行政院國家科學委員會補助專題研究計畫期中進度報告
廣義的機會式通訊：無線行動網路中之競爭、合作與感知-子計畫四:
感知無線行動網路之合作式媒體存取控制協定設計與用戶/基地台
選取研究
計畫類別：整合型計畫
計畫編號：NSC 96－2628－E－009－004－MY3
執行期間：96 年 8 月 1 日至 97 年 7 月 31 日
計畫主持人：
王蒞君教授
共同主持人：
計畫參與人員：
劉維正、葉俥榮、陳顥
成果報告類型(依經費核定清單規定繳交)：精簡報告
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□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、
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□涉及專利或其他智慧財產權，□一年□二年後可公開查詢
執行單位：
國立交通大學電信工程學系
中 華 民 國 97 年 5 月 29 日
中文摘要
在這一部份的研究中，我們考慮了結合 解碼轉送 (decode-and-forward, DF) 與網路 編碼 (network coding) 的合作式通訊系統 (cooperative communication systems)。對於我 們 提 出 的 合 作 式 網 路 編 碼 (cooperative network coding, CNC) 協定，我們推導了中 斷機率 (outage probability) 和分集-多工權 衡 (diversity-multiplexing tradeoff, DMT)。 我們的結果顯示中繼節點不但可以提供合 作式的分集增益 (diversity gain)，也可以提 供合作式的多工增益 (multiplexing gain)。 關鍵詞：合作式通訊，解碼轉送，分集多工 增益，網路編碼，中斷機率 Abstract
In this work we consider a decode-and-forward (DF) cooperative communications system combined with the network coding. We derive the outage probability and diversity-multiplexing tradeoff (DMT) for the proposed cooperative network coding (CNC) protocol. Our results show that the relay nodes not only can provide cooperative diversity gain, but also cooperative multiplexing gain.
Keywords: Cooperative communications, decode-and-forward,
diversity-multiplexing tradeoff, network coding, outage probability.
1 Introduction
Cooperative communication attracts a great deal of interests recently. Relay terminals in a cooperative communication system can help the transmitter send information to the receiver. This is similar to a virtual multiple-input multiple-output (MIMO) system because the terminals in a cooperative network form a virtual antenna array. Clearly, cooperative communication systems can
provide diversity gains similar to the MIMO techniques.
Many cooperative communication protocols were proposed to improve diversity gain, such as orthogonal amplify and forward (OAF) [11], nonorthogonal amplify and forward (NAF) [2], space-time coded (STC) cooperative diversity protocols [12, 14, 16], dynamic decode-and-forward (DDF) [2], enhanced static decode-and-forward (ESDF), and enhanced dynamic decode-and-forward (EDDF) [17].
However, how to provide multiplexing gain by taking advantage of relays has not received much attention so far. Combining the network coding with the cooperative communications, or called the cooperative network coding (CNC) [1, 3-10, 13, 15, 18], have a potential to exploit the multiplexing gain in many relay nodes (virtual antennas). The diversity-multiplexing tradeoff (DMT) analysis of CNC has not been seen in the literature. Thus, it motivates us to derive the diversity-multiplexing tradeoff of the cooperative network coding.
The rest of our paper is organized as follows. In Section 2, we describe our system model and introduce the cooperative network coding protocol. We analyze the outage probability and diversity-multiplexing tradeoff of the cooperative network coding protocol in Section 3. Numerical results are shown in Section 4. We give our conclusions in Section 5.
2 System Model and CNC Protocol
Figure 1 shows the system model for the cooperative network coding with one relay node. Terminals A and B transmit and receive users' data, and relay R forward data. Denote the channel gains between nodes X and Y as hXY, where X ,Y∈{A,B,R}. In
] [ ] [ = ] [ 2 2 n h x n z n yR BR b + R
addition to additive white Gaussian noise, the radio channel effect experienced at each terminal is assumed to be independent and identically distributed (i.i.d.) complex normal random variables with zero mean and unit variance.
XY
h respectively, where is the transmitted signal which contains information b from
] [n
b
x
B . In the third phase, the received signals at A and B are [ 3 yA [ 3 yB ] z 3 n + A ]+zB3 ] [n xc [ =hARxc [ =hBRxc ] [n ] [n n ] n ], n
Consider the half-duplex terminals which cannot transmit and receive data simultaneously. As shown in the figure, terminals A and B can directly communicate with each other.
and
respectively, where is the signal which contains information c=a⊕b from
R . We model as zero-mean mutually
independent, circularly symmetric, complex Gaussian random sequences with variance
, where ] [n , ,B zXi {A
In the figure the cooperative network coding protocols are illustrated for the case with one relay node. In phase (1) and (2), A and B transmit information and , respectively. Then
a b
R decodes out and
in the binary form and compute , where is the bitwise exclusive or (XOR) operator. In phase (3), terminal a b b a⊕ ⊕ R broadcasts the mixed information b 0 N X∈ R} and i∈{1,2,3}. 3.2 Parameterizations
In this subsection, we define signal to noise ratio (SNR), multiplexing gain r , and diversity gain d for the proposed cooperative network coding system. The SNR is defined as
a⊕ to A and B .
Then can obtain information via the operation and terminal
A b b a = b a ) ( ⊕ ⊕ B
can obtain information via the operation . Thus the relay node here play a decode-and-forward (DF) [11] role. a a b = b) a ( ⊕ ⊕ 3 Diversity-Multiplexing Tradeoff of CNC Protocol
3.1 Equivalent Signal Models
To begin with, the signals received by
B and R in the first phase are modeled as
] [ = ] [ 1 1 n h x z n yB AB a[n]+ B (1) and ], [ = ] [ 1 1 n h x z n yR AR a[n]+ R (2)
respectively, where is the transmitted signal which contains information a from
] [n xa
A . Similarly, in the second phase, the
received signals at A and R is represented
as ] [ = ] [ 2 2 n h x z n yA AB b[n]+ A (3) and , } | ] [ := SNR 0 2 N n xk XY } , { ,Y R X {| E h ,B A where ∈ , and . is the expectation of a random variable
} , , {a b c k∈ } { E Z Z .
Denote R as the data rate on each
channel, where R can be a function of
if the communication system applies the channel-driven rate adaptation scheme. The multiplexing gain
SNR r is defined as . SNR log ) SNR ( lim := SNR R r ∞ →
Note that the base of the function is in this paper.
log e
Let be the system outage probability as a function of SNR, which is defined as the probability of the maximum average mutual information
) SNR ( out P I between input
and output being less than the data rate R ,
i.e., ], < [ P := ) SNR ( out I R P
where denotes the probability of an event
] [ P E
E . The diversity gain d is then
defined as . SNR log )] SNR ( [ log lim := out SNR P d ∞ → − (1) 3.3 Diversity-Multiplexing Tradeoff Analysis
The analysis of the outage probability of the cooperative network coding protocol with one relay node at high SNR regime is given by the following theorem and proof:
Theorem 1 The outage probability of the CNC protocol with one relay node at high SNR regime is characterized by
1, 1) ( 2 1 ) (2 ) (0, 2 1 = 2 1 CNC out Γ − − + + − s e s s K s s P s (11) where , is the plica function, is the modified Bessel function of the second kind and the first order. SNR / =e3R/2 s a z ta e tdt z − − ∞
∫
Γ 1 = ) , ( ) ( 1 z KProof. First, the maximum average
mutual information of the CNC protocol with one relay node can be seen as the sum of that of two different decode-and-forward protocols [11]: ), | | SNR (1 log { min 3 1 = 2 CNC hBR I + )]} | | | (| SNR [1 log + hAR 2 + hAB 2 ), | | SNR (1 log { min 3 1 2 AR h + + (12) )]}. | | | (| SNR [1 log + hBR 2 + hAB 2
To ease the notation, let and . Then , | =| , | =|hAR 2 y hBR 2 x , , 2 | =|hAB z y
x and are i.i.d. exponential random variables with unit mean. The outage
probability can be computed as
z CNC out P ] < [ P = ICNC R ] , [ P ] , | < [ P = ICNC R y≤x+z x≤ y+z y≤x+z x≤y+z ] , > [ P ] , > | < [ PICNC R y x z x y z y x z x y z + + ≤ + + ≤ + ] > , [ P ] > , | < [ P ICNC R y≤x+z x y+z y≤x+z x y+z + ] > , > [ P ] > , > | < [ P ICNC R y x+z x y+z y x+z x y+z + . =: 4 1 = i i i q p
∑
(2) Then ⎥⎦ ⎤ ⎢⎣ ⎡ + + + R x y p log(1 SNR )< 3 1 ) SNR (1 log 3 1 P = 1 ] 3 < )] SNR )(1 SNR [(1 log [ P = + y + x R ]. < ) SNR )(1 SNR [(1 P = + y + x e3RWhen SNR is high, 1+SNRy and x
SNR
1+ can be approximated as SNR and , respectively. Then
y x SNR ] < SNR [ P = 2 3 1 R e xy p ] SNR / < [ P = xy e3R 2 dxdy e x y s xy − − ∞ ∞ ⋅
∫
∫
0 01 < 2 = ), (2 2 1 = − sK1 s where ⎩ ⎨ ⎧ . false is expression the if 0, , true is expression the if 1, = 1 E E EOn the other hand,
. 2 1 = 1 = , 0 0 0 1 e dxdydz q x y z z y x z x y − − − + ≤ + ≤ ∞ ∞ ∞ ⋅
∫
∫
∫
Similarly, we have ), (0, ) (1 1 = = 2 3 2 p s e s s p − + −s + Γ , 4 1 = = 3 2 q q and 0. = 4 qCombining the above equations into (2), we get the desired result.
According to the definition of the diversity gain in (1), we can find that the diversity-multiplexing tradeoff achieved by the cooperative network coding protocol with one relay node is characterized by
r r dCNC( )=2−3 (21) for 3 2 < < r 1 2× 0 . 4 Numerical Results
Figure 2 illustrates the diversity-multiplexing tradeoff comparison of the upper bound (UB), the CNC protocol for one relay node, selection decode-and-forward (SDF) [11], and DF. The upper bound is for a MISO system, which is the best situation that an one-relay cooperative communications system could achieve.
From this figure, we can see that the CNC protocol improves both diversity gain and multiplexing gain compared with the DF protocol. The maximum diversity and multiplexing gain that the CNC protocol can achieve are 2 and 2/3, respectively, while the maximum diversity and multiplexing gain of the DF protocol are 1 and 1/2, respectively. Furthermore, the CNC protocol also outperforms the SDF protocol, which is an enhanced version of DF. Hence, we can conclude that using network coding at the relay node can improve not only diversity gain but also multiplexing gain.
5 Conclusions
In this paper, we investigate the diversity-multiplexing tradeoff for the cooperative network coding protocol which integrates the concept of DF relay transmission of cooperative communications with the information mixing of network coding. The proposed CNC protocol is
suitable for two users which can transmit information to each other. We give a theorem to show our outage probability analytical result with proof and DMT comparison for our CNC protocol with upper bound, SDF, and DF. We find that the CNC protocol improves both diversity and multiplexing gain compared with the DF protocol.
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Figure 1: The system model and proposed CNC protocol, where phase (1): A sends a to B and R ; phase (2): B sends
to and b A R; phase (3): R broadcasts to and b ⊕ a A B. 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Multiplexing Gain r Diversity Gain d(r) UB CNC SDF DF Figure 2: Diversity-multiplexing tradeoff comparison of the upper bound (UB), cooperative network coding (CNC), selection
decode-and-forward (SDF), and decode-and-forward (DF).
中文摘要 我們推導出多重天線廣播系統在使用 強制歸零 (zero-forcing) 前處理技術下的系 統容量分析。近年來，多重天線廣播系統是 一種廣泛討論的多重天線技術。其好處為透 過前處理技術，可以同時傳送多筆不同資訊 給不同的使用者。其中，強制歸零前處理則 是一種簡單可實現卻擁有很好效能的一種 技術。而過去已有許多對於多重天線廣播系 統容量方面的研究，但大多着重於當使用者 數目趨近無限大的容量 scaling。也就是去探 討不同前處理技術在使用者數目趨近無限 大時，其容量與理想的容量上界是否擁有相 同的容量上升斜率(當訊雜比增加時)。在我 們的研究中，我們探討在有限使用者下，使 用強制歸零前處理技術之多重天線廣播系 統的系統容量。我們探討了沒有考慮使用者 排程(或等同於隨機排程)以及考慮了使用者 排程的情況。其中，針對使用者排程的情 況，我們發展了一種虛擬使用者方式的容量 近似分析。而分析的結果也很接近模擬的結 果。 關鍵詞：多重天線系統、強制歸零前處理、 排程技術、多重天線廣播系統 Abstract
Besides delivering high data rates in a point-to-point scenario, multi-input multi-output (MIMO) antenna techniques can
broadcast personalized data to multiple users
in the point-to-multipoint scenario. Zero-forcing beamforming (ZFB) is a suboptimal but simple MIMO broadcast technique, which basically decouple the MIMO channel into many parallel single-input single-output (SISO) channels. In this article, we first derive the closed-form expression for the sum rate of the ZFB MIMO broadcast system with random user selection. Secondly, under the condition with finite users, we develop a virtual user approach approximation method for estimating the sum rate of the ZFB MIMO broadcast system with exhaustive user selection. Our results indicate
that the proposed analysis method can accurately estimate the optimal sum-rate throughput of ZFB.
Keywords: MIMO systems, zero-forcing beamforming, scheduling, MIMO broadcast channels.
1. Introduction
In addition to enhancing the data rates in the point-to-point communication
environment [1], multiple-input-multiple-output (MIMO)
techniques play an important role in the point-to-multiple multiuser environment. More specifically, a MIMO system can transmit personalized data streams to multiple users concurrently. This kind of MIMO transmission is sometimes called the MIMO broadcasting technique [2]. However, it is a bit misleading to use the term broadcasting since broadcasting is used to send the same data to all the users in the system. To clarify, the term MIMO broadcast here implies that
different personalized data streams are
transmitted to a group of selected users.
Scheduling plays a key role in the multiuser MIMO system. Taking advantage of independent statistics in fading channels of multiple users, scheduling techniques can provide another form of diversity --
multiuser diversity [3]. If the system selects
one user at the time, this kind of selection principle is called time division multiple access ZFB. (TDMA)-based scheduling. Unlike the TDMA principle, the MIMO broadcast system select a group of users and thus can achieve higher data rates, but it requires a huge amount of feedback information during scheduling.
Capacity analysis of multiuser MIMO broadcast channels with independent information is a very hot research area [2, 4-8]. The capacity region of the MIMO broadcast channel was derived in [4] [9] [10]. When the
complete channel state information (CSI) is available at the transmitter, the sum rate of MIMO broadcast systems can be maximized by resorting to dirty paper coding (DPC) [2] [4] [9]. Although DPC is the optimal rate-achieving scheme, the complexity issue and the huge requirement of feedback information motivate a new line of research for other suboptimal MIMO broadcast transmission strategies, such as zero-forcing dirty-paper coding, orthogonal random beamforming and zero-forcing beamforming (ZFB) [6-8].
According to [6], ZFB with optimal user scheduling can achieve the same slope of throughput against SNR in dB as that for the capacity-achieving DPC strategy. In [8], it was shown that ZFB combined with multiuser scheduling can achieve the capacity asymptotically when the number of users approaches to the infinity. Because it is simpler than DPC and its asymptotical sum rate is the same as DPC, ZFB becomes an attractive MIMO broadcast technique. However, the performance analysis of ZFB MIMO broadcast system is still an open issue.
In this paper, we aim to evaluate the sum-rate performance of ZFB MIMO broadcast system with a finite number of users, rather than study the scalability of sum rate with an extremely large number of users as other existing work. We first develop analytical expressions for the sum rate of the ZFB MIMO broadcast system with random user selection. We also provide a virtual user approach approximation technique for evaluating the sum rate of the ZFB MIMO broadcast system with exhaustive user selection (the optimal selection policy).
The rest of this paper is organized as follows. In Section II, we describe the system model of MIMO downlink system and review ZFB scheme briefly. In Section III, we
evaluate the sum rate of ZFB with random user selection first. Then, we propose an approximate analysis for the exhaustive user selection at both low and high SNRs in Section IV. Numerical results are presented in Section V. Finally, we give our concluding remarks in Section VI.
2. Background 2.1 System Model
Consider a MIMO downlink channel with a single base station and K users. The base station and each user terminal is equipped with transmit antennas and single receive antennas, respectively. The base station can transmit different data streams up to users simultaneously. Let be the transmitted signal vector, the received signal vector, and
T M 1 × T M 1 × T M T M T M ∈ x Χ ∈ y Χ H
the MT×MT channel matrix. Denote as the circular complex additive white Gaussian noise vector with covariance matrix , where
represents the transpose conjugate operation and it is assumed for simplicity. Then the received signal can be expressed as
1 × T M E ∈ n Χ T M I 2 σ = 2 T = ] σ nn [ (⋅)T 1 , =Hx n y + (1)
where the entries of H∈ΧMT×MT are
Rayleigh fading channel element. Assume that all users experience independent fading and the transmit power is constrained by
T T P x x E[ ]= . 2.2 Zero-Forcing Beamforming
The ZFB scheme aims to invert the channel matrix to create orthogonal channels between the transmitter and the receivers without the receiver's cooperation. The transmitted signal vector with beamforming weights can be written as x =Wu , where W is the
T T M
M × beamforming matrix and is the input signal vector. Denote
1 ×
∈ΧMT u
} , {1,K K ⊂ Σ = H( W ( ZFB R Σ + ] [x , a subset of user indices to which a base station intends to transmit information and the channel matrix corresponding to . Then the beamforming matrix becomes
T M |= |Σ W ( T )H( H( ) Σ Σ ( log [ = ∈
∑
i Σ i b(
[
) H(Σ Σ ) T ) Σ , )]+ i b μ { max x i)
In general, the number of users is greater than the number of transmit antenna, i.e.,
. Thus, it is necessary to combine ZFB with scheduling in a multiuser MIMO system. It is shown that ZFB can achieve optimal throughput asymptotically for large
T
M K >
K when the selected users are searched by an
exhaustive method [8]. Denote the maximal sum rate with exhaustive user selection as .
1 −
(2) In [2], under the assumption of perfect CSI at the transmitter side, the sum rate of the MIMO system with ZFB is given by
). ( max = |= :| } , {1, Σ Σ Σ K MT ZFB max ZFB R R K ⊂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ T M K T M K = T T M M )
]
(3) (8)where represents and the effective channel gain of -th subchannel is
,0} This is a combinatorial optimization problem
to select the best one from
combinations so that the explicit performance closed-form is difficult to be found. The goal of this paper is to develop an analytical approach for evaluating the sum rate of downlink ZFB MIMO broadcast systems. We consider user selection approaches: random user selection and exhaustive user selection. . 1 ii i 1 T − ) )H(Σ Σ A = i b H( ij A] , j) (4)
Note that [ represents the ( -th entry of the matrix and μ is the water level satisfies the following criterion
. = ]+ PT )] ( log [ μ0bi
[
log(μ0bi)]
+ ) ( log 0 0 f z μ i b 1 [ i i∈ b −∑
μ Σ (5)2.3 Sum Rate with Long-term Power
Constraint 3. Sum Rate Analysis With Random User Selection
Consider the sum rate performance of ZFB subject to a long-term power constraint. The average throughput is
Consider a simple round-robin (RR) scheduling policy, which selects users in turns and does not exploit the multiuser diversity gain. In this case, it is just like there are users with the Rayleigh fading channel vector. From observation on a point-to-point ⎥ ⎦ ⎤ + ) ( dzz i b 0 ⎢ ⎣ ⎡ ∈
∑
= ) ( i ZFB E R Σ Σ ∈∑
= i E Σ× MIMO system with ZF receiver, the effective channel gain in (4) has the same form as the ZF receiver's substream effective channel gain. Due to the same statistics we can see the system as a virtual , = 1/ i
∫
μ∑
∞ ∈Σ (6)where is the probability density function (PDF) of and ) (z f i b μ is the solution of the water-filling equation with respect to the long-term power constraint
T T M
M × MIMO system with ZF receiver. Therefroe, the PDF of effective channel gain b in (4) can be obtained through the PDF of the ZF receiver's substream SNR. According to [11] [12], the substream SNRs { for an i T M i i}=1 γ MT MR = ]+⎥ ⎦ ⎤ . = 1 1 [ 0 T i i i P b b E + ∈ ⎥⎦ ⎤ − ⎢ ⎣ ⎡ −
∑
μ Σ i 0 E ∈ ⎢⎣ ⎡∑
μ Σ (7) × 2.4 Problem FormulationMIMO system with ZF receiver under the equal power allocation principle are independent and identically distributed (i.i.d.)
distributed random variables with degrees of freedom, i.e.
2 χ 2(MR −MT +1) 0, , )! ( = ) ( / ≥ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − γ γ ρ ρ γ γρ γ T M R M T T R T M T M M M e M f (9) where represents the average received SNR and 2 / = σ ρ PT γ ρ γi = bi/MT = T M
(i ). Note that the substream SNR performance in (9) is under the assumption of independent decoding [12]. The PDF of the effective channel gain can be obtained from (9) by letting and be an exponential distribution with parameter one. With the unordered i.i.d. effective channel gain , the long-term water-filling equation in (7) becomes T M , T M i i b}=1 1, = K { i b R M = dz e z M b E T z i T M i − ∞ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −
∫
∑
1 = 1 0 1/ 0 1 = μ μ μ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + − 0 1/ 0 1 = μ μ μ i T e E M (10) , =PTwhere is the exponential integral function. The resulting average sum rate of ZFB with random user selection is given by tdt e x E t x i( )= / − ∞ −
∫
−[
]
+∑
log( ) = 0 1 = i T M i ZFB E b R μ dz e z MT∫
∞ log( ) −z = 0 0 1/μ μ , 1 0, = 0⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ Γ μ T M ( wher 11)e μ0 be the solution of the water-filling equation for the long-term power constraint in
(10) and (a,x)= ∞ta−1e−tdt is the incomplete gam t ) ( = ) (0,x Ei x x
∫
Γma function [13]. Note tha − − Γ for x>0 when x is real. Sum Rate tra T M TDM scenario, th 4 Analysis With the observation i ere are Exhaustive n [1 User Selection Motivated by 4], we T M nslate the MIMO broadcast system with
T
M transmit antennas at the base station and users with single antenna into the MIMO A-based scheduling system with MT transmit antennas at the base station and receive antennas per user. In the reformulated
! T e ore, )!M receiv for the ( = T K M ⎟⎟⎠ − ⎞ ⎜⎜ ⎝ ⎛
each of which has M
⎟⎟ ⎠ ⎞ T combinations cheduling on br ! K K
antennas and a ZF MIMO receiver as shown in Fig 1. The maximal sum rate can be obtained by selecting the best transmission
K
reformulated scenario adopt the scalar feedback, meaning that only a scalar value is sent back from each receiver to base station. From the result in [14], the max-max and max-min scheduling schemes can approach the maximal sum rate for a scalar feedback MIMO system at low SNR and high SNRs, respectively. The basic principles for approximating the MT×1 antenna system for
T
M users based oadcasting by the
T M T M T Furtherm virtual users, pair from ⎜⎜ ⎝ ⎛ M the TDMA-based s T
M × antenna system for one user based DMA scheduling are described as follows:
• At the on the T
low SNR re
property of the logarithm function gion, because the
e x x) log (1
log2 + ≈ 2 for x≈0 , the ideal the ma sum rate for policy for achieving ximal
the point-to-point TDMA-based scheduling is to find a user having the maximal strongest subchannel and to allocate all power only to the strongest subchannel, i.e., a user with the best effective channel gain will most likely be selected. This principle coincides with the max-max scheduling scheme.
• At the high SNR region, the property of the logarithm function is log2(1+x)≈log2x for 1x? . Therefore, the
will aused by improving all subchannels with suitable scheduling gains and be allocated corresponding power. It is implies that no subchannel will be omitted in each scheduling run. From [14] and [15], the max-min scheduling scheme provides uniformly scheduling gains for all subchannels and has close rate performance compared to the maximal sum rate. Thus we use it to approach maximal sum rate in high SNR.
4.1 L
maximal sum rate be c
ow SNR Region
ing algorithm selects th
subchannel among virtual users at
each time slot. Denote t of all
virtual user (
chooses the target user according to
(12)
After determining the target user , we The max-max schedul
e target user with the maximal strongest K
subchannel effective channel gains for k th K
Z k
b }
{ from all users, the transmitter
. max arg = * k b k ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ T M T M i=1 the se ⎜⎜ ⎝ ⎛ T M = as ordered ef orm k i b } { Z , K
ith the inf
⎟⎟ ⎠ ⎞ k =1, T ) and fective ation of k M T M k T M b b1: ≤ K≤ : channel gains. W k T M T M : =1 * k have T k T M i max T M i b fori M b~: = :* =1,K, (13) where the supersc t max denotes the
i T =1
rip
M i:
max-max scheduling. Based on the order statistics analysis proposed in [11], we can obtain the PDFs of max MT
b~ } { as follows: , 1 = ) ( 1 : ~ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ZMT T M b T T M max T M T M b b e f (14) and −bM Te ZM ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − − −
∑
− 1 1 0 = 1 : ~ 1 1)! ( 1)! = ) ( a i i i ZM b f i a i max T M i b i b i T M a a Z T M a e M (1 2 1 ) 1) ( 0 = 2 ) 1 − + ++ − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅∑
( 1)! ( M M T T T T a Z 2 ( − :MT T M k . 1 1) ( 3 2 3 2 1 1 0 = 3 + + − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ + + − −∑
i M a aaa a T M a (15) As a result, from nd (15), we havT M i: 1 3 − − a i T (14) a e all the PDFs of b~max for i=1,L,M . obtain the sum-rate throughput and long-term power constraint equation, respectively.
T
Applying (14) and (15) to (6) and (7), we can
.2 High SNR Region
x-max scheme, the m
4
Different from the ma
ax-min scheduling selects the target user according to the maximal weakest subchannel among the Z virtual users. Based on the information of {bk }Z , the base station
slot according to ma arg = * k k T M =1 : 1
arranges the transmission during each time
(16)
Once the target user is selected, we have . x 1: k T M k b * k
T k T M i min T M i b fori M b~: = :* =1,K, (17) where the superscript min indicates the
max-min scheduling. Similarly, we can get the PDFs of MT i min T M i b: }=1 ~
{ based on the analysis in [11] as follows:
(
1)
, = −b1MT − −b1MT Z−1 Te e ZM f ( 1) : 1 ~ min T M b b (18) and∑
∏
− − + − 1 0 = 1 2 = 2)! ( 1) ( = Z a T i j T i j M ZM f : ~min( i) T M i b b 1) ( 2 2 1 1) ( 2 1 2 1 2 1 2 = 2 + + − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − ⋅ + +∑
T T T a a i i a M M a a M a i a Z[
+1)−e−bi(MT+a1MT)]
. ⋅ T M i min T M i b: }=1 2 (MT a i b e− − (19)After obtaining all the PDFs of {~ , we can obtain the sum rate by applying (18) and (19) to (6) and long-term power constraint (7).
5 Numerical Results
First, we verify the assumption that the PDF of the ZFB multiuser scheduling system can be approximated by the PDF of the ZF receiver according to (9). Figure 2 depicts the PDF of unordered the effective channel gain for by simulations and the analytical result from (9). For simulations, four users are selected randomly from the entire group of the users to obtain its value of
. We repeat this procedure 10,000 times to get the simulated PDF of . As shown in the figure, the PDF of can be closed matched by the PDF of the ZF receiver of (9). Figure 3 shows the sum rate of the ZFB with random user selection for transmit antennas
and . One can see that the sum rate obtained by simulations matches the value obtained by
(11) well. i b i b 4 = T M i b i b 3 = T M 4
Figure 4 shows the sum rate of ZFB in the low SNR region −20: 0 dB by the exhaustive user selection algorithm with users. The sum rate for the multiuser scheduling with the ZF-based MIMO receivers according to the max-max scheduling approach is also shown in the figure for comparison. From the figure, the sum rate of the ZF based on the max-max selection criterion is very close to that of the exhaustive search specially from
10 = K 20 − to 5
− dB. Furthermore, for comparison, the sum rate with random user selection is also shown in the figure. For SNR dB, the exhaustive search can provide the sum rate of 1 nats/Hz/sec, while the random user selection can only provide 0.4 nats/Hz/sec.
5 =−
By contrast, Fig. 5 compares the sum rates of the exhaustive user selection with approximately max-min approach in the high SNR region dB. It is shown that the sum rate performance of the max-min approach match the simulation result well. For SNR dB, the sum rate of ZFB with exhaustive search is about 4 nats/Hz/sec and that of ZFB with random user selection is 2 nats/Hz/sec.
20 0:
5 =
Basically, Figs. 4 and 5 show that the gains of multiuser diversity is significant even if the degrees of freedom is merely . More importantly, one can observe that at low SNRs a ``max-max" scheduling strategy is close to being optimal in an achievable sum rate sense, while at high SNRs, the ``max-min" scheduling strategy is not far from being optimal.
10 =
K
6. Conclusion
In this paper, we evaluated the sum rate of ZFB MIMO broadcast systems in the Rayleigh fading channel. An analytical expression for the sum rate of the ZFB MIMO
broadcast systems with random user selection is presented. Since the closed-form expression for the ZFB MIMO with exhaustive user selection is difficult to obtain, we develop an approximation method based on the maximal sum rate of the MIMO TDMA-based scheduling systems with the max-max and max-min scheduling at the low SNR and high SNR regions, respectively. Our results show that the proposed analytical method can accurately estimate the maximum sum rate of the ZFB MIMO broadcast system with exhaustive user selection. Besides the sum rate issue, the relationship of link reliability and coverage performance is also importance in practice. The future research direction will explore the cell coverage performance for MIMO broadcast systems.
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Figure 2: The PDF of unordered effective channel gain bi Figure 5: Sum rates of ZFB with exhaustive user selection and approximate analysis by the max-min approach for K = s with the number of transmit antennas MT igh SNR region 0:
10 user
3 = in h
20 dB.
Figure 3: Sum-rate capacity of the ZFB with random user selection for the number of transmit antennas MT =3 and 4.
Figure 4: Sum rates of ZFB with exhaustive user selection and approximate analysis by the max-max approach for
users with the number of transmit antennas in low SNR
region dB. 10 = K 3 = T M 0 20: −
中文摘要 我們從系統設計的角度來看多重天線 廣播技術。不同於過去及第二部分著重於系 統的容量分析，在此我們探討多重天線廣播 系統的涵蓋範圍。我們從分析多重天線廣播 系統的傳輸中斷機率，進而分析出傳輸連線 能提供可信賴傳送的涵蓋範圍。其中，可信 賴傳送代表著連線能在一定機率下，達到所 需的訊雜比臨界值。在這一部份我們著重於 兩種著名的傳送前處理技術：強制歸零髒紙 編碼前處理及強制歸零前處理。當沒有考慮 使用者排程時，我們提供了兩種前處理技術 的連線中斷機率與涵蓋範圍的分析。當考慮 了使用者排程，我們對強制歸零髒紙編碼前 處理進行分析，並對強制歸零前處理做模擬 的比較。我們也探討了不同系統參數如傳送 功率與通道衰減因子對於連線中斷機率與 涵蓋範圍的影響。 關鍵詞：多重天線系統、強制歸零前處理、 強制歸零髒紙編碼前處理、多重天 線廣播系統 Abstract
We consider the downlink of a multiuser multi-input multi-output (MIMO) broadcast channel under a single cell structure. To study the achievable link coverage performance of zero-forcing beamforming (ZFB) and zero-forcing dirty-paper coding (ZF-DPC) when the channel state information (CSI) is available to the transmitter. First we develop analytical closed-form expressions for the link outage probability and coverage reliability of baseline ZFB and ZF-DPC when no multiuser scheduling involved. We find that the coverage performance of ZFB can only approach to that of the weakest link of ZF-DPC under predetermined required SNR and outage probability. Secondly, for exploring the achievable cell coverage, we discuss the strongest link performance of both broadcast beamforming schemes under multiuser scheduling. Under a framework of analysis for ZF-DPC and simulation for ZFB,
we show that a soft coverage enhancement can be easily done by using scheduling techniques without extra hardware power consumption.
Keywords: MIMO systems, zero-forcing beamforming, zero-forcing dirty paper coding, coverage, MIMO broadcast channels
1. Introduction
Multiple-input multiple-output (MIMO) systems can significantly increase the spectral efficiency by exploiting the spatial degrees of freedom created by multiple antennas. In point-to-point MIMO system, it is well known that the capacity increases linearly with the minimum of the number of transmit and receive antennas [1] [2]. In the MIMO broadcast channels, [3] shows that higher capacity can be obtained by exploiting the spatial multiplexing of transmit antennas to multiple users simultaneously rather than to maximize the capacity of a single-user link.
Capacity analysis of multiuser MIMO broadcast channels is a very hot research area [3-9]. When the complete channel state information (CSI) is available at the transmitter, the maximum sum rate of MIMO broadcast systems can be achieved by dirty paper coding (DPC) [3] [4]. Although DPC is the optimal rate-achieving scheme, the applicability is limited due to its computational complexity and the need for full CSI at the transmitter (CSIT). It motivates a new line of research for other suboptimal MIMO broadcast transmission strategies, such as zero-forcing dirty-paper coding (ZF-DPC), zero-forcing beamforming (ZFB), orthogonal random beamforming and orthogonal linear beamforming [6-9]. Those suboptimal schemes can achieve the same throughput of DPC asymptotically when the number of users approaches to the infinity. However, The capacity gain of multiuser MIMO broadcast
system is highly dependent on the availability of CSIT.
Due to practicality, finite rate feedback have become a popular research area recently. Limited feedback was first considered for point-to-point MIMO system in [10] [11]. While in point-to-point case, CSIT contributes little to achieving the multiplexing gain but it is crucial for broadcast channels. For MIMO broadcast channels, the feedback load per user must be scaled with both the number of transmit antennas as well as the system SNR in order to achieve the full multiplexing gain with near-perfect CSI performance [12].
Besides dealing with practical feedback problems, some references begin to apply broadcast transmission strategies from a downlink single-cell setup to multi-cell scenarios [13] [14]. The main goal of using broadcast techniques combining with base station cooperation for multi-cell environment is to mitigate inter-cell interference for improving spectral efficiency. In [14] [15], a network coordination conception is proposed based on ZFB and ZF-DPC schemes. [16] analyzed the sum-rate performance of multi-cell cooperative ZFB under a Wyner downlink channel setup which is a simplified cellular model proposed in [17].
From both limited feedback transmission and coordinated network researches, we know that broadcast transmission techniques may play an important role in future increasingly high-speed wireless environment. However, most of the works focus on sum-rate sense performance. Based on development of coordinated network, we find that some research directions begin to move from single-cell to multi-cell setup. The link quality and achievable link coverage of broadcast transmission techniques are still open issues for baseline single-cell setup. In this paper, we aim to derive analytical closed-form
expressions for the link outage probability and coverage reliability of the single-cell multiuser MIMO broadcast system. We focus on the two popular schemes: ZF-DPC and ZFB.
2. Background 2.1 System Model
We consider a single-cell multiuser MIMO broadcast system with a base station and K user. The base station is equipped with transmit antennas, but all
t
N K user terminals
with single receive antenna. Since the base station has transmit antenna, users can be selected among
t
N Nt
K users for
simultaneous transmission with different data streams. Denote Σ⊂{1,K,K}, a subset of user indices to which a base station intends to transmit different information.
t N |= |Σ ] [ = 1 t N w w W K
Using linear spatial processing at the transmitter. Denote as the
linear beamforming weight matrix and
T t N ] i u t N u P u P , , [ = 1 1 K
u the input signal vector, where represents the power allocated for i th antenna, represents uncorrelated unit-power signal symbol and
represents the transpose conjugate operation. Then the transmitted signal vector
i P T ) (⋅ i u i i P w t N i Wu x
∑
1 = = = t N . Let the received signal vector, and the1 × t ) ∈ N y Χ G(Σ t N
× channel matrix corresponding to Σ . Denote as the circular complex additive white Gaussian noise vector with covariance matrix . Then the
received signal can be expressed as
1 × ∈ Nt n Χ t N T I nn E[ ]=σ2 , = = x n g x n y G(Σ) + H(Σ) + (1) where
, 0 0 0 0 0 0 = 2 1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ t N g g g g L M O M M L L (2)
and depicts the large-scale slowly-varying behavior of the local average channel gain between the th transmit antenna and the corresponding user terminal. For a user at a distance of
i
g
i
R from the base station, can
be written as [18] i g ], [ log 10 = 10R g0 dB i − + log10g 10 μ (3) where μ is the path loss exponent and is a constant subject to certain path loss models. Assume that all users experience independent flat fading and the entries of are Rayleigh fading channel element. The transmit power is constrained by
0 g t N t N× ) H(Σ T T P x x E[ . ∈Χ = ]
2.2 Zero-Forcing Dirty-Paper Coding
In [3], an intuitive suboptimal solution of the weight matrix based on QR-type decomposition have been proposed. Let
be the QR-type decomposition obtained by applying Gram-Schmidt orthogonalization to the rows of
W LQ = ) H(Σ ) H(Σ ,
where is a lower triangular matrix and has orthonormal rows. By letting , the corresponding system model in (1) is given by L Q T Q W= t i j j j j i i j i i i i i i l g Pu l g Pu n N y = , ,=1, , < , +
∑
+ K (4)Note that the particular choice of the weight matrix nulls out the interference from users with indices
T
Q
W=
i
j > . The remaining
interference terms from terminals with indices
i j <
i P
P
are taken care of by successive application of DPC. For simplicity, we consider equal power allocation, that is, or . Therefore, the
rate of i th link for ZF-DPC is , where t N / = T f i=1,K,Nt ) (1 log = ) | | (1 log + li,i i +γi 2 ρ i ρ is the average receive SNR shown as follows
, /10 μ 10 2 0 σ i t g T R = 2 σ t i Tg P N P (5) = N ρi i
and γ is the effective receive SNR. The term can be viewed as effective channel gain of th stream link.
2 = , | |lii W i T ) H(Σ 2.3 Zero-Forcing Beamforming
The ZFB scheme [3] aims to invert the channel matrix to create orthogonal channels between the transmitter and the receivers without the receiver's cooperation. The beamforming weight matrix is
. )−1 T ) Σ (H( )H(Σ (6)
Then, the corresponding system model in (1) is given by n + =gH( )H( ) (H( )H( ) ) u y Σ Σ T Σ Σ T −1 , =gu+n (7) and the th receive signal i i = i i.
, 2 T t N t N P ≤P + Π i i W i n u P g y +
Due to the transmit power constraint , we have the following relation:
T P ≤ 1 Πw i w [(HH T E[x x] K i i i , 1 2 ] ) = − wΠ Π 1 2 P Π T +Πw i … (8) where is the th column of and
. Equation (8) implies that ZFB incurs an excess transmission power penalty due to the required interference cancellation power of weight matrix . Note that we can express data rate of th link of ZFB as W ) ˆ (1 log = ) (1 2 2 σ σi i i i i P b g P g + + log log(1 ), = +γi (9) where is the transmit power allocated to the i th stream link so that
i i i P Pˆ =ΠwΠ2 i P
becomes effective transmit power loading. Hence, can be the effective channel gain. Under the assumption of equal power allocation, the transmit power
so that the average receive SNR
2 1/ = ΠiΠ i b w t N 2 /σ i T i P Pˆ = / ˆ = ρi giP 3. Definition
3.1 link Outage Probability
To begin with, we first define the link outage probability which reflects how reliable a system can support the corresponding link quality. For a single-input single-output (SISO) system in flat fading channel, link outage is usually defined as the probability that the effective received SNR is less than a predetermined value γth , i.e.
} < { th = r
out P
P γ γ [19]. The link outage for a point-to-point spatial multiplexing MIMO system in a flat fading channel is defined as the event when the effective receive SNR of any substream is less than γth [20] [21]. As for a point-to-multipoint MIMO broadcast system, we can define link outage probability of individual stream link as same as SISO case , i.e. = r{ i <γth}. i out P P γ out P ) (1−Pout
3.2 Link Coverage Reliability
With being the link outage probability, we define to be the link coverage reliability for its corresponding link radius associated with the required SNR. That is, for a user terminal at the link radius with link coverage reliability (1−Pout), the probability of the effective received SNR being higher than the required threshold γth is no less than
. Note that we concern the link reliability likely of high percentile, such as 90% or even higher, in this paper. For a point-to-point MIMO system, the data stream
with the lowest SNR will dominate the cell coverage performance due to the more likely outage link. As for point-to-multipoint MIMO broadcast system, all stream links represent different individual users so that the cell coverage will be determined by the strongest link.
4. Link Outage and Coverage Analysis 4.1 ZF-DPC without scheduling (or random selection)
In this section, we analyze the baseline performance of ZF-DPC without user selection. To begin with, we first analyze the effective received SNR of individual stream link i (denoted by γi ) with the help of following lemma shown in [3]:
Lemma 1 Let H∈Χr×t have i.i.d. entries
(0,1) ΧΝ : LQ = H 2 , | =| ii i l d 2 1) 2(t−i+ i d : Ξ f
, and let be the i th diagonal
element of in the decomposition
. Then, the random variables
are statistically independent and
, where denotes the central
Chi-squared distribution with degrees of
freedom, whose probability density function
(PDF) is . i i l, 2 2a Ξ /( 1 − − a e za z L ) (z QR 1)! − a 2 =
So that the PDF of effective channel gain is i d , , 1, = )! ( = ) ( t t z i t N i d i N i N e z z f K − − − ….. (10)
and the cumulative distribution function (CDF) of di can be written as dx i N e x z F t x i t N z i d )! ( = ) ( 0 − − −
∫
dx i N e x t x i Nt z ( 1) 1 = + − Γ −∫
∞ − − ) (1−Pout 1) ( ) 1, ( 1 = + − Γ + − Γ − i N z i N t t ),=1−Q(Nt −i+1,z (11)where is the complete
gamma function, is the upper incomplete gamma function and
dt e t x
∫
∞ x− −t Γ 1 0 = ) ( x a Γ( , ) ta e tdt x − − ∞∫
1 = ) ( ) , ( = ) , ( a x a x a Q Γ Γis the regularized gamma function. Note that Γ(n)=(n−1)!
n t N d1≥ i i i d for a positive integer . Form (10), we know that so that the first channel
row vector results in the strongest link.
d d2 ≥K≥
The CDF of i th link's effective receive SNR γ = ρ is ). 1, ( 1 = ) ( = ) ( i t i i d i F Q N i F ρ γ ρ γ γ γ − − + 0 > th (12) Thus, for a given threshold γ , the link outage probability of th link of ZF-DPC is i
} < { = r i th i out P P γ γ ) ( = th i r F γ ). 1, ( 1 = i th t i N Q ρ γ + − − (13)
(13)To derive cell coverage from (13), we first introduce the inverse of the
regularized incomplete gamma function which is shown as follows i ZFDPC R ). , ( = ) , ( =Q a z z Q 1 a x x ⇒ − (14)
By substituting (5) and (14) into (13), the link coverage is given by
μ σ γ 1 1 2 /10 0 ) 1,1 ( 10 = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + − − i out t th t g t i ZFDPC Q N i P N P R (15) for i=1,K,Nt
4. 2 ZFB without scheduling (or random selection)
Alternately, we analyze the baseline performance of ZFB without user selection in which selects users randomly and does not exploit the multiuser diversity gain. In this
case, it is just like there are users with the Rayleigh fading channel vector. From observation on a point-to-point t N K = t N N × t
MIMO system with ZF receiver, the effective channel gain has the same form as the ZF receiver's substream effective channel gain. Due to the same statistics we can see the system as a virtual MIMO system with ZF receiver. According to [22], the substream SNRs { for an i b t t N N × t N i i}=1 γ Nt×Nr
MIMO system with ZF receiver under the equal power allocation principle are i.i.d. Chi-squared distributed random variables with
r
N −Nt
2( +1) degrees of freedom, i.e. 0, )! ( = ) ( / ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − ρ γ ρ γ γρ γ t N r N i t r i i N N e f i i i b , γ ≥ (16) γ ρ where = r t i ) ( = th i r i out F P γ . The PDF of unordered are i.i.d. exponential distribution with parameter one from (16) in the case. Therefore, the link outage probability of th stream link for ZFB is
t N i i b}=1 { N N = ) ( = i th i b F ρ γ ). ( exp 1 = i th ρ γ − − i ZFB R (17)
As a result, the link coverage can be written as follows: . , 1, = 1 1 log 10 = 1 2 /10 0 t i out t g t i ZFB i N P R K μ ⎥ ⎥ ⎦ ⎤ ⎢ ⎡ ⎟⎟ ⎞ ⎜⎜ ⎛ − t N i = (18) th P Nγ σ ⎢⎣ ⎝ ⎠
Note that the link coverage of all ZFB stream links is equal to that of ZF-DPC's weakest link as shown in (15) for under the same link outage probability constraint.
4.3 ZF-DPC with greedy scheduling
strongest stream link which has the largest radius will determines the cell range. In [5], the authors have proposed a greedy scheduling algorithm for selecting Nt out of K users
to form and ordering those selected channel row vectors in the Gram-Schmidt orthogonalization, aiming to maximize the throughput. ) H(Σ (23) , )] , ( [1 = K t z N Q −
Therefore, the CDF of the strongest link's effective receive SNR γ is ~1
The strongest link will be determined by the first selected user's channel row vector. The effective channel gain , where
represents the channel row vector
of th user. Note that is a sum of squared magnitudes of circularly symmetric, zero-mean, unit-variance complex Gaussian random variables. Therefore,
with PDF * 1,k = k k d h h k d1, 1,k d t N k × ∈ 1 Χ h k Nt 2 2Nt Ξ : . 1)! ( = ) ( 1 1, − − − t z t N k d N e z z f (19) ) According to the greedy selection algorithm,
the selected user k* is
. max arg = 1, } , {1, * k K k d k K ∈ (20
Thus, the effective channel gain of the strongest link for greedy scheduling algorithm is 1 ~ d . max = = ~ 1, } , {1, * 1, 1 k K k k d d d K ∈ (21)
From order statistics, the PDF of d~1 is
), ( )] ( [ = ) ( 1, 1 1, 1 ~ z K F z f z f k d K k d d − (22)
and the CDF of d~1 can be written as
K k d d z F z F ( )=[ ( )] 1, 1 ~ , )] = ) ( 1 1 ~ K F ( )=[1 ( , 1 1 ~ t d Q N F ρ γ ρ γ γ γ − (24)
and the link outage probability is . )] , ( [1 = ~ 1 1 th K t out Q N P ρ γ − 1 (25) ~
To derive link coverage RZFDPC of the strongest link from (25), we use again the inverse of the regularized incomplete gamma function and get the following closed form:
. ) ] ~ [ ,1 ( 10 = 1 1 1 1 2 /10 0 μ σ γ ⎥⎥⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − K out t th t g t ZFDPC Q N P N P ~1 R 2 = th (26) 5 Numerical Results
In this section, we present numerical examples to illustrate achievable link outage and link coverage performances of both ZF-DPC and ZFB in multiuser MIMO broadcast systems. For ZFB with optimal user selection policy, exhaustive users search, the explicit performance closed-form is difficult to be found due to an optimization problem involved in itself. Therefore, we use simulation results to show the performance of strongest stream link for ZFB with exhaustive search. We first assume a predetermined value
γ dB, noisepower=−103 dBm, 32
=
0 −
g , 3μ= and . Figure 1 shows the simulative and analytical link outage performances of both ZF-DPC and ZFB without scheduling for user terminals at distance 3 = t N 1 =
R km from the base station. Similar to the analytic results shown before, the link outage of all ZFB stream links performs equally to that of ZF-DPC's weakest link under certain radius. Especially, we can find there is a diversity-like performance behavior between stream links of ZF-DPC.
