Time-frequency hop codes based upon extended quadratic

I . INTRODUCTION

Time-Frequency Hop Codes Based Upon Extended Quadratic Congruences

JEROME R. BELLEGARDA EDWARD L. TITLEBAUM University of Rochester

Time-frequency hop codes are developed based upon an extension of the theory of quadratic congruences. These codes can be used for coherent multiuser echolocation and asynchronous spread spectrum communication systems. They represent a compromise between Costas codes, which have nearly ideal autobut not so good cross-amhiguity properties, and linear congruential codes, which have nearly ideal cross- but unattractive autoambiguity properties. Extended quadratic congruential (EQC) code words are shown to have reasonably good auto- and cross-ambiguity properties across the whole class of code sets considered. A uniform upper hound is placed on the entire cross-amhiguity function surface, and hounds are placed on the position and amplitude of spurious peaks in the auto-ambiguity function. These hounds depend on time-bandwidth product and code length exclusively, and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical auto- and cross-ambiguity functions are given to illustrate the performance of the new codes.

Manuscript received November 17, 1987; revised February 23, 1988. IEEE Log No. 24683. This work was supported by the SDIOilST and managed by the Office of Naval Research under contract N00014-86-K-051 I . Authors’ current addresses: J.R. Bellegarda, IBM Research, T.J. Watson Research Center, Yorktown Heights, NY 10598; E.L. Titlebaum, Dep’t. of Electrical Engineering, University of Rochester, Rochester, NY 14627. 0018-925118811100-0726 $1.00 8 1988 IEEE 126

Coherent active radar and sonar echolocation systems often use time-frequency hop pulse train signals. Since range measurement or high target resolution is virtually always desired, these signals must be chosen in such a way that their auto-correlation functions exhibit a narrow mainlobe and adequately small sidelobes; these pulse compression characteristics are necessary to determine precisely the time of arrival of the received signal. When either or all of the transmitter, target, or receiver is in motion, these properties should extend to the ambiguity function for the signal set considered. In particular, one would like the auto-ambiguity function to assume the ideal “thumb tack” shape required to perform reliable target and/or channel scattering function measurements [ll. On the other hand, if several active echolocation systems view the same target complex, signals from one system may be interpreted as echoes or outputs from the other systemts). A similar situation arises in asynchronous spread spectrum communications if crosstalk occurs between two or more of the signals (code words) considered. These interference problems are typical of such multiuser environments. To achieve jamming resistance or low probability of intercept, it is necessary to use a sequence of time-frequency hop codes with small cross-correlation functions between any two elements of the sequence: as the exact time of arrival of the received signal is unknown a priori, this property is required to minimize the output of the matched filter for the correct signal in cases where spurious codes are present within the received signal. When motion is involved, one would like to place a uniform upper bound on the entire cross-ambiguity function surface for each code set considered. The need for finding code words which have simultaneously good auto-ambiguity functions and small mutual cross-ambiguity functions is therefore well motivated. Unfortunately, there seems to be qualitative evidence in the literature that a tradeoff is involved between these two entities. On the one hand, timefrequency hop pulse trains based upon Costas arrays [2], such as Welch-Costas codes [3, 41, are known to have nearly ideal auto-ambiguity functions but, even in the best circumstances, not very good cross-ambiguity properties [ 5 ] . Furthermore, for any given Costas set, the behavior of the cross-ambiguity function between two elements is heavily dependent on the pair of codes considered, which makes it necessary to find a “good” pair of codes if one is to use the class of Costas codes in practical multiuser situations. On the other hand, time-frequency hop codes based upon the theory of linear congruences [6] exhibit excellent cross-ambiguity properties across the whole class of codes but are unattractive from the point of view of their auto-ambiguity function. In particular, significant sidelobes can appear within the strip parallel to the time

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS

VOL. 24, NO. 6

NOVEMBER 1988

axis of the ambiguity function, i.e., for zero Doppler. This makes them unsuitable for performing reliable scattering function measurements. A class of time-frequency hop codes is presented which is best described as a compromise between these two coding schemes. It does not attain the nearly ideal auto-ambiguity properties of Costas codes, nor the nearly ideal cross-ambiguity properties of linear congruential (LC) codes, but comes close in both accounts. It is therefore especially well suited for application in multiuser environments. This compromise is based upon an extension of the theory of quadratic congruences . applied by Titlebaum and Sibul [7] to the related context of high-efficiency multicomponent signals. The extension guarantees that all elements of each available code set span all available frequencies; thus, unlike with the quadratic congruence placement of [7], each code word possesses maximum pulse compression capabilities. The paper is organized as follows. Section I1 defines the code word sets and presents some of their numbertheoretic properties. In Section I11 we introduce the concepts of (cyclic) placement difference and of coincidence, or zero placement difference. The properties of the new codes are used in Section IV to obtain a uniform upper bound on the cross-correlation between any two elements of any code set considered. These results can readily be extended to (narrowband) autoambiguity functions. In Section V, we derive similar bounds on the entire cross-ambiguity function surface for any code set. Finally, in Section VI we discuss the design tradeoffs involved in the new formalism. Examples of typical auto- and cross-ambiguity functions are given to illustrate how the performance of the new codes compares with those of Costas and LC codes.

Definition I . The class of extended quadratic congruence placement operators regroups all operators y given by

f o r some integers a and b, members of the set J N = (1, 2 , ..., N - I}. In this definition, expressions of the form y = [x]N should be read, " y is congruent to x, modulo N." Since N is assumed to be an odd prime, J N forms an abelian group under multiplication modulo N ; hence, jN= JN+ (0) = (0, 1, . . . , N - 1) forms a finite field of order N . (For convenience, a brief summary of the concepts of congruence and finite field is provided in the Appendix.) Without loss of generality, we, by convention, take 0 yk 5 N - 1. The integers a and b are specified shortly. Observe first that the choice a = b = 1 entails the quadratic congruence placement discussed in [7]. As an example, the code arrays generated for N = 5 and for N = 7 (with a = b = 1 ) are shown in Fig. 1. Notice the

t

'k 4 3

(a)

2 1

0

I I . CODE WORD SETS

0

We consider a rectangular pulse of length T seconds, divided into N equal segments of length TIN seconds. Throughout this work, N is restricted to be an odd prime; from a practical point of view, this condition induces little loss of generality, and considerably simplifies the analysis. Let B be the approximate (radian) bandwidth of the signal, so that each code word occupies a timebandwidth product of approximately 2BT. In each segment of the pulse (time slot) we place one, and only one, sine wave whose frequency belongs to the set of N (radian) frequencies specified by

1

2

3

4

3

4

k

a=b=l

'k

t

6

5 4

(b) 2

~k

=

WO

+y

B k

7

~

k = 0, 1, ..., N - 1

(1)

where the initial frequency wo is sufficiently high to ensure that we are dealing with analytic signals. The (ordered) set of integers y k , k = 0, 1, ..., N - 1, also denoted by { yk)fZd, is obtained through an extended quadratic congruence placement operator, a member of the following class. BELLEGARDA & TITLEBAUM: TIME-FREQUENCY HOP CODES

1

0 0

1

2

5

6

k

a=b=l Fig. I . N x N grid representation of ( ( X , yk): 0 5 k 5 N - l}. for a) N = 5 , and b) N = 7. Code arrays generated with a = b = I (standard quadratic congruence placement of [7] ). 721

symmetry with respect to the center of the arrays, which demonstrates that the placement operator of [7] induces a many-to-one correspondence over J N . In other words, the set { Y k } f z d is not a permutation of the set 5,, a serious drawback for the type of application considered here: see, e.g., [2]. In the sequel, we investigate proper choices of integers a and b which ensure that the corresponding placement operator induces a one-to-one correspondence over Intuitively, this ought to be possible since each placement defined in (2) is essentially composed of two “independent” operators nominally corresponding to 1 5 k < N d a n d 2N d s k 5 N - 1. To gain further insight into this issue, it is helpful to seek a closed-form expression for (2). From the recursive computation of (2), the following is easily seen.

sN.

1) F o r O < k < v ,

(3)

2) F o r y 5 k s N - 1 ,

( N - I )/2

1

/=o

1

=

k ( k + 1)

N2

, -

1

k ( k + 1)

These three steps, in effect, prove the following Lemma.

LEMMA1. The definition (2) can be rewritten in closed form as

\L

2

THEOREM 1. The sequence of integers { yk}fzl/ defined in ( 2 ) is a permutation of the set J N = { I , 2 , ..., N - I} i f and only i f a and b are not both quadratic residues (QR) or quadratic nonresidues (QNR) of the odd prime N. Each such permutation is uniquely defined by the ordered pair ( a , b). The proof of this result, left to the Appendix, involves the basic properties of the QRs and QNRs of a prime number, and requires some familiarity with the Legendre symbol. Although the main definitions are recalled in the Appendix, the reader is referred to a number-theoretic text, e.g., [8 or 91, for a more thorough treatment of these classical concepts. The significance of the above theorem is that we can parameterize any (extended quadratic congruence) placement operator inducing a one-to-one correspondence over JN by an ordered pair (a, b ) such that a and b are not both QR or QNR. This remark leads to a new class of time-frequency hop codes. Definition 2. The class of extended quadratic conguiential (EQC) codes regroups all code words parameterized by ordered pairs ( a , b ) with a and b not both QR or QNR. As there are QRs and QNRs, there are exactly 2 ( v ) 2 EQC code words, which, in contrast with the code arrays of [7] (and, incidentally, any code obtained through ( 6 ) with a and b both QR or both QNR), are always nonsymmetric. For reasons to become clear in Section IV, we organize these EQC code words in +N sets containing N - 1 code arrays each, with the property that each unequal pair of code words among the N - 1 code arrays of each set has one, and only one, intersection modulo N (at k = 0 by construction). In general, each such set contains the ordered pairs (al, bl), ( b l , a d , (a2, b d , (b2, a d , ..., ( U ( N - 1)/2. b ( N - 1)/2)9 ( b ( N - 1)/23 ai) for Some sequences of all distinct QR or QNR a, and all distinct QNR or QR hi;

v

N

(4) 3) Moreover, for k

At this point, we are able to prove the existence of extended quadratic congruence placement operators inducing a one-to-one correspondence over JN.

ifO