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(Article begins on next page) To appear in IEEE TRANS. INFORM.THEORY 1 Gray Coding for Multilevel Constellations in Gaussian Noise Erik Agrell, Johan Lassing, Erik G. Str¨om, and Tony Ottosson
Abstract—The problem of finding the optimal labeling (bit-to- The BEP of the systems considered herein was shown in [5] symbol mapping) of multilevel coherent PSK, PAM, and QAM to be a function of two quantities; the average distance spec- constellations with respect to minimizing the bit error probabil- trum (ADS), derived from the constellation labeling, and the ity (BEP) over a Gaussian channel is addressed. We show that using the binary reflected Gray code (BRGC) to label the signal communication channel. We established in [5] the somewhat constellation results in the lowest possible BEP for high enough artificial result that the BRGC is the optimum labeling for PSK signal energy-to-noise ratios and analyze what is “high enough” and PAM with respect to certain properties of the ADS, and the in this sense. It turns out that the BRGC is optimal for PSK and question whether the BRGC also yields minimum BEP over a PAM systems whenever the target BEP is at most a few percent, practical channel with a finite (and in some cases even for in- which covers most systems of practical interest. New and simple closed-form expressions are presented for the BEP of PSK, PAM, finite) SNR was left unanswered. In this paper, the optimality and QAM using the BRGC. criterion is therefore changed from the one in [5] to the more relevant requirement that the optimal labeling should minimize Index Terms— binary reflected Gray code, bit error probabil- ity, bit error rate, constellation labeling, digital modulation, Gray the BEP of the communication system. We assume an AWGN mapping, optimal labeling, PAM, PSK, QAM. channel and show that the minimum achievable BEP is, indeed, obtained by using the BRGC as long as the signal energy-to- noise ratio is higher than a finite threshold, which depends on I.INTRODUCTION the modulation scheme and the size of the constellation. This paper addresses the problem of selecting an optimal The paper is organized as follows. In Section II, the prelim- labeling with respect to minimizing the bit error probability inaries are presented and a proof method is outlined for the op- (BEP) in digital communication systems with coherent symbol timality of the BRGC. Section III presents general BEP expres- detection over an additive white Gaussian noise (AWGN) chan- sions for each of the three studied modulation formats, which nel [1]. We study transmission of equally likely, statistically in- hold for arbitrary labelings. In Section IV, we derive a par- dependent bits using multilevel phase shift keying (PSK), pulse ticularly useful partitioning of the set of all possible labelings amplitude modulation (PAM), and quadrature amplitude mod- (competingwith the BRGC on being the optimal labeling). This ulation (QAM) systems; with the binary reflected Gray code partitioning is used in Sections V–VII to prove the optimality (BRGC), other Gray codes, and other labelings; for finite and of the BRGC for PSK, PAM, and QAM, respectively. In Sec- infinite signal-to-noise ratios (SNR). Only uncoded transmis- tion VIII, the analysis from Section III is continued and spe- sion (or more precisely, coding without redundancy) is con- cialized to the case of BRGC labelings, resulting in explicit, sidered. The corresponding problem in systems with error- closed-form expressions for the ADS and BEP of the BRGC. correcting codes is considered in, e.g., [2]. Finally, conclusions and comments are given in Section IX. It is established engineering knowledge that labeling signal constellations with Gray codes (in particular, the BRGC [3])isa way to reduce the BEP for the systems considered in this paper. II. PRELIMINARIES There exist, however, a multitude of nonequivalent Gray codes. Some of the most central definitions and notation for the pa- The theoretical question whether the BRGC is the best way to per are collected in this section. The proof method is also out- label the constellations has so far been an open question even in lined and two important analytical results, central to the proofs, the asymptotic case of infinite SNR, although this is sometimes are given. stated as a fact in the literature [4]. That the use of the BRGC (or even Gray codes) is not optimal for all SNR’s for at least some constellation sizes can be demonstrated by explicit eval- A. Definitions and Notation uation of the BEP for various labelings and modulation forms The presented work deals with binary labelings and in this (see [5, Fig. 8] for a 64-QAM example). subsection we introduce the nomenclature and definitions that Manuscript submitted June 29, 2005; revised May 30, 2006. The material in are used in the discussion. this paper was published in part in Tech. Rep. R001/2005, Dept. of Signals and A binary labeling λ of order m Z+ is defined as a se- Systems, Chalmers University of Technology, Sweden, Jan. 2005. m ∈ E. Agrell, E. G. Str¨om, and T. Ottosson are with the Communication quence of M = 2 distinct vectors (labels or codewords), m Systems group, Department of Signals and Systems, Chalmers Univer- λ = (c0, c1,..., cM−1), where each ci 0, 1 . A rectan- sity of Technology, 41296 G¨oteborg, Sweden (e-mail: [email protected]; ∈ { +} + gular binary labeling λ of order (m1,m2) Z Z consists [email protected]; [email protected]). J. Lassing is with Qam- m1+m2 ∈ × com Technology, Gruvgatan 8, 42130 V¨astra Fr¨olunda, Sweden (e-mail: jo- of all vectors (labels) in 0, 1 , arranged in a matrix of m1 { } m2 [email protected]). dimension M1 =2 by M2 =2 . 2 To appear in IEEE TRANS. INFORM.THEORY
TABLE I index T (λ) of a PSK labeling λ Λm as THE BINARY REFLECTED GRAY CODES OF ORDERS m = 1, 2, 3, AND 4. ∈ T (λ) , min k Z+ : d¯(k, λ) = d¯(k,β ) . (1) { ∈ 6 m } β1 β2 β3 β4 0 00 000 0000 From [5, Th. 5] we know that 1 01 001 0001 d¯(k, λ) > d¯(k,β ) (2) 11 011 0011 m 10 010 0010 at k = T (λ). The definition (1) also applies to PAM if d¯(k, λ) 110 0110 is replaced with h¯(k, λ) (defined in Section III-B). 111 0111 The set of all critical indices is the critical index set 101 0101 100 0100 Ψm , T (λ) : λ Λm . (3) 1100 { ∈ } 1101 In addition, it will be convenient to have a designator for the set 1111 of labelings for which T (λ)= i, 1110 Λ (i) , λ Λ : T (λ)= i . (4) 1010 m { ∈ m } 1011 Although obvious from (3) and (4), we explicitly state that 1001 1000 Λm = Λm(i) i∈[Ψm A binary labeling λ′ of order m is said to be optimal for and Λ (i) Λ (j)= ∅ for i = j signal energy-to-noise ratio γ if m ∩ m 6 since these relations are central to the proof method as de- P (λ′,γ) P (λ, γ) b ≤ b scribed in the next subsection. for all labelings λ of order m, where Pb (λ, γ) is the bit error probability, which is defined for PSK, PAM, and QAM systems B. Outline of Proof Method in Section III, and γ is defined as Es/N0, where Es is the sym- Before proceeding to the details, we give an outline of the bol energy and N0/2 is the two-sided power spectral density of proof method that will be used. Using the definitions and the the AWGN. notation introduced in the previous subsection, the aim of this Throughout the paper, we discuss a particular class of label- paper is for each m and each modulation form to establish a ings; Gray codes. A binary Gray code of order m is a binary range of γ for which labeling with M = 2m distinct labels, where adjacent labels differ in only one of the m positions. If we impose the addi- Pb(βm,γ) min Pb(λ, γ) (5) ≤ λ∈Λm tional requirement that the first and the last labels differ in a single position, the labeling is said to be a cyclic binary Gray that is, for what signal energy-to-noise ratios the labeling βm code. Analogously, a rectangular binary Gray code is a rect- will result in the lowest BEP among all possible labelings. We ∗ angular labeling where adjacent labels, horizontally as well as define the optimality threshold γm for order m as the smallest vertically, differ in only one bit. value such that (5) holds for all γ γ∗ . ≥ m Among the cyclic Gray codes, we are especially interested in We will address (5) by using the equivalent formulation the binary reflected Gray code (BRGC) [3,5], and we denote the BRGC of order m by βm. For reference, we have listed βm 0 min min Pb(λ, γ) Pb(βm,γ) . (6) ≤ i∈Ψm λ∈Λm(i) − for m = 1,..., 4 in Table I. With a two-dimensional BRGC we mean the direct product of two BRGC’s. The BRGC can We will, for each i Ψ , lowerbound the expression inside the be transformed into a number of labelings that yield exactly the m brackets in (6) and∈ establish a range of γ for which the bound same BEP by means of some trivial operations such as inter- ∗ is non-negative. This yields an upper bound γˆm on γm, which changing bit positions. The role of such operations was dis- is computed separately for PSK, PAM, and QAM. cussed in some depth in [5], and we will not say more about them here. We loosely refer to the class of all such labelings as the BRGC. C. Two Lemmas of Monotonicity The ADS of the labeling λ, denoted by d¯(k, λ) for PSK, es- In order to find the range of γ for which (5) is valid, we sentially tells us the average number of bits that differ in two make use of two results from calculus, which are derived in labels separated by k symbols in the constellation. The pre- this subsection. cise definitions depend on the constellation and follow in Sec- Lemma 1: For constants a and b such that 0 a ′ • f (x) > 0 for x> 0, and ′′ • f (x) > 0 for x> 0, then ∆(x) is a strictly increasing function in x for x 0. Proof: Since f ′′(x) > 0 for x> 0, we havefor 0≥ a 0 ≤ √Es sin a bx Γ(a,γ) 0 < f ′′(t) dt = f ′(bx) f ′(ax). ax − Z O a Since f ′(x) 0 for x> 0, we have ≥ ′ a ∆ (x) √Es f ′(bx) f ′(ax) f ′(bx) f ′(ax)= − ≤ − b b for 0 a 1 2 1 2 M−1 − 2 x − 2 x 1 ′ e ′′ xe d¯(k, λ) , d c , c (8) Q (x)= and Q (x)= M H l (l+k) mod M − √2π √2π l=0 X we have f ′(x) > 0 for all x and where ck is the m-bit binary label assigned to the kth constella- 1 2 1 2 c c − 2 x − 2 x tion point and the Hamming distance dH( j , k) is the number ′′ xe e f (x)= Q(x) . of positions in which cj and ck differ. The ADS denotes the av- 2 √2πQ(x) " x√2π − # erage number of bits that differ between binary labels assigned 1 2 to constellation points separated by k steps in the PSK constel- Now, as − 2 x √ is a well known upper bound on e /x 2π Q(x) lation. If it is clear from the context which labeling λ is con- for [6, p. 98], we conclude that for , we have x > 0 x > 0 cerned, we will simply write d¯(k) for the ADS. The crossover ′′ . Applying Lemma 1 to , we find that f (x) > 0 f(x) probability P (k,γ) is the probability that the received signal Q(ax) vector is found in a decision region belonging to a signal point ∆(x)= log Q(bx)+log Q(ax) = log − Q(bx) k steps away (clockwise along the PSK circle) from the trans- mitted signal point. is a strictly increasing function for x> 0 and 0 a 6γ In order to address (6), we need to find the critical set Ψm (23) µ(γ)= 2 2 . defined in (3). We will rely on a method called labeling expan- sM1 + M2 2 − sion, which is a way to constructa labeling λm of order m from The labeling λ is now rectangular (see Section II-A) and a labeling λ of order m 1 [5]. For a labeling λ that is m−1 − m−1 expanded into λm the following relations hold for m 2 and M1−1 M2−1 Z ≥ , 1 c′ c′ k g¯(k,l,λ) dH( i,j , i+k,j+l). ∈ M1M2 i=0 j=0 ¯ ¯ X X d(4k, λm)= d(2k, λm−1)+ f1 (28) To exploit the symmetry of the constellation, as was done for d¯(4k +2, λm)= d¯(2k +1, λm−1)+ f2 PAM in (15), we form an ADS with two components t¯ and t¯0 ¯ ¯ ¯ d(k, λm−1)+ d(k +1, λm−1) by averaging components of g¯ in groups of four. This yields d(2k +1, λm)= + f3 (29) 2 ∞ ∞ 4 where f1, f2, and f3 are functions of k and m, but independent Pb(λ, γ)= t¯(k,l,λ) (k, µ) (l, µ) m1 + m2 P P of λm−1. The same relations, with different f1, f2 and f3, hold k=1 l=1 ¯ X X∞ for h(k, λ). An important property of labeling expansion is that 4 (0, µ) + P t¯0(k, λ) (k, µ) (24) expanding βm−1 gives βm. Furthermore, we derive the follow- m1 + m2 P k=1 ing property of expanded cyclic labelings from [5, Lemma 3]. X An analogous relation for h¯(k, λ), not explicitly stated here, can where for any integers k and l (dropping the dependence on λ be derived from [5, Lemma 3b]. to simplify the notation) Lemma 3: A labeling λ of order m 2 is an expanded ¯ ≥ ¯ 1 cyclic Gray code if and only if d(1, λ) = d(1,βm) and t¯(k,l) , [¯g(k,l)+¯g(k, l)+¯g( k,l)+¯g( k, l)] (25) ¯ ¯ 4 − − − − d(3, λ)= d(3,βm). 1 Proof: First, that a labeling λ is a cyclic Gray code if t¯ (k) , [¯g(0, k)+¯g(0, k)+¯g(k, 0)+¯g( k, 0)] . (26) ¯ ¯ 0 4 − − and only if d(1, λ) = d(1,βm) follows from the definition of a cyclic Gray code. Second, that a cyclic Gray code λ of order The expression (24) is suitable for analytically comparing the m 3 is an expandedcyclic Gray code if and only if d¯(3, λ)= performance of various labelings, as will be done in Section ≥ d¯(3,βm) was proved in [5, Lemma 3]. The case m = 2 is VII. To numerically evaluate the BEP of a given labeling, how- trivial. ever, the infinite summations can be replaced by a finite number The critical index set depends on the modulation form and of terms, because the order m, but the method used to find the critical index set for PSK and PAM is the same. We derive the critical index set t¯(k,l)= t¯(min k,M 1 , min l,M 1 ) { 1 − } { 2 − } for PSK in detail and only point out the essential differences t¯ (k)= t¯ (min k, max M ,M 1 ). 0 0 { { 1 2}− } in the derivation of the critical index set for PAM. QAM is not The calculations, which are not detailed here, follow in perfect treated in this section, as it will be shown in Section VII that analogy with (20)–(22). the concept of critical indices is not needed to analyze the per- So far, the expressions in this section hold for arbitrary QAM formance of two-dimensional BRGC’s. labelings. In the special case when the QAM labeling is the direct product of two PAM labelings, the expressions can be A. The Critical Index Set for PSK simplified further. Indeed, it can be shown that for any product Theorem 4: For PSK constellations, the critical index sets labeling λ1 λ2, the two-dimensional ADS components (25)– of orders satisfy ∅, , and × m 4 Ψ1 = Ψ2 = Ψ3 = 1 (26) get the particularly simple forms Ψ = 1, 3 . ≤ { } 4 { } Proof: It is trivial that Ψ1 = ∅. For m 2, we consider t¯(k,l)= h¯(k, λ1)+ h¯(l, λ2) ≥ the set Λm(1) of labelings not having the cyclic Gray property. ¯ ¯ h(k, λ1)+ h(k, λ2) For all m 2, there is at least one labeling in this set, so defi- t¯ (k)= . 0 2 nitely 1 ≥Ψ . ∈ m 6 To appear in IEEE TRANS. INFORM.THEORY TABLE II TABLE III THE NUMBER OF BINARY GRAY CODES AND BINARY CYCLIC GRAY THE CRITICAL INDEX SET Ψm FOR PSK AND PAM ASAFUNCTIONOF m. CODES THAT DO NOT HAVE IDENTICAL ADS AS A FUNCTION OF THE ORDER m.THE TABLE, WHICH WAS OBTAINED BY COMPUTER SEARCH, m Ψm (PSK) Ψm (PAM) DOESNOTCOUNTTHESAMEENTITIESAS [5, TAB.I], ALTHOUGH THE 1 ∅ ∅ NUMERICAL VALUES AGREE FOR m ≤ 4. 2 1 1 3 {1} {1,}3 m cyclic Gray Gray 4 {1,}3 {1, 3,}5 1 1 1 5 {1, 3,}5 {1, 3, 5,}9 2 1 1 6 {1, 3, 5,}9 {1, 3, 5, 9,}17 3 1 3 7 {1, 3, 5, 9,}17 {1, 3, 5, 9, 17,}33 4 9 131 8 {1, 3, 5, 9, 17,}33 {1, 3, 5, 9, 17, 33,}65 { } { } The set Λm(k) contains for k 2 only cyclic Gray codes. the expanded labeling λm and result in a critical index However, all cyclic Gray codes have≥ d¯(2) = 2. Therefore, all T (λ )=2T (λ ) 1. cyclic Gray codes have identical ADS’s for k = 1 and 2 and, m m−1 − hence, Λm(2) = ∅ for all orders m. For Λ (3), we turn to column 2 of Table II, where the num- In summary, Ψm = 1, 3 2i 1 i Ψm−1 for m 5, m { }∪{ − | ∈ } ≥ ber of cyclic Gray codes that do not have identical ADS is which is equivalent to (30). listed. Since there is only one cyclic Gray code for m 3, The PSK critical index sets Ψm are listed in the second col- ≤ umn of Table III for m =1,..., 8. Λm(3) = ∅ for m 3, which completes the proof that Ψ =Ψ = 1 . ≤ 2 3 { } For m = 4, we conclude from Lemma 3 and the fact that B. The Critical Index Set for PAM the Gray code of order m = 3 is unique that there is only one For PAM, the critical index set is derived in a similar way; the expanded Gray code of order 4, which is the BRGC. Hence, differenceis that the ADS is defined by (18) and we excludethe Λ (i) = ∅ for i 4. On the other hand, there exist several 4 cyclic requirement on the Gray codes. The change of definition Gray codes of order≥ 4 that are not the BRGC, see Table II, and for the ADS results in different values for h¯(k) for k = 1, 2 consequently not expanded, which proves that Λ (3) = ∅. 4 6 compared to the PSK case, but the conclusion that h¯(k) is iden- Theorem 5: For , is obtained from by m 5 Ψm Ψm−1 tical for all Gray codes of order m 3 for k =1, 2 is still valid adding another element≥ to the set, namely, [5, Lemma 2b]. ≥ From column 3 of Table II, we see that in the class of not Ψ =Ψ 2(max Ψ ) 1 . (30) m m−1 ∪{ m−1 − } necessarily cyclic Gray codes, there are three classes of Gray codes of order m = 3 that do not have identical ADS’s. This Proof: From the previous proof, 1 Ψ and 2 / Ψ for m m means that the critical index sets are not the same for PAM as m 2. To establish when 3 Ψ , we∈ recall from∈ Lemma 3 m for PSK. Indeed, they are given by the following two theorems. that≥Λ (3) is the set of Gray codes∈ that are not expanded Gray m Theorem 6: Ψ = ∅, Ψ = 1 , and Ψ = 1, 3 . codes. We will prove that this set is nonempty for m 4 by 1 2 { } 3 { } Theorem 7: For m 4, Ψ is given by (30). showing that it includes the class of balanced Gray codes.≥ Such m We omit the proofs,≥ which are analogous to the proofs of labelings exist for all orders, see [12] and [13, pp. 14-15]1 and Theorems 4–5. The critical index sets of PAM are listed in the they have the property that the M bit transitions in a cyclic list third column of Table III for m =1,..., 8. of the M labels are distributed as evenly as possible among the m bit positions. To be precise, no more than M/m +2 transi- tions occur in any one position of a balanced Gray code. In an V. THE OPTIMAL PSK LABELING expanded Gray code, on the other hand, half of the transitions At this point, we have established the foundation required to occur in the same bit [5] (cf. Table I). Since M/2 > M/m +2 address the proof of the optimality of β . In this section, we for m 4, we conclude that for m 4, balanced Gray codes m ≥ ≥ use the results in Section IV to derive sufficient conditions on are not expanded, Λm(3) = ∅, and 3 Ψm. 6 ∈ the signal energy-to-noise ratio for which βm is optimal for an Now, for T (λ) 4, we are dealing with the class of cyclic ≥ M-PSK system. Gray codes for which d¯(k) is identical to the ADS of βm for k = 1, 2, and 3. From Lemma 3, we know that all such la- belings of order m can be constructed by expansion of a Gray A. The Bounding Ratio code of order m 1. Hence, their ADS’s can be calculated The procedure we use is to compare βm to all labelings in using (28)–(29).− From the recursions we find that the critical Λm. This is done by finding, for each i Ψm, a signal energy- index T (λ ) of a labeling of order m 1 will propagate to ∈ m−1 − to-noise ratio γ such that 1The perhaps earliest proof of the existence of balanced Gray codes for all m Pb (βm,γ) min Pb (λ, γ) (31) is attributed to T. Bakos [14]. ≤ λ∈Λm(i) Submitted June 29, 2005; revised May 30, 2006; final version, Sept. 19, 2006 7 and showing that the highest of these values taken over all i with equality. The inequality is valid for all γ γ (i). ∈ ≥ m Ψm provides a γ above which βm yields the lowest possible Proof: The left-hand side of (34) is equal to one for γ =0 BEP over the Gaussian channel. and a continuous function of γ for γ 0. To complete the For this purpose, we define the bounding ratio for PSK as proof, we will show that it is also strictly≥ increasing and un- bounded. From the implicit definition in (9), M−i−1 P (k,γ) R(i,γ) , k=i+1 (32) 2P (i,γ) (2 max Ψm + 1)π P max bi = bmaxΨm = m . whose significance is given by the following lemma. i∈Ψm 2 Lemma 8: For any i Ψm, a sufficient criterion for (31) is m−3 Since maxΨm =2 +1 for m 4 and maxΨ3 =1, that ∈ ≥ 2 1 3 π R(i,γ) . bi + m π< ≤ M(m 1) ≤ 4 2 2 − Proof: Let ǫ , 2/M and ω , m 1. Define for any m 3 and i Ψm. From Lemma 2, we see that for − ≥ ∈ 0 ai TABLE IV THE M -PSK MAXIMALOPTIMALITYTHRESHOLDS γˆ FOR m = 2,..., 10 50 FROM (37) AND LEMMA 9, THECORRESPONDINGBITENERGY-TO-NOISE m = 10 RATIO γˆb , γ/mˆ , AND THE BEP WHEN THE BRGC IS USED AT γ =γ ˆ. 40 m γˆ [dB] γˆb [dB] Pb(βm, γˆ) 30 2 0.5 [dB] −∞ −∞ ) 3 3.8 1.1 0.147 i ( − m 4 10.6 4.6 0.090 γ 20 5 16.7 9.7 0.070 6 23.6 15.8 0.049 10 7 30.3 21.9 0.036 8 37.0 28.0 0.027 9 43.6 34.0 0.021 0 m =3 m =2 10 50.1 40.1 0.016 1 3 5 9 17 33 65 129 257 i Fig. 2. The function γm(i) for i ∈ Ψm from m = 2 (bottom, PAM only) to VI. THE OPTIMAL PAM LABELING m = 10 (top pair of curves). Bullets (•) mark M-PSK and crosses (×) mark In this section, we apply the methods used to prove optimal- M-PAM. ity of βm for M-PSK to systems using M-PAM. A. The Bounding Ratio For γ γm(i), the denominator is at least M(m 1)/2 by Lemma≥ 9. Lemma 8 completes the proof. − The proof method for the PAM case is very similar to that Corollary 11: For any M-PSK constellation, the optimal la- of the PSK case; the main difference lies in the evaluation of beling at asymptotically high SNR is the BRGC. the crossover probabilities. We again use the inequality (31), In Figure 2, the function γ (i) is shown for i Ψ and this time using (15) for the BEP expression, in order to find m ∈ m m = 3, 4,..., 10. The interpretation of γm(i) is the follow- a signal energy-to-noise ratio threshold above which βm gives ing. Consider a labeling λ Λ (i), i.e., a labeling for which the lowest possible BEP of all labelings. ∈ m T (λ) = i. For γ>γ (i), β will result in a lower BEP ac- Lemma 13: For any i Ψm, m m ∈ cording to (7) over a Gaussian channel, irrespective of the ADS ∞ k=i+1 (k, µ) 1 values of λ for k>i. If γ <γm(i), there may exist label- P (38) (i, µ) ≤ (m 1)(M 1) ings λ Λm(i) such that the BEP is lower than for βm even P ∈ P − − though d¯(i,βm) < d¯(i, λ). For example, we see from Figure 2 is a sufficient criterion for Pb(βm,γ) minλ∈Λm(i) Pb(λ, γ). that for m =4, any cyclic Gray code will give lower BEP than Proof: We rewrite (18) for k 1≤as any non-Gray labeling for γ 9.7 dB. Compared to the cyclic ≥ ≥ Gray codes in Λ (3), β gives a lower BEP than all these label- M−2 M−1 4 4 1 ings for γ 10.5 dB. For 9.7 dB γ 10.5 dB, the optimal h¯(k)= d (c′ , c′ )+ d (c′ , c′ ) ≥ ≤ ≤ 2M H j j+k H j j−k labeling may be different from β , but it must be a cyclic Gray j=0 j=1 4 X X code. (39) We define the maximal optimality threshold of order m 3 as ≥ and observe that each of the 2M 2 terms is between 1 and m, inclusively. Hence, − γˆm = max γm(i) (37) i∈Ψ m 1 1 1 h¯(k) 1 + ω and we let, formally, γˆ2 = . Clearly, γˆm is an upper bound − M ≤ ≤ − M −∞ ∗ on the corresponding optimality threshold γm. The main theo- rem for PSK now follows from Theorem 10. where ω , (m 1)(1 1/M). Furthermore, the resolution ¯ , − − Theorem 12: The BRGC is optimal for any M-PSK constel- of h(i) is ǫ 1/M, which can be shown by considering the following two cases separately. If , the terms of ¯ in lation at γ γˆm. i = 1 h(i) The maximal≥ optimality threshold (as seen in Figure 2) is (39) are pairwise equal, and if i > 1, λ is a Gray code and the ¯ given in Table IV for m = 2, 3,..., 10. We note that for this sum for h(i) contains an even number of odd values. In both cases, 2Mh¯(i) is even. range of m, it is only for m = 4 that γˆm = γm(1). From 6 Therefore, the ADS of any labeling λ Λ (i) satisfies the column Pb(ˆγ,βm), computed as detailed in Section VIII, m h¯(k, λ) hˇ (k), where ∈ we conclude that the maximal optimality thresholds are indeed ≥ i quite low from a practical viewpoint; the BRGC is the optimal h¯(k,β ), k =0, 1,...,i 1 labeling whenever the targeted BEP P is less than 1.6 % and m b ˇ , ¯ − m 10, which covers most M-PSK systems of practical inter- hi(k) h(k,βm)+ ǫ, k = i ≤ est. h¯(k,βm) ω, k = i +1,i +2,... − Submitted June 29, 2005; revised May 30, 2006; final version, Sept. 19, 2006 9 If now (38) holds, then TABLE V THE M -PAM MAXIMALOPTIMALITYTHRESHOLDS γˆ OBTAINED FROM ∞ (37) AND LEMMA 14, THECORRESPONDINGBITENERGY-TO-NOISE RATIO 0 (i, µ)ǫ (k, µ)ω ≤P − P γˆb , γ/mˆ , AND THE BEP WHEN THE BRGC IS USED AT γ =γ ˆ. k=i+1 ∞ X = (k, µ) hˇ (k) h¯(k,β ) m γˆ [dB] γˆb [dB] Pb(βm, γˆ) P i − m k=1 2 –2.6 –5.6 0.277 X∞ ∞ 3 7.3 2.5 0.146 (k, µ)h¯(k, λ) (k, µ)h¯(k,β ) 4 15.2 9.2 0.090 ≤ P − P m k=1 k=1 5 22.4 15.4 0.061 X X = Pb(λ, γ) Pb(βm,γ) 6 29.3 21.5 0.044 − 7 36.0 27.5 0.032 for any labeling λ Λ (i). ∈ m 8 42.6 33.6 0.025 As for the PSK case, it is possible to sharpen these bounds in 9 49.1 39.6 0.019 ˇ ¯ many ways, e.g., by letting hi(k) = h(k,βm) for k M 1, 10 55.6 45.6 0.015 but as such improvements appear to influence the overall≥ BEP− very little, we use the above bounds for simplicity. where µ is the positive root of Q(µ) 3Q(3µ)+2Q(5µ). B. BRGC Optimality Thresholds for M-PAM ∗ − For any γ < γ2 , the NBC is the best labeling, while for The theorems in this Section are analogous to similar theo- ∗ γ >γ2 , of course, the BRGC is the best one. At the thresh- rems in Section V-B, but the proofs are simpler, thanks to the old γ∗ = 5.22 dB, the BEP is P (β ,γ∗)=0.337, to be 2 − b 2 2 attractive properties of the Gaussian Q-function. compared with Pb(β2, γˆ2)=0.277. Lemma 14: For any m 2 and i Ψm, there exists a unique γ = γ (i) that satisfies≥ ∈ m VII. THE OPTIMAL QAM LABELING Q ((2i 1)µ(γ)) Not surprisingly, a similar technique as in the previous two − 1 + (m 1)(M 1) (40) Q ((2i + 1)µ(γ)) ≥ − − sections can be applied to rectangular QAM constellations. We will show that the same results hold: the two-dimensional for i 1 with equality. The inequality is valid for all γ ≥ ≥ BRGC is optimal for high enough SNR, and finite thresholds γm(i). are obtained above which the BRGC is better than any other Proof: The left-hand side of (40) is equal to one for γ =0 labeling. However, the QAM case is different from PSK and and a continuous function of γ for γ 0. From Lemma 2 and ≥ PAM in two respects. Firstly, the critical indices are irrelevant; the relation (16) between µ and γ, the left-hand side of (40) is if we only determine when Gray codes are better than non-Gray strictly increasing in γ for a given i. It can be made arbitrarily codes for QAM, then earlier results can be used to establish that large, as shown in (35). the BRGC is the best of all Gray codes. Secondly, the maximal Theorem 15: P (λ, γ) P (β ,γ) for every m 2, i b ≥ b m ≥ ∈ optimality threshold turns out to be much higher than for PSK Ψm, γ γm(i), and λ Λm(i). and PAM. Proof:≥ The theorem∈ follows immediately from Lemma The starting point is the BEP expression (24) and the ADS 13, (17), and Lemma 14. components t¯(k,l,λ) and t¯ (k, λ). We will upperbound and Corollary 16: For any M-PAM constellation, the optimal la- 0 lowerbound these components for Gray and non-Gray label- beling at asymptotically high SNR is the BRGC. ings, respectively. This allows us to lowerbound the difference The solutions to (40) for i Ψ are shown in Figure 2 for m in BEP for the two classes of labelings. Particular attention is m = 2, 3,..., 10. The maximal∈ optimality thresholds, again paid to t¯ (1, λ), which is the value where Gray codes differ defined as in (37), are listed in Table V along with the result- 0 from non-Gray labelings. It can be shown that for any rectan- ing P (β , γˆ ), computed as in Section VIII. In analogy with b m m gular Gray code λ of order (m ,m ), Theorem 12, the main result of this section is stated as a theo- G 1 2 rem, which follows immediately from Theorem 15. ¯ 1 1 , ˆ Theorem 17: The BRGC is optimal for any M-PAM con- t0(1, λG)=1 t1 − 2M1 − 2M2 stellation at γ γˆm. , ≥ t¯0(k, λG) m1 + m2 tˆ2, k 2 The last column of Table V indicates that the theorem holds ≤ ≥ , for most M-PAM systems of practical interest (m 10 and t¯(k,l,λG) m1 + m2 tˆ3, k,l 1 ≤ ≤ ≥ Pb 1.5 %). For m = 2, we also compare the upper bound ≤ ∗ and for any non-Gray labeling λNG with m1 2 and m2 2, γˆ2 with the optimality threshold γ2 , which can be calculated ≥ ≥ exactly. To do this, we first generate all distinct labelings (in t¯ (1, λ ) tˆ + ǫ the sense of having different ADS’s) of order m =2. There are 0 NG ≥ 1 1 1 three such labelings: the BRGC, the natural binary code (NBC), t¯0(k, λNG) 1 = tˆ2 ω2, k 2 and another non-Gray labeling. We calculate their differential ≥ − 2M1 − 2M2 − ≥ 1 ADS’s (22) and equate pairwise their BEP’s (21) to find all in- t¯(k,l,λ ) 1 = tˆ ω , k,l 1 ∗ 2 NG 3 3 tersections between the BEP curves. The result is γ2 =5µ /2 ≥ − M1M2 − ≥ 10 To appear in IEEE TRANS. INFORM.THEORY TABLE VI Proof: From [2], we know that the only way to assign THE M × M QAM MAXIMALOPTIMALITYTHRESHOLDS γˆ FROM a Gray code to a rectangular QAM constellation is by taking THEOREM 18, THECORRESPONDINGBITENERGY-TO-NOISE RATIO the direct product of two PAM constellations, each labeled by γˆb =γ/ ˆ (2m), AND THE BEP WHEN THE BRGC IS USED AT γ =γ ˆ. a Gray code. The BEP of such direct product constellations is given by (27), which is minimized for any γ γˆ when λ = ≥ 1 m γˆ [dB] γˆb [dB] Pb(βm βm, γˆ) and , according to Theorem 17 and the definition × βm1 λ2 = βm2 2 12.6 6.5 0.022 (42). We conclude that the two-dimensional BRGC is optimal 3 21.5 13.7 2.7 10−3 for QAM whenever γ γˆ. 4 29.2 20.2 3.9 · 10−4 The maximal optimality≥ thresholds γˆ of square constella- 5 36.4 26.4 6.3 · 10−5 tions are listed in Table VI. They were obtained by numerically 6 43.4 32.6 1.1 · 10−5 solving (41). The corresponding BEP of the two-dimensional 7 50.1 38.7 2.0 · 10−6 BRGC, evaluated as in Section VIII, is also listed. It is interest- 8 56.8 44.7 3.8 · 10−7 ing to observe that the upper thresholds are much higher than · −8 9 63.3 50.8 7.5 10 the corresponding values for PSK and PAM, and γˆ = γG for all 10 69.8 56.8 1.5 · 10−8 orders m in the range of the tables. It is still safe to conclude · that the BRGC is asymptotically optimal even for QAM, but we cannot claim that the BRGC is optimal in the SNR range ǫ , 3/(2M M ), ω , m + m 1+1/(2M )+1/(2M ), of practical interest. This is partly due to the fact that the used 1 2 2 1 2 − 1 2 and ω3 , m1 + m2 1+1/(M1M2) are all positive. bounding technique appears to be weaker for QAM than for Theorem 18: Let − PSK and PAM, but also to the fact that non-Gray codes are in- deed more competitive in two dimensions. Specifically, the two , Q(3µ) t(µ) (2ǫ + ω3)Q(µ) + (ǫ + ω2) . (41) most likely symbol errors for QAM require that the norm of Q(µ) the noise vector exceeds µ and µ√2, respectively, whereas the If t(µ(γG)) = ǫ, where the relation between µ and γ for QAM corresponding values for PAM are µ and 3µ. Therefore, sacri- is given by (23), then Pb(λG,γ) Pb(λNG,γ) for any Gray ficing the Gray property, which implies that more bit errors are ≤ code λG, any non-Gray labeling λNG, and any γ γG. associated with the most likely error pattern, carries a heavier ≥ Proof: From (24), the differencein BEP between any non- penalty for PAM than QAM. Gray labeling and any Gray code can now be lowerbounded as ∗ If the optimality threshold γPAM is known for an M-PAM ∗ 4 constellation, then γˇ = 2γPAM is a lower bound on the opti- Pb(λNG,γ) Pb(λG,γ) ǫ (0, µ) (1, µ) mality threshold for an M M QAM constellation via (27). − ≥ m + m · P P 1 2 × ∞ ∞ ∞ Specifically, we conclude from the 4-PAM results in Sec. VI- B that the two-dimensional BRGC is the best product label- ω2 (0, µ) (k, µ) ω3 (k, µ) (l, µ) − P P − P P ing (but not necessarily the best labeling) for 4 4 QAM at k=2 k=1 l=1 X X X dB, for which × , and 4 γ > γˇ = 2.21 Pb(β2 β2, γˇ)=0.337 = ǫ(1 2Q(µ))(Q(µ) Q(3µ)) that the product− of two NBC’s is the best× product labeling2 at m + m − − 1 2 γ < γˇ. This lower bound, however, is still far from the upper 2 bound for m =2 in Table VI and we do not know where γ∗ lies ω2(1 2Q(µ))Q(3µ) ω3Q (µ) − − − in this interval. 4 (ǫ t(µ))Q(µ). VIII. BEP OF SYSTEMS WITH BRGC LABELINGS ≥ m1 + m2 − Now that the BRGC has been shown to minimize the BEP of By Lemma 2 and the monotonicityof Q(µ), t(µ(γ)) is a contin- multilevel PSK, PAM, and QAM transmission over the Gaus- uous, monotonically decreasing function for γ 0. It ranges sian channel for large enough SNR, we evaluate this minimum from t(0) = 2ǫ + ω + ω /2 to lim t(µ(≥γ)) = 0 (see 2 3 γ→∞ BEP in the three cases. This is achieved by deriving closed- (35)). Thus, there is a unique positive value γ for which G form expressions for the differential ADS of the BRGC and t(µ(γ )) = ǫ. For any γ γ , t(µ(γ)) ǫ and thus G ≥ G ≤ utilizing the general BEP expressions given in Section III. Pb(λG,γ) Pb(λNG,γ). So far we≤ have shown that all Gray codes are better than all In [7], it was shown thatif an M-PSK constellation is labeled by βm, the ADS is given by non-Gray codes in rectangular QAM systems with γ γG, but which Gray code is the best of them? This question≥ can be m ¯ m i answered without further analysis by exploiting known results. d(k,βm) = tri(2 , k)+ tri 2 , k (43) Theorem 19: Let i=2 X 2 2 for all integers k. The function tri(N, k) is a periodic triangular , M2 1 M1 1 γˆ max γ , 1+ − γˆ 1 , 1+ − γˆ 2 G M 2 1 m M 2 1 m function of period N, defined by 1 − 2 − (42) k k tri (N, k) , 2 N − N where γˆm are the maximal optimality thresholds for PAM. Then the BRGC is optimal for any M M QAM constellation at 2 1 × 2 A third labeling that attains the same BEP at γ =γ ˇ is the product of one γ γˆ. BRGC and one NBC. ≥ Submitted June 29, 2005; revised May 30, 2006; final version, Sept. 19, 2006 11 where the function x rounds x to the closest integer (ties are same conclusion cannot be drawn for QAM, possibly because rounded arbitrarily)⌊ ⌉ the derived upper bounds on γ∗ are too loose. To calculate the BEP in the form (13), we need the differ- The paper includes new closed-form expressions for the BEP ential ADS of the BRGC. Since d¯(k) is a sum of triangular of the three modulation formats with BRGC labelings (Section sequences, ∆(¯ k) is a sum of piecewise constant functions. In VIII). These expressions, which we believe are the simplest particular, available for the purpose, have the additional benefit of sepa- rating the influence of the labeling on the BEP from that of the 2 tri (N, k) tri (N, k 1) = ( 1)⌊2(k−1)/N⌋ (44) constellation geometry. Analogous BEP expressions for arbi- − − N − trary labelings are also given (Section III). where x denotes the integer part of x. Combining (14) with (43) and⌊ (44),⌋ we obtain 1−m −i APPENDIX ( 1) (k−1)2 m−1 ( 1) (k−1)2 ∆(¯ k,β )= − ⌊ ⌋ + − ⌊ ⌋ (45) PROOF OF THEOREM 20 m 2m−1 2i i=1 X For labelings that are symmetric in the sense that for all integers k. We believe that (45), combined with (13), is d (c , c ) = d (c , c ) for all l = 0,...,M 1, the simplest published form for the exact BEP of M-PSK with H 0 l H M−1 M−1−l (22) can be simplified to − the BRGC. The BEP of PAM constellations with the BRGC can be com- M−k−1 puted using either (15) or (21). Since the former method turns 1 δ¯(k, λ)= (d (c , c ) d (c , c )) . out to yield somewhat complicated expressions [15, Pt. E], we M H l l+k − H l+1 l+k l=0 treat in this paper only the latter method, which is based on the X (47) differential ADS. Thus, the BEP is given by (21) in combina- tion with the following theorem, which is proved in the Ap- Define a mapping f : 0, 1 Z such that f(0) = 1 and pendix. Another expression for the BEP of PAM constellations f(1) = 1. If the components{ } → of a label c 0, 1 m are was given in [11, eq. (9)–(10)], with a more complicated proof. i denoted c− ,c ,...,c , then ∈ { } Theorem 20: The differential ADS of a PAM constellation i,m i,m−1 i,1 labeled with the BRGC of order m is, for 1 k 2m 1, m ≤ ≤ − 1 d (c , c )= (1 f(c )f(c )) . H j l 2 − j,i l,i m i=1 1 1 k 1/2 (k−1/2)/2i X δ¯(k,β )= − ( 1) . m 2i − 2m 2i − ⌊ ⌋ i=1 With this notation, (47) can be written as X (46) The two-dimensional BRGC is the direct product of two one- m M−k−1 1 dimensional BRGC’s. Thus, the BEP of a rectangular QAM δ¯(k, λ)= f(c ) (f(c ) f(c )) . 2M l+k,i l+1,i − l,i system with the BRGC is simply given by (27) in combination i=1 l=0 with (21) and (46). A recursive method to compute the same X X (48) BEP was given in [16]. From Table I or by induction on m, it is easily verified that IX. CONCLUSIONS AND COMMENTS the labels (c0,..., cM−1) of the BRGC satisfy We have addressed the problem of finding an optimum signal − constellation labeling with respect to minimizing the BEP for (l+1/2)2 i f(cl,i) = ( 1)⌊ ⌉ M-PSK, M-PAM, and M M QAM under the assumptions − 1 × 2 of a Gaussian channel, equally likely and statistically indepen- for l = 0,...,M 1 and i = 1,...,m. Furthermore, since dent transmitted bits, and coherentmaximum likelihood symbol the BRGC possesses− the required symmetry, (47) and (48) hold detection. The result is that for the asymptotic case when the and the differential ADS of the BRGC is signal energy-to-noise ratio γ approaches infinity, the BRGC gives the lowest possible BEP among all Gray codes (and other m M−k−1 1 −i labelings), for all three modulation types. δ¯(k,β )= ( 1) (l+k+1/2)2 m 2M − ⌊ ⌉ The BRGC is in fact the optimal labeling for a significant i=1 X Xl=0 range of values for γ. In particular, the BRGC is shown to be −i −i ( 1) (l+3/2)2 ( 1) (l+1/2)2 . optimal as long as γ γˆ, where γˆ is an upper bound on the · − ⌊ ⌉ − − ⌊ ⌉ optimality threshold γ≥∗ (defined as the smallest SNR for which h i the BRGC yields the smallest possible BEP). Numerical val- The bracketed expression is nonzero only when l = (n + ues of γˆ are given, and by evaluating the BEP at the thresholds, 1/2)2i 1 for some integer n. When l goes from 0 to M k 1, the conclusion is drawn that when the BEP is below a few per- then n −goes from 0 to nˆ , (M k)/2i 1/2 . (If the− brack-− cent, the BRGC is the optimum labeling for PSK and PAM. The eted expression is zero for all l in− the interval,− then nˆ = 1 and − 12 To appear in IEEE TRANS. INFORM.THEORY the sum over n below should be interpreted as zero.) Thus, [16] L.-L. Yang and L. Hanzo, “A recursive algorithm for the error probability evaluation of M-QAM,” IEEE Communications Letters, vol. 4, no. 10, m nˆ pp. 304–306, Oct. 2000. 1 −i δ¯(k,β )= ( 1)n+ 1/2+(k−1/2)2 m 2M − ⌊ ⌉ i=1 n=0 Erik Agrell received the M.S. degree in electrical en- X X −i −i gineering in 1989 and the Ph.D. degree in informa- ( 1)n+ (1+2 )/2 ( 1)n+ (1−2 )/2 tion theory in 1997, both from Chalmers University · − ⌊ ⌉ − − ⌊ ⌉ of Technology, Sweden. h m i From 1988 to 1990, he was with Volvo Technical Development as a Systems 1 −i = (ˆn + 1)( 1) 1/2+(k−1/2)2 Analyst, and from 1990 to 1997, with the Department of Information Theory, 2M − ⌊ ⌉ Chalmers University of Technology, as a Research Assistant. In 1997–1999, i=1 he was a Postdoctoral Researcher with the University of Illinois at Urbana- X −i −i ( 1) (1+2 )/2 ( 1) (1−2 )/2 Champaign and the University of California, San Diego. In 1999, he joined · − ⌊ ⌉ − − ⌊ ⌉ the faculty of Chalmers University of Technology as an Associate Professor, hm i holding a position first at Chalmers Lindholmen University College and since 1 −i = (ˆn + 1)( 1) (k−1/2)2 . (49) 2001 in the Department of Signals and Systems. His current research interests M − ⌊ ⌋ include geometrical aspects of coding theory, in particular lattices and block i=1 codes, bit-to-symbol mappings in digital communication systems, and coding X and modulation for fiber-optic channels. For any 0 <ǫ< 1 and any integers x and y, x/y +1/2 = Dr. Agrell served as Publications Editor for IEEE TRANSACTIONS ON IN- (x + ǫ)/y . Applying this identity to nˆ and letting⌊ ǫ = 1⌋/2, FORMATION THEORY from 1999 to 2002. ⌊we obtain ⌉ M k +1/2 nˆ = − 1 2i − which substituted into (49) yields (46). ACKNOWLEDGMENT Johan Lassing was born in Bor˚as, Sweden, in 1973. We thank two anonymous reviewers for their suggestions, He received the M.S. degree in engineering physics in 1995 and the Ph.D. degree in communication systems which improved the presentation of the paper significantly. in 2005, both from Chalmers University of Technol- ogy, Sweden. From 1997 to 1999 he was with the Swedish space system developer Om- REFERENCES nisys Instruments AB and was part of the development of the scientific mea- surements satellite ODIN until its launch in 2001. In 2001 he started Qamcom [1] J. G. Proakis, Digital Communications, McGraw-Hill, 3rd edition, 1995. Technology AB, a company devoted to communication systems equipment im- [2] R. D. Wesel, X. Liu, J. M. Cioffi, and C. Komninakis, “Constellation plementation, currently with 14 employees. Dr. Lassing currently works as the labeling for linear encoders,” IEEE Transactions on Information Theory, CEO of Qamcom Technology AB. vol. 47, no. 6, pp. 2417–2431, Sept. 2001. His main research interests lie in the fields of modulation and coding theory. [3] F. Gray, “Pulse code communications,” U. S. Patent No. 2632058, Mar. 1953. [4] G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation mappings for two-dimensional signaling of nonuniform sources,” IEEE Transactions on Communications, vol. 51, no. 3, pp. 400–408, March 2003. [5] E. Agrell, J. Lassing, E. G. Str¨om, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3170–3182, Dec. 2004. [6] S. Verd´u, Multiuser detection, Cambridge University Press, 1998. [7] J. Lassing, E. G. Str¨om, T. Ottosson, and E. Agrell, “Computation of the exact bit error rate of coherent M-ary PSK with Gray code bit mapping,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1758–1760, Erik G. Strom¨ (S’93–M’95–SM’01) received the Nov. 2003. M.S. degree from the Royal Institute of Technology [8] C. E. Shannon, “Probability of error for optimal codes in a Gaussian (KTH), Stockholm, Sweden, in 1990, and the Ph.D. channel,” Bell System Technical Journal, vol. 38, pp. 611–656, May 1959. degree from the University of Florida, Gainesville, in [9] J. Lassing, E. Str¨om, T. Ottosson, and E. Agrell, “The exact symbol and 1994, both in electrical engineering. He accepted a bit error probabilities of coherent M-ary PSK,” in Proc. IEEE Interna- postdoctoral position at the Department of Signals, tional Symposium on Information Theory, Yokohama, Japan, p. 11, June Sensors, and Systems at KTH in 1995. In February 2003. 1996, he was appointed Assistant Professor at KTH, [10] M. K. Simon and M. S. Alouini, Digital Communications over Fading and in June 1996 he joined the Department of Signals Channels. A Unified Approach to Performance Analysis, John Wiley & and Systems at Chalmers University of Technology, Sons, 2000. G¨oteborg, Sweden, where he is now a Professor in [11] K. Cho and D. Yoon, “On the general BER expression of one- and two- Communication Systems since June 2003 and head of the Communication Sys- dimensional amplitude modulations,” IEEE Transactions on Communi- tems Group since 2005. He received the Chalmers Pedagogical Prize in 1998. cations, vol. 50, no. 7, pp. 1074–1080, July 2002. Since 1990, he has acted as a consultant for the Educational Group for Individ- [12] G. S. Bhat and C. D. Savage, “Balanced Gray codes,” The Electronic ual Development, Stockholm, Sweden. He is a contributing author and asso- Journal of Combinatorics, vol. 3, no. 1, 1996. paper no. R25. ciate editor for Roy Admiralty Publishers’ FesGas-series, and was a co-guest [13] D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 2. editor for the special issue of the IEEE JOURNAL ON SELECTED AREAS IN Addision-Wesley, 2005. COMMUNICATIONS on Signal Synchronization in Digital Transmission Sys- [14] C. Savage, “Comments by the second author on R25 of volume 3 (1), tems, 2001. Dr. Str¨om was a member of the board of the IEEE VT/COM 1996,” The Electronic Journal of Combinatorics, 2005. Swedish Chapter 2000-2006. His research interests include communication [15] J. Lassing, “On the labeling of signal constellations,” Ph.D. disserta- theory in general, and constellation labelings, channel estimation, synchroniza- tion, Dept. of Signals and Systems, Chalmers University of Technology, tion, multiple access, and wireless sensor networks in particular. He has pub- G¨oteborg, Sweden, Jan. 2005. lished more than 70 conference and journal papers. Submitted June 29, 2005; revised May 30, 2006; final version, Sept. 19, 2006 13 Tony Ottosson was born in Uddevalla, Sweden, in 1969. He received the M.Sc. in Electrical Engi- neering from Chalmers University of Technology, G¨oteborg, Sweden, in 1993, and the Lic. Eng. and Ph.D. degrees from the Department of Information Theory, Chalmers University of Technology, in 1995 and 1997, respectively. Currently he is a Professor at the Communication Systems Group, Depart- ment of Signals and Systems, Chalmers University of Technology. During 1999 he was also working as a Research Consultant at Ericsson Inc, Research Triangle Park, NC, USA. From 1995 to 1998 he participated in the European FRAMES (Future Radio wideband Multiple Access System) project both as a co-worker and activity leader of the area of coding and modulation. Dr. Ottos- son has served as Associate Editor for IEEE Vehicular Technology from 2000 to 2004. His research interests are in communication systems and information theory. Specific topics are modulation, coding, CDMA, multiuser detection, combined source-channel coding, joint decoding techniques, synchronization, cross-layer interaction and scheduling. Dr. Ottosson has published more than 90 journal and conference papers, and holds several patents.