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Crest Factors of Complementary-Sequence-Based Multicode MC-CDMA Signals Byoung-Jo Choi and Lajos Hanzo

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Crest Factors of Complementary-Sequence-Based Multicode MC-CDMA Signals Byoung-Jo Choi and Lajos Hanzo 1114 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 6, NOVEMBER 2003 Crest Factors of Complementary-Sequence-Based Multicode MC-CDMA Signals Byoung-Jo Choi and Lajos Hanzo Abstract—The crest factor properties of binary phase-shift be bounded by 4 dB upon carefully selecting the code sets keying modulated two- and four-code assisted multicarrier and upon employing high-rate crest-factor reduction coding, code-division multiple-access (MC-CDMA) employing comple- provided that the number of simultaneously used codes is mentary-sequence-based spreading sequences are characterized. More specifically, a complementary-sequence pair, a comple- higher than or equal to four. When is less than four, the mentary-sequence-based subcomplementary code pair, and a MC-CDMA signals employing orthogonal Gold (OGold) Sivaswamy’s complementary code set are studied. It was found codes, Frank codes [8], and Zadoff–Chu codes [10], [11] that the corresponding crest factors are bounded by 3 dB, which resulted in considerably lower crest factors. Even though corresponds to the crest factor of a single sine wave. This low crest Zadoff–Chu codes indeed do produce an ideal crest factor of factor resulted in a lower power loss and a lower out-of-band power spectrum due to clipping when the time-domain signal was less than 3 dB in the context of single-code MC-CDMA, the subjected to a typical nonlinear power amplifier, in comparison to crest factor values of neither Zadoff–Chu codes, Frank codes, those of Walsh code and orthogonal Gold code-spread MC-CDMA nor those of OGold codes are bounded by 3 dB in the context of signals. two- and four-code-based MC-CDMA. It is widely recognized Index Terms—Complementary set, crest factor (CF), Golay that complementary sequences, also known as Golay codes codes, multicarrier code-division multiple-access (MC-CDMA), [17], produce crest factors bounded by 3 dB [18], [19], when Shapiro–Rudin sequence, subcomplementary set. they are applied to single-code MC-CDMA as in our context of interest [20], [21]. Ochiai and Imai [21] also reported that I. INTRODUCTION the employment of orthogonal complementary sequences in the context of multicode MC-CDMA resulted in low crest HE fundamental problem of reducing the dynamic range factors, which was shown using simulation studies. However, T of the sum of trigonometric series has been studied in the crest factor properties of Shapiro–Rudin-based codes in the numerous fields, such as in mathematics [1], [2], in radar engi- context of two- and four-code MC-CDMA signal have not been neering [3], [4], in measurement science [5], and in the context documented in the literature to the best of our knowledge. of orthogonal frequency-division multiplexing (OFDM) [6]. In this letter, we characterize the advantageous crest factor As a result, various binary and polyphase codes, such as Barker (CF) properties of three complementary-sequence-based codes [7], Shapiro–Rudin sequences [1], Frank codes [8], spreading sequences [17], namely those of a complemen- Newman phases [2], Schroeder phases [9], and Zadoff–Chu tary-sequence pair [17], a complementary-sequence-based codes [10], [11] have been investigated in order to arrive at a subcomplementary pair [3], and those of a Sivaswamy’s com- low crest factor. plementary set [3], in the context of both two- and four-code The crest factors of multicarrier code-division multiple-ac- binary phase-shift keying (BPSK) modulated MC-CDMA cess (MC-CDMA) [12], [13] signals have also been studied in schemes in order for transmitting two and four bits simultane- [14] and [15], where each user is assigned a different subcarrier ously. The scenario investigated can also be encountered in the offset. The envelope power of the multicode MC-CDMA signal context of multipulse radar signals [4]. was also analyzed in [16] and it was shown that the combined The letter is organized as follows. Section II introduces the aperiodic correlation properties of the spreading sequences em- multicode MC-CDMA system model considered. Section III ployed characterize the envelope power, which can be viewed presents the crest factor properties of three complementary-se- as a direct extension of Schroeder’s early findings [9] in the quence-based spreading sequences. The effects of clipping in context of a single-code scenario. A desirable set of spreading terms of the output power reduction and the out-of-band spu- sequences designed for multicode MC-CDMA is expected to rious power of multicode MC-CDMA signals employing com- exhibit low values of aperiodic correlations, while exhibiting plementary-sequence-based spreading sequences are presented orthogonality between the sequences. While investigating the in Section IV in comparison to those of other spreading se- crest factor properties of multicode MC-CDMA employing quences, followed by our conclusions in Section V. various spreading sequences, it was found in [16] that the crest factors of multicode MC-CDMA employing Walsh codes can II. SYSTEM MODEL AND ENVELOPE POWER The simplified transmitter structure of the multicode Manuscript received November 17, 2001; revised May 20, 2002; accepted June 4, 2002. The editor coordinating the review of this paper and approving it MC-CDMA system considered is portrayed in Fig. 1. Each for publication is C. Tellambura. of the number of input bits is spread The authors are with the Department of Electronics and Computer Sci- with the aid of an -chip spreading code selected from the ence, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]). set . Then the resultant number of Digital Object Identifier 10.1109/TWC.2003.819040 modulated spreading codes are superimposed and each of 1536-1276/03$17.00 © 2003 IEEE IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 6, NOVEMBER 2003 1115 Fig. 1. Multicode MC-CDMA transmitter schematic. the composite chips is mapped to one of the subcarriers. that , where is Finally, -point inverse fast Fourier transform (IFFT)-based the aperiodic crosscorrelation between the sequences and modulation is used for generating the modulated in-phase ( ) , defined as . When and quadrature-phase ( ) signals of Fig. 1. The system model the two sequences are identical, i.e., we have , the described above may correspond to the downlink scenario -transform of the corresponding autocorrelation function of an MC-CDMA base station or to the uplink scenario of can be represented as . mobiles transmitting multiple bits combined with time-division Then, a pair of equally long sequences is said to be a multiple-access or orthogonal frequency-division multiple-ac- complementary pair, if and only if [17], [19] cess. Although the attractive family of quadrature amplitude (3) modulation schemes [22] can be readily used for generating the information symbols from the source data bits, here on the boundary of the unit disc in the complex plane, i.e., at we limit the symbol mapping scheme to BPSK. Furthermore, . we limit our choice of the spreading sequences to those having elements satisfying . The normalized complex A. Two-Code Case envelope of a BPSK modulated multicode MC-CDMA signal Let and be two spreading sequences of length , em- may be represented for the duration of a symbol period as ployed for transmitting the information bits and , respec- tively. Then, the normalized complex envelope of the BPSK (1) modulated two-code MC-CDMA signal of (1) can be expressed as where is the number of simultaneously transmitted symbols and is the subcarrier separation parameter [12] invoked for mapping the -chip spread information symbol to subcar- riers that are sufficiently far apart in the frequency domain, in order to experience independent fading of the different (4) subcarriers. The CF of the multicode MC-CDMA signal is defined (5) as the ratio of the peak to the root mean square amplitude [18] (2) where the superscript in was used for denoting the number of simultaneously transmitted codes and we used . where and Lemma 1: The crest factor of the two-code BPSK . The CF values are often represented in terms of decibels as MC-CDMA signal employing a complementary pair of . Sometimes the peak factor (PF) or peak-to-mean sequences as its spreading sequence is bounded by 3 dB. envelope power ratio (PMEPR) is used instead of the CF, which Proof: Let and be the pair of complementary se- are defined as . quences of length . Then, the envelope power of given in (5) can be expressed as III. COMPLEMENTARY SEQUENCE-BASED SPREADING SEQUENCES (6) Let us denote the -transform of a sequence as , which is de- fined as . Then, we can readily verify (7) 1116 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 6, NOVEMBER 2003 where we used the property of a pair of complementary B. Four-Code Case sequences given in (3) and the inequality relationship of So far we have concentrated our attention on the specific . Since cases, when two symbols are transmitted simultaneously. Let us the average power of a two-code BPSK modulated MC-CDMA now extend our discussions to the scenario where four spreading signal is , the corresponding PF is bounded by two. sequences are used simultaneously. Let be We would like to point out that although Lemma 1 was a set of four spreading sequences of length , employed for presented in the context of BPSK modulation for the sake transmitting the information bits , , , and , respec- of consistency with other lemmas, it can be readily shown tively. Then, the normalized complex envelope of the BPSK that Lemma 1 holds also for the more generic class of -ary modulated four-code MC-CDMA signal of (1) can be phase shift keying modulated signals. Furthermore, it can be expressed as shown1 that when and are two arbitrary complementary sequences having a length of , which are not necessarily a complementary pair, the corresponding crest factor is bounded (14) by 6 dB in the same scenario as that in Lemma 1.
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