Cyclical Coincidence Arrays and Exact pdfs Representing Interference for Some CDMA Systems (1)

ZORAN K O S n 6 AT&T Labs-Research, Holmdel, New Jersey 07733

GORDANA PAVLOVI~ Lucent Bell Laes, Middletown, New Jersey 07748

EDWARD L. TITLEBAUM Electrical Engineering Department, University of Rochester, Rochester - ,New York 14627

Abstract. A new class of coincidence (hit) arrays called cyclical hit arrays is defined. Cyclical hit arrays give insight into pre- viously unknown properties of several sets of number theoretic frequency-hop patterns, based on Welch-Costas and Quadratic Congruence codes. Three theorems related to cyclical hit arrays are proved. The theorems are used to determine exact prob- ability distribution functions of random variables which represent the number of coincidences between two arbitrary patterns from the same set of codes. The number of coincidences expresses interference levels between two users of code-division mul- tiple-access (CDMA) systems which utilize the considered patterns. The obtained results can be used for computing the exact expressions for the total interference and for error probabilities in several CDMA systems.

1. INTRODUCTION

One of the important issues in designing code-division multiple-access (CDMA) systems is the construction of codes that represent direct sequence signature sequences, frequency-hopping (FH) patterns or time-hopping pat- terns which are mutually orthogonal, pseudo-orthogonal or have good correlation or coincidence (*) properties. Two desirable properties of code-sequences for CDMA systems, which are treated in this paper, are

i) that interference between two (or more) sequences is computable and

ii) that interference between two code-sequences is the same regardless of which pair from the whole sequence set is considered.

In this paper we define a cyclical hit array, which pro- vides means of evaluating properties of some specific classes of number theoretic signature sequences in CDMA systems. We first recall the definition of FH placement operators for quadratic congruence and Welch-Costas codes [ 11. We propose restrictive rules for

( I ) This paper was presented in part at The IEEE International Communications Conference. Chicago, Illinois. June 1992. This work was supported in part by the grant from SDIOnST and managed by the Omce of Naval Research under contract number N00014-86-K-05 I I .

(2) In this paper term "coincidence (array)" can be equated to the term '%dimensional correlation function".

designing FH patterns needed for existence of cyclical coincidence arrays which have desirable properties. We next show how the proposed cyclical hit arrays enable us to determine exactly the distribution of the number of hits at various positions of an array for the considered number theoretic based FH patterns. By using map- pings from frequency-hopping patterns into time-hop- ping patterns and CDMA optical codes, one can show that exact probability density functions for random vari- ables which represents interference between two users in any of those systems can be derived. These pdfs directly correspond to the pdfs of the number of hits in an appropriate cyclical hit array. Furthermore, based on obtained results it is possible to compute the exact pdf for the interference at a receiver caused by a number of simultaneous signals from other users, as well as the exact probability of error caused by interference. Previously, Gaussian approximations were most often used for interference computation in CDMA systems.

2. CLASSICAL DEFINITION: COINCIDENCE ARRAY

A formalism of (classical) coincidence (hit) array has been introduced in [ 11 by Bellegarda and Titlebaurn for use in coherent active radar and sonar echo location systems. This formalism is based on the concept of coincidence, or hit [2], between two frequency-hop pat- terns. The collection of all possible hits, together with their location is recorded in time-frequency space,

Vol. 8. No. 2 March - April 1997 149

which produces the hit array associated with the two patterns considered.

2.1. Frequency Hop patterns and coincidence arrays

A Frequency-Hop (FH) pattern is a form of represent- ing a given frequency-hop signal. The FH pattern presents the frequency ordering in a FH signal that is determined via a sequence of integers y(k), 1 S y ( k ) S N, k = 1, ..., N called the placement operator. In the most general case, the placement operator maps the set IN+, = (0, 1 ,..., N } into its subset (into itself if the placement operator is full [ 11). The cross-coincidence array (cross- hit array) [ l ] associated with two frequency-hop codes (patterns) A and B, each of size N x N , is a (2N - 1) x (2N - 1) grid with values hk,l where -N + 1 5 k, I 5 N - 1. The value of the grid at the position k, 1 is obtained as

hk,t = ~ 8 [ y 2 ( i + k ) - y i ( i ) - I ] ( m o d N ) , U

i=L

- ( N - l ) < k , l < N - I , L = m a x ( I , l - k ) , ( 1 )

U = min(N, N - k ) , 6 ( x ) = 1 if x = O { 0 otherwise

and y , (k ) and y 2 ( k ) are the placement operators corre- sponding to frequency-hop patterns A and B respective- ly. The event for which 6 ( x ) = 1 is called the hit (coin- cidence). The auto-coincidence array is obtained from the previous definition when A equals B, and it is an odd symmetric grid. Examples of frequency -hop pat- terns and their coincidence arrays are shown in Fig. 1.

Code A Code B

Auto-Coincidence of A Cross-Coincidence of A and 8 Fig. 1 - Two frequency-hop patterns and their classical coincidence arrays.

2.2. Deficiencies of classical coincidence arrays

For use in signal design and system analysis in coher- ent active radar and sonar echolocation systems, the classical coincidence arrays recalled above were suffi- cient. In these systems, a single frequency-hop signal would interfere with another frequency-hop signal in a non-modular way: when frequency or time shift

Fig. 2 al) The classical design of quadratic congruence frequency-hop patterns - both index k and frequency values form group structures; a2) The proposed design of quadratic congruence frequency-hop patterns - both index k and frequency values form field structures; b l ) The classi- cal design of Welch-Costas frequency-hop patterns - both index k and frequency values form group structures; b2) The proposed design of Welch-Costas frequency-hop patterns - index k values form a group whereas frequency values arc represented in a field.

occurred, only limited segments of signals would over- lap. Thus, the rules of the frequency hop pattern design were relatively loose. For instance, in the case of Welch Costas codes (Fig. 2, case al), the frequency-hop pat- tern used for sonar does not include a zero frequency offset. The absence of a zero frequency offset disables the modular computation (which is non-consequential for sonar systems). However, several FWCDMA com- munication systems were proposed [ 3 ] in which fre- quency hop patterns are “wrapped around”, modularly shifted in time and frequency, with the goal of encoding both a users address and a message to be sent over the channel. Moreover, some optical CDMA and time-hop- ping CDMA signature sequences have been recently designed based on the mappings from frequency hop patterns with similar properties [4, 51. For these codes, classical coincidence arrays do not make possible the computation of the probability density function repre- senting the interference between sequences. In the sequel we show how to obtain an exact expression for such a pdf, for Welch Costas and Quadratic congruence based codes, after presenting restrictive FH pattern design rules and a cyclical coincidence array definition.

2.3. Restrictive design of frequency hop patterns

In this subsection some restrictive rules for the design of frequency-hop patterns are shown. These rules need to be followed in order to facilitate the computation of cyclical coincidence arrays which have analytically computable probability density functions representing coincidence (hit) distribution in a given array. We con- sider Welch-Costas (WC) FH codes and quadratic con- gruence (QC) FH codes. The topic of restrictive design of Costas arrays has been addressed in the paper by Drumheller and Titlebaum [6].

The placement operator for Welch-Costas FH codes is given by y r c ( k ) I aRk (mod N), k = 1, 2 ,..., N - 1, where a determines the code-sequence from the family, R is a primitive root of an odd prime number N , and k determines the element of a code-sequence. The place- ment operator for a class of QC FH codes is given by y f c ( k ) = ak2 (mod N), k = 1, 2 ,..., N.

Observe the difference in the ranges of parameter k

I50 E7T

for WC and QC codes. The computation of the values of the coincidence array yields elements that belong to the complete residue set of N. Thus x (from 6(x ) in coinci- dence array definition), as well as indices i and i + k can take values which are congruent to zero [= 0 (mod N)]. Whatever the algebraic structure of the considered fre- quency-hop patterns is, the set of values of x (in the coin- cidence array definition) generates a multiplicative field with a congruent additive operation. We conclude that all frequency hop patterns (targeted for “modular” use) should span the range of “frequencies” from 0 to N - 1, even when some frequencies are not used for transmis- sion. This is illustrated in Fig. 2 for QC and Welch Costas codes. The change of parameter k used in place- ment operators follows different modularity rules from the value of the frequency. Also, this rule is different for different types of codes. For example, in the case of QC codes 1 5 k 5 N, whereas in the case of Welch-Costas codes 1 5 k I N - 1. Fig. 2 bl) seems to be a natural rep- resentation of WC FH codes (since both parameter k and placement operator values create algebraic groups). It is also the form used by most previous authors. In Fig.2 b2) it appears as if an unnecessary empty row has been added. This additional row is, in order to complete the field structure, crucial for providing good properties which will be derived in the following sections on cycli- cal coincidence arrays. These rules imply that some FH patterns will not be representable as square arrays such as assumed in most previous papers [ 1,2]. Their size can be represented as N, x N,.

3. CYCLICAL COINCIDENCE (HIT) ARRAYS

Classical coincidence arrays may be viewed as two- dimensional aperiodic correlation functions. We define cyclical coincidence arrays such that they are analog to periodic correlation functions. The number of hits in a classical hit array is always smaller or equal to the num- ber of hits in a cyclical hit array (due to its periodicity).

Definition: A cyclical cross-coincidence array asso- ciated with two frequency-hop codes (patterns) A and B, each of size N, x N,, is a N, x N2 grid with values hk,, where 0 I k I N, - 1 and 0 S 15 N2 - I. The value of the grid at position k,l is obtained as

N , - 1

‘ k , l = 2 6[[Y2((i-k)(modNI))- i=O

YI (i(modN,)) - q(modNz,}* (2) 0 5 k 5 N, - 1 , 0 5 1 I N2 - 1,

1 if x = O i 0 otherwise 6 ( x ) =

where y , ( k ) and y , (k ) are the placement operators for frequency-hop patterns A and B respectively. The cycli-

I Fig. 3 - The process of forming; a) classical hit array and b) cyclical hit array by overlapping frequency-hop patterns.

cal auto-coincidence array is obtained from the previous definition when A equals B.

Fig. 3 shows the difference in generating classical and cyclical coincidence arrays pictorially. In Fig. 4 we show an example of the relationship between a coinci- dence array and the appropriate cyclical coincidence array. Four highlighted square segments (of size 4 by 4 in the example) of a classical array are merged into a single (4 by 4) square segment of a cyclical coincidence array, by summing up the total number of hits in appro- priate array positions. To determine values of parame- ters N, and N,, one needs to look into specific classes of codes. For QC FH codes, N, = N2 = p where p is an odd prime number. For WC FH codes, N, = p - 1, and N2 = p where p is an odd prime.

Cross-Coincidence of A and 8 Fig. 4 - The transformation of a classical cross-coincidence array into a cyclical cross-coincidence array.

3.1. Properties of cyclical coincidence arrays

When restrictive requirements which we have imposed on the design of frequency-hop patterns hold, interesting properties of cyclical coincidence are true. We next present these properties in form of theorems. None of these properties can be obtained for classical coincidence arrays. As a consequence of discovering these properties we were able to derive exact probabil- ity distribution functions for the number of coincidences between two arbitrary code-sequences from considered code-sets. A pictorial representation of the statements of the theorems which follow is given in Fig. 5 .

Theorem 1: In a cyclical auto-coincidence array of a QC FH code designed over a prime p, there is one posi- tion with p hits, p? - p positions with 1 hit, and p - 1 positions with zero hits.

Vol. 8. No. 2 March - April I997 151

oc codes Auto-Ccinc

Qc Codes Auto-Coinc

WC Codes WC Codes Cross-Coinc Cross-Coinc

Fig. 5 - Pictorial representation of “hit distribution” theorems for p = 5. (Reciprocal prim. roots)

Proof: For QC FH patterns designed honoring the modularity requirement in both k and 1 (as proposed in the previous sections of this paper), equation y r (k + h) + v - y? ( k ) E 0 (mod p ) , k = 0, 1, 2 ,..., p deter- mines the number of hits for chosen parameters a,, k, h, v (3). It can be rewritten as

a, ( k + h ) 2 + v - u l kZ=O(modp)

2 a , h k + a l h ~ + v ~ O ( m o d p ) , k = 1,2 ,..., p (3)

w e observe:

- If and only if h = v = 0, the equation has p incon- gruent solutions. Thus there is a single position in a cyclical coincidence array with p hits.

- For a given index k, there is only one value of y Q C

(k). There are p - 1 values of k. Thus, a FH pattern shifted with respect to itself only by v (h = 0) for v # 0 never produces a hit (produces a zero hit). It fol- lows that there must be at least p - 1 positions with zero hits.

- The total number of hits in a coincidence array has to be p2 (since each of the two patterns has p ele- ments). It is also known [ 11 that except for h = v = 0 the bound on the height of a hit in a QC code auto- coincidence array is 1 (4). The number of positions in a coincidence array not considered by the two previous bullet items is p2 - p . Also, the number of hits not distributed by the two previous items is p2 - p . Thus there are p2 - p single hits to be distributed among the same number of array positions. With this all hits and array positions are exhausted. 0

(9 Here h represents a shift in k-direction and v represents a shift in minus 1 direction.

The equations used for computing the bound on the maximal num- ber of hits in [ I ] are congruence equations which follow modular arithmetic. That means that the bound is valid for the cyclical coinci- dence arrays presented here. The bound is valid but not an accurate one for classical coincidence arrays.

Theorem 2: In a cyclical cross-coincidence m y of two QC FH codes designed over an odd prime p . there are exactly @2 - p)/(2) array positions with two hits, @2 - p ) / ( 2 ) positions with zero hits, andp positions with one hit.

Proof: For QC FH patterns designed following the modularity requirement in both k and I, equation yf; ( k + h) + v - y c ( k ) 8 0 (mod p ) , k = 0, 1, . . . , p - 1 deter- mines the number of hits for chosen parameters a l , a2, k, h, v. It can be rewritten as a, ( k + h)z + v - a2 k2 I 0 (mod p ) , k = 1, 2, ... , p , and the square cari be completed to obtain (k + A)2 = B (rnodp). Let us f i x the value of h: We then obtain equation (kJ2 = B(v) (modp), where the value of v goes through the complete residue set of prime p . Function B(v) goes trough the complete residue set as well. It is known in number theory [7] that out of possible p values of B(v), @ - 1)/(2) are quadratic residues, and that @ - 1)/(2) values are quadratic nonresidues. For E ( v ) values which are quadratic residues, equation ( k J 2 s B(v)(mod p ) has two solutions: thus two hits occur in the appropriate cyclical coincidence array position (deter- mined by h and v). For values of B(v) which are quadratic nonresidues, equation (k32 = B(v)(mod p ) has no solu- tions: thus zero hit occurs in the appropriate cyclical coin- cidence array position. In the special case when B(v) = 0, there is a single solution k ’ = 0 thus one hit occurs in the appropriate cyclical coincidence array position. We can vary h through the complete residue set of prime p as well. For every h, the same properties follow with the change of v: as a result the total number of array positions with zero hits, one hit, and two hits for all combinations of h and v is p times higher than the number of hits for a fixed value of h. In total, for all combinations of h and v, there are thus @* - p ) / ( 2 ) positions with 2 hits, Q2 - p ) / ( 2 ) positions with zero-hits, and p positions with sin-

13 gle hits in a cyclical cross-coincidence array.

Theorem 3: In a cyclical auto-coincidence array of a Welch-Costas FH code designed over an odd prime p , there is a single position with p - 1 hits, @ - 1) x @ - 2) positions with one hit, and 2 p - 3 positions with zero hits.

Proof: We have established the rules of designing WC FH patterns, by which the size of a FH pattern (thus its cyclical coincidence array) is (p - 1) x p . The equa- tion that determines the number of hits for the chosen parameters a, k, h, v is aR(k+h) + v - aRk = 0 (mod p ) . One can deduce:

-There are @ - 1)2 hits to be distributed within a whole coincidence array (since each of the FH patterns has only @ - 1) elements).

- If and only if h = v = 0, the equation has p - 1 incongruent solutions (since k goes through the set { 1, 2, . . . , p - 1 1). This results in a single position withp - 1 hits.

- From the fact that two frequencies in a FH pattern never occupy the same time-slot, it follows that there are at least p - 1 positions with zero hits. Also, from the fact that WC FH codes are full [ 11

152 ETT

(no frequency in a pattern is repeated more than once), it follows that there must be another p - 2 positions with zero hits. Both facts follow from the bijective nature of the mapping from k to a R‘.

-The total number of hits has to be (p - 1)2, of which we have so far not distributed (p - 1)2 - (p - 1) =p2- 3 p + 2. Also, it is known that except for h = v = 0 the bound on the number of hits at any array position is 1 (5). The total number of posi- tions in a coincidence array not considered so far is @ - 1)p- 1 - (p - 1) - (p - 2) =p2 - 3 p + 2 = (p - 1) (p - 2 ) . Thus, there are (p - 1) (p - 2) hits of size 1 to be distributed among (p - 1 ) (p - 2) array positions. With this, we have exhausted all array positions as well as all available hits. 0

4. APPLICATIONS

The results of the previous sections can be used for computing the performance of various CDMA systems that use the considered codes. The distribution of the number of coincidences between two FH patterns can be mapped into the distribution of the random variable that represents interference between two CDMA system users. This can be done for FH systems, as well as time- hopping and optical CDMA systems [ S ] . This interfer- ence pdf is exucr. From there one can also compute the exact distribution of the random variable that represents the interference by many CDMA system users to a sin- gle user. Finally, the exact error probability due to multi-user interference can be computed (6). This work has been reported elsewhere [ 5 ] . As an illustration, Fig. 6 shows the distribution of the interference between two users for cases when no data is encoded on top of signa-

0.1 7

0.01

0.001 :

1 b

0.0001 I I I I I 0 2 4 6 8 10 12

Threshold

Fig. 7 - Probability of error versus matched filter threshold for an optical CDMA system.

ture sequences, and when (1,O) data is encoded on top of signature sequences. The results of the error compu- tation for one relevant optical CDMA system [5] are shown in Fig. 7.

5. CONCLUSION

We define a new concept of cyclical coincidence arrays. These arrays are valid for a restricted class of “modular” frequency-hop patterns for which the rules of construction are presented. For quadratic congruence codes and for Welch-Costas codes, we prove three theo- rems which express properties of cyclical coincidence arrays. From these properties an important previously unknown set of results is obtained: the exact distribu- tions of the number of hits in auto and cross coinci- dence arrays for the considered codes. These distribu- tions directly represent probability density functions of interference between two CDMA system users. The obtained results are applicable to the problem of com- puting the exact interference from many CDMA users to a single user and thus to the computation of the exact probability of error in various CDMA systems which utilize the considered codes.

Manuscript received on June 23, 1995. Fig. 6 - Probability distribution functions of random variables npre- senting interference without and with data encoded on top of signa- ture sequences.

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(J) The equations used for computing the bound on the maximal num- ber of hits are congruence equations which follow modular arithme- tic. That means that the bound is valid for the cyclical coincidence arrays presented here. The bound is valid but not an accurate one for classical coincidence m a y s .

( 6 ) The reader should note that some frequency-hopping communication systems perform equaly well regardless of the presence of one hit or more then one hit in any particular coincidence array position. For such systems the results of the work presented here are inconsequential.

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