國
立
交
通
大
學
電控工程研究所
博 士 論 文
應用於多輸入多輸出系統之最佳收發器設計
Optimal Transceiver Design for MIMO Systems
研 究 生：李建樟
指導教授：林源倍 教授
應用於多輸入多輸出系統之最佳收發器設計
Optimal Transceivesr Design for MIMO Systems
研 究 生：李建樟 Student：Chien-Chang Li 指導教授：林源倍 Advisor：Yuan-Pei Lin
國 立 交 通 大 學
電控工程研究所
博 士 論 文
A DissertationSubmitted to Institute of Electrical Control Engineering College of Electrical Engineering
National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Doctor
in
Department of Electrical Engineering
July 2010
Hsinchu, Taiwan, Republic of China
應用於多輸入多輸出系統之最佳收發器設計
學生：李建樟
指導教授：林源倍
國立交通大學電控工程研究所博士班
摘要
本論文包含兩個部分，在第一個部分我們考慮應用在多輸入多輸出系統的收發器以
及位元分配的設計。在早期的文獻當中，收發器的設計都是針對某一個特定的位元
配置而設計出來的。在近期的研究中，位元配置也會被考慮在收發器的設計中。在
這個部分中，我們考慮的問題是當總功率以及錯誤率有條件限制時，如何最大化傳
輸速率。在這問題之下，我們提出了同時設計收發器和位元配置的方法。一般而言，
我們推算出來的最佳位元配置都不會是正整數。可是這並不符合實際情況，所以在
本篇論文我們也有考慮當位元配置是限制在整數的時候。首先我們先推導出最大化
傳輸速率系統和最小化功率系統之間的關係。然後利用這個關係我們可以找到當位
元配置被限制在整數的時候，最大化傳輸速率的最佳解。我們也用了很多模擬結果
來驗證我們的推導。
在第二個部分，我們考慮的是應用於多載波系統上的傳送與接收窗框之設計。
對於多載波系統而言,在設計傳送端和接收端的時候，傳送端濾波器和接收端濾波
器的頻譜響應非常重要。對於傳送端而言，如果頻譜響應表現不理想，會導致功率
能量的散溢，並且影響到其他的傳輸系統。對於接收端而言，如果頻譜響應表現不
理想，受到外界電台的雜訊干擾將會非常嚴重。為了改善頻譜響應，我們會使用所
ii
謂的傳送與接收窗框。在這個部分，我們使用小波理論來設計傳送與接收窗框。我
們會引入所謂的子濾波器。我們發現使用小波理論設計的系統在對抗電台雜訊干擾
以及抑制功率能量的散溢有較為出色的表現。
Optimal Transceiver Designs for MIMO Systems
Student: Chien-Chang Li
Advisor: Yuan-Pei Lin
Institute of Electrical Control Engineering
National Chiao Tung University
Abstract
This dissertation consists of two parts. In the first part, we consider the joint design of transceiver and bit allocation for input multiple-output (MIMO) channels. In the literature, there have been many re-sults on designing transceivers for MIMO channels. In these rere-sults, the transceivers are designed for a given bit allocation or designed with real bit allocation. In this thesis, first we jointly optimize the transceiver with real-valued bit allocation for maximizing bit rate over MIMO channels. The optimal transceiver and bit allocation are obtained in a closed form. Second we consider the connection between the power-minimizing and rate-maximizing problems with bit allocation. We will show that if a transceiver is optimal for the power-minimizing problem, it is also optimal for the rate maximizing problem and the converse is true. The result holds whether the bit allocation is integer-constrained or not. Based on the duality, we develop algorithms to find the optimal solution for the power-minimizing problem and rate-maximizing problem with integer bit constraint.
In the second part, we consider the design of transmitting and receiv-ing windows for multicarrier systems. For multicarrier systems, frequency characteristics play an important role in the design of the transmitting and receiving filters. To improve frequency separation, windowing techniques are often used. As these are frequency based characteristics, a filterbank
representation provides a natural and useful way for formulating the prob-lem. In this thesis, we propose a unified filterbank approach to design the transmitting and receiving windows for multicarrier systems. Using the filterbank approach, the frequency separation among the subchannels can be improved.
誌謝
在交大已經過了六年多，在電控所的這幾年學到了很多東西，不管是在
專業領域方面或是做人處事。要特別感謝的是林源倍老師的耐心教導，
我才能順利的完成博士學位。也感謝我家人的支持，家人總是在我需要
的時候給我鼓勵。另外也要感謝實驗室的學長禮涵、彥棋、孟良、同學
沛如、雅雯給我的支持。感謝裕彬、尹劭以及紹倫常常找我打棒球以及
探索清大鬼故事之旅。感謝學弟宗堯、鈞麟帶給我對於我課業以及人生
上不一樣的看法。感謝學妹素卿、芳儀帶起了我減重的想法並且嚴格的
督促我，讓我的生活增添很多樂趣。感謝學弟懿德、士軒、人予、君維、
士傑陪我在實驗室度過歡樂的時光並且培養團隊合作的默契。感謝立人
學長、尚澕學長、學妹虹君、宛真、雅萱給我課業上的指導。因為有你
們生活才會變得更有活力。最後要感謝的是擔任我的口試委員的各位老
師。系上的董蘭榮老師、電工系的陳紹基老師、清大電機系的王晉良以
及祁忠勇老師,還有台大電機系的蘇炫榮老師。謝謝各位老師撥空來參
加我的口試並給予指導。學生的生涯到此算是告一段落，但是真正的挑
戰才要開始。在未來我將會帶著大家的祝福繼續往前走。
Contents
Abstract (in Chinese) i
Abstract (in English) iii
Acknowledgement (in Chinese) v
1 Introduction 1
1.1 Transceiver designs for MIMO Systems . . . 1
1.2 Multicarrier System . . . 3
1.3 Chapter Outline . . . 6
2 Overview of MIMO Systems 9 2.1 Systems Model . . . 9
2.2 ZF and MMSE Receivers . . . 10
2.3 Symbol Error Rate . . . 12
2.4 Bit Allocation . . . 13
2.5 Summary . . . 13
3 Rate-Maximizing Zero-Forcing Transceivers with Bit Allocation 14 3.1 Problem Formulation . . . 14
3.2 Optimal Zero-Forcing Transceiver . . . 15
3.3 Simulations . . . 21
4 Optimal MMSE Transceivers with Bit Allocation to Maximize
Bit Rate 24
4.1 Preliminaries . . . 24
4.2 Problem Formulation . . . 26
4.3 Optimal Transceiver Design . . . 27
4.4 Simulations . . . 37
4.5 Summary . . . 40
5 On the Duality of Transceiver Designs for MIMO Channels 41 5.1 Power-minimizing and Rate-maximizing Transceiver design . . . 42
5.2 Transceiver design with integer bit allocation . . . 47
5.3 Optimal solution for transceiver design with bit allocation . . . . 52
5.3.1 Optimal solution of Apow and Arate . . . 52
5.3.2 Optimal solution of Apow,int and Arate,int . . . 53
5.4 Simulation . . . 55
5.5 Summary . . . 59
6 Overview of Multicarrier Systems 61 6.1 DFT Based Multicarrier System . . . 61
6.2 Filterbank Representation . . . 64
6.3 Transmitted Power Spectrum . . . 68
6.4 Radio Frequency Interference . . . 70
6.5 Summary . . . 71
7 Receiver Window Designs for Radio Frequency Interference Sup-pression for Multicarrier Systems 72 7.1 Receiver Windowing in Multicarrier System . . . 73
7.2 Filterbank Representation of Receiver Windowing in Multicarrier System . . . 76 7.3 Informed Window . . . 77 7.4 Uninformed Window . . . 78 7.5 Simulations . . . 80 7.6 Summary . . . 83
8 A Filterbank Approach to Window Designs for Multicarrier Sys-tems 86 8.1 System Model . . . 87
8.2 Receivers with Subfilters . . . 88
8.3 Implementation of Receiving Bank with Subfilters . . . 89
8.4 Window coefficients bk . . . 90
8.5 Design of Receiver Subfilters . . . 91
8.6 Transmitter with Subfilters . . . 93
8.7 Implementation of Transmitting Bank with Subfilters . . . 95
8.8 Design of Transmitter Subfilters . . . 97
8.9 Simulations . . . 98 8.10 Summary . . . 100 9 Conclusion 101 Appendix 105 A Proof of Lemma 5.1 . . . 105 B Proof of Lemma 5.2 . . . 107 C Proof of Lemma 5.3 . . . 109 D Proof of Lemma 5.4 . . . 112 Bibliography 114
List of Tables
5.1 (a) Minimal power P (B0) for Apowwhen B0 = 2, 4, 6, 8, 10 bits. (b)
Maximal bit rate for Arate when the power constraint P0 = P (B0). 56
5.2 (a) Maximal bit rate for Arate when P0 = 2, 4, 8, 16, 32 dB. (b)
Minimal power P (B0) for Apow when B0 = B(P0). . . 57
5.3 Transmit power of Apow (without integer bit allocation), Apow,int
(ZF), and Apow,int (MMSE) when the target bit rate is B0 =
2, 4, 6, 8, 10, 12 bits. . . 60 5.4 Bit rate of Arate (without integer bit allocation), Arate,int(ZF), and
Arate,int(MMSE) when the target bit rate is P0 = 2, 4, 6, 8, 10, 12, 14, 16 dB. . . 60
List of Figures
1.1 Multicarrier system. . . 4
2.1 MIMO communication system. . . 9
3.1 Transmission bit rates for a fixed error rate. . . 22
3.2 Bit error rate performance. . . 23
4.1 Transmission bit rates for a fixed error rate. . . 38
4.2 Bit allocation for the channel in (4.61) when P0/N0 = 20 dB. . . 39
4.3 Bit error rate performance. . . 40
5.1 (a) Maximal bit rate as a function of power constraint for Arate,int. (b) Minimal transmit power as a function of target bit rate for Apow,int. . . . 50
5.2 Maximal bit rate B∗(P 0) for Arateas a function of power constraint P0 without integer constraint. . . 58
5.3 Minimal transmit power P∗(B 0) for Apow as a function of target bit rate B0 without integer constraint. . . 59
6.1 Block diagram of the DFT based multicarrier system. . . 61
6.2 Matrix forms of the transmitter and receiver for the DMT system. 65 6.3 Filterbank representation of the DMT system. . . 66
6.4 The magnitude response of the transmitting and receiving filters for M = 4 and ν = 2.(a) the transmitting filters, and (b) the
receiving filters. . . 67
6.5 Filter bank representation of the receiver with windowing. . . 70
7.1 (a) An example of receiver window; (b) receiver windowing. . . . 74
7.2 Receiver with windowing in the multicarrier system. . . 74
7.3 Frequency response of receiving windows. . . 75
7.4 Frequency response of receiving windows. . . 81
7.5 Subchannel interference power of the DMT system with window-ing. (a) Informed window, uninformed window, window in [71], and Hanning window. (b) Informed window, Blackman window, and Kaiser window with shape parameter β = 5. . . 83
7.6 Subchannel SINRs of the DMT system with windowing. (a) In-formed window, uninIn-formed window, window in [71], and Hanning window. (b) Informed window, Blackman window, and Kaiser window with shape parameter β = 5. . . 84
8.1 The receiving bank with subfilters. . . 88
8.2 The transmitting bank with subfilters. . . 94
8.3 Efficient DFT implementation of the transmitting bank. . . 96
8.4 Time-domain illustration of transmitter windowing. . . 96
8.5 The subchannel SINRs. . . 99
List of Notations
• Boldfaced lower-case letters represent vectors and boldfaced upper-case let-ters are reserved for matrices.
• The notation X† denotes the conjugate transpose of X.
• The notation XT denotes the transpose of X.
• The notation IM is used to represent the M × M identity matrix.
• The notation [X]kkdenotes the k-th diagonal element of the matrix X. The
notation xk denotes the k-th element of the vector x.
• The notation diag(g) denotes a diagonal matrix with the elements of g on its diagonal.
• The notation ⌊z⌋ denotes the largest integer that is less than or equal to z. • The notation E[x] denotes the expectation of a random variablex.
Chapter 1
Introduction
This thesis consists of two parts. Part I is on the design of bit allocation and transceiver for MIMO channels. Part II is on the design of transmitting and receiving windows for multicarrier systems. The introduction of parts I and II are given in section 1.1 and 1.2 respectively.
1.1
Transceiver designs for MIMO Systems
Multiple-input multiple-output (MIMO) channels arise in applications such as wireless communication systems that use multiple antennas, multicarrier commu-nication systems, and also telephone cables that consist of many twisted pairs. They represent a way to model a wide variety of scenarios. In this part, we focus on the transceiver design with bit allocation for MIMO channels. The design of the MIMO transceivers can be formulated as the optimization problem of an objective function based on the performance of each subchannels.
Transceiver designs for a given bit allocation. For a given bit allocation, many criteria have been considered in the transceiver designs for MIMO channels, e.g., [1]-[17]. Optimal transceivers that maximize the mutual information are proposed in [1]-[5]. Transceiver designs that minimize mean-square error (MMSE) are considered in [6]-[9]. Optimal transceivers that minimize the bit error rate
power are proposed in [14][16]. Using the MMSE receiver, unified frameworks for designing MIMO systems with a power constraint are proposed in [17]. A number of useful objective functions can be considered in this framework. For example, the optimal MMSE transceivers that maximize the bit rate and mutual information can be designed using this unified approach.
Transceiver designs with real-valued bit allocation. In [1]-[17], the transceivers are designed for a given bit allocation. Recently, bit allocation is also incorporate in the design of the MIMO system [18]-[27]. Optimal transceivers with bit allocation that minimize the transmit power are proposed in [18][22]. Op-timal transceiver with bit allocation designs that use the bit rate maximization criterion are addressed in [23]-[25]. Transceiver designs that consider a num-ber of design criteria are proposed in [26, 27]. For example, power-minimizing transceiver, rate-maximizing transceiver, capacity-maximizing transceiver and BER-minimizing transceiver can be obtained using [26, 27]. For the transceivers designs in [18]-[27], a smaller transmit power or a higher bit rate than the cases without bit allocation can be achieved. Hence bit allocation plays an important role in the power minimization and rate maximization problems. However, the bit allocation obtained in these designs are not integers in general. The MIMO transceiver design for minimizing transmit power or maximizing bit rate with integer bit constraint is not solved and still open.
Integer bit allocation for multicarrier systems. For the multicarrier systems, integer bit allocation has been considered [29]-[36]. The problem of designing integer bit allocation for maximizing bit rate and minimizing transmit power in multicarrier systems is considered in [29]. Algorithms for allocating integer bits to minimize transmit power is proposed in [30]-[31]. In [32], an optimal bit loading algorithm is presented for minimizing BER in multicarrier system. In [33], a bit loading algorithm is proposed to increase the noise margin (additional amount of noise that the system can tolerate). Problems of finding
the integer bit allocation for maximizing a concave function is considered in [34], where it is shown that a greedy algorithm can be used to find the optimal solution. In [35], an efficient bit loading algorithm is proposed to minimize an arbitrary convex objective function. The algorithm proposed in [34] and [35] can be used to find the optimal integer bit allocation for both the power-minimizing problem or rate-maximizing problem when a ZF transceiver is given. In [36], an integer bit allocation is proposed to maximize the transmission bit rate in the presence of intercarrier interference. An integer bit allocation for minimizing the quantization error of multiple sources is proposed in [37]. The algorithms in [29]-[37] can be used to find integer bit allocation only when a transceiver is given.
1.2
Multicarrier System
Multicarrier system has attracted considerable attention in recent years as a prac-tical technology for high-speed data transmission over frequency selective chan-nels [45]-[47]. The discrete Fourier transform (DFT) based multicarrier system has been recognized as a very cost-effective realization of multicarrier transceivers. Several important applications of multicarrier system have been found in discrete multitone (DMT) systems such as asymmetric digital subscriber lines (ADSL) [48] and very high speed digital subscriber lines (VDSL) [49][50], and orthogo-nal frequency division multiplexing (OFDM) systems such as wireless local area network [51] and digital video broadcasting (DVB) [52]. A generic multicarrier system is shown in Fig. 1.1. F0(z), F1(z), · · · , FM −1(z) are the transmitting
fil-ters and H0(z), H1(z), · · · , HM −1(z) are the receiving filters. In second part of
this thesis, we consider the design of the transmitting and receiving filters of the multicarrier system. In the design of the multicarrier system, the frequency char-acteristics of the transmitting and receiving filters are important considerations. The stopband attenuation of the transmitting and receiving filters determines
Figure 1.1: Multicarrier system.
ventional multicarrier system, the transmitting and receiving filters come from rectangular windows. Since the sidelobes of the rectangular window are large, the conventional multicarrier system has a poor frequency separation in both the transmitter side and the receiver side.
Frequency characteristic at the transmitter. At the transmitter side, poor frequency separation leads to significant spectral leakage. In some applica-tions, the PSD (power spectral density) of the transmit signal is required to have a large roll-off in certain frequency bands. Hence poor frequency separation could pose a problem in such applications. For example in some wired transmission ap-plication, the PSD of the transmitted signal needs to fall below a threshold in the transmission bands of the opposite direction to avoid interference [48, 49]. The PSD should also be attenuated in amateur radio bands to reduce interference to radio transmission or egress [49].
Frequency characteristic at the receiver. In some applications such as VDSL and ADSL transmission, the multicarrier systems share its spectrum with different types of radio transmission, for example, amplitude-modulation stations and amateur radio [48, 49, 50]. These radio signals can be coupled into telephone wires and interfere with the VDSL signal at the receiving side. This type of noise in a VDSL transmission system is known as radio frequency interference
(RFI) ingress [53]. At the receiver side, the spectral roll-off of the receiving filters determine how the tones are affected by RFI interference. A faster roll off means the effect of RFI diminishes faster and fewer neighboring tones are affected. Poor frequency separation results in poor out-of-band rejection. Since the receiving filters of the conventional multicarrier system come from the rectangular window, many neighboring tones can be affected by the RFI ingress.
Improving the Frequency characteristics. In the literature, many meth-ods have been proposed to improve the frequency characteristics of the trans-mitter. To improve the spectral roll-off of the transmitted signal, a number of continuous-time pulse shaping filters have been proposed, [54]-[59]. Usually continuous-time pulse shapes are designed based on an analog implementation of transmitters and a digital implementation is not admitted [60]. Discrete-time windows have been considered in [61]-[63]. The design of overlapping windows for OFDM with offset QAM (quadrature amplitude modulation) over ISI free channels are studied fully in [62, 63]. More recently, transmitting windows with the cyclic-prefixed property have been proposed in [64, 65] for egress control. Windows that are the inverse of a raised cosine function are optimized in [64], to minimize egress emission. To compensate for the transmitter window, the corresponding receiver requires post-processing equalization [64, 65]. Per-tone windows are proposed in [66] for shaping transmitted spectrum. The shaping of spectrum allows more tones to be used for transmission.
At the receiver side, windowing is also often applied to improve the frequency characteristics. Commonly used windows include Hanning window and Blackman window [75]. In [67], Muschallik use Nyquist windows, which have the property that shifts of the window in the time domain add to a constant, to improve the reception of OFDM systems. Optimal Nyquist windows are considered in [68] to mitigate the effect of additive noise and carrier frequency offsets. To improve RFI suppression, receiver windowing is proposed first in [69] by Spruyt et al. For the
suppression of sidelobes without using extra redundant samples, it is proposed in [70] to use windows that introduced controlled IBI, later removed using decision feedback. To minimize the RFI and channel noise, the receiver windowing is proposed in [71]. The optimal window can be found using the statistics of the received RFI and noise [71]. A combination of raised-cosine window and per tone equalizer are proposed to suppress RFI interference in [72]. However, the channel information is required in these designs. in [73], channel-independent windows are designed by minimizing the sidelobe energy. In this case, ISI (inter symbol interference) is introduced and post processing is required to cancel ISI. Using statistics of channel noise and RFI, a joint design of the TEQ and the receiving window for maximizing bit rates is given in [74].
1.3
Chapter Outline
The designs of transceivers with bit allocation for MIMO channels are discussed in Chapter 2-Chapter 5 and the designs of transmitting and receiving windows for the multicarrier system are discussed in Chapter 6-Chapter 8. Details of the research contributions in each chapter are as follows.
Chapter 2
In this section, we introduce the MIMO systems. We consider both the ZF and MMSE receivers. For the QAM modulation, symbol error rate and bit allocation are also given in this chapter.
Chapter 3
In this chapter, we consider the design of the zero-forcing transceivers for MIMO channels. We jointly optimize the transceiver and bit allocation to maximize the transmission rate for a given target error rate and transmit power. Using the high bit rate assumption, we can simplify the optimization problem and the optimal transceiver can be easily found by the Hadamard inequality and the P oincar´e
Chapter 4
In this chapter, we consider the rate maximizing problem in chapter 4 but the receiver is MMSE. In this design, we do not use the high bit rate assumption. We jointly optimize the transceiver and bit allocation to maximize the bit rate subject to a given error rate and a given transmit power. There are no constraints on the transceiver or the bit allocation. Using the majorization theorem [42], the optimal transceiver and bit allocation can be obtained in a simple close-form and the optimal solution diagonalizes the channel into parallel independent subchannels.
Chapter 5
In this chapter, we study the connections between the power minimization and rate maximization problem. For the problems without integer bit constraint, we will show that these two problems have the same solution. However, the result does not generalize completely to the case with an integer constraint on bit allocation. We show that the power minimization and rate maximization criterion yield the same solution if the statement of problems are modified slightly. Moreover, we also show how to find the optimal solution of the power-minimizing problem and rate-maximizing problem with the integer bit constraint.
Chapter 6
In this chapter, we will give an overview of the multicarrier system. We will derive the filterbank representation of the multicarrier system. We also study the spectral leakage at the transmitter and RFI interference at the receiver. Chapter 7
In this chapter, we design the receiving windows to improve the frequency sep-aration among the receiving filters. We will consider both the case when the statistics of the interference is available to the receiver (informed receiver) and the case when it is not (uninformed receiver). The frequency responses of the proposed windows achieve a good trade-off in spectral roll-off between high
fre-quency and low frefre-quency than that of rectangular window, Hanning window, Blackman window, Kaiser window and the window design method in [71]. As a result, fewer tones will be dominated by RFI interference. The proposed win-dows in both cases are channel independent and can be obtained in a closed form solution.
Chapter 8
In this chapter, we propose a unified filterbank framework for the design of win-dows for multicarrier systems. The approach is more general than the conven-tional windowing technique. We will use the so-called subfilters to enhance the frequency selectivity of the transmitting and receiving filters while maintain-ing the orthogonality among the subchannels. For the transmitter side spectral leakage can be reduced and for the receiver side RFI can be further suppressed. When the subfilters form a DFT bank, they can be tied nicely to the conventional windowing such as in [65], [71], and chapter 2. The windows can be optimized through the design of subfilters and frequency separation among the subchannels can be considerably improved.
Chapter 2
Overview of MIMO Systems
MIMO systems arise in many different scenarios such as wired-line systems or multi-antenna wireless systems. In this chapter, we will give an overview of the MIMO communication systems.
2.1
Systems Model
A generic MIMO communication system is shown in Fig. 2.1. The MIMO channel is modeled by a P × N memoryless matrix H. The P × 1 channel noise q is additive white Gaussian noise with variance N0. The transmitter matrix F is of
size N ×M. The receiver matrix G is of size M ×P . The input of the transmitter is s, an M × 1 vector of modulation symbols.
M P N M
H
F
G
s
s
x
q
Figure 2.1: MIMO communication system.
The symbols are assumed to be zero mean and uncorrelated; hence the auto-correlation matrix Λs= E[ss†] is a diagonal matrix with [Λs]kk = σs2k, where the
is P = E{x†x} = Tr(FΛsF†) = M −1 X k=0 [F†F]kkσ2sk, (2.1)
where x is the transmitter output. The output of the receiver is given by
ˆs = GHFs + Gq, (2.2)
where e = Gq is the error vector. Defined the error vector as
e = s − ˆs. (2.3)
The mean-squared error (MSE) matrix is given by
E = E[ee†], (2.4)
and the error variance of the k-th subchannel is σ2
ek = [E]kk.
2.2
ZF and MMSE Receivers
In this section, we will introduce the ZF and MMSE receivers for a given trans-mitter.
ZF Receiver. The zero-forcing condition is given by
GHF = IM. (2.5)
To achieve zero-forcing, the rank of F, H, G must be larger than or equal to M. In Lemma 2.1, we will show that without loss of generality we can choose G as the pseudo inverse of HF.
Lemma 2.1 It is no loss of generality to choose G as the pseudo inverse of HF.
That is,
In this case, the MSE matrix becomes
E = N0GG†= N0(F†H†HF)−1. (2.7)
Proof: Suppose (G, F) is a transceiver pair that satisfies the zero-forcing condi-tion in (2.5). Let G′ be the pseudo inverse of HF, i.e.,
G′ = (F†H†HF)−1F†H†. (2.8)
Then we have G′HF = IM. Define ∆ = G − G
′
. Since (G, F) and (G′, F) are
both zero-forcing, we have ∆HF = 0. It follows that ∆G′†
= 0. When we use G, the noise variance at k-th subchannel is given by
N0[GG†]kk= N0[(G
′
+ ∆)(G′ + ∆)†]kk≥ N0[G
′
G′†]kk, (2.9)
where we have used ∆G′†
= 0 and [∆∆†]
kk> 0. Therefore, we will have smaller
subchannel noise variances when we replace G with G′
. Using (2.6), we have the
MSE matrix as in (2.7). △△△
For the ZF case, the receiver in (2.6) and the MSE matrix in (2.7) depend on the channel matrix H, the transmitter F, and noise variance N0.
MMSE Receiver. For the MMSE case, the receiver is obtained by minimizing the mean square error [27], i.e.,
G = arg min
G E[e
†e]. (2.10)
Let y be the signal received at the receiver, i.e., y = HFs + q. Using the orthog-onality principle [44], we can find G by solving
E[ey†] = 0. (2.11)
Then the MMSE receiver is given by
Substituting (2.13) into (2.4), the MSE matrix becomes
E = Λs− ΛsF†H†[HFΛsF†H†+ N0IP]−1HFΛs. (2.14)
For the MMSE case, the receiver in (2.13) and the MSE matrix in (2.14) depend on the channel matrix H, the transmitter F, noise variance N0, and the signal
autocorrelation matrix Λs. If σs2k = 0 for some k, using (2.14) we have σ
2 ek = 0.
Reduced system of the MMSE and ZF transceiver. For both the ZF and MMSE transceivers, the signal power σ2
si assigned to the i-th subchannel may be
equal to zero and thus si = 0. In this case the autocorrelation matrix Λs is not
invertible. Suppose Mr subchannels are assigned nonzero power. Let sr be the
Mr× 1 vector obtained by deleting the entries of s that are assigned with zero
power. Let Fr be the N × Mr matrix obtained by deleting the columns of F that
correspond to the subchannels assigned with zero power. Then the transmitter output x is
x = Fs = Frsr. (2.15)
When we consider the transmitter Fr with input sr, the transmitter output is
the same as the original system. Hence we can consider only the subchannels assigned with nonzero power. Let Λr be the Mr× Mr diagonal matrix obtained
by deleting the columns and rows of Λs with zero power. Since Λr is invertible,
the reduced Mr× Mr MSE matrix becomes
Er = [N
−1
0 F†rH†HFr+ Λ−1r ]−1, for the MMSE receiver;
N0(F†rH†HFr)−1, for the ZF receiver. (2.16)
2.3
Symbol Error Rate
For the QAM modulation, suppose the power allocation Λs, transmitter F, and
error rate ǫk of the k-th subchannel can be approximated by [45] ǫk ≈ 4 1 −2b1k/2 Q s 3βk (2bk − 1) , (2.17) where βk= σ 2 sk/σ 2
ek, for the ZF receive;
σ2 sk/σ
2
ek − 1, for the MMSE receiver.
(2.18) The function Q(x) is the area under a Gaussian tail, i.e., Q(x) = (1/√2π)R∞
x e −u2/2
du.
2.4
Bit Allocation
Suppose the power allocation Λs and the transmitter F are given. Then βk in
(2.18) can be determined. For the QAM modulation, equation (2.17) relates the error rate to βk. It can be used to obtain the number of bits that can be loaded
on the k-th subchannel for a given βk and target symbol error rate ǫk [38]. By
rearranging the terms in (2.17), we get bk = log2 1 + βk Γk , (2.19)
where Γk = 13[Q−1(ǫk/4)]2. The total number of bits that can be transmitted in
one block is B = M −1 X k=0 bk = M −1 X k=0 log2 1 + βk Γk . (2.20)
2.5
Summary
In this section, we gave an overview of a generic MIMO communication system. We have introduced the ZF and MMSE receivers when the transmitter and power allocation, and channel are given. We also introduced the symbol error rate and bit allocation when the QAM modulation is used.
Chapter 3
Rate-Maximizing Zero-Forcing
Transceivers with Bit Allocation
In this chapter, we will jointly design the transceiver and bit allocation for max-imizing bit rate for the ZF transceiver. Using the high bit rate assumption, we can simplify the optimization problem. The solutions are obtained in two steps. Firstly, we design the optimal bit and power allocation for a given transceiver and a given power constraint. Secondly, we design the optimal transceiver that maximizes the bit rate based on the optimal bit and power allocation. In the second step, the optimal transceiver can be easily found by the Hadamard in-equality and the Poincar´e separation theorem. The optimal transceiver and bit allocation can be obtained in a closed form.
3.1
Problem Formulation
Suppose the target error rate of all the subchannels are equal to ǫ. Using the high bit rate assumption, i.e., 2bk ≫ 1, the bit allocation b
k in (2.19) is approximated by bk= log2 σ2 sk σ2 ekΓ . (3.1)
We will see in section 3.2 that such an assumption facilitates the derivation of the optimal transceiver. Using the high bit rate assumption, the problem of
maximizing bit rate subject to a zero-forcing constraint and a total transmit power constraint P0 can be formulated as
maximize G, F, {σ2 sk} B =PM −1 k=0 log2 σ2 sk σ2 ekΓ , σ2 ek = N0[GG †] kk subject to Tr(FΛsF†) ≤ P0 GHF = IM σ2 sk ≥ 0, k = 0, 1, · · · , M − 1. (3.2)
In section 3.2, we derive the optimal bit allocation and transmitter for the rate maximization problem.
3.2
Optimal Zero-Forcing Transceiver
First, we will find the power allocation that maximizes the bit rate for a given zero-forcing transceiver. To this end, we use the Karush-Kuhn-Tucker (KKT) condition [77]. Let σ2∗
sk be a local maximum for the optimization problem in
(4.6). Then there exists constants α and µk, for k = 0, 1, · · · , M − 1 such that:
1. α ≤ 0. 2. µk ≤ 0, for k = 0, 1, · · · , M − 1. 3. ∂ ∂σ2 sk PM −1 k=0 log2 σ2 sk σ2 ekΓ +α(Tr(FΛsF†)−P0)+ M−1 P k=0 µk(−σ2sk) σ2 sk=σ2∗sk = 0. 4. α(Tr(FΛsF†) − P0) σ2 sk=σ2∗sk = 0. 5. µk(−σ2sk) = 0 σ2 sk=σ2∗sk = 0, for k = 0, 1, · · · , M − 1.
By solving the above conditions, the optimal power allocation is given by σs2k = P0
M[F†F] kk
From (3.3), we can see that the power allocation depends only on the transmitter for the given P0 and M. Using (3.3), the bit rate is given by
B = M −1 X k=0 log2 P0 MΓ[F†F] kkσ2ek (3.4) = log2 M −1 Y k=0 P0 MΓ[F†F] kkσe2k . (3.5)
Next, we will design the optimal zero-forcing transceiver that maximizes the bit rate in (3.5). Suppose the P × N channel matrix H has rank K. Let the singular value decomposition of H be
H = U Λ 0
0 0
V†, (3.6)
where the K × K diagonal matrix Λ contains the nonzero singular values of H. The P × P matrix U and the N × N matrix V are unitary. We assume that the elements of Λ are in nonincreasing order and K ≥ M so that solutions of zero-forcing transceivers exist.
Lemma 3.1 Without loss of generality, we can express F to be of the following
form:
F = V A
0
, (3.7)
for appropriate K × M matrix A of rank M.
Proof: Suppose (G, F) is a transceiver pair that satisfies the zero-forcing condi-tion. As V is unitary, F can always be represented as
F = V A
A1
, (3.8)
where A is a K × M matrix, A1 is an (N − K) × M matrix, and the notation T
denotes the transpose. Define a new transceiver F′ as
F′ = V A
0
Then we have
GHF′ = GHF. (3.10)
Therefore, when we replace the transmitter by F′
, the new system still satisfies the zero-forcing condition GHF = IM. As the receiver is not changed, the new
system has the same subchannel noise variances and hence the same bit rate
performance. Now, let us compare the transmit power of F and F′ for the same
Λs. The transmit power when we use F is
Tr(FΛsF†) = Tr(AΛsA†) + Tr(A1ΛsA†1). (3.11)
Note that the transmit power with F′is Tr(F′ΛsF
′†
) = Tr(AΛsA†) ≤ Tr(FΛsF†).
This means a transmitter of the form in (3.7) is no loss of generality. △△△
Using Lemma 3.1 and Lemma 2.1, the receiver G is given by
G = (A†Λ2A)−1[ A†Λ 0 ]U†, (3.12)
where A is the matrix given in (3.7). In this case, the noise variance at the k-th subchannel becomes
σe2k = N0[GG†]kk = N0[(A†Λ2A)−1]kk. (3.13)
Note that the transmitter and receiver in (3.7) and (3.12) have the same form as those in [28] and [14]. The transceivers in [28] and [14] are designed for minimizing the transmit power for a given bit allocation, while we jointly design the optimal transceiver and bit allocation for maximizing the transmission rate. Lemma 3.1 lead us to conclude that the matrix A in (3.7) is the only part of the transceiver left to be designed. Using the expression of F in Lemma 3.1 and the expression of σ2
ek in (3.13), the bit rate in (3.5) becomes
B = log2 ( P0 MN0Γ )M1 Φ , (3.14) where Φ =QM −1
Optimal structure of A. Applying the Hadamard inequality [41], we have Φ = M −1 Y k=0 [A†A]kk[(A†Λ2A)−1]kk (3.15) ≥ det[A†A]det[(A†Λ2A)−1]. (3.16)
The equality holds if and only if the matrix A satisfies the following two con-ditions: 1) A†A is diagonal, and 2) A†Λ2A is diagonal. The first condition
means that the columns of A are orthogonal, while the second means that the columns of ΛA are orthogonal. As ΛA is orthogonal, we can express it as
ΛA = QD, for some K × M unitary matrix Q such that Q†Q = I
M, and
some M × M nonsingular diagonal matrix D. As Λ is nonsingular, we can write
A = Λ−1QD. Then the product of the two determinant quantities in (3.16)
becomes det[A†A]det[(A†Λ2A)−1] = det(Q†Λ−2Q). Hence the bit rate in (3.14)
is simplified to B = log2 ( P0 MN0Γ )M 1 det(Q†Λ−2Q) . (3.17)
Note that the bit rate in (3.17) is independent of D. Without loss of generality, we can choose D to be any M × M nonsingular diagonal matrix. For example, we can choose D = IM. To achieve the maximal bit rate, we need to find Q that
minimizes det(Q†Λ−2Q).
Optimal Q: We can find Q with the help of the Poincar´e separation theorem [41], which is presented below for convenience.
Poincar´e separation theorem [41]: Let B be an n × n Hermitian matrix and C be an n × r unitary matrix with C†C = Ir. Then we have ρi(B) ≤ ρi(C†BC) ≤
ρn−r+i(B), i = 0, 1, · · · , r − 1, where the notation ρi(X) denotes the i-th smallest
eigenvalue of X.
By the Poincar´e separation theorem, we have [Λ−2]
ii ≤ ρi(Q†Λ−2Q), i =
0, 1, · · · , M − 1, where we have used the property that the diagonal elements of Λ are in nonincreasing order. Since the diagonal matrix Λ is nonsingular,
we know that Λ−2 is positive definite and [Λ−2] ii > 0 for i = 0, 1, · · · , K − 1. Therefore, we have det(Q†Λ−2Q) = M −1 Y i=0 ρi(Q†Λ−2Q) (3.18) ≥ M −1 Y i=0 [Λ−2]ii = det(Λ−2M), (3.19)
where ΛM is an M × M diagonal matrix whose diagonal elements consist of the
M largest singular values of H. The inequality in (3.19) becomes an equality when we choose
Q = IM
0
. (3.20)
With this choice of Q and D = IM, we have
A = Λ−1QD = Λ −1 M 0 . (3.21)
Using (3.7) and (3.12), the optimal transceiver is given by
F = V Λ −1 M 0 , G = [ IM 0 ]U† . (3.22)
The maximal bit rate in (3.17) is given by b = log2[( P0
M N0Γ)
Mdet(Λ2
M)].
Substitut-ing (3.22) into (3.3) and (3.13), we have σs2k = P0
M[Λ
2
M]kk and σe2k = N0. (3.23)
Using (2.19), the number of bits allocated to the k-th subchannel becomes bk = log2 1 + ( P0 MN0Γ )[Λ2M]kk . (3.24)
We can see that more bits are assigned to subchannels that correspond to larger singular values of the channel.
Remarks:
1. Note that if we choose D = ΛM, we have
F = V IM
0
, G = [ Λ−1M 0 ]U†. (3.25)
In this case, σ2
sk = P0/M and all subchannels are assigned the same power.
The bit allocation and bit rate are the same as the case when we choose D = IM. This is because the signal to noise ratio σs2k/σ
2
ek is not affected by
D. Therefore, bit assignment and hence bit rate performance will be the same.
2. In (4.50), the bits are not integers in general. We can use truncation, i.e., ˜bk = ⌊bk⌋, where the notation ⌊z⌋ denotes the largest integer that
is less than or equal to z. Zero bits may be assigned to some subchannels (˜bk = 0 if P0[Λ2M]kk < MN0Γ) and the power allocated to these subchannels
is wasted. In this case, we will remove the worst subchannel and compute bit and power allocation in the remaining subchannel. We continue like this until all the power is used by subchannels with nonzero bits. The iterative bit allocation algorithm is given below.
Integer bit allocation algorithm:
Let M0 be the number of subchannels that will be assigned nonzero bits.
Initially, set M0 = M.
(a) Compute ξk = P0[Λ
2 M]kk
M0N0Γ for k = 0, 1, · · · , M0− 1.
(b) If ξk ≥ 1 for k = 0, 1, · · · , M0 − 1, then go to step (c). Else, if ξk < 1
for some subchannels, set M0 = M0− 1 and go to step (a).
(c) Compute the bit allocation by bk = ⌊log2(1 + ξk)⌋ for 0 ≤ k < M0.
3.3
Simulations
In the simulation, we evaluate the performance of the proposed method. The number of subchannels M is 4. The channel used is a 4 × 4 MIMO channel (P = N = 4). The elements of H are complex Gaussian random variables whose real and imaginary parts are independent with zero mean and variance 1/2. The
noise vector q is assumed to be complex Gaussian with E[qq†] = I
4. QAM
modulation is used for the input symbols. Optimal zero-forcing transceiver in (3.22) is used for the proposed method. Although the high bit rate assumption (bk ≫ 1) is used in the derivation of the optimal transceivers, the assumption is
not used in the computation of transmission bit rate in the simulations. We will use the integer bit allocation in remark 2 instead.
Fig. 3.1 shows the transmission rates for different transmit power to noise ratio (P0/N0). The symbol error rates are 10−2 for all the subchannels. The
transmis-sion rates are averaged over 106 random channel realizations. For comparison, we
have also shown the results of three zero-forcing systems: the maximum signal to noise ratio (MSNR) transceiver in [8], the unit noise variance (UNV) transceiver in [14], and the SVD-waterfilling solution in [3], and also the results of two op-timal transceivers in [13] that are designed using a minimum mean square error (MMSE) criterion and a maximum mutual information (MMI) criterion. Both of the MMSE [13] and MMI [13] systems use MMSE reception. In the UNV [14] and MSNR [8] systems, as all the subchannels have the same signal to noise ratios, the same bits are assigned for all subchannels. For the MMSE and MMI systems, we use the bit loading method mentioned in equation (46) of [13]. Fig. 3.1 shows that the proposed method can achieve a bit rate considerably higher than MMSE [13], UNV [14], and MSNR [8], and slightly better than MMI [13] and SVD-waterfilling [3]. We should note that, although the proposed system is zero-forcing, it is still better than the two MMSE systems in [13], in which the noise statistics is also
taken into consideration. In Fig. 3.2, we plot the bit error rates averaged over 106 random channel realizations when the total number of bits per block is fixed
to eight for the same six systems. For the proposed method, we compute the bit allocation that is obtained when P0/N0 = 12 dB (the corresponding bits per
block is eight in Fig. 3.1) and the same bit allocation is used in generating the plot in Fig. 3.2. Similarly, we allocate bits for the other five system such that the number of total bits is eight. In Fig. 3.2, we can see that the proposed method has the smallest bit error rate. The bit error rate of the proposed zero-forcing system is even smaller than the MMI [13], which use a MMSE receiver.
0 2 4 6 8 10 12 0 2 4 6 8 P 0/N0 (dB)
bits per block
proposed method SVD−waterfilling [3] MMI [12] MMSE [12] UNV [13] MSNR [7]
Figure 3.1: Transmission bit rates for a fixed error rate.
3.4
Summary
In this chapter, we have designed the transceiver over an MIMO channel for maximizing transmission rate. The bit allocation and transceiver were jointly optimized subject to a total power constraint for a fixed error rate. Using a high bit rate assumption, we showed that we can simultaneously obtain the optimal bit allocation and transceiver easily. We have demonstrated through simulations
0 5 10 15 20 25 10−6 10−4 10−2 100 P 0/N0 (dB) Bit error rate MSNR [7]
UNV [13] MMSE [12] MMI [12]
SVD−waterfilling [3] proposed method
Figure 3.2: Bit error rate performance.
that the proposed method can indeed achieve a higher transmission rate although the high bit rate assumption is not used in the computation of bit allocation.
Chapter 4
Optimal MMSE Transceivers
with Bit Allocation to Maximize
Bit Rate
In chapter 3, we have designed the ZF transceiver for maximizing bit rate using the high bit rate assumption. In this chapter, we will design the rate-maximizing transceiver for the MMSE receiver. In this design, we do not use the high bit rate assumption as in chapter 3. We will find the optimal solution using the majorization theory. We will show the optimal MMSE receiver is in fact zero-forcing. Based on the optimal solution, we can also develop an algorithm to find the optimal integer bit allocation.
4.1
Preliminaries
In this chapter, we will use the majorization theorem to solve the optimization problem. Some related notation and results from [42] are given in this section. Definition 4.1 [42] Let x, y be n × 1 vectors, and the elements of x and y be
in nonincreasing order. We say x is majorized by y (or y majorizes x) if
Pk l=0xl ≤ Pk l=0yl, 0 ≤ k < n − 1 Pn−1 l=0 xl = Pn−1 l=0 yl, (4.1) and it is denoted by x ≺ y.
Definition 4.2 [42] A real valued function φ defined on an n-dimensional space Ω is said to be Schur-convex on Ω if
φ(x) ≤ φ(y), (4.2)
whenever x ≺ y, for all x, y ∈ Ω.
Proposition 4.1 [42] Let X be an N × N Hermitian matrix with diagonal
ele-ments denoted by the vector d and eigenvalues denoted by the vector λ. Then we
have
d ≺ λ. (4.3)
Proposition 4.2 [42] Schur-convex linear combination. Let f (x0, x1, · · · , xP −1) =
P −1
X
l=0
alg(xl), (4.4)
where x0 ≤ x1 ≤ · · · ≤ xP −1 and assume the following conditions:
1. aP −1 ≥ aP −2 ≥ · · · ≥ a1 ≥ a0 ≥ 0.
2. dg(x)dx ≤ 0 (g(x) monotone decreasing). 3. d2dxg(x)2 ≥ 0 (g(x) convex ).
Then f (x0, x1, · · · , xP −1) is Schur-convex on {x0, x1, · · · , xP −1}.
Proposition 4.3 [42] Let ai, bi, i = 1, · · · , n, be two sets of numbers. Let the
nonincreasing arrangement of ai and bi be ˆai and ˆbi respectively, i.e., ˆa1 ≥ ˆa2 ≥
· · · ≥ ˆan and ˆb1 ≥ ˆb2 ≥ · · · ≥ ˆbn. Then we have n X i=1 aibi ≥ n X i=1 ˆ aiˆbn−i+1. (4.5)
4.2
Problem Formulation
In this section, we will formulate the problem of designing the optimal transceiver for maximizing bit rate. Assume the symbol error rates (SER) are the same for all the subchannels. The problem of maximizing bit rate subject to a total transmit power constraint P0 can be formulated as
maximize G,Λs b =PM −1 k=0 log2 1 + σ2 sk σ2 ek − 1 /Γ subject to Tr(FΛsF†) ≤ P0. (4.6) The following Lemma shows that without loss of generality we can assume the diagonal elements of Λs is either 0 or 1.
Lemma 4.1 For the bit rate maximization problem in (4.6), there is no loss of
generality to assume that σ2
sk ∈ {0, 1}.
Proof: Suppose the system (F, Λs) is optimal for the bit rate maximization
problem in (4.6). In general, Λs is not invertible. If σs2k = 0 for some k, using
(2.14) we have σ2
ek = 0. Let Mr be the number of nonzero elements in Λs. From
section 2.2, we can consider only the subchannels assigned with nonzero power. Let the reduced transmitter be Fr and the reduced power allocation be Λr, where
Λr is invertible. Now, consider a new MMSE system ( ˜F, ˜Λs) which is given by
˜
Λs = IMr, and
˜
F = FrΛ1/2r . (4.7)
Then the transmit power when we use ˜F and ˜Λs is given by
Tr( ˜F ˜ΛsF˜†) = Tr(FrΛrF†r). (4.8)
Clearly, the transmit power of the new system is the same as the original system. Now let’s compare the bit rate of the original system with the new system. For the new system, the MSE matrix is ˜E = Λ−1/2r ErΛ−1/2r , and hence the k-th
subchannel signal to noise ratio is ˜ σ2 sk ˜ σ2 ek = 1 σ2 ekσ −2 sk = σ 2 sk σ2 ek , (4.9)
which is the same as the original system. As the bit loading formulation in (2.20) depends only the subchannel signal to noise ratios, we can conclude that the bit rate of the new system is the same as the original system. Therefore, without loss of generality we can assume σ2
sk ∈ {0, 1}. △△△
In the next section, we will assume σ2
sk ∈ {0, 1} and find the optimal MMSE
transceiver that maximize the transmission rate in (4.6).
4.3
Optimal Transceiver Design
Suppose the P × N channel matrix H has rank K. Let the singular value decom-position of H be
H = U Λ 0
0 0
V†, (4.10)
where the K × K diagonal matrix Λ contains the nonzero singular values of H in nonincreasing order, i.e., λ0 ≥ λ1 ≥ · · · ≥ λK−1. The P × P matrix U and the
N × N matrix V are unitary.
Lemma 4.2 Without loss of generality, the transmitter can be expressed as
F = V A
0
, (4.11)
for appropriateK ×M matrix A. For the choice of F in (4.11), the reduced MSE matrix Er in (2.16) is given by
Er = [N0−1Ar†Λ2Ar+ IMr]
−1, (4.12)
where Ar is obtained by removing the columns of A that correspond to the
sub-channels assigned with zero power.
Proof: As V is unitary, F can always be represented as
F = V A
A1
where A is a K × M matrix and A1 is an (N − K) × M matrix. Define a new transmitter F′ as F′ = V A 0 . (4.14)
Then we have HF = HF′ Using (2.14), we can see that the new MSE matrix E′
is equal to E, i.e., E′
=E. Therefore, the new system has the same subchannel error variances and hence the same bit rate performance. Now, let us compare the transmit power. The transmit power when we use F is
Tr(FΛsF†) = Tr(AΛsA†) + Tr(A1ΛsA†1). (4.15)
Note that the transmit power with F′
is Tr(F′ΛsF
′†
) = Tr(AΛsA†) ≤ Tr(FΛsF†). (4.16)
This means a transmitter of the form in (4.11) is no loss of generality. We can verify that when the transmitter is as in (4.11), the reduced MSE matrix is as given in (4.2).
△△△ Note that the transmitter in (4.11) has the same form as in (3.7) for the ZF case. Using Lemma 4.2, the problem is reduced to the design of the matrix A only. The following Lemma shows that the optimization of A can be further simplified to that of a unitary matrix and a diagonal matrix.
Lemma 4.3 When the the transmission rate in (4.6) is maximized, the MSE
matrix E is diagonal. Then the matrix A in lemma 4.2 is of the form
A = Λ−1QD, (4.17)
for some K × M unitary matrix Q such that Q†Q = IM, and some M × M
diagonal matrix D.
Proof: Let g(x) = log2(1 + (x−1 − 1)/Γ) and x > 0. Then we have ∂g
∂2g
∂x2 ≥ 0 for x > 0. Suppose (Λs, F) is optimal for (4.6). Let Mr denote the
number of nonzero elements in the optimal power allocation. Let (Fr, Λr) be
the reduced system obtained for the optimal solution. Without loss of general-ity, we can assume {σ2
er,k} is in nondecreasing order
1. Using Proposition 2, the
transmission rate b({σ2 er,k}) = M −1 X k=0 log2 1 + 1 σ2 er,k − 1 /Γ = M −1 X k=0 g(σ2 er,k) (4.18) is a schur-convex function on {σ2
er,k}. Suppose Er is not diagonal. Let the
eigen-value decomposition of Er be Er= TΛeT†, where T is unitary and the diagonal
elements of Λe, denoted by λe,k, are in nondecreasing order. Now consider a new
transmitter ˜F = FrT. The new transmission power Tr( ˜F ˜F†) is the same as the
case when we use Fr. Using (2.16), the MSE matrix of the new system is given
by ˜ E = [N0−1F˜†H†H ˜F + I]−1 (4.19) = [N0−1T†Fr†H†HFrT + I]−1 (4.20) = T†ErT (4.21) = Λe (4.22)
The new subchannel noise will be decorrelated when we use ˜F and the subchannel noise variances are λe,0, · · · , λe,M −1. By Proposition 1, we have that {λe,k} ≻
{σ2
ek}. Then by Definition 2, we have
b({λe,k}) ≥ b({σe2k}). (4.23)
That is, a higher bit rate can be achieved when the subchannel noise are
decor-related. This is a contradiction, so Er must be diagonal, which implies E is
1Assume Fr is optimal for the problem in (4.6) and σ2
er,k = [Er]kk is not in nondecreasing order. Let the new transmitter be ˜F= FrP, where P is a permutation matrix. Then the new MSE matrix ˜E is ˜E = PT
ErP. Let P be chosen such that ˜σe2k = [ ˜E]kk is in nondecreasing order. We can verify that the transmit power and bit rate of new system are the same as the case when we use Fr. Therefore, it is no loss of generality to assume σ2 in nondecreasing
diagonal. Using the expression of Er in (4.12), we know that A†rΛ2Ar is
diago-nal. Since A†
rΛ2Ar is diagonal, the columns of ΛAr are orthogonal. Let Mr be
the number of nonzero elements in Λs. We can express ΛAr as
ΛAr = Q0D0, (4.24)
for some K ×Mr unitary matrix Q0 such that Q†0Q0 = IMr, and some nonsingular
Mr× Mr diagonal matrix D0. Rearranging (4.24), we can write Ar as
Ar = Λ−1Q0D0. (4.25)
Note that Ar is obtained by removing some columns of A. Since the columns
removed from A do not affect the transmit power and bit rate, without loss of generality we can assume these columns are zero vectors. Hence A can be expressed as
A = Λ−1QD, (4.26)
where Q is a K × M unitary matrix such that Q0 can be obtained by removing
the columns of Q and D is an M × M diagonal matrix whose diagonal elements
consists of the diagonal elements of D0 and zero. △△△
Using the expression of the matrix A in (4.17), the transmit power can be written as Tr(FΛsF†) = Tr(AΛsA†) = Tr(D†Q†Λ−2QDΛ s) = PM −1 k=0 σs2k|dk| 2[Q†Λ−2Q] kk, (4.27) where dkdenotes the k-th diagonal element of D. In this case, the k-th subchannel
error variance is σ2ek = ( 1 N−1 0 |dk|2+1, if σ 2 sk = 1; 0, if σ2 sk = 0, (4.28)
which depends on dk only but not Q. The bit allocation becomes
bk = log2 |dk|2 N0Γ + 1 (4.29)
Hence the problem in (4.6) becomes maximize Q, |dk|2 b =PM −1 k=0 log2 |dk|2 N0Γ + 1 subject to PM −1 k=0 |dk|2[Q†Λ−2Q]kk≤ P0, |dk|2 ≥ 0, for k = 0, 1, · · · , M − 1. (4.30) In (4.30), dk and the unitary matrix Q are the only free parameters left to be
determined. As the subchannel noise variances do not depend on the unitary matrix Q, changing Q affects only the transmission power but not the bit rate. The following lemma shows us how to find the optimal Q.
Lemma 4.4 One optimal choice of Q for the problem in (4.30) is
Q = IM
0
. (4.31)
In this case, the transmit power can be written as
M −1 X k=0 |dk|2[Q†Λ−2Q]kk = M −1 X k=0 |dk|2[Λ−2M]kk, (4.32)
where ΛM is an M × M diagonal matrix whose diagonal elements consists of the
M largest singular value of H.
Proof: We first establish a lower bound on the expression of the transmit power in (4.27) for any given Q and dk. That is,
M −1 X k=0 |dk|2[Q†Λ−2Q]kk ≥ M −1 X k=0 |dk|2[Λ−2M]kk, (4.33)
where ΛM is an M × M diagonal matrix whose diagonal elements consists of the
M largest singular value of H. The lower bound can be achieved by choosing Q as in (4.31). To prove (4.33), for convenience, we extend the M-point sequence dk to a K-point sequence ˜dk by zero padding, i.e.,
˜
dk = dk, 0 ≤ k ≤ M − 1
Without loss of generality, we can assume that |dk| is in nonincreasing order2,
and thus so is | ˜dk|. Let Q1 be a K ×(K −M) matrix such that the K ×K matrix
Q0 = [ Q Q1 ] is unitary. Then the transmit power in (4.27) can be rewritten
as M −1 X k=0 |dk|2[Q†Λ−2Q]kk= K−1 X k=0 | ˜dk|2[Q†0Λ−2Q0]kk = K−1 X k=0 | ˜dk|2αk, (4.35)
where αk = [Q†0Λ−2Q0]kk. Let {˜αk} be the nondecreasing arrangement of {αk}.
Then by Proposition 4.3 we have
K−1 X k=0 | ˜dk|2αk≥ K−1 X k=0 | ˜dk|2α˜k. (4.36)
Now, let us define a function φ as
φ({˜αk}) = − K−1
X
k=0
| ˜dk|2α˜k. (4.37)
Note that the function φ({˜αk}) is schur-convex on {˜αk}. To see this, let g(x) =
−x, x > 0. Because ∂g∂x ≤ 0 and ∂2g
∂x2 ≥ 0 for x > 0, by Proposition 4.2 we
know that φ({˜αk}) = PK−1k=0 | ˜dk|2g( ˜αk) is schur-convex on {˜αk}. Let γk be the
k-th eigenvalues of Q†0Λ−2Q
0 in nondecreasing order, i.e., γk = [Λ−2]kk. By
Proposition 4.1, we know {γk} ≻ {˜αk} , which implies
φ({˜αk}) ≤ φ({γk}) (4.38)
as φ is schur-convex. This means
K−1 X k=0 | ˜dk|2α˜k≥ K−1 X k=0 | ˜dk|2γk. (4.39)
Using (4.36), (4.39), and the facts that the last K − M elements of {dk} are zeros
and γk = [Λ−2]kk, we have the inequality in (4.33). Now, we will use (4.33) to
2For the case that {|dk|} is not in nonincreasing order, let D′
= PDPT
and Q′ = QPT , where P is the permutation matrix such that |d′k| is in nonincreasing order. We can verify the new transmission rate and the new transmit power for D′ and Q′ are the same as the case when we use D and Q. Therefore, it is no loss of generality to assume that {|dk|} is in nonincreasing order
show that one optimal choice of Q is as given in (4.31). Suppose Q∗ and d∗ k are
optimal. Then using (4.33) we have
M −1 X k=0 |d∗k|2[Q∗†Λ−2Q∗]kk≥ M −1 X k=0 |d∗k|2[Λ−2M]kk. (4.40) Suppose PM −1
k=0 |d∗k|2[Q∗†Λ−2Q∗]kk > PM −1k=0 |d∗k|2[Λ−2M]kk. Consider the new ˜Q
and ˜dk given by ˜ Q = IM 0 , and ˜dk = PM −1 k=0 |d∗k|2[Q∗†Λ−2Q∗]kk PM −1 k=0 |d∗k|2[Λ −2 M]kk 1/2 d∗k. (4.41)
The transmit power of the new system is
M −1 X k=0 | ˜dk|2[ ˜Q†Λ−2Q]˜ kk = M −1 X k=0 |d∗k|2[Q∗†Λ−2Q∗]kk, (4.42)
which is the same as the optimal solution. Since ˜dk > d∗k, the bit rate of the
new system is is larger than that of the optimal system. This contradicts the assumption that Q∗ and d∗
k are optimal for the problem in (4.30). Hence for the
optimal solution Q∗ and d∗
k, the equality in (4.40) must hold and one optimal
choice of Q∗ is as given in (4.31). △△△
Using the expression of transmit power in (4.32), the problem in (4.30) can be simplified as maximize {|dk|2} PM −1 k=0 log2 |dk|2 N0Γ + 1 subject to PM −1 k=0 |dk|2[Λ−2M]kk ≤ P0, |dk|2 ≥ 0, k = 0, 1, · · · , M − 1. (4.43)
To solve (4.43), only |dk|2 remain to be designed. We can use the
Karush-Kuhn-Tucker (KKT) condition [77]. Let |d∗
k|2 be a local maximum. Then there exists
constants α and βk, for k = 0, 1, · · · , M − 1 such that:
3. ∂|d∂ k|2 PM −1 k=0 log2 1 + |dk|2 N0Γ +α(PM −1 k=0 |dk|2[Λ −2 M]kk−P0)+ M −1 P k=0 βk(−|dk|2) |dk|2=|d∗k|2 = 0. 4. α(PM −1 k=0 |dk|2[Λ −2 M]kk− P0) |dk|2=|d∗k|2 = 0. 5. βk(−|dk|2) = 0 |dk|2=|d∗k|2 = 0, for k = 0, 1, · · · , M − 1. By solving condition 2, we have
1 (|d∗
k|2+ N0Γ) loge2
+ α[Λ−2M ]kk− βk= 0. (4.44)
Suppose α = 0. Since |d∗
k|2, [Λ−2M]kk, and loge2 are all positive, using (4.44) we
have βk = 1 (|d∗ k|2+ N0Γ) loge2 > 0. (4.45)
This contradicts condition 2. Hence we have α < 0. As α 6= 0, condition 3 is reduced to
M −1
X
k=0
|d∗k|2[Λ−2M]kk= P0. (4.46)
If βk < 0, then using condition 5 we have |dk|2 = 0. If βk = 0, by (4.44) we have
|d∗k|2 = −1
α loge2[Λ−2
M]kk − N
0Γ, (4.47)
Thus the optimal |d∗
k|2 is given by |d∗k|2 = −1 α loge2[Λ−2M ]kk − N 0Γ + , (4.48)
where (x)+ = max(x, 0), and the constant α is chosen to satisfy M −1 X k=0 −1 α loge2 − N0Γ[Λ −2 M]kk + = P0. (4.49)
The solution in (4.44) is the so-called “water-filling” solution. The number of bits allocated to the k-th subchannel is given by
bk = log2 |d∗ k|2 N0Γ + 1 . (4.50)
From (4.48), we see that for subchannels that correspond to larger singular values of the channel, {|d∗
k|2} is larger and more bits are assigned. Once the optimal
{|d∗
k|2} is obtained by (4.48), the bit allocation in (4.50) can be determined. Using
the choice of Q in (4.31), the matrix A becomes
A = Λ −1 M 0 D. (4.51)
Substituting (4.51) into (4.13), the optimal transmitter is given by
F = V Λ −1 M 0 D, (4.52)
Using the optimal transmitter in (4.52), the optimal receiver in (2.13) becomes
G = ˜D[ IM 0 ]U†, (4.53)
where ˜D is a diagonal matrix whose diagonal elements is [ ˜D]kk =
d∗ k
1 + N0−1|dk|2
. (4.54)
In the optimal solution, only dk depends on the transmit power P0 and the given
error rate. The unitary matrices V, U and the diagonal matrix ΛM depend only
on the channel matrix H. When the optimal transceiver is applied, the output of the receiver is given by
ˆs = GHFs + Gq (4.55)
= Ts + n, (4.56)
where n = Gq and T = GHF. The autocorrelation of n is N0D ˜˜D†, which is
a diagonal matrix. The overall transfer function T is diagonal and the diagonal element is
[T]kk= |dk| 2
1 + N0−1|dk|2
Let Tr be the Mr × Mr diagonal matrix obtained by removing the rows and
columns of T that correspond to the zero diagonal elements. Then Tr is the
overall transfer function of the reduced system, i.e.,
ˆsr = Trsr+ Grq. (4.58)
Since Tr is diagonal, for the same transmitter Fr and signal autocorrelation
matrix Λr, we can a ZF receiver that achieve the same bit rate. Consider a ZF
receiver given by Gr,zf = T−1r Gr. We have Gr,zfH ˜F = IMr. The unbiased signal
to noise ratio of the k-th subchannel for the new system is 1 N0[Gr,zfG†r,zf]kk = 1 N0[T−1r GrGr†T−†r ]kk (4.59) = |[Tr]kk| 2 N0[GrG†r]kk , (4.60)
which is the same as the optimal solution. Thus the bit rate of the ZF system is the same as the optimal solution. This implies the solution of the MMSE transceiver is the same as the ZF transceiver.
In general, the bit allocation obtained in (4.50) is not integer. To obtain the solution with integer bit allocation, we can use the results of [34]. The results in [34] shows the greedy algorithm is optimal when a transceiver with diagonal structure is given. The algorithm is shown below:
Greedy algorithm for integer bit allocation:
Suppose the power constraint P0 is given. Initially, set b0 = b1 = · · · = bM −1= 0.
Define the power increase of k-th subchannel as △pk= N0Γ[Λ−2M]kk(2bk+1− 2bk).
1. Compute △pk for k = 0, 1, · · · , M − 1.
2. Find the index i such that △pk is minimal. Set bi = bi+ 1.
3. Computed the transmit power P =PM −1
k=0 N0Γ(2bk− 1)[Λ−2M]kk. If P < P0,
go to step 1. If P = P0, the optimal bit allocation is {b0, · · · , bM −1}. If
4.4
Simulations
In this section, we evaluate the performance of the proposed method. The number of subchannels M is 4. The channel used is a 4 × 4 MIMO channel (P = N = 4). The elements of H are complex Gaussian random variables with zero mean and unit variance. The noise vector q is assumed to be complex Gaussian with E[qq†] = I
4. QAM modulation is used for the input symbols. In the following
examples, we will use the optimal transceiver in (4.52) and (4.53). In (4.50), the bits are not integers in general. For integer bit allocation, we will use the greedy algorithm to find the optimal solution.
Example 1. Fig. 4.1 shows the transmission rates for different transmit power to noise ratio (P0/N0). The symbol error rates are 10−2 for all the subchannels. The
transmission rates are evaluated for 106 channel realizations. For comparison, we
have also shown the results of five more systems: the bit rate maximizing forcing transceiver in chapter 3, the forcing transceiver in [18], the forcing maximum signal to noise ratio (MSNR) transceiver in [8], and the zero-forcing unit noise variance (UNV) transceiver in [14], and also the results of the optimal transceivers in [13] that using a maximum mutual information (MMI) criterion. In the UNV [14] and MSNR [8] systems, as all the subchannels have the same signal to noise ratios, the same bits are assigned for all subchannels. For the MMI systems, we use the bit loading method mentioned in equation (46) of [13]. For the system in [18], we use the bit allocation in (24) of [13] and then truncate it to be integer. Fig. 4.1 shows that the proposed method can achieve a higher bit rate. For example, when P0/N0 = 12 dB, the number of bits that
can be transmitted is 9 per block for the proposed system, 8 for the system in chapter 3, 7.8, 6, 3, and 2 respectively for MMI [13], [18], UNV [14] and MSNR [8] systems. The proposed method can achieve a higher bit rate the other five systems.
0 2 4 6 8 10 12 0 2 4 6 8 10 P 0/N0 (dB)
bits per block
proposed method chapter 3 MMI [12] [16] UNV [13] MSNR [7]
Figure 4.1: Transmission bit rates for a fixed error rate.
To better illustrate the advantage of the proposed method, we show in Fig. 4.2 the bit allocation when the data blocks are sent over a fixed channel for P0/N0 =
20 dB. The channel matrix H in this case is given by
H =
−0.31 − 0.36i 0.28 − 0.43i −0.91 − 0.73i 0.05 + 0.64i
−0.12 + 0.17i 0.53 − 0.86i −1.65 + 0.94i 0.03 − 0.21i
−0.15 + 0.17i 1.26 + 0.22i 0.64 − 0.30i 1.57 + 0.73i
0.38 + 0.05i 0.86 − 0.95i −1.30 − 0.10i −0.05 − 0.24i
. (4.61)
In the proposed method, the bits are allocated according to the subchannel signal to noise ratios, and 16 bits per block can be transmitted for this channel. 15 and 14 bits per block can be transmitted respectively for the transceiver in chapter 3 and MMI [13]. For UNV [14] and MSNR [8], all subchannels carry the same number of bits. The number of bits that can be transmitted in each block are eight and four respectively.
Example 2. In Fig. 4.3, we plot the bit error rates for a fixed transmission rate. The total number of bits per block is fixed to eight for the same five systems in example 1. For the proposed method, we compute the bit allocation that is obtained when P0/N0 = 11 dB (the corresponding bits per block is eight for the
0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 P 0/N0 (dB)
bits per block
proposed method chapter 3 MMI [12] UNV [13] MSNR [7]
Figure 4.2: Bit allocation for the channel in (4.61) when P0/N0 = 20 dB.
the plot in Fig. 4.3. Similarly, we allocate bits for the other five system as in example 1 such that the number of total bits is eight. The bit error rates are evaluated for 106 channel realizations. In Fig. 4.3, we can see that the proposed
