Peak-to-Average Power Ratio Analysis in Multicarrier DS-CDMA

Transcription

1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 3, MAY Peak-to-Average Power Ratio Analysis in Multicarrier DS-CDMA Xi-Kai Zhao and Xian-Da Zhang, Senior Member, IEEE Abstract Multicarrier direct-sequence code-division multiple access (MC-DS-CDMA) is an attractive technique for achieving high-bit-rate wireless data transmission. Although the potentially large peak-to-average power ratio (PAPR) is an important factor for its application, there is no theoretical PAPR analysis of the transmitted signal in an MC-DS-CDMA system in the literature. The purpose of this paper is to fill such a gap. We derive a general formula for the complementary cumulative distribution function (CCDF) of the PAPR of an MC-DS-CDMA signal, giving a possible lower bound of CCDF. To achieve the lowest bound of CCDF, a criterion is proposed for optimal interleaver design in an MC-DS-CDMA system. By using two interleaver design methods as examples, simulation results demonstrate the effectiveness of the new criterion in interleaver design. Index Terms Code-division multiple access (CDMA), frequency spreading, high-speed wireless communication, multicarrier modulation, peak-to-average power ratio. I. INTRODUCTION FOR WIRELESS communications, it is desirable to support high data rate applications such as image and video. The direct-sequence code-division multiple access (DS-CDMA) system is a strong candidate for third-generation (3G) mobile radio-communication systems because it has higher frequency efficiency than conventional time division multiple access (TDMA) or frequency division multiple access (FDMA) system. However, its capacity is limited by interchip interference (ICI) and multiple-access interference (MAI). On the other hand, with a multicarrier modulation technique, the entire bandwidth is subdivided into several narrow-band subchannels operating at lower data rates. If the number of subcarriers is chosen appropriately, there is only flat fading that has no ICI in each subchannel. Orthogonal frequency division multiplexing (OFDM), which is the standard for digital audio/video broadcasting (DAB/DVB), is a successful example of achieving a high data rate with multicarrier modulation. Recently, there has been considerable interest in a combined scheme of OFDM and CDMA. The CDMA systems combined with OFDM are mainly categorized into two types. One is a frequency spreading system referred to as a multicarrier (MC-) CDMA or OFDM-CDMA Manuscript received November 12, 2001; revised July 2, 2002 and December 6, This work was supported by the National Natural Science Foundation of China under Grant no X.-K. Zhao is with the Department of Automation, the State Key Laboratory of Intellegent Technology and Systems, Tsinghua University, Beijing , China. X.-D. Zhang was with the Key Laboratory for Radar Signal Processing, Xidian University, Xi an , China. He is now with Department of Automation, the State Key Laboratory of Intellegent Technology and Systems, Tsinghua University, Beijing , China ( zxd-dau@mail.tsinghua.edu.cn). Digital Object Identifier /TVT system [1] [3] in which multiple copies of the same data symbol, each multiplied by one chip of a user-spreading code, are transmitted on different subcarriers. That is, the spreading is done in the frequency (subcarrier) domain. Another is a time spreading system in which serial-to-parallel (S/P) converted data symbols are direct-sequence spread-spectrum (DS-SS) modulated, respectively, using a user-spreading code in the time domain. These DS signals are transmitted in parallel on different subcarriers, as shown in Fig. 1. This system is called a multicarrier (MC) DS-CDMA system [4] [6]. In recent years, there has been considerable interest in MC-DS-CDMA systems due to their bandwidth efficiency, frequency diversity, lower speed parallel type of signal processing, and interference rejection capability in high data-rate transmission [3] [9]. For MC-DS-CDMA systems, the transmitted signal in a multicarrier modulation system is the sum of signals in each subcarrier, so its envelope and the transmitted power may vary greatly. That is, like other multicarrier transmission systems such as OFDM or discrete multitone (DMT), MC-DS-CDMA suffers a high peak-to-average power ratio (PAPR). In a practical RF transmitter, PAPR determines the dynamic linear-range requirement for power amplifier. A potentially large PAPR will result in at least the following three disadvantages [10]: a) if the peak power exceeds the dynamic range of the amplifier, the transmitted signal will be clipped and, thus, give rise to intersubcarrier modulation and out-of-band radiation; b) to prevent spectral growth of the transmitted signal, the transmission amplifier must operate in its linear region (i.e., with a large input backoff), where the conversion from dc to RF power is highly inefficient; c) the required large-linear range directly translates into significantly more expensive devices, limiting widespread industrial applications. Great efforts have been devoted to the reduction of PAPR in multicarrier modulation systems [11] [18]. In a broadcasting transmission scheme such as OFDM or DMT, several methods such as nonlinear block coding [11], partial-transmit sequence (PTS) [12], and selective mapping (SLM) [13] are proposed for PAPR reduction. Recently, the PAPR for MC CDMA or OFDM CDMA systems has been widely studied, see e.g., [16] [18]. In all of the PAPR reduction techniques [11] [18], mutual independence is assumed over parallelly transmitted signal in each subcarrier. However, this may not be the case in an MC-DS-CDMA system as described in [6]. In such a system, the high-speed serial data stream is first S/P converted to parallel substreams; each substream will then modulate several subcarriers to achieve frequency diversity. The subcarriers modulated by the same substream are called identical-bit /03$ IEEE 转载

2 562 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 3, MAY 2003 Fig. 1. Transmitter diagram of the MC-DS-CDMA system, P = 2. subcarriers [6] and the number of identical-bit subcarriers for a substream is called the frequency-diversity number. If this number is greater than one, then the parallel signals transmitted over the subcarriers will not be independent any more. In this sense, the PAPR analysis for MC-CDMA or OFDM-CDMA systems is not available for MC-DS-CDMA systems. As a matter of fact, the dependence over the parallel transmitted signal will result in severe PAPR problem, as will be shown in this paper. To our best knowledge, there is no PAPR analysis for MC- DS-CDMA signals in the literature, although MC-DS-CDMA is a very attractive technique for high bit-rate transmission in a radio environment and the potentially large PAPR may limit its application. The purpose of this paper is to evaluate the PAPR effect introduced by the identical-bit subcarriers in an MC-DS-CDMA system and to propose a way to mitigate this effect while retaining the ability to achieve frequency diversity. The remainder of this paper is organized as follows. A general MC-DS-CDMA system model is described and the PAPR problem in such a system is stated in Section II. In Section III, we derive a formula that describes the statistical property of the PAPR in MC-DS-CDMA with frequency diversity. Based on this formula, we obtain the lowest PAPR achievable in the system. The sufficient condition to achieve this PAPR is also presented. In Section IV, we analyze the PAPR of two frequency-diversity methods in MC-DS-CDMA. Simulation results are presented in Section V to demonstrate the effectiveness of our theoretical analysis. Finally, the paper is concluded in Section VI. II. SYSTEM MODEL Our MC-DS-CDMA model is the same as [6] [9], except that we add a frequency interleaver to accommodate different frequency-diversity methods. The block diagram of the transmitter of the MC-DS-CDMA system is shown in Fig. 1. It is assumed that is the original symbol stream of the th user taking values from a QPSK constellation equally. After S/P operation, this stream is converted into parallel substreams. Each substream is then spread by the th user s spreading code to achieve multiple access ability. Usually, each substream will modulate subcarriers to get frequency diversity. These subcarriers are called identical-bit subcarriers [6]. Thus, the total number of subcarriers in our MC-DS-CDMA system is equal to. The process of assigning different subcarriers for each substream is called interleaving and the corresponding assignment algorithm is referred to as an interleaver. In fact, any interleaver can be viewed as a 1 vector that contains a permutation of the index sequence. After interleaving, an inverse discrete Fourier transform (IDFT) operation is performed to implement the multicarrier modulation [20]. If users are assumed to be active and the processing gain is, then the base-band signal at the transmitter for the th chip is given by where denotes the absolute subcarrier index for the th identical-bit subcarrier used to transmit the th substream and is the interleaver for the th user at the time of interest. For a common equally spaced interleaver (ESI) [6] [9],, i.e.,. In (1), is the average energy transmitted in each subcarrier during a chip interval and is the angular frequency of the th subcarrier specified by where is the chip interval in the subcarriers and is the lowest subcarrier frequency. It is assumed that if the transmitted signal is sampled at the Nyquist rate and, then the digital version of is represented as for. (1) (2) (3)

3 ZHAO AND ZHANG: PEAK-TO-AVERAGE POWER RATIO ANALYSIS IN MULTICARRIER DS-CDMA 563 III. PAPR ANALYSIS The PAPR of any continuous-time signal is defined as the ratio of the peak envelope power to the average envelope power, i.e., where is the average envelope power of. For our MC-DS-CDMA model shown in Fig. 1, the PAPR of the discrete-time transmitted signal is given by where is the instantaneous power of the transmitted signal at the sampling point and is the average power specified by Here, represents the expectation operation. Since the PAPR of a transmitted signal is a random variable, it is meaningless to determine the exact value of PAPR and its statistical information is preferable. Usually, we are interested in the probability of PAPR exceeding a certain level, called the complementary cumulative distribution function (CCDF) of the PAPR, namely Especially, if the CCDF curve of system A is lower than that of system B, we say that the PAPR of system A is lower than that of system B. (4) (5) (6) (7) Since the autocorrelation function of the transmitted signal sequence and its power spectrum form a Fourier transform pair, the PAPR property of the foregoing sequences can be estimated by the sidelobe of the autocorrelation function of the sequences (in the frequency domain). If the sidelobe has large values, then the PAPR of the sequence is also large. Based on this observation, a simple but effective peak-reduction scheme is proposed by Ochiai and Imai [16] for OFDM CDMA. A natural question to ask is whether the above CCDF formula of transmitted signal in an MC-CDMA system is directly applicable for an MC-DS-CDMA system. The next subsection will answer this question. B. CCDF for MC-DS-CDMA Systems To answer the above question, let us consider mutual independence of signal samples. In OFDM or DMT, since no frequency diversity is incorporated, the parallel substreams transmitted in each subcarrier will be independent. However, if some kind of forward error-correction (FEC) coding, such as convolutional or turbo coding, is used, there will be some dependency between subcarriers. But this dependency will be very small as long as the number of subcarriers is reasonably large, since the coded source is S/P converted into parallel substeams and the dependency among the substeams is weakened. Due to this fact, one can ignore the dependency between subcarriers introduced by FEC and assume that the parallel substeams are independent after S/P operation. This implies that the signal samples are also independent in OFDM or DMT since they are the IDFT of the parallel substreams. In MC-DS-CDMA, however, due to introduction of frequency diversity, independent substeams are mapped into subcarriers, so some degree of dependence must exist among these parallel subcarriers. In other words, after IDFT operation, at different times are usually of dependence upon each other. It should be stressed that we can make be independent of if designing an interleaver advisably. To see it, let us analyze the dependence of. As shown in (3), the transmitted signal is the sum of independently identically distributed (i.i.d.) complex random variables. If is large enough, then we can regard as a complex Gaussian variable by the central limit theorem [21]. It is well known that the dependence is completely determined by the correlation function for a Gaussian random process, while the correlation between different times is defined by A. CCDF for MC-CDMA Systems For a downlink MC-CDMA or OFDM-CDMA system, by using the approximation approach in [14] and[15], it is known that under the assumption of the real and imaginary parts of -point IFFT output samples having mutually independent Gaussian probability distribution function with, probability of any of samples exceeding certain PAPR is given by (8) where and are identical-bit subcarriers, respectively. Hence, when designing an interleaver, it is expected that a), i.e., the transmitted signals at different times are independent, which may suppress the redundance of the transmitted information; b) the bit error rate (BER) is as small as possible; c) the transmitter has as low PAPR as possible. (9)

4 564 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 3, MAY 2003 Let be the fading amplitude, where the index indicates the subchannel that has the largest fading amplitude among all the subchannels of the th user (mobile). In this paper, the BER conditioned on is defined by [8] Theorem 4.1: If the transmitted signal is independent of for, then the CCDF of PAPR satisfies the inequality (14) where When the average sample power is equalized, i.e., (15) and is the energy per bit ( is the processing gain), and represents the energy of the additive Gaussian white channel. In the next section, we further discuss how to make via interleaver design. Here, we assume that the transmitted signal is independent of for all. Under this assumption, the instantaneous transmitted power is independent of for as well. Since is an independent complex Gaussian signal, must be an independent exponentially distributed random process whose cumulative density function is given by [21] the CCDF attains to its lower bound, i.e., the equality occurs in (14). Proof: Define an auxiliary function as (16) where is a constant. It can be easily verified that the second order derivative of is nonnegative, i.e.,. That is, is a convex function. By the famous Jensen s inequality for convex functions, we have (17) where is called the average sample power, defined as (11) Combining (16) and (17) and let, we get (12) Using, we can deduce the CCDF of PAPR as (18) Since, we can rewrite (18) as (13) This is a general CCDF formula for MC-DS-CDMA with frequency diversity. It shows that the CCDF is determined by the average sample power if the transmitted signal is independent at different times. IV. LOWER BOUND OF CCDF In this section, we first deduce the lower bound of CCDF of PAPR for MC-DS-CDMA and then compare the CCDFs for the two existing interleaver design methods. A. Lower Bound of CCDF For our MC-DS-CDMA model shown in Fig. 1, the following theorem reveals the lower bound of its CCDF. or (19) Substituting (19) into (13), we have (14) immediately. Clearly, when the condition (15) is satisfied, the equality holds in (14). Theorem 3.1 provides a criterion for optimal interleaver design, which gives the lowest PAPR. This criterion is to design an interleaver such that the equal sample power condition (14) is satisfied. In the following, we demonstrate this criterion using two existing interleavers as examples.

5 ZHAO AND ZHANG: PEAK-TO-AVERAGE POWER RATIO ANALYSIS IN MULTICARRIER DS-CDMA 565 B. CCDF Performance for Equal Space Interleaver In an MC-DS-CDMA system, a frequency interleaver is used to assign subcarriers among subcarriers to each substream. A commonly used interleaver [6] [9] is the so-called equal space interleaver (ESI), which assigns the th, th,, and th subcarriers to the th substream, where. The basic idea behind this ESI is to choose the subcarriers as apart as possible so that the correlation between them are as weak as possible. Hence, when, the multiple substream type system, becomes the single symbol stream type system. For the ESI method, if PSK modulation is used, then. Moreover, the spreading code. Case 1: For, from (9) it follows that (See equation (20) at the bottom of the page.) Using the summation formula of a geometric series, we have and (21) (22) where we have used for any integer. Substitute the above two results into (20) to get This implies that when the ESI is used, the transmitted signal is independent of. Case 2: When, (9) gives the average sample power of the transmitted signal, namely where we have used (21) and (22), respectively. Equation (23) shows that the ESI method does not satisfy the condition (15) in Theorem 4.1 and, thus, the CCDF of PAPR of transmitted signal cannot attain to its lower bound. As a matter of fact, due to large variations in average sample power, a large CCDF of PAPR will be suffered in such a frequency diversity method. However, if, i.e., no frequency diversity is introduced and the parallelly transmitted substream in each subcarrier is independent, then the average sample power will be equal and the MC-DS-CDMA system will have the lowest PAPR. C. CCDF Performance for Adaptive Subchannel Allocation Method Another alternative for interleaver design is proposed by Kim et al. [8] who call it the adaptive subchannel allocation (ASA) method, which is available for the single-symbol stream type MC-DS-CDMA in forward link. In their method, the th mobile estimates the fading amplitudes of all subchannels and feeds back the index of the subchannel having the largest fading amplitude to the base station. Then, the base station allocates the th user s DS waveform into the subchannel with the index provided by the mobile. Clearly, this method can be easily extended to the multiple substream type MC-DS-CDMA: the mobile estimates the fading amplitudes of all subcarriers and feeds back the indexes of subcarriers, which have, respectively, largest subcarrier fading amplitudes to the base station, while no power is allocated to the remaining subcarriers. Following the indexes, the base station allocates the th user s th substream into the th subcarrier. For an MC-DS-CDMA system with the ASA method, its transmitted signal is given by (24) where is the subcarrier selected to transmit the th substream. In this kind of frequency-diversity method, the correlation function of is given by otherwise. ; (23) (25) (20)

6 566 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 3, MAY 2003 The probability of each subcarrier selected to transmit the substream is the same and is given by th TABLE I SIMULATION PARAMETERS. (26) When since is the discrete random variable, from (25) and (26) it follows that (27) By using the summation formula of a geometric series, we get since for all. Hence, (27) becomes for all. In conclusion, the Gaussian signal is independent of if the ASA method is used in interleaver design. On the other hand, when (25) gives the average sample power (28) Substituting (23) and (28) into (6), respectively, we know that the ESI and ASA methods have the same average power. From (28) and Theorem 4.1, we conclude that for the ASA method the CCDF of PAPR attains to its lower bound (29) in contrast to the ESI method. By [8], we know that the BER performance of the ASA is better than that of ESI. Therefore, the ASA is clearly a better way to achieve both frequency diversity and PAPR reduction, as compared with the ESI. We point out that the above PAPR analysis was not made by Kim et al. [8], who proposed the ASA method. It should be stressed that our PAPR analysis is available for any potentially new interleaver as well, which can evaluate any new interleaver from the viewpoint of PAPR. Generally speaking, we have the following two basic criterions for evaluating an interleaver: 1) the output sequence after frequency diversity should be as statistically independent as possible before IDFT in order to get the lowest CCDF of PAPR for the transmitted signal; 2) the diversity gain should be as high as possbile in order to get the lower BER. As an example, when each substream is assigned to get one randomly selected subcarrier, these selected subcarriers are statistically independent such that (24) (26) hold and, thus, such an interleaver would also achieve the CCDF lower bound in (29). However, this method for interleaver design is clearly worse in transmit diversity because it does not utilize the best subcarriers available and will result in worse BER performance, as compared with the ASA method. V. NUMERICAL RESULTS The above theoretical analysis on the PAPR of MC-DS- CDMA system was verified by numerical simulations. All the simulations were considered in the synchronized downlink. Key simulation parameters are listed in Table I. To effectively implement the ASA method, a Rayleigh-fading multipath channel was simulated by the Clarke model [22]. Figs. 2 and 3 show complimentary cumulative distribution functions (CCDFs) of PAPR for the ESI and ASA methods when the frequency diversity number and, respectively. The dotted lines are the theoretical results given by (13) and the solid lines are the simulation results. In order to better approximate the true PAPR, the CCDF should be computed by oversampling. Note that symbol-spaced sampling will miss some of the singal peaks and result in optimistic results for the PAPR [18]. In simulations, we took the oversampling factor equal four for computation reduction. As a matter of fact, oversampling with a smaller oversampling factor will still miss some of the signal peaks and result in optimistic simulation results, which seem to be better than the theoretical result, as shown in [13, Fig. 2] and Figs. 2 and 3 in this paper. It should be noted that such a phenomenon may be more serious for a larger PAPR since more signal peaks may be missed. It can be seen from these figures that as is increasing, the CCDF curve for the ESI is enlarging, while the curve for the ASA is unchanged. We can also see that considerable PAPR reduction gain can be achieved by the ASA method. For example, if the CCDF level is taken as and diversity

7 ZHAO AND ZHANG: PEAK-TO-AVERAGE POWER RATIO ANALYSIS IN MULTICARRIER DS-CDMA 567 Fig. 2. results. CCDF of PAPR for the ESI and ASA with diversity number P = 2, where the dotted lines are the theoretical results, while the solid lines are the simulation Fig. 3. results. CCDF of PAPR for the ESI and ASA with diversity number P =4, where the dotted lines are the theoretical results, while the solid lines are the simulation number, the corresponding for the ASA is about 11.5 db, while that for the ESI is 16.5 db. VI. CONCLUSION In an MC-DS-CDMA system, the PAPR of the transmitted signal is an important factor for system implementation since the potentially large PAPR has limited its application. In this paper, we have presented the PAPR analysis for MC-DS-CDMA systems. First, we have deduced a general formula for the CCDF of the PAPR of the transmitted signal in an MC-DS-CDMA system, giving a possible lowest bound of the CCDF. To achieve this lowest bound, a criterion has been presented for optimal interleaver design, i.e., the output sequence of the interleaver should be statistically independent. By using the ESI and the ASA as two examples, simulation results have demonstrated the

8 568 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 3, MAY 2003 effectiveness of our theoretical analysis on PAPR. It is noted that the new criterion for optimal interleaver design can be used for evaluating any potentially new interleaver design method from a viewpoint of PAPR reduction. REFERENCES [1] N. Yee, J. P. Linnarts, and G. Fettweis, Multi-carrier CDMA in indoor wireless radio networks, IEICE Trans. Commun., vol. E77-B, pp , [2] K. Fazel, S. Kaiser, and M. Schnell, A flexible and high performance cellular mobile communications system based on orthogonal multi-carrier SSMA, Wireless Personal Commun., vol. 2, pp , [3] S. Hara and R. Prasad, Overview of multicarrier CDMA, IEEE Commun. Mag., vol. 35, pp , Dec [4] Q. Chen, E. S. Sousa, and S. Pasupathy, Performance of a coded multicarrier DS-CDMA system in multi-path fading channels, Wireless Personal Commun., vol. 2, pp , [5] S. Kondo and L. B. Milstein, Performance of multicarrier DS CDMA systems, IEEE Trans. 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New York: Prentice-Hall, Xi-Kai Zhao was born in Wuhan, China, in He received the B.S. and M.S. degrees from Northwestern Polytechnical University, Xi an, China, in 1996 and 1998, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2001, all in electrical engineering Since October 2001, he has been a System Level Design Specialist with Synopsys Inc., Shanghai, China. His research interests include statistical signal processing and signal processing for wireless communications. Xian-Da Zhang (SM 93) was born in Jiangxi, China, in He received the B.S. degree in radar engineering from Xidian University, Xi an, China, in 1969, the M.S. degree in instrument engineering from Harbin Institute of Technology, Harbin, China, in 1982, and the Ph.D. degree in electrical engineering from Tohoku University, Sendai, Japan, in From August 1990 to August 1991, he was a Postdoctoral Researcher with the Department of Electrical and Computer Engineering, University of California at San Diego. In 1992, he was with the Department of Automation, Tsinghua University, Beijing, China, as a full Professor. From April 1999 to April 2002, he was with the Key Laboratory for Radar Signal Processing, Xidian University, Xi an, China, as a Specially appointed Professor awarded by the Ministry of Education of China and the Cheung Kong Scholars Programme. He has published over 20 papers in several IEEE Transactions and is the author of five books. He holds four patents. His current research interests include signal processing with applications in radar and communications and intelligent signal processing.