Unitary Precoder with Constant Modulus Elements

5.3 Unitary Precoder with Constant Modulus Elements

This section describes the CUBF matrices by applying a framework for the description of complex Hadamard matrices.

5.3.1 Description of Complex Hadamard Matrices

In this section, we provide the mathematical framework for the construction of unitary beamforming matrices with constant modulus entries G_cubf. We first introduce various definitions that we will use later to parametrizeG_cubf. Definition 5.1. A square matrix A of size M where the entries are of equal modulus|aij|²= _M¹; i, j= 1, . . . , M, is called normalized Hadamard matrix if

AA^H=IM. (5.8)

INPUT: H,P,σ² OUTPUT: G^?_ubf

Step 1: # Initialization j= 0,G^(j)_ubf=IM andµ= 1

Step 2: # Gradient of sum rate on Euclidean space Γ^(j)= [c₁h1h^H₁g^(j)₁ , . . . , c_KhKh_K^Hg^(j)_K] with

c^(j)_k =

ξk+ 1−(ν^(j)_k )²−1

Step 3: # Gradient direction in Riemannian space Φ^(j)=Γ^(j)(G^(j)_ubf)^H−G^(j)_ubf(Γ^(j))^H

Step 4:

Evaluate ¹₂<(trΦ^(j)(Φ^(j))^H) if small enough, then STOP and G^?_ubf =G^(j)_ubf

Step 5: # Determine the rotation matrices R^(j)= exp(µΦ^(j)),T^(j)=R^(j)R^(j)

Step 6:

while |Rsum(G^(j)_ubf)−Rsum(T^(j)G^(j)_ubf)| ≥ ^µ₂<(trΦ^(j)(Φ^(j))^H) do

R^(j)=T^(j),T^(j)=R^(j)R^(j),µ= 2µ end while

Step 7:

while|Rsum(G^(j)_ubf)−R_sum(R^(j)G^(j)_ubf)|< ^µ₄<(trΦ^(j)(Φ^(j))^H)) do

R^(j)= exp(µΦ^(j)),µ= ^µ₂ end while

Step 8: # Update

G^(j+1)_ubf =R^(j)G^(j)_ubf, j=j+ 1, go toStep 2

Table 5.1: Self-tuning Riemannian steepest descent algorithm from [79, Table II] applied to compute sum rate maximizing unitary precoderG^?_ubf.

The set of normalized complex Hadamard matrices of sizeMis denotedHM. In the unnormalized case: AA^H=MI_M.

Definition 5.2. [81, Definition 2.2] The complex Hadamard matrices{A,A} ∈˜ HM are equivalent, written A ∼= ˜A, if there exist diagonal unitary matrices Dr,Dc and permutation matricesPr,Pc such that¹

A=DrPrAP˜ cDc. (5.9)

There are M! row and column permutation matrices Pr and Pc, respectively.

The equivalence class of A∈ HM under the equivalence relation (5.9)is QM(A) ={B∈ HM|A∼=B}. (5.10)

1In this definition transposition and complex conjugate are excluded since they are mean-ingless in the application of precoding.

We denote GM =HM/∼=, the set of equivalence classesGM.

In the next section, we will present the set G_M of equivalence classes for several dimensionsM.

5.3.2 Equivalence Classes

Interestingly, the complete set of equivalence classesGM is only known forM <

6. The problem of findingall equivalence classes for dimensionsM ≥6 remains unsolved and a catalog of known equivalence classes can be found in [81]. In the following we give a short overview of the (unnormalized) equivalence classes for M = 2, . . . ,5.

M = 2

There is only one equivalence class G₂={Q₂(F₂)} with F2=

1 1 1 −1

(5.11) The real Hadamard matrix coincides with the discrete Fourier transform (DFT) matrixF₂, where FM of sizeM is defined as

FM(m, n) =e^{−i2πM(m−1)(n−1)}; m, n= 1,2, . . . , M. (5.12) M = 3

There exists only one equivalence classG₃={Q₃(F₃)}equal to the DFT matrix F₃ defined in (5.12).

M = 4

Here, there exists a continuous family of equivalence classes with one free pa-rameterG4={Q4(Q^o₄(θ)); θ∈[^π₂,³₂π)}, whereQ^o₄(θ) is defined as

Q^o₄(θ) =









1 1 1 1

1 −1 e^iθ −e^iθ

1 1 −1 −1

1 −1 −e^iθ e^iθ









. (5.13)

Note that the real Hadamard matrix Q^o₄(π) and the DFT matrixF4∼=Q^o₄(^π₂) are special cases of (5.13).

M = 5

All complex Hadamard matrices are equivalent to the DFT matrix, i.e., G₅ = {Q5(F5)}.

In the next section, we will apply the framework of describing all complex Hadamard matrices through the equivalence relation (5.9) to parametrize the CUBF matrices which are a special case of complex Hadamard matrices.

5.3.3 Parametrization of CUBF Matrices in MISO BC

In general, the set of CUBF matrices is equal to the set of normalized com-plex Hadamard matrices HM. The description of HM is solely given by the equivalence relation (5.9) and the equivalence classes (5.10) and can be used to parametrize the CUBF. However, depending on the objective function, some pa-rameters in the general description become obsolete. If the beamforming matrix G_cubf is intended to modify the SINR of each user in (5.1) (and hence the sum rate) the diagonal unitary matrixDc in (5.9) can be omitted since it does not affect the SINR (5.1). Consequently, the diagonal unitary matrix Dr in (5.9) takes the form

Dr= diag([1, e^iϕ1, . . . , e^iϕM−1]) (5.14) withϕi∈[0,2π), i= 1,2, . . . , M−1.

Remark 5.1. One may remark that the equivalence relations in (5.9) involve continuous parameters (phases in the diagonals) and discrete parameters (per-mutations). One may think of counting the number of continuous parameters by subtracting from the2M²real entries the number of real constraints imposed by CUBF: M² due to unitarity, (M −1)² for the constant element magnitudes (suffices to apply to a (M−1)×(M−1) sub-matrix), andM (for a first row of all 1’s). One ends up with M −1 degrees of freedom, which correspond to the matrixD_r. The curiosity is the unexpected appearance ofθ inQ^o₄(θ). The explanation is that counting the obvious constraints must lead to redundancies.

The appearance of the additional free parameters can be explained as follows.

(Unnormalized) complex Hadamard matrices can in fact be constructed recur-sively as follows: [83] V(A,B) =

A B A −B

whereA andBare itself complex Hadamard and hence allow equivalence transformations as in (5.9). Now, for A they do not need to be applied since they can equivalently be applied to V.

However, since Bappears both asBand−B, not all equivalences onBappear in V. AtM = 4, we can take A=B=F2, but the one such equivalence that needs to be allowed at the level ofBisDBwithD= diag([1e^iθ]). So we obtain forM = 4: V(F₂,DF2).

IfM = 4 another construction of CUBF matrices via the Householder trans-formation exists which is used in 3GPP LTE [75]. The set V of all CUBF matrices generated by the Householder transformation is

V =

V=I_M −2uu^H u^Hu

u∈C^M×1; |u_i|= 1; u₁= 1

. (5.15)

The construction of a CUBF matrix via the Householder transformation de-scribes only a subset of all possible CUBF matrices. In fact, V ⊂ Q^o₄(π) ⊂ H4. To prove V ⊂ Q^o₄(π) ⊂ H4, observe that V ∼= Q^o₄(π) since we have Q^o₄(π) = 2PrD1D^HVDD1Pc with D = diag(u), D1 = diag([1,−1,−1,−1]), Pc= [e1,e2,e4,e3] andPr= [e1,e3,e2,e4], whereei is theith column of IM. Hence,V is the subset ofH4 that stems from the unique real equivalence class

d0(k, m) =|akm|²+|bkm|² d1(k, m) = 2|akm| · |bkm| δkm=∠bkm−∠akm

d2(k, m) =d1(k, m) cosδkm

d3(k, m) = 2d1(k, m) sinδkm

Table 5.2: Auxiliary Variables

Q^o₄(π). Thus, restrictingG_cubf ∈ V leads to a significant performance loss as we show by simulation in Section 5.5.

In the next Section, we derive an optimization algorithm that computes the sum rate maximizing CUBF matrix G^?_cubf.