - 1 U r U l " " " U r - 1 " " " Ur " " �9

and t means the transpose.

uj_ 1]

Proof. Suppose x = [xi] e R" satisfies (1). Then, for all k = O, 1 . . . . .

xi(k + 1) = [x i + 1 (k)]R, i = l, 2 . . . . . n -- 1,

x,(k + 1) = [ ~ aixi(k)]R.

Since (1) sustains a limit cycle of period r, for some k = k o, k = k o + 1 , . . . , there are integer vectors U1, U2 . . . . . U r, U z r U i, such that for Uj = [ui~],

j ~ - 1 , 2 , � 9 r , u i 2 ~ u i + 1, 1 , u i 3 ~--- u i + - i , 2 , � 9 , U i r ~ u i + 1, r - 1 and uil = ui+ 1, , ,

i = 1, 2 . . . . . n - 1. These equations give the relations between the elements of the vectors in the limit cycle. The i row of Uj is equal to the i + 1 row of Uj_ 1 and t h e i r o w of U 1 is equal to t h e i + 1 row of Ur fo r i = 1,2 . . . . , n - 1 . Using these relations the set of vectors in (i) for r > n is obtained. Fo r r < n, starting with ui2 = ui+ 1, 1 and eliminating u~2 using the second equat ion and then eliminating u~3 using the third equat ion and continuing this procedure results in u~+, = u~l, i = 1, 2 . . . . . n - r. This equat ion gives the relation between the elements in U 1 . Using this relation, U1 in (ii) of the conclusion is established�9 The second subscript, "1 ", has been omitted since all elements have the same second subscript. The vectors U2, . . . , U, are established using the relations between the elements of the vectors. This completes the proof. Note that for n = r, (i) and (ii) are equivalent. [ ]

An n-order direct-form digital filter which has a period-1 limit cycle U satisfies, by Theorem 1, U = up where u is an integer and p ~R", p t = [1 1 . . . 1]. That is, all the elements of U are equal. If the limit cycle is period-2, then, by Theorem 1, U] = [u I u 2 .. . u,] and U~ =

[u2 u lu~ u3 "" u.-1]. A bound on each element in the vector comprising a limit cycle is

established in the following theorem. In the corollary a simple special case is given.

156 BOSE AND BROWN

Theorem 2. I f the system described by (1) sustains a period-r limit cycle, then each element ui of the vectors comprising the limit cycle satisfies

1 lui[ _< 27~Y~ [Aljl,

where A = det A,, Als is the 1,j cofactor in A, and A, is given in (4)for r <_ n and in (5) for r > n.

Proof. Since (1) sustains a limit cycle of period-r, for some k = ko, ko + 1, . . . , there are integer vectors U1, U 2 . . . . . Ur, U~ ~ Us, such that

lAUd] R = Ui+l , i = 1 , 2 , . . . , r , (2)

and i + 1 is modulo r. Suppose r _< n and define As = ~ as+vr where j = 1, 2 , . . . , r, as+pr = 0

whenever j + pr > n and either s = [_n/rl if r does not divide n or s = n/r - 1 if r divides n. With this nota t ion and U 1 = [ui] as in (ii) of Theorem l, the last row of (2) is

AlUi+A2ui+l+' . ' ' k -ArUi+r_l )R=Ui+j , i = 1 , 2 . . . . . r,

j is any number among 1, 2 . . . . . r and all the subscripts are modulo r. F r o m the meaning of the R opera tor it follows that

- - �89 <-- A l U i + "'" + A j u i + j - 1 + (As+l - 1)ui+s

q- A j + 2 u i + j + 1 q- . . . . .1- A r u i + r _ 1 <~ �89

or in matrix form,

- ( �89 <_ A,U < �89 (3)

w h e r e A , e R ~ • • 1 .-. 1 ] , U = E u i ] , a n d

A 1 .. . A s A s + l - 1 Aj+z -- . A r ]

A = AS..,i,.i . . . . Aj~.I . . . . . . . . Aj . . . . . . AS.+),~. I.. ,'.'.'.,, A t _ ) / . (4) /

A2 "'" A s + l - 1 As+2 A j + 3 "'" A1 ]

When r > n the limit cycle is as in (i) of Theorem 1 and the last row of (2) is

(alul + azu i+ 1 + . . . + anu i +n-1 )R = Ui+n,

i = 1, 2 . . . . . r and all subscripts are modulo r. With r > n, the inequalities in (3) hold and

al a2 "'" a, - 1 0 ... 0

0 a 1 " ' " a n - 1 a n - - 1 . . " 0 A = (5)

a : a 3 . . . . 1 0 0 al

LIMIT CYCLES IN N-ORDER DIRECT FORM DIGITAL FILTERS 157

Equat ion (3) is writ ten as

]A, U I < l p ,

where the absolute value 1. ] is the absolute value of each element in the matrix. Since all the elements of [ A , 11 are nonnegative,

[Aul [ [AuU[ < (1 ) [Aul lp

and since

it follows that

IUI = ]A,-1A, U[ <_ IAff-l[ IAuU[

[UI ~ (�89

The matr ices in (4) and (5) are r -order Toepli tz matr ices and satisfy the propert ies of circulants [10]. In par t icular JAu] = rAi+~.j_~], cr = 1, 2 . . . . . r, where i + cr and j - cr are modu lo r and A u is the i,j cofactor in det A,. I t follows that the sum of the elements in each row of [A~- 1 ] is the same. Thus, for any i = 1, 2 . . . . , r,

1 lull < 2 1 ~ ]Aljl,

where A = det Au. This completes the proof. [ ]

Corol lary 1. I f all nonzero elements in At~ 1 o f Theorem 2 have the same sign, then

lu, t <__ (�89 a j - 11-1

Proof. Since all nonzero elements in A , 1 have the same sign, either [A u 1[ = A u I or --Au-1. Using the fact that the sum of each row of A, in (4) or (5) is a = ~ a j - 1,

A.p = ~p.

Since the system (1) is stable, ~ = 0 and

A~ lp = or lp.

If I A ( 1 [ = A~- 1, ~ > 0 and

I UI _< (�89 = (1)~-%

If [A~-ll = - A ~ - t , ~ < 0 and

I UI ___ -(�89 = _ ( 1 ) ~ - lp.

Combin ing these two inequalities results in the conclusion. [ ]

In the following theorem a bound on the sum of the elements in the vectors compris ing a limit cycle is established.

158 BOSE AND BROWN

(i)

(ii)

Theorem 3. I f the system described by (1) sustains a period-r limit cycle where u~, i = 1, 2, . . . , r, are the elements in the vectors comprising the limit cycle and Y', ui # 0, then

(�89 IZ u'l <- tZ a j - 1]'

1 - (�89 ~ ~ aj <_ l + (�89

Proof. Adding the r rows of (3) and noting that the sum of the elements in each column of (4) and of (5) is ~ aj - 1 obtains

-( �89 <_ (Y, aj - 1)(y~ uO ~ (�89

Since (1) is stable, ~ a# # 1. Therefore

I(Z aj - 1)(y~ uOI _< (�89

and since

it follows that

[E u,I = ](E a j - 1)- 1(~ a j - 1)( E u,)l

g ](Eaj- 1)-1l I ( ~ a j - 1)(~ uO]

I(Z u,)l _< (�89

To establish (ii) the foregoing inequality is written as

which is the same as

(1)r I(Y~ " j - 1)1 _< IZ u,~

-(�89 (�89 IE ui~l <- E a j - 1 <_ ]E uil"

Since each u i is an integer and I~ ui[ r 0,

lY, u,I-> a

and therefore

1 -- (�89 < s < 1 + (�89

This completes the proof. []

3. Accessibility of limit cycles

The following classifications of limit cycles have been suggested in [2]:

(1) Inaccessible limit cycles are those which only appear if the filter is started with initial conditions pertaining to that cycle.

LIMIT CYCLES IN N-ORDER DIRECT FORM DIGITAL FILTERS 159

(2) Accessible limit cycles are those which can be reached from initial conditions that do not pertain to that limit cycle.

To the above list may be added a definition of the access time of a limit cycle:

(3) The access time of a limit cycle is the number of samples required for the system to enter that limit cycle.

In the next theorem a simple sufficient condition is presented for the existence of a limit cycle with a specified access time.

Theorem 4. Suppose (1) sustains a limit cycle o f period r. I f there exists k o integer vectors Vii, i = 1, 2 , . . . , ko, such that V i r Vj, i r j, Vko is a vector of the limit cycle, and

- ( � 8 9 j = 0 , 1 . . . . , k o - 1,

then, for x(O) = V o, the system sustains the limit cycle starting at k = k o.

Proof. From the meaning of the R operator and the hypothesis, it follows that ifx(O) = Vo, the system satisfies x(j) = Vj,j = 1, 2 . . . . , ko. Since Vko is a vector of the limit cycle and Vii r Vj, the limit cycle is accessible in k o steps. This completes the proof. []

Any two integer vectors x(k) and x(k + 1) satisfying

x(k + 1) = [Ax(k)JR,

where A is in the direct form, are related as x t ( k ) - - I x 1 x2 "" x ,] and xt(k + 1) = Ix2 x3 "" x, x ,+l] . That is, except for xl all elements in x(k) are shifted up one row in x(k + 1) and a new element x, + 1 appears in row n of x(k + 1). This relationship is useful when working with direct form Systems.

Consider a second-order direct-form digital filter with the system matrix

A = --0.5 0.9 "

It is not difficult to show that this system sustains the period-6 limit cycle: U i = [ 1 0], U ~ = [ 0 - 1 ] , U ~ = [ - 1 - 1 ] , U ~ , = [ - 1 0], U ~ - [ 0 1],

A =

- 0 . 5 0.9 - 1 0 0 ]

J 0 - 0 . 5 0.9 - 1 0

0 0 - 0 . 5 0.9 - 1 .

- 1 0 0 - 0 . 5 0.9

0.9 - 1 0 0 - 0 . 5

u~ = [1 13. From Theorem 2, in this case [ui[ = 1, and

160 Bose AND BROWN

Therefore

1 lu, I ~ ~ - ~ IAlj[ = 1.498

and Theorem 2 is satisfied. To illustrate Theorem 4, suppose that it is required to find x(0) such that the system enters the limit cycle with an access time of three samples and x(3) = U1. In accordance with Theorem 4, first find an integer vector V 2 such that

- ( � 8 9 < A V 2 - U , < (�89

and V 2 is not equal to any of the vectors corresponding to the limit cycle. In this case V~ = [2 1]. Next find 1/1 such that

- ( � 8 9 < AV1 - 1/2 < (�89

with the same restriction applying. The only choice is Vta = [2 2]. Finally, find x(0) such that

- ( �89 < Ax(O) - Vt < (�89

It can be shown that x2(0 ) = 2 and - 1.4 < x~(0) < 0.6 satisfy the above. With any choice of x(0) in this range, the system enters the period-6 limit cycle starting at x(3). Figure 2 illustrates this example. It should be noted that an algorithm based on Theorem 4 can be easily implemented in a computer.

X 2

-1

2 �84

1-

i i ' l i ' ~---

--1 0 1 2 X I

Figure 2. Period-6 limit cycle with an access time of three samples.

LIMIT CYCLES IN N-ORDER DIRECT FORM DIGITAL FILTERS 161

4. Conclusion

Several new results concerning zero-input limit cycles due to rounding in n- order direct-form digital filters have been established. Various properties of the elements of the vectors comprising period-r limit cycles are given and the relationship between such elements and the elements of the state matrix is investigated. A general relationship has been established between the initial state of the digital filter and the access time of the limit cycle. This suggests that a digital filter sustaining a limit cycle may be used to implement a discrete oscillator with a prescribed transient time.

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