直接序列分碼多重進接無線通訊系統之分析及設計

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(1)國立交通大學電子工程學系電子研究所. 博士論文. 直接序列分碼多重進接無線通訊系統之分析及設計 Analysis and Design of Direct-Sequence Code-Division Multiple Access for Wireless Communications. 研究生：林郁男指導教授：林大衛. 中華民國九十四年七月.

(2) 直接序列分碼多重進接無線通訊系統之分析及設計. 研究生：林郁男. 指導教授：林大衛博士. 國立交通大學電子工程學系暨電子研究所. 摘要. 直接序列分碼多重進接(DS-CDMA)技術已被廣泛的應用於第二代行動通訊系統並成為下一代通訊系統的主要技術之一。本論文的主要內容在深入地分析傳統技術之效能，並提出一個可增進其效能的片碼間插(chip-interleaving)技術。首先，我們針對傳統系統在經迴旋碼(convolutional codes)之通道編碼後的檢測結果進行理論上的分析。基於隨機展頻碼(random-spreading codes)的假設，我們提出一個新穎且簡單的分析結果。我們也分析使用短展頻碼(short-spreading codes) 和長展頻碼(long-spreading codes)的差別，分析結果顯示雖然在未經通道編解碼時，展頻碼的週期長度並未對檢測結果造成影響，但是由於長展頻碼有較佳的干擾隨機性(randomness)，其在經通道編碼的系統中效能明顯優於短展頻碼；而此差異在淡化通道下，相對較不明顯。此外，我們也發現直接序列分碼系統雖然能經耙狀接收器(rake receiver)及位元間插器(bit interleaver)分別得到路徑分集 (path diversity)及時間分集(time diversity)，進而在淡化(fading)通道中有較好的表現，但是其效果仍較未經淡道化通道時來得差。因此，仍需進一步的分集技術。. i.

(3) 多重用戶干擾(multiple access interference)是影響分碼多重進接系統容量的重要因素。因此，我們接著討論應用平行式干擾消除(parallel interference cancellation) 檢測器以處理多重用戶干擾問題的系統。針對硬式(hard-limiting)干擾消除器，我們研究其在同步(synchronous)系統中的效能極限。經由分析我們發現，此檢測器在第兩級後的效能即未有明顯的差異，而當僅有兩用戶時，只要一級的干擾消除就夠；我們也驗證此效能的限制是來自於用戶間相互的干擾影響。在接下來的部份，我們研究片碼技術。我們首先提出一個使用片碼間插技術的直接序列分碼系統(CIDS-CDMA)。此系統由於能額外提供內符碼分集(intra-bit diversity)，在淡化通道下有很穩健的效能。當合併使用位元間插技術時，此系統最大的分集度是展頻係數(spreading factor)和迴旋碼之自由距離(free distance)的乘積。我們從理論上分析此系統在未經和經過通道編碼的效能。分析顯示，檢測對象的移動速度及其片碼間插深度幾可完全決定其檢測效果的好壞，其他干擾用戶的情形並未對其有明顯的影響。我們也發現在此系統中，展頻碼的週期長度在完美片碼間插情形下，並未如同傳統系統一樣影響檢測結果。在第四個部份，我們基於 3GPP 的寬頻分碼多重進接系統(WCDMA)設計了片碼間插寬頻分碼系統(CI-WCDMA)。片碼間插技術應用於實數分支(I-branch)以傳遞用戶資料。虛數分支(Q-branch)則用以傳送前導序列(pilot sequence)而在接收端利用傳統非片碼間插技術進行通道估測。在此系統中，我們亦應用了數種不同的非線性平行式干擾消除器。經由模擬驗證，此系統在淡化通道下的確有極為優異的效能表現。最後，我們研究應用多展頻碼(multicode)的片碼間插分碼系統於慢速淡化(slow fading)通道中。經由適當的零填補(zero-padding)，此系統在非同步(asynchronous) 及多重路徑 (multipath) 反射的情形下仍能維持展頻碼間的同步相關性 (synchronous correlation)，進而減少干擾。我們提出一個基於高登序列(gold sequences)的設計範例，其能在檢測品質及系統容量中有較好的效能取捨。和最近所提出的片碼間插區塊展頻系統(chip-interleaved block-spread CDMA)相較，我們所提出的系統在通道編碼的系統中有較佳的效能。. ii.

(4) Analysis and Design of Direct-Sequence Code-Division Multiple Access for Wireless Communications. Student: Yu-Nan Lin. Advisor: Dr. David-W. Lin. Department of Electronic Engineering & Institute of Electronics National Chiao Tung University. Abstract Direct-sequence code division multiple access (DS-CDMA) has been widely implemented in second generation cellular communications systems and become one of the most promising techniques in next generation system. In this thesis, we study the performance of conventional DS-CDMA systems and propose a chip-interleaved system which can greatly improve the performance. First, we concentrate on the analysis of conventional DS-CDMA and begin the discussion with convolutional coded systems. Random-spreading codes are assumed. A novel and relatively simple theoretical result is presented. The performance difference between short- and long-code spreading is also studied and the results show that due to the randomness of interference over symbols, the system with long-code spreading performs much better than with short-code spreading although they have similar performance when channel coding is not applied. The results also show that although DS-CDMA can gain diversity from rake receiving (path diversity) and bit interleaver (time diversity), it still degrades a lot in fading channels and hence the need for other diversity sources. Secondly, we consider the use of Parallel Interference Cancellation (PIC) in combating multiple access interference (MAI), which is the limiting factor to the capacity of DS-CDMA. The performance of hard-limiting PIC in synchronous systems is investigated. We show that hard-limiting PIC in such case has performance iii.

(5) limit at the second stage. In addition, for two-user, one stage is enough. And the performance limit comes from the mutual influence between desired and interfering users. The use of chip-interleaving in DS-CDMA is studied. We present a chip-interleaved DS-CDMA (CIDS-CDMA) which is resistant to channel fading by leveraging intra-bit diversity. When combined with bit-interleaving, this system can, in the limit, provide a diversity order equal to the product of spreading factor and the convolutional code’s free distance. Thus, CIDS-CDMA is much resistant to channel fading. Both channel coded and uncoded systems are theoretically analyzed. The analysis shows that the fading rate and chip-interleaving depth of the desired user itself almost single-handedly determines the resulting performance and the interfering users’ fading speeds do not affect the performance significantly. It also reveals that in the case of perfect chip-interleaving, the spreading code period has no influence on the performance. Fourthly, we proposed a chip-interleaved WCDMA (CI-WCDMA) based on several 3GPP WCDMA system features. In this system, data symbols are transmitted through I-branch while Q-branch is used for the transmission of pilot sequence. Channel estimation is done in Q-branch before chip de-interleaving as conventional systems. Several nonlinear PIC detectors are considered as the receiving structures. The result shows that the proposed system has a great improvement over conventional DS-CDMA system in fading channels. Lastly, we demonstrate a multicode CIDS-CDMA which can improve the performance of DS-CDMA in asynchronous and multipath channels by effecting the synchronous correlations among the spreading codes in such conditions. Slow fading channels are considered. An example using gold sequences is presented and shows a better trade-off between performance and capacity. Compared to the recently proposed chip-interleaved block-spread CDMA (CIBS-CDMA), the presented scheme can attain better performance in channel-coded systems.. iv.

(6) 誌. 謝. 這篇論文的完成，首先要感謝我的指導老師林大衛教授多年的教誨與指導。從老師身上，我學得許多從事研究所需具備的精神及態度，尤其是老師嚴謹的研究精神、對問題的堅持及執著更是深深地影響了我。在此，我要再次對林大衛老師表達我十二萬分的謝意與敬意。同時，也要感謝實驗室的學長姐、同學、學弟妹們，不論是在學業上的討論、一起合作計劃時的配合，以及在生活上的相互幫忙，都使我這段求學過程格外不同。尤其要謝謝峰誠學長，在工作站的管理及撰寫程式上所給予莫大的幫助。此外，也要謝謝崑健和俊榮在許多方面所給予的幫助及討論。最後，我要獻上最誠摰的感謝給予我的父母及家人，有你們在我背後的支持及鼓勵，我才能完成這篇論文。願與你們一同分享這份喜悅。. v.

(7) Contents 1 Introduction 1.1 Motivation and Discussed Topics . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of The Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contributions of The Thesis . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview of DS-CDMA Systems 2.1 System Model . . . . . . . . . . . . . . . . . . . . 2.1.1 Transmitted Signal . . . . . . . . . . . . . 2.1.2 Channel Model . . . . . . . . . . . . . . . 2.1.3 Received Signal . . . . . . . . . . . . . . . 2.2 Correlation Properties of Spreading Codes . . . . . 2.3 Detection of DS-CDMA . . . . . . . . . . . . . . 2.3.1 Simple Synchronous Systems . . . . . . . 2.3.2 Simple Asynchronous Systems . . . . . . . 2.3.3 Rake Receiving for Multipath Transmission 2.4 Random Spreading Codes . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 3 Analysis of Bit-Interleaved DS-CDMA with Convolutional Coding 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis of Synchronous Transmission . . . . . . . . . . . . . . 3.2.1 Perfectly Power-Controlled Channels . . . . . . . . . . 3.2.2 Rayleigh Fading Channels . . . . . . . . . . . . . . . . 3.3 Performance in Asynchronous and Multipath Channels . . . . . 3.3.1 Perfectly Power-Controlled Channels . . . . . . . . . . 3.3.2 Rayleigh Fading Channels . . . . . . . . . . . . . . . . 3.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . 3.4.1 Synchronous Transmission . . . . . . . . . . . . . . . . 3.4.2 Asynchronous and Multipath Channels . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 1 1 3 4. . . . . . . . . . . .. 6 6 6 7 8 8 9 10 10 11 13 14. . . . . . . . . . . .. 15 15 17 18 24 27 28 29 31 31 32 33. 4 Analysis of Hard-Limiting Parallel Interference Cancellation (PIC) for Synchronous CDMA Communication 37 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 vi.

(8) 4.3. 4.4 4.5. Performance Analysis . . . . . . . . . 4.3.1 Conventional GA Analysis . . 4.3.2 Error Bit Based Analysis . . . 4.3.3 Case of Two Users . . . . . . 4.3.4 Case of More Than Two Users Numerical Results . . . . . . . . . . . Summary . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 5 A Chip-Interleaved DS-CDMA System Resistant to Channel Fading 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Analysis of Synchronous Uncoded Systems . . . . . . . . . . . . . . . 5.3.1 Implications of Chip Interleaving Depths of Interferers . . . . . 5.3.2 Implications of Chip Interleaving Depth of Interfered User . . . 5.4 Analysis of Synchronous Coded Systems . . . . . . . . . . . . . . . . 5.4.1 Simple CIDS-CDMA . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Bit-and-Chip-Interleaved DS-CDMA . . . . . . . . . . . . . . 5.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Analysis of Coded Systems over Asynchronous and Multipath Channels 5.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 5.6.1 Uncoded Systems . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Coded Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Chip-Interleaved WCDMA with Parallel Interference Cancellation 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 CI-WCDMA Signaling . . . . . . . . . . . . . . . . . . . . . . . 6.3 Receiving of CI-WCDMA Signals . . . . . . . . . . . . . . . . . 6.3.1 Rake-Like Receiver . . . . . . . . . . . . . . . . . . . . . 6.3.2 PIC Detector . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Uncoded Systems . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Coded Systems . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 7 Multicode Chip-Interleaved DS-CDMA Concerning Interference 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 CIDS-CDMA Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Transmitted Signal . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Received Signal . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Effecting Synchronous Code Correlation After Multipath Propagation 7.3.1 An Example Using Gold Sequences . . . . . . . . . . . . . . 7.4 Comparison with CIBS-CDMA . . . . . . . . . . . . . . . . . . . . 7.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Comparison with Conventional DS-CDMA . . . . . . . . . . vii. . . . . . . . . .. . . . . . . . . .. . . . . . . .. 40 40 41 41 45 49 51. . . . . . . . . . . . . . .. 53 53 55 56 57 59 60 60 63 66 67 69 69 71 75. . . . . . . . . .. 76 76 77 78 78 81 83 83 84 87. . . . . . . . . .. 88 88 89 89 91 92 93 94 95 95.

(9) 7.6. 7.5.2 Comparison with CIBS-CDMA . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96 98. 8 Conclusions and Future Research Topics. 101. A The. 104.

(10) . Distribution. B Three-Dimensional Channel Modeling B.1 Introduction . . . . . . . . . . . . B.2 3-D Modeling . . . . . . . . . . . B.3 Measured Doppler Spectra . . . . B.4 Examples . . . . . . . . . . . . .. . . . .. viii. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 106 106 106 108 109.

(11) List of Figures 2.1. A simple implementation of rake receiver. . . . . . . . . . . . . . . . . .. 3.1. The average SINR of perfectly power-controlled channels as function of multipath number, where , dB, , and . . . . Average BER of synchronous BIDS-CDMA under perfect power control as function of user number, where , dB, and long-code spreading employs an infinite code period. . . . . . . . . . . . . . . . . . Average BER of synchronous BIDS-CDMA under perfect power control as function of the spreading code period in number of bits, where , dB, and . . . . . . . . . . . . . . . . . . . . . . Average BER of synchronous BIDS-CDMA in Rayleigh fading as function of user number, where , dB and long-code spreading employs an infinite code period. . . . . . . . . . . . . . . . . . . . . . . Average BER of asynchronous systems as function of multipath number, where , and dB. PC = perfect power control; RF = Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average BER of asynchronous systems as function of user number, where , and dB. PC = perfect power control; RF = Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2. 3.3. 3.4. 3.5. 3.6.

(12) . .

(13) .

(14) . # $ . . . !" . %. " . &. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9. Structure of PIC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reconstructor for th user in stage . . . . . . . . . . . . . . . . . . . Error region for initial stage PIC. . . . . . . . . . . . . . . . . . . . . Error region for first stage PIC. . . . . . . . . . . . . . . . . . . . . . Error region for second stage PIC. . . . . . . . . . . . . . . . . . . . One-stage PIC performance for with SF = 8. . . . . . . . . . . One-stage PIC performance for with SF = 32. . . . . . . . . Performance analyses for the first stage at SF = 32 and SNR = 10 dB. Two-stage PIC performance at SF = 32 and SNR = 10 dB. . . . . . .. 5.1 5.2 5.3. A simple chip-interleaved system. . . . . . . . A CIDS-CDMA signal example at , CDFs of pairwise error probability of different Rayleigh fading at , , and limited operation. . . . . . . . . . . . . . . . .. 34. 35. 35. 36. 36 39 40 44 44 45 50 50 51 52. . . . . . . . . . . . . . . , and chips. . . DS-CDMA schemes in , under interference. . . . . . . . . . . . . .. 55 56. (' ) *,+-.' /0

(15) ix. 30. . . . . . . . . ..

(16)

(17) . . . . . . . . . .. 11. 67.

(18) 5.4 5.5 5.6 5.7 5.8 5.9 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.1. 7.2. 7.3 7.4. . BER performance of CIDS-CDMA in the case of two users, as a function . . . . . . . . . . . of coherence time of user 1, at spreading factor BER performance of users with different maximum Doppler shifts versus the interleaving depth , where and SNR dB. . . . . . . . Average BER of synchronous CIDS-CDMA systems, where and dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average BER of asynchronous CIDS-CDMA systems as function of multipath number, where , and dB. . . . . . . . . . . Average BER of different asynchronous DS-CDMA systems in multipath Rayleigh fading at , and dB. . . . . . . . . . . . . Average BER of with different interleaver sizes in Rayleigh fading, where and dB. . . . . . . . . . . . . . . . . . . . . . . . . . . .. ). . . . '. .

(19) $ . The proposed CI-WCDMA signaling scheme. . . . . . . . . . . . . . . . An example of CI-WCDMA signal. . . . . . . . . . . . . . . . . . . . . A rake-like receiver for CI-WCDMA signals. . . . . . . . . . . . . . . . Illustration of semi-hard detector. . . . . . . . . . . . . . . . . . . . . . . Performance of Hard limiting PIC in CI-WCDMA and WCDMA as function of user number with SNR= dB. . . . . . . . . . . . . . . . . . . . Performance of different types of PIC in CI-WCDMA with SNR= dB. . Performance of different PIC detectors in channel coded WCDMA systems under quasi-static channels with .. . . . . . . . . . . . . . . Performance of Hard limiting PIC in channel coded WCDMA systems under correlated channels with SNR= dB. . . . . . . . . . . . . . . . .. .

(20) . . . 72 73 73 74 74 75 78 79 79 82 85 85 86 86. $. . BER performance of conventional DS-CDMA and CIDS-CDMA under Gold-sequence spreading, for different user numbers at , , and SNR dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 BER performance of conventional DS-CDMA and CIDS-CDMA under Gold-sequence spreading, for different user numbers at , , and SNR dB under fading channels. . . . . . . . . . . . . . . . . . 99 BER performance of different CDMA systems before channel decoding at fully loaded case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 BER performance of different CDMA systems after channel decoding at fully loaded case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. . $. ' . . B.1 Geometry for 3-D scattered environment. . . . . . . . . . . . . . . . . . 107 B.2 Autocorrelation functions under different angle spreads. . . . . . . . . . . 108 B.3 Autocorrelation functions under different angle positions. . . . . . . . . . 110. x.

(21) List of Tables 3.1. Maximum Path Distance in BER Estimation for the Example Code . . . .. 23. 5.1. SINRs That Govern Pairwise Sequence Error Probabilities in Rayleigh Fading Channels for Different Systems Under Interference-Limited Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67. xi.

(22) Chapter 1 Introduction Nowadays, mobile communication has become part of our daily life. Direct-sequence code division multiple access (DS-CDMA) has been widely employed in the secondgeneration mobile communication systems. In CDMA, signal is spread with a spreading code, which is unique for each user, before transmission. Spreading process is operated in a chip rate much higher than the actual data rate and hence, results in noise-like transmitted signal. With spreading codes known in the receiver side, user signal can be extracted by despreading operation. Therefore, CDMA is more efficient in bandwidth usage compared to time-division multiple access (TDMA) and frequency-division multiple access (FDMA). In addition, wireless communications are normally subject to channel fading. For CDMA systems, multipath energy can be captured through rake-combining. Hence, there is path diversity in CDMA and makes it more robust in fading channels. Due to its success in the second-generation communication, a technique based on wideband CDMA (W-CDMA) has been accepted as one of the air interface technologies for next generation communication system [15],[22],[25],[30].. 1.1 Motivation and Discussed Topics Ideally, orthogonal sequences can be used as the spreading codes for CDMA systems. However, when the system is asynchronous or suffers frequency-selective fading, the orthogonality between users cannot be preserved at receiver side. Hence, in reality, multiple 1.

(23) access interference (MAI) has been a limiting factor to the capacity of DS-CDMA systems. In the receiver side, various multiuser detection (MUD) technologies have been proposed to reduce MAI. In [18],[35],[48], decorrelating or zero-forcing detector is proposed to project desired user’s signal into the space orthogonal to that spanned completely by the interfering users. Although decorrelating detector removes MAI perfectly, it suffers noise enhancement problem. MMSE linear detector is proposed to balance between MAI cancellation and noise enhancement [35],[45],[74]. When a receiver, for example the base station, knows all the users’ spreading codes and channel state information (CSI), interference cancellation (IC) detector can be applied. The basic idea behind IC detector is simple: regenerate interference and then cancel them out. Interference regeneration is based on tentatively made decision and have more reliable regenerated interference, the IC detectors usually operates in multi-stage. There are different ways of canceling regenerated interference. Successive interference cancellation (SIC) [43],[51] cancels interference in an serial order such that the strongest interferer is the first canceled and the weakest one is the last canceled. On the other hand, parallel interference cancellation (PIC) [49],[71],[76] cancels all the reconstructed interference simultaneously. In this thesis, PIC is the main topic among various MUD techniques. Recently, some have pay attention on the transmitter design for low MAI system. One research direction is to design new spreading sequences having low correlation values when the relative delay between sequences is within a range, which is named interference free window (IFW). Thus, there is reduced MAI as long as the maximum asynchronous delay plus multipath delay spread is less than IFW. LS-codes [10],[41],[62] used in LASCDMA is an example. Not designing new spreading code sets, another research direction is to effect spreading code properties in frequency-selective channels as in flat fading channels. Block precoding is the common technique to this problem as in [26],[80]. In this direction, we will study the use of chip-interleaving.. 2.

(24) Wireless communications are usually subject to channel fading and diversity is the common strategy to remedy this impairment. In DS-CDMA systems, rake receiver is employed to lever path diversity and bit interleaver is applied for time diversity. However, later we will see that the performance of conventional DS-CDMA still degrades a lot due to channel fading. In regarding of this, chip-interleaving is considered as another way of levering diversity.. 1.2 Organization of The Thesis Basic operation of DS-CDMA is described in Chapter 2. We then study the performance of channel coded DS-CDMA over multiuser and multipath channels in Chapter 3. Random-spreading codes are assumed. Two channel conditions are considered. One is the Rayleigh fading channel and the other one is the modeling of perfect power control. By approximating the correlation among the spreading codes (rather than the ensuing interference) as Gaussian, we are able to inspect the influence of spreading code periods and obtain the analytical results with relatively simple expressions. The analysis shows that, due to the randomness in multiple access interference (MAI), long-code systems outperform short-code systems when channel coding is included. In Chapter 4, we consider the systems with hard-limiting PIC. While PIC normally involved nonlinear functions, previous analyses mainly addressed linear PIC. We find that the performance of purely hard-limiting PIC may not improve after the second stage and for two-user case, one stage usually suffices. Approximate expressions for numerical evaluation of hard-limiting PIC are also proposed and shown to be accurate. In the second part, we study the technique of chip-level interleaving both in slow and fast fading channels. The case in fast fading channels is discussed in Chapter 5. While practical channel coded systems use bit interleaving to effect inter-bit diversity, chip-level interleaving provides intra-bit diversity gain. We analyzes the performance of chip-interleaved DS-CDMA (CIDS-CDMA) without and with channel coding in Rayleigh 3.

(25) fading channels. It is concluded that, though there exists path diversity already, chipinterleaving still improves the performance greatly. We design a chip-interleaving scheme based on several 3GPP WCDMA system features, such as the use of Q-branch to transmit control bits and I-branch to transmit data bit in a QPSK-like modulation scheme in Chapter 6. Different types of PIC detector is used in such system. The proposed chip-interleaved WCDMA (CI-WCDMA) is shown to have significant performance advantage compared to simple WCDMA in transmission over multipath fading channels. Finally, the use of chip-interleaving in slow fading channels to alleviate MAI is addressed in Chapter 7. We show that while conventional DS-CDMA cannot preserve periodic correlation values of spreading codes over asynchronous and multipath channels, CIDS-CDMA can do this. A gluttonous code arrangement of Gold sequences is given as an example to show the superiority of this scheme.. 1.3 Contributions of The Thesis The main contributions of this thesis are: 1. We present the theoretical analysis for short- and long-code DS-CDMA systems over various channel conditions with random-spreading codes. 2. The performance of hard-limiting PIC is theoretically analyzed and we find that for two-user case, one stage PIC is enough while for more users case, the performance may not improved after second stage PIC. 3. We theoretically analyze the performance of chip-interleaved CDMA with convolutional coding in fading channels. 4. A practical design of chip-interleaving technology on the WCDMA system is proposed.. 4.

(26) 5. We demonstrate a multicode chip-interleaved DS-CDMA which can have reduced MAI and ISI over multipath channels.. 5.

(27) Chapter 2 Overview of DS-CDMA Systems In this chapter, we describe the basic operation of DS-CDMA systems. Signal model for uplink transmission is introduced, while it is of no difficulty to derive the downlink results. The detection results for some basic channels are also reviewed.. 2.1 System Model 2.1.1 Transmitted Signal Consider DS-CDMA systems employing BPSK modulation. The baseband signal trans-. %. mitted for the th user is given by. + . . with. +. . . . . . . . . . . . + +.

(28) . +.

(29) . . . "!#$. . . *. . !. . (2.1). * . (2.2). * are the bit and the chip periods, respectively, + is the user is the spreading waveform of the % th user during the th bit period, and datum, + + is the corresponding spreading code, * * is the spreading a square pulse of chip duration that is normalized so that . factor, and where. *. %. . and. &

(30) '. . &

(31) ' +,! $. )(. *. -(. .0/. *. 13 24 $. . . A spreading code with (ideally) infinitely long period is termed a long code. On the other hand, a short code has its code period equal to the spreading factor and has . +. &

(32) 7! . . 6. +. &

(33) ' 56!#. .

(34) 2.1.2 Channel Model. %. (2.3) + + +

(35) is the Dirac delta function, + is where $ + is the maximum number of multipaths, is the time-varying path coefficient with its variance as . In the path delay, and + +. Rayleigh fading channels, + is complex Gaussian random process. In addition, the path coefficients from different paths are independent and we let + 0 . When + for convenience. perfect power-controlled channels are considered, we let +. Assume user. is subject to a multipath channel with the impulse response being . . . . . . . . *3. . . *. . . Based on Clarke’s 2-D scattering model [11], which assumes scattered waves travel horizontally and with uniformly distributed incident angles, the first-order and secondorder correlation functions can be described as:. + + +

(36) ! #" %$ + $# + +

(37) ! & " #" (' + )

(38) + +*-, . +

(39) ! (2.4) where ! & / is the maximum Doppler frequency shift, & is the mobile velocity, / is

(40) is the zero-order Bessel function of the first kind. the radio wavelength, and

(41). .

(42). . *. *. *. . . . *. . . . . . . . . . . . /. . + + 0 21

(43). Based on Clarke’s model, Jakes derived the well-known Jakes’ model for simulation. + 3 as +4 0 +4 3 . of Rayleigh fading channels and generate each path coefficient, . . *. . . . 5 #687 )9;:< >= 9;:< >? A@ B9 :< C= ;9 :< >? ! D 5 687 )<-EGF >= 9;:< >? @ B<-EHF >= 9 :< C? D ? ! , ? ? I9;:< ! , and where ' ) , = K J LM ON QP . . . . . . . . . . . . . . . . . . . . . . . . . . . (2.5). L. .! /. !. 7. . !. 7. (2.6).

(44) With. ). , this sum-of-sinusoids simulator can serve as a simple but accurate model. and various modifications of Jakes’ model have been proposed in the literatures [40],[52],[54],[79]. Recently, more characteristics of wireless channel have been studied. For example, 3-D channel model, which considers the vertical incident angles as well, is addressed in [12],[24],[78], the distribution of incident angles is discussed in [24], Zhao et al. studied the path number distribution in [77], and modeling of wideband channel is presented in [17],[32],[37]. More information about the channel model is addressed in Appendix B.. 2.1.3 Received Signal The received signal of DS-CDMA at the base station is the composition of all users’ signals and is given as. . . 5 + + + +4 (2.7) + + where

(45) is the total number of users, @ + the normalized signal amplitude, + the . *. . . . . . . *. . asynchronous delay uniformly distributed over . power spectral density (PSD) equal to. + +. SNR is given by. /.

(46). *. . . *. , and . . . *. . *. white Gaussian noise with. . The signal-to-noise ratio (SNR) or average. .. 2.2 Correlation Properties of Spreading Codes Let .

(47) 7! . and

(48) 7! be two different binary sequences of period. odic (or even), and odd correlation functions of

(49) 7! and [58],. and.

(50)

(51) . . . . . . . . . . . . .

(52) .

(53)

(54) . . . 3.

(55).

(56) 7! &

(57) 7!

(58)

(59) 7!

(60) !#.

(61) . %.

(62). . . 8. .

(63) . . are given by, respectively.

(64) !#.

(65) . . . . The aperiodic, peri-. .

(66) . . .

(67) . . .

(68) . P . . . . (2.8) (2.9). (2.10).

(69) Most code designs are based on the periodic correlation. For example, Gold sequences have a 3-value periodic function given by for

(70) for for . . where. . . . . . . . . . . . . . . . . . .

(71). . of values values of

(72)

(73) values of. . . (2.11). is the degree of primitive polynomials generating Gold sequences of period !. 0. . , . . when. . . is odd, and !. when !. is even; the periodic correla-. . . tion functions when ! is even for the small set of Kasami sequences take on the values in the set and in the set for the large set of Kasami sequences. More correlation properties of pseudo. . . . . . . . . . . . . random sequences can be found in [60]. In simple synchronous channels, the zero-delay. ) governs the performance of DS-CDMA systems. However,. periodic correlation ( . later we will see that the detection in asynchronous systems are actually controlled by the aperiodic correlation. In such a condition, most sets of codes, including the Gold, the Kasami, and the m-sequences, have performance close to that of random codes which are composed of independent binary random variable taking values. . and. . with equal. probability [8],[33]. The investigation of the aperiodic correlation analytically is difficult and hence the need for numerical approaches.. 2.3 Detection of DS-CDMA In the following, we discuss the detection results of DS-CDMA under different channel conditions. To simplify the discussion, we discuss the system that uses short spreading codes. The systems with long-code spreading will be discussed in the Chapter 3. Assume perfect channel state information (CSI) is available.. 9.

(74) 2.3.1 Simple Synchronous Systems When the transmission is synchronous (. ), the received signal becomes . +. 6.

(75) '. . . . . . . 5 + +. . 2.

(76) ' . (2.12). + . . . . . . &

(77) '. 2. . * D 5 +

(78) +. + . . . . *. . after despreading becomes 2 . . where. . . . . . %. The decision signal for user. . + 5 + + P . . +! ) with ideal channel condition ( + . .

(79). +. . . . 2. . . * . . + .

(80) ' .

(81) . * . . . (2.13). (2.14). denotes the noise term after despreading. From the result, it is clear that the decision signal heavily relies on the design of spreading codes. As orthogonal spreading codes (. + ), such as Walsh-Hardmard sequences, are used, the signals from other. users are perfectly removed and in such case, DS-CDMA can serve as an ideal multiple access scheme.. 2.3.2 Simple Asynchronous Systems Next, we consider the asynchronous transmission over ideal channels. After despreading, the decision signal becomes. . +. 6 + * + !D 5 + + + 5

(82) + + .

(83) '. . . . . . 2. . *. . 2. &

(84) ' . where. +.

(85) . . . . . %. . 2. . .

(86) ' . . + . . . . . . . *3. . 2. 10. . .

(87) . * + . %. . . (2.15). (2.16).

(88) Integrate & Dump. (αk0 )* ( ). a(kh)(t−τ k −τ^ k,0 ) Integrate & Dump. r (t ). yk[h ] (αkl )* ( ). a(kh)(t−τ k −τ^ k,l ) Integrate & Dump. (α(kL k))*. ak(h)(t−τ k −τ^k,L k). Fig. 2.1: A simple implementation of rake receiver.. %. denotes the MAI induced to user Define. + as +4 B

(89) . . . . . . . N . +. . and. . + +. + ( . . !. . . is similarly defined as previous..

(90) . !.

(91) . . when !. +. and . by user. .

(92) . . . .

(93) . .

(94) . . . . !. . . . !. !. ( . ( + . + ( . . . . + . . + . . !.

(95) '. .

(96) . B

(97) . !. .

(98) '. . . .

(99) . . . . . . !. . !. . . (2.17). + + ( (2.18). .

(100) '. .

(101) '. . . . . . . . . !. !. . . . Since +4. addresses the asynchronous correlation between users, + + + where + + +4 * with we can compute +. + . Knowing that +

(102) is the aperiodic cross-correlation between +. when. !. . . !. . and . . . . . . .

(103) '. . . . !. . . !. . . . . . !. . . . &

(104) 7! . 24.

(105) 7! . , we can see that both the even correlation (when the consecutive bits have the. same phase) and the odd correlation (when the consecutive bits have the opposite phases) affect the amount of MAI.. 2.3.3 Rake Receiving for Multipath Transmission Wideband signal normally undergoes frequency selective fading channels. One advantage of wideband transmission is the availability of frequency diversity although it also faces 11.

(106) intersymbol interference (ISI) problem. For DS-CDMA systems, with spreading and despreading process, a simple rake receiver [55] can both attain the diversity and remedy the ISI. Figure 2.1 shows a simple structure of rake receiver. Each finger of the rake receiver is synchronized to one desired path and acquires the signal form that path. Then, the despread signals form all the fingers are maximal-ratio-combined (MRC) to form the decision signal.. + . Consider a slow fading channel that the path coefficients remain unchanged during the . detection and simply refer them to . With multiuser signals temporarily ignored, the

(107) despread signal at th finger can be expressed as * 2 %

(108) ' . . . +. + + * + 5 5 +

(109) +

(110) + + P (2.19) +

(111) +

(112) + + + * denotes the interpath interference (IPI). With + +

(113) . , it is of no difficulty to obtain + + + + + . Unlike 2. . &

(114) ' . . . + +. where. . . and !. . . . . . . . . .

(115) ' . . . . . . . . . .

(116) . . !. . . !. . . . . . . . 24. MAI, the IPI comes from the non-ideal aperiodic autocorrelation of spreading codes. The decision signal after combining all the fingers’ output becomes. + 6 + *

(117) + D P . .

(118) . . . . . (2.20).

(119) . . In the case that the spreading factor is large or the autocorrelation of spreading codes is much lower, we can neglect the IPI and obtain.

(120) 5 + + + +.

(121) '. . . . . + *

(122) D 6 + + P . . &

(123) ' . . . . .

(124) . (2.21). Clearly, a simple rake receiver can fully capture the multipath energy and hence gain the path diversity. In practical application, IPI and MAI cannot be fully removed due to the non-ideal spreading codes properties and therefore, the rake receiver does not provide optimal decision result. However, its simplicity of implementation still draws great attention. 12.

(125) 2.4 Random Spreading Codes In general, long spreading codes are well modeled as random codes and can randomize the interference more efficiently. Short spreading codes, on the other hand, have deterministic correlation properties which can be utilized to effectively cancel interference. However, for asynchronous channels such as uplink transmission, the real correlation function regulating MAI is the aperiodic correlation. Since aperiodic correlations are hard to trace analytically, random codes also act as an appropriate model for short-code systems with asynchronous transmission. To analyze the performance of DS-CDMA, MAI and IPI are usually modeled as Gaussian random variables and hence can be fully characterized by their variance. With this approximation, decision results can be evaluated by computing the variance of these Gaussian variables. Under assumption of random spreading codes, the variance of can be computed as. . + .

(126)

(127) (

(128). !. . .

(129). .

(130). . . !. !. . . . . . !. . !. . . . . P. . P. . +4. . . !. . (2.22). , the result can be further computed as. (2.23) + P Hence, when the asynchronous delays, + , are uniformly over * and users’ datum As is uniformly distributed over.

(131). . . . . !.

(132). . . . are independent, the variance of MAI term in (2.15) can be derived as . . +

(133) + '

(134) + P . . .

(135) '. . . (2.24). The signal-to-interference-plus-noise ratio (SINR) in simple asynchronous channels can be expressed as. + . . . . . . . 13. . + ( + P . . /. (2.25).

(136) SINRs under other channel conditions can be computed in a similar way. As shown in (2.25), when a user increases the transmission power, other users’ SINRs will be degraded. Therefore, DS-CDMA is a MAI-limited system.. 2.5 Summary In this chapter, we introduced the basic operation of DS-CDMA systems. Due to the non-ideal correlation functions, we have seen that the capacity of DS-CDMA is limited by MAI. Thus, the need for advanced interference techniques is self-evident. Though Gaussian approximation is quite accurate over a large operation range, we will later see that it loses some information for channel coded systems.. 14.

(137) Chapter 3 Analysis of Bit-Interleaved DS-CDMA with Convolutional Coding This chapter presents the performance analysis for the bit-interleaved DS-CDMA (BIDSCDMA) with convolutional coding. Random spreading codes are assumed. By approximating the correlation among the spreading codes (rather than the ensuing interference) as Gaussian, we obtain novel and relatively simple results for the analysis. The performance difference between short- and long-code systems is also discussed and the results showed that with randomized interference over bits, long-code spreading has better channel decoding results.1. 3.1 Introduction Theoretical analyses of DS-CDMA system performance frequently assume use of random spreading codes. Besides the well-known Gaussian approximation to MAI, many have attempted more accurate analytical characterization of the performance of unchannel-coded conventional DS-CDMA under random-code spreading. In [38], the density function of the MAI is studied extensively, from which arbitrarily tight upper and lower bounds on bit error rates (BERs) can be obtained. Based on [38], some have proposed simplified methods for performance calculation [9],[29],[46],[47]. Comparatively, there are fewer 1. Part of this chapter has been published in “Novel Analytical Results on Performance of Bit-Interleaved and Chip-Interleaved DS-CDMA with Convolutional Coding,” IEEE Trans. Vech. Technol., vol. 54, no. 3, pp. 996–1012, May 2005.. 15.

(138) publications devoted to the performance analysis of channel-coded DS-CDMA. For longcode systems, an approximate analysis is simpler than for short-code systems due to the random correlations among spreading codes. For short-code systems, the analysis is more difficult. In [72], the authors employ computer simulation to obtain the histogram of signal-to-interference ratios (SIRs). However, from the simulation results it is somewhat difficult to derive simple intuitive insights concerning the SIR distribution. Similarly in [68], simulation results are used to determine the probability density function (pdf) of SIRs numerically. In [50], many observations of the difference between short-code and long-code systems with and without channel coding are made. But the analysis is through Monte Carlo simulation where the BERs can only be calculated as the spreading sequences are known and the distribution of SIRs can only be known through extensive computation. The analysis presented by Viterbi in [73] does not consider the interference statistics over symbols and hence, only holds for long-code systems. In this chapter, we consider an analytical approach. By approximating the correlation among the spreading codes (rather than the ensuing interference) as Gaussian, we obtain novel, simple results concerning the transmission performance under various conditions. Unlike the conventional Gaussian approximation, the results are quite accurate for shortcode systems under channel coding. Another motivation of the present work comes from the following observation concerning the underlying mechanism that leads to the performance difference between longcode and short-code spreading in channel-coded DS-CDMA. With random spreading codes, the distribution of their correlations in a symbol interval is the same regardless of whether long-code or short-code spreading is used. In the absence of channel coding, this results in a similar average error performance for both kinds of spreading. In the presence of channel coding, however, we have a different picture. With long-code spreading, the correlations among different users’ spreading codes change from symbol to symbol. In maximum-likelihood decoding, it is the total MAI in the Viterbi decoding delay (or the. 16.

(139) total MAI in the span of a codeword in the case of block coding) that affects the error performance. Thus if the channel code’s minimum Hamming distance is large, then by the law of large numbers, all users will be subject to a total MAI of substantially similar statistics and have similar error performance in equal noise. With short-code spreading, on the other hand, the correlations of the spreading codes remain the same over time. Hence the law of large numbers is not at work in the temporal direction as it is in a long-code system. Thus the MAI has a greater variance. Or, from another viewpoint, a proportion of the users will experience high MAI over an extended period of time. This results in poor decoding performance for these users as well as a poorer average error performance over all users when compared with a long-code system. Since this effect comes from the correlation among spreading codes, bit interleaving (the conventional way to deal with bursty errors) does not give a fully satisfactory solution.. 3.2 Analysis of Synchronous Transmission We start the discussion with synchronous transmission over flat fading channels. The received signal is, as previous, given by . . (3.1) @ + + + the channel coefficients, where @ is the normalized signal amplitude, + the transmitted DS-CDMA signal as described previwhite Gaussian noise, and + . . . . *. *. . . . *. . *. (for all % ) for convenience. In Rayleigh fading. In the former case, we let + are time-varying, zero-mean, complex-valued Gaussian random the latter case, + variables, and we let + for all % . In both cases, the value or the expected ously in Chapter 2. Two kinds of channel are considered: perfectly power-controlled and . *. . value of the received signal power for each user is simply vantaged position. The SNR or average SNR is given by. and no user is at a disad-. . /. . . The Rayleigh. fading case may be viewed as a system with perfect long-term power control. Although 17.

(140) in synchronous short-code systems, other kinds of spreading codes (such as the orthogonal codes) may yield lower interference, random codes have advantage in addressing the influence of asynchronism. Indeed, uplink transmission is usually asynchronous. Hence we assume random codes in the following discussion. Assume the receiver employs conventional matched filtering and despreading. With-. . out loss of generality, take user as the desired user. To simplify the analysis, assume the path coefficients, within. *.

(141) %. . . + , remain unchanged during one bit and denote + . * . . the coefficient. . Assume CSI can be estimated without error. Then the decision. signal for the th bit is given by

(142) '.

(143) . 6+

(144) . . . . .

(145) '. 2. . . . . *. . %. 2. !D *. . P. (3.2). We analyze the performance in the two channel conditions in two subsections.. 3.2.1 Perfectly Power-Controlled Channels Consider first the case of perfectly power-controlled channels, where the channel coefficients have a constant amplitude but random phases. Without loss of generality, let. .

(146) '. and let +

(147) '. where. +.

(148) . for . @

(149)

(150) .

(151) .

(152) '.

(153) &

(154) '. . &

(155) '. . (3.4).

(156) '. being the correlation between . as. (3.3). . . . is given by .

(157) ' #

(158) '. @ + + 9;: < + +. is the MAI, with . % . Then the decision signal for bit. .

(159) '.

(160) '. . +. and . !#. . &

(161) '. . + +. and + being the value of + (assumed constant during the bit period), and *

(162) is white, we get is the noise in the despread signal. Since .

(163) .

(164) '. . . ! &

(165) '. . defined. !#. (3.5). ! . *.

(166) . . #

(167) . . . . 2. . 2. . . . 18. . %. . .

(168) '. . (3.6) ..

(169) Short-Code Spreading. +. becomes independent of . Hence we drop the time. &

(170) ' and

(171) all as random variables and index . Conventional analysis takes , With short-code spreading,. models.

(172) .

(173) . &

(174) '. + 9 :< +. +. as zero-mean Gaussian. Thus, with random spreading codes, the variance of. is given by. .

(175) '. .

(176) . P. (3.7). For analysis of average uncoded BER, this expression is applicable to both long- and short-code systems and leads to quite accurate results. For channel-coded systems, however, the decoding requires observing the received signal over a time interval spanning multiple bit periods. The corresponding performance analysis thus needs the statistics of interference (e.g., joint pdf) over such a multi-bit period, but the above variance only characterizes the interference statistics in individual bit periods. In a short-code system, suppose a user is assigned a spreading code that has high correlation values with other users’ spreading codes. Then this user’s signal will suffer from high interference. In a synchronous system, this condition will persist until the assigned spreading code are changed. (In an asynchronous system, the condition may change when the relative delays among the users are changed.) Bit interleaving does not help in this situation. Therefore, the transmission performance is worse than that predicted using (3.7) with the usual assumption of independent interference in different bit periods. The mathematics below provides more insights. While our discussion concentrates on convolutional coding with soft-decision Viterbi decoding, the situation with block coding can be addressed in a related fashion. Assume. . the modulator maps channel coder output values and to. . and , respectively. The . following analysis is based on uniform error property (UEP), which means that the error probability is the same for all the transmitted codewords. When used with BPSK or QPSK over AWGN channels, binary linear codes satisfy UEP. Since convolutional codes belong to linear codes, it suffices to consider the all-zero code sequence only. The probability of 19.

(177) erroneously decoding it to a trellis path that remerges with the all-zero path and differs. . from it in. coded bits can be expressed as . . . . Prob. . . . ..

(178)

(179) . (3.8).

(180) where the index runs over the set of bits in which the two paths differ. The superscript. has been used to simplify the notation because these time. Term . . .

(181).

(182) . bits may not be consecutive in. a pairwise error probability for convenience. By (3.3),.

(183)

(184)

(185)

(186) . @

(187) @

(188) . . . . sian random variable if. . . @. + + 9;:< + + . .

(189) P + , the quantity 9 :< +

(190) + . .

(191).

(192) .

(193) . . .

(194).

(195) . can be modeled as a Gaus-. is large. And the conditional variance of. .

(196)

(197) + + 9;:< +.

(198) + @ +

(199) P. . . (3.9).

(200) . . . . . . Conditioned on a set of . as. . . . . can be computed. . . .

(201) . . . . . . (3.10). Therefore, the resultant signal-to-interference-plus-noise ratio (SINR) conditioned on can be expressed as. . . "

(202)

(203) . . . . (3.11). . and the conditional pairwise error probability given by where . . " . is the Gaussian . no expectation over. ". (3.12). function. The main difference in (3.10) from (3.7) is that. + is taken and hence the impact of constant correlations among the 20. .

(204) spreading codes over the remerging distance is not overlooked. Notice that since the interference is modeled as Gaussian here, UEP still holds and the discussion of all-zero code sequence is enough for tracing the performance.. To find the unconditional error probability, we need the pdf of . . .. With random. + observe the binomial distribution given by (3.13) P;P P P. spreading codes, the code correlations . . + is By the central limit theorem, + is approximately Gaussian when is large. Then Gaussian the sum-square value of

(205) random variables of zero mean and unit variance.. is a standard central random variable with

(206) degrees of freedom, where Thus . . . . . . . . . . . . . . . the pdf of a standard central. with. $. . random variable with ! degrees of freedom is. . . . . . . . . . . . . . . . ( . (3.14). being the gamma function defined as. . . $. . Though Gaussian approximation of. . N P . . . (3.15). $. + is not exact in the tail part of the pdf, it does not. have major effect on the accuracy of the analysis, as later numerical results will demonstrate. Thus, the unconditional pairwise error probability can be obtained as . " . " P. . . . . . . (3.16). Often, BER is more useful than the pairwise error probability, but is less easy to obtain. Conditioned on , an upper bound on the BER is given by. . . where. =. . . . . =

(207) " .

(208)

(209). . is the weight spectrum and . (3.17). is the minimum free distance. Exact weight. spectra of some low-rate convolutional codes can be found in [13]. Since (3.17) is a. 21.

(210) union bound, it seriously overestimates the corresponding BER when the pairwise error probability is high. As a result, its average, i.e., . . .

(211)

(212). =

(213) " . . . .

(214)

(215). =

(216) " . (3.18). merely gives a very loose bound on the average BER. Burr [7] presents approximate weight distributions of convolutional codes that are useful for BER below. . . . Since the. pairwise error probabilities in our problem may spread over a greater range, the approximations in [7] are not tight enough. To get a tighter bound, we simulate transmission over additive white Gaussian noise (AWGN) channels under different SNRs and find the least. be the. number of weight spectrum terms required to bound the simulated BER. Let . resulting maximum path distance that need be considered to bound the decoded BER at SNR. (. where. ". at decoder input. Then the tighter bound on average BER can be evaluated as

(217).

(218) (3.19)

(219)

(220) . =

(221). . " . . . . is as given in (3.11). Further, we find that, when the summation in (3.19). contains more than one term, a better approximation to the BER can be obtained if we replace. =. by. = . for the maximum-distance term. This is used in our numerical results. /. presented later.. for the rate-1/2, constraint length-7 convolutional code with # and . It has . Since numerical results generator vectors show that the influence of N is hardly significant, we simply ignore them. Table 3.1 lists . . . . The techniques to obtain the unconditional pairwise error probability and the BER bound from the conditional pairwise error probability are mostly similar in all the other conditions discussed in subsequent sections. Hence we shall omit their discussion unless the difference warrants it.. 22.

(222) Table 3.1: Maximum Path Distance in BER Estimation for the Example Code. . . (dB)

(223). . . . . .

(224). .

(225). P P ' . . .

(226). Long-Code Spreading For long-code spreading, consider first a system where the period of spreading codes is an. ! where + different correlation denotes the modulo operation. And there are a total of

(227). integer. . times the spreading factor. Thus +. &

(228) '. . !# 5

(229) . . !. . . values between any user’s spreading code and the other interfering users’ spreading codes. If these values occur the same number of times within the remerging distance of the channel code’s trellis, the conditional variance of the interference in (3.9) is changed to 4 . (3.20) . . . + + P correlation values do not repeat an equal number of times The case where the

(230) . . . . 4 . . . within the remerging distance of the channel code’s trellis requires tedious mathematics to characterize precisely. Later simulation results will show that the performance is only. slightly different from that calculated using the last equation. is a random variable, but with Like , the quantity. . . .

(231) degrees . 4 . of freedom. Hence the conditional SINR is given by. . . 4. 4. .

(232) . . 4. . . P. (3.21). From this the corresponding pairwise error probabilities (conditional and unconditional) and BER can be obtained in a way similar to that for short-code spreading. We note that the unconditional pairwise error probability is given by . . . . . . . . . (ideal long-code spreading has . . For large. 4. . . . 4 . ),. . . . . . . 4 . 4 . . . . . P. has very concentrated. pdf (result of the law of large numbers) that decays faster than the Gaussian 23. (3.22). function.

(233) for values of . . . . . 4 .

(234) . . . away from. . 4. . 4. . .

(235) . . . 4 . . In this case, . . . . . . . . . P. (3.23). . Thus, in the limit, we obtain the same result as that obtained under conventional Gaussian approximation.. 3.2.2 Rayleigh Fading Channels Now consider Rayleigh fading channels. We assume a fully bit-interleaved, quasi-static condition where the fading coefficients for different users at different bit times are uncorrelated and stay unchanged during a bit period. Consequently, the maximal-ratio combined signal in the convolutional decoder can be expressed as. .

(236).

(237) . . where. @

(238)

(239)

(240)

(241) . . . . 3 . @ + +

(242) H +

(243)

(244) * + . . . . . . . . . is the MAI and. . . .

(245). .

(246) . . . 2. . . . *. 2. the variance of can be computed as. . . .

(247).

(248) . (3.25). .

(249) .

(250) . . is the additive noise. Since

(251) '.

(252) . (3.24). . .

(253).

(254) . . *. . (3.26). . .

(255) P . . .

(256) . The variance of , however, depends on which type of spreading code is used.. 24. (3.27). (3.28).

(257) Short-Code Spreading Under short-code spreading, we have . . . @ + + +

(258)

(259) * +

(260) P . . . Rewrite. +

(261)

(262) * +

(263) .

(264) .

(265) .

(266) . . (3.29).

(267) .

(268)

(269) +

(270) + +

(271)

(272) +

(273) B +

(274) P.

(275). .

(276) .

(277) .

(278) . Then.

(279) . . . .

(280) . (3.30).

(281) . (3.31) @ + +

(282)

(283) B +

(284) P

(285) Since + is uniformly distributed in ! under the Rayleigh fading assumption and +

(286) with equal probability, +

(287) is Gaussian. Given

(288) and + , is the. . . . . .

(289) . .

(290) .

(291).

(292) .

(293) . . (.

(294) .

(295) . combination of Gaussian random variables and is hence also Gaussian. Taking expecta

(296) tion over the “relative fading coefficients”

(297) , we get .

(298)

(299) (3.32) . +. . . + +

(300) # due to the earlier assumption that where we have used the fact that +. + . Consequently, we can express the SINR as

(301)

(302) " . +

(303)

(304) + +

(305) , ' @

(306) ' + + @ ,

(307)

(308) " P (3.33) " Note that the numerator of is a (normalized) random variable of degree and . .

(309) . /. *.

(310) . . . .

(311) . .

(312) . . . /. . /. /. . .

(313) . . . /. /. . . the denominator is a (normalized). . random variable of degree. 25.

(314) . plus a constant..

(315)

(316) where and , respectively, with . . In Appendix A, we show that when random variables of degrees dependent.

(317). number, and is a constant, the pdf of. /. /