IET Communications Research Article
Half-ZF beamforming scheme for downlink two-user multiple input single output-based non-orthogonal multiple access systems
ISSN 1751-8628 Received on 8th January 2017 Revised 25th April 2017 Accepted on 21st May 2017 E-First on 17th July 2017 doi: 10.1049/iet-com.2017.0018 www.ietdl.org
Wenbo Cai1 , Guocheng Lv1, Ye Jin1 1State
Key Laboratory of Advanced Optical Communication Systems and Networks, Peking University, Beijing, People's Republic of China E-mail:
[email protected]
Abstract: The authors investigate the beamforming (BF) problem to minimise the power consumption for the downlink 2-user multiple input single output-based non-orthogonal multiple access system. Inspired from the conventional zero-forcing (ZF) scheme, they propose the Half-ZF scheme which only mitigates the interference introduced by the signal of the cell-interior user through the ZF method. Meanwhile, the interference introduced by the signal of the cell-edge user is removed through the SIC technology. We derive the closed-form expression for the BF vectors of the two users under the Half-ZF scheme. The optimal power consumption for the Half-ZF scheme is analysed and the corresponding solution of the BF factors are also obtained. Compared with the conventional ZF scheme, both the Half-ZF scheme and the existing quasi-degradation (QD) scheme can achieve lower power consumption than the ZF scheme. However, the Half-ZF scheme can achieve this advantage under more relaxed conditions. Consequently, the Half-ZF scheme consumes lower power than the QD scheme when combined with the ZF scheme in practical systems.
1 Introduction Non-orthogonal multiple access (NOMA) has been recognised as a promising technology for the fifth generation (5G) mobile communication system, as it can improve the spectral efficiency, support more user access requirements and reduce the latency [1– 3]. For the extensive application of the multiple input multiple output (MIMO) technology in the mobile communication system, the application of MIMO to NOMA (MIMO-NOMA) has become a research hot-point for NOMA [4–9]. Furthermore, the beamforming (BF) problem under the minimisation of power (MinPower) consumption criteria is a key point [4, 10–13] which is usually discussed in the multiple input single output-based NOMA (MISO-NOMA) system (the MISO-NOMA system is a special case of the MIMO-NOMA system). For the two-user MISONOMA system, the optimal power consumption can be obtained through the solving of the dual convex problem [11]. However, the solving process of the corresponding BF vectors requires high computational complexity and only numerical result can be obtained. Therefore, the research on the closed-form expression for the low complexity and efficient BF schemes is in hotspot, e.g. recently, the quasi-degradation (QD) scheme [10, 11, 13] has been discussed. When combined with the conventional zero-forcing (ZF) scheme, the hybrid scheme of the QD scheme and the ZF scheme (QD + ZF) has been considered as an efficient BF scheme [10, 13]. However, the performance of the QD + ZF scheme is seriously limited because the quasi-degraded property between the channel state information (CSI) of the two users is hard to be satisfied especially when the number of antennas at the base station is large [11]. In this paper, we further study the BF problem in the downlink 2-user MISO-NOMA system. We propose a new BF scheme (the so-called Half-ZF scheme) and also drive the corresponding closed-form BF vector expressions. According to the simulation results, the Half-ZF scheme can achieve lower power consumption than the QD scheme especially when the correlation coefficient between the CSI of the two users is small (the number of antennas at the base station is large) or the minimum data rate requirements for the two users are large. The inspiration point of the Half-ZF scheme comes from the conventional ZF scheme. In the downlink 2-user MISO-NOMA system, UE-1 (the cell-edge user) treats the interference introduced by the signal of UE-2 (the cell-interior user) as noise when IET Commun., 2017, Vol. 11 Iss. 10, pp. 1633-1640 © The Institution of Engineering and Technology 2017
decoding the signal of UE-1. UE-2 will first remove the interference introduced by the signal of UE-1 through the successive interference cancellation (SIC) technology and then decode the signal of UE-2. Since the existing joint design methods for the BF vectors of UE-1 and UE-2 requires high computational complexity (e.g. the dual convex problem-based scheme) or has limited performance (e.g. the QD + ZF scheme), we propose the Half-ZF scheme in this paper. Under the Half-ZF scheme, UE-1 removes the interference introduced by the signal of UE-2 through the ZF method, UE-2 removes the interference introduced by the signal of UE-1 through the SIC technology. Since only the interference introduced by the signal of UE-2 is removed through the ZF method, we name this scheme as the Half-ZF scheme. Therefore, the BF vector for UE-2 is the same as the conventional ZF scheme, we only need to design the BF vector of UE-1 to save more power consumption. The key point for the Half-ZF scheme is the calculation of the BF factors for UE-1. In the two-user MISO-NOMA system, the effective rate for the signal of UE-1 is the minimum achievable rate at the terminal of UE-1 and UE-2. Thus, we propose two subschemes under the Half-ZF scheme, i.e. the Half-ZF-1 scheme and the Half-ZF-2 scheme. Under the Half-ZF- g, g = 1, 2 scheme, the effective rate for the signal of UE-1 is determined by the terminal of UE- g, g = 1, 2. We derive the optimal closed-form solution for the BF factors of UE-1 under both the Half-ZF-1 scheme and the Half-ZF-2 scheme. We analyse the constraint conditions for the Half-ZF- g, g = 1, 2 scheme to consume lower power than the ZF scheme. Compared with the QD scheme, the constraint condition for both the Half-ZF-1 scheme and the Half-ZF-2 scheme are more relaxed. We also analyse the power consumption for the hybrid scheme of the Half-ZF-1 and Half-ZF-2 schemes. When the correlation coefficient of the CSI of the two users, i.e. ρ, satisfies ρ 2 ≤ 1/2, the hybrid scheme indeed achieves the same performance as the Half-ZF-1 scheme. When ρ 2 > 1/2, the hybrid scheme can achieve better performance than both the Half-ZF-1 and Half-ZF-2 schemes. Being similar to the QD + ZF scheme, we also consider the hybrid scheme of the Half-ZF- g, g = 1, 2 scheme and the ZF scheme, i.e. the Half-ZF-1 + ZF scheme and the Half-ZF-2 + ZF scheme. Under the same user selection (US) result, both the HalfZF-1 + ZF scheme and the Half-ZF-2 + ZF scheme can consume lower power than the QD + ZF scheme. When considering the 1633
matched US scheme for each BF scheme, respectively, the advantage for the Half-ZF- g, g = 1, 2 + ZF scheme over the QD + ZF scheme can be further enhanced. This paper is organised as follows: Section 2 describes the system model for the two-user MISO-NOMA system. Section 3 proposes the Half-ZF- g, g = 1, 2 BF scheme. Section 4 analyses the power consumption for the Half-ZF- g, g = 1, 2 BF scheme and their hybrid scheme. Section 5 presents the simulation results. Conclusions are drawn in Section 6. Notations: Vectors and matrixes are expressed in bold italics type in this paper. AH denotes the conjugate transpose matrix of A. a∗ denotes the conjugate of variable a. ∠a represents the calculation of the phase for variable a. 𝒞𝒩( ) represents a complex Gaussian variable. 𝔼{ ⋅ } represents the statistical average of a random variable. ∥ ∥2 represents the norm square of a vector. represents the absolute value of a scalar.
2 System model
R1, g = log2 1 +
2
∑ wgsg,
(1)
2
hH g w2 + N 0
,
g = 1, 2.
min ∥ w1 ∥2 + ∥ w2 ∥2 s . t . Rg ≥ Rg, min, g = 1, 2,
(4)
where ∥ w1 ∥2 + ∥ w2 ∥2 is the total power consumption. Rg, min is the minimum data rate requirement for UE‐g, g = 1, 2 which is determined by the QoS of UE‐g, g = 1, 2. Considering the data rate expressions in (3), Problem (4) can be transformed as min w1Hw1 + w2Hw2 w1Hh1h1Hw1 ≥ r1N 0+r1w2Hh1h1Hw2
s.t.
w1Hh2h2Hw1 ≥ r1N 0+r1w2Hh2h2Hw2
(5)
w2Hh2h2Hw2 ≥ r2N 0, where rg = 2
Rg, min
− 1, g = 1, 2.
g=1
where x is the aggregate signal of UE-1 and UE-2. sg is the signal 2
of UE‐g, g = 1, 2, and 𝔼{ sg } = 1. wg is an N T × 1 vector representing the BF vector of UE‐g, g = 1, 2. The received signal at the terminal of UE‐g, g = 1, 2 can be written as yg = hH g x + zg 2
3 Proposed BF scheme It requires high computational complexity to directly solve Problem (5) especially when N T is large. Chen et al. [11] have proposed a lower bound for the power consumption by solving the dual problem of Problem (5). However, to obtain the BF vectors of wg, g = 1, 2 corresponding to the dual result, we need to solve the following Karush–Kuhn–Tucker (KKT) equation
(2)
= hH g ∑ wlsl + zg,
H w1Hh1h1Hw1 − r1wH 2 h1 h1 w2 − r 1 N 0 = 0
l=1
where yg is the received signal. hg is the channel fading coefficient of UE‐g, it is an N T × 1 vector. For the Rayleigh fading property, the elements of hg are independent and identically distributed complex Gaussian variables. Each element experiences a block fading channel [14], i.e. hng ∼ 𝒞𝒩(0, σ2g); n = 1, 2, …, N T , where
hng is the nth element of hg, σ2g is the mean channel gain of UE‐g which can be considered as the path loss of UE‐g. zg is the background white complex Gaussian noise. We assume that both users have the equivalent background noise power, denoted by N 0, i.e. zg ∼ 𝒞𝒩(0, N 0), g = 1, 2. In this paper, we define Δ = σ22 /σ21. Because UE-1 is a cell-edge user and UE-2 is a cell-interior user, we have Δ > 1. In practical systems, there are more than two users in a cell. UE-1 and UE-2 are paired through US at each time slot. For the cochannel interference, hardware complexity and processing delay requirements of the SIC decoder at the user terminal [15, 16], we assume that ∥ h2 ∥2 /∥ h1 ∥2 ≥ Δ which is guaranteed through US. Under the downlink NOMA scheme, the signal for UE-1 is decoded before the signal of UE-2. Thus, the achievable data rate for the two users can be shown as [17] R1 = min {R1, 1, R1, 2} R2 = log2 1 +
H
2
h2 w2 , N0
w1Hh2h2Hw1 − r1w2Hh2hH 2 w2 − r 1 N 0 = 0 w2Hh2h2Hw2 − r2N 0 = 0 Agwg = 0, g = 1, 2,
(3)
(6)
where H A1 = I N − λ1h1hH 1 − λ2 h2 h2 T
H A2 = I N + λ1r1h1h1H + λ2r1h2hH 2 − λ3 h2 h2 , T
I N is an N T dimensional unit matrix. λi, i = 1, 2, 3 are the Lagrange T
multipliers, their optimal solution can be obtained by solving the dual problem of Problem (5) which is formulated as max N 0r1λ1 + N 0r1λ2 + N 0r2λ3 s . t . Ag ⪰ 0, g = 1, 2
(7)
λi ≥ 0, i = 1, 2, 3. Problem (7) can be optimally solved through convex optimisation tools. The optimal solution of Problem (7) can be treated as the lower bound of the power consumption, i.e. PLB = N 0r1λ1 + N 0r1λ2 + N 0r2λ3,
where Rg is the effective data rate for UE‐g, g = 1, 2. R1, g is the achievable rate for the signal of UE-1 at the terminal of UE‐g, g = 1, 2 which can be expressed as 1634
2
In this paper, we consider a quality-of-service (QoS) optimisation problem to minimise the total power consumption under given minimum data rate requirement for both users. The power consumption minimisation optimisation problem can be formulated as
We consider the two-user (denoted by UE-1 and UE-2) downlink MISO-NOMA system with N T antennas at the base station and N R (N R = 1) antennas at each user terminal. UE-1 is a cell-edge user and UE-2 is a cell-interior user. The transmitted signal at the base station can be expressed as x=
hH g w1
(8)
where PLB is the power consumption lower bound. PLB is achievable if and only if wg, g = 1, 2 satisfy the KKT equations in (6). However, the solving process for (6) requires high computational complexity and only numerical result can be obtained. Thus, the existing papers [10, 11] have given several low complexity BF schemes, such as the ZF scheme, the QD scheme IET Commun., 2017, Vol. 11 Iss. 10, pp. 1633-1640 © The Institution of Engineering and Technology 2017
and their hybrid scheme, which can be shown with closed-form expressions. In this paper, we propose a new low complexity BF scheme, i.e. the Half-ZF scheme, which can achieve better performance than these low complexity schemes above. Inspired from the formulation of the ZF scheme and the QD scheme [10], we assume a new BF scheme as the following formulation: h1 h + α12 2 ∥ h1 ∥ ∥ h2 ∥ h h w2 = α21 1 + α22 2 , ∥ h1 ∥ ∥ h2 ∥ w1 = α11
(9)
where α11 and α12 are the BF factors for UE-1. α21 and α22 are the BF factors for UE-2. Under the conventional ZF scheme, we actually attempt to find the solution of the BF factors in (9) with the following constraints: h1Hw1w1Hh1 = r1N 0 H hH 2 w2w2 h2 = r 2 N 0
h2Hw1 = 0
(10)
as
ρ∗ (1 − ρ 2) ρ
βr1(r2 + 1)N 0 r1 N 0 ρ∗ − . 2 ∥ h2 ∥ 1 − ρ 2 ∥ h1 ∥2
(14)
2
h1Hw1 = ηr1N 0 2
(15)
Being similar to (12), by assuming ∠α11 = ∠α12 ρ, we can get r1(r2 + 1)N 0 ∥ h2 ∥2 ηr1N 0 α11 2 + α12 2 ρ 2 + 2 α11 α12 ρ = . ∥ h1 ∥2 α11 2 ρ 2 + α12 2 + 2 α11 α12 ρ =
as
hH 1 h2 . ∥ h1 ∥∥ h2 ∥
2
h2Hw1 = r1(|h2Hw2| + N 0) = r1(r2 + 1)N 0 η ≥ 1,
(11)
where ρ is the correlation coefficient between the channel coefficients of the two users, i.e. ρ=
α12 =
For the Half-ZF-2 scheme, we assume R11 ≥ R12. Thus, when R12 = R1, min, we have R11 ≥ R1, min. Consequently, by introducing a new factor η which will be optimised in the next section, we formulate the following equation:
2
r2N 0 h1 h2 ρ = − + , 1 − ρ 2 ∥ h2 ∥ ∥ h1 ∥ ∥ h2 ∥2
r1 N 0 βr1(r2 + 1)N 0 1 ρ − 1 − ρ 2 ∥ h1 ∥2 1 − ρ 2 ∥ h2 ∥2
3.2 Half-ZF-2 BF scheme
The constraints h2Hw1 = 0 and h1Hw2 = 0 in (10) will seriously limit the power consumption performance of the ZF scheme. However, since the signal of UE-1 can be removed through the SIC technology at the terminal of UE-2 in the NOMA system, we can omit the constraint h2Hw1 = 0 but retain the constraint h1Hw2 = 0. Under this inspiration, we call the corresponding BF scheme as the Half-ZF scheme. According to the ZF scheme, the BF factor for UE-2 is given as r2 N 0 (∥ h1 ∥2h2 − (h1Hh2)h1) ∥ h1 ∥ ∥ h2 ∥2(1 − ρ 2)
α11 =
According to (14), α11 and α12 are both related to the value of β. In the power consumption analysis section, we will find that the minimisation of the power consumption for the Half-ZF-1 scheme is indeed an optimisation problem for β. We will also find that these assumptions, i.e. ∠α11 = ∠α12 ρ = 0, do not affect the power consumption for the Half-ZF-1 scheme.
h1Hw2 = 0.
w2 =
By further assuming the phase of α11 as zero, we can solve (13)
(16)
By further assuming the phase of α11 as zero, (16) can be solved
α11 =
ηr1N 0 r1(r2 + 1)N 0 1 ρ − 1 − ρ 2 ∥ h1 ∥2 1 − ρ 2 ∥ h2 ∥2
α12 =
ρ∗ (1 − ρ 2) ρ
r1(r2 + 1)N 0 ηr1N 0 ρ∗ − . 2 2 ∥ h2 ∥ 1 − ρ ∥ h1 ∥2
(17)
For the BF vector of UE-1, we propose two solutions under the Half-ZF scheme which are shown in the following sections.
Being similar to the Half-ZF-1 scheme, the minimisation of the power consumption for the Half-ZF-2 scheme is indeed an optimisation process for η which is shown in the next section.
3.1 Half-ZF-1 BF scheme
4 Power consumption analysis
For the first Half-ZF scheme, i.e. the Half-ZF-1 scheme, we consider the following equation:
Under the ZF and QD schemes [10], the power consumption can be given as
2
h1Hw1 = r1N 0 2
2
h2Hw1 = βr1 h2Hw2 + N 0 = βr1(r2 + 1)N 0
(12)
β ≥ 1, where β is a new unknown factor which needs to be optimised. We actually have assumed that R11 ≤ R12. Thus, when R11 = R1, min, we have R12 ≥ R1, min. Consequently, β is restricted to be larger than 1 in (12). Under the BF formulation of w1 in (9) and with an extra assumption ∠α11 = ∠α12 ρ, we can transform (12) as βr1(r2 + 1)N 0 ∥ h2 ∥2 rN α11 2 + α12 2 ρ 2 + 2 α11 α12 ρ = 1 0 2 . ∥ h1 ∥
α11 2 ρ 2 + α12 2 + 2 α11 α12 ρ =
IET Commun., 2017, Vol. 11 Iss. 10, pp. 1633-1640 © The Institution of Engineering and Technology 2017
(13)
r1 N 0 r2N 0 1 1 + 1 − ρ 2 ∥ h1 ∥2 1 − ρ 2 ∥ h2 ∥2 rN 1 + r1 r2N 0 PQD = 1 0 2 + , ∥ h1 ∥ 1 + r1(1 − ρ 2) ∥ h2 ∥2 PZF =
(18)
where PZF and PQD are the power consumption for the ZF and QD schemes, respectively. The PQD is feasible if and only if ∥ h2 ∥2 1 + r2 r2 ρ 2 − 2 ≥ 2 2 . ∥ h1 ∥ ρ (1 + r1(1 − ρ 2))
(19)
We analyse the power consumption for Half-ZF-1 and HalfZF-2, respectively.
1635
4.1 Power consumption for Half-ZF-1 According to the BF factor solution in (11) and (14), we define the power consumption for Half-ZF-1 as PHZF, 1 which can be derived as
r2 + 1 ∥ h2 ∥2 r2 + 1 < < . 4ρ2 ∥ h1 ∥2 ρ2
PZF − PHZF, 1 = f (1)
PHZF, 1 = ∥ w1 ∥2 + ∥ w2 ∥2
r2 N 0 1 + α11 2 + α12 2 + 2 α11 α12 ρ 1 − ρ 2 ∥ h2 ∥2 r2N 0 r1 N 0 1 = + (1 − ρ 2) α12 2 2 2 + 1 − ρ ∥ h2 ∥ ∥ h1 ∥2 = PZF − f (β),
∥ h2 ∥2 r2 + 1 < ∥ h1 ∥2 4ρ2
1, we can always find a value of λ to ensure PHZF, 1 ≤ PZF. Furthermore, f (λ) will achieve its maximum value at the point of λ0 =
r1 N 0 ρ2 . 1 − ρ 2 ∥ h1 ∥2
(23)
To find the optimal value of λ where PZF − PHZF, 1 is larger than zero and maximised, we need to confirm relationship of λ0, λ1 and 1. If λ0 > 1, the maximum value of PZF − PHZF, 1 is f max. If λ1 < 1, there is no feasible region for λ to ensure PZF − PHZF, 1 ≥ 0. If λ0 < 1 < λ1, the the maximum value of PZF − PHZF, 1 will be achieved at the point λ = 1 where f (1) =
r1N 0 (r2 + 1) 1− ρ2
(28)
To confirm the real value of δ1 and δ2, we need to guarantee τ1 = (1 − ρ 2)
∥ h1 ∥2(r2 + 1) ≤ 1. ∥ h2 ∥2
If τ1 ≤ 1 is satisfied, it is obvious that δ1 ≤ δ2. Thus, when δ2 ≥ 1, we can always find a value of δ to ensure PHZF, 2 ≤ PZF. The maximum value of φ(δ) is achieved at the point of
φmax = φ(δ0) = −
λ1 < 1 λ0 < 1 < λ1 .
∥ h1 ∥2(r2 + 1) ∥ h ∥2(r2 + 1) − 1 − (1 − ρ 2) 1 2 ∥ h2 ∥ ∥ h2 ∥2
∥ h1 ∥2(r2 + 1) . ∥ h2 ∥2
The maximum value φmax is given as
For summary, we can get
PZF − PHZF, 1 = f (1) f max
To ensure PHZF, 2 ≤ PZF, we also need to find the feasible region for η to make φ(η) ≥ 0. By assuming δ = η, φ(δ) is also a quadratic linear equation for δ. By assuming φ(δ) = 0, the solution for δ can be expressed as
δ0 = ρ
(r2 + 1) 2ρ − . ∥ h2 ∥2 ∥ h1 ∥2∥ h2 ∥2
1
Thus, whether the Half-ZF-1 scheme can achieve lower power consumption than the ZF scheme, it is determined by the value of ∥ h1 ∥2, ∥ h2 ∥2, ρ and the data rate requirement of UE-2, i.e. r2. The power decrement is also related to the the data rate requirement of UE-1, i.e. r1. For more clearly, we can rewrite (24) as
r1(r2 + 1)N 0 r1 N0 + . 2 2 ∥ h2 ∥ 1 − ρ ∥ h1 ∥2
(29)
We can also optimise the value of δ to maximise PZF − PHZF, 2. When δ0 ≥ 1, τ1 < 1, i.e. ∥ h1 ∥2(r2 + 1) ∥ h1 ∥2(r2 + 1) 2 (1 − ρ 2) < 1 < ρ , 2 ∥ h2 ∥ ∥ h2 ∥2 and δ2 > 1, the maximum value of PZF − PHZF, 2 is φmax. Here, we define τ0 =
1636
(25)
∥ h1 ∥2(r2 + 1) 2 ρ . ∥ h2 ∥2
IET Commun., 2017, Vol. 11 Iss. 10, pp. 1633-1640 © The Institution of Engineering and Technology 2017
When δ2 ≤ 1 or τ1 > 1, there is no feasible value for δ to guarantee PZF − PHZF, 2 ≥ 0. When δ0 < 1 < δ2, the the maximum value of PZF − PHZF, 2 will be achieved at the point δ = 1 where φ(1) = 2
ρ r2 + 1 r1N 0 1 r1(r2 + 1)N 0 − . 1 − ρ 2 ∥ h1 ∥2∥ h2 ∥2 1 − ρ 2 ∥ h2 ∥2
In summary, we can get 1
PZF − PHZF, 2 = φ(1) δ0 < 1 < δ2 and τ1 < 1. φmax
(30)
τ1 < 1 < τ0 and δ2 > 1
Being similar to the transformation from (24) to (25), we can also show the constraints in (30) as the same way in (25). As shown in (30), the constraint δ2 > 1 can be transformed as ∥ h2 ∥2 / ∥ h1 ∥2 > (r2 + 1)/4|ρ|2 or (r2 + 1)(1 − |ρ|2) < ∥ h2 ∥2 / ∥ h1 ∥2 < (r2 + 1)|ρ|2. The constraint τ1 < 1 can be transformed as ∥ h2 ∥2 / ∥ h1 ∥2 > (r2 + 1)(1 − |ρ|2). The constraint δ0 < 1 (τ0 < 1) can be transformed as (r2 + 1)|ρ|2 < ∥ h2 ∥2 / ∥ h1 ∥2. Moreover, we can always have r2 + 1/4|ρ|2 > (1 − |ρ|2)(r2 + 1) for any value of |ρ|2. Therefore, (30) can be rewritten as (r2 + 1)/4|ρ|2 or 2 2 (r2 + 1)(1 − |ρ| ) < ∥ h2 ∥ / ∥ h1 ∥2 < (r2 + 1)|ρ|2. Compared with the condition for PZF − PQD ≥ 0 in (19), it is hard to say which scheme has more relaxed condition. However, we can confirm that the Half-ZF- g, g = 1, 2 scheme has a more relaxed condition than the QD scheme when R1, min = R2, min = 1 bps, i.e. r1 = r2 = 1. Under this condition, we have r2 + 1 r2 ρ 2 r +1 − − 2 2 2 2 2 ρ 4ρ (1 + r1(1 − ρ )) 3 ρ2 2 − 2 2ρ (2 − ρ 2) 12(1 − ρ 2) + ρ 4 = > 0. 2 2 ρ 2(2 − ρ 2)
=
PZF − PHZF, 1/2 =
According to the result in (32), we can say that the Half-ZFg, g = 1, 2 scheme has a more relaxed condition than the QD scheme under the special case where R1, min = R2, min = 1 bps. In versus, when ∥ h2 ∥2 /∥ h1 ∥2 is given, the condition for the value of ρ 2 is also shown through simulation in the following section. We can obtain the same conclusion that the Half-ZF- g, g = 1, 2 scheme has a more relaxed condition than the QD scheme under the special case where R1, min = R2, min = 1 bps.
∥ h2 ∥2 r2 + 1 < ∥ h1 ∥2 4ρ2 r2 + 1 ∥ h2 ∥2 r2 + 1 < < , (33) 4ρ2 ∥ h1 ∥2 ρ2 r2 + 1 ∥ h2 ∥2 < ρ2 ∥ h1 ∥2
where PZF − PHZF, 1/2 = f (1) = φ(1) represents that both Half-ZF-1 and Half-ZF-2 schemes can be adopted when r2 + 1 ∥ h2 ∥2 r2 + 1 < < . 4ρ2 ∥ h1 ∥2 ρ2 When r2 + 1 ∥ h2 ∥2 < , ρ2 ∥ h1 ∥2 the Half-ZF-1 scheme should be adopted. Otherwise, the Half-ZF cannot obtain lower power consumption than the ZF scheme. (2) ρ 2 > 1/2 case: Under this case, we can get PZF − PHZF, 2 = φmax in (31). Thus, we have (see (34)) We can find that the Half-ZF-2 scheme should be adopted when (r2 + 1)(1 − ρ 2)
