使用雙極化天線的空間調變技術：訊號映射、通道估測及訊號偵測

國

立

交

通

大

學

電信工程研究所

碩

士

論

文

使用雙極化天線的空間調變技術：

訊號映射、通道估測及訊號偵測

Spatial Modulation Using Dual-Polarized Antennas:

Signal Mapping, Channel Estimation and Data Detection

研 究 生：林可涓

指導教授：蘇育德 教授

使用雙極化天線的空間調變技術：

訊號映射、通道估測及訊號偵測

Spatial Modulation Using Dual-Polarized Antennas:

Signal Mapping, Channel Estimation and Data Detection

研 究 生：林可涓 Student：Ke-Jyuan Lin

指導教授：蘇育德 Advisor：Dr. Yu T. Su

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

Submitted to the Institute of Communications Engineering in partial Fulfillment of the requirements

for the Degree of Master of Science

in

Communications Engineering at the

National Chiao Tung University

July 2012

Hsinchu, Taiwan, Republic of China

使用雙極化天線的空間調變技術：訊號映射、通道估測及訊號偵測

學生：林可涓

指導教授

蘇育德

國立交通大學電信工程研究所碩士班

摘

要

空間調變 (SM) 是近年來被提出的一種提升頻譜效率及降低傳送接收端複

雜度的技術。 這是由於 SM 可避免傳統 MIMO 技術中通道間交互干擾 (ICI) 的

問題，且大量減少硬體上射頻鏈路 (RF chain) 的高花費。

大多數對於 MIMO 訊號的偵測，傳送或是接收端需要知道通道資訊 (CSI)，

此通道資訊可透過某種估計演法得到。在本文中，提出了兩種基於 SM 系統下，

隨時間變動的通道估測方法，此方法可以省掉領航信號 (pilot) 的額外負擔因此

而提升系統效能。

因為共置 (co-located) 雙極化天線的使用可以節省空間和花費，因此我們也

探討使用雙極化天線的SM技術及其訊號偵測的方法，其中所提出的次佳化的偵

測器複雜度只需要將近最大似然偵測法 (ML) 的一半，且效果比基於匹配濾波器

(MF-based) 的偵測法更好。在使用雙極化天線及有空間相關性的通道模型中，

我們也引進了一個較一般性的通道模型並提出相對應的通道估測方法。

由於 SM 系統在傳送訊號時，一次只使用單根傳送天線，利用 Alamouti 的

時空編碼可以使 SM 系統更具彈性。本文中提出了在 SM 系統下差分的 Alamouti

時空編碼技術，此法接收端不須要通道資訊即可解碼。

針對所提出的方法，我們透過了模擬結果來檢驗其效能並與現有的方法做比

較。

i

Spatial Modulation Using Dual-Polarized Antennas:

Signal Mapping, Channel Estimation and Data

Detection

Student : Ke-Jyuan Lin Advisor : Yu T. Su

Institute of Communications Engineering National Chiao Tung University

Abstract

Spatial modulation (SM) technique has received intensive interest recently for its capability to improve the spectral efficiency and lower the transceiver complexity. This is because SM induces zero inter-channel interference and requires a single RF chain only.

To detect or encode SM signals, channel state information (CSI), which is to be obtained by a channel estimator at either the transmit or the receive side, is needed for spatial identification. The first SM-related issue investigated in this thesis concerns channel estimation in a correlated time-varying channel. We propose superimposed-pilot-assisted and decision-directed spatial channel estimation schemes. These schemes improve the system throughput by either removing or reducing the pilot overhead.

Since the use of co-located dual-polarized antennas offers a space- and cost-effective alternative for multiple antenna systems, we then review the feasibility of using such antenna arrays and present a new SM scheme which takes advantage of the channel decorrelation inherited in a dual-polarized antenna array. A suboptimal detector which needs only half of the ML detector complexity is proposed. This suboptimal detector performs much better than the MF-based method. For a dual-polarized antenna array

based SM system, we suggest a general channel model which takes into account the spatial correlation and introduce a model-based spatial channel estimator.

Finally, to avoid the CSI requirement, we propose an Alamouti coding based differ-ential space time block coded SM (DSTBC-SM) scheme.

For each proposed scheme, we provide computer simulation results to demonstrate and verify its superiority against the existing solutions.

誌

謝

對於論文得以順利完成，首先感謝指導教授 蘇育德博士。老師的教誨使我

對於通訊領域的研究有更深入的了解，也教導我們許多書上學不到的知識與人生

道理，讓我受益匪淺。並感謝口試委員蘇賜麟教授、林茂昭教授及呂忠津教授給

予的許多意見，以補足這份論文的缺失與不足之處。

我也非常感謝實驗室的劉彥成學長，在我研究上有問題時，能給予我建議及

討論，使我的研究能夠順利完成，從中學習到的經驗是很珍貴的。另外也感謝實

驗的成員們，在這兩年內的互相支持與鼓勵。

最後，要感謝我的家人及朋友，他們總是在背後默默的關心與支持，使我有

動力可以努力往前進，在此僅獻上此論文，以代表我最深的敬意。

iv

Contents

Chinese Abstract i

English Abstract ii

Acknowledgement iv

Contents v

List of Tables vii

List of Figures viii

1 Introduction 1

2 Preliminaries 5

2.1 Conventional MIMO System Model . . . 5

2.2 Spatial Modulation (SM) Schemes and Their Detections . . . 6

2.2.1 Spatial Modulation Transmission . . . 7

2.2.2 Optimal Detector . . . 9

2.2.3 Sub-Optimal Detector . . . 10

2.3 Spatial-Correlated MIMO Channel Models . . . 10

2.3.1 Conventional Correlated Channel Model . . . 11

2.3.2 Kronecker Model . . . 11

2.3.3 Virtual Channel Representation and Weichselberger Model . . . 12

2.3.4 Model-Based Correlated Channel . . . 13

3 Estimation of Spatial-Modulated Time-Varying Channels 15 3.1 Conventional Pilot-Based Channel Estimation Methods . . . 15

3.1.1 Least Square Channel Estimation . . . 16

3.1.2 Minimum Mean Square Error Channel Estimation . . . 16

3.2 Decision-Directed SM Channel Estimation . . . 17

3.3 SM Channel Estimation With Superimposed Pilots . . . 19

3.4 Simulation Results . . . 22

4 Spatial Modulation Using Dual-Polarized Antenna Arrays 29 4.1 Dual-Polarized MIMO Channel Models . . . 29

4.2 Dual-Polarized Spatial-Correlated (DPSC) Channel Model . . . 33

4.3 Time-Varying DPSC Channel Estimation . . . 35

4.4 Data Detection in SM-DPSC Channels . . . 35

4.5 Simulation Results . . . 38

5 Space-Time Block-Coded Spatial Modulation (STBC-SM) System 48 5.1 STBC-SM System . . . 48 5.2 Differential STBC-SM Scheme . . . 52 5.3 Simulation Results . . . 54 6 Conclusion 57 Appendix A 59 Bibliography 61

List of Tables

2.1 SM mapping table for 3 bits/transmission . . . 8 4.1 An SM mapping table for the dual-polarized system for 3 bits/transmission 37 5.1 STBC-SM mapping table for 2 bits/transmission . . . 50

List of Figures

2.1 A MIMO system model. . . 6 2.2 A MIMO-SM system model. . . 7 3.1 Periodical pilot signal insertion in transmit data. . . 16 3.2 BER performance of SM detectors with decision-directed channel

estima-tion. . . 23 3.3 NMSE performance of SM detectors with decision-directed channel

esti-mation. . . 24 3.4 BER performance of MF-based detector with decision-directed channel

estimation under various mobility. . . 25 3.5 NMSE performance of MF-based detector with decision-directed channel

estimation under various mobility. . . 25 3.6 BER performance of ML detector with decision-directed channel

estima-tion under various mobility. . . 26 3.7 NMSE performance of ML detector with decision-directed channel

esti-mation under various mobility. . . 26 3.8 BER performance of superimposed channel estimation with different

su-perimposed pilot symbol energy. . . 27 3.9 BER performance of different superimposed pilot symbol energy versus

SNR. . . 27 3.10 BER performance of superimposed channel estimation method with

3.11 BER performance of superimposed channel estimation method with ML

detector under different mobile velocity. . . 28

4.1 A co-located dual-polarized MIMO system model. . . 30

4.2 Depolarization mechanisms. . . 31

4.3 A dual-polarized SM system model. . . 36

4.4 Comparison of BER performance for SM under conventional MIMO chan-nel and dual-polarized MIMO chanchan-nel. . . 39

4.5 Comparison of BER performance for SM and VBLAST and Alamouti scheme under dual-polarized channel for 8 bits/transmission. . . 40

4.6 NMSE comparison of channel estimation for dual-polarized spatial-correlated MIMO channel with different modelling order, AS=2 and 15. . . 41

4.7 Comparison of BER performance of SM for MF-based detector, subopti-mal detector and ML detector under perfect CSI. . . 42

4.8 The effect of different mu (inverse of CPR) value with different detectors on BER performance. . . 43

4.9 The effect of different chi (inverse of XPR) value with different detectors on BER performance. . . 43

4.10 BER performance of decision-directed channel estimation method with MF-based detector in DPSC channel under different mobile velocity. . . . 44

4.11 NMSE performance of decision-directed channel estimation method with MF-based detector in DPSC channel under different mobile velocity. . . . 44

4.12 BER performance of decision-directed channel estimation method with low complexity ML detector in DPSC channel under different mobile ve-locity. . . 45

4.13 NMSE performance of decision-directed channel estimation method with low complexity ML detector in DPSC channel under different mobile ve-locity. . . 45

4.14 BER performance of decision-directed channel estimation method with ML detector in DPSC channel under different mobile velocity. . . 46 4.15 NMSE performance of decision-directed channel estimation method with

ML detector in DPSC channel under different mobile velocity. . . 46 4.16 BER performance of superimposed channel estimation method based on

three detectors in DPSC under different mobile velocity. . . 47 5.1 Comparison of BER performance for STBC-SM with perfect CSI,

DSTBC-SM and STBC-DSTBC-SM with channel estimation error. . . 55

Chapter 1

Introduction

Multiple-input and multiple-output (MIMO) techniques have drawn intensive atten-tion and R&D efforts in the past decade as they promise to offer a system capacity which is linearly proportional to the minimum of the transmit and receive antennas numbers [1]-[3] in a richly scattered environment. The extra spatial degrees of freedom is specially welcome for wireless communication system designers who are always looking for novel transmission techniques to achieve both high data rate and high spectral efficiency.

Many existing techniques have been developed to exploit the promised MIMO ca-pacity. One simple yet efficient scheme to achieve the full diversity (array) gain is the class of space-time codes (STC) [4]. Besides achieving full diversity gain with low re-ceiver complexity, it also has high spectral efficiency of one symbol per channel use [5]. The Bell Labs layered space-time (BLAST) [6], on the other hand, demultiplexes data streams into a number of substreams which are then transmitted by different anten-nas, resulting in a data rate increase proportional to the number of transmit antennas. The multiplexing gain, however, is obtained by using equal number of receive antennas and complex signal processing at the receive side to eliminate the inter-(spatial)channel interference (ICI).

Another transmission technique which uses multiple antennas is spatial modulation (SM). SM is a simple scheme which avoids ICI, timing synchronization of multiple spatial data streams and reduces the cost of multiple radio frequency (RF) chains by allowing

one transmit (and receive) antenna to be active in any one transmission interval [7]. In addition, SM can exploit transmit antenna index to convey information to enhance spectral efficiency and capacity [11]. The low system complexity requirement, relative high spectral efficiency, and robust error performance in correlated channels [8] have made SM an attractive candidate for high rate transmissions. However, the SM system needs at least two transmit antennas and the transmit-receive wireless links have to be sufficiently dissimilar to each other to yield adequate performance [9]. When the only information is carried by the transmit antenna index, SM degenerates to the so-called space shift keying (SSK) modulation [10] which is easy to implement although the associated achievable data rate is rather limited.

Maximum likelihood (ML), matched-filter (MF)-based and sphere detection (SD) based receivers have been introduced for detecting SM signals [12], [13]. These detectors perform fairly well only when the channel state information (CSI) is perfectly known by the receiver. CSI is often obtained by using a pilot-assisted least square (LS) or minimum mean square error (MMSE) channel estimator [14]. In most cases, the channel is assumed to be either static (time-invariant) or block faded [15], [16], these known estimators yield poor performance valid for time-varying or correlated block-fading channels. In this thesis, we propose two channel estimation schemes for use in time-varying block-faded channels that takes advantage of the SM structure. The first scheme is a decision-directed one which uses the detected signals of previous blocks to update estimated channel coefficients. Since the selected transmit antenna is uniformly distributed, all CSIs would be updated in a sufficient long transmission period. The performance of this scheme has error floor in high SNR (signal-to-noise ratio) region due to the propagation of channel estimation error. We introduce a superimposed pilot CSI estimation scheme to overcome this shortcoming and improve the system performance without incurring extra pilot overhead.

pro-vide multiplexing and/or array gains, these antennas need to have a large physical sepa-ration to yield uncorrelated spatial channels. A space- and cost-effective alternative [17] can be obtained by using co-located orthogonally-polarized antennas. A MIMO channel using dual polarized antennas has a better channel condition as compared to that of the conventional one when fixing antenna numbers and the area that can accommodate antenna hardware. Moreover, the use of such antennas offers additional degree of free-dom with the same structure of antenna array [18]. The dual-polarized antenna system, however, possesses complicated depolarized properties because of the coupling effect be-tween different orthogonal polarizations as were shown in [19], [20], [21]; measurements of depolarization parameters can be found in [19].

We make use of the advantages inherited in dual-polarized antennas and propose a new scheme called dual-polarized spatial modulation (DPSM) to fully exploit multi antenna capabilities. This scheme convey information in the polarization of antenna, antenna index, and symbol transmitted. It can outperform conventional MIMO system in correlated channel because of the better channel condition. Moreover, we develop the ML detector and the MF-based detector of DPSM, and we propose a sub-optimal DPSM detector whose complexity is almost half of ML detector and its performance is much better than MF-based method. In addition, DPSM assuages the problem of poor SM performance when the transmit-to-receive wireless links are too much alike, i.e., the MIMO channel is too correlated or has a bad condition [9].

Spatial correlation is also considered in dual-polarized channels. In [19], [21], [22] the transmit and receive spatial correlations are assumed to be de-coupled, resulting in Kronecker-like model [23]. As spatial correlation between transmitter and receiver does exist [24], alternate correlated channel models are introduced to incorporate this joint correlation [25], [26]. However, these analytic models often call for the knowledge of second-order channel statistics that are not easy to obtain, [27] proposed a reduced-rank channel model and compact CSI representation to solve this inconvenience. and

generalizes the above-mentioned channel models. In this thesis, we combine this model with dual-polarization effects and propose a modified channel estimation method for this modified dual-polarized channel model.

Finally, we notice that space-time block-coded spatial modulation (STBC-SM) has been proposed recently [28] to improve the SM spectral efficiency. An Alamouti coded-based differential STBC-SM (DSTBC-SM), which dose not need CSI and performs well in slow time-varying channel, is studied.

The rest of this thesis is organized as follows. In Chapter 2 we present the transceiver structure of a typical SM system along with spatial correlated channel models. In Chapter 3, we propose two time varying channel estimation methods for SM systems and give simulated performance. Chapter 4 describes a dual-polarized antenna array based SM scheme and the associated CSI estimation methods for spatial correlated channels. Space-time-coded systems are introduced in Chapter 5. Our main contributions are summarized in Chapter 6.

The following notations are used throughout the thesis: upper case bold symbols denote matrices and lower case bold symbols denote vectors. IN is a N × N identity

matrix. (·)T_{, (·)}H_{, and (·)}† _{represent the transpose, conjugate transpose, and}

pseudo-inverse of the enclosed items, respectively. (·)−1 _{denotes the inverse of matrix. vec(·) is}

the operator that forms one tall vector by stacking columns of a matrix. While E{·}, |·|, and k · kF denote the expectation, absolute value, and Frobenius norm of the enclosed

Chapter 2

Preliminaries

2.1

Conventional MIMO System Model

When MIMO system is used in wireless communications, it is commonly referred to the uni-polarized multiple input multiple out antenna system. These antennas are spatial separated to yield an uncorrelated channel, thus they can provide diversity gain and increase the reliability of wireless links [1]. Moreover, under suitable channel fading conditions, spatially multiplexing gain can be achieved and increase the MIMO capacity. For conventional MIMO system, the MIMO channel between transmitter and receiver at time k is modeled as H(k, τ ) = G X p=1 Hp(k)δ(t − τp), (2.1)

where G is the maximum number of paths between the subchannel of a transmit and receive antenna pair. τp is the delay of the pth path , and δ denotes the Dirac delta

function. If we consider a narrowband flat-fading MIMO system with NT transmit

antennas and NR receive antennas, MIMO channel representation is reduced to a NR×

NT single-tap fading channel matrix. Then when data vector x(t) is transmitted, the

received signal is denoted as

y(k) = H(k)x(k) + z(k), (2.2)

where for rayleigh-fading spatial-uncorrelated channel, the elements of H(k) are indepen-dent and iindepen-dentically distributed, zero mean complex Gaussian random variables. z(k) is the additive white Gaussian noise (AWGN) vector, whose entries are of zero mean and with variance σz2. Figure 2.1 gives the brief MIMO system model,

݄ேೃǡଵ ݄ேೃǡே೅ ݄ଵǡே೅

ڭ

ġ

ڭ

ġ

ڭ

ġ

Figure 2.1: A MIMO system model.

2.2

Spatial Modulation (SM) Schemes and Their

De-tections

In MIMO system, many transmission techniques have been designed to improve spectral efficiency such as vertical Bell Laboratories layered space-time (V-BLAST) ar-chitecture. BLAST transmission systems suffer from high inter-channel interference resulting from the simultaneous transmissions on the same frequency for MIMO chan-nel. Spatial modulation (SM) is an innovative approach which can boost the spectral efficiency and further avoid ICI by using active transmit antenna indices as additional source of information. In the following subsections, we will discuss the transceiver design of SM.

2.2.1

Spatial Modulation Transmission

We consider the block fading case that channel gain remains unchanged within a block time and eliminate the time parameter t in this chapter. To begin with, data bits are partitioned into groups of m = log₂(MNT) bits where in each group the first log2NT

bits indicate the transmit antenna to be used and the remaining bits corresponds to an

M-QAM symbol. This M-QAM constellation is denoted as AM and the symbol in it

are transmitted by the indexed antennas. Figure 2.2 depicts the system model described above.

Input bits Antenna Index Transmit Symbol Antenna Index Transmit Symbol 000 001 010 011 100 101 110 111 Pilot Pilot

Pilot Pilot Pilot Data Pilot

ڭ

ڭ

ࡽ෡

܆෡

X

Q

H

Figure 2.2: A MIMO-SM system model.

Spatial modulation (SM) maps m × B data matrix Q to X, NT × B transmitted

signal matrix, where B denotes block size which should be equal or larger than NR. The

reason is that Matrix X , [x1, · · · xB] has only one nonzero element in each column

where xi is the signal vector transmitted at the ith time slot. At time slot i, the ℓth

entry of xi, xℓ,i, is the transmit symbol of antenna ℓ. The case when xℓi = 0 means

antenna ℓ is not used at the ith instant. One example of SM mapping rule is shown in Table 2.1.

Input bits Antenna Index Transmit Symbol Antenna Index Transmit Symbol 000 1 +1 1 +1 + j 001 1 -1 1 -1 + j 010 2 +1 1 -1 - j 011 2 -1 1 +1 - j 100 3 +1 2 +1 + j 101 3 -1 2 -1 + j 110 4 +1 2 -1 - j 111 4 -1 2 +1 - j Pilot Pilot

Pilot Pilot Pilot Data Pilot Data

ڭ ڭġ

ۿ෩ ෩κ

Table 2.1: SM mapping table for 3 bits/transmission

At the receiver side, the NR× B received signal matrix Y can be expressed as

Y = HX + Z, (2.3) where X = [x1, · · · , xB]; (2.4) Y = [y1, · · · , yB]; (2.5) Z = [z1, · · · , zB]; (2.6) H = [h1, · · · , hNT]. (2.7)

Matrix H describes the overall NR× NT channel matrix whose (m, n)-th element

hm,n is the channel response between the nth transmit antenna and the mth receive

antenna. The elements of H are independent and identically distributed (i.i.d.), zero-mean complex Gaussian random variables with unit variance, σ2

h = 1. In addition, Z is

the NR× B additive white Gaussian noise (AWGN) matrix, whose entries are of zero

mean and E{ zizHi } = σz2INR, observed at the receiver. We assume an average transmit power of Ex, i.e., Ex= 1 B E [k X k 2 F ] = 1 B E [ tr{ XX H } ], (2.8)

Through this thesis, we consider a quasi-static scenario, where channel remains un-changed over the period of B and is independent of both X and Z.

Because only one transmit antenna is active, say ℓth antenna, during symbol time i, alternatively we have

yi = hℓxℓi+ zi, (2.9)

where xℓi∈ AM and for i = 1, · · · , B, i = 1, · · · , NT

xi = [0, · · · , xℓ,i, · · · , 0]T ∈ CNT, (2.10) yi = [y1,i, · · · , yNR,i] T ∈ CNR_{, (2.11)} zi = [z1,i, · · · , zNR,i] T ∈ CNR_{, (2.12)} hj = [h1,j, · · · , hNR,j] T ∈ CNR_{. (2.13)}

2.2.2

Optimal Detector

Due to SM’s specific structure, its receiver is inherently of low complexity. With the assumption that the channel state information (CSI) H is known to the receivers, we introduce the single-stream-based maximum likelihood (ML) and matched filter (MF)-based detector respectively in this and the next subsection, respectively.

Since the channel coefficients are assumed equally likely, the optimal detector based on the ML principle is equal to maximizing the probability P (Y|H, X) [29],

P (Y|H, X) = 1 (πσ2 z)NR e− 1 σ2zkY−HXk 2 F . (2.14)

ML detection exhaustively searches over all transmit antenna index and constellation point pairs. It is often regarded as a high complexity detection technique. However, the complexity of SM’s ML detector is much reduced due to the fact that only one transmit antenna is used at a time. Therefore, the ML metric of each time can be expressed as

(ˆℓi, ˆxℓi) = arg max ℓ,xℓi

P (yi|H, xi) = arg min ℓ,xℓi ky

i− xℓihℓk2, (2.15)

whose search space is of order O(MNT). It is also called the single-stream-based ML

detector

2.2.3

Sub-Optimal Detector

For the MF-based detector, we need to normalize every column of H by its norm before estimation of transmit antenna index. The reason is as follows. First, let

¯ hj =

hj

khjk, j = 1, · · · , NT

. (2.16)

As Cauchy-Schwarz inequality suggests: |¯hHℓ hℓ| = khℓk = kh

jkkhℓk

khjk ≥ |¯h H

j hℓ|, ℓ = 1, · · · , NT, (2.17)

we can define the MF receiver as

gj = ¯hHj yi, i = 1, · · · , B, (2.18)

which reduces to khℓkxℓ,iplus noise if j = ℓ or noise only when yi = xℓihℓ+zi. Therefore,

we can estimate transmit antenna index by finding the maximum value of |gj|.

ˆ

ℓi = arg max ℓi∈{1,··· ,NT}

|gℓi,i|. (2.19)

Next, left-multiplying yi by the pseudo-inverse of h_ℓˆ_i (h†_ℓˆ_i = (h_ℓHˆ_ih_ℓˆ_i)−1hH_ℓˆ_i) and

quan-tizing this product to the constellation points with function Q(·) to recover transmit symbol ˆx_ℓˆ_i,i, i.e., ˇ g_ℓˆ_i,i = h†_ℓˆ_iyi = hH ˆ ℓiyi kh_ℓˆ_ik2 = gℓˆi,i kh_ℓˆ_ik , (2.20) ˆ x_ℓˆ_i,i = Q(ˇg_ℓˆ_i,i). (2.21)

2.3.1

Conventional Correlated Channel Model

In general, a full correlation matrix R that specifies the NRNT × NRNT mutual

correlation coefficients between all channel matrix elements is used to describe the spatial behavior of a general MIMO channel; specifically,

R_{, E{vec(H}H)vec(HH)H_}. (2.22)

For example, the spatial correlation matrix of a 2 × 2 MIMO channel can be described as R =     1 t∗ _r∗ _s∗ 1 t 1 s∗ 2 r∗ r s2 1 t∗ s1 r t 1     , (2.23)

where t and r are transmit and receive antenna correlation coefficients, and s1 def

= E{h1,1h∗2,2} and s2

def

= E{h1,2h∗2,1} are cross-channel correlation coefficients.

Consequently, a spatial correlated Rayleigh fading channel can be modeled as vec(HH_{) = R}1

2vec(HH

ω), (2.24)

where Hω is the i.i.d. complex Gaussian matrix with unit variance.

However, there are some difficulties in using this model. First, large number of transmit and receive antennas will make the number of correlation matrix elements, (NRNT)2, too large to compute. Moreover, physical propagation of the radio channel,

such as angle of arrival (AOA), direction of departure (DOD), and etc., could not be easily interpreted by this correlation matrix [24].

2.3.2

Kronecker Model

Kronecker model is commonly used when correlation between transmit and receive antennas are independent and can be separated that the spatial correlation matrix is given by the Kronecker product of those of the transmit and receive antennas, which is reasonable when the main scattering is locally rich at each transmitter and receiver side

[24], H = R 1 2 RHω(R 1 2 T)T, (2.25)

where NT × NT matrix RT and NR× NR RR represent spatial correlation of transmit

and receive antennas, respectively.

The separation statistic of Kronecker model implies that NRNT × NRNT correlation

matrix of H can be expressed as

R = RR⊗ RT. (2.26)

Since the strict assumption of separate correlation between transmitter and receiver side, it would not be appropriate to model a correlated channel where transmitter and receiver side have some correlation leading to capacity and error probability misfits.

2.3.3

Virtual Channel Representation and Weichselberger Model

Both [25] and [26] consider joint correlation at both ends of MIMO channel, so the correlated channel is modeled by basis matrices of two one-sided correlation matrix and one coupling matrix which contains the correlation between transmitter and receiver side. In [25], a virtual channel representation using predefined discrete Fourier transform (DFT) matrices is proposed to model the correlated channel. Specifically,

H = FR( ˜Ωvirt⊙ Hω)FHT, (2.27)

where FR and FT are respectively NR× NR and NT × NT are predefined DFT matrices

and ˜Ωvirt is the coupling matrix. However, this model is restricted to single polarized

uniform linear arrays (ULAs) only, we therefor introduce in the following a model that copes with this issue.

In [26], eigenbases of transmit and receive correlation matrices are used to model this spatial correlated channel, i.e.,

where UR and UT are the eigenbasis of the receive and transmit correlation matrix,

respectively. With ΛR and ΛT being the diagonal matrices comprise the eigenvalues,

which are nonnegative, of RR and RT, the eigen decompositions of RR and RT are

RR= URΛRUHR and RT = UTΛTUHT , (2.29)

˜

Ω is the element-wise square root of the coupling matrix Ω which is defined as

[Ω]i,j = E{|uHR,iHu∗T,j|2}, i = 1, . . . , NR, j = 1, . . . , NT. (2.30)

where uR,i and uT,j are the ith and jth column vector of UR and UT, respectively. From

this coupling matrix, the mean amount of energy that is coupled with an eigenvector of one side to that of the other can be identified a more general framework of channel model.

2.3.4

Model-Based Correlated Channel

The model-based correlated channel is introduced by [27]. Since the channel matrix

H can be decomposed via singular value decomposition (SVD), H = UΛVH_{, where U}

and V are NR× NR and NT × NT unitary matrices, respectively, and Λ is a NR× NT

diagonal matrix with non-negative entries. The two unitary matrices can be represented by predefined unitary matrices Q1 and Q2 as UP1 = Q1 and VP2 = Q2 where P1 and

P2 are unitary. As a result, we have

H = Q1P−11 Λ(P−12 )HQH2 = Q1CQH2 , (2.31)

where C is a complex random matrix, equation (2.29) can be regarded as a generalization of all the models mentioned above. To be specific, it is equivalent to the Kronecker model if C satisfies

vec(C) = (ΞT ⊗ ΞR)vec(Hω), (2.32)

where ΞT and ΞR are obtained by Gram-Schmidt orthonormalization with R

1 2 T = Q2ΞT and R 1 2

R = Q1ΞR, and is related to the Weichselberger model when

UT = Q2PHT and UR= Q1PHR, (2.33)

with PT and PR being the eigenbasis matrices of E{CCH} and E{CTC∗} whose

eigen-values are the same as those of E{HHH_{} and E{H}T_H∗_{}. Finally, if Q}

1 and Q2 are

chosen to be composed of columns of DFT matrices, this general model is compatible with the virtual channel representation by Sayeed [25].

The fact that second-order statistics is not required, because spatial correlation is captured by predetermined nonparametric regression, reduces the number of parameters needed to be estimated when modelling the channel H by (2.29). This provides a great complexity reduction when the dimension of H is large.

Chapter 3

Estimation of Spatial-Modulated

Time-Varying Channels

In this chapter, we first give a review of conventional pilot-based channel estimation techniques and present the proposed time-varying channel estimation methods for SM system.

3.1

Conventional Pilot-Based Channel Estimation

Methods

Considering a time-varying block-fading channel, its CSI is obtained by transmitting pilot signal which is known to the receiver. Assume pilot signal matrices XP(k)’s are of

size NT × B and the average pilot symbol energy equals to the data symbol energy. The

received signal at kth block time can be written as

YP(k) = H(k)XP(k) + ZP(k), (3.1)

where YP(k) is NR× B. H(k) is the NR× NT MIMO channel matrix at time k, and the

entries of AWGN matrix Zp(k) are i.i.d., zero mean complex Gaussian with variance σz2.

To estimate a time-varying channel, pilot signal blocks are inserted in transmit data stream periodically. Transmit data detection can thus be conducted with the estimated channel. However, if the channel changes rapidly, the period of pilot signal insertion

should be short to keep up with the change. This in turn reduces the power or spectral efficiency of the data signal. We illustrate this idea in Figure 3.1.

Table 1 SM mapping table

Pilot Data Pilot Data

Pilot Data Pilot Data Pilot Data Pilot Data

ڭ

ڭ

ࡽ෡

܆෡

Figure 3.1: Periodical pilot signal insertion in transmit data.

Two conventional pilot-based channel estimation methods which are discussed in the following.

3.1.1

Least Square Channel Estimation

The least square (LS) channel estimate of H(k) is found by minimizing the squared error kYP(k) − H(k)XP(k)k2 [29]. As a result, the estimated channel ˆHLS(k) can be

derived by

ˆ

HLS(k) = YP(k)XHP(k)(XP(k)XHP(k))−1 (3.2)

= H(k) + ZP(k)XHP(k)(XP(k)XHP(k))−1.

While the main advantage of the LS channel estimation method is its low complexity, it fails to take the statistics of the channel and noise into account. Such obliviousness can result in significant noise power enhancement offer channel equalization performance degradation. As LS estimation may be a suitable solution when channel and noise statistics are unknown, it is not sufficient for our requirement.

3.1.2

Minimum Mean Square Error Channel Estimation

In order to solve the noise enhancement problem, minimum mean square error (MMSE) channel estimation method is introduced. As its name suggests, if F minimizes

the mean square error E{kYP(k)F − H(k)k2}. Since the entries of H(k) are assumed to be unit-variance, we have ˆ HM M SE(k) = YP(k)XHP(k)(σ2zINT + XP(k)X H P(k))−1 (3.3)

Based on the abovementioned channel estimation methods, we propose two time-varying channel estimation methods for SM systems and discuss in the ensuing sections.

3.2

Decision-Directed SM Channel Estimation

The main idea of decision-directed channel estimation is to track channel variations using detected data symbols of previous blocks as pilots. This method saves the pilot signal overhead and thus retains the data rate. Due to the SM systems’ capability to avoid ICI, the utilization of detected data symbols can be a good approach to estimate channel. The reason is that it is proved the optimal channel estimation performance can be achieved by using unitary pilot signal matrices which help decoupling channel vectors at the receiver [30], [31]. This behavior can similarly be realized by SM transmit data matrices. Although it is likely that one or more transmit antennas remain inactive in one data block, in general, all channel coefficients would be updated for a sufficiently long transmission period since the selected transmit antenna index is a uniformly distributed random variable.

However, the problem of error propagation resides in all decision-directed channel estimation methods. An incorrectly detected symbol of a previous block may affect the channel estimation performance of its following blocks and produce more symbol errors that continue to propagation along the entire transmission. Hence, pilot signals are inserted periodically and the previous result of channel estimation is forgotten, and the forgetting factor α is introduced to compress the error propagation effect.

The initial guess of channel coefficients is derived by transmitting a full block of pilot signals and estimating with either LS- or MMSE- based methods. These coefficients can

then be updated with decision-directed channel estimation for SM. When MF-based detector is employed, as explained in Chapter 2, we first normalize the column vectors of the estimated channel matrix of the previous ((k − 1)-th) block time, i.e.,

¯ H(k − 1)def= " ˆ h1(k − 1) kˆh1(k − 1)k , · · · , hˆNT(k − 1) kˆhNT(k − 1)k # . (3.4)

Then the received signal of current instant (kth block time) is multiplied by the normal-ized channel matrix ¯_{H(k − 1) and we have}

gi(k) def =    g1,i(k) ... gNT,i(k)   = ¯H H (k − 1)yi(k), i = 1, · · · , B. (3.5)

The active transmit antenna index of ith instant of block k can be estimated by finding the maximum value of the MF output:

ˆ

ℓi(k) = arg max ℓi∈{1,··· ,NT}

|gℓi,i(k)|, (3.6)

and the carried data is detected via gℓi,i(k), ˆ x_ℓˆ_i(k) = Q gˆ_ℓi(k) kˆh_ℓˆ_i(k − 1)k ! . (3.7)

On the other hand, when ML detector is used, transmit antenna index and data are jointly estimated by

(ˆx_ℓˆ_i(k), ˆℓi(k)) = arg min (xi,ℓi)ky

i(k) − ˆhℓi(k − 1)xik

2_{. (3.8)}

Let ˆxi(k) = [0, · · · , 0, ˆx_ℓˆ_i(k), 0, · · · , 0]T, the estimated data matrix ˆX(k) can be denoted

as

ˆ

X(k) = [ˆx1(k) ˆx2(k), · · · , ˆxB(k)], (3.9)

which may not be full rank. Define L = {1, · · · , NT} as the set of transmit antenna

indices and ˆ_{ℓ(k) = {ˆℓ}1(k), · · · , ˆℓB(k)} the set of active antenna estimates in block k. We

flatten ˆ_{X(k) by removing its all-zero rows, which have indices belong to L \ ˆℓ(k), and} denote the result by ¯X(k).

The channel estimates for the columns indexed by ˆℓ(k) are thus ˆ

H[1, · · · , NR; ˆℓ(k)](k) def

= Y(k) ¯XH(k)( ¯X(k) ¯XH(k))−1. (3.10) While columns that are not updated in this block are kept unchanged and remains the same as the previous block, i.e.,

ˆ

H[1, · · · , NR; L \ ˆℓ(k)](k) = ˆH[1, · · · , NR; L \ ˆℓ(k)](k − 1). (3.11)

However, to ameliorate error propagation effect, we would like to update channel estimate not so abruptly. We define, instead,

ˆ

H(k) = (1 − α) ˆH(k) + α ˆ_{H(k − 1),} (3.12)

where ˆ_{H(k) is of the form as (3.10) and (3.12) and 0 ≤ α ≤ 1.}

We conclude this section by noting that the error-rate performance curve of this scheme has an error floor in high SNR region due to the possibility that not all channel coefficients are updated in each block. Therefore, superimposed pilot signals, which ensure every coefficient is updated, are introduced in next section to improve channel estimation performance.

3.3

SM Channel Estimation With Superimposed

Pi-lots

Superimposed pilot-assisted approaches add (superimposed) low power pilot signal onto the data signals before transmission. At the receiver, the channel can be estimated by these superimposed pilot signals. Unlike the traditional pilot-assisted methods, which do not send any information during the estimation phase, superimposed pilot signals do not cause any loss in transmit data rate. However, this is at the cost of decreasing effective SNR due to the additional power for pilots. In the thesis, we propose a channel estimation method, which combines the decision-directed channel estimation method and use of superimposed pilot signals, for SM schemes.

Note that the initial channel coefficient estimates are acquired by LS or MMSE esti-mation while after that the estimated coefficients are updated by the detected SM data matrices and imposed pilot signals. Since for an SM system only one transmit antenna is active at a time, other transmit antennas can transmit pilot signals simultaneously to improve the channel estimation quality. The power of superimposed pilots shall be carefully designed because of the following reasons: i) pilots with large powers may cause serious ICI in SM system; and ii) low power pilots do not give reliable estimates.

At the transmitter, data matrix X(k) and NT × B superimposed pilot matrix S(k)

are transmitted at the same time. The received signal is denoted as

Y(k) = H(k)(X(k) + S(k)) + Z(k), (3.13)

Due to the fact that the antenna selection is random, chances are one or more antennas are not active in a block duration. Thus, the transmit data matrix X(k) might not be full rank and inversible, making channel tracking fails. Nevertheless, this problem may be solved by designing the superimposed pilot matrix S(k) to be data-dependent. Specifically, the pilot signals are transmitted by transmit antennas which are not used in X(k) and its corresponding matrix S(k) has column vectors that span Ker{X(k)} (nullspace of X(k)).

Let Nx(k) be the matrix consists of a set of basis vector of Ker{X(k)}, where it

satisfies

X(k)Nx(k) = 0, (3.14)

and the size of Nx(k) is B × M withM =Nullity(X(k)).

Therefore,

Y(k)Nx(k) = H(k)(X(k) + S(k))Nx(k) + Z(k)Nx(k) (3.15)

= H(k)S(k)Nx(k) + Z(k)Nx(k).

In the following, we detail the design of the superimposed pilot signal matrix for SM systems. For block k, S(k) has (B-M) nonzero vector, 0, and M single-component

columns that have only one nonzero elements on the rows not used by X(k), where the positions of the nonzero vectors depend on the time instants in which transmit antennas are reused. For example, let both the number of transmit antennas and block size be 4. For a single transmit data matrix

X(k) =     0 0 0 0 0 0 0 0 0 x2 x3 x4 x1 0 0 0     or X(k) =     x1 0 0 x4 0 0 0 0 0 x2 x3 0 0 0 0 0     , (3.16)

the corresponding superimposed pilot signal matrix can be

S(k) =     0 0 S1 0 0 0 0 S2 0 0 0 0 0 0 0 0     or S(k) =     0 0 0 0 0 S2 0 0 0 0 0 0 0 0 0 S4     , (3.17)

where Si’s are the superimposed pilot signals with energy Es.

Due to the SM characteristic (3.15) and the structure of superimposed pilots (3.17), we proposed a channel estimation scheme for this system. It can be summarized by the following procedure:

Step 1: With the detected data matrix ˆX(k), derived by either MF-based or ML detector, we obtain the matrix Nˆx(k) corresponding to its null space by solving M

underdetermined systems of linear equations ¯

X(k) · Nˆx[1, · · · , B; m](k) = 0 (3.18)

for m = 1, · · · , M, where ¯X(k)def= ˆX[ˆ_{ℓ(k); 1, · · · , B](k).}

To simplify channel estimation (in the next step) and incorporate the sparse nature of transmit data matrix, we arrange Nˆx(k) into a special form such that

¯

S[L \ ˆℓ(k); 1, · · · , M](k) = EsIM, (3.19)

where ¯S(k) = S(k)Nˆx(k). The technique to achieve it is given in Appendix A.

Step 2: With Nˆx(k) superimposed pilots can be taken out as

¯

Y(k)def= Y(k)Nx(k) = H(k)S(k)Nx(k) + Z(k)Nx(k). (3.20)

Hence the LS channel estimations on the pilot positions can be obtained as ˆ H[1, · · · , NR; L \ ˆℓ(k)](k) = ¯Y(k) ¯P(k))H( ¯P(k) ¯P(k))H)−1. (3.21) Let ¯_{H[1, · · · , N}R; L\ˆℓ(k)](k) def = ˆ_{H[1, · · · , N}R; L\ˆℓ(k)](k) and ¯H[1, · · · , NR; ˆℓ(k)](k) def = 0, ¯

Y(k)def_{= Y(k) − ¯}H(k)S(k). (3.22)

The LS channel estimates for the remaining columns ˆℓ(k) are ˆ

H[1, · · · , NR; ˆℓ(k)](k) = ¯Y(k) ¯XH(k)( ¯X(k) ¯XH(k))−1, (3.23)

where ¯X is defined similarly in Section 3.2.

3.4

Simulation Results

In this section, the simulation results of the proposed two channel estimation meth-ods are presented. For a 4 × 4 MIMO system, we generate each independent path of the spatial uncorrelated time-varying channels by jake’s model [35]. The carrier frequency is 2GHz and sampling time Ts is 0.1ms of the duration 10s channel paths. Then the

time correlation function is defined as

E{hij(t1)h∗ij(t2)} = J0(2πfD(t1− t2)Ts). (3.24)

Setting the block length to 4 for the reason that at least 4 symbol times are needed to fully estimate the total number of channel coefficients. The frame size is 5 where the first block consist of pilot signals only. In addition, QPSK constellation is used.

In Figure 3.2 and Figure 3.3, we generate the time-varying channel at mobile velocity 30 km per hour, and compare the bit error rate(BER) and normalized minimum mean square error(NMSE) of decision-directed channel method based on MF-based and ML detector and perfect CSI under the same pilot overhead. It is shown that the performance of decision-directed channel estimation is better than that of only initial pilot estima-tion is used. Moreover, the decision-directed channel estimaestima-tion based on ML detector

outperforms MF-based detector. The BER and NMSE performance of decision-directed channel estimation method via different mobile velocity based on MF-based and ML detector are given in Figure 3.4, Figure 3.5, Figure 3.6, and Figure 3.7. It can be seen that there is error floor in high SNR region.

0 5 10 15 20 25 30 10−3 10−2 10−1 100 E b/N0 BER DD−MF−based perfect CSI Pilot DD−ML

Figure 3.2: BER performance of SM detectors with decision-directed channel estimation.

Superimposed channel estimation method is proposed to improve the error floor performance of decision-directed channel estimation, the energy of superimposed pilot signals affects the BER performance which is shown in Figure 3.8 where the MF-based detector is used and the mobile velocity is set to 30 km/hr. If the energy of superimposed signals is too large, it would cause the serious ICI to receiver and thus degrades the BER performance. However, if the energy is small, it could not have good channel estimation performance. Hence, the optimal energy of the superimposed signal is approximately 2/SNR which is shown in Figure 3.9. The BER performance of superimposed channel estimation method under different mobile velocity based on MF-based and ML detector are shown in 3.10 and 3.11, respectively. In addition, we can see from these simulation results that the use of superimposed signal can get better performance than that of

0 5 10 15 20 25 30 10−3 10−2 10−1 100 E b/N0 BER DD−MF DD−ML pilot

Figure 3.3: NMSE performance of SM detectors with decision-directed channel estima-tion.

decision-directed channel estimation method in high SNR region and in high mobile velocity.

0 5 10 15 20 25 30 10−4 10−3 10−2 10−1 100 E_b/N₀ BER 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.4: BER performance of MF-based detector with decision-directed channel es-timation under various mobility.

0 5 10 15 20 25 30 10−3 10−2 10−1 100 E b/N0 NMSE 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.5: NMSE performance of MF-based detector with decision-directed channel estimation under various mobility.

0 5 10 15 20 25 30 10−5 10−4 10−3 10−2 10−1 100 E_b/N₀ BER 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.6: BER performance of ML detector with decision-directed channel estimation under various mobility.

0 5 10 15 20 25 30 10−3 10−2 10−1 100 E_b/N₀ NMSE 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.7: NMSE performance of ML detector with decision-directed channel estimation under various mobility.

0 5 10 15 20 25 30 10−3 10−2 10−1 100 E_b/N₀ BER 1/20 Es 1/10 Es 1/5 Es 1/2 Es 1/30 Es

Figure 3.8: BER performance of superimposed channel estimation with different super-imposed pilot symbol energy.

0 5 10 15 20 25 30 10−4 10−3 10−2 10−1 100 E b/N0 BER 5/SNR 4/SNR 2/SNR 1/SNR

Figure 3.9: BER performance of different superimposed pilot symbol energy versus SNR.

0 5 10 15 20 25 30 10−3 10−2 10−1 100 Eb/N0 (dB) BER 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.10: BER performance of superimposed channel estimation method with MF-based detector under different mobile velocity.

0 5 10 15 20 25 10−4 10−3 10−2 10−1 100 E_b/N₀ BER 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 3.11: BER performance of superimposed channel estimation method with ML detector under different mobile velocity.

Chapter 4

Spatial Modulation Using

Dual-Polarized Antenna Arrays

4.1

Dual-Polarized MIMO Channel Models

In MIMO wireless communication systems, antenna spacings are usually required to be at least half a wavelength at subscriber units and ten wavelengths at base stations to achieve satisfactory performance. This condition restricts the implementation of MIMO systems on some space-limited devices. However, since orthogonal polarization can decrease the correlation of transmit antennas or receive antennas, the usage of co-located dual-polarized antennas can be a space- and cost-effective alternative. Figure 4.1 depicts a co-located dual-polarized MIMO system with antennas grouped into pairs.

For ideal dual-polarized antennas, cross-polar transmissions, from a vertically-polarized transmit antenna to a polarized receive antenna or from a horizontally-polarized transmit antenna to a vertically-horizontally-polarized receive antenna, equal to zero. Practically, there are two depolarization mechanisms that can cause polarization in-terference: cross-polar isolation (XPI) due to use of imperfect antennas and the depo-larization caused by the propagation channel which can be identified by the existence of cross-polar ratio (XPR). The interplay between both effects forms the global cross-polar discrimination (XPD) which quantifies the separation between two channels of different

ڭ

ġ

ڭ

ġ

Figure 4.1: A co-located dual-polarized MIMO system model.

polarizations. Besides, vertically-polarized electromagnetic wave is vulnerable to electric current on the ground, so co-polar ratio (CPR) can be taken into consideration in some cases. Figure 4.2 shows the depolarization mechanisms discussed above.

We first consider one dual-polarized transmit-receive antenna pair, Let pij def

= |hij|2(i, j ∈

{V, H}) be the instantaneous dual-polarized channel gain. The channel is a 2×2 matrix,

HP =  hV V hV H hHV hHH  , (4.1)

and depolarization parameters can be defined as (1) Cross-polar isolation:

Transmitter : XPIT def= E{pii}

E{pij}

(4.2) Receiver : XPIRdef= E{pii}

E{pji} (4.3) (2) Cross-polar ratio: XPR def= pvv phv = phh pvh (4.4)

݌



݌



Depolarizing

Medium

݌

݌



Figure 4.2: Depolarization mechanisms.

(3) Co-polar ratio:

CPR def= pvv phh

(4.5) Analytically, XPI effect at transmitter and receiver can be respectively modelled with coupling matrices Mt =  1 √χa,t √_χ a,t 1  , (4.6) Mr =  1 √χa,r √_χ a,r 1  , (4.7)

where χa,t and χa,r are the inverse of XPI at transmitter and receiver, respectively. If

the used dual-polarized antennas are in perfect condition, these scalars will equal to zeros. Note that this depolarization mechanism affect line-of-sight (LOS) and scattered components whereas XPR exists only in non-line-of-sight components.

While the above definitions concentrate only on the channel gains, to get a thorough understanding of the depolarization effect, the dual-polarized channel can be characterize by the correlation matrix E{vec(HH

P)vec(HHP)H}. The diagonal terms of this 4×4 matrix

are the average channel gains of each channel coefficient and the off-diagonal ones include the cross-polar correlations (XPC) between hiiand hij or hji, co-polar correlation (CPC)

between hvvand hhh, and anti-polar correlation (APC) between hvh and hhv. This model

is verified by several measurements to be a good approximation to the real dual-polarized channels. The corresponding parameters to various environment settings are given in [19] and references therein.

In the following, one dual-polarized antenna pair is extended to a MIMO system. As the measurement result [19] shown, in Rayleigh fading environment, the spatial correlation properties are independent of the polarization, i.e., beam pattern are similar for all antennas, the parameters concerning the polarization and spatial correlation can be decoupled in our model. It is especially true when the system is implemented in macrocells or microcells. It is explained in the following.

An NR× NT dual-polarized MIMO fading channel consists of NT/2 and NR/2

co-located dual-polarized transmit and receive antenna pairs. If all dual-polarized antennas are identically oriented, the joint transmitter-to-receiver direction spectrum would be equivalent for all co-located antennas. The matrix can thus be written as

˜ Hx = HNR 2 × NT 2 ⊗ Mr ˜ WMt (4.8)

where HNR/2×NT/2 is the spatial correlated Rayleigh fading channel and ˜W is the de-polarization matrix which is separated from the spatial correlation parameters. Matrix

˜

W models the differential attenuation and the correlated phase shifts between the dual-polarized channels. Specifically,

vec( ˜WH) =     1 √µχϑ∗ √_χσ∗ √_µδ∗ 1 √_{µχϑ µχ} √_µχδ∗ 2 µ√χσ∗ √_χσ √_µχδ 2 χ √µχϑ∗ √_µδ 1 µ√χσ √µχϑ µ     1/2 vec( ˜WH_ω), (4.9)

where µ and χ represent the inverse of CPR and XPR, σ and ϑ are the receive and transmit correlation coefficients between polarization vv and hv, hh and hv, vv and vh or hh and vh. δ1 and δ2 are respectively co- and anti-polar correlation coefficients. The

last term, ˜ Wω def =  e jΦ1 ejΦ2 ejΦ3 ejΦ4  , (4.10)

where Φk’s are uniformly distributed in [0, 2π).

As a result, a dual-polarized MIMO channel can be formed into

˜ Hx=       H1,1 H1,2 . . . H_1,NT 2 H2,1 H2,2 . . . H_2,NT 2 .. . ... ... ... HNR 2 ,1 HNR 2 ,2 . . . HNT 2 , NT 2       , (4.11)

where Hi,j is the dual-polarized channel matrix of the jth transmit and ith receive

antenna pairs: Hi,j =  hiV jV hiV jH hiHjV hiHjH  . (4.12)

where each entry hiPijPj is the channel coefficient between the Pi-polarized ith transmit antenna pair and Pj-polarized jthe receive antenna pair. In addition, we define the

column vectors of ˜Hx as ˜ Hx= h h1V, h1H, · · · hNT 2 V , hNT 2 H i . (4.13)

4.2

Dual-Polarized Spatial-Correlated (DPSC)

Chan-nel Model

Most of the previous proposals model spatial correlated channels by the Kronecker model which is not reasonable when joint correlation exists between transmitter and receiver. A more general model [27] has been discussed in Chapter 2 and is considered here to incorporate with the dual-polarized systems.

We assume the XPIs of all dual-polarized antenna pairs are infinite, i.e., the transmit and receive antenna in a pair are perfectly polarized. Matrices Mrand Mtbecome

iden-tity and are thus ignored in the following discussion. A dual-polarized MIMO channel is denoted as ˜ Hx= HNR 2 × NT 2 ⊗ ˜ W. (4.14)

It has been proven that 4.14 can be equally represented by ˜

Hx= QNR,KRCQ˜

H

NT,KT, (4.15)

where ˜C is complex random and QNR,KR and QNT,KT are NR × KR and NT × KT predefined unitary matrices, respectively with KR (≤ NR) and KT (≤ NT) being the

modelling orders to be discussed later [27].

Several types of functions can be chosen as (basis) vectors in the predefined unitary matrices [27]. In this thesis, polynomial basis functions [32] are used. Specifically, these polynomial functions are of degree D where the entries of the corresponding basis matrix D are specified as

[D]i,j = (i − 1)j−1, i, j = 1, 2, · · · , D, (4.16)

where D equals to NT when D is used to determine QNT,KT and NR to determine QNR,KR. Consider first the construction of QNT,KT. In order to satisfy the unitary property, by applying QR decomposition, we can obtain the orthonormal polynomial

basis matrix Q, i.e., D = QR. Then, we can choose the first KT columns of Q to

be the predefined matrices QNT,KT where modelling order KT can be determined by Akaike information criterion (AIC) or minimum description length (MDL) approach [33]. QNR,KR is obtained analogously.

Due to the property that co-located dual-polarized antennas experience the same spatial characteristics, we are able to reduce the degree of this basis matrix D to half of the original DPSC MIMO channel model is thus modified to

˜ Hx = (QNR 2 ,KR⊗ I2 )C(QHNT 2 ,KT ⊗ I 2), (4.17) C = ˜_{C ⊗ ˜}W, (4.18)

where the depolarization effect is coupled into C and the modelling orders KR≤ N₂R and

KT ≤ N₂T. With predefined QNR 2 ,KR

and QNT 2 ,KT

, identification of the unknown channel ˜

Hx is equivalent to the estimation of ˜C, which usually has fewer unknowns than those

4.3

Time-Varying DPSC Channel Estimation

Let XP(k) be NT× B, a full-rank pilot matrix used to estimate time-varying DPSC

channel ˜Hx. The received signal can be expressed as

YP(k) = H˜x(k)XP(k) + ZP(k) (4.19) = (QNR 2 ,KR⊗ I2 ) ˜C(k)(QHNT 2 ,KT ⊗ I 2)XP(k) + ZP(k). where QNT 2 ,KT , QNR 2 ,KR

, and XP(k) are known to the receivers. By a property of vector

operation, we have

vec(ABC) = (CT _{⊗ A)vec(B).} (4.20)

Due to the fact that (4.20),

vec(YP(k)) = (XP(k)H(QNT/2,KT ⊗ I2)) ⊗ (QNR/2,KR ⊗ I2) vec(C(k)) (4.21) +vec(ZP(k)),

the LS estimate of C(k) can be obtained as

vec( ˆC(k)) = (VHV)−1VHvec(YP(k)), (4.22)

where Vdef= XP(k)H(QNT/2,KT ⊗ I2)
⊗ (QNR/2,KR ⊗ I2).

Note that the decision-directed channel estimation technique proposed in Chapter 3 can also be utilized here by substituting XP(k) in (4.22) by ˆX(k)[ˆℓ(k); 1, · · · , B](k).

The detection algorithms for SM in dual-polarized channel are discussed in the next subsection.

4.4

Data Detection in SM-DPSC Channels

When co-located dual-polarized in the SM system, information is conveyed by the index of the transmit antenna pair and the specific polarization in that pair used and the symbol transmitted. Hence, m = log₂(NTM) bits are transmitted in each channel

use. The receiver’s task thus contains the used transmit antenna and polarization index estimation and transmitted symbol detection. The system model is depicted in Figure 4.3. A mapping rule for this SM system with two dual-polarized antenna pairs in both the transmitter and receiver and BPSK or QPSK modulated symbols is suggested in Table 4.1 HP- Ant. ேೃ ଶġ VP- Ant. ேೃ ଶ ġ HP- Ant. 1ġ VP- Ant. 1ġ HP- Ant. ே೅ ଶġ VP- Ant. ே೅ ଶ ġ HP- Ant. 1 VP- Ant. 1ġ

෩

ڭ

ڭ

ř

Œ

ʼn

ࡽ෩

Polar. Est. Ant. Est.ġ Symbol Det.ġ

Transmitterġ Receiverġ

Figure 4.3: A dual-polarized SM system model.

Three data detection techniques are proposed for the spatial modulated dual-polarized MIMO system. Since we only concentrate on its data detection in this section, CSI ˜Hx

is assumed known at the receiver and block index k is ignored. The received signal corresponding to transmitted signal X can be represented as

Y = ˜HxX + Z. (4.23)

The ML detector is simply

(ˆxi, ˆℓi) = arg min xi,ℓiky

i− ˜hx,ℓixik

Input bits Antenna Index Transmit Symbol Antenna Index Transmit Symbol 000 VP-1 +1 VP-1 +1 + j 001 VP-1 -1 VP-1 -1 + j 010 HP-1 +1 VP-1 -1 - j 011 HP-1 -1 VP-1 +1 - j 100 VP-2 +1 HP-1 +1 + j 101 VP-2 -1 HP-1 -1 + j 110 HP-2 +1 HP-1 -1 - j 111 HP-2 -1 HP-1 +1 - j Pilot Pilot

Pilot Data Pilot Data Pilot Data Pilot Data

ڭ ڭġ

ࡽ෡ ܆෡κ

Table 4.1: An SM mapping table for the dual-polarized system for 3 bits/transmission

The MF-based detector for (4.24) is similar to the traditional one (2.16)-(2.21) for (2.3) with following procedure:

¯ H = " ˜ hx,1 k˜hx,1k , · · · , h˜x,NT k˜hx,NTk # , (4.25) gi = H¯Hyi, (4.26) ˆ ℓi = arg max ℓi∈{1,··· ,NT} |gℓi,i|, (4.27) ˆ xℓi = Q( g_ℓˆ_i,i k˜h_x,ℓˆ ik ), i = 1, · · · , B. (4.28)

Prior to the introduction of the last detection method, we consider the following.

Since antenna polarization selection bares information for SM system in dual-polarized MIMO system, we shall give a few facts. Based on the Cauchy-Schwarz inequality, for a specific spatial channel i (the ith column of ˜Hx, hi), we have the following due to the

mismatch of polarization: |hH iVhiH| khiHk ≤ |hH iVhiV| khiVk = kh iVk (4.29)

which agree with our intuition. On the other hand, for two different spatial channel vectors, say hi and hj, the correlation between their portions corresponding to the same

polarization outweighs that corresponding to different polarizations, i.e., |hH iVhjH| khjHk ≤ |hH iVhjV| khjVk ≤ khiVk. (4.30) 37

This is because of the inability of a polarized antenna to receive signal of other polar-izations. In this way, the (horizontal) polarization of a signal passing through hiV can

be effective estimated prior to the antenna and symbol detection.

As a result, the polarization used can be detected before ML detection of antenna pair index and symbol. This suboptimal method effectively lower the complexity of detection algorithm with some performance loss. Specifically, first calculate (4.24) and find the MF output of vertical or horizontal polarized transmit antenna of each transmit antenna pair, ˆ nP = [ˆnV nˆH] = NT 2 X i=1 arg max nP∈{[1,0],[0,1]}   ¯hH_iVy · n_P,1+ ¯hH_iHy · n_P,2   (4.31)

where nP is the indicator vector whose position of value 1 represents the detected

po-larization in a specific transmit antenna pair and ˆnV and ˆnH count the total number

of detected polarization used in all transmit antenna pairs. Based on the majority vote algorithm, the used polarization of the transmit antenna is decided via

ˆ

P = arg max

P =V,H {ˆnV nˆH} (4.32)

Based on the result, the ML detector of the antenna pair used and symbol transmitted only needs to search over the specific ˆP -polarized channel vectors,

(ˆxi, ˆℓi) = arg min xi,ℓi=i ˆP

kyi(k) − ˜hx,ℓixik

2_{, i = 1, . . . , B. (4.33)}

This detection method reserves the high detection performance of the ML detector but needs only about half of the complexity required by the latter. As will be shown later, it outperform the MF-based detector.

4.5

Simulation Results

In this section, we investigate the performance of SM scheme using dual-polarized antenna. For simplicity and the measurement results in [19], we concentrate on the effect

of XPR and CPR. The values of µ and χ are set to 0.7 and 0.1 for all the simulation results except for Figure 4.8 and Figure 4.9, and the parameters of polarized correlation are set to zero. First, the BER performance of 2 × 2 SM comparing to 4 × 4 SM using dual-polarized antenna is given in Figure 4.4 where MF-based detector is used and CSI is assumed known to receiver. We can see from the result that the use of dual-polarized antennas can give the polarization diversity gain.

0 2 4 6 8 10 12 14 16 18 10−3 10−2 10−1 100 E b/N0 BER SM−BPSK SM−16 QAM DPSM−BPSK DPSM−16 QAM

Figure 4.4: Comparison of BER performance for SM under conventional MIMO channel and dual-polarized MIMO channel.

We also give a BER performance comparison between SM and V-BLAST and space time code based on Alamouti[4] under dual-polarized channel in Figure 4.5. Based on the same spectral efficiency which is 8 bits/transmission, SM outperforms V-BLAST and Alamouti scheme. V-BLAST in this simulation result uses QR-based detector and Alamouti and SM scheme use ML detector.

Then the normalized mean square error of dual-polarized spatial correlated channel estimation method is shown in Figure 4.6. In this figure, we use the 3GPP SCM model

0 5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 E_b/N₀ BER V−BLAST 16−QAM 4x2 Alamouti 256QAM 4x2 SM 32 QAM 4x8

Figure 4.5: Comparison of BER performance for SM and VBLAST and Alamouti scheme under dual-polarized channel for 8 bits/transmission.

to generate the co-located dual-polarized spatial correlated channel. Two channels with different angle spread (AS) 2 and 15 with mobile velocity 60 km/hr are adopted in the simulation result. The number of transmit and receive antennas setting to 8, we compare the NMSE performance of conventional MIMO channel estimation methods and modified dual-polarized channel estimation method. It can be seen from this figure that dual-polarized spatial correlated channel estimation method only use half of the basis to achieve the same estimation performance with conventional method.

The BER performance of proposed detectors for SM in dual-polarized system is shown in Figure 4.7. We consider a 4 × 4 MIMO channel using dual-polarized antennas, and QPSK modulated signals. From the result, ML detector performs the best of three detectors and the low-complexity sub-optimal detector outperforms MF-based detector where we assume CSI is known to receiver in this simulation result. We also investigate the influence on the three detectors of depolarization effect caused by propagation chan-nel. From Figure 4.8, fixed parameter χ to 0.5, when the value µ is too small, all the three detectors can not have adequate performance because the horizontal-polarized channel

0 5 10 15 20 25 30 10−3 10−2 10−1 100 101 E b/N0 NMSE KT=8, KR=8,AS=15 KT=4, KR=4, AS=15 (DP) KT=7, KR=8, AS=15 KT=6, KR=8, AS=15 KT=8, KR=8, AS=2 KT=7, KR=8, AS=2 KT=6, KR=8, AS=2 KT=4, KR=4, AS=2 (DP)

Figure 4.6: NMSE comparison of channel estimation for dual-polarized spatial-correlated MIMO channel with different modelling order, AS=2 and 15.

is too small than vertical and thus cause the signals transmit by horizontal-polarized antenna cannot be successfully detected. The effect of χ which denotes the inverse of XPR value with fixed µ = 0.5 on the detectors’s performance is given in Figure 4.9. The result shows that when the value of χ is too large which means that cross interference between polarization is large, the performance of the MF-based and the suboptimal detector degrade since the large χ value can make the channel vector of vertical- and horizontal-polarized transmit antenna become similar thus the value of MF output of both the vertical and horizontal polarized antenna are approximately equal. Therefore, the MF-based and suboptimal detectors cannot detect the polarization correctly.

The following part, we give the simulation result of different time-varying channel estimation methods in DPSC channel where t and r are assumed to be 0.3, and three detectors are investigated. In Figure 4.10–4.15, the BER and NMSE performance of decision-directed channel estimation method based on MF-based, suboptimal and ML

0 2 4 6 8 10 12 14 16 18 10−4 10−3 10−2 10−1 E_b/N₀ BER subopt. ML MRC

Figure 4.7: Comparison of BER performance of SM for MF-based detector, suboptimal detector and ML detector under perfect CSI.

detector are shown, we can see that using ML or suboptimal detectors can have better performance than that of MF-based detector. In addition, the performance loss increase when the mobile velocity increases. In Figure 4.16, BER performance of superimposed channel estimation method is given. It performs better than decision-directed channel estimation in high SNR and fast varying channel.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−4 10−3 10−2 10−1 100 µ BER MF−based Deteector Sub−ML Detector ML Detector

Figure 4.8: The effect of different mu (inverse of CPR) value with different detectors on BER performance. 0 0.2 0.4 0.6 0.8 1 10−4 10−3 10−2 10−1 χ BER MF−based Detector Sub−ML Detector ML Detector

Figure 4.9: The effect of different chi (inverse of XPR) value with different detectors on BER performance.

0 5 10 15 20 25 10−2 10−1 100 E b/N0 BER 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 4.10: BER performance of decision-directed channel estimation method with MF-based detector in DPSC channel under different mobile velocity.

0 5 10 15 20 25 10−2 10−1 100 101 E b/N0 NMSE 30 km/hr 40 km/hr 50 km/hr 60 km/hr 80 km/hr 100 km/hr

Figure 4.11: NMSE performance of decision-directed channel estimation method with MF-based detector in DPSC channel under different mobile velocity.