Study on modulation techniques free of orthogonality restriction

Science in China Series F: Information Sciences 007 SCIENCE IN CHINA PRESS Springer Study on modulation techniques free of orthogonality restriction CAO QiSheng & LIANG DeQun Department of Information Engineering, Dalian Maritime University, Dalian 11606, China The Nyquist rate is a limit of transmission for traditional modulation methods from orthogonality restriction. Nonorthogonal modulation techniques (NMT) is proposed, which removes the orthogonality restriction, and as a result, higher bandwidth efficiency than the traditional methods can be achieved. First, the symbol error rate of NMT is introduced by using estimation theory. Then the relation between bandwidth efficiency and signal to noise ratio is discussed. Finally, a design instance of NMT is present and numerical experiment is made. This study explores for new modulation methods and points out a widened direction for modulation theory and applications. digital communications, Nyquist rate, nonorthogonality, bandwidth efficiency 1 Introduction The network technique is applied more and more widely. Data, voice, and video transmission applications are more welcome than ever. The current access technique cannot satisfy subscribers strong and continuous desire for high-performance service of digital communications. This situation raises the development of various communication services, such as ADSL and VDSL [1], bluetooth [], mobile communications of 3G [3] and 4G [4], Wi-Fi [5], WiMax [6], UWB [7], etc. However, the development of modulation methods becomes relatively pale by comparison. For example, the multi-carrier theory, fundament of ADSL, VDSL, 3G, and 4G, was first proposed early in ref. [8]. Another example, FSK and PSK, introduced much earlier, are still used to overcome multi-path fading in the mobile and long-distance wireless communications of today. Since modulation method is fundamental in digital communications, if a breakthrough is achieved in its research, there will be a huge impact on the progress of digital communications. Then, what is the key to achieve breakthrough and create new modulation methods? We find that orthogonality restriction limits the performance of traditional modulation methods. Some examples are as follows: OFDM system has high out-of-band interference and is sensitive to fre- Received June 16, 006; accepted April 9, 007 doi: 10.1007/s1143-007-0064-z Corresponding author (email: stab@newmail.dlmu.edu.cn) Supported by the National Natural Science Foundation of China (Grant No. 607017) www.scichina.com www.springerlink.com Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896

quency dispersion, because the orthogonality restriction forbids the system to use better-localized pulseshaping except rectangular function; in baseband communication system, the orthogonality restriction contained in Nyquist criterion makes it difficult to design spectrally efficient baseband waveform; moreover, the restriction limits the symbol rate of baseband system to Nyquist rate [9]. Certainly, it is the orthogonality restriction that prevents the traditional methods from achieving better performance. We ever proposed a modulation method called phase-offset overlapped wave technique [10], which is nonorthogonal in nature and has a much higher transmission rate than the traditional. In this study, the method is extended to a general form, nonorthogonal modulation technique (NMT), and some theoretical, and simulation results are given. NMT fundamentals In traditional digital communication systems, the transmitted signal can be represented by a linear combination of finite number of orthogonal bases [11,1] : K 1 i i= 0 yt () = xg(), t 0 < t < T, (1) i where ( ) gi () t is a set consisting of K real functions, i = 0,1,, K 1. Eq. (1) can be rewritten in vector form as follows: y = Gx, () yt is the transmitted signal, x i denote digital information symbols, and { } T where x = [ x0 x1 xk 1] and G = [ g0 g1 g K 1]. The definition of x depends on the modulation constellations employed, and {g i } is chosen according to the practical purpose. Without loss of generality, we assume g i = 1. An illustration of K-dimensional orthogonal signal space (with K=3) is shown in Figure 1, and two designs of {g i } are shown in Figure. Obviously, the greater K is, the more information the transmitted signal conveys. However, it is pointed out in ref. [13] that, given a duration of T seconds and a bandwidth of W hertz (No practical system is absolutely band-limited. A frequently used definition is fractional power containment bandwidth, e.g. 99% power bandwidth. The conclusion of ref. [13] and our study are both based on it), the dimension of orthogonal function set {g i }, or K, is limited to WT. It indicates that if each g i conveys an independent symbol, the limit of symbol rate K/T is W (Nyquist rate). Figure 1 Signal space with K=3. Figure Two {g i } designs. (a) Baseband; (b) multi-carrier. 890 CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896

Once the orthogonality restriction on {g i } is removed, the limit of transmission efficiency is hence broken through, which is the main idea of this study. NMT, like traditional methods, can be written as eq. (). The difference is the restriction on {g i } reduces to linear independence, and orthogonality is not required. It means that NMT is a parent set of traditional modulation methods. In this study we focus on the NMT of amplitude modulation type, where each x i is independently and equiprobably chosen from M evenly spaced numbers between A and A. That is to say, there are M K constellation points symmetrically arranged in a K-dimensional cube. Based on the constellations, the bit rate and transmitted signal power of this NMT can be computed, respectively, as follows: K Rb = log M, (3) T K 1 1 T 1 T 1 AK M+ 1 Ey [ ( t)] = E( y y) = E( x Rx ) = Exx ( i j) ri, j =, (4) T T T i, j= 0 3T M 1 T where R= { r i, j } = G G, Exx ( i j) = Aδij( M+ 1) 3( M 1), and δ ij stands for Kronecker delta function. Using eqs. (3) and (4), the average bit energy of this NMT is obtained: Ey [ ( t)] A M+ 1 Eb = =. (5) Rb 3log M M 1 Consider the demodulation problem of the NMT mentioned above. Let y = y+ n, where y represents the received signal through AWGN channel, and n represents the AWGN with one-sided power spectrum density n 0. The optimal solution of this problem is the one that minimizes the squared error (MMSE): x = arg min y Gx, (6) x X where X is the constellations points set. As MMSE solution is hard to find, a practical method is suggested. Solve the over determined equations of eq. () with least square method, then we obtain the solution x, 1 T 1 x = G y = R G y = x+ R G T n, (7) 1 T where G = R G is the Moore-Penrose inverse matrix of G. Then, the demodulation is done by making an M-level threshold decision on every element of x. Now, we study the error performance of eq. (7). Define the error vector as e= e e e = x x. Its mean and deviation values are [ ] T 0 1 K 1 1 T E = E = () e R G () n 0, n E( ee ) = R G E( nn ) GR = R T 1 T T 1 0 1 Thus, we have Ee ( i ) = ψ in0, where ψ i stands for the ith element on the primary diagonal of R 1. Considering R is a positive matrix whose elements on the primary diagonal are all one, it can be proved, according to the generalization of the Bergstrom inequality [14], that ψ i 1 stands, where the equality holds if {g i } is an orthogonal set. It follows that, compared with traditional methods, the nonorthogonality of NMT enlarges the error deviation Ee ( i ) by ψ i times.. (8) CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896 891

Having this conclusion and eq. (5), we finally yield the NMT symbol error rate (SER) formula (omit the derivation for reasons of brevity): K 1 M 1 3log M Eb Ps = erfc, (9) KM i= 0 ψ i ( M 1) n 0 where Eb n 0 is the normalized SNR (signal to noise ratio). In the case of orthogonal {g i }, eq. (9) becomes the tradition amplitude modulation SER formula [15] : M 1 3log erfc M Eb Ps =. (10) M M 1 n 0 From eqs. (9) and (10), we can observe that the error performance of NMT must be worse than the traditional if they choose the same value for M. However, we cannot assert NMT is inferior to the traditional. The reason is that, since NMT removes the orthogonality restriction, it could achieve a higher symbol rate than the traditional. To compare their performance fairly, first we should draw a curve of spectrum efficiency versus SNR for either of them, and then put the two curves into the same figure. 3 Spectrum efficiency analysis In this section, transmission performance in terms of spectrum efficiency versus SNR is examined for both NMT and the traditional methods. According to eq. (3), the spectrum efficiency of NMT can be written as follows: Rb K = log M, (11) W WT where W denotes the bandwidth occupied by NMT transmitted signal. In additional, using the S N = E n R W, eq. (9) can be rewritten as follows: equality ( )( ) b 0 b K 1 M 1 3WT S Ps = erfc. (1) KM i= 0 Kψ i ( M 1) N Next, we shall establish a relation between Rb W and S N based on eqs. (11) and (1). Observe that erfc( ) decreases monotonously, and define ψ = max( ψ i ) ; then we have 3WT S Ps < erfc. (13) Kψ M N Trying to make M appear on the left side of eq. (13), we have WT S M >, (14) λψ K N where [ ] λ erfcinv( P s ) 3 and erfcinv( ) stands for the inverse function of erfc( ). The right side of the above inequality is a lower bound of M. Conservatively, substitute this lower bound for M in eq. (11) and finally derive 89 CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896

Rb log 10 K S λψ K 10lg, (15) W 10 WT N db WT where ( S N ) db means SNR in decibel. Consider the traditional modulation system that has the Nyquist rate. For this ideal system (we also use this term to describe such system in the following text), we have KT= W and ψ = 1. Then, eq. (15) can be simplified to Rb log 10 S 10lg( λ). (16) W 10 N db As we know, Shannon channel capacity formula C = W log ( 1+ S N) is derived from ideal system [16]. Similarly, we can approximate the formula as C log 10 S, (17) W 10 N db where C/W is the normalized channel capacity. Some conclusions are given below: (i) Each of eqs. (15) (17) indicates that the relation of Rb W or C/W versus ( S N ) db approximates a linear function. (ii) If the NMT of interest can achieve a symbol rate over Nyquist rate, i.e., KT> W, the corresponding linear function given by eq. (15) has greater slope and x-intercept than eq. (16). (iii) The two linear functions given by eqs. (16) and (17) have equal slope. If we assume λ > 0.5 (or P s < 0.0833, which is true for most of practical applications), then the x-intercept of the former line is greater than the latter. (iv) Based on (i) (iii), we depict the three linear functions in Figure 3. Note that R b /W or C/W given by eqs. (15) (17) is a lower bound of the actual value. However, these approximations do not substantially alter the relation among the lines in Figure 3. It can be observed from Figure 3 that NMT s R b /W eventually exceeds ideal system, even C/W, with the increasing of (S/N) db. Figure 3 Illustration of eqs. (15) (17), P s <0.0833. It should be stressed that we do not mean NMT challenges the Shannon theorem, though its R b /W exceeds C/W. In fact, it is unfair to compare them because the former is obtained based on a given error probability, whereas the latter is based on error-free transmission (If NMT s P s needs to be arbitrarily close to zero, the linear function given by eq. (15) will have arbitrarily great x-intercept, and then exceeding C/W is out of question). However, C/W can be a reference for the comparison between NMT and traditional modulation methods. After all, for any finite small P s, NMT can exceed C/W ultimately, but traditional methods cannot. 4 Design example and experiments There are various designs of G that satisfy NMT s definition. A simple one is presented in Figure 4, which is an expansion of Figure (a). In this design, g i is an its -delayed version of a baseband CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896 893

pulse with duration of T d. Then, we have T ( K 1) T s T d = +. We also notice that T R= G G is a symmetrical Toeplitz matrix, which can be denoted as R = Toep( r0, r1,, rk 1), and r, r,, r are the elements on R s first row. 0 1 19 Figure 4 An illustration of {g i } design. As for the above NMT example, we expect KT> W, that is K K KTd = = ρ >, ρ =. (18) WT W[( K 1) Ts + Td] WTd ( K 1) Ts + Td Roughly, 99% energy bandwidth of the baseband pulse can be regarded as an approximation of W. In this sense, WT d in eq. (18) becomes the time-bandwidth product of the pulse, which is a constant for a given pulse type. The purpose of our numerical experiments is to obtain NMT s R b /W and the required E b /n 0 when M =, 4,8, and bit error rate P b =1E-5. Here, we use Gray codes, so if P 1, we have Pb Ps log M [17]. The design of this experiment and its results are present as follows: (i) We choose truncated discrete-time Gaussian function f [ n ] as the baseband pulse. Its time-bandwidth product, or WT d, is 1.19. In order to satisfy eq. (18), we let T d =7, T s =4 and K=0. f [n] is shown in Figure 5(a). Then, we can derive G, R= Toep( r0, r1,, r19), and R 1, where r0, r1,, r19 and ψ i are shown in Figure 5(b) and (c), respectively. With these, and considering eq. (9) and the relation between P b and P s given above, we obtain the theoretical values of E b /n 0 to meet P b =1E-5, as listed in Table 1. (ii) Based on each pair of ( M, Eb n 0 ) in Table 1, we stimulate NMT with 1E8 symbols tested and calculate the actual R b /W and P b. The results are listed in Table. Here, W is obtained by measuring the 99% power bandwidth of NMT transmitted signal. (iii) The values of P b in Table are close to 1E-5, so the data in Tables 1 and are what we need. Plot them in Figure 6, including the theoretical R b /W curve of ideal system based on P b =1E-5 for comparison. Apparently, when (E b /n 0 ) db is over 34 db, NMT exceeds ideal system. s 894 CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896

Figure 5 f [n], R and ψ i. Table 1 Theoretical E b /n 0 for different M, P b =1E-5 M 4 8 16 3 64 (E b /n 0 ) db 6.67 30.51 34.86 39.58 44.54 49.69 Table Experiment results based on Table 1 P b 1.0E-5 9.87E-6 1.0E-5 9.95E-6 9.90E-6 1.01E-5 R b /W 4.4 8.83 13.4 17.65.07 6.47 Figure 6 R b /W of experimental NMT and theoretical ideal system, P b =1E-5. 5 Discussion As a nonorthogonal modulation method, NMT has many practical and theoretical problems need to be solved. At the end of this paper, we present some preliminary discussion. We observe from Figure 6 that the proposed NMT design has no advantage unless E b /n 0 is above 34 db or so, which suggests to us in the following: First, appropriate applications for NMT are those where high SNR condition is met, like short distance wired transmission, e.g. VDSL or computer bus communications. We have performed simulations on these applications and achieved positive results. Second, more effective NMT designs are needed. Specifically, the loss of SNR due to nonorthogonality should be reduced. We have applied NMT to multicarrier systems. Different from the traditional, the subcarriers here are nonorthogonal. We will introduce it in other papers. Throughout this paper, we adopt fractional power bandwidth definition. For traditional modulation methods, it often implies that the loss of out-of-band signal power can be ignored. However, is it the same for NMT? Whether and how this loss will counteract the intrinsic advantage of NMT? On the other hand, to promote bandwidth efficiency is not the only benefit of using CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896 895

nonorthogonality. The design of {g i } becomes flexible because of the elimination of orthogonality restriction, which is helpful to solve the problems that are hard for traditional methods, for example, time-frequency dispersion. These need our further study. In conclusion, our study on NMT is a significant attempt to explore new modulation methods, though it has many challenges to overcome. Logically, nonorthogonality is a far broader concept than orthogonality, thus nonorthogonal modulation methods certainly will be the future and promising direction for the development of modulation theory. 1 ANSI T1E1.417. Spectrum Management for Loop Transmission Systems, 003 Bluetooth SIG. Bluetooth Specification Version 1.1, 001 3 Chen C, Song W T, Luo H W. A review on wireless transmission technology for IMT-000 mobile communication system. Telecommun Eng (in Chinese), 1999, 4: 49 54 4 Zhu J K. Technical challenge and revolution of future mobile communications. Acta Elect Sin (in Chinese), 004, 3(1): 6 10 5 Xia W, Li K, Xu J Y. Solutions of bluetooth system on the coexistence of bluetooth and Wi-Fi. Appl Res Comput (in Chinese), 004, 7: 14 19 6 IEEE Standard 80.16. A Technical Overview of the Wireless MAN Air Interface for Broadband Wireless Access, 00 7 Compiling Committee of UWB Special Issue. Ultra wide-band radio technology. J Commun (in Chinese), 005, 6(10): 6 8 Weistain S B, Ebert P M. Data transmission by frequency-division multiplexing using the discrete Fourier transform. IEEE Trans Commun Tech, 1971, COM-19: 68 634 9 Nyquist H. Certain topics of telegraph transmission theory. AIEE Trans,198, 47: 617 644 10 Liang D Q, Liang W H, Sun C N. The phase-offset overlapped wave technique. J Elect (in Chinese), 003, 0(): 11 17 11 Couch II LW. Digital and Analog Communication Systems (in Chinese). Beijing: Tsinghua University Press, 1998. 148 151 1 Proakis J. Digital Communications. New York: McGraw-Hill, Inc., 1995. 163 167 13 McEliece R J. The Theory of Information and Coding (in Chinese). Beijing: Publishing House of Electronics Industry, 004. 85 14 Compiling Committee of Modern Applied Mathematics Handbook. Modern Applied Mathematics Handbook: Modern Applied Analysis (in Chinese). Beijing: Tsinghua University Press, 1998. 386 387 15 Fan C X, Zhang F Y, Xu B X. Communication Theory (in Chinese). Beijing: National Defense Industry Press, 1995. 71 16 Shannon C E. Communication in the Presence of Noise. Proc IRE, 1949, 37(1): 10 1 17 Sklar B. Digital Communication: Fundamentals and Applications (in Chinese). Beijing: Publishing House of Electronics Industry, 00. 181 896 CAO QiSheng et al. Sci China Ser F-Inf Sci Dec. 007 vol. 50 no. 6 889-896