Adjusting the antenna phases of a block based on (2.4) ∼ (2.8), the main beams of different blocks can be tuned towards the directions of different users, offering spatial division to support multiple access (SDMA) of users. Despite the array gain provided by (2.4), the signal to interference-plus-noise ratio (SINR) of SDMA also depends on the antenna pattern of (2.1) and the beam patterns of adjacent users.
Suppose that all patch antennas in Fig. 2.3(a) are used to support a single user. The maximum achievable array gain in this case is 20 log10(64)= 36 dB when.
Ψx+ (−1)pβx,(p,q)= 2c5π (2.9)
Ψy+ (−1)qβy,(p,q) = 2c6π, {c5, c6} ∈ Z (2.10)
Scaling the power by 1/MN for each element antenna of the M × N PAA, the maximum effective array gain is 10 log10(64)= 18 dB..
On the other hand, if each block in Fig. 2.3(a) serves a user with an 4 × 4 antenna array, the maximal array gain now reduces to a smaller value of 20 log10(16) = 24 dB.. Except for the smaller array gain, each user’s signal is also interfered by the signals of adjacent users. Fig. 2.2(b) shows the array factors for the 2-user SDMA based on the
partition in Fig. 2.3(a). The two beams are pointed toward the elevation angle of π4 and the azimuth angles of 0 and π, respectively. As can be seen from the figures, the side beams of adjacent users overlap with the main beam of the user of interest, making it difficult to maintain the SINR in practice. To provide a better control for the SINRs of users supported with PAA, a hybrid approach of beamforming (HBF) is introduced in the next section to take the advantage of baseband beamforming techniques.
Chapter 3
SDMA Using Hybrid Beamforming
The SDMA method introduced in Section 2.1 is based on phase adjustment with the phase shifter of each element antenna. However, adjusting only the phases of the radio signals sometimes may not be able to achieve the desired SINR for the user of interest, as the beam direction of the user might be severely jammed by the side beams of other users. To overcome this difficulty, baseband beamforming techniques can be used to jointly steer the beam patterns and suppress the interference for all users. More specifically, in addition to steering the main beam towards the direction of interest, the baseband array factor can be nulled as well in the directions of other users’ main beams.
However, it is impractical to apply a baseband beamforming weight for each element antenna of the 8 × 8 PAA. Taking into account the implementation cost, each partition of PAA is driven by a common baseband beamforming weight, while each antenna is still equipped with an individual phase shifter. To distinguish the array factor B(φ, θ) formed with the baseband beamforming weights of a user from the array factor A(φ, θ) obtained by tuning the phase of the radiated wave of each antenna, we refer to B(φ, θ) as the baseband array factor (BAF) in contrast to the array factor A(φ, θ) tuned in the radio-frequency (RF) band.
Now we consider this hybrid type of baseband and RF beamforming for the simple
partition shown in Fig. 2.3(a). Suppose that the RF array factor (RAF) for differ-ent blocks of a user are the same and pointed to the desired direction of interest, the composite beam pattern of HBF is given by
H(φ, θ) , B(φ, θ)A(φ, θ)E(φ, θ) (3.1)
where A(φ, θ) is the array factor of the 4 × 4 antenna arrays. In the extreme case of Fig.
2.3(a) that the entire PAA is used to support a single user, the BAF is given by
B(φ, θ) = X1
r=0
X1 s=0
w(r,s)ej4rΨxej4sΨy. (3.2)
where Ψx , kdxcos φ sin θ and Ψy , kdysin φ sin θ. The enlarged distances between the adjacent effective antennas make 4kdx = 4kdy = 4π in (3.2) as dx = dy = λ/2, which in turn results in the periodic baseband beam pattern of B(φ, θ) shown in Fig. 2.3(b).
The angular coordinate corresponds to the elevation angle θ and the radial coordinate represents the normalized beamforming gain. Due to the periodic pattern, the product of B(φ, θ) and A(φ, θ) will yield significant sidelobes on both sides of the main beam.
For clarity, the RAF A(φ, θ) of the 4 × 4 block is also shown in Fig. 2.3(b). Since the patch antenna has a fixed radiation pattern, its pattern is not shown in the figure.
The strong sidelobes of HBF with the partition in Fig. 2.3(b) will cause sever in-terference in SDMA. To suppress the sidelobe while still be able to benefit from the advantage of HBF, we consider an alternative partition in Fig. 3.1(a). Patch antennas of the same color belong to a block and are driven by the same baseband beamforming weights. Thus, the PAA is still partitioned into four blocks in Fig. 3.1(a). With this partition, the distances between two element antennas increase to (2dx, 2dy) in both the x and y axes. While the largest distances between any two effective antennas become (dx, dy) of the distances between the blue and the yellow blocks. Consequently, the
2dx
Figure 3.1: Configurations of the rearranged planar antenna arrays respect to baseband beamforming weights and polar plot of the beamforming pattern of the hybrid scheme in rearranged PAA when θ = π/2.
Fig. 3.1(b) shows the polar plots of the BAF and RAF according to the partition in Fig 3.1(a) when θd = π/2. The RAF still bears the same form of (2.4) except that the parameters dx and dy now become 2dx and 2dy, respectively. On the other hand, the BAF now becomes the form of
B(φ, θ) =
Despite the periodic pattern of RAF, it clearly shows that the product of B(φ, θ) and A(φ, θ) will form a sharper and stronger mainbeam along the desired direction. This makes the configuration in Fig. 3.1(b) more suitable for joint beam steering and inter-ference suppression in SDMA.
3.1 HBF based on the MD beamforming
According to the configuration of Fig. 3.1(a), we consider a partition to implement HBF for two-user SDMA. The antennas in blue and yellow colors of Fig. 3.1(a) belong to user one, and the antennas in green and orange belong to user two. That is two BF weights are employed for each user. The resultant BAF for user one and two are given
by
B1(φ, θ) = w(1,0)+ w(1,1)ejΨx+jΨy (3.4) B2(φ, θ) = w(2,0)ejΨx + w(2,1)ejΨy. (3.5)
Now to steer the main beam towards the direction of the user of interest and, in the mean time, to suppress the interference in the direction of the other user, a typical method is the so-called maximum directivity (MD) BF [30].
The MD BF basically constructs the baseband BF weights by superposition of the steering vectors
s1(φ, θ) , [1 ej(Ψx+Ψy)]T (3.6) s2(φ, θ) , [ejΨx ejΨy]T (3.7)
of user one and two in (3.4) and (3.5), respectively. Specifically, the BAFs are expressed as
B1(φ, θ) , X2
i=1
bi[1 e−j(Ψxi+Ψyi)]s1. (3.8)
B2(φ, θ) , X2
i=1
ci[e−jΨxi e−jΨyi]s2, (3.9)
where Ψxi , kdxcos φisin θi and Ψyi , kdysin φisin θi, and {φi, θi} is the desired beam direction of user i. Substituting the constraints of
Bm(φi, θi) =
1, i = m 0, i 6= m
, i, m ∈ {1, 2}. (3.10)
back into (3.8) and (3.9) yields the coefficients bi and ci. Furthermore, equating the
Bm(φ, θ) respectively for m ∈ {1, 2} in (3.4) ∼ (3.9) results in the baseband BF weights
w(1,r)= X2
i=1
bie−jr(Ψxi+Ψyi), (3.11)
w(2,r)= X2
i=1
cie−j(1−r)Ψxie−jrΨyi r ∈ {0, 1}. (3.12)
Though simple and straightforward, the MD BF does not offer the degrees of freedom to control the total power of BF and more, importantly, the signal to interference-plus-noise ratio (SINR) in SDMA. In the next section, we refine the design of HBF based on the concept of convex optimization.
Chapter 4
Multiuser hybrid beamforming based on convex optimization
Our goals for the design of HBF for SDMA are twofold: one is to minimize the overall power consumption subject to the signal quality constraint of each user, the other is to look for the best SINR for each user subject to the total power constraint. To meet the design objectives, we consider a number of design criteria from the perspective of convex optimization. They are classified in two categories and described in the following subsections.
4.1 HBF based on the constrained minimization of power
A widely used approach for power minimization is the linear constrained minimum power (LCMP) method [29]. To minimize the power consumption of BF and, in the mean time, null the interference in the beam direction of the user of interest, we first apply the LCMP subject to (s.t.) constraints similar to that of the MD beamforming in
4.1.1 Power minimization based on LCMP
Let ui(t), i ∈ {1, 2} be the transmitted signal of user i, with E[|ui(t)|2] = 1. The baseband transmitted signal for the two-user SDMA can be modeled as
x(t) = s1u1(t) + s2u2(t). (4.1)
where the steering vectors s1 and s2are defined in (3.6) and (3.7), respectively. To design the BF weight vector wi for user i such that the output power and the interference to the beam direction of the other user are both minimized, the LCMP is formulated as
arg min
wi
wiHSxwi (4.2)
s.t. wiHC = ei (4.3)
where Sx , E{x2(t)}, C , [si(φ1, θ1), si(φ2, θ2)] with {φi, θi} being the desired beam direction of user i, and ei is a 1 × 2 basis vector with 1 in the ith position and the others zero.
The above optimization problem can be easily solved by making use of the Lagrange multiplier as below
J = wHi Sxwi+£
wHi C − eHi ¤
λ + λH£
CHwi− ei¤
(4.4)
with λ , [λ1, λ2]T. The resultant optimal BF weight vector for user i is given by
wHi = eHi [CHS−1x C]−1CHS−1x . (4.5)
4.1.2 Power minimization based on SOCP
The BF weight vectors obtained with the LCMP method are essentially carried out individually. The output powers for user one and two are not jointly minimized. In addition, we often are more interested in searching for BF weights that can guarantee the SINR for each user. To design BF weights that fulfill the above goals, we adopt an alternative approach making use of the standard second order cone programming (SOCP) for convex problem.
To formulate the design for BF weights that minimize the power consumption and maintain the SINR, we first define the SINR for SDMA using HBF. For convenience of expression, we rewrite the composite beam pattern (3.1) for user i as
Hi(φ, θ) = wiHsi(φ, θ)Ai(φ, θ)E(φ, θ)
= wiHgi(φ, θ), i ∈ {1, 2}. (4.6)
Following the above notations, the SINR with respect to (w.r.t) the i-th user at its desired beam direction (φi, θi) is defined as
SINRi(φi, θi) = kwHi gi(φi, θi)k2
Σj6=ikwHj gj(φi, θi)k2+ σn2, (4.7)
where σn2 is the noise variance. Given the SINR definition, now minimizing the total baseband power subject to SINR constraints can be formulated as
P(γ0) =
arg minwi
P2
i=1kwik2
s.t. miniSINRi(φi, θi) ≥ γ0
i ∈ {1, 2}
(4.8)
where γ0 > 0 is the lower bound on the SINR.
(QoS) characterized by γ0. Setting a real-valued variable p, (4.8) can be rewritten as
The SINR constraints are actually not convex. To apply the convex optimization scheme, the constraints need to be reformulated.
Since the BF vector with an arbitrary phase rotation is still optimal as long as the BF vector itself is already optimal. Without the loss of generality, we can constrain wHi gi to be nonnegative real [21]. Define vectorization operation. The constraint can be rewritten as
µ
As a result, the optimization problem now becomes
which is of the standard form of SOCP, thus can be solved efficiently by using the CVX toolbox [24].