適用於直接序列碼分多重擷取系統多用戶偵測之部分平行式干擾消除：效能分析與新演算法

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國立交通大學

電信工程學系

博士論文

適用於直接序列碼分多重擷取系統多用

戶偵測之部分平行式干擾消除：效能分析

與新演算法

Partial Parallel Interference Cancellation for

DS-CDMA Multiuser Detection: Performance

Analysis and New Algorithms

研究生：謝雨滔

指導教授：吳文榕

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Partial Parallel Interference Cancellation for DS-CDMA

Multiuser Detection: Performance Analysis and New

Algorithms

適用於直接序列碼分多重擷取系統多用戶偵測之

部分平行式干擾消除：效能分析與新演算法

研究生：謝雨滔 Student: Yu-Tao Hsieh

指導教授：吳文榕博士 Advisor: Dr. Wen-Rong Wu

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Department of Communication Engineering

College of Electrical Engineering and Computer Science

National Chiao Tung University

in Partial Fulfillment of the Requirement

for the Degree of

Doctor of Philosophy

In

Communication Engineering

July 2004

Hsinchu, Taiwan, Republic of China

中華民國九十三年七月

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適用於直接序列碼分多重擷取系統多用戶偵測之部分

平行式干擾消除：效能分析與新演算法

研究生：謝雨滔

指導教授：吳文榕教授

國立交通大學電信工程學系博士班

摘要

平行式干擾消除法乃是針對直接序列碼分多重擷取系統一簡單而有效之多用戶偵測器。然而其效能表現可能因前幾階不可靠之干擾消除而降低，因此就有部分平行式干擾消除法的發展，此法乃利用部分消除因子來控制欲消除之干擾量，而提高系統效能。雖然部分消除因子佔有關鍵地位，然其完整的最佳解尚未有深入探討。本論文重點即在於針對不同形式之部分平行式干擾消除法，求得其最佳消除因子值，並進行效能分析。在論文的第一部份，吾人考慮一個二階式軟決策部分平行式干擾消除接收機，利用最低位元錯誤率的條件，吾人導證出完整的部分消除因子解，其中包括了週期碼、非週期碼系統，並適用於白高斯通道，與多重路徑通道。實驗結果顯示，經由理論求得之最佳部分消除因子值與實際值相當接近。此利用最佳部分消除因子值之二階式部分平行式干擾消除法不僅優於二階全平行式干擾消除法，亦優於三階全平行式干擾消除法。在論文的第二部分，吾人

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分析二階適應性盲蔽型硬決策部分平行式干擾消除法。在此架構中，經調適過而被用作最佳消除因子之權重值，乃是由最小均方理論訓練而得。吾人推導出最佳權重值、權重值之平均誤差、及其均方差值。根據這些理論結果，吾人得到每個使用者之輸出信號均方差及位元錯誤率。步階值在最小均方理論的收斂行為中，扮演著關鍵角色，對部分平行式干擾消除法的系統效能也影響甚鉅。藉著所推導之輸出信號均方差，吾人可以求得最佳步階值。在論文的最後一部份，吾人針對適應性盲蔽型硬決策部分平行式干擾消除法，提出一改善方法，其主要概念在於減低最小均方理論中所訓練之權重值的數目，並且進行權重值之後續濾波處理，使得最終多餘的均方差能因此減低。吾人也推導改良理論之輸出均方差與位元錯誤率。實驗結果證實所提出之改良理論表現優於傳統部分平行式干擾消除法，而理論分析結果也相當準確。

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Partial Parallel Interference Cancellation for

DS-CDMA Multiuser Detection: Performance

Analysis and New Algorithms

Student: Yu-Tao Hsieh

Advisor: Dr. Wen-Rong Wu

Department of Communication Engineering

National Chiao Tung University

Hsinchu, Taiwan 30050

Abstract

Parallel interference cancellation (PIC) is considered a simple yet effective multiuser detector for direct-sequence code-division multiple-access (DS-CDMA) systems. However, its perfor-mance may deteriorate due to unreliable interference cancellation in the early stages. Thus, a partial PIC detector in which partial cancellation factors (PCFs) are introduced to control the interference cancellation level has been developed as a remedy. Although PCFs are crucial, complete solutions for their optimal values are not available. In this dissertation we focus on the determination of optimal PCFs and performance analysis for various partial PICs. In the first part of the work, we consider a two-stage soft-decision partial PIC receiver. Using the minimum bit error rate (BER) criterion, we derive a complete set of optimal PCFs in the sec-ond stage. This includes optimal PCFs for periodic and aperiodic spreading codes in additive white Gaussian channels and multipath channels. Simulation results show that our theoretical

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optimal PCFs agree closely with empirical ones. Our two-stage partial PIC using derived opti-mal PCFs outperforms not only a two-stage, but also a three-stage full PIC. In the second part of the work, we analyze the performance of a two-stage adaptive blind hard-decision partial PIC. In this scheme, the adapted weights serving as optimal PCFs are trained using the least mean square (LMS) algorithm. We derive the analytical results for optimal weights, weight error means, and weight error variances. Based on these results, we also derive the output mean square error (MSE) and BER for each user. The step size known to be a critical parameter in the LMS algorithm controls the LMS convergence behavior and partial PIC performance. Us-ing the output MSE criterion, we can then optimize the step size. Simulation results indicates that our analytical results can well match with empirical ones. In the final part of the work, we propose an improved adaptive blind hard-decision partial PIC and analyze its performance. The main idea is to reduce the number of active weights in the LMS algorithm and to perform weight post filtering such that the resultant excess MSE can be reduced. We also derive the output MSE and BER for the proposed algorithm. Simulation results verify that the proposed algorithm outperforms the conventional partial PIC approach and analytical results are accurate.

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Acknowledgements

I would like to thank my advisor, Dr. Wen-Rong Wu, for his constant support, encouragement, and guidance, in my research work. His enthusiasm and persistence in discovering in new avenues of research impressed me deeply. I really appreciate his efforts in improving my paper organization and writing skills. I would also especially thank him for his helpful advice and experience not only on academic fields but also in my building my future career.

I am grateful to my colleges in WTSP Lab for building a comfortable environment. I also would like to thank my parents and my friends for their support and love.

Finally I would like to give a special thanks to my beloved wife, Amy, for being with me whatever has happened to me in my pursuit to the phd program. I would like to thank her for being always believed me that I can accomplish my dream, and being a forever supporter in every aspect of my life.

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6 Conclusions 114 Appendix 117 A Periodic Code System Optimal PCFs for Asynchronous AWGN Channels 117 B Expressions for Expected Terms in (3.50)-(3.51) 121 C Optimal PCFs under Fading Channels 127 D Supplemental Derivation for Analytical Results in Chapter 4 129 D.1 Two-user Scenario . . . 129

D.2 -user Scenario . . . 130

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List of Figures

2.1 BER performance comparison of multiuser receivers for the first user (

dB, and). . . 15

2.2 BER performance comparison of multiuser receivers ( , and). . 16

2.3 BER performance comparison of multiuser receivers (random codes, , and dB.) . . . 17

2.4 Interference estimate functions. (a) Soft-decision function (b) Hard-decision function (c) Null-zone function (d) Hyperbolic tangent function (e) Unit-clipper function (f) Modified unit-clipper function. . . 18

2.5 Block diagram for an SIC receiver. . . 19

2.6 Block diagram for a general two-stage partial SPIC receiver. . . 20

2.7 Block diagram for a two-stage coupled partial HPIC receiver. . . 21

2.8 Block diagram for a two-stage decoupled partial SPIC receiver. . . 22

2.9 LMS algorithm for two-stage adaptive blind partial HPIC receivers. . . 23

2.10 BER performance comparison for different multiuser receivers (, dB, and power balanced). . . 24

3.1 General partial SPIC receiver structure. . . 29

3.2 Performance comparison for HPIC and SPIC ( , , and dB); The optimal PCFs for the partial HPIC were obtained by trial and error and those for the SPIC were obtained from (3.27). . . 41

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3.3 Performance comparison for the coupled and decoupled structures (three users

with

,, and 8 dB); The optimal PCFs for the coupled

structure were obtained by trial and error, and those for the decoupled structure

were obtained from (3.38). . . 42

3.4 BER of the partial SPIC detector versus (aperiodic AWGN channels, and power balanced). . . 43

3.5 Optimal PCF versus number of users (Gold codes, asynchronous AWGN chan-nels, dB, and power balanced). . . 44

3.6 Optimal PCF versus number of users (aperiodic codes, multipath channels, dB, and power balanced). . . 45

3.7 BER versus number of users (Gold codes, asynchronous AWGN channels, dB, and power balanced). . . 46

3.8 BER versus number of users (aperiodic spreading codes, multipath channels, dB, and power balanced). . . 47

3.9 BER with channel estimation error (aperiodic spreading codes, multipath chan-nels,=6, dB, and power balanced). . . 48

4.1 LMS algorithm for two-stage adaptive blind partial HPIC receivers. . . 50

4.2 Optimal weight comparison for two power-balanced users. . . 82

4.3 Optimal weight comparison for five power-balanced users. . . 83

4.4 Optimal weight comparison for 15 power-balanced users. . . 84

4.5 Weight mean comparison for two power-balanced users ( , and dB). . . 85

4.6 Weight mean comparison for five power-balanced users ( , and dB). . . 85

4.7 Weight mean comparison for 15 power-balanced users ( , and dB). . . 86

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4.8 Weight error power comparison for two power-balanced users ( , and dB). . . 86

4.9 Weight error power comparison for five power-balanced users (

, and dB). . . 87

4.10 Weight error power comparison for 15 power-balanced users (

, and dB). . . 87

4.11 Optimal step-size comparison for different user numbers. . . 88

4.12 Second-stage BER comparison for power-balanced cases (

). . . 88

5.1 Adaptive blind partial HPIC receivers. . . 92

5.2 Functions used in the proposed algorithm. (a) Weight selection function. (b)

Weight post filtering function. . . 96

5.3 Flow chart for the proposed algorithm. . . 97

5.4 Probability density function for adapted weights from LMS algorithm. . . 102

5.5 Second stage parameter optimization for the proposed algorithm. (Weight

se-lection is not performed). . . 103

5.6 Second stage parameter optimization fo the proposed algorithm (

). 104

5.7 Second stage BER performance for the proposed algorithm ( ,

, and dB). . . 105

5.8 Second stage BER performance comparison vs. user numbers (

, , , and dB). . . 106

5.9 Third stage BER performance comparison vs. user numbers (the parameter

setting is the same as that in Figure 5.8 for all stages). . . 107 5.10 Fourth stage BER performance comparison vs. user numbers (the parameter

setting is the same as that in Figure 5.8 for all stages). . . 108 5.11 Fifth stage BER performance comparison vs. user numbers (the parameter

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5.12 Second stage BER performance comparison vs.

ratios (

, and the

parameter setting is the same as that in Figure 5.8 for all stages). . . 110

5.13 Fifth stage BER performance comparison vs.

ratios (

, and the

parameter setting is the same as that in Fig. 5.8. . . 110 5.14 Second stage BER performance comparison for the weakest user (power-imbalanced,

dB, and the parameter setting is the same as that in Fig. 5.8). . . 111

5.15 Fifth stage BER performance comparison for the weakest user (power-imbalanced,

dB and the parameter setting is the same as that in Fig. 5.8). . . 111

5.16 Fifth stage BER performance comparison for single-path rician fading channels

( dB, , and

for all stages). . . 112

5.17 Fifth stage BER performance comparison for two-ray multipath fading channels

( dB, , and

for all stages). . . 112

5.18 Fifth stage BER performance comparison vs. channel estimation errors (

, dB, , , and

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List of Tables

2.1 Required information for different multiuser receivers. . . 25

4.1 Sets offor all decision and bit patterns . . . 68

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Chapter 1 Introduction

§ 1.1 Background

Since G. Marconi first used radio for wireless communication in 1897, many new methods have been developed. In the 1960s and 1970s, Bell Laboratories developed the cellular concept in wireless communication systems. At the same time, the semiconductor industry has also experienced enormous progress such that design and manufacture of low-cost radio frequency devices appeared feasible. These result in the today’s exponential growth in cellular radio and personal communication systems throughout the world. As known, the most critical resource in wireless communication is the spectrum. In order to support as many users as possible on a limited spectrum, multiple access techniques have been developed. The progressive multiple access schemes also witness the development of the advanced techniques, which raises to deal with the increasing demands for both voice and data service, accompanied by the performance guarantee under diverse environments and stringent device specification.

The first-generation (1G) mobile cellular system was developed in early 1980’s and de-ployed in mid 1980’s. The 1G system used the frequency division multiple access (FDMA) as the multiple access scheme. The well known standards include the advanced mobile phones system (AMPS) in the United States, the total access communications system (TACS) in

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Eu-rope, and the NTT system in Japan. Due to the use of the cell structure, frequencies can then be reused and handover among cells is required between cells. In the 1G era, the service content consists the voice data only. The rapidly increasing demand for higher system capacity has soon pushed the development of an improved system, the second generation (2G) system. The 2G system claimed to support at least three-folded capacity than the 1G system. Most of 2G systems adopted time division multiple access (TDMA) as the multiple access scheme. The scheme uses non-overlapping time-slots to transmit data of different users. Since more users can use the same frequency band, its efficiency is higher than the pure FDMA systems. Since then the cellular system becomes digital, and the advanced signal processing techniques, such as the voice compression, error control coding, and encryption, were incorporated. The repre-sentative standards are the IS-136 in the United States and the GSM in Europe. Specifically, the GSM system enjoys a great success. At the end of 2003, the GSM system has a total subscribers over one billion in more than 170 networks over the world. In additional to the aforementioned capacity advantage, the 2G communication system also provides low-rate data services. It uses voice activity detectors and insert data in the unused slots. This enables the packet-based data services, such as e-mail and internet browsing. Typical standards include generation packet radio service (GPRS) and enhanced data rate for GSM evolution (EDGE). Yet, there is another 2G system that uses a totally different multiple access scheme, the code division multiple ac-cess (CDMA) [1]. In conventional multiple acac-cess methods, the transmission is partitioned into dedicated channels in frequency and/or time domain such that the interference among users can be avoided. In the CDMA system, however, orthogonal codes are used as the user signatures. These codes, when transmitted, occupy the same frequency band and same time period. The CDMA belongs to the spread spectrum communication technique and its required bandwidth is wider than the TDMA system. Conventional CDMA systems were used in military applica-tion since it has the advantages of high tolerance for jamming or unintenapplica-tional interference, as well as low detectability [2]. The major advantage of commercial CDMA is to provide higher capacity (than TDMA and FDMA). The first commercial CDMA system was developed by

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Qualcomm and referred to as the IS-95 in United States.

Although the 2G system can support data service, its data-rate is low. As a result, the third-generation (3G) standard, which is supposed to supply a transmission rate of 2M bits per second, were developed. It turns out that most standard bodies chose CDMA as the multiple access scheme. This includes cdma2000, WCDMA, and TD-SCDMA [3]. As mentioned, in CDMA all users share the same frequency bands and time slots, and thus the main factor lim-iting the system capacity is the interference from other users. Hence suppression of cochannel interference becomes a major challenge for CDMA systems. The major distinction between CDMA and other multiple access schemes is the virtual code space in which users can be iden-tified when sharing the same time slots and frequency bands. There are two major classes of spreading codes utilized in the CDMA system based on the correlation property between codes. The first class is the orthogonal codes, which are Walsh codes in general. The other class be-longs to the pseudonoise(PN) code. When Walsh codes are used, there will be no interference between users. However, there are several reasons for which the PN codes are preferred in real-world applications. Firstly, the number of Walsh codes is limited (the number of active users is limited). Secondly, the orthogonal property only holds in synchronous transmission and additive white Gaussian noise (AWGN) channel. In the uplink transmission or multipath environments, code orthogonality can not be hold. The PN sequence has the property that the

normalized auto-correlations equal for all time lags, where is the processing gain.

This makes the receiver more robust to the coherent interference in multipath environments. Although the CDMA receiver inherently has the interference suppression property, however, as the user number increases, interference (due to non-orthogonal codes) becomes stronger and stronger. The performance is then degraded accordingly. The interference from other users is generally referred to as multiple access interference (MAI). In order to combat the MAI, some signal processing techniques have been proposed and these include

Source and channel coding / interleaving

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Multiuser detection

This dissertation focus on the third one, the multiuser detection (MUD) algorithms.

§ 1.2 Multiuser Detection

The significant progress of the MUD development was due to the work of S. Verdu. He pro-posed a multiuser receiver utilizing the maximum-likelihood criterion [4] and showed a great performance enhancement. However, He also showed that the computational complexity grows exponentially with the user number. The high computational complexity adversely affects its real-world applications. Thus, a variety of low-complexity suboptimum receivers were then proposed [5]-[7],[8].

The first category of suboptimal receivers is the linear receiver. It performs MUD through a linear transformation of the matched filter outputs. The rationale behind this approach is simi-lar to that of equalization in TDMA systems [9]. The decorrelating detector (or decorrelator), being a linear receiver, uses the correlation matrix inverse as the transformation matrix [10]. It can completely eliminate the MAI and achieve the near-far resistance close to the optimal receiver. Another feature of the decorrelator is that the algorithm does not require the receive signal powers (for each user) nor the noise variance. However, it may enhance noise and thus the performance is degraded when signal to noise ratio (SNR) is low . The linear minimum mean square error (LMMSE) detector, an improvement to the decorrelator, gives a compro-mise between interference suppression and noise enhancement [11],[12]. Leveraging the linear property, the linear receivers lends the performance analysis feasible [10],[13], [14]. Although the linear approaches are much more simpler than the optimal one, they may require matrix

inversion operations. The computational complexity is on the order of

where is the

user number. In [15] and [16], iterative algorithms, which do not require matrix inversion, were proposed to obtain the decorrelator and linear MMSE receivers. These iterative methods were shown to have a close relationship with the soft-decision interference cancellation methods that

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will be described later. The other strategy reducing the complexity of these detectors is the use of adaptive algorithms, which includes the least mean square (LMS) algorithm [17],[18], [19] [20], the recursive least square (RLS) algorithm [21], and the Kalman filtering algorithm [22].

In addition to the aforementioned linear detectors, another category of interest is the subtrac-tive type multistage interference cancellation method. Cancellation of this type involves only vector operations making it a good candidate for real-world implementation. For a particular desired user, the subtractive-type canceller estimates interference from other users, regenerates it, and cancels it from the received signal. This canceller is usually implemented with a mul-tistage structure. The temporary data decision for a stage is obtained from its previous stage. The successive interference cancellation (SIC) cancels interference from other users one by one [23],[24],[25], while the parallel interference cancellation (PIC) cancels it all at one time [26], [27], [28]. A hybrid of PIC and SIC is also possible [29]. To have best performance, signal power ranking is necessary in SIC. The strongest user usually has lowest probability of decision errors and cancellation of its interference gives the most significant result. For these reasons, SIC works well where users have unbalanced powers. However, SIC requires addi-tional complexity for power ranking and the longer processing delay. By contrast, PIC cancels the interference disregard to the interference power distribution and is more suitable for power-balanced systems.

As mentioned, the subtractive-type of MUD estimates the interference from other users and then subtract it from the received signal. Each interference estimate involves bit estimation and spread signal regeneration. According to how the transmit bits are estimated, an interference cancellation algorithm can be classified as linear or nonlinear [30], [31], [32], [33]. For each stage, the simplest bit estimate is the soft-decision operated on the previous stage output (for each user) [34], [30], [35]. This bit estimate gives a linear receiver. It has been shown that the soft-decision PIC (SPIC) can converge to the decorrelator when the number of stage is infinite [32]. In practice, a two-stage SPIC may approximate the decorrelator well [37]. Due to the linear property, we can use the Gaussian approximation [38] or an improved Gaussian

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approximation [34] to carry out SPIC performance analysis. The analysis was extended to include the scenario when the timing and phase errors were present [39]. Although simple, some undesirable properties were reported that the SPIC may perform worse than the matched filter when the correlation between user signals exceeds a certain threshold [40]. The analysis for SIC can also be found in [25].

The other commonly used bit estimate is the hard-decision. In this approach, channel infor-mation is generally required. The hard-decision PIC (HPIC) was investigated in [26], [41], [42] while the hard-decision SIC was investigated in [64]. The HPIC operated in a multipath fading channel was considered in [43],[44]. Theoretical analysis for this type of interference cancel-lation appears more complicated due to the non-linear decision operation. A two-stage HPIC was analyzed in [45]. Other performance criteria such as the signal-to-interference-noise-ratio (SINR) or the capacity were discussed in [46]. The decision function is not limited to be soft or hard. In [47], the hyperbolic tangent function was used as the decision function. This function can reflect the reliability of interference estimate more faithfully. Note that the hard-decision and soft-decision functions are special cases of the hyperbolic tangent function. The null-zone decision function was also studied for PIC [48], [49] and SIC [50]. Other decision functions can be found in [31], [51].

One problem in the PIC approach is that the interference estimates may not be reliable in early stages. In other words, interference cancellation does not necessarily reduce interference. To alleviate this problem, partial PIC was then developed. Partial cancellation factors (PCFs) ranging from 0 to 1, were introduced to control the signal cancellation level. The partial HPIC approach was first proposed in [30]. Since the interference estimate reliability is different, the PCF is usually different for each stage. It has been shown that the performance of the linear MMSE receiver can be achieved using partial SPIC through proper choice of PCFs [52],[53]. The PCF optimization for multistage SPIC has also been considered in [54]. It was shown that partial SPIC can converge to the decorrelator with very few stages. It was also shown that the partial SPIC can be seen as a realization of the steepest descent MMSE optimization method

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where the PCF acts as the step size in each stage. The bias reduction in the partial SPIC was further analyzed in [55] and [56].

Let the number of user be. Thus, a specific user has interfering users. To have

best cancellation result, we then require PCFs. However, we have total users to

consider. Thus, a general partial PIC require PCFs. As we can see, the computational

complexity of the general partial PIC is high. In order to reduce the computational complexity, two simplified structures were developed; we refer to them as the coupled and the decoupled

structure. In these two structures, only PCFs are involved. The difference of these two

structures lies in the position where the PCF is inserted. In the coupled structure, PCFs are inserted (multiplied) after each regenerated user signal. For a specific user, the interference

estimate is just the summation ofweighted regenerated signal. Thus, the estimate involve

PCFs. For the decoupled structure, the regenerated signals are first summed

and then a PCF is inserted (multiplied). Thus, for a specific user, the interference estimate only involve one PCF. In partial HPIC, the coupled structure is usually employed and only the approximated optimal PCFs are available for a two-stage processing [57]. The derived PCFs is obtained by minimizing the MSE between signal outputs and desired data. The approximate optimal PCFs for partial HPIC with timing error can be found in [58] while the optimal PCFs supports the multicode transmission was reported in [59]. The PCFs for coded systems with HPIC were investigated in [60]. The coupled partial SPIC has been considered in [61] and the closed-form results applied to a power balanced control scenario were derived. Besides the theoretical solutions, the LMS adaptive algorithm was also used to search optimal PCFs for partial HPIC [62],[63]. Due to its special architecture, this approach does not need training sequence. We call it a adaptive blind partial HPIC algorithm. It was found that this partial HPIC outperforms non-adaptive ones. The LMS algorithm was also utilized to track the channel information in hard-decision SIC [64].

The MUD algorithms are by no means limited to those described above. However, other algorithms either require higher computational complexity, or consider special operation

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condi-tions (no user information for example). The objective of this work is to study low-complexity MUD algorithms that are suitable for real-world implementation. As mentioned, the interfer-ence cancellation method only involves vector operations making its computational complexity lower than others’. We will then focus on this type of MUD. For other MUD related works, please see [65].

§ 1.3 Objective and Overview

As mentioned, the PIC performance may deteriorate due to unreliable interference cancellation in the early stages. Thus, the partial PIC detector in which partial cancellation factors (PCFs) are introduced to control the interference cancellation level has been developed as a remedy. It is apparent that these PCFs are crucial. However, complete solutions for their optimal values are not available. Also, performance analysis is only available for limited scenarios. In this dissertation we focus on the determination of optimal weights and performance analysis for various partial PICs. There are three main parts in this work. In the first part of the work, we consider a two-stage decoupled soft-decision partial PIC receiver. The reason to consider this architecture has manifold. Firstly, it is known that the value of PCFs will approach to one when the number of stage is greater than two [55]. Thus, there is no need to consider a higher stage structure. Secondly, theoretical analysis is much more simpler for a two-stage structure. The analysis is also simpler for the decoupled SPIC. The performance of the partial SPIC is similar to that of other structures (for example, the coupled partial HPIC). Using the minimum bit error rate (BER) criterion, we derive a complete set of optimal PCFs for the second stage. This includes optimal PCFs for periodic and aperiodic spreading codes in additive white Gaussian channels and multipath channels. Simulation results show that our theoretical optimal PCFs agree closely with empirical ones. Our two-stage partial PIC using derived optimal PCFs outperforms not only a two-stage, but also a three-stage full PIC.

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partial HPIC receiver. This is known to be a difficult problem and the corresponding result is not reported in literature. In this scheme, the adapted weights serving as optimal PCFs are trained using the least mean square (LMS) algorithm. The analysis difficulty arises from the nonlinear operation involved in the decision process and its interaction with the LMS algorithm. Although there exist many theoretical results for the LMS algorithm, most of them consider the steady-state performance and are valid only for the small step size scenario. This cannot be applied in the problem considered here. This is because the sample size available is small and a large step size must be used. Also the weights will not converge at the end of each bit interval and the LMS algorithm is still in its transient-state. Note that the input to the LMS algorithm depends on the decision in the previous stage and this complicates the problem furthermore. We derive the analytical results for optimal weights, weight error means, and weight error variances. Based on these results, we also derive the output mean square error (MSE) and BER for each user. The step size known to be a critical parameter in the LMS algorithm controls the LMS convergence behavior and partial PIC performance. Using the output MSE criterion, we are able to obtain an optimal step size. Simulation results indicates that our analytical results can well match with empirical ones.

In the final part of the work, we propose an improved adaptive blind multistage hard-decision partial PIC and analyze its performance. It is well known that the LMS is a stochastic gradient descent algorithm and its excess MSE is proportional to the number of filter taps and the step size value. The main idea here is to reduce the number of active weights in the LMS algorithm and reduce the adapted weight variance such that the resultant excess MSE can be reduced. To implement this idea, we include a decision making mechanism before adaptation and a weight post filtering function after adaptation. We also derive the output MSE and BER for the proposed algorithm. Simulation results verify that the proposed algorithm outperforms the conventional partial PIC approach and analytical results are accurate.

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§ 1.4 Organization of the Dissertation

The organization of this dissertation is described as follows. In Chapter 2 we survey significant contributions in multiuser detection. The optimal and several suboptimal multiuser receivers are described.

Chapter 3 presents a two-stage partial SPIC multiuser receiver with a decoupled structure. We derive the optimal partial cancellation factors (PCFs) based on the minimum BER criterion. We consider periodic and periodic code scenarios, the AWGN as well as multipath channels.

Chapter 4 focuses on the analysis of a two-stage adaptive partial HPIC receivers. In this regard, the LMS algorithm is used to obtain optimal PCFs. We derive the optimal weights and analyze the weight error mean and weight error variance for one and two-user cases. We then

extend the results to the general-user case. Due to the difficulty of the problem, we are only

able to obtain approximate results. However, simulations show that our results are accurate. We also use our theoretical results to optimize the step size used in the LMS algorithm.

In Chapter 5, we propose an improved adaptive blind multistage HPIC receiver. We show that the convergence rate of the LMS algorithm can be accelerated and the performance can be enhanced. Based on the convergence analysis given in Chapter 4, we also analyze its perfor-mance and derive theoretical MSE and BER.

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Chapter 2 Multiuser Detection

At the time of introduction of CDMA, it was argued that interference from other users (after despreading) has the statistical property just as the noise. Thus in the receiving end each user can use a matched filter to demodulate its own signal independently. It is simple to see that the interference level is proportional to the number of users and their signal strength. This is referred to as the single-user detection scheme. The performance of the matched filter will be greatly affected when the near-far effect arises. In this case, the weak user signals may be over-whelmed by strong user signals. In this regard, using power control to balance the receiving powers among users seems the most efficient way. However, the challenges for power control is the requirement of fast and accurate power adjustment to maintain the received levels within a fraction of one dB error from the possible dynamic range up to 90 dB. In addition, differ-ent services may have differdiffer-ent transmission rates and powers making power control difficult. Multiuser detection (MUD) was developed to alleviate this problem. In MUD, all users are demodulated simultaneously. Signal from other users are not treated as interference any more. Application of the MUD algorithm greatly improves the system performance and at the same time eliminates the precise power control requirement. In this chapter, several MUD techniques are briefly reviewed. In Section 1, the optimal receiver are described. Section 2 presents the lin-ear suboptimal receivers, which include the decorrelator and the LMMSE receiver. In Section

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3, we described the interference cancellation methods that include SIC, PIC, and partial PIC.

§ 2.1 Optimal Multiuser Receiver

Consider a synchronous CDMA system for the AWGN channel with users. The received

signal at a certain bit interval can be represented by

(2.1) where

is the received signal for the th user,

and

are the channel and data bit for

theth user,

is the normalized spreading waveform, and is the bit interval length. The

AWGN is denoted by . The mean and the variance of is zero and

, respectively. The maximum likelihood (ML) solution for the input bits maximizes the likelihood function shown below. (2.2) where

. The log-likelihood function, which is equivalent the likelihood function, is used more frequently. The log-likelihood function is shown to be

(2.3) wherediag

,is the correlation matrix with the entry

given by (2.4) and

is the matched filter output vector with its element given by

(2.5)

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The the matched filter output vector can be written as (2.6) where ,

, is a noise related vector. It is well known

that the optimal solution maximizing requires an exhausted bit search. This

combi-natorial problem is shown to be NP hard and the required computation complexity is on the

order of

. Although the ML criterion and the minimum BER criterion are different, their

solutions are close especially for high SNR ratios. When the asynchronous transmission is considered, it has been shown that the complexity of the optimal receiver, implemented by a

matched filter bank followed by the Viterbi algorithm, remains

. The ML receiver

re-quires the information of the signal amplitudes, signature waveforms, and signal delays for all users. When the criterion of minimum BER is utilized, the optimum detection, implemented

with the backward-forward dynamic programming, still requires the complexity of

. In

this case, the variance of background noise is also necessary. These requirements along with the high computational complexity makes the optimal receiver infeasible for real-world imple-mentation.

§ 2.2 Linear Suboptimal Receivers

The optimal MUD has been regarded as powerful yet complicated. The suboptimal MUD was developed to reduce the complexity while still provide performance gain. In this section, we describe the suboptimal linear multiuser receivers. The linear receiver performs a linear

trans-formation on the received signal vector. The first linear multiuser receiver is called the

decor-relating detector or simply the decorrelator, whose name stems from the fact that the detector simply inverts the correlation matrix in (2.6). Let

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Then, the receiver output is given by (2.8)

As shown, the interference from the MAI is eliminated completely. However, noise becomes colored and its level may be enhanced. When the noise level dominates the MAI, the perfor-mance of the decorrelator is degraded. The decorrelator is also the joint ML solution for the simultaneous estimate of channel gains and transmission bits. The solution can be found by the minimization of a least-squares criterion.

(2.9)

In contrast to the optimal MUD, the decorrelator does not require user signal amplitudes. In addition, it was shown that the near-far resistance is equal to the that of the optimal receiver. The fluctuation of the interference powers do not have any influence on the performance of the decorrelator.

Another commonly cited suboptimal linear receiver is the LMMSE receiver whose transfor-mation matrix is defined by

(2.10)

After some matrix manipulation, we can obtain the transformation matrix for the LMMSE mul-tiuser receiver as (2.11)

Comparing the decorrelator with LMMSE receivers, we can observe that the LMMSE receiver

becomes the decorrelator as

approaches zero. On the other hand, the LMMSE receiver will

degenerate into the matched filter when noise

approaches infinity. This means that LMMSE multiuser receiver performs a compromise between noise enhancement and interference cancel-lation. When the LMMSE receiver is used, the signal amplitudes as well as the noise variance have to be known, in addition to the signal spreading codes and received signal delays.

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−10 −5 0 5 10 15 10−4 10−3 10−2 10−1 100 a 2/a1 ( dB) BER Conv. receiver Decorrelator LMMSE Optimal ML

Figure 2.1: BER performance comparison of multiuser receivers for the first user (

dB, and).

It can be observed that the matrix inversion is required in linear receivers. In order to

reduce the computational burden, adaptive implementation was proposed. Let

, . Rewrite (2.9) as ² (2.12) where and

are the chip-sampled sequences of and

, respectively, and

is the processing gain. The estimate of gives the channel gains and bits, which are those in

(2.8). The adaptive implementation of the decorrelator does not involve the complicated matrix inversion operation. Similarly, the LMMSE receiver can have an adaptive implementation. Let

. It can be easily shown that the LMMSE receiver output is

(2.13) where

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0 1 2 3 4 5 6 7 8 9 10 10−2 10−1 100 E b/N0 BER Conv. receiver Decorrelator LMMSE Optimal ML

Figure 2.2: BER performance comparison of multiuser receivers ( , and).

derive . Thus, (2.14)

Note that some transmission bits are required for training. The MUD performance measure includes the BER, the asymptotic multiuser efficiency, and the near-far resistance [7]. We have carried out some simulations to evaluate the performance of the receivers described above. Figure 2.1 shows the result for BER vs. interference power. Here, the user number is two, the

code correlation is, and

dB (

). Note that the-axis of the figure is

the power ratio of the two users. The conventional receiver suffers from the interference from the second user, and its performance degrades rapidly when the normalized interference power increases up to 5 dB. The ML receiver has the best near-far resistance among the four detectors. The decorrelator exhibits a constant near-far resistance in all interference power ratios. The LMMSE receiver is degenerated to the conventional receiver when the interference is weak while to the decorrelator when the interference is strong; it performs very similarly to the ML

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2 3 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 User number BER Conv. receiver Decorrelator LMMSE Optimal ML

Figure 2.3: BER performance comparison of multiuser receivers (random codes, , and

dB.)

receiver in weak interference. Figure 2.3 shows the BER vs.

for ten equicorrelated users

( ). The single-user receiver suffers from MAI and perform poorly in most cases. The

linear receivers perform similarly to the ML receiver when the number of users is small.

§ 2.3 Interference Cancellation Methods

The interference cancellation scheme first estimates interference from other users and then

can-cels it from the received signal. Let

be the interference cancelled signal for User. We

then have (2.15)

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(a) _(b) (c) (d) (e) (f) j y y_j j y j y j y j y j g gj gj j g gj gj j a a_j j a a_j aj j a − j a − j a − j a − j a −

Figure 2.4: Interference estimate functions. (a) Soft-decision function (b) Hard-decision func-tion (c) Null-zone funcfunc-tion (d) Hyperbolic tangent funcfunc-tion (e) Unit-clipper funcfunc-tion (f) Modi-fied unit-clipper function.

where

represents the interference estimate of

. The number of interference cancelled

in (2.15) depends on the algorithm used. For description simplicity, we assume a two-stage

cancellation scheme such that

, where is a decision function. Commonly used

decision functions are summarized in Fig. 2.4. Note that channel gains are assumed to be known. The second stage output is obtained by

(2.16)

The decision functions in Fig. 2.4 are further described below.

(a) Soft-decision function:

(b) Hard-decision function: sgn

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Matched filter bank Select Max Regeneration ( ) + -z_k k=k+1 1ˆ ( ) k j j= s t

∑

{a ii| =k k, +1,",K}

Figure 2.5: Block diagram for an SIC receiver.

(c) Null-zone function:

(d) Hyperbolic tangent function:

where

represents the power of interference and noise for theth user.

(e) Unit-clipper function:

(f) Modified unit-clipper function:

where

In the following, we describe the basic types of interference cancellation schemes, namely, SIC and PIC.

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First stage Second stage y₁ y₂ y₃ 1 ˆ ( ) s n 2 ˆ ( ) s n 3 ˆ ( ) s n 1 z 2 z 3 z 2 , 1 C 1 , 2 C 1 , 3 C 2 , 3 C 3 , 1 C 3 , 2 C Respreading Matched Filter Matched Filter Matched Filter Respreading Respreading Matched Filter Matched Filter Matched Filter 1 ˆ b 2 ˆ b 3 ˆ b ( ) r n

Figure 2.6: Block diagram for a general two-stage partial SPIC receiver.

§ 2.3.1 Successive Interference Cancellation

The SIC cancels one user interference from the received signal at a time. Since only one inter-ference needs to be estimated and subtracted in each stage, the strongest user signal is then the best candidate. It’s structure is depicted in Figure 2.5. Assume that the received signal powers

are ranked as

, and the interference cancelled signal for user

at theth stage is obtained as (2.17) where

, for allis the initial receive signal. The SIC output at theth stage is then

(2.18)

Although the SIC is simple to apply, there are some drawbacks listed below.

Since the user is detected successively, the subsequent users will experience less

interfer-ence. To make all users have similar performance, transmission power for each user will be different. A proper power profile may not be easy to obtain. In addition, the power ordering operation requires additional computational complexity.

The interference resulted from the erroneous cancellation will propagate to all the users

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First stage Second stage 1 1 a C 2 2 a C 3 3 a C Respreading Matched Filter Matched Filter Matched Filter Respreading Respreading Matched Filter Matched Filter Matched Filter (2) 1 ˆ b (2 ) 2 ˆ b ( 2) 3 ˆ b (1) 1 ˆ b (1) 2 ˆ b (1) 3 ˆ b 1 ˆ ( ) s n ( ) r n 1 y 2 y 3 y 2 ˆ ( ) s n 3 ˆ ( ) s n

Figure 2.7: Block diagram for a two-stage coupled partial HPIC receiver.

A SIC scheme needs at least stages for a -user environment. This will greatly

in-crease the detection delay especially when the user number is large.

§ 2.3.2 Parallel Interference Cancellation

The PIC cancels interference from all other users at the same time. In contrast to SIC, the PIC has lower detection delay and does not have the power assignment problem. It has been shown that the PIC has superior performance over the SIC in an power balanced scenario. Conventional PIC receivers permit a full cancellation of the MAI. One problem associated with this full PIC is that the MAI estimate may not be reliable in the earlier canceling stages. This makes the PIC less effective when the number of users is large. As a remedy, the partial PIC detector has been proposed in which partial cancellation factors (PCFs) are introduced to control the interference

cancellation level. As shown in 2.6, a complete partial PIC requires PCFs for one

stage where is the number of users; the computational complexity is high. Simplified partial

PICs have been proposed, in which only PCFs are needed. Two structures are commonly

used for simplified partial PICs; we call them the coupled and decoupled structures. In the

coupled structure, each user output is influenced by all PCFs [62] as seen in Figure 2.7,

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First stage Second stage Respreading Matched Filter Matched Filter Matched Filter Respreading Respreading Matched Filter Matched Filter Matched Filter (2) 1 ˆ b (2) 2 ˆ b (2) 3 ˆ b 1 ˆ ( ) s n ( ) r n 1 y 2 y 3 y 2 ˆ ( ) s n 3 ˆ ( ) s n 1 C 2 C 3 C

Figure 2.8: Block diagram for a two-stage decoupled partial SPIC receiver.

in Figure 2.8. The partial HPICs mentioned in Chapter 1 all use the coupled structure. A MSE criterion, as shown below, has been proposed to optimize PCFs [57].

! ! ! ! ! ! " " (2.19) where"

is the error probability for the

th user. As we can see, each PCF can be determined

independently. From (2.19), we can observe that when the data bits are all correctly detected, the optimal PCFs will approach unity. On the other hand, when the data bits are all erroneously

detected, ("

), the optimal PCFs will approach zero. This is intuitively appealing.

Although simple, the optimal PCFs in (2.19) are not accurate for short codes. Thus, its real-world application is limited.

The optimal PCF obtained by theoretical calculation may not be efficient when the channel is time-varying. There exist an adaptive partial HPIC that can overcome this problem [62]. This adaptive HPIC is blind in the sense that no training sequence is required. Due to its simplicity

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r(n) e(n) + ˆ( ) r n x1(n) 1( )n χ 2( )n χ ( ) K n χ 1 ˆ b 2 ˆ b ˆ K b LMS weight update equation 1( ) w n 2( ) w n ( ) K w n Matched Filter x₂(n) x_K(n) x1(n) x₂(n) x_K(n)

First stage _{Second stage} Matched

Filter

Matched Filter

Figure 2.9: LMS algorithm for two-stage adaptive blind partial HPIC receivers.

and robustness, the LMS algorithm was used as the adaptive algorithm. A typical block diagram for a two-stage HPIC is shown in Fig. 2.9. The weights are trained using the LMS algorithm

which minimizes a MMSE criterion defined as (for theth stage)

# (2.20) where

is the optimal weight vector at the

th stage, and # $ (2.21)

The weight after trained, $

, acts as each user’ PCF. Note that this is a system

identifi-cation problem. The LMS update equation for the th stage (with stages of interference

cancellation) is formulated as % (2.22) %

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5 10 15 20 25 30 10−2 10−1 100 User number BER Conv. receiver LMMSE SIC Full HPIC Full SPIC Partial HPIC (C k=0.6) partial SPIC (C k=0.3)

Figure 2.10: BER performance comparison for different multiuser receivers (,

dB, and power balanced).

where

is the input vector. The interference

estimate for theth user in the-stage is given by

$ (2.23)

Then the-stage output from the adaptive blind partial HPIC can be obtained

(2.24)

Note that the adaptive blind partial HPIC is different from the work in [17], since this scheme does not require the training sequence. The optimal weights are optimized in one bit interval; its adaptation is on the chip-level.

As to the partial SPIC, both the coupled and decoupled structures have been studied. In this dissertation, we focus on the decoupled structure which is shown in Figure 2.8. The reason to consider this structure is that the PCF optimization is simpler and its performance is comparable

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Table 2.1: Required information for different multiuser receivers.

SU ML MBER DEC LMMSE AMMSE IC

Desired user’s signature

Desired user’s timing

User amplitude noise variance Others’ signature Others’ timing Training data SU : Single-user receiver ML : Maximum-likelihood receiver

MBER : Minimum BER receiver

DEC : Decorrelator

LMMSE : Linear mean square error receiver

AMMSE : Adaptive LMMSE

IC : Interference cancellation receiver

to other structures. We have carried out simulations to compare performance of various

two-stage PIC with LMMSE receivers. The result is shown in Figure 2.10 (,

dB,

and power balance is assumed). The LMMSE performs the best among all multiuser receivers. The SIC has only minor advantage over the single-user receiver. This is because in the power balanced scenario, the power ranking does not have advantages. The full HPIC performs better than SIC. Note that the full SPIC perform poorly when the user number increases. Partial PICs with optimal PCFs perform much better and the partial SPIC performs similarly to the LMMSE receiver. In Table 2.1 we summarize requirement information for various MUD methods.

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Chapter 3 Optimal Two-stage Partial SPIC Receivers

§ 3.1 Introduction

In this chapter, we focus on a two-stage partial SPIC receiver with a decoupled structure. Our motivation for using two-stage processing is that it requires low computational complexity and is particularly suitable for real-world implementation. As indicated in [55] that in higher stage processing, the PCFs will approach unity for stages greater than two. In other words, the PCFs in the second stage will dominate system performance. We first consider the additive white Gaussian noise (AWGN) channel and derive optimal PCFs for systems employing periodic codes. The criterion for optimization is the bit error rate (BER). We then extend the result to systems with aperiodic spreading codes. Finally, we consider optimal PCFs with multipath channels. Simulations show that the performance of our theoretical optimal PCFs is close to that of empirical ones. In addition, the optimal two-stage partial SPIC outperforms not only the two-stage full SPIC, but also the three-stage full SPIC. The remainder of this chapter is organized as follows. In Section 2, we describe the two-stage full and partial SPIC receiver structures. In Section 3 and Section 4, we derive optimal PCFs with periodic and aperiodic codes, both in AWGN and multipath channels. Simulation results are presented and discussed in Section 5.

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§ 3.2 System Model

Consider a synchronous CDMA system accommodating users. Let denote the received

signal (for a certain bit interval),

theth user’s transmitted signal, and additive white

Gaussian noise. The equivalent baseband received signal can be described as

(3.1) where and are the

th user’s amplitude and data bit,

denotes its signature waveform,

and is the bit period. The signature waveform can be expressed as

(3.2) where

is the binary spreading chip sequence for User , is the

processing gain,

is a rectangular pulse waveform with support

and unit magnitude. Note

that

is the chip period.

The first stage of a PIC receiver is the conventional matched filter bank. The output can be represented as (3.3) where

is a correlation coefficient and

is the noise term after despreading. They are defined

as (3.4) and (3.5)

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It can be seen that the output metric in (3.3) consists of three parts: the desired signal, MAI, and

. The conventional detector makes a decision based on

. Thus, MAI is treated as another

noise source. When the number of users is large, MAI will seriously degrade the system per-formance. A PIC, being a multiuser detection scheme, was proposed to alleviate this problem.

Let

be an interference-subtracted signal (for User) given by

(3.6) where

is a regenerated signal for User. For SPIC, this signal is obtained by

(3.7)

Thus, the output signal in the second stage is then

(3.8)

Finally, the symbol data is detected as

. In principle, the interference cancellation

procedure in (3.6)-(3.8) can be repeated with multiple stages to obtain better performance. It is apparent from (3.3) and (3.7) that the regenerated signal is noisy. Thus, fully cancelling the regenerated interference may not yield best results. One solution to this problem is to partially cancel the interference. This idea is implemented by modifying (3.6) as

! (3.9) The constants!

’s are called the partial cancellation factors (PCFs) for User

and their

am-plitudes should reflect the fidelity of the interference estimate. The structure of a partial SPIC receiver with three users is shown in Figure 3.1.

Generally, PCFs are needed for a two-stage partial PIC. It is apparent that the

computational complexity of the partial PIC is high when the number of users is large [on the

order of

]. Two simplified structures, whose complexities are on the order of , were

investigated in the literature. The first one corresponds to the case in which!

!

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First stage Second stage y₁ y₂ y₃ 1 ˆ ( ) s n 2 ˆ ( ) s n 3 ˆ ( ) s n 1 z 2 z 3 z 2 , 1 C 1 , 2 C 1 , 3 C 2 , 3 C 3 , 1 C 3 , 2 C Respreading Matched Filter Matched Filter Matched Filter Respreading Respreading Matched Filter Matched Filter Matched Filter 1 ˆ b 2 ˆ b 3 ˆ b ( ) r n

Figure 3.1: General partial SPIC receiver structure.

In this case, all regenerated signals are first weighted and then summed. Thus, each regenerated interference signal in (3.9) has an individual PCF and the signal to be estimated is a function of all PCFs. We call this structure the coupled structure. The other structure is one in which

!

. In this case, all regenerated signals are summed first and then weighted. Thus, there

is one PCF for the signal to be estimated. We thus call this structure the decoupled structure. A thorough discussion of both structures is not available in the literature. Optimal PCFs have only been derived for the coupled structure under power balanced scenarios [61]. In what follows, we focus on a two-stage partial SPIC receiver with a decoupled structure. Primary simulation results (in Section 5) show that both PIC structures with optimal PCFs perform similarly.

§ 3.3 Optimal PCFs for AWGN Channels

In this section, we derive the optimal PCFs for a two-stage partial SPIC under an AWGN chan-nel. For ease of description, we only give the results associated with synchronous transmission. Periodic and aperiodic spreading codes are both considered.

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§ 3.3.1 Periodic Code Scenario

Assuming perfect chip synchronization, we first sample the received continuous-time signal in

(3.1) with period . Let

be the received signal sample

vector,

be the th user’s spreading sequence vector, and

be the noise sample vector. From (3.1), we have

(3.10)

Thus, we can obtain the matched filter output as

(3.11) Note that

is a discrete version of the correlation term

shown in (3.4). Similarly,

is a discrete version of the noise-related term

in (3.5). For notational simplicity, we still use

to represent and to represent

. Thus, (3.11) can be re-written as

(3.12)

For the second stage of a partial SPIC (with the decoupled structure), the regenerated signal

for Useris ! (3.13)

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where

. The second stage output is then

! ! ! ! ! ! (3.14)

The bit error probability for User, denoted as"

, can be written as " " " " (3.15)

In (3.15), we assume that the occurrence probabilities for

and

are equal, and the

error probabilities for

and

are also equal. As we can see, there are three terms

in (3.14). The first term corresponds to the desired user bit. If we let

, it is a deterministic

value. The second term corresponds to noise interference which is Gaussian distributed. The third term corresponds to the interference from other users and each interference is Binomial distributed. Note that correlation coefficients in (3.14) are small and CDMA systems are usually operated in low signal-to-noise ratio (SNR) environments. The variance of the third term is then

much smaller than that of the second term. Thus, we can assume that

conditioned on

is Gaussian distributed. The error probability is then

" (3.16)

whereis the Q-function and

(3.17) (3.18)

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Note that the expectations in (3.17) and (3.18) are operated on interfering user bits and noise. Let and . Evaluating (3.17), we obtain ! (3.19) where (3.20)

Similarly, we obtain the variance as

! ! (3.21)

where the coefficients of

are represented by (3.22) (3.23) (3.24)

The optimal PCF for Usercan be found as

! ! ! ! (3.25)

Substituting (3.19) and (3.21) into (3.25) and simplifying the result, we have the following equation. ! ! (3.26)

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We have two possible solutions now. The first solution for the first parenthesis is trivial since it

makes the squared mean value

in (3.19) zero. The optimum PCF is then

! (3.27)

We also derived optimal PCFs for an asynchronous CDMA system. The results are summarized in Appendix A. In what follows we discuss some special cases to give a better understanding of the optimal PCF characteristics. Let the correlations between any two user spreading codes be

equal (

for ) and the power control be perfect (

and

). The optimal

PCF can then be expressed as

! (3.28)

As we can see from (3.28), the optimal PCF is smaller whenoris larger, because when the

correlations between user codes are higher and the number of users is larger, the MAI is larger and the regenerated signal is unreliable. As a result, the PCF should be smaller. Also, when the

user power is larger or the noise is smaller (is larger), the optimal PCF is larger. If we assume

that the noise is much smaller than the signal power ( ), the optimal PCF can be further

simplified to ! (3.29)

Now the optimal PCF is independent of the transmission signal power. The bit error perfor-mance would also be saturated in this interference-limited region. From (3.28), we can also see

that when the noise is large (), the optimal PCF tends to be small (!

). Note that the

effect of the processing gain is reflected in the receiving SNR. If is larger, the receiving

SNR will become smaller.

§ 3.3.2 Aperiodic Code Scenario

In commercial CDMA systems, the users’ spreading codes are often modulated with another code having a very long period. As far as the received signal is concerned, the spreading code

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is not periodic. In other words, there will be many possible spreading codes for each user. If we use the result derived above, we then have to calculate the optimum PCFs for each possible code and the computational complexity will become very high. Since the period of the modulating code is usually very long, we can treat the code chips as independent random variables and

approximate the correlation coefficient,

, as a Gaussian random variable. As a result, the

expectations in (3.17) and (3.18) can be further operated on

. This greatly simplifies the

optimal PCF evaluation. We now rewrite (3.16) as

" (3.30) where

denotes the expectation operator over the spreading code setand

and

are the expected squared mean and variance of

, respectively, given the

&th possible code in

. Letting' , considering

as a Gaussian random variable, and evaluating (3.17)

and (3.18), we have ! (3.31) where (3.32) and ! ! (3.33) where ' (3.34) ' (3.35)

適用於直接序列碼分多重擷取系統多用戶偵測之部分平行式干擾消除：效能分析與新演算法

國 立 交 通 大 學

電信工程學系

博 士 論 文

適用於直接序列碼分多重擷取系統多用

戶偵測之部分平行式干擾消除：效能分析

與新演算法

Partial Parallel Interference Cancellation for

DS-CDMA Multiuser Detection: Performance

Analysis and New Algorithms

研究生： 謝 雨 滔

指導教授： 吳 文 榕

Partial Parallel Interference Cancellation for DS-CDMA

Multiuser Detection: Performance Analysis and New

Algorithms

適用於直接序列碼分多重擷取系統多用戶偵測之

部分平行式干擾消除：效能分析與新演算法

研究生：謝雨滔 Student: Yu-Tao Hsieh

指導教授：吳文榕 博士 Advisor: Dr. Wen-Rong Wu

國立交通大學

電信工程學系

博士論文

A Dissertation

Submitted to Department of Communication Engineering

College of Electrical Engineering and Computer Science

National Chiao Tung University

in Partial Fulfillment of the Requirement

for the Degree of

Doctor of Philosophy

In

Communication Engineering

July 2004

Hsinchu, Taiwan, Republic of China

中華民國九十三年七月

適用於直接序列碼分多重擷取系統多用戶偵測之部分

平行式干擾消除：效能分析與新演算法

研究生：謝 雨 滔

指導教授： 吳 文 榕 教授

國立交通大學電信工程學系博士班

摘要

Partial Parallel Interference Cancellation for

DS-CDMA Multiuser Detection: Performance

Analysis and New Algorithms

Student: Yu-Tao Hsieh

Advisor: Dr. Wen-Rong Wu

Department of Communication Engineering

National Chiao Tung University

Hsinchu, Taiwan 30050

Abstract

Acknowledgements

Table of Contents

List of Figures

List of Tables

Chapter 1

Introduction

§ 1.1

Background

§ 1.2

Multiuser Detection

§ 1.3

Objective and Overview

§ 1.4

Organization of the Dissertation

Chapter 2

Multiuser Detection

§ 2.1

Optimal Multiuser Receiver

§ 2.2

Linear Suboptimal Receivers

§ 2.3

Interference Cancellation Methods

∑

§ 2.3.1

Successive Interference Cancellation

§ 2.3.2

Parallel Interference Cancellation

Chapter 3

Optimal Two-stage Partial SPIC Receivers

§ 3.1

Introduction

國立交通大學

博士論文

研究生：謝雨滔

指導教授：吳文榕

指導教授：吳文榕博士 Advisor: Dr. Wen-Rong Wu

研究生：謝雨滔

指導教授：吳文榕教授