# Grassmannian Constellation Based on Antipodal Points and Orthogonal Design and Its Simplified Detecting Algorithm

Authors: Li Peng; Dacong Hu; Lin Zhang; Zhen Qin
Journal: Journal of Computer Networks and Communications
 (2017)

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## Abstract

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.


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## Body

## 1. Introduction

Grassmannian constellation is a set of unitary space time modulation (USTM) signal matrices defined on Gassmann manifold presented by Hochwald and Marzetta [1] and Zheng and Tse [2] 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 [3] and algebraic structural schemes [4–8]. This study concentrated on the random and algebraic orthogonal [4] 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 Section2, 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 [4] 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 withM transmit and N 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 T and change to a new realization in the next interval. A system model [2] is given as follows:(1)Y=ρMX·H+W,where X∈CT×M and Y∈CT×N are, respectively the transmitted and received signal matrices, H∈CM×N is a fading coefficient matrix and W∈CT×N is an additive noise matrix, of which the elements of both are drawn from the i.i.d. standard complex Gaussian distribution CN(0,1), and ρ 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 [1]: {X}=TΦbb=1B in which each matrix satisfies Φb∗Φb=IM and all Φb’s are points on a Stiefel manifold, or the subspace Ωb spanned by column vectors of T×M matrix Φb is uniformly distribution in Grassmann manifold GT,M; that is, Ωb∈GT,M [2]. Let a set Φbb=1B denote a USTM constellation which contains BT×M complex unitary matrices Φb.As the coefficients ofH are unknown to both receiver and transmitter, the noncoherent ML detector [1] is introduced: (2)Φ^ML=argmaxXi=TΦi,Φi∈Φbb=1BTrY∗XiXi∗Y,where Tr(·) is the trace operation of a matrix and ·∗ is the complex conjugate transpose.Let vector setsui and vi be two principal vectors corresponding to two M-planes U,V∈GT,M. The principal angles θ1,θ2,…,θM∈[0,π/2] between U and V are defined as cos⁡θi=maxu∈U⁡maxv∈V⁡u·v=ui·vi for i=1,2,…,M, subject to u·u=v·v=1, ui·uj=0, vi·vj=0(1≤j≤i-1) [9]. The chordal Frobenius distance measure is defined as follows ([3] and references therein):(3)dU,V=2M-2TrΣU∗V=∑i=1M4sin2⁡12θi,where ΣU∗V denotes a diagonal matrix formed by the singular values of the matrix U∗V.


## 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 manifoldGT,M and each signal point is denoted as a unitary matrix Φ, Φ- is defined as an antipodal point of Φ if Φ- is the orthogonal complement of Φ on GT,M. Then how can one decide whether the two matrices on GT,M are the antipodal matrix? This can be done with the following lemma.Lemma 1.
LetU,V be two T×M unitary matrix on GT,M. U and V become a pair of antipodal matrices if and only if either U∗V=0M or UV=QT and QT∗QT=IT, where IT is a T×T identity matrix and 0M is a M×M full-zero matrix.Proof.
U
∗
V
=
0
M implies that each of M column vectors of U is orthogonal to each of M column vectors of V; that is, U and V are orthogonal and complement each other. Since U,V are two T×M unitary matrices, they are used to construct a T×T matrix QT=UV. QT∗QT=IT indicates that T column vectors of U and V span a basis of Euclid space CT, so U and V are orthogonal and complement each other.Construction 1 (a framework of USTM).
LetC=Φbb=0B-1 denote a constellation and Φb∈C⊂GT,M. Let φij∈C denote a complex element at the ith row and the jth column of Φb for i=1,2,…,T and j=1,2,…,M. If for any positive integer B, each code word Φb of C has the structure (4)Φb=φ11φ12⋯φ1Mφ21φ22⋯φ2M⋮⋮⋱⋮φT1φT2⋯φTMforb=0,1,2,…,B-1and satisfies the following constraints:(1)
For allb,Φb∗Φb=IM, where IM is a M×M identity matrix;(2)
All points ofC=Φbb=0B-1=C1∪C2 are partitioned into two parts C1=Φbb=0B/2-1 and C2=Φbb=B/2B-1 in such a way that there exist B/2 one-to-one antipodal points between C1 and C2 but there is no antipodal point in each of C1 and C2;(3)
The degree of freedom of elements in eachΦb∈C⊂GT,M is dim⁡(GT,M)=M(T-M) [2];(4)
Allφαβ≠0 are unknown for α=1,2,…,T and β=1,2,…,M.Then the constellation set C=Φbb=0B-1 is called a framework of the full diversity USTM constellation based on antipodal points on GT,M.


### 3.2. The Construction of Antipodal Constellation with Orthogonality

According to the analysis on how to chooseT and M [1, 2], the simplest case of T=2M=4 was considered. A framework of the 4×2 unitary matrix Φb on G4,2 was built similar to (4), and the degree of freedom of its elements was M(T-M)=4. Therefore, let φk=Akejϕk∈C, k=1,2,3,4 be four independent complex elements of Φb, where j is an imaginary unit. Then a unitary matrix with uncertain eight values Ak, ϕk, k=1,2,3,4, is formed as follows:(5)Φb=1A12+A22+A32+A42A1ejϕ1A2ejϕ2-A2e-jϕ2A1e-jϕ1A3ejϕ3A4ejϕ4-A4e-jϕ4A3e-jϕ3=aφ1φ2-φ2∗φ1∗φ3φ4-φ4∗φ3∗=faAkejϕkk=14=faφkk=14.The expression (5) defines a function f:φ1,φ2,φ3,φ4→Φb; that is, four complex numbers are mapped into a unitary matrix similar to (5) which is a point of the USTM constellation, or f:Ak,ϕkk=14→Φb, where φk∈C and Ak,ϕk∈R.The optimal packing method stated was used to determineC=Φbb=0B-1 with all points like (5). That is, for the fixed T=2M=4 and B, design a packing C in G4,2 of cardinality C=B so that its minimum distance similar to (3) is as large as possible. In fact, a complex number set of φk=Akejϕk∈C needs to be obtained in order to form a constellation C=Φbb=0B-1 on G4,2 so that the minimum Frobenius chordal distance minφk∈C⁡dΦuφk,Φvφk of (3) is maximized. The optimal packings of B points on G4,2 require the solution of the following optimization problem:(6)maxAk,ϕk∈Rminu,v∈B⁡dΦuAk,ϕk,ΦvAk,ϕkSubjecttoΦuAk,ϕk,ΦvAk,ϕk∈C=Φbb=0B-1,ΦuAk,ϕk≠ΦvAk,ϕk,u≠v∈0,1,…,B-1.If the complex elementsφk∈C(k=1,2,3,4) 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 Ak,ϕk∈R 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 Ak,ϕk∈R. 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 parametersAk,ϕk∈R, 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 [3] 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 [4]. Hence, let ϕk=0,π/2,π,3π/2 which means that an orthogonal constraint is imposed on each point of the constellation C=Φbb=0B-1 and which also means that the parameter ϕk 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 Ak has some regular pattern so that all points of the constellation can be denoted by the expression like the orthogonal design of [4] rather than by the way of enumeration.Let{Φ} be initialized into an empty set, Φ be the size of Φ, and B be the total of the constellation points. Our searching scheme is described as follows:(a)
Select an initial point. Due toφαβ=φk≠0, Ak=1 and ϕk=0 is stipulated for k=1,2,3,4. Place the initial point Φb=0=Φ0 and its antipodal point Φ-b=0=ΦB/2 into Φ.(b)
Determine a step length. Letγ=1/m∈[0,1] be a step length of increasing Ak and σ=2π/n∈[0,2π] be a step length of increasing ϕk, where m and n are positive integers. For orthogonal scheme, let n=4; then σ=π/2 which implies that all principal angles θ1,θ2,…,θM between any two points Φu,Φv are orthotropic each other; equivalently, ϕk∈{0,π/2,π,3π/2}. We selected the amplitude value of Ak by fixed m=2,4,8. So two step lengths of γ and σ provide a coarse grid, which can avoid the curse of dimensionality and reduce time- consuming of the grid search.(c)
Select the distance thresholddth in accordance with the practical cases. The choice of dth=0.8 was given by considering the Frobenius chordal distance distribution of the orthogonal design [4], such as dth=0.7321 shown in Figure 1.(d)
Searching method: findφk=Akejϕk, k=1,2,3,4 with Ak and ϕk modified by γ and σ in order to generate a Φβ∈C like (5). Calculate d=mini=1,…,Φ,Φi∈Φ⁡d(Φi,Φβ). If d>dth, then place Φβ into Φ; otherwise, discard Φβ. If Φβ is placed into Φ, then its antipodal point Φ-β is also placed into Φ. Repeat (d) until Φ=B.Figure 1
The distance distribution of the searching orthogonal constellation with the minimum distancedmin=0.8.About the solution of the above optimization problem, there arePmn=mn!/mn-4! candidate matrices like (5). Hence the calculating complexity of the search algorithm is determined by m and n.We used the grid search technique to do the following tests. Form=2,4,8 and n=4, let dth=0.8, we obtained the same results. There are 24 matrix points of satisfying the orthogonal condition within the range of (mn)!/(mn-4)! unitary matrix points, and they constitute a set Θ of C2416=735471 constellations in which each constellation contains 16 matrix points with the minimum distance dmin=0.8. By observing these 24 matrix points, we find out the following orthogonal constellation with the minimum Frobenius chordal distance dmin=0.8376.(7)Ωk=11022-22ejkπ/2ejk±1π/2-e-jk±1π/2e-jkπ/2for k=0,1,2,3, Ωk contains eight points and its antipodal points set, denoted as {Ω-k}, has the following form:(8)Ω-k=11011-112ejkπ/22ejk±1π/2-2e-jk±1π/22e-jkπ/2for k=0,1,2,3; Ω-k also contains eight points. So the searching orthogonal constellation is (9)Φbb=015=Ωk∪Ω-k.Whendth≥0.8376, we cannot search any constellation by using the presented algorithm. Note that when dth=0.8376, 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).Algorithm 1:Algorithm of searching the antipodal orthogonal constellation.
Input:B, m, n, dth;
Output: a constellationC=Φbb=0B-1;
Point(Ak(b),ϕk(b)){
a
=
1
A
1
b
2
+
A
2
b
2
+
A
3
b
2
+
A
4
b
2;
Φ
b
=
f
a
A
k
e
j
ϕ
k
k
=
1
4 of (5)}// function of calculating points
A
k
(
0
)
=
1; ϕk(0)=0; // for k=1,2,3,4
Φ
0
=
Point
(
A
k
(
0
)
,
ϕ
k
(
0
)
);
Φ
0
→
{
Φ
}; ΦB/2→{Φ}// initial the constellation set
γ
=
1
m
;
σ
=
2
π
n;
ForAk(b)=1,Ak(b)=Ak(b)+γ, Ak(b)≤2{{{{
// four for-cycle to calculate A1,A2,A3,A4
For ϕk(b)=0,ϕk(b)=ϕk(b)+σ,ϕk(b)<2π{{{{
// four for-cycle to calculate ϕ1,ϕ2,ϕ3,ϕ4
Φb=PointAkb,ϕkb;
For l=0,l++,l<Φ{
Ifd(Φb,Φl)>dth;
Φb→C=Φbb=0B-1;Φb+B/2→C=Φbb=0B-1}
If|{Φ}|<B
b=b+1}}}}}}}}
Φ
=
B
ReturnC=Φbb=0B-1


## 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 manifoldGT,M and each signal point is denoted as a unitary matrix Φ, Φ- is defined as an antipodal point of Φ if Φ- is the orthogonal complement of Φ on GT,M. Then how can one decide whether the two matrices on GT,M are the antipodal matrix? This can be done with the following lemma.Lemma 1.
LetU,V be two T×M unitary matrix on GT,M. U and V become a pair of antipodal matrices if and only if either U∗V=0M or UV=QT and QT∗QT=IT, where IT is a T×T identity matrix and 0M is a M×M full-zero matrix.Proof.
U
∗
V
=
0
M implies that each of M column vectors of U is orthogonal to each of M column vectors of V; that is, U and V are orthogonal and complement each other. Since U,V are two T×M unitary matrices, they are used to construct a T×T matrix QT=UV. QT∗QT=IT indicates that T column vectors of U and V span a basis of Euclid space CT, so U and V are orthogonal and complement each other.Construction 1 (a framework of USTM).
LetC=Φbb=0B-1 denote a constellation and Φb∈C⊂GT,M. Let φij∈C denote a complex element at the ith row and the jth column of Φb for i=1,2,…,T and j=1,2,…,M. If for any positive integer B, each code word Φb of C has the structure (4)Φb=φ11φ12⋯φ1Mφ21φ22⋯φ2M⋮⋮⋱⋮φT1φT2⋯φTMforb=0,1,2,…,B-1and satisfies the following constraints:(1)
For allb,Φb∗Φb=IM, where IM is a M×M identity matrix;(2)
All points ofC=Φbb=0B-1=C1∪C2 are partitioned into two parts C1=Φbb=0B/2-1 and C2=Φbb=B/2B-1 in such a way that there exist B/2 one-to-one antipodal points between C1 and C2 but there is no antipodal point in each of C1 and C2;(3)
The degree of freedom of elements in eachΦb∈C⊂GT,M is dim⁡(GT,M)=M(T-M) [2];(4)
Allφαβ≠0 are unknown for α=1,2,…,T and β=1,2,…,M.Then the constellation set C=Φbb=0B-1 is called a framework of the full diversity USTM constellation based on antipodal points on GT,M.


## 3.2. The Construction of Antipodal Constellation with Orthogonality

According to the analysis on how to chooseT and M [1, 2], the simplest case of T=2M=4 was considered. A framework of the 4×2 unitary matrix Φb on G4,2 was built similar to (4), and the degree of freedom of its elements was M(T-M)=4. Therefore, let φk=Akejϕk∈C, k=1,2,3,4 be four independent complex elements of Φb, where j is an imaginary unit. Then a unitary matrix with uncertain eight values Ak, ϕk, k=1,2,3,4, is formed as follows:(5)Φb=1A12+A22+A32+A42A1ejϕ1A2ejϕ2-A2e-jϕ2A1e-jϕ1A3ejϕ3A4ejϕ4-A4e-jϕ4A3e-jϕ3=aφ1φ2-φ2∗φ1∗φ3φ4-φ4∗φ3∗=faAkejϕkk=14=faφkk=14.The expression (5) defines a function f:φ1,φ2,φ3,φ4→Φb; that is, four complex numbers are mapped into a unitary matrix similar to (5) which is a point of the USTM constellation, or f:Ak,ϕkk=14→Φb, where φk∈C and Ak,ϕk∈R.The optimal packing method stated was used to determineC=Φbb=0B-1 with all points like (5). That is, for the fixed T=2M=4 and B, design a packing C in G4,2 of cardinality C=B so that its minimum distance similar to (3) is as large as possible. In fact, a complex number set of φk=Akejϕk∈C needs to be obtained in order to form a constellation C=Φbb=0B-1 on G4,2 so that the minimum Frobenius chordal distance minφk∈C⁡dΦuφk,Φvφk of (3) is maximized. The optimal packings of B points on G4,2 require the solution of the following optimization problem:(6)maxAk,ϕk∈Rminu,v∈B⁡dΦuAk,ϕk,ΦvAk,ϕkSubjecttoΦuAk,ϕk,ΦvAk,ϕk∈C=Φbb=0B-1,ΦuAk,ϕk≠ΦvAk,ϕk,u≠v∈0,1,…,B-1.If the complex elementsφk∈C(k=1,2,3,4) 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 Ak,ϕk∈R 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 Ak,ϕk∈R. 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 parametersAk,ϕk∈R, 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 [3] 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 [4]. Hence, let ϕk=0,π/2,π,3π/2 which means that an orthogonal constraint is imposed on each point of the constellation C=Φbb=0B-1 and which also means that the parameter ϕk 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 Ak has some regular pattern so that all points of the constellation can be denoted by the expression like the orthogonal design of [4] rather than by the way of enumeration.Let{Φ} be initialized into an empty set, Φ be the size of Φ, and B be the total of the constellation points. Our searching scheme is described as follows:(a)
Select an initial point. Due toφαβ=φk≠0, Ak=1 and ϕk=0 is stipulated for k=1,2,3,4. Place the initial point Φb=0=Φ0 and its antipodal point Φ-b=0=ΦB/2 into Φ.(b)
Determine a step length. Letγ=1/m∈[0,1] be a step length of increasing Ak and σ=2π/n∈[0,2π] be a step length of increasing ϕk, where m and n are positive integers. For orthogonal scheme, let n=4; then σ=π/2 which implies that all principal angles θ1,θ2,…,θM between any two points Φu,Φv are orthotropic each other; equivalently, ϕk∈{0,π/2,π,3π/2}. We selected the amplitude value of Ak by fixed m=2,4,8. So two step lengths of γ and σ provide a coarse grid, which can avoid the curse of dimensionality and reduce time- consuming of the grid search.(c)
Select the distance thresholddth in accordance with the practical cases. The choice of dth=0.8 was given by considering the Frobenius chordal distance distribution of the orthogonal design [4], such as dth=0.7321 shown in Figure 1.(d)
Searching method: findφk=Akejϕk, k=1,2,3,4 with Ak and ϕk modified by γ and σ in order to generate a Φβ∈C like (5). Calculate d=mini=1,…,Φ,Φi∈Φ⁡d(Φi,Φβ). If d>dth, then place Φβ into Φ; otherwise, discard Φβ. If Φβ is placed into Φ, then its antipodal point Φ-β is also placed into Φ. Repeat (d) until Φ=B.Figure 1
The distance distribution of the searching orthogonal constellation with the minimum distancedmin=0.8.About the solution of the above optimization problem, there arePmn=mn!/mn-4! candidate matrices like (5). Hence the calculating complexity of the search algorithm is determined by m and n.We used the grid search technique to do the following tests. Form=2,4,8 and n=4, let dth=0.8, we obtained the same results. There are 24 matrix points of satisfying the orthogonal condition within the range of (mn)!/(mn-4)! unitary matrix points, and they constitute a set Θ of C2416=735471 constellations in which each constellation contains 16 matrix points with the minimum distance dmin=0.8. By observing these 24 matrix points, we find out the following orthogonal constellation with the minimum Frobenius chordal distance dmin=0.8376.(7)Ωk=11022-22ejkπ/2ejk±1π/2-e-jk±1π/2e-jkπ/2for k=0,1,2,3, Ωk contains eight points and its antipodal points set, denoted as {Ω-k}, has the following form:(8)Ω-k=11011-112ejkπ/22ejk±1π/2-2e-jk±1π/22e-jkπ/2for k=0,1,2,3; Ω-k also contains eight points. So the searching orthogonal constellation is (9)Φbb=015=Ωk∪Ω-k.Whendth≥0.8376, we cannot search any constellation by using the presented algorithm. Note that when dth=0.8376, 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).Algorithm 1:Algorithm of searching the antipodal orthogonal constellation.
Input:B, m, n, dth;
Output: a constellationC=Φbb=0B-1;
Point(Ak(b),ϕk(b)){
a
=
1
A
1
b
2
+
A
2
b
2
+
A
3
b
2
+
A
4
b
2;
Φ
b
=
f
a
A
k
e
j
ϕ
k
k
=
1
4 of (5)}// function of calculating points
A
k
(
0
)
=
1; ϕk(0)=0; // for k=1,2,3,4
Φ
0
=
Point
(
A
k
(
0
)
,
ϕ
k
(
0
)
);
Φ
0
→
{
Φ
}; ΦB/2→{Φ}// initial the constellation set
γ
=
1
m
;
σ
=
2
π
n;
ForAk(b)=1,Ak(b)=Ak(b)+γ, Ak(b)≤2{{{{
// four for-cycle to calculate A1,A2,A3,A4
For ϕk(b)=0,ϕk(b)=ϕk(b)+σ,ϕk(b)<2π{{{{
// four for-cycle to calculate ϕ1,ϕ2,ϕ3,ϕ4
Φb=PointAkb,ϕkb;
For l=0,l++,l<Φ{
Ifd(Φb,Φl)>dth;
Φb→C=Φbb=0B-1;Φb+B/2→C=Φbb=0B-1}
If|{Φ}|<B
b=b+1}}}}}}}}
Φ
=
B
ReturnC=Φbb=0B-1


## 4. Simplified Maximum Likelihood Detecting Algorithm Based on Antipodal Point

The antipodal constellation has the following feature.Lemma 2.
LetC=C1∪C2=Xbb=1B/2∪X-bb=1+B/2B be an antipodal constellation defined on GT,M, X∈C1=Xbb=1B/2 be a T×M transmitted signal matrix, and X-∈C2={X-b}b=1+B/2B be an antipodal point ofX. IfY is a T×N received signal matrix, then X, X- and Y satisfy(10)TrY∗XX∗Y+TrY∗X-X-∗Y=TrY∗Y.Proof.
AsX,X-∈C⊂GT,M are a pair of antipodal points, according to Lemma 1, there are [XX-]=QT and X∗X-∗∗=QT∗, where QT is a T×T unitary matrix. Thus XX∗+X-X-∗=XX-X∗X-∗=QTQT∗=IT is obtained. From here, (10) is deduced from the left to the right as follows: (11)Left=TrY∗XX∗Y+TrY∗X-X-∗Y=TrY∗XX∗+X-X-∗Y=TrY∗XX∗+X-X-∗Y=TrY∗QTQT∗Y=TrY∗ITY=TrY∗Y=Right
This completes the proof.It can be observed from (10) that the matrix Xi∈C1 determined by the maximum value of Tr(Y∗XiXi∗Y) matches with the matrix X-i∈C2 determined by the minimum value of Tr(Y∗X-iX-i∗Y). Therefore, the following two lemmas are self-evident.Lemma 3.
Leta=Tr(Y∗Y) and bi=Tr(Y∗XiXi∗Y) which forms a set bi. bmax=max⁡{bi∣i=1,…,B/2} and bmin=min⁡{bi∣i=1,…,B/2} is calculated. By Lemma 2, b-max=a-bmin and b-min=a-bmax are obtained.For the sake of obtaining the signal matrices in the subsetC1=Xbb=1B/2 corresponding to bmax and bmin of the set bi, let smax and smin denote the indexing indicator of max⁡bi and min⁡bi, respectively. ID denotes taking the index of each of all elements for the set bi.Lemma 4.
s
max
=
ID
max
⁡
{
b
i
} indicates that smax is an index of the maximum value bmax in the set {bi}. Similarly, we have smin=IDmin⁡bi, s-max=IDmax⁡b-i, and s-min=IDmin⁡b-i. By Lemmas 2 and 3, smax=s-min and smin=s-max are obtained.The simplified ML detecting criterion is as follows.Theorem 5.
LetC=C1∪C2=Xbb=1B/2∪X-bb=1+B/2B be the transmitted signal constellation and Y be the received signal matrix. Calculate a=Tr(Y∗Y) and bi=TrY∗XiXi∗Y for i=1,2,…,B/2. According to Lemmas 3 and 4, we obtain bmax, smax, bmin, smin, and b-max=a-bmin. Ifbmax>b-max, the detector outputs X^=Xsmax∈C1 as an estimate of the transmitted signal X∈C; If bmax<b-max, then X^=Xsmin∈C2 is given as an estimate of X∈C.Proof.
The result is obtained from Lemmas2, 3, and 4 at once.
The complexity analysis of the simplified scheme: the ML detector of (2) requires B points (matrices) to take part in the calculation of Tr(Y∗XbXb∗Y), while the antipodal simplified ML algorithm of Theorem 5 only requires B/2 points to take part in the calculation of Tr(Y∗XbXb∗Y); in addition, looking for the maximum and minimum of {bi} requires the use of t comparison operations with B/2<t<B; this is such that bmax and bmin are two nonempty sets initialized by the first two bi’s and if bi=Tr(Y∗XiXi∗Y)>bmax, then put bi into bmax; else ifbi=Tr(Y∗XiXi∗Y)<bmin, then put bi into {bmin}; otherwise calculate the nextbi and the current bi is discarded for i=1,2,…,B/2. In one word, the calculated amount of Tr(Y∗XbXb∗Y) descends by half but there is no performance loss. Refer to [10] regarding the other details of the detector.


## 5. The Algebraic Orthogonal Design Constellation

In Section3.2, 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 [4] 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. [4] presented the following Algebraic orthogonal (AO) scheme of USTM: (12)Xk,l=EUk,l=11-11s0s1-s1∗s0∗=11-11ej2π/Qkej2π/Ql-e-j2π/Qle-j2π/Qk,where (k,l)∈F×F, F=0,1,…,P-1, and B=P2. An AO USTM constellation is denoted as CAO={Xk,l∣(k,l)∈F×F}. Let X-k,l=E∗U-k,l∗∗=1-1s2s411s3s5∗. By [Xk,lX-k,l]=QT and QT†QT=IT of Lemma 1, it is easy to verify that CAO has the antipodal feature. According to the known elements of Xk,l and the antipodal relation between Xk,l and X-k,l, elements of X-k,l are calculated as s2=-s0=-ej2π/Pk, s3=-s1=-ej2π/Pl, s4=s1∗=e-j2π/Pl, and s5=-s0∗=-e-j2π/Pk. For the case of P=4 and B=P2=16, we determine two distributions of antipodal points for CAO, and they are given as scheme one:(13)CAO1=Xk,lk=0P/2-1l=0P-1=X0,0,X0,1,X0,2,X0,3,X1,0,X1,1,X1,2,X1,3CAO2=Xk,lk=P/2P-1l=0P-1=X2,2,X2,3,X2,0,X2,1,X3,2,X3,3,X3,0,X3,1and scheme two:(14)CAO1=Xk,lk=0P-1l=0P/2-1=X0,0,X1,0,X2,0,X3,0,X0,1,X1,1,X2,1,X3,1CAO2=Xk,lk=0P-1l=P/2P-1=X2,2,X3,2,X0,2,X1,2,X2,3,X3,3,X0,3,X1,3.Thus the general method of forming a pair of antipodal points in CAO was derived as follows.Theorem 6.
In the setCAO={Xk,l∣(k,l)∈F×F}=CAO1∪CAO2, given a constellation point Xk,l∈CAO1, then its antipodal point is X(k+P/2)(modP),(l+P/2)(modP)∈CAO2.Proof.
FromdU,V=2M-2Tr⁡ΣU∗V of (3), it is known that if U∗V=0, then dU,V=2M attains its maximum value. Thus, by calculating Xk,l∗X(k+P/2)(modP),(l+P/2)(modP)=0, we obtain d(Xk,l,X(k+P/2)(modP),(l+P/2)(modP))=2M=2.


### 5.2. The Antipodal Simplified ML Detector ofCAO

The orthogonal USTM design [4] also has the relation similar to (10) in Lemma 2. Let X∈CAO1=Xk,lk=0P/2-1l=0P-1orX∈CAO1=Xk,ll=0P/2-1k=0P-1 and X-∈CAO2={X(k+P/2)(modP),(l+P/2)(modP)}; then CAO={Xk,l∣(k,l)∈F×F}=CAO1∪CAO2. Knowing that the shape of the transmitted signal with normalizing factor of 1/2 is a T×M unitary matrix X=E∗U∗∗ in CAO and its antipodal point is X-=E∗U-∗∗, if E is invariable via transmission, then the received signal matrix is Y=E∗V∗∗, where (15)E=11-11,U=s0s1-s1∗s0∗,U-=-s0-s1s1∗-s0∗,V=y1y2y3y4.Lemma 7.
LetCAO=CAO1∪CAO2 be a algebraic orthogonal USTM constellation with antipodal points on GT,M, and X=E∗U∗∗∈CAO1 and X-=[E∗U-∗]∗∈CAO2. IfY=[E∗V∗]∗ is a received matrix when X is a transmitted matrix, then U, U-, and V satisfy the constraint:(16)Tr⁡V∗UU∗V+Tr⁡V∗U-U-∗V=Tr⁡V∗V.Proof.
The proof is similar to Lemma2.Similar to Theorem5, the algebraic orthogonal USTM [4] has the following antipodal simplified ML detector.Lemma 8.
Leta=4Tr⁡(V∗V) and bi=Tr⁡(V∗UiUi∗V). Calculate bmax=max⁡{bi∣i=1,…,B/2} and bmin=min⁡{bi∣i=1,…,B/2}. By Lemma 7, b-max=a-bmin and b-min=a-bmax are obtained.Theorem 9.
LetCAO=CAD1∪CAO2 be the transmitted signal constellation and Y be the received signal matrix. Calculate a=4Tr⁡(V∗V) and bi=Tr⁡V∗UiUi∗V for i=1,2,…,B/2. By Lemmas 8 and 4, obtain bmax, smax, bmin, and smin. If bmax>a-bmin, then the detector outputs X^=Xsmax∈CAO1 as an estimate of the transmitted signal X∈CAO, else if bmax<a-bmin, then it outputs X^=Xsmin∈CAO2 as an estimate of X∈CAO.Obviously, the calculating quantity ofTr⁡(V∗UU∗V) in (16) is half of that of Tr⁡(Y∗XX∗Y) in (10) because the sizes T×M of X and Y in Tr⁡(Y∗XX∗Y) for nonorthogonal design decrease by half the sizes of M×M of U and V in Tr⁡(V∗UU∗V) for orthogonal design.


### 5.3. The Indexing Simplified ML Detector ofCAO

Zhao et al. [4] presented the other simplified detecting algorithm for the algebraic orthogonal USTM, here called the indexing simplified detecting algorithm with k and l.(17)X∗Y2=Tr⁡Y∗XX∗Y=2+12Tr⁡V∗V+12Tr⁡V∗Θk+Θk∗V+12Tr⁡V∗Ψl+Ψl∗V=2+12V2+y2∗-y3ej2π/Pk+y2-y3∗e-j2π/Pk+y2∗-y3ej2π/Pl+y2-y3∗e-j2π/Pl,where(18)Θk=s000s0∗,Ψl=0s1-s1∗0.Notice that Tr⁡{Y∗XX∗Y} is decomposed into Tr⁡{V∗Θk+Θk∗V} and Tr⁡{V∗Ψl+Ψl∗V}, where Tr⁡{V∗Θk+Θk∗V} is only related to the first index k, while Tr⁡{V∗Ψl+Ψl∗V} is only related to the second index l. Based on this decomposition and the exhausted expansion up to the level of elements of the matrix, the simplified ML detector [4] can further be simplified as follows: (19)X^ML=arg⁡maxXk,l∈CAO⁡Xk,l∗Y2=arg⁡maxk,l=0,1,…,P-1⁡Uk,l∗V2=Xk^ML,l^ML,where k^ML and l^ML are computed as follows:(20)k^ML=arg⁡maxk=0,1,…,P-1⁡Tr⁡V∗Θk+Θk∗V=arg⁡maxk=0,1,…,P-1⁡y1∗+y4ej2π/Pk+y1+y4∗e-j2π/Pkl^ML=arg⁡maxl=0,1,…,P-1⁡Tr⁡V∗Ψl+Ψl∗V=arg⁡maxl=0,1,…,P-1⁡y2∗-y3ej2π/Pl+y2-y3∗e-j2π/Pl.The simplified approach of Zhao et al. [4] is required to calculate Tr⁡{YY∗Ak}, P times and Tr⁡{YY∗Bl}, P times. The aforementioned simplified approach only needs to calculate the k^ML and l^ML expressions formed by two complex multiplications and three complex additions, P times, respectively, ignoring matrix operations.


## 5.1. The Antipodal Feature of Orthogonal Design

Zhao et al. [4] presented the following Algebraic orthogonal (AO) scheme of USTM: (12)Xk,l=EUk,l=11-11s0s1-s1∗s0∗=11-11ej2π/Qkej2π/Ql-e-j2π/Qle-j2π/Qk,where (k,l)∈F×F, F=0,1,…,P-1, and B=P2. An AO USTM constellation is denoted as CAO={Xk,l∣(k,l)∈F×F}. Let X-k,l=E∗U-k,l∗∗=1-1s2s411s3s5∗. By [Xk,lX-k,l]=QT and QT†QT=IT of Lemma 1, it is easy to verify that CAO has the antipodal feature. According to the known elements of Xk,l and the antipodal relation between Xk,l and X-k,l, elements of X-k,l are calculated as s2=-s0=-ej2π/Pk, s3=-s1=-ej2π/Pl, s4=s1∗=e-j2π/Pl, and s5=-s0∗=-e-j2π/Pk. For the case of P=4 and B=P2=16, we determine two distributions of antipodal points for CAO, and they are given as scheme one:(13)CAO1=Xk,lk=0P/2-1l=0P-1=X0,0,X0,1,X0,2,X0,3,X1,0,X1,1,X1,2,X1,3CAO2=Xk,lk=P/2P-1l=0P-1=X2,2,X2,3,X2,0,X2,1,X3,2,X3,3,X3,0,X3,1and scheme two:(14)CAO1=Xk,lk=0P-1l=0P/2-1=X0,0,X1,0,X2,0,X3,0,X0,1,X1,1,X2,1,X3,1CAO2=Xk,lk=0P-1l=P/2P-1=X2,2,X3,2,X0,2,X1,2,X2,3,X3,3,X0,3,X1,3.Thus the general method of forming a pair of antipodal points in CAO was derived as follows.Theorem 6.
In the setCAO={Xk,l∣(k,l)∈F×F}=CAO1∪CAO2, given a constellation point Xk,l∈CAO1, then its antipodal point is X(k+P/2)(modP),(l+P/2)(modP)∈CAO2.Proof.
FromdU,V=2M-2Tr⁡ΣU∗V of (3), it is known that if U∗V=0, then dU,V=2M attains its maximum value. Thus, by calculating Xk,l∗X(k+P/2)(modP),(l+P/2)(modP)=0, we obtain d(Xk,l,X(k+P/2)(modP),(l+P/2)(modP))=2M=2.


## 5.2. The Antipodal Simplified ML Detector ofCAO

The orthogonal USTM design [4] also has the relation similar to (10) in Lemma 2. Let X∈CAO1=Xk,lk=0P/2-1l=0P-1orX∈CAO1=Xk,ll=0P/2-1k=0P-1 and X-∈CAO2={X(k+P/2)(modP),(l+P/2)(modP)}; then CAO={Xk,l∣(k,l)∈F×F}=CAO1∪CAO2. Knowing that the shape of the transmitted signal with normalizing factor of 1/2 is a T×M unitary matrix X=E∗U∗∗ in CAO and its antipodal point is X-=E∗U-∗∗, if E is invariable via transmission, then the received signal matrix is Y=E∗V∗∗, where (15)E=11-11,U=s0s1-s1∗s0∗,U-=-s0-s1s1∗-s0∗,V=y1y2y3y4.Lemma 7.
LetCAO=CAO1∪CAO2 be a algebraic orthogonal USTM constellation with antipodal points on GT,M, and X=E∗U∗∗∈CAO1 and X-=[E∗U-∗]∗∈CAO2. IfY=[E∗V∗]∗ is a received matrix when X is a transmitted matrix, then U, U-, and V satisfy the constraint:(16)Tr⁡V∗UU∗V+Tr⁡V∗U-U-∗V=Tr⁡V∗V.Proof.
The proof is similar to Lemma2.Similar to Theorem5, the algebraic orthogonal USTM [4] has the following antipodal simplified ML detector.Lemma 8.
Leta=4Tr⁡(V∗V) and bi=Tr⁡(V∗UiUi∗V). Calculate bmax=max⁡{bi∣i=1,…,B/2} and bmin=min⁡{bi∣i=1,…,B/2}. By Lemma 7, b-max=a-bmin and b-min=a-bmax are obtained.Theorem 9.
LetCAO=CAD1∪CAO2 be the transmitted signal constellation and Y be the received signal matrix. Calculate a=4Tr⁡(V∗V) and bi=Tr⁡V∗UiUi∗V for i=1,2,…,B/2. By Lemmas 8 and 4, obtain bmax, smax, bmin, and smin. If bmax>a-bmin, then the detector outputs X^=Xsmax∈CAO1 as an estimate of the transmitted signal X∈CAO, else if bmax<a-bmin, then it outputs X^=Xsmin∈CAO2 as an estimate of X∈CAO.Obviously, the calculating quantity ofTr⁡(V∗UU∗V) in (16) is half of that of Tr⁡(Y∗XX∗Y) in (10) because the sizes T×M of X and Y in Tr⁡(Y∗XX∗Y) for nonorthogonal design decrease by half the sizes of M×M of U and V in Tr⁡(V∗UU∗V) for orthogonal design.


## 5.3. The Indexing Simplified ML Detector ofCAO

Zhao et al. [4] presented the other simplified detecting algorithm for the algebraic orthogonal USTM, here called the indexing simplified detecting algorithm with k and l.(17)X∗Y2=Tr⁡Y∗XX∗Y=2+12Tr⁡V∗V+12Tr⁡V∗Θk+Θk∗V+12Tr⁡V∗Ψl+Ψl∗V=2+12V2+y2∗-y3ej2π/Pk+y2-y3∗e-j2π/Pk+y2∗-y3ej2π/Pl+y2-y3∗e-j2π/Pl,where(18)Θk=s000s0∗,Ψl=0s1-s1∗0.Notice that Tr⁡{Y∗XX∗Y} is decomposed into Tr⁡{V∗Θk+Θk∗V} and Tr⁡{V∗Ψl+Ψl∗V}, where Tr⁡{V∗Θk+Θk∗V} is only related to the first index k, while Tr⁡{V∗Ψl+Ψl∗V} is only related to the second index l. Based on this decomposition and the exhausted expansion up to the level of elements of the matrix, the simplified ML detector [4] can further be simplified as follows: (19)X^ML=arg⁡maxXk,l∈CAO⁡Xk,l∗Y2=arg⁡maxk,l=0,1,…,P-1⁡Uk,l∗V2=Xk^ML,l^ML,where k^ML and l^ML are computed as follows:(20)k^ML=arg⁡maxk=0,1,…,P-1⁡Tr⁡V∗Θk+Θk∗V=arg⁡maxk=0,1,…,P-1⁡y1∗+y4ej2π/Pk+y1+y4∗e-j2π/Pkl^ML=arg⁡maxl=0,1,…,P-1⁡Tr⁡V∗Ψl+Ψl∗V=arg⁡maxl=0,1,…,P-1⁡y2∗-y3ej2π/Pl+y2-y3∗e-j2π/Pl.The simplified approach of Zhao et al. [4] is required to calculate Tr⁡{YY∗Ak}, P times and Tr⁡{YY∗Bl}, P times. The aforementioned simplified approach only needs to calculate the k^ML and l^ML expressions formed by two complex multiplications and three complex additions, P 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. Figure1 shows the distance spectrum of an searching orthogonal constellation with the minimum chordal Frobenius distance dmin=0.8 which consists of the first 16 points obtained by the grid search algorithm via setting the threshold dth=0.8. This fist constellation belongs to the set Θ of C2416 constellations (see the last in Section 3.2 ). In the set Θ, the best constellation is of (9) with the minimum distance dmin=0.8376 and its distance spectrum is shown in Figure 2. The algebraic orthogonal constellation like (12) does not belong to the set Θ because its minimum Frobenius chordal distance is dmin=0.7321 and its distance spectrum is shown in Figure 3.Figure 2
The distance distribution of the searching orthogonal constellation with the minimum distancedmin=0.8376.Figure 3
The distance distribution of the algebraic orthogonal constellation with the minimum distancedmin=0.7321.In Figure4, 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 10-5, the best searching orthogonal constellation with the minimum distance dmin=0.8376 yields a SNR gain of about 2 dB over the algebraic orthogonal constellation of [4] when the number of receive antennas N=2. In order to compare our testing results with those from the algebraic orthogonal constellation of [4], Figure 4 also shows the case that the number of receive antennas is N=1. At a symbol error probability of 10-3, the orthogonal constellation system for two receive antennas yields an SNR gain of about 8 dB over the system for one receive antenna.Figure 4
The performance comparisons between two searching orthogonal constellations and an algebraic orthogonal constellation for two cases with one and two receive antennas.


## 7. Conclusion

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 [4] 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.



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