應用於相關性多輸入多輸出系統之通道建模、估測及前置編碼

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國立交通大學

電信工程學系

博士論文

應用於相關性多輸入多輸出系統

之通道建模、估測及前置編碼

Channel Representation, Estimation and

Precoding for Correlated MIMO Systems

研究生：陳彥志

指導教授：蘇育德博士

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應用於相關性多輸入多輸出系統

之通道建模、估測及前置編碼

Channel Representation, Estimation and Precoding

for Correlated MIMO Systems

研究生：陳彥志

Student:

Yen-Chih

Chen

指導教授：蘇育德博士 Advisor:

Dr.

Y.

T. Su

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Communication Engineering

Hsinchu, Taiwan

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研究生：陳彥志指導教授：蘇育德博士國立交通_{大學電信工程學系博士班} 摘摘摘要要要有別於傳統單一天線系統，多輸入多輸出(MIMO)技術由於能大幅提升通道傳輸容量，因此已被納入目前許多重要的無線通訊標準之中。藉由在傳輸端與接收端設置多根_{天線，我們可以透過分解通道矩陣來創造許多平行通道並用以同時傳輸多個資料} 流。為了充份發揮多天線系統的優勢，特別是要進行高速資料傳輸時，精準的通道狀態資訊通_{常是不可獲缺的。然而，隨著天線個數的增長，估測及處理龐大通道矩陣的} 工作顯得益發困難。在本論文中，我們提出一種簡潔有效的通道矩陣表示法，藉由此表示法我們可以減少在描述通道矩陣時所需的參數數量，同時也減輕後續信號的運算複_{雜度。在中至高度相關的多天線傳輸環境下，使用本文所提出的通道表示法將可大} 幅減少通道狀態資訊的參數個數，同時維持良好的資訊品質。基於所提之通道描述，我們發展了一種遞迴最小平方法來估測幾種典型的多輸出入通道。所得到的通道估測值呈現一個緊緻的形式，該形式將有助於簡化許多需要利用通道矩_{陣估測值進行的後置信號處理程序。此外，藉由調整通道估測器中的一個模型} 階數，我們可以在演算法的估測準確性和計算複雜度之間取得平衡。值得一提的是，由於估測器中維_{度縮減特性所帶來額外的雜訊消除效果，我們將可得到比傳統最小平} 方估測法更優越的均方誤差表現。我們對所提的通道估測器之相關性能也作了理論分析並就不同通道環境進行數值模擬用以評估該通道估測器的效能並證明理論的正確性。為了能_{充份發揮所提通道模式及估測器所帶來的好處，我們更進一步利用該模式發} 展新型的回饋前置編碼系統。由於在設計前置編碼器時引入前述的通道模式，我們可大幅減少回饋前置編碼系統所需的回授頻寬，並降低建構前置編碼器及後置等化器的

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計_{算複雜度。相較於傳統上使用完整即時通道資訊的前置編碼系統，我們的系統只在} 非_{常高訊號雜音比時造成極輕微的性能損失，然而其在縮減回授頻寬和簡化計算複雜} 度上所帶來的好處卻是相對可觀。為了進一步評估系統因為模型通道模式的誤差所帶來_{可能的效能損失，我們在數學上推導了數個效能上界，用以評估訊號接收的均方誤} 差及回授訊號的資訊品質。同樣地，我們也提供了相關的數值結果用以驗証系統效能並証_{實所推導的效能上界的確可以準確預測系統的效能趨勢。}

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Channel Representation, Estimation and Precoding

for Correlated MIMO Systems

Student: Yen-Chih Chen Advisor: Y. T. Su

Department of Communications Engineering National Chiao Tung University

Abstract

Multiple-input multiple-output (MIMO) technology has been included in many in-dustrial standards to achieve significant throughput enhancement compared with conven-tional single antenna systems. By using multi-element antennas at both transmit and receive sides, multiple data streams can be transmitted simultaneously through parallel spatial modes. To realize the advantages of MIMO systems, accurate channel state infor-mation (CSI) is indispensable, especially for high rate transmissions. With the increase of antenna number, the task of estimating or processing a MIMO channel matrix becomes more and more difficult. In this thesis, we propose an efficient channel representation such that the number of required parameters is reduced and the computation complexity can be lessened as well. For medially to highly correlated MIMO environments, the proposed representation can lead to significant parametric dimension reduction while maintaining good CSI quality.

Based on the proposed channel representation, we develop iterative least squared (LS) schemes to estimate several typical MIMO channels. The reduced-rank CSI representa-tion is very useful for many post-channel-estimarepresenta-tion operarepresenta-tions that require processing the instantaneous channel matrices. Depending on the specified modelling order, the pro-posed channel estimators offer tradeoff between identification accuracy and computational complexity. Moreover, the dimension-reduction induced noise rejection effect enables the proposed model-based estimators to achieve superior mean squared error (MSE) perfor-mance over certain SNR region when compared with that of the conventional LS approach.

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Theoretical analysis and numerical simulations of MSE performance are provided to assess the estimators’ performance and validate the analytical predictions.

Taking advantage of the proposed compact CSI representation, we proceed to develop a model-based feedback precoded system. By incorporating our new channel representation into the precoder design, the resulting precoded system provides significant reductions on the feedback bandwidth and the computational complexity needed for constructing the precoder and equalizer matrices. Numerical results show that compared with the conven-tional approaches that need full knowledge of instantaneous CSI, our proposal suffers only negligible performance degradation at very high SNR region. The reductions on comput-ing complexity and feedback channel bandwidth, nevertheless, are significant. To assess the performance of our model-based approach, we establish several bounds regarding the reception error and feedback information loss. Simulated results are compared with these analytical bounds to verify that performance trends can indeed be accurate predicted.

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誌

誌謝

謝

本論文得以完成，首先要感謝恩師蘇育德教授。在漫長的博士班生涯中，老師對於學術領域的_{專注、熱情與不斷學習的態度，激勵著我在研究的道路上前進。除了學} 術之_{外，老師對於學生們不管是在生活上的幫助、經濟上的支援或精神上的開導，都} 使我深感慶_{幸能在老師的照顧下完成學位。此外，感謝在博士班口試期間蒞臨指導的} 口試委員，教授們專業的意見和想法使得本論文得以更加完整。感謝實驗室同儕鄭延修、李昌明、林淵斌、劉人仰在博士班就讀期間給予我學術上的交流和生活上的關照，我從你們身上學習到許多，同時也得到許多美好的生活回憶。感謝最親愛的老婆翠_{如，妳無怨無悔的支持和愛，是我身心上最大的動力來源，} 我_{們一起經歷的磨練終於結為甜美的果實，獻上我最真摯的愛和感激。感謝父母親、} 家人_{及岳父岳母一直以來的鼓勵和照顧，暖暖的親情護佑著我完成攻讀博士學位的夢} 想。最後，謹以此論文獻給所有關心過我的人，在未來的人生道路上，我會更加努力的。

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List of Figures

2.1 “One-ring” model with M transmit antennas at BS and N receive antennas at MS. D: distance from BS to MS. R: radius of the scatterer ring. φ: angle of departure. ∆: angle spread at BS. dT_{: antenna spacing at BS. d}R_:

antenna spacing at MS. . . 5 3.1 MSE performance of Algorithm B as a function of SNR with different

mod-elling orders; solid curves: AS=2◦_{, dotted curves: AS=15}◦_{. . . 31}

3.2 The effect of the modelling order on Algorithm B’s MSE performance in a channel generated by the model described in [1] with AS=2◦_{. . . 32}

3.3 The effect of modelling order on Algorithm B’s MSE performance in a chan-nel generated by the model described in [1] with AS=15◦ _{and f}

dTs=0.031772. 33

3.4 Comparison of theoretical and simulated MSE performance of Algorithm

B in a channel generated by the model described in [1]; AS=15◦ _and

fdTs=0.031772. . . 33

3.5 The effect of the modelling order on the MSE performance of Algorithm B in a channel generated by IEEE 802.11 TGn channel model A; AS=15◦_,

and fdTs=0.0022. . . 34

3.6 The effect of the modelling order (KT) on the MSE performance of

Algo-rithm B in a 3GPP-SCM channel; AS=15◦ _{and f}

dTs=0.02844. . . 35

3.7 MSE performance comparison of Algorithm A (−−) and Algorithm B (−); AS=15◦_{. . . 35}

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3.8 The effect of the update period on the MSE performance of Algorithm B. Channel-1 is based on [2] with fdTs=0.015886 while Channel-2 is based on

[3] with fdTs = 0.0022. AS=2◦, Toc = 1; both Toc and Tow are measured in

EIs. . . 36 3.9 The effect of the angle spread on the MSE performance of Algorithm B in a

channel generated by the model described in [2]; L = 12 and fdTs=0.015886. 37

3.10 MSE performance of Algorithm B in an SCM channel; AS=8◦_{, L = 12 and}

fdTs=0.02844. . . 38

4.1 Basic structure of a general MIMO transceiver. . . 43 4.2 CSI compression rate of the proposed transceiver; M = number of transmit

antennas, KT = modelling order of the transmit spatial correlation. . . 45

4.3 Proposed structure of the model-based MIMO transceiver. . . 48 4.4 MSE performance of DCT-based transceiver; angel spread = 4◦_{, AOD =}

45◦_{, L = 2. . . 55}

4.5 MSE performance of DCT-based transceiver; angel spread = 15◦_{, AOD}

= 45◦_{, L = 2. . . 56}

4.6 MSE upper bounds of DCT-based transceiver; angel spread = 10◦_{, AOD}

= 45◦_{, L = 2. . . 58}

4.7 (a): angle spread = 2◦_{, AOD = 45}◦_{, and (b): angle spread = 4}◦_{, AOD}

= 45◦_. _{: distance between signal subspaces of perfect CSI and}

model-based CSI using a rank-1 approximation; △ : perturbation bound of (4.36). ◦ : perturbation bound of (4.37). . . 59

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Chapter 1 Introduction

Increasing demand for higher wireless system capacity has catalyzed several ground-breaking transmission techniques, among which is the multiple-input/multiple-output (MIMO) technology that has attracted the great part of recent attention. It has been shown that in comparison with conventional single antenna systems, significant capacity gains are achievable when multi-element antennas (MEA) are used at both the transmit and receive sides [4],[5]. Spatial multiplexing techniques, for example, the BLAST (Bell-labs Layered Space-Time) system, was developed to attain very high spectral efficiencies in rich scattering environments.

Ideal rich-scattering environments decorrelate channels between different pairs of trans-mit and receive antennas so that maximum number of orthogonal subchannels is available. In practice, however, spatial correlations do exist and should be considered when design-ing a MIMO receiver for evaluatdesign-ing the corresponddesign-ing system performance [6]. Spatial correlation depends on physical parameters such as antenna spacing, antenna arrange-ment, and scatters’ distributions. Antenna correlations reduce the number of equivalent orthogonal subchannels, decrease spectral efficiency, making it more difficult to detect the transmitted data [4].

A coherent MIMO receiver requires an accurate channel estimate to perform critical operations and provide satisfactory performance. Not only is reliable channel estimation mandatory in guaranteeing signal reception quality but it is also needed in designing an adequate precoder at the transmit side to achieve maximum throughput or minimum bit error rate in feedback MIMO systems. Various pilot-assisted MIMO channel estimators

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have been proposed [7, 8]. Unfortunately, few estimators are specifically designed for correlated MIMO channels and those few exploited only channel’s time and frequency correlation characteristics by approximating the time- and/or frequency-domain responses by an analytic model [8, 9]. These analytic model based approaches can do without the channel information like covariance functions and signal-to-noise ratio which are required by most estimators and are to be obtained by on-line measurements. However, they fail to take into account and the advantage of the spatial structure of such channels which has significant impact on the system performance and should also be explored. The spatial correlation structure instead was often used to analyze the system capacity [10], to design beamformer [11] or pilot sequences [7, 12].

We present novel pilot-assisted channel estimation schemes on the basis of the pro-posed new general MIMO channel representation which does not require information of second-order channel statistics. Spatial and time covariance (or correlation) functions are described by nonparametric regression and the influence of the mean angle of departure (AoD) is related to other channel parameters via a regression model. This representation admits a reduced-rank channel model and compact channel state information (CSI) repre-sentation, making possible reduced feedback channel bandwidth requirement. It results in separable descriptions of channel correlations and mean AoD for correlated MIMO systems and enables us to develop efficient algorithms to identify the realistic channel responses. Although a model-based scheme inevitably induces a modelling error [9]-[13], as will be shown in Chapter 3, our algorithms are capable of describing realistic correlated MIMO channels with negligible modelling errors. The estimated CSI can be efficiently exploited for use in many channel estimation related operations such as MIMO data detection and optimal MIMO transceiver designs.

Optimal MIMO transceiver designs based on CSI at the transmitter (CSIT) have been thoroughly studied under several performance measures such as minimum mean squared error (MMSE) or maximum mutual information (MMI) [14]-[11]. When CSI is available at both ends of a link, conventional precoding-eigen-beamforming schemes can adapt to the channel condition to optimize the reception performance in the correlated

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environment. However, in practice, downlink CSI (from base station to mobile unit) is often not available at transmit site and has to be estimated unless the channel transfer function can be assumed to be identical in both directions. Oftentimes, the downlink receiver has to send the information back to the transmitter through a feedback channel. It is thus critical that one control the amount of feedback information as the feedback channel usually has a very limited bandwidth.

To lessen the feedback load, several transmitter precoding/eigen-beamforming schemes based on partial channel information such as channel mean feedback and channel covari-ance feedback were proposed to reduce the feedback cost [15]-[16]. Mean feedback relies on the proposition that CSI resides in the mean of the distribution with white covariance. Therefore, only for very slowly faded channels can mean feedback adequately capture the channel behavior. On the other hand, covariance feedback models the channels as random vectors with zero mean and non-white covariance, which are only hold for rapid fading environments. Both feedback scenarios relies on imperfect long term statistical models and thus cannot well represent the instant or short term channel variations. Moreover, prior knowledge of channel statistics are often needed to compute the approximated feed-back information. Generally, systems using statistical feedfeed-back come with a non-negligible performance loss compared with those using instantaneous channel realization.

Based on the proposed model-based channel representation, the instantaneous CSI is represented in a more compact form and estimated accordingly. With this efficient CSI estimation, we present a framework of transceiver design to render the advantage of the proposed model-based structure. For correlated MIMO channels, the proposed precoding scheme provides an alternative to reduce the requirement of instantaneous CSI feedback, while retaining or even improving the reception performance. Several perfor-mance bounds regarding reception error and feedback information loss are established to assess the system performance.

The rest of this thesis is arranged as follows. After a brief review of the typical space-time antenna setup and a general received MIMO signal model, we derive two new models [17] for spatial-correlated block-faded narrowband MIMO channels and their

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rela-tions with some established analytic models in Section 2.3. We then propose single-block based iterative least squares (LS) channel estimators in Section 3.1 while the extension that takes the time-correlation and frequency-selective cases into account are given in Section 3.2 and 3.3, respectively. In Section 3.4, we analyze the mean squared error (MSE) of the proposed channel estimation algorithms. Numerical examples using indus-trial standard approved channel models are given in Section 3.5 to validate the proposed channel models and to demonstrate the effectiveness of our algorithms. In Chapter 4, we develop the basic framework of transceiver designs based on the reduced rank CSIT. Section 4.1 quickly reviews the channel representation proposed in Section 2.3 as the foundation for the proposed MIMO eigen-beamforming system. Section 4.2.1 and Section 4.2.2 give a brief review of some conventional MIMO precoder/beamforming systems with feedback CSI. Section 4.2.3 make use of the proposed channel representation to establish a nonparametric CSIT. With the nonparametric CSIT, the proposed eigen-beamforming design is developed and discussed in Section 4.2.4. Performance analysis of the proposed beamforming method is given in Section 4.3. In Section 4.5, we provide several numer-ical and simulation examples by using some well-established industrial channel models. Conclusion and remarks are given in Section 4.6. Chapter 5 summarizes the studies in this thesis and suggests some interesting research subjects under the framework of the proposed nonparametric scheme.

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Chapter 2 MIMO Channel Representation

2.1 MIMO System

In this thesis, we focus on the clustered channel model. In such a MIMO setup, MS is surrounded by local scatterers and waveforms impending the receive antennas are richly scattered. On the other hand, BS is often unobstructed by local scatterers and has a mean angle of departure (AOD) with respect to the receiver cluster. The clustered channel setup is typical in urban environments, and has been validated through field measurements. A typical “one-ring” model is shown in Fig. 2.1, ∆ denotes the azimuthal angle spread at the BS and φ denotes the mean AOD between BS and MS.

D f D R dT dR BS at antennas transmit M MS at antennas receive N

Figure 2.1: “One-ring” model with M transmit antennas at BS and N receive antennas at MS. D: distance from BS to MS. R: radius of the scatterer ring. φ: angle of departure. ∆: angle spread at BS. dT_{: antenna spacing at BS. d}R_{: antenna spacing at MS.}

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2.2 Modelling Spatial-Correlated MIMO Channels

2.2.1 System Setup

Consider a cellular MIMO system in which the base station (BS) and a mobile station (MS) are equipped with linear arrays of M and N antennas, respectively. Independent data streams x(t) = [x1(t), x2(t), x3(t), · · · , xM(t)]T are transmitted from the BS at time

t, where xm(t) denotes the source signal of the mth transmit antenna and the superscript

T denotes vector (matrix) transposition. At the MS, the received baseband signals are given by y(t) = [y1(t), y2(t), y3(t), · · · , yN(t)]T, where yn(t) is the signal received by the

nth receive antenna at time t. With a sampling interval of △t seconds, the corresponding ith transmit and receive sample vectors are xi = x(i△t), and yi = y(i△t), respectively.

2.2.2 Wireless MIMO Channels

A general MIMO channel between BS and MS antennas is modelled as H(t) =

G

X

l=1

Hlδ(t − τl), (2.1)

where G is the maximum number of paths associated with any sub-channel between a transmit and receive antenna pair, τl is the delay of the lth path, and δ denotes the

Dirac delta function. The complex channel gain matrix associated with the lth path is given by Hl = [hlij], for 1 ≤ i ≤ N, 1 ≤ j ≤ M, where hlij is the complex

sub-channel gain between the jth transmit and ith receive antennas. For a narrowband fading channel, (2.1) is reduced to a single-tape fading matrix and the received vector waveform is y(t) = H(t)x(t) + n(t), where H(t) is an N × M complex channel matrix and n(t) a zero mean additive white Gaussian noise (AWGN) vector with covariance matrix E{nnH_{} = N}

0IN. We first consider the block fading case in which the channel

gain matrix remains unchanged within a block of B symbol intervals and eliminate the time parameter t in related expressions. Section IV will discuss the case which takes the time-correlation among blocks into consideration.

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2.2.3 Spatial-correlated block fading channels

Many analytic models for spatial-correlated MIMO channels have been proposed in the literatures. The Kronecker model [6] assumes separable statistics at transmitter and receiver so that the spatial correlation matrix Φ of vec(H), vec(·) being the stacking operator, is given by the Kronecker product (⊗) [18] of those of the transmit (ΦT) and

receive (ΦR) antennas, Φ = ΦR ⊗ ΦT = Φ

1

2(Φ12)H, where the “square root” matrix Φ12

has a similar decomposition Φ12 = Φ 1 2

T⊗ Φ

1 2

R. The separable statistics assumption yields

H= Φ12

RHwΦ

1 2 T

T , (2.2)

where Hw is an N × M channel matrix whose entries are i.i.d. complex zero-mean,

unit-variance Gaussian random variables.

Although the Kronecker model is mathematical tractably, many measurement and theoretical results reveal that this separable model in general leads to misfits for capacity and error probability due to the smaller number of degrees of freedom (DF) [19],[20]. The Kronecker model has been generalized by Sayeed [21] and, more recently, by Weichsel-berger et al. [22] who considered joint correlation of both link ends and suggested the following analytic model

H= UR

˜

Ω_{⊙ R}UT_T, (2.3)

where UT and UR are the eigenbases of the one-sided correlation matrices at the transmit

and receive sites, respectively. Operator ⊙ denotes the Hadamard product operation [18]. R denotes a random matrix whose elements are i.i.d. zero-mean, unit-variance complex Gaussian random variables. ˜Ω is the element-wise square root of the coupling matrix in which each entry specifies the mean amount of energy coupled with an eigenvector of the transmitter to that of the receiver. The Weichselberger model provides a more general framework of canonical modelling [22],[23],[24], where (2.3) can be represented by the following canonical form,

H= URHindUTT. (2.4)

Hind has independent, but not necssarily identically distributed entries. The Kronecker

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model by the following equations H = Φ 1 2 RHwΦ 1 2 T T canonical (2.4) = URD 1 2 RHwD 1 2 T | {z } : Hind UT_T (2.5) Weichselberger (2.3) = UR((diag(D 1 2 R)diag(D 1 2 T)T) | {z } : ˜Ω ⊙Hw)UTT, (2.6)

where ΦT = UTDTUHT and ΦR = URDRUHR denote the eigen decomposition of

correla-tion matrices at transmitter and receiver, respectively. (2.5) follows from the isotropicity of an i.i.d. random matrix under an unitary transformation. Note, the DF of ˜Ωin (2.6) is N + M, while ˜Ω of the general Weichselberger model in (2.3) has DF equal to NM. The small number of DF explains the deficiency of the Kronecker model as described above and is mainly due to the lack of modelling the cross-correlation between transmit-ter and receiver sides. In the following, we will develop a channel representation which takes the Kronecker, Weichselberger and canonical model as special cases, and is useful for reduced-rank processing.

2.3 Channel Representation

An N ×M matrix H always admit the singular value decomposition (SVD), H = UΛVT, where U is an N × N unitary matrix, V is an M × M unitary matrix, and the diagonal matrix Λ is N × M with non-negative entries. When H is random, its SVD component matrices are random and depend on the sample (matrix) value of H. As U and V can be transformed into two predefined unitary matrices QR and QT by UP1 = QR and

VP2 = QT, with both transforms P1 and P2 being unitary, we have

H= QRP−11 Λ(P−12 )TQTT = QRCQTT (2.7)

and the only random component is C. For the Weichselberger model, the predefined matrices are eigenbases of the one-sided correlation matrices while Sayeed’s virtual channel representation uses the DFT bases.

Let Φ12

T def

= [φT(i, j)], where φT(i, j) represents the root spatial correlation between ith

and jth transmit antennas. As the M column vectors of Φ

1 2

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subspace, we have

Φ1/2_T = QTΛT, (2.8)

where QT is an unitary matrix and the coefficient matrix ΛT can be obtained by the

Gram-Schimdt orthonormalization procedure. The above equation implies φT(i, j) =

PKT

k=1λ j

kqk(i), where qk(i) is the ith element of the kth basis vector, λjk is the projection

of the jth column on qk.

Using a similar decomposition for Φ1/2_R leads to

Φ1/2 = (QTΛT) ⊗ (QRΛR) = (QT⊗ QR) (ΛT⊗ ΛR) ,

where we have invoked the identity [18],

(A1⊗ B1)(A2⊗ B2) · · · (Ak⊗ Bk) = (A1A2· · · Ak) ⊗ (B1B2· · · Bk). (2.9)

From the canonical representation, vec(H) = Φ12vec(H_w), we obtain

vec(H) = (QT⊗ QR) (ΛT⊗ ΛR) vec(Hw) def

= (QT⊗ QR) vec(C). (2.10)

The identity

vec (ABD) = DT _{⊗ A}vec (B) (2.11)

implies vec(H) = vec QRCQTT

, and so H = QRCQTT, which is the same as (2.7).

We summarize the above derivation on the relation between the proposed analytic model with the Kronecker, Sayeed, and Weichselberger models in

Proposition 2.1. _{An N × M MIMO channel matrix H, can always be expressed as}

H= QRCQTT (2.12)

where C is a complex random coefficient matrix, QR and QT are predefined unitary

ma-trices. The above model is equivalent to the Kronecker model if the matrix C satisfies the separable correlation condition

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where ΛT and ΛR are coefficient matrices that depend on the spatial correlations among

the transmit and the receive antenna arrays, respectively. (2.12) is related to the Weich-selberger model via

UT = QTPHT, UR = QRPHR (2.14) PH_RΓRPR = E CCH , PH_TΓTPT = E CTC∗ (2.15)

where PT, PR are unitary matrices and ΓR, ΓT have the same eigenvalues of the matrices

EHHH and EHTH∗ , respectively. When the predefined matrices are the same as U_R and UT, C has the special form ˜Ω⊙ R. Moreover, (2.12) is equivalent to the virtual

representation of Sayeed if columns of QR and QT are DFT basis vectors and entries of

C are independent complex Gaussian random variables.

[1] suggested and [25] verified through field measurements that the mean direction of arrival (DoA) can be embedded in the channel model by pre-multiplying the channel matrix H by a diagonal matrix which is a function of the DoA. We can derive a similar model by invoking the fact that if W is a diagonal matrix with unit modulus entries and V is unitary then both VW and W−1V are also unitary, to obtain the alternative representation (2.16).

Corollary 2.1. An equivalent channel matrix for stationary frequency-flat MIMO channel is given by

H= QRCQ

T

TW (2.16)

where QT_TW= QT

T and W is a diagonal matrix with unit modulus entries.

Several remarks and observations on the channel models (2.12) and (2.16) are given below.

R1. The Kronecker model requires that C has the special structure (2.13) while the Weichselberger, Sayeed and canonical models demand that the entries of C be in-dependent (but not identical) random variables. In contrast, the proposed model does not impose any constraint on the coefficient matrix C and is valid for arbitrary block-faded H.

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R2. The Weichselberger model is perhaps more convenient to generate the matrix chan-nel H and for evaluating the chanchan-nel capacity of correlated MIMO chanchan-nels as the coefficient matrix has independent entries. It is also useful to analyze MIMO system performance. However, it is not suitable for channel estimation applications because the number of parameters, including the unknown eigenbases, is even larger than that of H.

R3. For practical correlated MIMO channels, which are of particular concern, the entries of H are not i.i.d. but correlated random variables and H admits reduced-rank representation. That is, although H is likely to be of full rank, one can approximate it by reduced-rank unitary matrices (so is the coefficient matrix), ignoring the weaker eigenmodes. The rank-reduction is most obvious for typical urban macro-cellular environments in which an MS is surrounded by local scatterers, and waveforms impending the receive antennas are richly scattered, while the BS is not obstructed by the local scatterers [6][26]. Appendix A shows that, if the angle spread (AS) ∆ is not too large, the diagonal matrix W

W= diag [w1, w2, · · · , wM] , (2.17)

has entries of the form wi = exp

h

−j2π(i−1)dλ sin φ

i

, d being the inter-element dis-tance, that bear the mean AoD information. As will become clear later, the sep-arability of channel correlation and angle information characterizations has some useful implications.

R4. Given predefined bases QR, QT, or QT, the statistic properties of the corresponding

coefficient matrix is completely determined by those of H. Identification of the unknown channel H is equivalent to the estimation of C or the pair (C, W), which usually has a lower rank and much smaller number of entries than those of H for the link environment of interest. Thus, using model (2.12) or (2.16) reduces the number of parameters to be estimated and enhances the performance. Moreover, as the bases in both (2.12) and (2.16) are pre-defined, these two models can be easily extended to time-varying block fading and frequency-selective fading environments.

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R5. There are several classes of basis functions to choose from. The Taylor and Weier-strass arguments and the results of [27] suggest the use of polynomial bases. If we use polynomials of degree P as basis functions in expanding a spatial correlation function of length P , the corresponding basis matrix PP has entries

[P]_i,j _{= (i − 1)}j−1, i, j = 1, 2, . . . , P, (2.18) Although the column vectors in (2.18) form a basis, they are not orthogonal. Fur-thermore, these vectors have different norms, which might result in numerical insta-bility. By applying the QR decomposition to the corresponding PP [28], we obtain

an orthonormalized polynomial basis matrix Po. The basis matrices QM,KT and

Q_N,K_R of (3.2) or QL,KL of (3.23) are obtained by selecting the first KT, KR or KL

columns of the corresponding Po.

R6. For a fixed base one needs to determine the modelling orders, KT and KR.

Ei-ther the Akaike information criterion (AIC) and the minimum description length (MDL) approach can be used to determine the optimal modelling orders that trade-off the system complexity and performance [29]. Time domain modelling order KL

discussed in Section IV can also be similarly determined. Depending on the appli-cation scenario, these order parameter values can be obtained by an one-shot open loop estimate or should be periodically updated.

R7. The model (2.16) is especially useful for channel estimation application because, as will be shown in the next section, it allows very efficient (in terms of convergence rate) channel estimation algorithms that iteratively estimate C and W separately, and, at low SNR’s, the reduced-rank model gives performance superior to that of the full-rank model. Furthermore, for a small-to-medium AS, which occurs quite often in cellular downlinks, the extracted AoD information can be feedback for downlink beamforming.

R8. Our simulation experiments indicate that, when the AS △ becomes large, the rank of C increases and there is no dominant spatial angle. The steering matrix W

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becomes an identity matrix which gives no AoD information and (2.16) degenerates to (2.12).

R9. The proposed channel representation for single-block frequency-flat MIMO channels, i.e., (2.16), can also be extended to the cases of variant frequency-flat and time-variant frequency-selective fading channels by properly modelling the time-domain correlation. These extensions are given in Section 3.2 and Section 3.3 respectively.

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Chapter 3 Model-based MIMO Channel

Estimation

This chapter presents novel schemes for estimating correlated input multiple-output (MIMO) fading channels. Our schemes are based on an analytic correlated block fading model and its time-variant extension which encompass the popular Kronecker model and the more general Weichselberger model as special cases. Both static and time-variant models offer compact representations of spatial- and/or time-correlated channels. When the transmit antenna array is such that the associated MIMO channel has a small angle spread (AS), which occurs quite often in a cellular downlink, our models admit reduced-rank channel representations. They also enable us to develop effective estimators and provide compact channel state information (CSI) descriptions which are needed in feedback systems and for many post channel estimation applications. The latter has the important implication of reduced feedback channel bandwidth requirement and lower post-processing complexity.

We propose iterative algorithms for estimating static and time-variant MIMO chan-nels. The proposed models make it natural to decompose each iteration into two succes-sive stages that are responsible for estimating the correlation coefficients and the signal direction, respectively. Both spatial- and time-correlated fadings are considered. The mean-squared error (MSE) performance of our estimators are analyzed as well. Using popular industry-approved standard channel models, we verify through simulations that our algorithms yield offer good MSE performance which, in many practical cases, is better than that achievable by a conventional least-square estimator.

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3.1 Single-Block Based Channel Estimation

In this section we consider estimation schemes which are based on a single block of ob-servation without taking into account the (time-)correlation among blocks. We propose two iterative schemes in which an iteration consists of two phases. The first phase is responsible for the estimation of the coefficient matrix, C, while the directional matrix, W in (2.16), is estimated in the second phase. Both tentative estimates are updated as one proceeds with each new iteration until the stopping criterion is met. The two schemes differs in the second phase only.

Consider the M × B matrix X = [x1, x2, · · · , xB] formed by B length-M input symbol

vectors, where B ≥ M. Assuming H remains static during a B-block period, we express the received sample block, Y = [y1, y2, · · · , yB] as

Y = HX + N, (3.1)

where N = [n1, n2, · · · , nB] is the corresponding noise matrix whose entries are i.i.d.

zero mean complex Gaussian random variables. In estimating H, X is assumed to be composed of either the pilot vectors or some decision feedback results. Substituting two known unitary matrices QM,KT and QN,KR with ranks KT(≤ M) and KR(≤ N) for QT

and QR in (2.16), we want to find the optimal solution {Wopt, Copt} to the problem

arg min

W,C kY − QN,KRCQ T

M,KTWXk

2 _(3.2)

We express the corresponding optimal (least-squares) channel estimate in terms of Wopt

and Copt

H_opt = QN,KRCoptQ

T

M,KTWopt (3.3)

so that (3.1) can be rewritten as

Y = HoptX+ ∆HX + N

def

= HoptX+ eN, (3.4)

where eN represents the sum of the modelling error ∆HX due to the reduced rank repre-sentation and the AWGN vector, N.

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To derive an iterative algorithm for obtaining the joint directional and channel solution {Wopt, Hopt}, we assume that, at the (i − 1)th iteration,

Y = bH_i−1X+ ∆ bH_i−1X+ eN (3.5)

where ∆ bHi−1 def

= Hopt− bHi−1 is the residual error at the end of the (i − 1)th iteration,

and consider the estimation of the channel (coefficients) and AoD in two separate phases.

3.1.1 Phase I - Coefficient Estimation

Assume that the directional matrix in this phase is optimum, i.e., W = Wopt . From

(3.1) and (3.3), we have vec(Y) = (WoptX)TQM,KT ⊗ QN,KR vec(C) + vec(N). (3.6)

Substituting the definition Z def= ((WoptX)TQM,KT) ⊗ QN,KR into (3.6), we have the LS

solution

vec( bC) = (ZHZ)−1ZHvec(Y)def= F (Wopt). (3.7)

While the optimal directional matrix estimate is not available, we replace it by the tenta-tive estimation from the previous iteration, Wi−1. vec( bC) is then obtained by computing

F (Wi−1) instead, and the corresponding tentative estimate is denoted by bCi. Initially,

we can arbitrarily set W0 to be an identity matrix.

3.1.2 Phase II - Direction Estimation

Similar to Phase I, we begin with the assumption that the coefficient matrix in this estimation phase is optimum. The directional information is to be obtained by estimating a diagonal matrix W with unit modulus entries; see (2.17). Setting

Gdef= QN,KRCoptQ

T

M,KT (3.8)

and invoking (3.3), we have bHi−1 = GcWi−1. As Copt is unavailable, Copt is replaced by

the previous estimate bC_i−1 in computing G during the ith iteration. In the following, we propose two algorithms to estimate the phase of the unit modulus diagonal entries of W.

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3.1.2.1 Algorithm A - Maximum Matching Output

To estimate cWi in diagonal form, we start with the following lemma whose proof is given

in Appendix B.

Lemma 3.1. _{For two matrices A and B of size N × M and M × E respectively, and an} arbitrary vector c of size M × 1, the following identity holds.

vec (A · diag(c) · B) =(1E⊗ A) ⊙ BT ⊗ 1N

c, (3.9)

where “diag” denotes the diagonal operation used to translate a vector into a diagonal matrix, with its diagonal terms being the elements of the original vector.

Combined with matrix G defined in (3.8), (3.4) is rewritten as

Y = GWoptX+ eN. (3.10)

Let woptbe the column vector that consists of the diagonal elements of Wopt, i.e., wopt(i) =

W_opt_{(i, i), for any 1 ≤ i ≤ M. Then, by Lemma 3.1, we have} vec (Y) = (1B⊗ G) ⊙ XT ⊗ 1N

w_opt+ vecNe (3.11)

and the LS estimate of woptis given by bwLS = T†·vec(Y), where

(1B⊗ G) ⊙ XT ⊗ 1N

def

= T.

In order to extract the steering vector bw, we introduce v(θ)def= 1, v(θ), . . . , vM −1_(θ)T_,

where v(θ) = exp_−j2πd_λsin(θ). The AoD information bφ is retrieved by maximizing the matching output b φ = arg max −π≤θ≤πRe n P_{( b}_w_LS₎H _v(θ)o_, _(3.12)

where P(·) is defined by the following phase extraction operator, P _[a₀_ejb0_{, a}

1ejb1, · · · , aKejbK]

def

= [1, ej(b1−b0)_{, · · · , e}j(bK−b0)_],

for {ai}Ki=0∈ RK+1 and {bi}Ki=0 ∈ [0, 2π). (3.13)

Once bφ is available, it is straightforward to obtain cW = diag(v(bφ)). Solving (3.12) over [0, 2π) can be accomplished by using the conventional line searching algorithm.

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Computing bwLS in (3.12) involves a pseudo-inverse operation of matrix T, and is thus

computational expansive. Thanks to the special structure of T, a training matrix with orthogonal rows can be used to bypass the calculation of pseudo-inversion. Note that

THT = (BGHG_{) ⊙ (NX}∗XT), (3.14)

the right-hand side of (3.14) will become a diagonal matrix with nonnegative real elements if X∗_XT _{= BI. Such an orthogonality is guaranteed provided that the optimal training}

matrix for LS estimator is used [7]. Under the assumption of orthogonal training matrix, we have

P_{( b}_w_LS_{) = P T}†_{· vec(Y)}_{= P T}H _{· vec(Y)}def_{= P}ebw_LS_, _(3.15) where ewb_LS doesn’t require the cumbersome matrix inversion. The AoD information can thus be obtained simply by substituting ew_b_LS _{for b}w_LS in (3.12).

3.1.2.2 Algorithm B - Root Finding Method

An alternative way to find the optimal phase is to convert (3.12) into a root finding problem. Note that the elements of wopt are of geometric progression, i.e., they form a

row vector of a Vandermonde matrix. Hence if we define the correlation polynomial

P (z)def= P( bw_LS)H_z_{− M,} _(3.16)

where z = [1, z, . . . , zM −1_{] and let Z be the set of its zeros in the complex plane, then}

solving (3.12) is equivalent to b z = arg min z∈Z |(|z| − 1)| and bφ = sin −1 −Arg{bz}λ 2πd (3.17) and the directional matrix is reconstructed by cW_{= diag(bz), where bz = [1, b}_{z, · · · , b}zM −1_].

Unlike Algorithm A whose solution accuracy relies on the resolution the numerical search algorithm used, this algorithm gives the exact analytic solution once (3.16) is solved. Sim-ilar to Algorithm A, ew_b_LS _{can be substituted for b}w_LS in (3.16) to simplify the computation. Since the object function in (3.2) is jointly convex with respect to C and W and the proposed algorithms have the form of a nonlinear Gauss-Seidel algorithm, the con-vergences of our algorithms are guaranteed [30]. All the simulation examples reported

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in Section 3.2 converge and achieve the theoretical performance lower bound derived in Section 3.4.

The computation complexity of the proposed algorithm is dominated by the LS oper-ations in Phase I. The flop count of the LS operation in Phase I is O(BK2

T), KT ≤ M

while the conventional LS estimator needs O(BM2_{) flops [31]. The complexity of Phase}

II is mainly contributed by the product of T and vec(Y), and is in general much less than that of Phase I, thanks to Eq. (3.15) and the special structure of T. Therefore, the total complexity of the proposed algorithm is less than that of the conventional LS. Moreover, except for static channels, the estimates for both W and C need to be updated periodically. Let each B−symbol interval be called an estimation interval (EI). Since the mean AoD usually change much slower than the channel coefficients (gains) variation, updating frequencies for W and C can and should be different, i.e., if the two estimates are updated every Tc

o and Tow EIs, respectively, then Tow ≫ Toc (see Fig. 3.8 of Section

3.5). This dual updating frequency option is unique to our approach and implies that Phase II may be disabled most of the time while Phase I needs single iteration per update EI, hence our algorithm can be computational more efficient than the conventional LS approach for many non-static channels.

The major advantage of our channel model and estimator lies not only in the computa-tional efficiency of the channel estimator but also in the compactness of CSI representation which is needed in a feedback system and that of post processing operations. As men-tioned in R3 and R4 in the previous chapter, a small KT is often sufficient to accurately

describe a MIMO channel with high transmit spatial correlation. For any post channel es-timation operation associated with H, e.g., taking pseudo-inverse or eigen-decomposition of H, the computing load is reduced as it involves the KR× KT coefficient matrix and

the estimated AoD instead of the original N × M channel matrix.

3.1.3 Order Determination for Block-Fading Channels

The remark R4 in Chapter 2 tells us that we can choose to use low-rank bases to closely approximate H provided that the modelling orders KT and KRare properly selected. We

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tends to give robust and reliable results, especially for small sample size. Taking into account the proposed channel estimator, the AIC-based order determination scheme is given by [29], [KT, KR] = arg min 1≤KT≤M,1≤KR≤N N log RSSKT,KR N + 2(KT + KR). (3.18)

where RSSKT,KR is the squared error of (3.2) associated with the modelling orders KT

and KR. Instead of using the instant sample error, we can use the time-average squared

error in calculating the AIC solution to obtain a more reliable estimate.

Since the channel statistics varies much slower than the instantaneous channel strength, the update period of the modelling order is much longer than that of the instant channel estimate; reducing the overhead required by order estimation. Similar order determination scheme can be used to estimate the modelling order for time-correlated fading channels, provided that RSSKT,KR of (3.2) is calculated using the time-correlated model (3.24) given

in the next section and the frequency-selective case (3.38) discussed in Section 3.3. More-over, the optimal time domain modelling order can also be determined by incorporating KL discussed in the next section into the degrees of freedom in AIC’s formula.

3.2 Channel Estimation with Time Correlation

Con-sideration

We now extend our investigation to the case that considers the time correlation among blocks. Similar to our spatial modelling approach, we use a set of orthonormal basis functions to describe a snap shot of a fading channel’s time domain behavior. We assume an equally spaced pilot-block arrangement. The issue of the optimal pilot arrangement that minimizes the MSE or bit error rate (BER) was addressed in [12] and [32].

Assume the two leading pilot symbol vectors of two consecutive pilot block is T symbol intervals away. The receive signal block at time nT can be written as

Yn= HnXn+ Nn (3.19)

where Yn = Y(nT ) and Xn = X(nT ) are the N × B receive matrix at time nT and the

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entry represents the link gain between the ith transmit and the jth receive antennas at time nT .

We consider the time-variant behavior of a MIMO channel within a fixed observation window of L blocks (EIs). The received sample blocks from nT to (n + L − 1)T can be cascaded into the matrix

Yn,L def = [Yn, Yn+1, . . . , Yn+L−1] . (3.20) Using (2.11), we obtain vec(Yn,L) = XTn,L⊗ IN · vec (Hn,L) + vec (Nn,L) (3.21) where vec(Hn,L) def = vec(Hn)T, . . . vec(Hn+L−1)T T , vec(Nn,L) def = vec(Nn)T, . . . vec(Nn+L−1)T T , and XT_n,Ldef=      XT_n 0 _{· · ·} 0 0 XT_n+1 ... 0 ... ... . .. ... 0 0 _{· · · X}T_n+L−1     .

Substituting (2.12) for each Hn and assuming the eigenbases QT and QR remain invariant

during an estimation period, we obtain

vec(Hn,L) = (IL⊗ QT⊗ QR) Γn,L. (3.22)

Each component of the vector Γn,L= [γn, γn+1, · · · , γn+L−1]T is itself an (NM)×1 column

vector γn= γ1n, γ2n, · · · , γ(N M )n

T

that represents the complex fading coefficients for all NM MIMO subchannels at time nT and, γpn, 1 ≤ p ≤ NM, are independent.

The stacked vector, γ(p) = γpn, γp(n+1), · · · , γp(n+L−1)

T

, represents a finite-duration sample of the complex random process associated with the pth subchannel [25]. Such a process can also be expanded by a set of smooth functions [8, 33], and thus its estimation can be obtained by using a method similar to that developed in the previous section. Similar to the approach used in Section 2.2.3, we can first apply the orthogonal transform γ(p) = QLbpn, where QLis an L × L orthogonal matrix, and bpn is the transform domain

coefficient vector. Then, the time domain channel correlation can be approximated by using the reduced basis matrix QL,KL

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where KLdenotes the time domain modelling order, and cpn is a KL×1 coefficient vector.

By using (2.16), (3.22) and the approximation (3.23), we decouple the signal part of (3.21) into the product of two modelling domains - space and time domains

vec( ¯Yn,L) ≈ XTn,L⊗ IN QL,KL⊗ (W T_Q T) ⊗ QR ccoef ≈ XT_n,L_{⊗ I}_N Q_L,K_L_{⊗ (W}TQ_T,K_T_{) ⊗ Q}_R,K_R˜ccoef def = ((WLXn,L)TQeT,KT) ⊗ QR,KR ˜ccoef (3.24) where WL def = (IL⊗W), eQT,KT def

= QL,KL⊗QT,KT and QT,KT and QR,KR are composed of

KT and KR column vectors of QT and QR, respectively. W is the steering matrix defined

in (2.17). Since the mean AoD usually varies slowly with respect to a sub-channel’s coherent time, we assume that W remains the same during a period of L data blocks. Similar to the narrowband model (2.16), we do not impose the implicit Kronecker structure and Gaussian assumption on ˜ccoef.

As (3.24) can be obtained by replacing X, Y, W, vec(C), QM,KT, and QN,KR in (3.6)

by Xn,L, Yn,L, WL, ˜ccoef, eQT,KT, and QR,KR, we conclude that both spatial and time

correlations can be described by similar models. Hence, the two-phase iterative estimation scheme developed in Section 3.1 can be extended to estimate the coefficient vector ˜ccoef,

and the directional matrix WL in (3.24). In the following, we describe two-phase channel

estimation schemes with time correlation consideration.

3.2.1 Phase I - Coefficient Estimation

Following an argument similar to that used in Section 3.1, we assume that the directional matrix WL is optimal in the coefficient estimation phase and define

e

Z def= (WL,optXn,L)TQeT,KT

⊗ QR,KR. (3.25)

The LS estimate of ˜ccoef is

b˜ccoef= (eZHZ)e −1ZeHvec (Yn,L) def

= eF (WL,opt), (3.26)

which is a function of the optimal directional matrix WL,opt. At the ith iteration, since the

optimal directional matrix is not available, the tentative estimation, WL,i−1, is substituted

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3.2.2 Phase II - Direction Estimation

Similar to the single-block based case, we propose two AoD estimation algorithms. Again, we assume the optimal coefficient vector is available, i.e., ˜ccoef = ˜ccoef,opt, when estimating

the directional information.

Define a new matrix eG def= QR,KRCecoef,optQe

T

T,KT, where eCcoef,opt is a KR × KLKT

matrix derived from ˜ccoef,opt by eCcoef,opt(i, j) = ˜ccoef,opt(KR(j − 1) + i), 1 ≤ i ≤ KR, 1 ≤

j ≤ KLKT. We rewrite the received matrix in vector form

vec(Yn,L) = vec e GWLXn,L + eNn,L = XT_n,L_{⊗ e}Gvec(IL⊗ W) + eNn,L, (3.27)

where eNn,L represents the sum of the modelling error associated with eG and the AWGN

term Nn,L.

3.2.2.1 Algorithm A - Maximum Matching Output

If W is constrained to be a diagonal matrix, i.e., W = diag(w), then IL⊗W = diag(1L⊗

w) and therefore vec(Yn,L) = vec e G_{· diag(1}L⊗ w) · Xn,L + eNn,L. (3.28)

From Lemma 3.1, we have vecGe _{· diag(1}L⊗ w)) · Xn,L

=1_BL_{⊗ e}G_{⊙ X}_n,LT _{⊗ 1}_N(1L⊗ IM)w def

= eTw. (3.29) Similar to Algorithm A presented in the previous subsection, the LS estimate of wopt is

b

wLS = eT† · vec(Yn,L). To improve the estimate and reconstruct a steering vector bw,

we analogously define a steering vector v(θ) def= _{1, v(θ), · · · , v}M −1_(θ)T_{, where v(θ) =}

exp(−j2πd

λsin(θ)). The AoD information bφ can be retrieved by

b

φ = arg max

−π≤θ≤πRe

P_{( b}_w_LS₎H_v(θ) _, _(3.30)

where P denotes the phase extraction operator defined by (3.13). Having obtained bφ, we then proceed to compute cWL= IL⊗ V(bφ), where V(bφ) = diag(v(bφ)).

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Also, the pseudo-inverse operation eT†is not necessary if orthogonal training matrix is used for Xn, i.e., XnXHn = BI for each n. Following the discussion given in Section 3.1.2,

for orthogonal training matrices, we have

P_{( e}_T†_{· vec (Y}_n,L_{) ) = P(} def = ewbLS z }| { e TH _{· vec (Y}n,L) ), (3.31)

and bwLS can be replaced by ewbLS in (3.30).

3.2.2.2 Algorithm B - Root-Finding Method

The root-finding approach for the block fading case can be used as well. It is easy to see that (3.30) is equivalent to searching for the root of the correlation polynomial P (z) which is the closest to the unit circle, i.e.,

b

z = arg min

z ||z| − 1|, subject to P (z) def

= P( bwLS)Hz− M = 0 (3.32)

and then retrieving the AoD information from bz = exph_−j2πd_λsin(bφ)i. The directional matrix is to be reconstructed by cW_L = IL⊗ diag(bz), where bz = [1, bz, . . . , bzM −1]. Also, for

orthogonal training matrices, ewbLS can be substituted for bwLS to skip the pseudo-inverse

computation.

The total complexity per block of the proposed algorithm, like the single-block based case in Section 3.1, is smaller than that of the conventional LS estimator. Given a fixed iteration number, the flop count of the proposed algorithm is decided by Phase I and is of the order O(BK2

TL), while the conventional LS estimator needs O(BM2L) flops. Thus,

we can save the computational complexity up to the ratio KT2

M2. Moreover, if the operating

scenario allows the use of the dual updating frequencies option and Tw

o ≪ Toc, the total

complexity can be reduced further. For slowing time-variant channels, the required time domain modelling order, KL, is small, the number of channel representation parameters is

reduced from LMN to KLKTKR+1. Such a reduction yields compact CSI representation

and benefits many post channel estimation operations involving H, as was discussed at the end of last section.

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3.3 Channel Estimation for Frequency-Selective

Time-Varying Fading Channels

For estimating a correlated frequency-selective time-varying fading channel, approaches used in the previous sections are extended to accommodate frequency-selective character-istics. Assume that the power delay profiles for the sub-channels between transmit and receive pairs are independent but of the same form. With a sampling interval of T , the receive signals at time nT can be written as

Yn= D−1

X

d=0

H(d)_n X(d)_n + Nn (3.33)

where Yn = Y(nT ) is the N × B receive vector at time nT , X(d)n = X(nT − dT ) is the

M × B transmit block, at time (n − l)T , and B denotes the length of training block. H(d)n

is the N × M matrix whose (i, j)th entry represents the fading coefficient of the dth delay path, at time nT for the channel between the ith transmit and jth receive antennas.

In the estimation of the time-variant fading MIMO channel, an observation window is included to take into account the channel variation. For an observation window of size L, the stacked receive sample vector from time nT to (n + L − 1)T can be expressed as

Y_n,Ldef= [Yn, Yn+1, . . . , Yn+L−1] . (3.34)

Applying (2.11) to the stacked version of (3.33), we have vec(Yn,L) = XTn,L⊗ IN · vec (Hn,L) + vec (Nn,L) (3.35) where vec(Hn,L) def = vec(Hn)T, . . . vec(Hn+L−1)T T , vec(Hn) def = vec(H(0)_n )T_{, · · · , vec(H}(D−1)_n )TT , vec(Nn,L) def = vec(Nn)T, . . . vec(Nn+L−1)T T , and XT_n,Ldef=       X(0)n T , · · · , X(D−1)n T 0 _{· · ·} 0 0 X(0)_n+1T _{, · · · , X}(D−1)_n+1 T ... 0 ... ... . .. ... 0 0 _{· · · X}(0)_n+L−1T _{, · · · , X}(D−1)_n+L−1T       .

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Substituting (2.12) for each Hnand assuming the eigenbases QTand QR remain invariant

during an estimation period, we obtain

vec(Hn,L) = (ILD ⊗ QT⊗ QR) Γn,L. (3.36)

Each component of the vector Γn,L =

h γn(0) T , · · · , γn(D−1) T , · · · , γn+L−1(0) T , · · · , γn+L−1(D−1) TiT

is itself an (NM) × 1 column vector γn(d) =

γ_1n(d), γ(d)_2n_{, · · · , γ}_{(N M )n}(d) T that represents the complex fading coefficients of the dth path for the pth MIMO subchannel at time nT , p ∈ {1, 2, · · · , NM}, and d ∈ {0, 1, · · · , D − 1}.

The stacked vector, γ(p) = hγpn(d), γ(d)_p(n+1), · · · , γ_p(n+L−1)(d)

iT

, represents a finite-duration sample of the complex random process associated with the dth delay path which has a fixed Doppler spectrum [25]. Such a process can also be expanded by a set of smooth functions [33],[8], and thus its estimation can be obtained by using a method similar to that developed in the previous section. Similarly, as described in Section 2.2.3, we can first apply the orthogonal transform γ(p) = QLb(d)pn, where QL is a full-rank orthogonal

matrix, and b(d)pn is the transform domain coefficient. Then, the time domain channel

correlation is approximated in the following equation by using the reduced bases matrix QL,KL,

γ(p) ≈ QL,KLc

(l)

pn, and Γn,L ≈ (QL,KL⊗ IDM N) · ccoef, (3.37)

where KLdenotes the time domain modelling order, and c(d)pn is a KL×1 coefficient vector.

By using (3.35), (3.36) and the approximation (3.37), we decouple the signal part of (3.35) into the product of two modelling domains - space and time domains

vec(Yn,L) ≈ XTn,L⊗ IN (ILD⊗ ULS⊗ UR) (QL,KL⊗ IDM N) ccoef = XT_n,L_{⊗ I}_N (QL,KL⊗ ID) ⊗ (W T_Q T,KT) ⊗ QR,KR ˜ccoef def = ((WDLXn,L)TQeT,KT) ⊗ QR,KR ˜ccoef (3.38) where WDL def = (ILD⊗W), eQT,KT def

= QL,KL⊗ID⊗QT,KT, ˜ccoef= (IKLD ⊗ PT⊗ PR) ccoef.

Wis the steering matrix defined in (2.17). Since the mean AoD usually varies slowly with respect to a sub-channel’s coherent time, we assume that W remains the same during a period of L data blocks. Similar to the narrowband model (21), we do not impose the implicit Kronecker structure and Gaussian assumption on ˜ccoef.

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As (3.38) can be obtained by replacing X, Y, W, vec(C), QM,KT, and QN,KR in

(3.6) by Xn,L, Yn,L, WDL, ˜ccoef, eQT,KT, and QR,KR, we conclude that both block fading

and time variant frequency selective fading channels can be described by similar models. Hence, the two-phase iterative estimation scheme developed in Section 3.1 can be extended to estimate the coefficient vector ˜ccoef, and the directional matrix WDL in (3.38). In the

following, we describe a two-phase channel estimation approach for frequency-selective time-variant MIMO fading channels.

3.3.1 Phase I - Coefficient Estimation

Following an argument similar to that used in Section 3.1, we assume that the directional matrix WDL is optimal in the coefficient estimation phase and define

e

Zdef= (WLD,optXn,L)TQeT,KT

⊗ QR,KR. (3.39)

The LS estimate of ˜ccoef is

b˜ccoef = (eZHZ)e −1ZeHvec (Yn,B) def

= eF (WLD,opt), (3.40)

which is a function of the optimal directional matrix WLD,opt. At the ith iteration, since

the optimal directional matrix is not available, we substitute the tentative estimation at (i − 1)th iteration, WLD,i−1, for WLD,opt.

3.3.2 Phase II - Direction Estimation

The two AoD estimation algorithms established in Section 3.2 can be directly extended here for wide-band MIMO channels. Again, we assume the optimal coefficient vector is available, i.e., ˜ccoef= ˜ccoef,opt, when estimating the directional information.

Define a new matrix eGdef= QR,KRCecoef,optQe

T

T,KT, where eCcoef,optis a KR× DKLKT

ma-trix derived from ˜ccoef,opt by eCcoef,opt(i, j) = ˜ccoef,opt(KR(j − 1) + i), 1 ≤ i ≤ KR and 1 ≤

j ≤ DKLKT. We rewrite the received matrix in vector form

vec(Yn,L) = vec e GWDLXn,L + N′ n,L = XT_n,L_{⊗ e}Gvec(IDL⊗ W) + N′n,L, (3.41) where N′

n,L represents the sum of the modelling error associated with eG and the AWGN

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If W is constrained to be a diagonal matrix, i.e., W = diag(w), then IDL ⊗ W =

diag(1DL⊗ w) and therefore

vec(Yn,L) = vec e G_{· diag(1}DL⊗ w) · Xn,L + N′ n,L. (3.42)

From Lemma 3.1, we have vecGe _{· diag(1}DL⊗ w)) · Xn,L = 1BL⊗ eG ⊙ XTn,L⊗ 1N (1LD⊗ IM)w def = eTDw. (3.43)

Here, we can extend the proposed two direction estimation algorithms developed in Section 3.2.2 to extract the AoD information for frequency-selective channel, simply by replacing the eT in (3.29) with eTD in the above equation.

3.4 Performance Analysis

In analyzing the MSE performance ǫdef= En_{kH − b}H_k2_F

o

= En_{kvec(H) − vec( b}H_)k2₂ o

. (3.44)

of the proposed bH, we first make the optimistic assumptions that the optimal orthogonal pilot matrix [7] for conventional LS channel estimator is used and the directional matrix estimate cW is perfect.

Notations

For notational simplicity and when there is no danger of ambiguity, H and W in this section denote the channel and directional matrices of (3.1)/(3.2) or (3.21)/ (3.24) for single-block based or time-correlated based estimators, and Xp represents X in (3.6) or

Xn,L in (3.24). Furthermore, QT and QR denote either the modelling bases QM,KT and

QN,KR in (3.6), or eQT,KT and QR,KR in (3.24).

Then (3.44) can be expressed as

ǫ(Xp; W) = E n kvec(H) − vec(QRCQb TTW)k22 o = E_{kvec(H) − ΨΩ}zvec(HXp+ N)k22 (3.45) where Ψdef= (WT_Q T) ⊗ QR and Ωz def

= (bZHZ)b −1_Z_bH_{, b}_Z_{being the LS estimate of Z defined}

in 3.1.1, i.e.,

b

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As HXp and N are statistically independent, the MSE can be separated into two terms

which are contributed by modelling error (reduced-rank basis matrices) and AWGN, re-spectively. ǫ(Xp; W) = E kvec(H) − ΨΩzvec(HXp)k22 + E_kΨΩzvec(N)k22 def = ǫh(Xp, W) + ǫn(Xp, W). (3.47)

Define the following projections

PW def = WTQT(QTTW∗X∗pXTpWTQT)−1QTTW∗X∗pXTp ⊗ QRQTR ˜ PW def = WTQT(QTTQT)−1QTTW∗ ⊗ QRQTR.

The first term on the RHS of (3.47) becomes

ǫh(Xp; W) = Ek(I − PW) vec(H)k22 = tr_{(I − ˜}PH_W_{)(I − ˜}PW)Rh = χ X k=1 λkk(I − ˜PW)fkk22 (3.48) where Rh = E

vec(H)vec(H)H _{is the channel correlation matrix and f}

kis Rh’s

eigenvec-tor associated with the eigenvalue λk, λ1 ≥ λ2 ≥ · · · ≥ λχ; χ being the degree of freedom of

H. For the single-block based case, χ = NM and it is equal to NML when the estimator considers the time correlation effect. (3.48) is valid since the orthogonal training matrix X_p _{is used. Let 1 < K ≤ χ be the rank of the dominant signal subspace of the channel} covariance matrix. Then Rh =

Pχ

k=1λkfkfkH ≃

PK

k=1λkfkfkH, with λk ≪ 1 for K < k ≤ χ.

Since k(I − ˜PW)fkk22 ≤ 1, we have

Pχ

k=K+1λkk(I − ˜PW)fkk22 ≤

PKs

k=K+1λk ≪ 1. Let the

compound modelling order Ks be equal to KTKR and KTKRKL for the two cases under

investigation. If Ks is chosen to be larger than K, the rank of Rh, i.e., K < Ks ≤ χ,

and the basis matrices QT and QR span the dominant signal subspace of Rh, then the

matrix ˜PW is a projection operator whose range lies mostly in the space spanned by

{fk}, 1 ≤ k ≤ K and we conclude that k(I − ˜PW)fkk22 def

= | ˜P⊥

Wfkk22 ≪ 1, for 1 ≤ k ≤ K.

Therefore, the modelling error ǫh is negligible in this case. On the other hand, if the

modelling order is not enough to span the signal subspace, there is under-modelling error contributed by those non-negligible terms λkk(I − ˜PW)fkk22 which will dominate the mean

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As for the MSE due to thermal noise–the second term on the RHS of (3.47), we can show that ǫn(Xp, W) = E kΨPzvec(N)k22 = tr N0 B P˜W = N0 B Ks, (3.49)

where we have invoked the facts that (i) the training signal Xp and the noise N are

independent, (ii) unitary pilot matrix is used, X∗

pXTp = BI and (iii) elements of N is i.i.d.

complex white Gaussian noise with variance σ2

n = N0. (3.49) implies that thermal noise

induced MSE can be reduced by using a small modelling order. In Section 3.5 (Figs. 3.3 -3.5), we find that this noise-reduction effect is significant in low SNR environments where thermal noise dominates the MSE performance while the modelling error of (3.48) dominates in high SNR region.

If cW is not perfect and W = cW+ ∆W, then b

Zdef= Z + ∆Z = ((WXp)TQT) ⊗ QR+ ((∆WXp)TQT) ⊗ QR. (3.50)

The coefficient vector estimation vec( bC) can be approximated up to the first order of ∆Z as [34]

vec( bC_{) ≃ vec(C) − Z}†∆Zvec(C) + Z†_{vec(N) + (Z}H_Z)−1_∆ZH_P⊥

Zvec(N) − Z

†_∆ZZ†_vec(N),_(3.51)

where P⊥

Z = I − Z(ZHZ)−1Z. The above equation indicates that, besides the terms that

have to do with the noise N, the coefficient vector estimation error is determined by the projection error ∆Z. Hence, when the projection error ∆W is small (and thus ∆Z is small), vec( bC) is a good approximation of vec(C) at high SNR region.

3.5 Numerical Results and Discussion

Simulation results reported here use the reference MIMO channel model [2], the IEEE 802.11 TGn channel model [3], and the SCM model [35]. The former two are stochastic models whose spatial correlation matrices are generated by the power azimuth spectrum (PAS) at the BS and MS, respectively. The SCM model generates the channel coefficients according to a set of selected parameters (e.g., AS, AoD, AoA, etc.). It is a popular parametric stochastic model whose spatial cross correlations are functions of the joint

應用於相關性多輸入多輸出系統之通道建模、估測及前置編碼

國 立 交 通 大 學

電信工程學系

博 士 論 文

應用於相關性多輸入多輸出系統

之通道建模、估測及前置編碼

Channel Representation, Estimation and

Precoding for Correlated MIMO Systems

研 究 生：陳彥志

指導教授：蘇育德 博士

應用於相關性多輸入多輸出系統

之通道建模、估測及前置編碼

Channel Representation, Estimation and Precoding

for Correlated MIMO Systems

研究生：陳彥志

Student:

Yen-Chih

Chen

指導教授：蘇育德 博士 Advisor:

Dr.

Y.

T.

Su

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Institute of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy

in

Communication Engineering

Hsinchu, Taiwan

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國立交通大學

博士論文

研究生：陳彥志

指導教授：蘇育德博士

指導教授：蘇育德博士 Advisor:

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