Complex Spreading Sequences with a Wide Range of ... - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

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Complex Spreading Sequences with a Wide Range of Correlation Properties Ian Oppermann and Branka S. Vucetic, Member, IEEE Abstract— This paper presents a new family of complex valued pseudo-random sequences for use in coded multiple access communication systems. The family offers a very wide range of values for both auto-correlation (AC) and cross-correlation (CC) functions, allowing great flexibility in the selection of characteristics of sequence sets. Based on the measure described in this paper for the mean-square aperiodic AC and CC values of a set of sequences, the correlation properties of sets of these sequences are compared to well-known sequence sets, and it is shown that sets from the new family of sequences have superior qualities. Tables of parameters for various sequence sets are presented to enable the construction and comparison of sets from this new family. Index Terms—CDMA, communication theory and signals, polyphase sequences, spread spectrum technology and applications.

I. INTRODUCTION

O

NE OF THE MOST significant limitations in direct sequence (DS) code-division multiple access (CDMA) communications over a finite bandwidth channel is the interference introduced by competing users. Such multiple access interference (MAI) reduces the practical channel capacity and leads to degraded performance due to increased bit-error rate (BER) for individual receivers. In order to limit this interference, asynchronous CDMA systems require the use of sets of code waveforms (signatures) with good cross-correlation (CC) properties. By making the sequences in the set as “orthogonal” as possible [1], the amount of interference experienced at each receiver due to other users is minimized. It is important to note that because many users may be operating in the system at any time, the CC properties of all sequences in the set should be considered when determining the average performance. For this reason, this paper concentrates on the average properties of sequences in a set. In recent times, the definition of “good” signatures for CDMA systems has been moving away from the measure of the maximum value of side lobes of periodic correlations, to the mean-square (MS) value of the aperiodic CC [1]–[3]. This measure is far more reasonable in the context of CDMA systems in which the most meaningful performance parameter, average BER [4], [5], is dependent on the average Paper approved by R. Kohno, the Editor for Spread Spectrum Theory and Applications of the IEEE Communications Society. Manuscript received February 23, 1996; revised June 4, 1996 and July 17, 1996. This is work was supported in part by the Australian Telecommunications and Research Electronics Board (ATERB) and an Australian Research Council (ARC) grant. This paper was presented in part at GLOBECOM, San Francisco, CA, November 1994. The authors are with the Department of Electrical Engineering, The University of Sydney, Sydney, NSW 2006, Australia. Publisher Item Identifier S 0090-6778(97)01978-8.

interference produced by the aperiodic CC, rather than the worst case error associated with the peak interference values between individual users. These works and others indicate that the level of MAI experienced by a given pair of users is heavily dependent on the value of the MSCC function of their respective signature sequences. For these reasons, analysis and results presented in this paper do not touch on maximum peaks of CC functions. Rather, results and discussion are based on values of MSCC. The paper also focuses on the aperiodic correlation functions since this combines both the odd and even periodic correlation functions [6], [7]. The auto-correlation (AC) properties of a set of sequences are important for spectrum spreading as well as for initial timing synchronization and tracking. Sequences with flat spectra are important in CDMA systems to ensure uniform distribution of energy over the entire frequency range considered. Ideally, the AC of a sequence should approximate a delta function in the time domain. This indicates that the sequences should have a wide and flat frequency spectrum. It has been noted in many publications [4], [6] that the AC properties of a set of sequences come at the expense of CC properties and vice versa. Thus, sequences in sets that have very good CC properties will have poorer AC properties and so, necessarily, will have frequency spectra that are less wide-band and flat. The sequence sets presented in this paper are large, with a wide range of achievable CC values corresponding to a wide range of MAI values between users in a CDMA system. They also allow for sets containing sequences with flatter, more uniform frequency spectra for a specified level of MAI. In fact the range of MSCC values achievable with sets from this new family of sequences is by far superior than any other known large set of sequences. In this paper, a number of sets from the new family of sequences have been examined and the correlation properties are compared to those of sets of binary Gold sequences [6] along with the complex Frank–Zadoff–Chu [8], [9] (FZC) sequences. The performance of the new sequences are also examined relative to new poly-phase sequences, such as the EOE-Gold Sequence Family [7]. It will be shown that for all of these well-known sequence families, it is possible to find a set from the new family of the same or similar length that has better correlation properties. II. MEAN-SQUARE CORRELATION MEASURE FOR A SEQUENCE SET In digital communication systems, one of the most common performance measures is the BER a given user will experience.

0090–6778/97$10.00  1997 IEEE

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The BER in a multiple access environment is dependent on the modulation technique used, but invariably it is closely related to the signal-to-noise power ratio (SNR) available at the receiver. Pursley [3] showed that it is possible to express the average SNR at the receiver output of a binary phase shift keying (BPSK), asynchronous, DS-CDMA system using a correlator receiver for the th user as a function of the average interference parameter (AIP) for the other users and the power of the additive white Gaussian noise present in can the channel. The SNR for the th user, denoted as be expressed in the form

A similar measure, denoted by the AC function

(1)

is the bit energy, is the one-sided noise power where spectral density, and is the sequence length. The AIP, for a pair of sequences and is defined as (2) For complex valued sequences, the expression for can be defined as (3) Following the derivation in [1] and [10], however, the AIP, for complex sequences may be well approximated as (4) where is the aperiodic correlation function of two sequences in the set. For a pair of sequences, and the aperiodic correlation is defined in [6] as

may be defined for

(7) where (8) This value allows for comparison of the AC properties of the set on the same basis as the CC properties. These two measures have been used as the basis for comparing the sequence sets in this paper. It should be noted that for a pair of sequences from a set of length evaluating (6) and (7) requires and calculations, respectively, of the term. If the calculation itself requires complex multiplications, then the complexity of (6) and (7) may be estimated at and respectively. A. Figure of Merit As has been mentioned, the price for being able to select sets with good CC properties is a degradation in the AC properties. This has a direct effect on the spectral characteristics of the sequences. The spectral characteristics considered in this paper are determined from the nonperiodic Fourier transform of the time-domain signal. If the MS AC value of a sequence is high, the spectrum will not be wide-band and flat. In order to determine quantitatively how significant this degradation is for a given sequence, a Figure of Merit (FOM), is used. Sequences with a low FOM have narrow or irregular frequency spectrum. The FOM for a sequence is defined in [14] (derived from the work of Golay [15]) in the form (9)

(5) represents the complex conjugation operation. where Thus, the SNR degradation of the th user due to the interference from the other users depends strongly on the squared absolute value of the aperiodic CC of the th sequence with every other sequence in the set. By normalizing each of the CC values by the sequence length it is possible to make a comparison of the AIP for sequences of different lengths. In order to evaluate the performance of a set of sequences as a whole, the average MS value of the CC for every sequence in the set, denoted by may be calculated and used as a measure of the set performance [11], [12] (6)

this is nothing more Apart from a scaling factor of than the inverse of the MS AC value for a given sequence. Therefore, the FOM may be extended to the whole set as each sequence has the same absolute value of AC. III. NEW SEQUENCE FAMILY Let be the sequence length. Let take integer values that are relatively prime to such that The set of sequences is defined by while the th element of a given sequence is defined by

(10) and and are any real numbers. For where the purposes of this paper, the triple specifies the sequence set. The values for this triple may be chosen arbitrarily, however the choice of specific values determines the

OPPERMANN AND VUCETIC: COMPLEX SPREADING SEQUENCES

367

characteristics of the sequences and the correlation properties of the set. Due to the general nature of the sequence definition, this family of sequences will also include some more specific sequence families. The FZC sequences are an example of such a family. They may be generated by the triple An element of the set of FZC sequences is therefore defined as (11) As may be determined from (10), the sequences from the new family are constant amplitude complex numbers. Each sequence in the set will differ only in phase terms. It is possible to show that, if in (10) is set to one, each sequence in the set will have the same AC function magnitude.1 The AC functions of each sequence in any given set (defined by will differ only in phase terms. For the purposes of this paper, only sets with will be considered. The maximum number of sequences in the set for any combination of parameters is determined by Euler’s totient function [13]. When is prime, the maximum set size of is achieved. As it is desirable to have the largest set possible, only prime values of will be considered. IV. CORRELATION PROPERTIES One of the advantages of this new family of sequences is the enormous range of correlation properties that may be obtained. It is possible to generate sets of sequences that have values or values that approach the lower bound for this family of sequences. It is also possible to have many combinations of correlation properties which, for a given value, offer better values that well-known sequences sets. Some of the parameters for these sets are provided in tables in later sections. The sets with the lowest FOM also have very low CC values. These sets have characteristics that are very narrow-band, indicating that the use of such sets in DS CDMA will result in a system that effectively has frequency division multiple access characteristics. According to (1), their use will also result in very low interference between simultaneous users in such a system. Examples of the frequency spectra for sequences from a set with very low FOM are given in Fig. 1. From this figure, it is possible to see that the overlap between the frequency spectra is very low. This manifests in the time domain as low levels of CC. At the other extreme of correlation values, it is also possible to obtain sequence sets with low values (and so high FOM values) from the new family of sequences. Associated with selecting sets with decreasing is an increase in for the set. Results presented in the next section indicate that, in the best case, the value of increases inversely to the decrease in Examples of the frequency spectra and AC for sequences from a set with very high FOM are given in Figs. 2 and 3, respectively. The following theorems and propositions are given to describe quantitatively the bounds on the correlation properties 1 The proof of this statement is given as part of Theorm 1 in the Appendix of this paper.

Fig. 1. Frequency = 5:1e

0 2) (

Solid line—M

spectra

of

four

low-FOM

sequences

(FOM

= 31; M = 1; 2; 3; and 4; p = 1:0; m = 1:0; n = 2:0). = 1; dash—M = 2; dot—M = 3; dot–dash—M = 4:

N

Fig. 2. Frequency

spectra

of

two

= 8:6) (N = 31; M = 1 and 5; Solid line—M = 1; dash—M = 5:

high-FOM p

sequences

(FOM

= 1; m = 1:075; n = 2:0).

of the new sequence sets. The proofs of these are given in the Appendix. Theorem 1: Aperiodic AC function of sequences. Given a set of sequences as defined in (10), the AC function of a sequence from the set is given by

(12)

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Fig. 3. Magnitude of AC function of high-FOM sequence (FOM = 8:6) (N = 31; M = 1; p = 1; m = 1:075; n = 2:0).

The first exponential term in (12) is independent of the index variable which refers to a particular element in the sequence being correlated. Therefore, this term is fixed for all and so becomes a multiplicative constant for the correlation value at a given value of This implies that for a given position in the AC function, the summation magnitude will only vary with the second exponential term. The second exponential term in (12) is independent of the particular sequence chosen. Therefore, it is possible to state that, for a given sequence set, the AC function magnitudes are independent of the particular sequence chosen. The AC function of each element in the set will differ only in phase. Following similar reasoning, it is also possible to infer from (12) that the magnitude of the AC function is also independent of the value of the parameter chosen. Other parameters in the set do change with , however. Fig. 4 shows the effect of modifying on the maximum value of a small subset peak in the CC functions and the of sequences from a particular set. Theorem 2: Aperiodic CC function of sequences. Given a set of sequences as defined in (10), the CC function of a pair of sequence and from the set is given by

Fig. 4. Magnitude of CC parameters of high-FOM sequence set (FOM = 8:6) (N = 31; p = 1; n = 2:0): Solid line—max peak of all set CC functions; dash—RCC value.

(14) Theorem 3: Lower bound on the aperiodic mean-square CC function. Given a set of sequences as defined in (10), the MSCC value of a pair of sequence and from the set is bounded by (15)

and for sets. Theorem 4: Relative value of Given a set of sequences as defined in (10), the and values of the set, as defined by (6) and (7), respectively, are related by (16) or, put in terms of the FOM described in (9), (17) where

is given by

(13) (18) where is the size of the sequence set. In Section V, it will be seen that (17) provides a very tight bound on the maximum obtainable FOM for a set of these sequences.


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continuous phase, full set 3 ) Gold : o—FZC seq set; seq aet; .—new seq family; solid line—boundary of (19).

Fig. 6. Sequence set FOM versus RCC continous phase sequences, six-element subset (N = 31; 0:5 m1 :1 ; 1 :0 n 5:0): o—FZC seq set; 3 ) Gold seq set; .—new seq family; solid line—boundary of (20).

V. NUMERICAL RESULTS

It has been stated in Section I that a set of sequences with a very low value typically has a high value [4], [6]. With this new family of sequences, this typically signifies large values of the maximum peak in the magnitude of the CC function of individual elements in the set. Large peaks in the CC functions can lead to problems in practical systems resulting in high levels of instantaneous bit error in conventional matched filter systems and add to the difficulty associated with sequence synchronization. In order to limit the effect of high peaks in the CC function of the large sequence sets, subsets were investigated with sequences from the set, which excluded pairs that formed high CC peaks. The subset size considered was Fig. 6 shows the versus FOM for 31-b, six-element subsets with and The sixelement subsets with the highest and lowest values for the FZC and Gold sequence sets are also shown. As can be seen, the new family of sequences still offers higher FOM with lower CC values than the other families. The equation for the boundary plotted for the 31-chip-length subset is given in

Fig. 5. Sequence set FOM versus (N = 31; 0:5

m

R

CC

1 1 1 0 5 0) : ;

:

n

:

Fig. 5 shows a plot of FOM versus for sets of 31b sequences, with 30 sequences in the set, with and These sets and their relative positions were found by calculating the values for (6) and (7) over the range of and given for small, discrete increments of each parameter. The figure therefore does not represent all the possible sets that may be produced. Fig. 5 also shows the relative position of randomly selected subsets of Gold (30 elements) [6] and FZC [8] (30 elements) sequence sets of the same length. It is obvious from this figure that the range of possible correlation values is very large indeed. Taking the conventional binary Gold sequences as a reference, sets of length 31 have been found that have values approximately three times lower and other sets have a FOM over nine times higher. The bound plotted for the set of sequences of length 31 is given by (19) ” refers to the set of sequences of size where “ Points along this boundary represent the best trade-off of values for values. As may be seen from Fig. 5, (19) forms a very tight bound for the achievable FOM as a function of It may be seen that the set of Gold sequences lies very close to this boundary, indicating that the set represents as good a combination of CC and AC that is possible to achieve with the full set of the new family of sequences. The position of the FZC set behind the boundary, however, indicates that for the given value of it is possible to find a set from the new family with much higher FOM and vice-versa.

(20) ” refers to the set of sequences of size where “ Table I gives some of the relevant parameters for sets and subsets of sequences with extremes of values. The “CC Pk” column gives the amplitude of the maximum peak in the CC as a proportion of the amplitude of the main peak in the AC. The “AC Pk” column gives the maximum peak value of the AC function side-lobes. It should be noted that large values in this column indicate that the AC function is very wide rather that indicating there are large peaks in the AC function apart from the central lobe.

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MAX FOM

AND

TABLE I MIN CC SEQUENCE SETS

R

TABLE II PARAMETERS FOR SELECTED SEQUENCE SETS

TABLE III PARMAETERS FOR SELECTED SEQUENCE SETS

MAX FOM

AND

MIN

TABLE IV CC SUBSETS FOR KNOWN SETS

R

N = 31

N = 61

Tables II and III provide the parameters for sequence sets with selected values of The sets described in this paper are the result of computer search with various combinations of The subsets listed in these tables are produced by , taking the sequence elements from the whole set, of size

which have the lowest peak CC values. Similar to Table I, Table IV gives the maximum and minimum obtainable values and for subsets of the well-known sequence for sets. The subsets are taken as the same size as those used for the new family of sequences. Comparing the values in Tables II and III to the well-known sets in Table IV, it is possible to see that the new family of sequences offers a very wide range of sets with excellent and combinations. The EOE-Gold sequences described in [7] and the complexvalued sequences described in [16] have been designed with the intention of minimizing the value of the largest peak in the CC function of any pair of sequences in the set. They are included in this paper for completeness. When these sets were evaluated in terms of the and values, it was found that they have very similar values of and to the Gold sequences. For this reason, only the values for sets and subsets of the 31-b–length sequences in [7] are presented. In order to evaluate the performance of conventional correlator receiver systems based on each of these sequence sets, (1) was evaluated to determine the required user (bit energy to thermal noise ratio) to achieve a specific value of SNR (signal power to MAI plus thermal noise ratio) at the decision device of a BPSK system. The level of SNR specified at the receiver was set at 6 dB, roughly that required to allow a BER of 10 for BPSK operating over an AWGN channel. Fig. 7 shows the input required for a user to achieve a SNR of 6 dB at the receiver in the presence of increasing MAI using BPSK over an AWGN channel. The sequence sets from the new family considered are listed in Table V. The “Size” column indicates the size of the subset chosen to perform the evaluation. For the Gold and FZC sets, a subset of nine elements was selected. Subset selection was performed by removing set elements which formed large CC peaks as described above. The sequence length in each case is 31 chips. As may be seen, the increase in input SNR required to maintain the desired SNR at the receiver increases dramatically as the number of users increases for all sets. The performance of systems based on FZC (solid) and gold (dot) degrades so rapidly, that the required input SNR tends to infinity as the


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TABLE VI MINIMUM MSCC VALUES FOR SPECIFIC SEQUENCE PAIRS

while the th element of a given sequence

is defined by

(21)

Fig. 7. Input Eb =No required to achieve SNR of 6 dB at receiver for BPSK system with correlator receiver over AWGN channel for various sequence sets, N = 31: TABLE V PARAMETERS FOR EVALUATED SEQUENCE SETS

N

= 31

number of simultaneous users approaches eight. The system based on sequence set C3 is able to support eight users at an input value of of approximately 22 dB. As sets with relatively smaller values of are considered, however, the maximum possible number of users that may be supported increases dramatically. This may be seen with the use of sets C2 and C1. VI. FINITE-PHASE ALPHABET SEQUENCES The results presented so far have been for sequence sets, which may take continuous phase terms from the range If sequence phase terms may take only discrete values, a slightly different set of results are obtained. It is expected that the finite number of phase values required to produce these sequences will allow them to be more readily implemented in real systems. This section presents the results from sequence sets and subsets that have phase terms corresponding to the th roots of unity. The calculation of the expected phase for any given element is carried out exactly as described in (10), however the final value of the phase is changed to the closest th root of unity. A set of sequences is described in a similar manner to that of (10) and is given below. be the sequence length. Let take integer values Let that are relatively prime to such that The set of sequences is defined by

and and are real numbers as used in where (10) and represents the “nearest neighbor” operation, which maps the continuous phase to the nearest allowable phase. This nonlinear change in the sequence generation process has several effects on the properties of the sequence family. One of the most significant is the result that the assumption of constant AC magnitude function for each element in the set no longer holds. The magnitude of the AC functions of elements of the set are, however, similar, and so the measure may still be applied. The bound presented in Theorem 4 on for continuous phase sets are still applicable. The discrete sequences still offer a comparable range of sets with excellent and combinations. Another difference is the MSCC value of a pair of sequences in a set defined by (21) is no longer bounded by (15). Table VI illustrates the minimum MSCC values found for pairs of sequences for specific sequence sets. This table shows that it is possible to find pairs of sequences that have MSCC values approximately half of that specified by the lower bound of Theorem 2. VII. CONCLUSIONS An inherent problem in the task of designing sequences for use in CDMA systems is the dual requirement of desirable AC characteristics and minimal CC values between sequences. These properties may well be viewed as mutually exclusive criteria, and at best, a compromise must be sought. This paper has presented a new family of constant amplitude, complex valued sequences that allow for such a compromise. Using the measure presented in this paper, the correlation properties of various sequence sets from this new family have favorably compared with well-known sets. It has been seen that for any given CC requirement, the new family can offer a sequence set with superior AC values to existing sets. It was also found that this family allows for the selection of sets with a greater range of correlation values and frequency characteristics than well–known sets. Tables of results have been presented, which give the parameters of various useful sets that have very low and values. The wide range of CC values available, coupled with the ease of generating large numbers of sets, means that it is possible to choose sequences that have the best combination of CC and AC properties for a given application.

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APPENDIX: PROOF

OF

THEOREMS

Theorem 1: Aperiodic AC function of sequences. Given a set of sequences as defined in (10), the AC function of a sequence from the set, is given by

Using the argument that follows Theorem 1, it is clear that the magnitude of the AC function depends only on the summation term in both parts of (29). The magnitude of the AC function for a given will be

(30) (22) Proof: If

and

are defined by (10) then (31) (23)

and (24) Using the definition of the aperiodic correlation function given by (5):

From (31), it is clear that the magnitude of the AC function is symmetric about the central, zero-shift, position. The proof of Theorem follows directly from the proof of Theorem 1. Theorem 3: Lower bound on the aperiodic mean-square CC function. Given a set of sequences as defined in (10), the MSCC value of a pair of sequence and from the set is bounded by (32) Proof: From [6], we can re-write the value of the MSCC function as

(25) As we are considering only the case where becomes

this (33) From the proof of Theorem 1, it is possible to see that for the sequence family being investigated, this equation becomes (34) (26) In order to find the minimum value of the left-hand side of (34), we require the additive term to be as large as possible and negative. We will therefore focus on the additive term

(35) (27) For the AC function,

From (29),

may be written as

so the above equations become

(28)

(36) (29)

As the first exponential term in each correlation function is independent of the index term, it is constant for a given


The

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less then one when is not an integer. So, the minimum summation CC value will be

term may therefore be re-written as

(37) Rearranging gives

A justification for the inequality used in the second step may be found in [17]. Alternatively, if we consider the case where and are both odd or both even, then (40) has the maximum negative value when is equal to (modulo The closest possible value is where is as above. Thus, the summation becomes

(38) The value of will be the largest when the second exponential term is large and positive i.e. when this term equals one. This occurs for Therefore the maximum negative value of becomes

(39) Taking the real part of

which is the same as in the previous case, so the proof proceeds in the same manner. and for sets. Theorem 4: Relative value of Given a set of sequences as defined in (10), the and values of the set, as defined by (6) and (7), respectively, are related by

gives (42) or, put in terms of the FOM described in (9), (43)

(40) where

is given by

is We can now consider two cases; first, when odd, and then when this term is even. Considering the first case, (34) becomes

(44) is the size of the sequence set. where Proof: Focussing on (34) in the proof of Theorem 3, (45)

(41) Equation (41) will be minimized when the summation term is as large as possible and negative. Due to the cosine function, this will occur when is zero (mod N). This cannot occur when thus the closest the cosine argument may come to zero is when is (mod N). The value of will be one when is an integer, and will be some fraction

We will use the same approach of focusing on the additive term (46) as given in (35).

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If the average value of (45) is taken and the result normalized by dividing by , for a given set of sequences of elements, we obtain

(51)

(47) The left-hand side of (47) is simply the equation may be rewritten as

Given that the constant summation terms for one of the sequences is conjugated, the arguments for the summation terms cancel to give

term, so this (52) Taking the real part of

gives

(48) To find the upper bound, we require the magnitude of be as large as possible and for to be negative. From (37), may be written as

to (53) So the right-hand side of (48) becomes

(49) (54)

Noting that the magnitude of the AC function for every sequence in a given set depends only on the exponential terms independent of the sequence number we may rewrite (37) as

Considering only the double summation term in Equation (54), we may write

(50)

(55)

From Theorem 1, it is known that the magnitude of the AC function of every sequence in the set is constant so (50) becomes

Simplifying the double summations by recalling that and collecting like terms gives

(56)


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So, (48) becomes

(57) (58) Rearranging (58) gives the final required inequality (59) ACKNOWLEDGMENT The authors would like to acknowledge with pleasure the many helpful discussions with Dr. P. Rapajic on this subject. REFERENCES [1] K. H. Kärkkäinen, “Mean-square cross-correlation as a performance measure for spreading code families,” in IEEE 2nd Int. Symp. on Spread Spectrum Techniques and Applications (ISSSTA ’92), pp. 147–150. [2] K. H. Kärkkäinen and P. A. Leppänen, “Comparison of the performance of some linear spreading code families for asynchronous DS/CDMA systems,” in IEEE Military Communications Conf. (MILCOM ’91), Maclean, VA, Nov. 4–7, 1991, pp. 784–790. [3] M. B. Pursley, “Performance evaluation of phase coded spread-spectrum multiple-access communication—Part 1: System analysis” IEEE Trans. Commun., vol. COM-25, pp. 795–799, Aug. 1977. [4] D. V. Sarwate, “Mean-square correlation of shift-register sequences,” Proc. Inst. Elec. Eng., pt. F, vol. 131, no. 2, Apr. 1984, pp. 795–799. [5] E. A. Geraniotis and M. B. Pursley, “Error probability for directsequence spread-spectrum multiple-access communications—Part 2: Approximations” IEEE Trans. Commun., vol. COM-30, pp. 985–995, May 1982. [6] D.V. Sarwate and M. B. Pursley, “Cross correlation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [7] H. Fukumasa, R. Kohno, and H. Imai, “Design of pseudonoise sequences with good odd and even correlation properties for DS/CDMA,” IEEE J. Select. Areas Commun., vol. 12, pp. 828–836, June 1994. [8] R. L. Frank, “Polyphase codes with good nonperiodic correlation properties,” IEEE Trans. Inform. Theory, vol. IT-9, pp. 43–45, Jan. 1963. [9] D. C. Chu, “Polyphase codes with good periodic correlation properties,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 720–724, July 1972.

[10] I. Oppermann and B. S. Vucetic, “Pseudo random sequences with good cross-correlation properties,” in Int. Symp. on Information Theory and Its Applications 1994 (ISITA ’94), Sydney, Australia, Nov. 1994, pp. 1001–1006. [11] I. Oppermann, P. Rapajic, and B. S. Vucetic, “Complex valued spreading sequences with good cross-correlation properties,” in Int. Symp. on Spread Spectrum Techniques and Applications (ISSTA ’94), Oulu, Finland, July 1994, pp. 500–504. [12] I. Oppermann and B. S. Vucetic, “Complex pseudo random spreading sequences with a wide range of correlation properties,” in Proc. GLOBECOM ’94, San Francisco, Nov. 1994, pp. 1738–1742. [13] D. V. Sarwate, “Bounds on cross-correlation and auto-correlation of sequences,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 720–724, Nov. 1979. [14] G. F. M. Beenker, T. A. C. M. Claasen, and P. W. C. Hermens, “Binary sequences with a maximally flat amplitude spectrum,” Phillips J. Res., vol. 40, pp. 289–304, 1985. [15] M. J. E. Golay, “The merit factor of long low autocorrelation binary sequences,” IEEE Trans. Inform. Theory, vol. 28, pp. 543–549, May 1982. [16] P. V. Kumar, A. Shanbhag and T. Helleseth, “Quaternary and binary sequences with low correlation,” in IEEE Information Theory Workshop 1995 (ITW ’95), Rydzyna, Poland, July 1995. [17] I. J. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products. New York: Academic, 1965.

Ian Oppermann was born in Maryborough, Qld, Australia, in 1969. He received the B.Sc and B.E degrees (electrical) in 1989 and 1991, respectively, from the University of Sydney, Australia. He is currently a Ph.D. candidate in the Communications Science and Engineering Laboratory at the University of Sydney. Before commencing postgraduate study, he was employed by OTC Australia (now Telstra). His research interests include wideband channel modeling, CDMA receiver structure, and spreading sequences for CDMA systems.

Branka S. Vucetic (M’83) received the B.S., M.S., and Ph.D. degrees from the University of Belgrade, Belgrade, Yugoslavia, in 1972, 1978, and 1982, respectively. In 1986, she joined Sydney University, where she is currently Associate Professor at the Department of Electrical Engineering and the Director of the Communication Science and Engineering Laboratory. Her specific research interests include digital communications, coding, modulation, channel modeling, and communication system simulation.