Abstract
Abstract-Quasi-orthogonal space-time block codes (QSTBC) achieve full diversity by constellation rotations. Several authors have introduced optimal rotation angles, found either by computer search or by analytical derivation. However, existing analytical methods do not seem general enough to analyze optimal rotations for arbitrary constellations, and some previous results seem to conflict. We present a novel method to exactly derive the coding gain of QSTBC as a function of the rotation angle and the minimum Euclidean distance of two-dimensional constellations such as the ones carved from lattices of squares and triangles, and phase-shift keying (PSK) constellations. The upper bound of coding gain for amplitude PSK (APSK) is also obtained. We find the whole range of optimal rotations for maximizing the coding gain of QSTBC. Simulation results confirm the theoretical analysis.
I. INTRODUCTION
Space-time block codes (STBC) have been extensively studied recently. One of the most well-known STBC is the Alamouti code [1] and it belongs to a general class of codes known as orthogonal STBC (OSTBC) [2] . Orthogonality enables single symbol maximum likelihood (ML) detection but results in low code rates. To improve the code rate, quasiorthogonal STBC (QSTBC) are independently proposed in [3] - [5] . QSTBC use OSTBC as building blocks and the code rate of QSTBC with 2m transmit (Tx) antennas equals the code rate of OSTBC with m Tx antennas. Therefore, QSTBC have higher code rates than OSTBC. In the codeword matrix of QSTBC, not all pairs of columns are orthogonal; however, the transmitted symbols can be ML decoded in pairs.
In order to achieve full diversity, half of the input symbols have to be rotated before encoding [6] , [7] . However, given a constellation, the optimum rotation angle must be chosen to minimize the symbol error rate (SER) or BER. Authors in references [6] - [11] provide some specific optimal rotation angles for quadrature amplitude modulation (QAM), phaseshift keying (PSK) and hexagonal constellations (or lattices of equilateral triangles) [12] . However, some results are not identical. For example, according to [10] , the optimal rotation angle is π/4 for 4QAM (and also higher order QAM in general), whereas it can be any value in the range from π/6 to π/3, the best one should be π/6 [13] , (or approximately π/6 [7] due to simulation results) or in the vicinity of 35 0 [6] . This problem motivates a general approach to derive optimal rotation angles for an arbitrary constellation.
In this paper, we present a simple method using the basic Euclidean geometry to derive not only the optimal rotations for QSTBC but also the coding gain of QSTBC as a function of rotation angle and the minimum Euclidean distance of two-dimensional constellations carved from lattices of squares (such as QAM), lattice of triangles (TRI) and phase-shift keying (PSK) constellations. An upper bound of coding gain for amplitude PSK (APSK) constellations is also obtained.
Since the optimal rotation angle is not unique, we propose an additional criterion to select the rotation angles, in which the number of signal points with minimum coding gain is minimized. Simulation results support this idea. To the best of our knowledge, ours is the first presented method that is general enough to analyze the coding gain of QSTBC with arbitrary constellations.
II. SYSTEM MODEL AND PRELIMINARY RESULTS
We first set some notations to be used in the paper. Superscripts (·) * and (·) † denote conjugate and conjugate transpose operations, respectively. The m × m identity matrix is I m . A signal constellation S is a finite set of possibly complex numbers. The minimum Euclidean distance of S is d min = min{|s −ŝ|∀s =ŝ; s,ŝ ∈ S} and the order or size of S is the number of elements of S. The set of positive integer numbers is N. A circularly complex Gaussian random variable x with mean m and variance σ 2 is denoted by x ∼ CN (m, σ 2 ). We consider data transmission over a quasi-static Rayleigh flat fading channel. The transmitter and receiver are equipped with m Tx and n Rx (receive) antennas. The channel gains h ik (i = 1, 2, ..., m; k = 1, 2, ..., n) between any pair of TxRx antennas are assumed CN (0, 1). We assume no spatial correlation at either Tx or Rx array. The receiver, but not the transmitter, completely knows the channel gains. Modulation symbols are drawn form a constellation S with zero mean and unit energy. They are encoded into space-time (ST) codeword C of size t × m, where the entry c li denotes the symbol transmitted from antenna i at time l (1 ≤ l ≤ t). The average transmitted power is constrained such that
The received signals y lk of the kth antenna at time l can be arranged in a matrix Y of size t × n. Thus, one can represent the Tx-Rx signal relation compactly as
The upper-bound of pair-wire error probability (PEP) derived by Tarokh et al. [14] is as follows:
where C andĈ are the transmitted and erroneous codewords, Γ is the minimum rank of a matrix ∆ C (∆ C = C −Ĉ) for all C =Ĉ, and λ 1 , λ 2 , , ..., λ Γ are non-zero eigenvalues of a product matrix P C = ∆ † C ∆ C . The diversity gain or diversity order d and coding gain G of ST codes are defined as
, respectively. The maximum achievable diversity order is d = mn, and in this case, the
Quasi-orthogonal STBC are constructed from OSTBC. In particular, a QSTB code for 2m Tx antennas employs OSTBC for m antennas as a building element. For example, a code proposed by Tirkkonen et al. [6] is as follows:
where
The scale factor λ guarantees the transmit power constraint.
Xu and Xia [10] show that
. With other QSTBC [3] , [5] , similar results can be obtained.
For two distinct QSTB codewords
We have the product matrix P Q as
where ∆a =
. To achieve full diversity, symbols s K+i (i = 1, 2, ..., K) are rotated, i.e., half of the input symbols are drawn from a constellation S and the other half are drawn from the rotated constellation R = e jφ S (j 2 = −1) where φ is the rotation angle [10] . Then det(P Q ) = 0 and QSTBC achieve full diversity. The coding gain of QSTBC with constellation rotation is given by [10] 
III. ANALYZING THE CODING GAIN OF QSTBC WITH TWO-DIMENSIONAL CONSTELLATIONS
In this section, we approach the problem from a new viewpoint. Different from Su and Xia [10] , who work with constellation S, we operate on the constellation generated by taking the differences of all signal points of S.
Let us introduce some basic notations on two-dimensional Euclidean vector space. Let us denote the vector connecting two arbitrary points U and V as − − → UV , the modulus of vector
If U overlaps the origin O and the location of V in the two-dimensional complex plane is v = V x + jV y where
We can use the vector notation to represent signal points in the two-dimensional plane. Let s andŝ be two elements of S, which can be labelled as two signal points S andŜ in the complex plane accordingly. Consider a constellation E, which contains the set of differences (s −ŝ) defined as:
We define another constellation F such that
Sets S, E and F may be called generating constellation, differential constellation and rotated differential constellation, respectively, where no confusion may arise. From (5) and (6), one has min e∈E, e =0 {|e|} = min
we obtain F = e jφ E. In words, constellation F is a rotated version of the constellation E with rotation angle φ. The coding gain of QSTBC in (4) is rewritten as
At this point, we have made no assumption on the structure of constellations E and F. Let E and F be two arbitrary signal points in the complex plane. Then | − − → OE| = |e|, | − − → OF | = |f |. Using vector addition and subtraction, we have
We define a quantity ω EF as
; then
where ϕ is the angle between vectors − − → OE and − − → OF ; (8) follows from basic trigonometry. We call ω EF the product distance because of its geometrical meaning. We further define the minimum product distance
ω EF . The subscript S emphasizes the association of MPD with the generating constellation S. The upper bound for minimum ζ-distance is given in [10] and [13] as d min,ζ ≤ d min . Equivalently, we can prove the upper bound of MPD. However, the proofs are omitted due to the lack of space. The reader is referred to [16] for full details.
Lemma 1: The MPD of QSTBC is bounded by
The coding gain of QSTBC is now proportional to the MPD Ω of two constellations E and F.
We distinguish two cases: constellation S has either lattice or non-lattice structure.
A. QSTBC with Lattice-Based Constellations
In general, any point S belonging to a two-dimensional lattice S can be represented in the complex plane as s = s 0 + k 1 ν 1 + k 2 ν 2 , where k 1 and k 2 are integers, and ν 1 and ν 2 are complex numbers that form a basis for the lattice [17] .
Proposition 1: If S has lattice structure, the constellation E defined in (5) has the following properties:
• Lattices S and E have the same basis.
• Lattice E includes the origin and is symmetric via the origin.
• If lattice S is finite, so is the lattice E. The proof is straightforward and is not given here. Proposition 1 shows the advantages of using the differential lattice instead of the original one. First, the lattice structure is preserved. Second, although some signal constellations may not be symmetric around the origin, their differential constellations are always include the origin and are symmetric about the origin.
We now examine two important lattices: Lattices of squares (including square and rectangular QAM) and lattices of equilateral triangles (TRI) (including hexagonal constellations) [12] . The optimal rotation angles for these constellations are discussed in [10] . However, we will show that the results of [10] are just special cases of our results.
Square Lattices
For square lattices, ν 1 = d min and ν 2 = e jπ/2 d min . We first study the basic 4-point square lattices, in particular, 4QAM plotted in Fig. 1 . Note that there is no restriction on the locations of the square lattices on the two-dimensional plane.
We have
min , and because of the symmetry of E, we first examine the case φ ∈ (0, π/2). One obtains
Therefore, the maximal value of Ω 4 QAM is d min , and is achieved if and only if the rotation angle satisfies π/6 ≤ φ ≤ π/3. Because of the symmetry of constellation R, any rotation angle φ such that
, n = 0, 1, 2, 3 also maximizes the MPD and hence maximizes the coding gain.
For higher order constellations, the lattices E and F are expanded outward. We have the following facts about their MPD.
Proposition 2: Let E and F be the signal points, each drawn from lattices E and F, where E and F are generated from a square lattice S by rules (5) and (6).
• If |e| = |f |, the product distance of E and F is lower bounded by d min . Fig. 1 . 4QAM, its differential and rotated differential constellations.
• Otherwise, their MPD is given by ω EF = (2 sin α) 1/2 |e|, where α is the angle between two vectors − − → OE and − − → OF . Thus, for square lattices, MPD is the product distance of the two signal points lying on the innermost circle, where |e| is minimum. Thus the MPD of any square lattices is upperbounded by the MPD of 4QAM. For 4QAM, we state the result as follows. . This result agrees with that found in [13] . For a higher order of QAM, we need to solve an additional constraint, which is more difficult for 32QAM or higher order. We can show analytically that the rotation φ = π/4 meets this additional constraint [16] . We have not shown analytically that φ = π/3, π/6 meet the constraint. Nevertheless, we can easily verify numerically that φ = π/3, π/6 are also optimal for square QAM constellations with at least up to 2 20 points.
Lemma 2: For 4-point square lattices with minimum Euclidean distance d min , the coding gain of QSTBC with rotation angle
φ ∈ [ kπ 2 , (k+1)π 2 ], k = 0, 1, 2, 3, is given by G Q = λ min{2| sin(φ − kπ 2 )| d 2 min , 2| cos(φ − kπ 2 )| d 2 min , d
Lattices of Equilateral Triangles
For the TRI lattice, ν 1 = d min and ν 2 = e jπ/3 d min . We present the main results for TRI lattices for brevity; however the derivations are similar to the case of square lattices. . Note that our Theorem 1 subsumes [10, Theorem 2], where only rotation angle φ = π/6 is provided. Using the same approach, with minimal effort, one can show that for PAM (one-dimensional lattice), the optimum rotation is π/2. This simple case is also briefly mentioned in [10] . Before concluding this section, we note that two optimal rotations for TRI lattices are φ = ±π/2 (k = 1, 4), and they do not require explicit multiplications; hence the encoding complexity is minimized without any loss of performance.
Theorem 1: For lattices of equilateral triangles with minimum Euclidean distance d min , the coding gain of QSTBC with rotation angle
φ ∈ [ kπ 3 , (k+1)π 3 ], k = 0, 1, ..., 5, is given by G Q = λ min{2| sin(φ − kπ 3 )| d 2 min , 2| sin[ (k+1)π 3 − φ]|d
B. QSTBC with Non-Lattice Constellations 1. M PSK Constellations

IV. SIMULATION RESULTS
This section verifies theoretical analysis by the simulations performed for a MIMO system with 4 Tx and 1 Rx antennas.
We verify the performance of different constellations with the same spectrum efficiency. Seven 8-point constellations with MPDs given in Table II are compared. Fig. 3 shows that the constellation with higher MPD has better SER performance at high SNR. Let N S denote the average number of neighbors at distance d min of constellations S. The performance of 8TRI-a and -b is only slightly better than that of the 8QAM rotated square. The reason is that N 8 TRI −a = 3.25 and N 8 TRI −b = 3.5 are higher than N 8 QAM rotated square = 2.5. This side effect also causes the performance of 8QAM square to be worse than that of 8QAM rectangular (N 8 QAM square = 2.5, N 8 QAM rect. = 2). Note that the constellation (4, 4) has the smallest MPD among the 8-point constellations (Table II) ; its SER should therefore be the worst. However, this prediction is not corroborated in Fig. 3 because the SNR is not high enough
V. CONCLUSION
We have introduced a general method to analyze the coding gain and optimal rotation angles for QSTBC with arbitrary two-dimensional signal constellations. Our new framework is general, unifies the existing results and delivers some new results. For lattices of equilateral triangles, we proposed optimal rotations of φ = ±π/2 to minimize the encoding complexity by eliminating the multiplications. Optimal rotations for PSK and various constellations are also presented. More important, the coding gain (or its upper bound) of QSTBC can be derived for any constellation. Thus, the performance of the codes can be accurately predicted.

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