穩健性多用戶波束成形設計: 近似方法、性能分析與分散式最佳化

行政院國家科學委員會專題研究計畫 期末報告

穩健性多用戶波束成形設計: 近似方法、性能分析與分散 式最佳化

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 101-2218-E-011-043-

執 行 期 間 ： 101 年 11 月 01 日至 102 年 09 月 30 日 執 行 單 位 ： 國立臺灣科技大學電子工程系

計 畫 主 持 人 ： 張縱輝

計畫參與人員： 碩士班研究生-兼任助理人員：曾義恆

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

公 開 資 訊 ： 本計畫涉及專利或其他智慧財產權，1 年後可公開查詢

中 華 民 國 102 年 10 月 28 日

中 文 摘 要 ： 在此計畫中，我們已針對機率約束的穩健性波束成形設計進 行了相關研究。特別是，我們考量所謂的干擾通道

(interference channel)，其中多個傳送端（如基地台）同 時與多個接收端在同一頻帶上進行通訊。所考量的環境為傳 送端擁有多個天線，且使用波束成形技術以傳送資料。然 而，傳送端只知曉接收端的通道分佈資訊(channel

distribution information)。針對此環境，我們考量在傳送 端個別功率約束下能提供接收端中斷條件約束(outage constrained)服務品質的穩健性波束成形設計。針對此問 題，我們提出基於代價之連續凸面最佳化方法（pricing based successive convex optimization），並且同時以理 論分析與電腦模擬評估所提方法的效能。此研究為與國立清 華大學之李威錆同學、林澤教授與祁忠勇教授的共同合作成 果。部分結果已投稿 IEEE ICASSP 2013。

中文關鍵詞： 干擾通道，波束成形，穩健設計，機率約束、代價賽局、凸 面最佳化

英 文 摘 要 ： 英文關鍵詞：

1

行政院國家科學委員會補助專題研究計畫 ■期中進度報告

□期末報告

（計畫名稱）

穩健性多用戶波束成形設計: 近似方法、性能分析與分散式最佳化

計畫類別：■個別型計畫 □整合型計畫 計畫編號：NSC 101-2218-E-011 -043 -

執行期間：101 年 11 月 1 日至 102 年 09 月 30 日 執行機構及系所：國立臺灣科技大學

計畫主持人：張縱輝 共同主持人：

計畫參與人員：曾義恆（碩士班研究生）

本計畫除繳交成果報告外，另須繳交以下出國報告：

□赴國外移地研究心得報告

□赴大陸地區移地研究心得報告

□出席國際學術會議心得報告及發表之論文

□國際合作研究計畫國外研究報告

處理方式：除列管計畫及下列情形者外，得立即公開查詢

□涉及專利或其他智慧財產權，□一年■二年後可公開查詢 中 華 民 國 101 年 12 月 27 日

附件一

2

國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價 值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性） 、是否適 合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□ 未達成目標（請說明，以 100 字為限）

□ 實驗失敗

□ 因故實驗中斷

□ 其他原因 說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：□已發表 □未發表之文稿 ■撰寫中 □無 專利：□已獲得 □申請中 □無

技轉：□已技轉 □洽談中 □無

其他：（以 100 字為限） 本計畫主要研究成果的一部分目前已投稿 IEEE ICASSP, 目前正整理所得結果投搞期刊論文。

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價 值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以 500 字為限）

本計畫已分別考慮單一基地台多用戶與多基地台多用戶環境下的穩健性多用 戶波束成形設計問題，同時提出相對應的近似演算法，與性能分析。特別是 針對干擾通道中波束成形設計問題，我們為相關領域中第一個考量非理想通 道下機率條件約束的模型，同時提出有效的近似方法與低複雜度實現方法。

本計畫所得之研究成果多為在學理上的創新，相信相關研究成果對未來通訊 產業發展能提供相關問題的有效解決方法，同時也對其它的擴展性問題(如認 知無線電，隨意網路等)提供有關的設計思想與最佳化工具。

附件二

3

英文摘要: In this project, we have conducted several researches regarding the chance constrained robust beamforming designs. Specifically, we have considered the interference channel where multiple transmitters (e.g., base stations) communicate with multiple receivers over a common frequency band. The considered scenario is that the transmitters have multiple antennae and transmit data by beamforming. However, the transmitters have only channel distribution information of the receivers. Under such circumstances, the proposed design is to provide an outage constrained quality of service to the receivers, subject to individual power constraints. Efficient approximation methods based on pricing based successive convex optimization are proposed and evaluated analytically and by computer simulations. This research is done with collaboration with Mr. Wei-Chiang Li, Prof. Che Lin and Prof. Chong-Yung Chi at National Tsing Hua University. Part of our research results have been submitted to IEEE ICASSP 2013.

Key Words: Interference channel, beamforming, robust design, probability constraints, pricing game, convex

optimization

中文摘要: 在此計畫中，我們已針對機率約束的穩健性波束成形設計進行了相關研究。特別是，我們 考量所謂的干擾通道(interference channel)，其中多個傳送端（如基地台）同時與多個接收端在同一頻 帶上進行通訊。所考量的環境為傳送端擁有多個天線，且使用波束成形技術以傳送資料。然而，傳送 端只知曉接收端的通道分佈資訊(channel distribution information)。針對此環境，我們考量在傳送端個別 功率約束下能提供接收端中斷條件約束(outage constrained)服務品質的穩健性波束成形設計。針對此問 題，我們提出基於代價之連續凸面最佳化方法（pricing based successive convex optimization），並且同 時以理論分析與電腦模擬評估所提方法的效能。此研究為與國立清華大學之李威錆同學、林澤教授與 祁忠勇教授的共同合作成果。部分結果已投稿 IEEE ICASSP 2013。

關鍵詞：干擾通道，波束成形，穩健設計，機率約束、代價賽局、凸面最佳化

主要研究成果：Outage Constrained Weighted Sum Rate Maximization for Multiuser MISO Interference Channel by Pricing- based Optimization (Submitted to ICASSP 2013)

This paper considers beamforming designs for weighted sum rate maximization (WSRM) in a multiple-input single-output interference channel subject to probability constraints on the rate outage. We claim that the outage probability constrained WSRM problem is an NP-hard problem, and therefore focus on devising efficient approximation methods. In particular, inspired by an insightful problem reformulation, a pricing-based sequential optimization (PSO) algorithm is proposed for efficiently handling the considered outage constrained WSRM problem. We show that the proposed PSO algorithm has semi-analytical beamforming solutions in each iteration, and hence can be efficiently implemented. Moreover, the PSO algorithm upon convergence can reach a Karush-Kuhn-Tucker (KKT) point of the original outage constrained problem. Simulation results to be presented show that the proposed PSO algorithm can yield promising sum rate performance and is computationally more efficient than the existing methods.

詳細研究目的、文獻探討、研究方法、結果與討論請見附檔之論文。

OUTAGE CONSTRAINED WEIGHTED SUM RATE MAXIMIZATION FOR MISO INTERFERENCE CHANNEL BY PRICING-BASED OPTIMIZATION

Wei-Chiang Li^?, Tsung-Hui Chang^†, Che Lin^?, and Chong-Yung Chi^?

? Institute of Communications Engineering & Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan 30013

†Department of Electronic and Computer Engineering

National Taiwan University of Science and Technology, Taipei, Taiwan 10607

E-mail:{weichiangli, tsunghui.chang}@gmail.com, {clin, cychi}@ee.nthu.edu.tw

ABSTRACT

This paper considers beamforming designs for weighted sum rate maximization (WSRM) in a multiple-input single-output interfer- ence channel subject to probability constraints on the rate outage.

We claim that the outage probability constrained WSRM problem is an NP-hard problem, and therefore focus on devising efficient ap- proximation methods. In particular, inspired by an insightful prob- lem reformulation, a pricing-based sequential optimization (PSO) algorithm is proposed for efficiently handling the considered out- age constrained WSRM problem. We show that the proposed PSO algorithm has semi-analytical beamforming solutions in each itera- tion, and hence can be efficiently implemented. Moreover, the PSO algorithm upon convergence can reach a point satisfying Karush- Kuhn-Tucker (KKT) conditions of the original outage constrained problem. Simulation results show that the proposed PSO algorithm not only can yield competing weighted sum rate performance, but also is computationally more efficient than the existing method [1].

Index Terms— Interference channel, weighted sum rate maxi- mization, outage probability, transmit beamforming

1. INTRODUCTION

Inter-cell cooperation has been recognized essential to improving the spectral efficiency of wireless cellular networks [2]. Consider a multiple-input single-output interference channel (MISO IFC) [3]

whereK multi-antenna transmitters simultaneously communicate withK single-antenna receivers over a common frequency band.

When instantaneous channel state information (CSI) is available at the transmitters, it has been shown that transmit beamforming is a Pareto optimal transmission strategy for the MISO IFC [3, 4].

However, practically finding such optimal beamformers is a diffi- cult task. In fact, it has been shown that beamforming design prob- lems for maximizing a class of commonly used rate utilities, (e.g., the weighted sum rate) are NP-hard in general [5]. Consequently, many research efforts have been devoted to investigate suboptimal but computationally efficient approximation algorithms [5, 6].

Considering that it is not always feasible to obtain instantaneous CSI, especially in fast fading scenarios, there are parallel works that study the MISO IFC with only channel distribution information (CDI) available at the transmitters [7, 8]. For example, assuming that each MISO channel is (circularly symmetric) complex Gaussian distributed, the authors in [7] characterized the structure of Pareto optimal beamformers for an ergodic achievable rate region. The This work is supported by National Science Council, R.O.C., under Grant NSC-99-2221-E-007-052-MY3 and Grant NSC 101-2218-E-011-043.

authors of [8, 9] instead considered an outage constrained scenario where the probability of the rate outage is constrained to be no larger than a predefined, usually small value. In particular, the works in [8, 9] studied the outage constrained achievable rate region for a two-user MISO IFC, and presented a numerical method for attaining the Pareto boundary. This method, however, has a complexity that increases exponentially with the number of users.

In this paper, we assume that only CDI is available at the trans- mitters, and study the beamforming design problem for weighted sum rate maximization (WSRM) under outage probability con- straints. The goal is to develop efficient algorithms for obtaining the outage constrained optimal beamformers. However, our complexity analysis shows that such outage constrained WSRM problem is intricate – it is NP-hard in general. We thereby focus on devising ef- ficient approximation methods. In particular, by carefully inspecting the constraint structure, we reformulate the original outage con- strained problem as a form that is analogous to the WSRM problem with instantaneous CSI in [5]. This intriguing connection inspires us to propose a pricing-based sequential optimization (PSO) algo- rithm [10] for efficiently handling the considered outage constrained WSRM problem. We show that the proposed PSO algorithm can improve the system sum rate from iteration to iteration, and, when upon convergence, can reach a Karush-Kuhn-Tucker (KKT) point of the original problem. Moreover, the subproblem involved in each iteration has semi-analytical solutions which can be implemented efficiently. The presented simulation results show that the PSO algo- rithm is computationally more efficient than the previously proposed distributed SCA (DSCA) algorithm in [1], though both methods can yield almost the same sum rate performance.

2. SIGNAL MODEL AND PROBLEM STATEMENT We consider a MISO IFC consisting ofK pairs of multiple-antenna transmitters and single-antenna receivers. Each transmitter is equipped withNt antennae, and communicates with its intended receiver using transmit beamforming. The transmit signal from transmitteri is given by wisi, wheresi∼ CN (0, 1) is the informa- tion signal for receiveri, and wi∈ C^Ntis the associated beamform- ing vector, fori = 1, . . . , K. Let hik ∈ C^Nt denote the channel vector between transmitteri and receiver k, for all i, k = 1, . . . , K.

Here, we assume that each hik ∼ CN (0, Qik) with Qik  0 (positive semidefinite) denoting the channel covariance matrix. The received signal at receiveri is given by

xi= h^H_iiw_isi+

K

X

k=1,k6=i

h^H_kiw_ksk+ ni, (1)

whereni∼ CN (0, σ²_i) is the additive noise at receiver i with vari- anceσ_i² > 0. Assume that each receiver i decodes the informa- tion signalsi by single user detection, i.e., treating the cross-link interference as noise. Then, the instantaneous achievable rate (in nats/sec/Hz) of theith transmitter-receiver pair is given by

ri

{hki}^K_k=1, {wk}^K_k=1

= log 1 +  h^H_iiw_i



2

P

k6=i|h^H_kiw_k|²+ σ²_i

! .

In this paper, we consider a scenario where the transmitters have knowledge of CDI only, i.e., the channel covariance matrices Qik, i, k = 1, . . . , K. Under such circumstances, given a transmission rateRi for the ith transmitter-receiver pair, receiver i may suffer from the rate outage, i.e., ri({hki}^K_k=1, {wk}^K_k=1) < Ri. Our goal is to provide quality of service guaranteed within a tolerable outage probability for the receivers, while maximizing the system throughput (i.e., the sum rate) at the same time. Specifically, given

i ∈ (0, 1) as the maximum tolerable outage probability for each receiveri, we consider the following beamforming design problem [1, 11]

max

w_i∈CNt,Ri≥0, i=1,...,K

K

X

i=1

αiRi (2a)

s.t.Probn

ri({hki}^Kk=1, {wk}^Kk=1) < Ri

o≤ i, (2b) kwik²2≤ Pi, i = 1, . . . , K, (2c) whereα1, . . . , αK > 0 are priority weights, and P1, . . . , PK >

0 are the power constraints. Notice that, in (2b), the rate outage probabilities are constrained no higher thani, for i = 1, . . . , K.

3. THE PROBLEM NATURE

It has been shown in [1] that each of the outage constraints in (2b) has an equivalent closed-form expression, given by

ρie

(2Ri −1)σ2i wHi QiiwiY

k6=i



1+(2^Ri− 1)w^H_kQ_kiw_k w^H

i Qiiw_i



≤ 1, (3)

whereρi , 1 − i, i = 1, . . . , K. As one can see, the outage constraint in (3) has a non-convex, complicated structure, and thus solving (2) seems to be a challenging task. Therefore, a fundamental question is whether the outage constrained problem (2) is truly a dif- ficult problem in terms of computational complexity. The following theorem gives the answer.

Theorem 1 Problem (2) is NP-hard in general.

In fact, one can show that problem (2) is NP-hard even whenNt= 1, i.e., when only the transmit powers (without beamforming direction) are optimized. The proof is to construct a polynomial time transfor- mation from the Max-Cut problem, which is known to be NP-hard [12], to problem (2), thereby implying the NP-hardness of the latter problem. Due to space limit, we leave the detailed proof to our fu- ture publication. It is worthwhile to note here that Theorem 1 can be regarded as an outage constrained counterpart of the complex- ity analysis result in [5, 13], where it was shown that the weighted sum rate maximization (WSRM) problem with instantaneous chan- nel state information (CSI) is NP-hard.

Theorem 1 implies that it is unlikely to globally solve prob- lem (2) in a polynomial-time complexity. Therefore, one may have

to consider approximation methods, in order to deal with instances wherein a large number of transmitter-receiver pairs exist. In the next section, we propose a pricing-based sequential (block coordi- nate) optimization method for efficiently handling the considered problem (2).

4. PROPOSED PRICING-BASED ALGORITHM 4.1. Equivalent Formulation

Due to the complex constraint structure in (3), it is not easy to apply general approximation methods to problem (2) (with (2b) replaced by (3)) in its current form. In view of this, we first present in this subsection an alternative formulation for problem (2) which, as one will see, can reveal useful insights for efficient approximation.

To this end, let us define Φi(x|{wk}k6=i), ρie^σi2xY

k6=i

(1 + (w_k^HQ_kiw_k)· x). (4)

Then, (3) can be written asΦi((2^Ri− 1)/wi^HQiiw_i|{wk}k6=i)≤ 1, i = 1, . . . , K. Furthermore, because bothPK

i=1αiRiandΦiare strictly increasing in(R1, . . . , RK), it must be true that (3) holds with equality for problem (2), i.e.,

Φi

 2^Ri− 1 w^H

i Qiiw_i    

{wk}k6=i



= 1, i = 1, . . . , K, (5) On the other hand, eachΦi(x|{wk}k6=i) is strictly increasing in x;

therefore, there exists a unique positive valueξi({wk}k6=i) such that Φi(ξi({wk}k6=i)|{wk}k6=i) = 1. As a result, constraint (5) holds if and only if

2^Ri−1 w^H

iQiiw_i= ξi({wk}k6=i), i = 1, . . . , K. (6) By (6), problem (2) can be concisely expressed as

max

w_i∈CNt i=1,...,K

K

X

i=1

αilog(1 + ξi({wk}k6=i)wi^HQiiw_i) (7a)

s.t.kwik²≤ Pi, i = 1, . . . , K, (7b) There are two interesting observations about (7). Firstly, one can observe that eachξi({wk}k6=i), though being implicit, characterizes the impact of cross-link interference plus noise on receiveri, for i = 1, . . . , K. Secondly, by comparing (7) with its instantaneous- CSI counterpart in [5, Eqn. (3)], an intriguing analogy between the two problems in mathematical formulation is observed. This motivates the use of a pricing-based sequential optimization (PSO) method, which was used in [10, 14] for the instantaneous-CSI case [5, Eqn. (3)], for handling the outage constrained problem (2) in the subsequent two subsections.

4.2. Pricing-based Sequential Optimization Algorithm

The proposed PSO algorithm for problem (2) is an iterative algo- rithm which optimizes the beamforming vectors w1, . . . , wK in a round-robin fashion. Specifically, in an iteration for optimizing wi, given a set ofw¯₁, . . . , ¯w_K (that are feasible to (7b)), we seek to updatew¯_iby solving the following problem

max

w_i

log(1 + ξi({ ¯w_k}k6=i)w^Hi Qiiw_i)−X

k6=i

πikw^H_i Q_ikw_i

s.t. kwik²≤Pi, (8)

whereξi({ ¯w_k}k6=i) is the unique solution to Φi(x|{ ¯w_k}k6=i) = 1¹. The terms−πikw^H_i Qikw_i,k 6= i, respectively weighted by the unit prices{πik}k6=i, in the objective function imply that the throughput of useri is maximized at the cost of the interference induced by transmitteri to the other receivers. For notational simplicity, let us denote

Iik, w^Hi Q_ikw_i ( ¯Iik, ¯w_i^HQ_ikw¯_i) for alli, k = 1, . . . , K. Moreover, define

U ({Ij`}<sup>K</sup>j,`=1),

K

X

i=1

αilog(1 + ξi({Iki}k6=i)Iii) (9)

as an alternative expression of the objective function of (7), where ξi({Iki}k6=i) , ξi({wk}k6=i) (by (4), (5) and (6)) for all i = 1, . . . , K. According to [10, 14], the unit prices are given by

πik= −1 αi

∂U ({Ij`}<sup>K</sup><sub>j,`=1</sub>)

∂Iik

    _I

j`= ¯I<sub>j`</sub>∀j,`

(10)

for allk 6= i. Specifically, by (9) and by applying the implicit func- tion theorem [15] for computing the gradient ofξk({Ijk}j6=k) with respect toIik, one can show thatπikhas an explicit form as πik=αk

αi

I¯kkξk({ ¯Ijk}j6=k) 1 + ¯Ikkξk({ ¯Ijk}j6=k)



σ²k+X

`6=i,k

I¯`k

1 + ¯I`kξk({ ¯Ijk}j6=k)



× 1+ ¯Iikξk({ ¯Ijk}j6=k)
+ ¯Iik

−1

. (11)

The optimization steps of the proposed PSO algorithm for handling problem (7) (i.e., problem (2)) is presented in Algorithm 1.

Algorithm 1 Proposed PSO algorithm for problem (7) 1: Given an initial set ofw¯₁, . . . , ¯w_Ksatisfying (7b);

2: Set ¯Iik := w¯^H_i Q_ikw¯_i ∀i, k = 1, . . . , K, and compute ξk({ ¯Ijk}j6=k), k = 1, . . . , K, by bisection;

3: repeat

4: fori = 1, . . . , K do

5: Compute the unit prices{πik}k6=iby (11);

6: Solve problem (8) by Proposition 2 below to obtain an optimal solution w^?_i, followed by updatingw¯_iwith w^?_i; 7: Update ¯Iik = ¯w^H_i Qikw¯_i,k = 1, . . . , K, and compute

ξk({ ¯Ijk}j6=k), k = 1, . . . , K;

8: end for

9: until the predefined stopping criterion is met.

10: Output( ¯w₁, . . . , ¯w_K) as an approximate solution to (7).

While Algorithm 1 seems to be a straightforward application of the pricing-based method in [10, 14] to problem (7), it is actually not obvious to see whether Algorithm 1 can reach any interesting point of problem (7). This is mainly becauseξ1({Ik1}k6=1), . . . , ξK({IkK}k6=K) are implicit. To answer the above question, let us analyze the relation between problem (8) and the original problem (7). The following lemma is needed in the subsequent analysis.

Lemma 1 For eachi ∈ {1, . . . , K} and k ∈ {1, . . . , K}\{i}, the individual ratelog(1 + ξk({Ijk}j6=k)Ikk) is strictly convex in Iik≥ 0 for any given {Ijk≥ 0}j6=i.

1Since Φi(x|{ ¯w_k}k6=i) in (4) is strictly increasing in x, and Φi(ξi({ ¯w_k}_k6=i)|{ ¯w_k}_k6=i) = 1. The value ξi({ ¯w_k}_k6=i) can easily be obtained via a bisection method.

The proof of Lemma 1 is presented in the Appendix. By Lemma 1, (10), and by the first-order condition of convex functions, we have

αklog 1 + ξk({ ¯Ijk}j6=k,i, Iik) ¯Ikk

≥ αklog(1 + ξk({ ¯Ijk}j6=k,i, ¯Iik) ¯Ikk)− αiπik(Iik− ¯Iik), for allk ∈ {1, . . . , K}\{i}. Hence, it follows from (9) and the above inequality that

U (Ii1, . . . , IiK, { ¯Ij1, . . . , ¯IjK}j6=i)

≥ αilog(1 + ξi({ ¯Iki}k6=i)Iii)− αi

X

k6=i

πik(Iik− ¯Iik)

+X

k6=i

αklog(1 + ξk({ ¯Ijk}j6=k,i, ¯Iik) ¯Ikk) (12) , U_LB⁽ⁱ⁾({Ii`}<sup>K</sup>`=1

{ ¯Ij1, . . . , ¯IjK}j6=i).

Since the sum of the first two terms on the right hand side of (12) is proportional to the objective function in (8), optimizing problem (8) for useri is equivalent to maximizing the lower bound U_LB⁽ⁱ⁾.

More importantly, one can check thatU_LB⁽ⁱ⁾is locally tight, in the sense that

U_LB⁽ⁱ⁾({ ¯Ii`}<sup>K</sup>`=1



{ ¯Ij1, . . . , ¯IjK}j6=i) = U ({ ¯Ij`}<sup>K</sup>j,`=1), (13) fori = 1, . . . , K. Therefore, using an argument similar to [14, Lemma 1], one can show that the weighted sum rateU ({ ¯Ij`}<sup>K</sup><sub>j,`=1</sub>) achieved byw¯₁, . . . , ¯w_Kin Algorithm 1 would be non-decreasing from one iteration to another. This, together with the fact that U ({ ¯Ij`}<sup>K</sup>j,`=1) is bounded due to the power constraints (7b), implies that U ({ ¯Ij`}<sup>K</sup>j,`=1) eventually will converge. Besides, U_LB⁽ⁱ⁾ has locally tight gradients, i.e.,

∂U_LB⁽ⁱ⁾({ ¯Ii`}<sup>K</sup><sub>`=1</sub>

{ ¯Ij1, . . . , ¯IjK}j6=i)

∂Iik

=∂U ({ ¯Ij`}<sup>K</sup><sub>j,`=1</sub>)

∂Iik

, (14) for allk, i = 1, . . . , K. This property can be exploited to show that Algorithm 1, upon the convergence of( ¯w₁, . . . , ¯w_K), attains a KKT point of problem (7). The detailed derivations are omitted here due to space limit. We summarize the above analyses in the following proposition.

Proposition 1 The weighted sum rateU ({ ¯Ij`}<sup>K</sup>j,`=1) achieved in each iteration of Algorithm 1 converges monotonically. Moreover, any convergent point of( ¯w₁, . . . , ¯w_K) is a KKT point of (7).

4.3. Efficient Implementation of PSO Algorithm

An important computational aspect of Algorithm 1 lies in how to ef- ficiently solve problem (8), which is not convex. A similar problem was studied in [10] for the WSRM problem with instantaneous CSI;

however, the approach to handling such problem is via linear ap- proximation, and hence is suboptimal. We herein provide the global optimal solution to problem (8) in a semi-analytical form.

The proposed approach is based on the popular semidefinite re- laxation (SDR) technique [16]. In particular, we relax the rank-one matrix wiw^H_i to a rank-unconstrained positive semidefinite matrix Wi 0, and consider the following convex problem

max

W_i0 log 1+ξi({ ¯Iki}k6=i)Tr(WiQii)
−Tr Wi

X

k6=i

πikQik

s.t.Tr(Wi)≤Pi. (15)

A key finding is that the SDR problem (15) has a rank-one optimal solution, as stated in the following proposition.

Proposition 2 Letµ^?_i ≥ 0 denote the optimal dual variable asso- ciated with the power constraint in (15), and let Ri = µ^?_iINt + P

k6=iπikQik, where INtis anNt× Ntidentity matrix. Then, there exists a principal eigenvector of R^−1/2_i QiiR^−1/2_i , denoted by ν^?_i, such that W^?_i = w^?_i(w^?_i)^His optimal to problem (15), where

w^?_i =pp^?_iR^−1/2_i ν_i^? (optimal to problem (8)) p^?_i = max



1− 1

λ^?_iξi({ ¯Iki}k6=i), 0

 ,

andλ^?_i is the maximum eigenvalue of R^−1/2_i QiiR^−1/2_i .

The proof of Proposition 2 is omitted here due to space limit.

It can be shown thatµ^?_i can be computed by simple bisection, and thus w^?_i can be obtained efficiently. More specifically, since the major computation load of computing w^?_i lies in matrix inversion and eigenvalue decomposition each having a complexity order of O(Nt³), the complexity order of solving (8) is roughly given by O(Nt³log(1/ε1)), where ε1 > 0 is the solution accuracy of the bisection search forµ^?_i. As a result, the overall complexity order of Algorithm 1 isκ1KO(N_t³log(1/ε1)), where κ1denotes the to- tal number of round-robin iterations (steps 3 to 8 in Algorithm 1).

Note that, for the DSCA algorithm in [1], the subproblem for each transmitter is implemented by interior-point methods. Hence, the DSCA algorithm has an overall complexity order ofκ2KO((N_t^6.5+ K^3.5) log((Nt+ K)/ε2)) where κ2is the total number of round- robin iterations andε2> 0 is the solution accuracy of interior-point methods [17]. One can see that the complexity order of PSO algo- rithm is lower than that of the DSCA algorithm. Thus, it is expected that the PSO algorithm is computationally more efficient than the DSCA algorithm, which will be verified by the simulation results.

5. SIMULATION RESULTS AND DISCUSSIONS For simplicity, it is set thatσ²1 =· · · = σK² , σ²andP1 =· · · = PK = 1. The tolerable outage probability is set to 10%, i.e., 1 =

· · · = K = 0.1. The channel covariance matrices {Qik}^K_i,k=1are randomly generated with full rank, and the maximal eigenvalues of {Qik}^Ki,k=1are normalized toλmax(Qii) = 1 and λmax(Qik) = η for allk 6= i; therefore, 0 ≤ η ≤ 1 reflects the strength of cross- link channels. Algorithm 1 stops when the difference between the weighted sum ratesU ({ ¯Ij,`}<sup>K</sup><sub>j,`=1</sub>) of two consecutive round-robin iterations is no larger than0.1% of that in the previous iteration. All simulation results are averaged over 500 realizations of{Qik}^Ki,k=1. We first examine the efficacy of the PSO algorithm (Algorithm 1) by comparing it with the DSCA algorithm in [1] and the naive maximum-ratio transmission (MRT) strategy. Figure 1 shows the average weighted sum rate versus1/σ²forK = 4 and Nt = 4. It can be seen that the PSO algorithm and the DSCA algorithm yield almost the same weighted sum rate performance, and they outper- form the MRT strategy.

In Fig. 2, we compare the average computation time (in seconds) of the PSO algorithm and the DSCA algorithm versus the number of usersK, for 1/σ² = 10 dB, η = 0.5, and Nt = 4 and 8. The sub- problems involved in the DSCA algorithm are handled by CVX [18].

Note that the computation time of the PSO algorithm increases al- most linearly withK, whereas that of the DSCA algorithm increases much faster withK. According to Fig. 2, the PSO algorithm is about 10³times faster than the DSCA algorithm.

6. APPENDIX: PROOF OF LEMMA 1

We show that∂ log(1 + ξk({Ijk}j6=k)Ikk)/∂Iikis strictly increas- ing inIik≥ 0, which implies that log(1+ξk({Ijk}j6=k)Ikk) is con-

0 5 10 15 20

0 0.5 1 1.5

1/σ² (dB) Average Weighted Sum Rate (bits/sec/Hz)

PSO Algorithm, η=0.2 PSO Algorithm, η=1.0 DSCA Algorithm, η=0.2 DSCA Algorithm, η=1.0 MRT, η=0.2

MRT, η=1.0

Fig. 1. Average achievable sum rate versus1/σ²forK = Nt= 4, andrank(Qki) = 4 for all k, i.

2 4 6

0 100 200 300 400 500 600 700

K

Average Time Consumption (secs)

DSCA, Nt=4 DSCA, N_t=8

2 4 6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

K

Average Time Consumption (secs)

PSO, Nt=4 PSO, N_t=8

Fig. 2. Average computation time of the PSO algorithm and the DSCA algorithm versusK, for Nt = 4, 8, 1/σ² = 10 dB, and η = 0.5.

vex inIik ≥ 0 [17], using the two properties of ξk({Ijk}j6=k): 1) ξk({Ijk}j6=k) is strictly decreasing in Iik≥ 0; 2) Iik·ξk({Ijk}j6=k) is strictly increasing inIik≥ 0.

To prove the first property, observe from (4) thatΦk(x|{Ijk}j6=k) is strictly increasing inx > 0 and Iik ≥ 0. Thus, the function ξk({Ijk}j6=k), which satisfies Φk(ξk({Ijk}j6=k)|{Ijk}j6=i) = 1 uniquely, is strictly decreasing inIik≥ 0. To show the second prop- erty, suppose thatI_ik⁰ < I_ik⁰⁰, and defineξ_k⁰ = ξk({Ijk}j6=i,k, I_ik⁰ ) andξ⁰⁰_k = ξk({Ijk}j6=i,k, I_ik⁰⁰). By the definition of ξk({Ijk}j6=k), we haveΦk(ξ⁰_k|{Ijk}j6=i,k, I_ik⁰ ) = Φk(ξ_k⁰⁰|{Ijk}j6=i,k, I_ik⁰⁰) = 1.

Moreover,ξ⁰_k > ξ⁰⁰_k by the first property. Therefore, the following chain holds

1 =ρkexp(σ²_kξ⁰_k)(1 + I_ik⁰ ξ_k⁰) Y

`6=i,k

(1 + I`kξ⁰_k) (by (4))

=ρkexp(σ²_kξ⁰⁰_k)(1 + I_ik⁰⁰ξ_k⁰⁰) Y

`6=i,k

(1 + I`kξ⁰⁰_k)

<ρkexp(σ²_kξ⁰_k)(1 + I_ik⁰⁰ξ_k⁰⁰) Y

`6=i,k

(1 + I`kξ⁰_k)

which impliesI_ik⁰ ξ⁰_k< I_ik⁰⁰ξ⁰⁰_k. By these two properties, and by (11) and the fact of∂ log(1 + ξk({Ijk}j6=k)Ikk)/∂Iik =−_α^αi

kπik(see (9) and (10)), one can verify that∂ log(1 + ξk({Ijk}j6=k)Ikk)/∂Iik

is strictly increasing inIik≥ 0. This completes the proof. 

7. REFERENCES

[1] W.-C. Li, T.-H. Chang, C. Lin, and C.-Y. Chi, “Coordinated beamform- ing for multiuser MISO interference channel under rate outage con- straints,” to appear in IEEE Trans. Signal Process.

[2] E. G. Larsson and E. A. Jorswieck, “Competition versus cooperation on the MISO interference channel,” IEEE J. Sel. Areas Commun., vol. 26, pp. 1059–1069, Sep. 2008.

[3] E. A. Jorswieck, E. G. Larsson, and D. Danev, “Complete characteriza- tion of the Pareto boundary for the MISO interference channel,” IEEE Trans. Signal Process., vol. 56, pp. 5292–5296, July 2008.

[4] X. Shang, B. Chen, and H. V. Poor, “Multiuser MISO interference channels with single-user detection: Optimality of beamforming and the achievable rate region,” IEEE Trans. Inf. Theory, vol. 57, pp. 4255–

4273, July 2011.

[5] Y.-F. Liu and Z.-Q. Luo, “Coordinated beamforming for MISO inter- ference channel: Complexity analysis and efficient algorithms,” IEEE Trans. Signal Process., vol. 59, pp. 1142–1157, Mar. 2011.

[6] R. Zhang and S. Cui, “Cooperative interference management with MISO beamforming,” IEEE Trans. Signal Process., vol. 58, pp. 5450–

5458, Oct. 2010.

[7] E. Bj¨ornson, R. Zakhour, D. Gesbert, and B. Ottersten, “Cooperative multicell precoding: Rate region characterization and distributed strate- gies with instantaneous and statistical CSI,” IEEE Trans. Signal Pro- cess., vol. 58, pp. 4298–4310, Aug. 2010.

[8] J. Lindblom, E. Karipidis, and E. G. Larsson, “Outage rate regions for the MISO IFC,” in Proc. Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 1-4, 2009, pp. 1120–1124.

[9] ——, “Outage rate regions for the MISO interference channel: Defini- tions and interpretations,” http://arxiv.org/abs/1106.5615v1.

[10] D. A. Schmidt, C. Shi, R. A. Berry, M. L. Honig, and W. Utschick,

“Distributed resource allocation schemes: Pricing algorithms for power control and beamformer design in interference networks,” IEEE Signal Process. Mag., vol. 26, pp. 53–63, Sep. 2009.

[11] J. Lindblom, E. G. Larsson, and E. A. Jorswieck, “Parameterization of the MISO interference channel with transmit beamforming and par- tial channel state information,” in Proc. Asilomar Conference on Sig- nals, Systems and Computers, Pacific Grove, CA, Oct. 26-29, 2008, pp.

1103–1107.

[12] R. M. Karp, “Reducibility among combinatorial problems,” in 50 Years of Integer Programming 1958-2008, M. J ¨unger, T. M. Liebling, D. Nad- def, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, and L. A. Wolsey, Eds. Springer Berlin Heidelberg, 2010, ch. 8, pp. 219–

241.

[13] Z.-Q. Luo and S. Zhang, “Dynamic spectrum management: Complex- ity and duality,” IEEE J. Sel. Topics Signal Process., vol. 2, pp. 57–73, Feb. 2008.

[14] C. Shi, R. A. Berry, and M. L. Honig, “Monotonic convergence of dis- tributed interference pricing in wireless networks,” in Proc. IEEE ISIT, Seoul, Korea, June 28-July 3, 2009, pp. 1619–1623.

[15] S. G. Krantz and H. R. Parks, The Implicit Function Theorem: History, Theory, and Applications. Boston, MA: Birkh¨auser, 2002.

[16] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Process.

Mag., vol. 27, pp. 20 –34, May 2010.

[17] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK:

Cambridge University Press, 2004.

[18] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 1.21,” http://cvxr.com/cvx, Apr. 2011.

1

國科會補助專題研究計畫項下出席國際學術會議心得報告

日期：2013 年 6 月 3 日

一、參加會議經過

筆者於 102 年 5 月 24 日(Friday)從桃園國際機場搭長榮班機 BR01 出發，於當地時間 5 月 24 日(Friday) 晚間 8:30 抵達 Vancouver 機場, 並且進住 YWCA Hotel. 以下為我參與會議的主要經過。

5/25-5/26 (Saturday and Sunday)：筆者稍微遊覽 Vancover 的城區與了解到達會議地點的路徑，及

其確切位置。其餘時間皆於旅館內休憩，調整時差。

5/27 (Monday)：筆者一早即到會議會場完成報到手續。相遇來自台灣、大陸及美國等地的學者朋

友，一行人前往 University of British Columbia 參訪並且參加了知名訊號處理領域教授 Prof. Yonina C. Eldar 在當地的 seminar talk: Defying Nyquist in Analog to Digital Conversion。

當日晚間筆者參與大會舉辦的歡迎晚宴。

5/28 (Tuesday)：Participated in

(a) 09:50 - 10:40：Plenary: PLEN1: Recent Developments in Deep Neural Networks (Geoffrey Hinton)

計畫編號 NSC 101-2218-E-011 -043

計畫名稱 穩健性多用戶波束成形設計: 近似方法、性能分析與分散式最佳化

出國人員

姓名 張縱輝 服務機構

及職稱

國立台灣科技大學電子工程系 助理教授

會議時間

2013 年 5 月 26 日 至

2013 年 5 月 31 日

會議地點 加拿大 溫哥華 (Vancouver, Canada)

會議名稱 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP-2013)

發表論文 題目

1. Outage constrained weighted sum rate maximization for MISO interference channel by pricing based optimization (Wei-Chiang Li, Tsung-Hui Chang, Che Lin, Chong-Yung Chi)

2. Optimal sensor placement for hybrid state estimation in smart grid

( Xiao Li, Anna Scaglione and Tsung-Hui Chang)

2

(b) 10:50 - 12:50：SPCOM-L1: Interference Management I SPCOM-P1: Sensor Networks I

SPCOM-P2: Sparse Signal Processing and Wireline Multicarrier Systems (c) 14:20 - 15:10：Plenary: PLEN2: Inverse Problems Regularized by Sparsity (Martin Vetterli)

(d) 15:30 - 17:30：筆者於 Session SPCOM-P4: Resource Allocation I 中 poster paper:

SPCOM-P4.12: OUTAGE CONSTRAINED WEIGHTED SUM RATE MAXIMIZATION FOR MISO INTERFERENCE CHANNEL BY PRICING-BASED OPTIMIZATION

Wei-Chiang Li; National Tsing Hua University

Tsung-Hui Chang; National Taiwan University of Science and Technology Che Lin; National Tsing Hua University

Chong-Yung Chi; National Tsing Hua University

5/29 (Wednesday)：Participated in

(a) 08:00 - 10:00：SPTM-L3: Learning in Distributed Networks SPCOM-L3: Resource Allocation II

SPCOM-P5: Interference Management III (b) 10:30 - 12:30：SPCOM-L4: Sensor Networks II

(c) 14:10 - 15:00: Plenary: PLEN3: Information measures and estimation theory (Sergio Verdú) (d) 15:20 - 17:20：SPCOM-L5: Sensor Networks III

SPCOM-P7: Relay Networks

SPCOM-P8: MIMO Communication (e) 18:30 - 22:00 : Banquet

3

5/30 (Thursday)：Participated in

(a) 08:00 - 10:00：SPCOM-P9: Encoding and Decoding SPCOM-P10: Localization and Tracking

(b) 10:30 - 12:30：筆者於 Session SPCOM-P11: Smart Grids, Social Networks 中 poster paper:

SPCOM-P11.10: OPTIMAL SENSOR PLACEMENT FOR HYBRID STATE ESTIMATION IN SMART GRID

Xiao Li; University of California, Davis

Anna Scaglione; University of California, Davis

Tsung-Hui Chang; National Taiwan University of Science and Technology

(c) 14:10 - 15:00：Plenary: PLEN4: The Splendor of Nature: Lessons in Adaptation and Learning over Networks (Ali H. Sayed)

(d) 15:20 - 17:20：SPTM-L6: Robust Methods for Detection and Estimation

5/31 (Friday)：Participated in

(a) 08:00 - 09:00：SPTM-P12: Detection Theory, Methods, and Applications SAM-P6: Sensor Network and Distributed Estimation (b) 10:30 - 12:30 : SPCOM-L6: Sparse Signal Processing

(c) 15:20 - 17:20：SPCOM-L7: Communication Systems

SPTM-P15: Sparse Modeling and Estimation Methods

6/1 清晨 2:20 由 Vancouver 機場搭機於台灣時間 6/2 日上午 5:30 抵達桃園國際機場。

二、與會心得

ICASSP 為訊號處理及通訊領域之最大國際會議，因此匯集了世界各地的知名學者及專家。筆者這 次全程參與各個時段大會所安排的議程。如同往年 ICASSP 的盛況，此次 ICASSP 的 poster sessions 吸引了大批學者前往，茟者記得在兩次的 poster 報告中，每一個成果都至少講了 7 到 8 次以上。

這也表示筆者的工作受到了相當的關注。此外，除了筆者過去已熟識的專家學者教授，此次 ICASSP 筆者也新認識了一些工作於相近領域的教授，並且對筆者的工作進行了一些討論交流。整體而言，

這次參與 ICASSP 的學習令人滿意，對我在一些新領域上的研究有諸多幫助，筆者特別感謝國科 會的支持與補助。

三、考察參觀活動(無是項活動者略)

無。

四、建議

無。

五、攜回資料名稱及內容

4

ICASSP-2013 會議論文集之隨身碟、會議手冊。

六、其他

致謝：感謝國科會補助 ICASSP-2013 旅費。

國科會補助計畫衍生研發成果推廣資料表

日期:2012/12/27

國科會補助計畫

計畫名稱: 穩健性多用戶波束成形設計: 近似方法、性能分析與分散式最佳化 計畫主持人: 張縱輝

計畫編號: 101-2218-E-011-043- 學門領域: 通訊

無研發成果推廣資料