Research Article | Open Access
Grassmannian Constellation Based on Antipodal Points and Orthogonal Design and Its Simplified Detecting Algorithm
This study presents a framework of the unitary space time modulation (USTM) constellation based on antipodal points over Grassmannian manifold. The antipodal constellation enables an intrinsic simplified ML detecting algorithm. The algebraic orthogonal USTM constellation is also an antipodal constellation which, apart from being adaptive to the antipodal simplified ML detector, also has another simplified ML detector based on its self-indexing features, and the latter is simpler because of getting rid of the matrix operation. A searching orthogonal USTM constellation based on the grid search algorithm is obtained under the presented framework and its minimum Frobenius chordal distance and simulation performance are be superior to those of the algebraic orthogonal USTM constellation.
Grassmannian constellation is a set of unitary space time modulation (USTM) signal matrices defined on Gassmann manifold presented by Hochwald and Marzetta  and Zheng and Tse  for robustness against very fast fading in high speed mobile channels in which learning the channel fade coefficients becomes increasingly difficult for both transmitter and receiver. There are many methods about how to construct the USTM constellation, mainly including derivative-based optimization searching schemes  and algebraic structural schemes [4–8]. This study concentrated on the random and algebraic orthogonal  USTM constellation having the feature of antipodal point on Grassmannian manifold and their simplified maximum likelihood (ML) detecting algorithm.
The content of the paper including its main contributions is organized as follows. In Section 2, the preliminary knowledge which will be used throughout this paper is described, including the system model, the noncoherent maximum likelihood (ML) detector, and the chordal Frobenius distance measure. In Section 3, we build a framework of USTM constellation based on the antipodal points. The optimal packing method of searching the orthogonal unitary matrices over Grassmannian manifold and the corresponding searching algorithm are investigated. Under the constraint of the framework and by using the grid searching algorithm, we obtain a set of the orthogonal unitary matrices which contains many constellations of satisfying antipodal feature and orthogonality. Among them, an orthogonal USTM constellation with the optimum distribution of chordal Frobenius distance is determined by two explicit expressions. In Section 4, a simplified ML detecting algorithm based on antipodal points is derived and discussed. In Section 5, we demonstrate the antipodal feature of the algebraic orthogonal USTM constellation from  and derive its simplified ML detecting algorithm based on antipodal points. Furthermore, we deduce the indexing simplified ML detector of the algebraic orthogonal USTM constellation which only needs to operate several complex-values and get rid of the matrix operation. In Section 6, we show the simulation testing results between the searching and algebraic orthogonal USTM constellations which indicate that the searching constellation is superior to the algebraic that in both chordal Frobenius distance spectrum and performance with regard to symbol error probability and signal noise ratio. We conclude with some remarks in Section 7.
2. Preliminary and System Model
Consider a system with transmit and receive antennas. The channels between antenna pairs are Rayleigh flat fading and independent of each other. The channel fading coefficients are constant in a coherence interval and change to a new realization in the next interval. A system model  is given as follows:where and are, respectively the transmitted and received signal matrices, is a fading coefficient matrix and is an additive noise matrix, of which the elements of both are drawn from the i.i.d. standard complex Gaussian distribution , and is the expected signal-to-noise ratio (SNR) at each receiver antenna.
The capacity-achieving space time modulation signal distribution at high SNR is modelled as a set of unitary matrices : in which each matrix satisfies and all ’s are points on a Stiefel manifold, or the subspace spanned by column vectors of matrix is uniformly distribution in Grassmann manifold ; that is, . Let a set denote a USTM constellation which contains complex unitary matrices .
As the coefficients of are unknown to both receiver and transmitter, the noncoherent ML detector  is introduced: where is the trace operation of a matrix and is the complex conjugate transpose.
Let vector sets and be two principal vectors corresponding to two -planes . The principal angles between and are defined as for , subject to , , . The chordal Frobenius distance measure is defined as follows ( and references therein):where denotes a diagonal matrix formed by the singular values of the matrix .
3. A Framework of Grassmannian Constellation Based on Antipodal Point
3.1. A Framework of USTM Constellation
A pair of antipodal points are defined as two points with the furthest distance on a sphere. Since the capacity-achieving USTM signal distribution at high SNR is isotropic on the Grassmannian manifold and each signal point is denoted as a unitary matrix , is defined as an antipodal point of if is the orthogonal complement of on . Then how can one decide whether the two matrices on are the antipodal matrix? This can be done with the following lemma.
Lemma 1. Let be two unitary matrix on . and become a pair of antipodal matrices if and only if either or and , where is a identity matrix and is a full-zero matrix.
Proof. implies that each of column vectors of is orthogonal to each of column vectors of ; that is, and are orthogonal and complement each other. Since are two unitary matrices, they are used to construct a matrix . indicates that column vectors of and span a basis of Euclid space , so and are orthogonal and complement each other.
Construction 1 (a framework of USTM). Let denote a constellation and . Let denote a complex element at the th row and the th column of for and . If for any positive integer , each code word of has the structure and satisfies the following constraints:(1)For all ,, where is a identity matrix;(2)All points of are partitioned into two parts and in such a way that there exist one-to-one antipodal points between and but there is no antipodal point in each of and ;(3)The degree of freedom of elements in each is ;(4)All are unknown for and .Then the constellation set is called a framework of the full diversity USTM constellation based on antipodal points on .
3.2. The Construction of Antipodal Constellation with Orthogonality
According to the analysis on how to choose and [1, 2], the simplest case of was considered. A framework of the unitary matrix on was built similar to (4), and the degree of freedom of its elements was . Therefore, let , be four independent complex elements of , where is an imaginary unit. Then a unitary matrix with uncertain eight values , , , is formed as follows:
The optimal packing method stated was used to determine with all points like (5). That is, for the fixed and , design a packing in of cardinality so that its minimum distance similar to (3) is as large as possible. In fact, a complex number set of needs to be obtained in order to form a constellation on so that the minimum Frobenius chordal distance of (3) is maximized. The optimal packings of points on require the solution of the following optimization problem:
If the complex elements of each unitary matrix (a constellation point) are referred as to the parameter of the model for the underlying system, such as USTM, then the parameters can be thought of the hyperparameter of the same system. The so-called hyperparameter optimization, also called model selection, is the problem of choosing a set of hyperparameters . Thus we need to solve the problem of hyperparameter optimization. The traditional way of performing hyperparameter optimization has been grid search algorithm, or a parameter sweep, which is simply an exhaustive searching through a manually specified subset of the hyperparameter space. From the above, we need to consider the following factors.
A grid search algorithm must be guided by some performance metric. Here the performance metric space is to maximize the minimum chordal Frobenius distance.
Since the parameter space may include real-valued or unbounded value spaces for our parameters , our searching scheme needs to be tuned for good performance on an unknown data set; then manually set bounds and discretization may be necessary before applying grid search.
Since grid search suffers from the curse of dimensionality and doing a complete grid search may also be time-consuming, we considered using a coarse grid first. If the searched constellation cannot satisfy the some predetermined threshold of Frobenius chordal distance, we will use the fine grid.
In fact, there are several optimal methods used by  which can obtain the constellations with the better distribution of the minimum Frobenius chordal distance. However, there are several motivations why we prefer the simple grid search approach. One is that we want to know whether there exists the other orthogonal structural constellation whose performance is superior to the performance of the algebraic structural orthogonal constellation . Hence, let which means that an orthogonal constraint is imposed on each point of the constellation and which also means that the parameter is discretized into a coarse grid. Thus it is natural to introduce the grid searching algorithm. Another is that we expect that the value distribution of has some regular pattern so that all points of the constellation can be denoted by the expression like the orthogonal design of  rather than by the way of enumeration.
Let be initialized into an empty set, be the size of , and be the total of the constellation points. Our searching scheme is described as follows:(a) Select an initial point. Due to , and is stipulated for . Place the initial point and its antipodal point into .(b) Determine a step length. Let be a step length of increasing and be a step length of increasing , where and are positive integers. For orthogonal scheme, let ; then which implies that all principal angles between any two points are orthotropic each other; equivalently, . We selected the amplitude value of by fixed . So two step lengths of and provide a coarse grid, which can avoid the curse of dimensionality and reduce time- consuming of the grid search.(c) Select the distance threshold in accordance with the practical cases. The choice of was given by considering the Frobenius chordal distance distribution of the orthogonal design , such as shown in Figure 1.(d) Searching method: find , with and modified by and in order to generate a like (5). Calculate . If , then place into ; otherwise, discard . If is placed into , then its antipodal point is also placed into . Repeat until .
About the solution of the above optimization problem, there are candidate matrices like (5). Hence the calculating complexity of the search algorithm is determined by and .
We used the grid search technique to do the following tests. For and , let , we obtained the same results. There are 24 matrix points of satisfying the orthogonal condition within the range of unitary matrix points, and they constitute a set of constellations in which each constellation contains 16 matrix points with the minimum distance . By observing these 24 matrix points, we find out the following orthogonal constellation with the minimum Frobenius chordal distance .for , contains eight points and its antipodal points set, denoted as , has the following form:for ; also contains eight points. So the searching orthogonal constellation is
When , we cannot search any constellation by using the presented algorithm. Note that when , the grid searching algorithm cannot obtain the constellation of (9), because this constellation does not contain the initial point of searching process (see Algorithm 1).
4. Simplified Maximum Likelihood Detecting Algorithm Based on Antipodal Point
The antipodal constellation has the following feature.
Lemma 2. Let be an antipodal constellation defined on , be a transmitted signal matrix, and be an antipodal point of. If is a received signal matrix, then , and satisfy
Proof. As are a pair of antipodal points, according to Lemma 1, there are and , where is a unitary matrix. Thus is obtained. From here, (10) is deduced from the left to the right as follows: This completes the proof.
It can be observed from (10) that the matrix determined by the maximum value of matches with the matrix determined by the minimum value of . Therefore, the following two lemmas are self-evident.
Lemma 3. Let and which forms a set . and is calculated. By Lemma 2, and are obtained.
For the sake of obtaining the signal matrices in the subset corresponding to and of the set , let and denote the indexing indicator of and , respectively. denotes taking the index of each of all elements for the set .
The simplified ML detecting criterion is as follows.
Theorem 5. Let be the transmitted signal constellation and be the received signal matrix. Calculate and for . According to Lemmas 3 and 4, we obtain , , , , and . If, the detector outputs as an estimate of the transmitted signal ; If , then is given as an estimate of .
Proof. The result is obtained from Lemmas 2, 3, and 4 at once.
The complexity analysis of the simplified scheme: the ML detector of (2) requires points (matrices) to take part in the calculation of , while the antipodal simplified ML algorithm of Theorem 5 only requires points to take part in the calculation of ; in addition, looking for the maximum and minimum of requires the use of comparison operations with ; this is such that and are two nonempty sets initialized by the first two ’s and if , then put into ; else if, then put into ; otherwise calculate the next and the current is discarded for . In one word, the calculated amount of descends by half but there is no performance loss. Refer to  regarding the other details of the detector.
5. The Algebraic Orthogonal Design Constellation
In Section 3.2, the searching orthogonal USTM constellation based on the antipodal points was provided. In this Section, the feature of the antipodal point for the algebraic orthogonal USTM constellation presented  will be verified first, followed by the discussion of its antipodal simplified ML detector and the indexing simplified ML detector.
5.1. The Antipodal Feature of Orthogonal Design
Zhao et al.  presented the following Algebraic orthogonal (AO) scheme of USTM: where , , and . An AO USTM constellation is denoted as . Let . By and of Lemma 1, it is easy to verify that has the antipodal feature. According to the known elements of and the antipodal relation between and , elements of are calculated as , , , and . For the case of and , we determine two distributions of antipodal points for , and they are given as scheme one:and scheme two:Thus the general method of forming a pair of antipodal points in was derived as follows.
Theorem 6. In the set , given a constellation point , then its antipodal point is .
Proof. From of (3), it is known that if , then attains its maximum value. Thus, by calculating , we obtain .
5.2. The Antipodal Simplified ML Detector of
The orthogonal USTM design  also has the relation similar to (10) in Lemma 2. Let and ; then . Knowing that the shape of the transmitted signal with normalizing factor of 1/2 is a unitary matrix in and its antipodal point is , if is invariable via transmission, then the received signal matrix is , where
Lemma 7. Let be a algebraic orthogonal USTM constellation with antipodal points on , and and . If is a received matrix when is a transmitted matrix, then , , and satisfy the constraint:
Proof. The proof is similar to Lemma 2.
Lemma 8. Let and . Calculate and . By Lemma 7, and are obtained.
Theorem 9. Let be the transmitted signal constellation and be the received signal matrix. Calculate and for . By Lemmas 8 and 4, obtain , , , and . If , then the detector outputs as an estimate of the transmitted signal , else if , then it outputs as an estimate of .
5.3. The Indexing Simplified ML Detector of
Zhao et al.  presented the other simplified detecting algorithm for the algebraic orthogonal USTM, here called the indexing simplified detecting algorithm with and .whereNotice that is decomposed into and , where is only related to the first index , while is only related to the second index . Based on this decomposition and the exhausted expansion up to the level of elements of the matrix, the simplified ML detector  can further be simplified as follows: where and are computed as follows:
The simplified approach of Zhao et al.  is required to calculate , times and , times. The aforementioned simplified approach only needs to calculate the and expressions formed by two complex multiplications and three complex additions, times, respectively, ignoring matrix operations.
6. Numerical Results
In order to compare the quality between the searching orthogonal constellations and the algebraic orthogonal constellation, we plot their distance spectrums about average number of constellation points versus chordal Frobenius distance distribution away from an initial point. Figure 1 shows the distance spectrum of an searching orthogonal constellation with the minimum chordal Frobenius distance which consists of the first 16 points obtained by the grid search algorithm via setting the threshold . This fist constellation belongs to the set of constellations (see the last in Section 3.2 ). In the set , the best constellation is of (9) with the minimum distance and its distance spectrum is shown in Figure 2. The algebraic orthogonal constellation like (12) does not belong to the set because its minimum Frobenius chordal distance is and its distance spectrum is shown in Figure 3.
In Figure 4, we demonstrate the corresponding performances of three constellations when used in noncoherent communication and operated on the additive white Gaussian noise (AWGN) channel, by plotting the curves between the symbol error probability versus the signal noise ratio. At a symbol error probability of , the best searching orthogonal constellation with the minimum distance yields a SNR gain of about 2 dB over the algebraic orthogonal constellation of  when the number of receive antennas . In order to compare our testing results with those from the algebraic orthogonal constellation of , Figure 4 also shows the case that the number of receive antennas is . At a symbol error probability of , the orthogonal constellation system for two receive antennas yields an SNR gain of about 8 dB over the system for one receive antenna.
We build a framework of generating a general USTM constellation based on full diversity and antipodal feature. Under the constraint of this framework, we search a set of the orthogonal constellations all of which are superior to the algebraic orthogonal constellation of  in both the distance spectrum and the performance of symbol error probability versus signal noise ratio. But the algebraic orthogonal constellation has a simpler ML detecting algorithm which is only the linear combination of complex elements of the received matrix without dependence on matrix operations.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported in part by the National Natural Science Foundation of China (Grant no. 61271261) and in part by the Basic Research Projects of Shenzhen (JCYJ20150616144425373).
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