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Euclidean and Space–Time Block Codes: Relationship, Optimality, Performance Analysis Revisited Alexander E. Geyer, Member, IEEE, Reza Nikjah, Member, IEEE, Sergiy A. Vorobyov, Senior Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE
Abstract—An equivalent model for a multiple-input– multiple-output communication system with space–time block codes (STBCs) is proposed based on a revealed connection between STBCs and Euclidean codes. Examples of distance spectra, signal constellations, and signal coordinate diagrams of Euclidean codes equivalent to simplest orthogonal STBCs are given. A new asymptotic upper bound for the symbol error rate (SER) of STBCs, based on the distance spectra of the equivalent Euclidean codes, is derived, and new general design criteria for signal constellations of the optimal code are proposed. Some bounds relating distance properties, dimensionality, and cardinality of STBCs with constituent signals of equal energy are given, and new signal constellations with cardinalities of 8 and 16 for Alamouti’s code are designed. A general methodology for performance analysis of STBCs is revisited. As an example of the application of this methodology, an exact evaluation of the SER of an orthogonal STBC is given. Namely, a new expression for the SER of Alamouti’s code with binary phase shift keying signals is derived. Index Terms—Euclidean codes, group codes, MIMO, optimal constellations, SER, signal coordinate diagrams, spherical codes, STBC.
I. I NTRODUCTION
A
NY space-time block code (STBC) [1]–[4] can be described mathematically by its corresponding code matrix and a constituent signal constellation. Hereafter the STBC signal constellation refers to the set of all realizable samples of the STBC matrix, each transmitted in a number of consecutive time slots, while the constituent signal constellation refers to the set of signals that constitutes the components of the STBC matrix. The cardinality of the former constellation will be denoted as M, and of the latter one, as L. The code matrices of orthogonal STBCs have been tabulated for many important cases [1], [2], [5]–[11] using the theory of complex orthogonal design [2] with the objective of adding multiplexing gain and combining
Manuscript received October 25, 2014; revised April 4, 2015; accepted May 26, 2015. Date of publication June 5, 2015; date of current version August 7, 2015. The associate editor coordinating the review of this paper and approving it for publication was G. Bauch. A. E. Geyer is with Huawei, Moscow 143441, Russia. R. Nikjah is with TEKTELIC Communications Inc., Calgary, AB T2E 8X2, Canada (e-mail:
[email protected]). S. A. Vorobyov is with the Department of Signal Processing and Acoustics, Aalto University, 02150 Espoo, Finland (e-mail:
[email protected]). N. C. Beaulieu is with the School of Information and Communication Engineering, and the Beijing Key Laboratory of Network System Architecture and Convergence, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2015.2442241
it with good minimum determinant for achieving the diversitymultiplexing tradeoff [12]. While the constructions of STBC signal constellations have been well investigated and extended to the designs of multi-block codes [13] and even the codes for multiple-access multiple-input multiple-output (MIMO) channels [14], the results on the optimal constituent signal constellations of STBCs are extremely limited [15]. In fact, almost all the investigations of STBCs are based on a restricted group of constituent signals which belong to a class of the constellations with independent signals that presumes the orthogonality of a code. It is, however, unknown whether such signal constellations are actually optimal. Moreover, no general results for guaranteeing the optimality of STBCs are available since the existing optimality criteria such as the rank criterion or more generally diversity-multiplexing tradeoff are addressing the STBC signal constellation only. It has been stressed, for example in [16], that general design criteria for optimal STBCs are unknown. Even the problem of finding constellations optimal in the sense of minimizing an average error probability of maximum-likelihood (ML) decoding on Rayleigh flat fading channels remains an open problem, which has been investigated in [16] only for the smallest possible constellations (up to M = 5). Particularly, orthogonal STBCs that are optimal in the sense of minimizing the symbol error rate (SER) of ML decoding have been designed only for constellations with M = 2 ∼ 5. In these cases, the SER minimization is equivalent to the minimization of the average error probability of ML decoding. Similarly, although the distance properties of STBCs have been investigated in some previous research works [17]–[19], the distance properties of STBC signal constellations have not attracted any attention. Indeed, existing results on the distance properties of orthogonal STBCs aim at verifying the resilience properties of the codes, where a multidimensional constellation is said to be resilient in flat fading if it retains its shape when its points are subject to the multiplicative distortion associated with fading coefficients [20]. However, it is specifically the full understanding of the distance properties of STBC signal constellations that can enable formulating requirements or design criteria for STBC signal constellations. In this paper,1 the aforementioned distance properties of STBCs with arbitrary signal constellations are analyzed. Based on this analysis, a new equivalent model for a communication
1 Some initial results have been reported in [21].
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GEYER et al.: EUCLIDEAN AND SPACE–TIME BLOCK CODES
system with orthogonal space-time block coding is proposed. The model exploits a connection found between the distance properties of STBCs and the distance properties of Euclidean codes, which allows viewing certain Euclidean codes as equivalent codes to STBCs. This connection brings insights into STBCs since Euclidean codes fall under the classic theory of error correcting codes [17]. This connection enables one to formulate a new general criterion for designing optimal STBCs with arbitrary constituent signals for the case of large signal-tonoise ratios (SNRs). Indeed, STBCs can be viewed as a subclass of error correcting codes having a specific design criterion that enables searching for new existence conditions for optimal STBC signal constellations with constant envelope constituent signals. Such conditions are based on a connection between the optimal STBC signal constellations with equal energies and a class of spherical codes [22]. As an example, we derive two new biorthogonal signal constellations with cardinalities M = 8 and M = 16 for the Alamouti code with constant energy signals. The model introduced for the STBC MIMO system enables one to develop a new performance analysis methodology. Existing results on performance analysis, including orthogonal STBC performance analysis, (see [23]–[30] and the references therein) aim at deriving exact solutions only for the SER of the constituent signals of the code, and there are no results on the exact solution for the true SER. Thus, such new methodology is of interest. As an example of applying this methodology, we derive a closed-form solution for the SER of the Alamouti STBC with constituent binary phase shift keying (BPSK) signals. To the best of the authors’ knowledge, this is a new expression for the SER of the Alamouti STBC with constituent BPSK signals. Moreover, this result is the only exact expression available for the SER of any orthogonal STBC. The remainder of this paper is organized as follows. In Section II, the distance properties of orthogonal STBC signal constellations are analyzed and a new equivalent model for a MIMO communication system with orthogonal space-time block coding on a quasistatic fading channel is given. Some examples of signal coordinate diagrams and distance spectra of some simplest STBCs are given. A new union bound on the SER of an STBC based on the distance properties of equivalent Euclidean codes is derived in Section III. Using this bound, a new general design criterion for optimal STBC constellations and a new design criterion for optimal constant envelope STBC signals as well as some existence conditions are formulated. As an example of applying this new design criterion, two new optimal biorthogonal constellations for the Alamouti code with M = 8 and M = 16 are designed. Moreover, a new general STBC performance analysis methodology based on the equivalent model for MIMO systems is described. A new closed-form solution for the SER of the Alamouti STBC with constituent BPSK signals is also derived. Section IV presents some numerical examples and is followed by some conclusions in Section V. II. N EW S YSTEM M ODEL AND E QUIVALENT C ODES A. STBCs With Arbitrary Constituent Signals An orthogonal STBC can be defined by an NT × NT code matrix Gu with orthogonal columns. The entries gui,j (i, j = 1,
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. . . , NT ) of Gu are the elements st,u (t = 1, . . . , Nt ; u = 0, . . . , M − 1) of the codewords (signal constellations) su = [s1,u , s2,u , . . . , sNT ,u ]T ,
u = 0, . . . , M − 1
(1)
as well as the complex conjugates s∗t,u (t = 1, . . . , NT ; u = 0, . . . , M − 1), linear combinations of st,u (t = 1, . . . , NT ; u = 0, . . . , M − 1) and s∗t,u (t = 1, . . . , NT ; u = 0, . . . , M − 1), or zeros. Here, [·]T is the matrix transpose and NT is the number of transmit antennas. The codewords (1) belong to a block code with J constituent 1- or 2-dimensional (1- or 2-D) signals st,u (t = 1, . . . , J; u = 0, . . . , M − 1) and NT − J zero signals st,u = 0 (t = J + 1, . . . , NT ; u = 0, . . . , M − 1) with J ≤ NT denoting the number of information bearing constituent signals of the STBC. Since the multidimensional signal constellations (1) belong to a block code constructed of modulated symbols from its alphabet, such a code corresponds to the so-called Euclidean code known from the classic theory of error correcting coding. This connection helps to define a complex structure of M-ary constellations belonging to orthogonal STBCs. Definition 1 [31]: The Euclidean code is a finite set of M points (codewords) in n-D Euclidean space Rn . The constituent signals st,u (t = 1, . . . , NT ; u = 0, . . . , M − 1) of a canonical orthogonal STBC [1], [2] use the same, typically L-PSK or L-QAM, modulation with L being the cardinality of the constellation. Note, however, that in the general case, the constituent symbols of the Euclidean code st,u (t = 1, . . . , NT ; u = 0, . . . , M − 1) can have arbitrary signal constellations including correlated constellations. Assuming a flat fading channel, the signal received by the jth receiving antenna (j = 1, . . . , NR ) can be expressed as rj = Gu hj + nj ,
j = 1, . . . , NR
(2)
where hj = [h1,j, . . . , hNT ,j ]T is the NT × 1 vector of fading channel coefficients, which are assumed to be independent identically distributed zero-mean complex Gaussian variables with variance ρ/2 per dimension and are constant over NT (or some multiple of NT ) time periods; nj = [n1,j, . . . , nNT ,j ]T is an NT × 1 noise vector consisting of independent samples of zeromean complex Gaussian random variables each of variance N0 /2 per dimension; and rj = [r1,j , . . . , rNT ,j ]T is the NT × 1 received signal vector at the jth receiving antenna. It can be observed from (2) that if the STBC codewords occur with equal probability, the average received SNR per antenna is given by 2 ρ M−1 u=0 Gu F (3) SNRR MNT N0 where · F is the Frobenius norm of a matrix [32]. B. Distance Properties and Equivalent Model According to the classic approach of analyzing any type of modulation or coding schemes, the distance properties (signal coordinate diagrams) of the signals under consideration should be first studied. To study the distance properties, we assume the noise-free case. Then the Euclidean distance between received noise-free codeword vectors Gu hj and Gt hj (u = t) of an j orthogonal STBC, denoted as du,t,OSTBC, can be expressed
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Fig. 1. Equivalent model of a communication system with an STBC and ML decoding.
in terms of the distance for the equivalent Euclidean code du,t,EC su − st as j
du,t,OSTBC = hj du,t,EC j = 1, . . . , NR ; u, t = 0, . . . , M − 1; u = t
(4)
where · denotes the Euclidean norm of a vector. j Indeed, the distances du,t,OSTBC are defined as j d Gu hj − Gt hj = (Gu − Gt )hj u,t,OSTBC
j = 1, . . . , NR ; u, t = 0, . . . , M − 1; u = t.
(5)
Using the orthogonality property, and the property that [18, p. 120] (Gu − Gt )H (Gu − Gt ) = su − st 2 INT
(6)
(·)H
denotes the Hermitian transpose, the distances (5) where can be written as in (4) by also noting that su − st du,t,EC . Here, INT is the NT × NT identity matrix. Moreover, using the orthogonality property, the norms of the received noise-free codeword vectors Gu hj (u = 0, . . . , M − 1; j = 1, . . . , NR ) can be computed as H Gu hj = hH j Gu Gu hj = su hj u = 0, . . . , M − 1; j = 1, . . . , NR .
(7)
On the other hand, let j be an NT × NT arbitrary uniH tary matrix, i.e., H j j = j j = INT . Now consider the new constellation hj j su , u = 0, . . . , M − 1. It can be observed that this constellation has exactly the same Euclidean distance properties (4) and (7) of the fading-inflicted constellation Gu hj , u = 0, . . . , M − 1. Then the following result can be formulated based on the distance properties (4), (7). Result 1: A communication system with NT transmitting and NR receiving antennas, orthogonal STBCs, and ML decoding of received signals is equivalent to the system given in Fig. 1 for Gaussian noise channels. Proof of Result 1: The statement of the result directly follows from the properties (4) and (7), and the fact that two codes (signals) with the same Euclidean distance properties provide the same performance with ML decoding in the Gaussian noise channel [33].2 2 The claim of Result 1 may not hold in channels with other types of noise, such as Laplacian noise [34].
Note that the matrix j is, in fact, a rotation matrix in NT dimensions. What can be seen from Result 1 and Fig. 1 is that the STBC effectively transforms the fading MIMO channel into an equivalent coded single-input multiple-output (SIMO) channel with the corresponding fading coefficients cj hj (j = 1, . . . , NR ). This SIMO channel is invariant to phase rotation in the sense that different arbitrary rotation matrices j (j = 1, . . . , NR ) give rise to the same ML performance. In addition, the resulting SNR in the equivalent model in Fig. 1 is not always equal to the original average SNR (3), if we exclude the zeros, i.e., non-information-bearing components, of su in the equivalent Euclidean code of the equivalent model. In this case, recalling the definition of J in Section II-A, we can show by energy conservation that the average SNR in the equivalent model is NT /J times the SNR (3). It is worth stressing that the system model in Fig. 1 is a special case of a receiver diversity system [23], [35]. Another example of diversity scheme that exploits the polarization diversity (channel polarization) is the system with the capacityachieving polar codes [36]. However, exploring this path is outside of the scope of this paper. We rather stress here that the proposed coded SIMO model is, however, also fundamentally different from the well known single-input single-output (SISO) model of [23] because (i) it represents the actual multidimensional structure of M-ary STBCs; (ii) it is applicable to arbitrary constituent signal constellations; and (iii) it allows using any existing receiver diversity schemes. Finally, it is also worth highlighting that based on Fig. 1 one can see that Euclidean codes are equivalent to orthogonal STBCs in the sense that the parameters of the Euclidean codes are the only optimization parameters for optimizing the corresponding MIMO communication system. Most of the known orthogonal STBCs belong to a subclass of canonical codes, i.e., to the codes based on signal constellations with uncorrelated constituent signals. However, this condition is extremely restrictive for designing good signal constellations, while it is not necessary or particularly appealing from the practical (decoder complexity) viewpoint. Even if the orthogonal structure of STBCs is destroyed by correlated constellations, symmetries can be used for decoder design or a sphere decoder can be applied. The complexity of channel estimation at the receiver is anyway higher than the decoder complexity. As a result, STBCs with different or correlated constellations for the constituent signals and their properties are essentially overlooked and have not been studied. We aim at correcting this deficiency in the existing literature by providing a detailed analysis and design criteria for such codes in Section III. Let us start by connecting the terminology used to describe the signal constellations of STBCs with the terminology used for describing error correction codes. Definition 2: The Euclidean code s˜ u = [˜s1,u , . . . , s˜K,u ] is called equivalent to an STBC with “proxy” Euclidean code su = [s1,u , . . . , sNT ,u ] if the Euclidean distance between two arbitrary codewords su and st of the orthogonal STBC coincide with the distance between s˜u and s˜t for all u, t ∈ {0, . . . , M − 1}. In other words, the distances between two codeword vectors Gu hj and Gt hj of the orthogonal STBC coincide with the distances
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between s˜u and s˜t for all u, t ∈ {0, . . . , M − 1}, that is, du,t,EC j and du,t,OSTBC satisfy (4) for all j = 1, . . . , NR . It follows from Definition 2 that if an STBC code matrix does not contain any zeros, the equivalent Euclidean code coincides with the “proxy” Euclidean code. The salient example is Alamouti’s code. However, if the code matrix contains zeros, the dimensionality of the equivalent Euclidean code is smaller. For example, the equivalent Euclidean code to the well known rate 3/4 orthogonal STBC with the code matrix [11] ⎤ ⎡ s1,u 0 s2,u −s3,u ⎢ 0 s1,u s∗3,u s∗2,u ⎥ ⎥ , u = 0, . . . , M−1 Gu = ⎢ ⎣−s∗2,u −s3,u s∗1,u 0 ⎦ −s2,u 0 s∗1,u s∗3,u (8) and the Euclidean code su = [s1,u , s2,u , s3,u , 0] is s˜ u = [˜s1,u , s˜2,u , s˜3,u ].
TABLE I D ISTANCE S PECTRUM OF THE E UCLIDEAN C ODE E QUIVALENT TO A LAMOUTI ’ S C ODE W ITH C ONSTITUENT BPSK S IGNALS
TABLE II D ISTANCE S PECTRUM OF THE E UCLIDEAN C ODE E QUIVALENT TO A LAMOUTI ’ S C ODE W ITH C ONSTITUENT QPSK S IGNALS
TABLE III D ISTANCE S PECTRUM OF THE E UCLIDEAN C ODE E QUIVALENT TO THE R ATE 3/4 C ODE W ITH C ONSTITUENT BPSK S IGNALS
C. Examples of Euclidean Codes Equivalent to Simplest STBCs Traditionally, only distance properties of STBC matrices have been investigated [17]–[19]. However, the results that would report the distance properties of signal coordinate diagrams are lacking even for the simplest STBCs. Thus, the distance properties of Euclidean codes equivalent to some simplest STBCs are studied here starting from the following definitions. Definition 3: A distance profile Du of a codeword su is a set of Euclidean distances du,j,EC between the codeword su and all other codewords st (t = 0, . . . , M − 1; t = u). Definition 4 [37]: A code has a uniform constellation if all its codewords have the same distance profile. This means that all sets of distances between any codewords of the code are the same, and therefore, the corresponding average error probabilities under ML decoding are the same for all codewords. Definition 5: If a code has a uniform constellation, the corresponding distance profile is called a distance spectrum. Let the normalized Euclidean distance be defined as du,t,EC d˜ u,t,EC E¯ EC
(9)
where E¯ EC is the average energy of a codeword of the Euclidean code. The normalized distance spectra of two Euclidean codes equivalent to the simplest Alamouti code with constituent BPSK and quadrature PSK (QPSK) signals are given in Tables I and II, respectively. In Table III the normalized distance spectrum of the rate 3/4 orthogonal STBC [11] and the equivalent Euclidean code with constituent BPSK signals is also given. The average energies of the codes are E¯ EC = 2E for the Alamouti code with constituent BPSK and QPSK signals and E¯ EC = 3E for the rate 3/4 code with constituent BPSK signals, where E is the energy of a constituent signal of the code. All these codes have uniform signal constellations in the equivalent Euclidean codes. The corresponding signal constellations and signal coordinate diagrams (graphical representations) of the Euclidean codes equivalent to STBCs with the spectra given in Tables I–III are illustrated in Figs. 2–4, where Fig. 2, Fig. 3, and Fig. 4 correspond to Tables I, II, and III, respectively. Note that the
Fig. 2. The Euclidean code equivalent to Alamouti’s code with M = 4 and constituent BPSK signals with Gray mapping. (a) The signal constellation. (b) The normalized signal coordinate diagram.
codes are represented using the notation introduced in [38] for 4-D group codes. In these figures, su (u = 0, . . . , M − 1) is the codeword transmitted for binary u. In Figs. 2(a), 3(a), and 4(a), the constellation points transmitted for su are labeled by su itself, for simplicity. Note that in all figures, a Gray mapping scheme is followed, where the Euclidean distance between codewords su and st is nondecreasing as the Hamming distance between binary u and binary t increases. For example for M = 8, s0 and s7 or s2 and s5 have the largest distance. Figs. 3(b) and 4(b) use the Schlegel diagram [39] to provide a geometrical representation of the signals and codes. The tesseract depicted in Fig. 3(b) is an example of the 4-D Euclidean code (group code) well defined in 4-D geometry [40]. Note that the Euclidean codes given in Figs. 2, 3, and 4 belong to the class of spherical codes [22] and are also group codes [40].
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Fig. 3. The Euclidean code equivalent to Alamouti’s code with M = 16 and constituent QPSK signals with Gray mapping. (a) The signal constellation. (b) The Schlegel diagram of the tesseract (hypercube) in the 4-D space showing a geometrical representation of the code.
ample, [2], [16]), the most natural one is the following definition which is commonly used for modulated/coded signals in communication systems. Definition 6: An STBC with given constituent signals and M codewords is called optimal for a given type of channel if it provides the smallest SER under ML decoding among all STBCs with the same number of codewords M and arbitrary constituent signals. Note that the design criteria for optimal codes are typically based on connecting the asymptotic SER behavior of a code for a given channel under a given decoding algorithm, with distance properties of this code. Thus, it is important to derive a general design criterion for the optimal signal constellations of STBCs by connecting the distance properties of the equivalent Euclidean codes to the asymptotic SER performance. Then the design criterion for the particular case of large SNR and a large number of antennas is analyzed and existence conditions for the optimal signal constellations of STBCs with constituent signals having constant energy are provided, exploiting the connection of such STBCs with the optimal spherical codes. A new biorthogonal constellation, which is an example of a signal constellation for the Alamouti code, is also given. A. Union Bound on SER of STBCs for Rayleigh Fading Channels The union bound on the SER of STBCs in Rayleigh fading channels is needed to connect the distance spectra of equivalent Euclidean codes to the asymptotic properties of STBCs. This connection can be then used for formulating design criteria for equivalent Euclidean codes (i.e., constituent multidimensional signals) of the optimal STBC for the Rayleigh fading channel. Different upper bounds on the SER of orthogonal STBCs have been derived in [4], [16], [27]–[29]. However, one of the most often used upper bounds on the SER of orthogonal STBCs is a union bound, which can be written for codes with uniform constellations as M−1 Pr(Gu → Gt ) (10) Prs,OSTBC ≤ t=0,t =u
Fig. 4. The Euclidean code with M = 8 equivalent to the rate 3/4 code with constituent BPSK signals. (a) The signal constellation. (b) A cube in the 3-D space, which is a geometrical representation of the code.
Typically, the canonical orthogonal STBCs are defined in the literature as STBCs with independent information-bearing PSK signals, as is the case with the examples given above. However, it should be noted that independent information-bearing signals are just the L-ary auxiliary ‘components’ of a spatial modulator generating the actual M-ary multidimensional signal constellation of the orthogonal STBC with correlated signals.
where Pr(Gu → Gt ) is the pairwise error probability (PEP) of the code, i.e., the probability of detecting Gt when Gu is transmitted. The closed-form solution for the PEP of ML decoding for orthogonal STBCs with arbitrary constituent signals in the Rayleigh fading channel is well known [15]. Indeed, the PEP is calculated as the expectation of Pr(Gu → Gt | H) over H, where H [h1 , . . . , hNR ] is the matrix of channel coefficients. Using our notation for the equivalent Euclidean codes, the PEP of ML decoding for orthogonal STBCs with arbitrary constituent signals from [15] can be rewritten as r K−1 1 − μ2u,t 1 μu,t 2r Pr(Gu → Gt ) = − (11a) 2 2 4 r r=0
III. O PTIMALITY AND P ERFORMANCE A NALYSIS A cornerstone of designing “optimal” codes is a proper definition of the design optimality criterion. Although there is a number of different definitions of STBC optimality (see, for ex-
where K NT NR and where ˜2 du,t,EC γ¯c μu,t 2 4 + d˜ u,t,EC γ¯c
(11b)
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where γ¯c ρ E¯ EC /N0 is the average received Euclidean codeto-noise ratio (cf. (3)). To analyze the asymptotic behavior of the PEP (11), the following new lemma is useful. Lemma 1: The PEP (11) satisfies the following identity r K−1 1 − μ2u,t 1 μu,t 2r − 2 2 4 r r=0
K K−1 K − 1 + r 1 + μu,t r 1 − μu,t = . (12) 2 2 r r=0
Proof: Substituting z = (1 + μu,t )/2 into the combinatorial identity [41, eq. (5.138)] n n n+r r 1 − 2z 2r r n z 1+ (z(1 − z)) = (1 − z) r r 2 − 2z r=1 r=0 (13) we immediately obtain (12). Substituting (12) in (10) yields the union bound for the SER of orthogonal STBCs with uniform signal constellations in the Rayleigh fading channel in the form M−1 1−μu,tK K−1 K −1+r 1+μu,tr Prs,OSTBC ≤ × . 2 2 r t=0,t =u
r=0
(14) B. Large SNR γ¯c 1 In the case of large SNR when γ¯c 1, approximating the terms (1 − μu,t )/2 and (1 + μu,t )/2 using the first component of their Taylor series yields (1 − μu,t )/2 ≈ 1/ d˜ 2 γ¯c and u,t,EC
(1 + μu,t )/2 ≈ 1. Furthermore, using the following combinatorial expression [35, eq. (14.4–17)] K−1 K − 1 + r 2K − 1 = (15) r K r=0
we can approximate (14) as Ps,OSTBC
2K − 1 ≤ CEC (K) γ¯c −K K
(16a)
where CEC (K)
M−1
−2K d˜ u,t,EC
(16b)
t=0,t =u
is the normalized distance spectrum constant (NDSC) of the STBC. It is interesting that the NDSC is a fixed parameter of the STBC for a given K. The NDSC is defined only by the distance properties of the equivalent Euclidean code and it does not depend on SNR. Therefore, an Euclidean code with minimal CEC (K) among all Euclidean codes with the same M, K, and dimensionality n is optimal in the sense that it provides the smallest SER at large SNR. Here, identical dimensionality ensures the same requirement for time/frequency resources and the same required number of transmitted bits per dimension, needed for fair comparison of the SER. The following theorem gives a more precise statement of the optimality.
Theorem 1: For a quasistatic fading channel, large SNR γ¯c 1, and a given K, an STBC with cardinality M is optimal if and only if the Euclidean code equivalent to this STBC has the minimal NDSC (16b) among all Euclidean codes with the same M and dimensionality. Proof: Both necessity and sufficiency follow directly from (16). If an STBC is optimal, it has the minimal NDSC. Otherwise, a code with a smaller NDSC achieves smaller SER according to (16). Conversely, if an STBC has the minimal NDSC, it is not outperformed by any other STBC, as the latter has an equal or larger NDSC, and thus SER, based on (16). It follows from Theorem 1 that for a quasistatic fading channel, large SNR, and given K, an STBC signal constellation is optimal if and only if the Euclidean code equivalent to this STBC has the minimal NDSC (16b) among all Euclidean codes with the same M and dimensionality. It can serve as the general criterion for designing optimal STBC signal constellations on quasistatic fading channels. This is a new general design criterion for optimal STBC signal constellations. Moreover, as also follows from (16b) and Theorem 1, the optimality of a Euclidean code equivalent to a STBC for a given number of receiving antennas NR is not a sufficient condition for optimality of the same code for a different number of receiving antennas. This is due to the nonlinear behavior of the NDSC (16b) with respect to NR . However, methods of design for Euclidean codes with minimal NDSC are not known. Note that the results embodied in (16) have not appeared before in the context of STBCs. As a result, Euclidean codes satisfying the conditions of Theorem 1 have not yet been investigated for any STBC. Moreover, there is no regular method of design for any class of Euclidean codes which are optimal according to any design criterion. C. Large SNR γ¯c 1 and Number of Antennas NT , NR 1 In this case, the NDSC (16b) can be approximated as −2K CEC (K) ≈ Nd˜ min,EC d˜ min,EC
(17)
where d˜ min,EC is the minimal normalized Euclidean distance of the Euclidean code equivalent to the orthogonal STBC and where Nd˜ min,EC is the number of codewords with the minimal distance d˜ min,EC . Approximation (17) simplifies bound (16a) to
2K − 1 −K −2K ˜ γ¯c . (18) Ps,OSTBC ≤ Nd˜ min,EC dmin,EC K The bound (18) is well known [30]), and thus, can serve as a check here. However, we use this bound to derive existence conditions for STBCs based on their connection with the equivalent Euclidean codes. Note from (18) that the dominant parameter for STBC optimality in the case of large SNR and large number of antennas is the minimal distance of the equivalent Euclidean code. Although the design criterion (18) is known, it has not been exploited before that such criterion coincides with the standard one for the error correcting codes optimal for the Gaussian channel. Thus, results for the optimal Euclidean codes known from the classic theory of error correcting coding can be used to define the existence conditions of the optimal STBC for large SNR and large number of antennas.
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An interesting special case of Euclidean codes is a spherical code, for which every symbol of the code has the same norm [22]. Since we are interested in designing optimal STBCs, the notion of optimality for spherical codes is of importance. The optimal spherical code [22] is the code with the maximal minimum normalized Euclidean distance d˜ min,EC among all spherical codes with the same cardinality M and dimensionality n. Note that the Euclidean codes equivalent to STBCs with constant energy constituent signals belong to the class of spherical codes. This leads to interesting connections. Particularly, the bounds obtained for the spherical codes can be used to define parameters of the asymptotically optimal STBCs. Some of the strongest and deepest results on the existence conditions of spherical codes with small dimensionality and 2 ≤ 4 were obsquared minimal Euclidean distance 0 < d˜ min tained by Rankin [42] (see also [22, Ch. 1.4]) and CoxeterBöröczky [43, p. 28]. Based on the results of Rankin and Coxeter-Böröczky for spherical codes, the following bounds for asymptotically optimal signal constellations of STBCs can be formulated for the case of large SNR and large number of antennas. Theorem 2 (Similar to the Coxeter-Böröczky Bound): For quasistatic fading channels at large SNR and for a large number of antennas, any asymptotically optimal STBC that uses constituent signals with equal energies in the n ≥ 2 dimensional Euclidean space, satisfies the conditions d˜ min = 2 sin α2 and M ≤ 2Fn−1(α)/Fn (α) where Fn (α) is the Schläfli’s α function Fn (α) = π2 1 arcsec(n−1) Fn−2 (β)dα, with sec(2β) = 2 sec(2α) − 2, F0 (α) = F1 (α) = 1, 0 < α ≤ π. Proof: An STBC with equal-energy constituent signals corresponds to a Euclidean spherical code. Under the asymptotic hypotheses of the theorem, the optimality of the orthogonal STBC corresponds to the maximality of the minimum Euclidean distance of the equivalent spherical code. This maximality condition is satisfied under the claims of the theorem, based on [43, p. 28]. Theorem 3 (Similar to Rankin’s First Bound): For quasistatic fading channels at large SNR and for a large number of antennas, any asymptotically optimal STBC that uses constituent signals with equal energies satisfies the inequality 2 ≤ 2M/(M − 1). d˜ min Proof: See the proof of Theorem 2, and we also refer to [42] and [22, Ch. 1.4]. Note that the Rankin’s first bound does not depend on the dimensionality of the code. Theorem 4 (Similar to Rankin’s Second Bound):For quasistatic fading channels at large SNR and for a large number of antennas, the largest M of an STBC that uses constituent signals with 2 ≤ 4. equal energies satisfies the inequality M ≤ n+1 for 2 < d˜ min Proof: See the proof of Theorem 2, and we also refer to [42] and [22, Ch. 1.4]. Theorem 5 (Similar to Rankin’s Third Bound): For quasistatic fading channels at large SNR and for a large number of antennas, the largest M of an STBC that uses constituent signals 2 = 2. with equal energies satisfies the inequality M ≤ 2n for d˜ min Proof: See the proof of Theorem 2, and we also refer to [42] and [22, Ch. 1.4].
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Fig. 5. A 4-D biorthogonal code with M = 8 for Alamouti’s code. (a) The optimal signal constellation. (b) The Schlegel diagram of the hexadecachoron (16-cell) in the 4-D space showing a geometrical representation of the optimal signal constellation.
The importance of Theorems 3–6 is especially stressed by the fact that these theorems provide the only known general bounds on M as existence conditions for asymptotically optimal STBCs using constituent signals with equal energies, assuming coherent receivers, quasistatic fading channels, large SNR, and a large number of antennas. Although STBCs are connected now to spherical codes, it is still worth noting that regular methods for designing spherical codes with constituent modulated signals are not known. Thus, the code design problem is still not a simple problem, but such connections allow us to exploit some results on the design of spherical codes, such as a number of results summarized in [22]. Moreover, an approach based on the theory of group codes [35], [38] can also be useful, although methods for regular design of group codes with optimal distance properties are not known either. A possible undesirable consequence of considering group codes is that the constituent signals of these codes have symmetric properties; this is a severe restriction for code design and can result in non optimal codes. Finally, it is noteworthy that some useful properties of group codes suitable for the signal constellations of orthogonal STBCs have been exploited in the literature on unitary code design [44], [45]. D. New M = 8 and M = 16 Biorthogonal Signal Constellations for the Alamouti Code As an example of code design based on our studies in this section, we consider biorthogonal spherical codes. Indeed, biorthogonal spherical codes can be constructed for almost any multidimensional space [22]. Consider a 4-D biorthogonal code with M = 8 and an 8-D biorthogonal code with M = 16. Such codes satisfy the Rankin’s third bound with equality. Therefore, the signal
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form of receiver diversity of the block coded signals. Using the model in Fig. 1, the bit error rate (BER), denoted as Prb , and SER, denoted as Prs , of an STBC with arbitrary constituent signals over Rayleigh fading channel can be evaluated based on the classic approach of estimating the error performance of a digital communication system over fading channels. That is, the BER and SER of the equivalent Euclidean code are evaluated by statistically averaging the conditional BER PrEC b (γb ) and SER PrEC (γ ) at the output of a coherent receiver of this code for b s the Gaussian channel over the joint probability density function (PDF) of the fading amplitudes fγb (γb ) as ∞ PrEC (19) Prb,STBC = b (γb ) fγb (γb ) dγb 0 ∞ PrEC (20) Prs,STBC = s (γb ) fγb (γb ) dγb 0
Fig. 6. Optimal signal constellation of an 8-D biorthogonal code with M = 16 for Alamouti’s code.
TABLE IV N ORMALIZED D ISTANCE S PECTRUM OF THE B IORTHOGONAL S PHERICAL C ODE E QUIVALENT TO A LAMOUTI ’ S C ODE W ITH B IORTHOGONAL 4-D C ONSTITUENT S IGNAL C ONSTELLATION AND M = 8
constellations of the new biorthogonal spherical codes depicted in Figs. 5(a) and 6 based on QPSK signaling can serve as examples of new asymptotically optimal signal constellations for Alamouti’s codes with M = 8 and M = 16. The spectrum and graphical representation of the code with M = 8 are given in Table IV and Fig. 5(b), respectively. The spectrum of the code with M = 16 is similar to that of the code with M = 8; only the number 6 in Table IV changes to 14. The signal coordinate diagram of the code with M = 16 (in 8-D space) has not been depicted as it is cumbersome. Note from Figs. 5(a) and 6 that codewords s0 , . . . , sM/2−1 are orthogonal, and are respectively the complements of sM−1 , . . . , sM/2 to ensure Gray mapping. Performance simulations for these two codes are presented in Section IV. E. Performance Analysis Methodology The existing results on orthogonal STBC performance analysis [23]–[30] aim at deriving exact expressions only for the SER of constituent signals of orthogonal STBCs, while there are no results on the SER of an STBC in the sense of the probability that a codeword (code matrix) is transmitted but another codeword is detected. However, it is the latter SER for all types of modulation and coding, including orthogonal STBCs, that is a common performance evaluation measure in communication systems. Performance analysis of orthogonal space-time coding MIMO communication systems in a fading channel can be performed using the proposed equivalent model given in Fig. 1. The significant difference (useful from the performance analysis viewpoint) of the model in Fig. 1 with the classic receiver diversity system [35], [46, Fig. 1]) is that Fig. 1 is, in fact, a
where γb is the total instantaneous SNR per bit at the output R of the ML receiver given by γb N j=1 γj with γj being the instantaneous SNR per bit in the jth channel. Towards evaluating (19) and (20), it is first required to obtain the PDF of the combined fading coefficient fγb (γb ), conditional EC BER PrEC b (γb ), and SER Prs (γb ) of the equivalent Euclidean code on the Gaussian channel. As also follows from (19) and (20), the main problem of performance analysis of STBCs based on the proposed model in Fig. 1 can be reduced to the evaluation of BER/SER of the corresponding Euclidean code over the channel with Gaussian noise. For example, the problem is reduced to the BER/SER evaluation of a 4-D Euclidean code in the case of the canonical Alamouti code with 2-D constituent signals and to the evaluation of a 6-D Euclidean code in the case of the rate 3/4 orthogonal STBC with 2-D constituent signals. Although this methodology for exact BER/SER evaluation for the multidimensional signal constellations of interest is straightforward after the model in Fig. 1 is introduced, it cannot be found in the available literature and appears for the first time here. Another general methodology for performance analysis of STBCs with arbitrary constituent signals has been formulated in [26]. However, our methodology based on the classic performance analysis approach and the equivalent model in Fig. 1 is more straightforward and appears to be significantly simpler than the approach of [26]. As an example of applying our performance analysis methodology, we derive a closed-form solution for the BER and SER of the Alamouti orthogonal STBC with constituent BPSK signals. The signal coordinate diagram of the equivalent Euclidean code for Alamouti’s code with constituent BPSK signals is given in Fig. 2(b). Note that this diagram coincides with the signal coordinate diagram of QPSK with Gray mapping. Therefore, the Alamouti scheme with constituent BPSK signals corresponds to the receiver diversity scheme of Fig. 1 with QPSK signaling and with the ML receiver simplifying to the maximal-ratio combining receiver. EC The expressions for PrEC b (γb ) and Prs (γb ) of a coherent receiver for QPSK with Gray mapping in the Gaussian channel are well known and can be found, for example in [15], as (21) (γ ) = Q 2γb PrEC b b PrEC 2γb − Q2 2γb (22) s (γb ) = 2 Q
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∞ 2 where Q(x) = √1 x e−t /2 dt is the Gaussian Q-function. 2π Also, the average SNR per bit is γb =
E¯ η NT N0 log2 M
(23)
R NT 2 where, as follows from Fig. 1, η N j=1 cj = i=1 NR 2, E ¯ 2Eb , and M = 4 since the signaling is quah i,j j=1 ternary. Then, the average SNR per bit can be expressed as γb =
NR NT Eb hi,j2 . NT N0
(24)
i=1 j=1
It has been shown in [35] (see also [23]) that the PDF of the average SNR (24) is given as fγb (γ ) =
1 −γ γ K−1 e γ¯b K (K − 1)!γ¯b
(25)
where K = NT NR and where γ¯b Eb /NT N0 . Substituting (21) and (25) into (19), the average BER of the Alamouti code with constituent BPSK signals can be expressed as ∞ 1 −γ γ K−1 e γ¯b Q 2γ dγ . (26) Prb = K (K − 1)!γ¯b 0 Moreover, after some computations, it can be derived that r K−1 1 − μ2b 1 μb 2r Prb = − 2 2 4 r
(27)
r=0
√ where μb γ¯b /(1 + γ¯b ). Note that the solution (27) is not new and has been derived by Bauch et al. in [23] based on the SISO model and later also verified in [25]–[27]. The average SER of Alamouti’s code with BPSK constituent signals has not been obtained previously. Substituting (22) and (25) into (20), the average SER can be expressed as
∞
Prs = 0
1 −γ γ K−1 e γ¯b K (K − 1)! γ¯b × 2Q 2γ − Q2 2γ dγ . (28)
Moreover, using [46, eqs. (2) and (6)] and performing some computations, the expression (28) can be rewritten as 2 Prs = π
π 4
0
cos2 θ cos2 θ + γ¯b
K
1 dθ + π
π 4
0
sin2 θ sin2 θ + γ¯b
K dθ. (29)
To the best of the authors’ knowledge, (29) is a new expression for the SER of Alamouti’s code with the constituent BPSK signals. Moreover, this is the only available exact expression for the SER of any orthogonal STBC. All other known results are for the SER of the constituent signals of the orthogonal STBC, that is, obviously, not the same and less descriptive of the system performance.
Fig. 7. The BERs of three codes, Alamouti’s code with constituent QPSK signals, the rate 3/4 STBC with constituent QPSK signals, and Alamouti’s code with constituent biorthogonal 4-D signals, for the Rayleigh fading channel.
IV. N UMERICAL E XAMPLES The performance of the new asymptotically optimal signal constellations for the Alamouti code from Section III-D are tested by simulation in this section. Figs. 7 and 8 show the simulated BER performances for the STBC designs based on the spherical codes presented in Section III-D. The receiver structure is ML decoding based on the equivalent model in Fig. 1. In fact, the signaling for the equivalent model is Mary biorthogonal [35]. All the simulations have been performed using 107 trials. The standard identically and independently distributed (i.i.d.) Rayleigh fading channel model has been assumed with a unit noise variance. In other words, an additive white Gaussian noise (AWGN) channel with slow, frequency non-selective Rayleigh fading has been considered, and the fading coefficients remain constant over the transmission of a code matrix. The average SNR in Figs. 7 and 8 encapsulates the combined effect of the transmission power and the channel path loss. This simplified channel model is chosen here to allow for the use of existing analytical results for the performances of the Alamouti and rate-3/4 STBCs, which are presented in Figs. 7 and 8 for comparison. Fig. 7 shows the BER performance of the STBC design of Section III-D based on 4-D spherical codes. It shows additionally the BERs of two other conventional schemes for comparison, Alamouti’s code, and the rate 3/4 orthogonal STBC [11], both with constituent QPSK signals. Note that in these codes, the constituent signals are independent (in contrast to the new design) so that the Alamouti code has M = 16, and the rate 3/4 code has M = 64. The BERs of these codes are known to be equivalent to the BER of QPSK signaling in Nakagami fading channels [47, Section 5.1]. The choice of these two codes for comparison is justified as follows. The new code based on 4-D spherical codes uses two transmitting antennas and its rate is 3/2 bits per 2-D degree of freedom (DoF). However, there is no conventional space-time code with the same rate that uses two transmitting antennas. Nonetheless, the Alamouti code with constituent QPSK signals uses two transmitting antennas, but
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TABLE V P ERFORMANCE T RADEOFFS IN T RANSITION F ROM A LAMOUTI ’ S C ODE W ITH C ONSTITUENT B IORTHOGONAL 4-D S IGNALS TO THE C ODE W ITH B IORTHOGONAL 8-D S IGNALS
Fig. 8. The BERs of three codes, Alamouti’s code with constituent QPSK signals, Alamouti’s code with constituent BPSK signals, and Alamouti’s code with constituent biorthogonal 8-D signals, for the Rayleigh fading channel.
its rate is 2 bits per 2-D DoF. The rate 3/4 code uses four transmitting antennas, but has the same rate as the new code. The label “rate 3/4” only refers to the fact that the system transmits three symbols in four time slots. It can be seen from Fig. 7 that the new design is superior to the conventional Alamouti scheme. However, it is outperformed by the rate 3/4 orthogonal STBC [11], especially for a smaller number of antennas. However, the new code uses half as many transmitting antennas as the 3/4 code, which translates into less complexity and smaller size. Fig. 8, in a manner similar to Fig. 7, exhibits and compares the BER performances of three codes, including the new design based on 8-D spherical codes, and two conventional schemes, Alamouti’s code with constituent BPSK signals, and Alamouti’s code with constituent QPSK signals. The QPSK Alamouti code here is the same as the one used for comparison in Fig. 7. The BER of the BPSK Alamouti scheme has been obtained using the performance analysis methodology of Section III-E. All the codes use two transmitting antennas, and their rates are respectively 1 bit, 1 bit, and 2 bits per 2-D DoF. Fig. 8 demonstrates the BER superiority of the new code over the two conventional schemes, including the BPSK Alamouti scheme which has the same rate. The superiority is augmented as the number of receiving antennas increases. The performances of the new designs based on 4-D and 8-D spherical codes are, however, not directly comparable. In transition from the code with n = 4 and M = 8 (Fig. 7) to the code with n = 8 and M = 16 (Fig. 8), the number of dimensions or DoFs is doubled, which means that twice as many time and/or frequency resources are needed. The impact of this transition is shown in Table V in terms of tradeoffs between the utilized time DoFs, utilized frequency DoFs, continuous-time transmitting power, SNR, bit rate, and BER. Four different cases of tradeoff are presented. Value x denotes in the table that the value of the corresponding quantity is multiplied by x as a result of the transition. The approximate
Fig. 9. The SER and BER of Alamouti’s code with constituent BPSK signals for the Rayleigh fading channel.
changes shown for the BER are obtained by a comparison between Figs. 7 and 8 for relatively large values of SNR. Finally, Fig. 9 shows the SER and BER performances of Alamouti’s code with BPSK constituent signals for Rayleigh fading. The figure exhibits both results based on the analysis presented in Section III-E, and simulation results. In the analytical approach, integral (29) has been evaluated numerically. Note that the simulation results are in excellent agreement with the analytical results where the union bound has been used. V. C ONCLUSION Based on the analysis of the distance properties of STBCs, an equivalent model for MIMO communication system with STBCs has been proposed and a class of Euclidean codes equivalent to STBCs has been introduced. Examples of distance spectra, signal constellations, and signal coordinate diagrams of Euclidean codes equivalent to some simplest orthogonal STBCs have been given. A new asymptotic upper bound on the SER of STBCs, which is based on the distance spectra of the introduced equivalent Euclidean codes, has been derived. Also, new general design criterion for the signal constellations of optimal STBCs have been proposed based on the classical approach to code design of improving the code’s distance properties. This criterion has been shown to lead to the same results for asymptotic cases of large SNR and large SNR and large number of antennas, when the non-vanishing determinant criterion was,
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for example, used. Exploiting the connection between STBCs and spherical codes, some bounds which link the distance properties, dimensionality, and cardinality of equal-energy STBC signals were given. Then, two new signal constellations with cardinalities 8 and 16 for Alamouti’s code were designed as an example of using the connection between STBCs and spherical codes. Finally, using the model introduced for MIMO communication systems with STBCs, a general methodology for performance analysis of STBCs was formulated. As an application example of this methodology, a new expression for the SER of Alamouti’s code with BPSK signals was derived. This result is the first example of exact SER analysis of STBCs. Another example of diversity scheme is the coded system that exploits the polarization diversity (channel polarization) and the corresponding capacity-achieving polar codes. The study of polar codes, from the diversity point of view taken in this paper, is an interesting topic for future research. R EFERENCES [1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 10, pp. 1451– 1458, Oct. 1998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [3] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp. 1169–1174, Jul. 2000. [4] W.-K. Ma, “Blind ML detection of orthogonal space-time block codes: Identifiability and code construction,” IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3312–3324, Jul. 2007. [5] O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory, vol. 48, no. 2, pp. 384–395, Feb. 2002. [6] W. Su and X.-G. Xia, “Two generalized complex orthogonal space-time block codes of rates 7/11 and 3/5 for 5 and 6 transmit antennas,” IEEE Trans. Inf. Theory, vol. 49, no. 1, pp. 313–316, Jan. 2003. [7] X.-B. Liang, “Orthogonal designs with maximal rates,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2468–2503, Oct. 2003. [8] X.-B. Liang, “A high-rate orthogonal space-time block code,” IEEE Commun. Lett., vol. 7, no. 5, pp. 222–223, May 2003. [9] W. Su and X.-G. Xia, “On space-time block codes from complex orthogonal designs,” Wireless Pers. Commun., vol. 25, no. 1, pp. 1–26, Apr. 2003. [10] W. Su, X.-G. Xia, and K. J. R. Liu, “A systematic design of highrate complex orthogonal space-time block codes,” IEEE Commun. Lett., vol. 8, no. 6, pp. 380–382, Jun. 2004. [11] G. Ganesan and P. Stoica, “Space-time block codes: A maximum SNR approach,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1650–1656, May 2001. [12] L. Zheng and D. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003. [13] C. Hollanti and H.-F. Lu, “Construction methods for asymptotic and multiblock space-time codes,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 1086–1103, Mar. 2009. [14] H.-F. Lu, C. Hollanti, R. Vehkalahti, and J. Lahtonen, “DMT optimal codes constructions for miltiple-access MIMO channel,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3594–3617, Jun. 2011. [15] H. Jafarkhani, Space-Time Coding: Theory and Practice. Cambridge, U.K.: Cambridge Univ. Press, 2005. [16] M. Gharavi-Alkhansari and A. B. Gershman, “On diversity and coding gains and optimal matrix constellations for space-time block codes,” IEEE Trans. Signal Process., vol. 53, no. 10, pp. 3703–3717, Oct. 2005. [17] H. Shulze, “Geometrical properties of orthogonal space-time codes,” IEEE Commun. Lett., vol. 7, no. 1, pp. 64–66, Jan. 2003. [18] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [19] M. Gharavi-Alkhansari and A. B. Gershman, “Constellation space invariance of orthogonal space-time block codes,” IEEE Trans. Signal Process., vol. 51, no. 1, pp. 331–334, Jan. 2005.
[20] D. M. Ionescu and Z. Yan, “Fading-resilient super-orthogonal space-time signal sets: Can good constellations survive in fading?” IEEE Trans. Inf. Theory, vol. 53, no. 9, pp. 3219–3225, Sep. 2007. [21] A. E. Geyer, S. A. Vorobyov, and N. C. Beaulieu, “Equivalent codes and optimality of orthogonal space-time block codes,” in Proc. Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, USA, Nov. 6–9, 2011, pp. 1559–1563. [22] T. Ericson and V. Zinoviev, Codes on Euclidean Spheres, 1st ed. Amsterdam, The Netherlands: North Holland, 2001. [23] G. Bauch, J. Hagenauer, and N. Seshadri, “Turbo processing in transmit diversity systems,” Ann. Télécommun., vol. 56, no. 7–8, pp. 455–471, 2001. [24] C. Gao, A. M. Haimovich, and D. Lao, “Bit error probability for spacetime block code with coherent and differential detection,” in Proc. IEEE VTC, vol. 1, Vancouver, BC, USA, Sep. 24–28, 2002, pp. 410–414. [25] M. Gharavi-Alkhansari and A. B. Gershman, “Exact symbol-error probability analysis for orthogonal space-time block codes: Two-and higher dimensional constellations cases,” IEEE Trans. Commun., vol. 52, no. 7, pp. 1068–1073, Jul. 2004. [26] H. Zhang and T. A. Gulliver, “Capacity and error probability analysis for orthogonal space-time block codes over fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 808–819, Mar. 2005. [27] H. Lu, Y. Wang, P. V. Kumar, and K. M. Chugg, “Remarks on space-time codes including a new lower bound and an improved code,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2752–2757, Oct. 2003. [28] F. Behnamfar, F. Alajaji, and T. Linder, “Tight error bounds for spacetime orthogonal block codes under slow Rayleigh flat fading,” IEEE Trans. Commun., vol. 53, no. 6, pp. 952–956, Jun. 2005. [29] S. Sandhu and A. Paulraj, “Union bound on error probability of linear space-time block codes,” in Proc. IEEE ICASSP, Salt Lake City, UT, USA, May 7–11, 2001, vol. 4, pp. 2473–2476. [30] M. Brehler and M. K. Varanasi, “Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers,” IEEE Trans. Inf. Theory, vol. 47, no. 6, pp. 2383–2399, Sep. 2001. [31] K. Zeger and A. Gersho, “Number of nearest neighbors in a Euclidean code,” IEEE Trans. Inf. Theory, vol. 40, no. 5, pp. 1647–1649, Sep. 1994. [32] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1990. [33] S. Lin, and D. J. Costello, Error Control Coding, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 2004. [34] H. Shao and N. C. Beaulieu, “An investigation of block coding for Laplacian noise,” IEEE Trans. Wireless Commun., vol. 11, no. 7, pp. 2362–2372, Jul. 2012. [35] J. G. Proakis, and M. Salehi, Digital Communications, 5th ed. New York, NY, USA: McGraw-Hill, 2008. [36] E. Arikan, “Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jul. 2009. [37] G. D. Forney, Jr., “Geometrically uniform codes,” IEEE Trans. Inf. Theory, vol. 37, no. 5, pp. 1241–1260, Sep. 1991. [38] L. H. Zetterberg and H. Brändstörm, “Codes for combined phase and amplitude modulated signals in a four-dimensional space,” IEEE Trans. Commun., vol. 25, no. 9, pp. 943–950, Sep. 1977. [39] H. S. M. Coxeter, Regular Polytopes, 3rd ed. New York, NY, USA: Dover, 1973. [40] I. Ingemarsson, “Group codes for the Gaussian channel,” in Lecture Notes in Control and Information Sciences, Topics in Coding Theory. New York, NY, USA: Springer-Verlag, 1989, vol. 128, pp. 73–108. [41] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA, USA: Addison-Wesley, 1994. [42] R. A. Rankin, “The closest packing of spherical caps in n dimensions,” in Proc. Glasgow Math. Assoc., 1955, vol. 2, pp. 139–114. [43] J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. New York, NY, USA: Springer-Verlag, 1998. [44] B. L. Hughes, “Optimal space-time constellations from groups,” IEEE Trans. Inf. Theory, vol. 49, no. 2, pp. 401–410, Feb. 2003. [45] X.-B. Liang and X.-G. Xia, “Unitary signal constellations for differential space-time modulation with two transmit antennas: Parametric codes, optimal designs, and bounds,” IEEE Trans. Inf. Theory, vol. 48, no. 8, pp. 2291–2322, Aug. 2002. [46] M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communication over generalized fading channels,” Proc. IEEE, vol. 86, no. 9, pp. 1860–1877, Sep. 1998. [47] M. K. Simon and M.-S. Alouni, Digital Communication Over Fading Channels, 2nd ed. New York, NY, USA: Wiley, 2005.
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Alexander (Alex) E. Geyer (M’98) was born in Nikolaev, Ukraine. He received the RadioCommunication Engineer degree and the Ph.D. degree from Odessa National Academy of Telecommunication, Odessa, Ukraine, in 1989. From 1989 to 1992, he was with Odessa National Academy of Telecommunication as an Assistant Professor; from 1992 to 2002, as an Associate Professor; and from 2001 to 2002, as a Department Head. From 2003 to 2004, he was with Telecommunication Research Center of Catalonia (CTTC) as a Senior Research Fellow. In 2000 and 2001, he was with Digital Communication Group, Institute of Experimental Mathematics, Essen University, as a DFG (Deutsche Forschungsgemeinschaft) and DAAD (German Academic Exchange Service) Research Fellow. From 2008 to 2009, he was with the University of Alberta as a Research Fellow. From 2010 to 2013, he was with Huawei’s Research Center as a Chief Algorithm Engineer. His research interests include digital communications, and communication and coding theory.
Reza Nikjah (S’05–M’10) received the B.Sc. degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 2000, the M.Sc. degree in communications engineering from Sharif University of Technology, Tehran, Iran, in 2003, and the Ph.D. degree in communications from the University of Alberta, Edmonton, AB, Canada, in 2011. Since 2011, he has been with TEKTELIC Communications Inc., Calgary, AB, where he is currently a Radio Systems Engineer. His research interests include wireless relaying networks, rateless coding, MIMO, OFDM, error detection and error correction coding, spread spectrum, interference management, and space–time coding.
Sergiy A. Vorobyov (M’02–SM’05) received the M.Sc. and Ph.D. degrees in systems and control from Kharkiv National University of Radio Electronics, Kharkiv, Ukraine, in 1994 and 1997, respectively. Since his graduation, he also held various research and faculty positions at Kharkiv National University of Radio Electronics, Ukraine; the Institute of Physical and Chemical Research (RIKEN), Japan; McMaster University, Canada; Duisburg-Essen University and Darmstadt University of Technology, Germany; and the Joint Research Institute between Heriot-Watt University and Edinburgh University, U.K. He has also held shortterm visiting positions at Technion, Haifa, Israel, and Ilmenau University of Technology, Ilmenau, Germany. From 2006 to 2010, he was with the University of Alberta, Edmonton, AB, Canada, as an Assistant Professor; from 2010 to 2012, as an Associate Professor; and in 2012, as a Full Professor. He is currently a Professor with the Department of Signal Processing and Acoustics, Aalto University, Espoo, Finland. His research interests include statistical and array signal processing, applications of linear algebra, optimization, and game theory methods in signal processing and communications, estimation, detection, and sampling theories, and cognitive systems. Dr. Vorobyov was a member of the Sensor Array and Multi-Channel Signal Processing Committee of the IEEE Signal Processing Society from 2007 to 2012. He is a member of the Signal Processing for Communications and Networking Committee since 2010. He has served as the Track Chair for Asilomar 2011, Pacific Grove, CA, the Technical Co-Chair for IEEE CAMSAP 2011, Puerto Rico, and the Tutorial Chair for ISWCS 2013, Ilmenau, Germany. He served as an Associate Editor for the IEEE T RANSACTIONS ON S IGNAL P ROCESSING from 2006 to 2010 and for the IEEE T RANSACTIONS ON S IGNAL P ROCESSING L ETTERS from 2007 to 2009. He received the IEEE Signal Processing Society Best Paper Award in 2004, the Alberta Ingenuity New Faculty Award in 2007, the Carl Zeiss Award (Germany) in 2011, the NSERC Discovery Accelerator Award in 2012, and other awards.
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Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively, all in electrical engineering. From September 1986 to June 1988, he was with the Department of Electrical Engineering, Queen’s University, Kingston, ON, Canada, as a Queen’s National Scholar Assistant Professor; from July 1988 to June 1993, as an Associate Professor; and from July 1993 to August 2000, as a Professor. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications, University of Alberta, Edmonton, AB, Canada, and in January 2001, the Canada Research Chair in Broadband Wireless Communications. In 2014, he was a Copernicus Visiting Scientist, University of Ferrara, Italy. His current research interests include broadband digital communications systems, ultrawide bandwidth wireless systems, ad hoc wireless networks and cooperative wireless networks, fading channel modeling and simulation, diversity systems, multiple-input–multiple-output (MIMO) systems, space–time coding, synchronization in interference channels, and cognitive radio. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 IEEE International Conference on Communications and as a Corepresentative to the Technical Program Committee of the 1993 IEEE International Conference on Communications and the 1996 IEEE International Conference on Communications. He was the General Chair of the IEEE Sixth Communication Theory Mini-Conference in association with GLOBECOM’97 and the Cochair of the Canadian Workshop on Information Theory in 1999 and 2007. He served as the Cochair of the Technical Program Committee of the Communication Theory Symposium of the 2008 IEEE International Conference on Communications. He has been an Editor for Wireless Communication Theory of the IEEE T RANSACTIONS ON C OMMUNICATIONS since January 1992, and was the Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of the IEEE C OMMU NICATIONS L ETTERS from November 1996 to August 2003. He also served on the Editorial Board of T HE P ROCEEDINGS OF THE IEEE from November 2000 to December 2006. He has served as the Senior Editor of Security and Communication Networks (Wiley InterScience) since July 2007 and also served as the Cochair of the Technical Program Committee of the 2009 IEEE International Conference on Ultra-Wideband (ICUWB). He was a Distinguished Lecturer of the IEEE Communications Society in 2014. He received the University of British Columbia Special University Prize in Applied Science in 1980, as the highest standing graduate in the Faculty of Applied Science; the Natural Science and Engineering Research Council of Canada E. W. R. Steacie Memorial Fellowship in 1999; the Médaille K.Y. Lo Medal from the Engineering Institute of Canada in 2004; the Thomas W. Eadie Medal from the Royal Society of Canada and the ASTech Outstanding Leadership in Alberta Technology Award from Alberta Science and Technology Leadership Foundation in 2005; the J. Gordin Kaplan Award for Excellence in Research from the University of Alberta in 2006; the Edwin Howard Armstrong Achievement Award from the IEEE Communications Society in 2007; the IEEE Canada Reginald Aubrey Fessenden Medal and the Canadian Award in Telecommunications Research in 2010; the Radio Communications Committee Technical Recognition Award in 2011; the (Inaugural) Signal Processing and Communications Electronics (SPCE) Technical Committee Technical Recognition Award in 2013; and the IEEE ComSoc Communication Theory Technical Committee Technical Recognition Award in 2014. He was elected a Fellow of the Engineering Institute of Canada in 2001, of the Royal Society of Canada in 2002, of the Canadian Academy of Engineering in 2006, and of the IET in 2012. He is listed on ISIHighlyCited.com and was an IEEE Communications Society Distinguished Lecturer in 2007 and 2008. He served as a Distinguished Lecturer for the IEEE Communications Society in 2014.
