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Blind multiuser detection for time-frequency spread multicarrier CDMA 2007

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Bl ind Multiuser Detection for Time-Frequency Spread Multicarrier C D M A by Wilson Tarn B . A . S c . University of British Columbia. Vancouver, Canada. 2005 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Electrical and Computer Engineering) T H E UNIVERSITY OF BRITISH C O L U M B I A October 2007 © Wilson Tarn, 2007 Abstract Multicarrier transmission technology has shown tremendous potential in realizing high data rates for next generation broadband wireless communication systems. Multicar- rier modulation schemes are robust to frequency selective fading which are inherent in broadband wireless channels. This thesis considers blind multiuser detection for the recently proposed time frequency spread multicarrier C D M A ( T F - M C - C D M A ) sys- tem in which the system uses both time and frequency domain spreading to achieve diversity in both time and frequency domains. The challenge for T F - M C - C D M A multiuser detection is to mitigate multiple access interference (MAI) in both the time and the frequency domain. The objective of this thesis is to develop and analyze the performance of blind multiuser detection algorithms for T F - M C - C D M A for three types of channel models: A W G N with M A I . slowly fading downlink Rayleigh multi- path channels, and the slowly fading downlink multipath channels with Doppler shift induced intercarrier interference (ICI). A new suboptimal decoupling technique for blind multiuser detection for T F - M C - C D M A in A W G N with M A I is proposed. It is found that the original T F - M C - C D M A blind multiuser detection problem can be suboptimally decoupled into two blind mul- tiuser D S - C D M A problems. These two problems can be solved separately using blind D S - C D M A multiuser techniques. Our effort focused on using blind linear multiuser detectors in which we investigated into four types of blind detection methods: blind direct matrix inversion (DMI) method, blind C M O E RLS method, blind subspace ii multiuser detection method using SVD and PASTcl adaptive subspace tracking. The suboptimal decoupling technique for blind multiuser detection for T F - M C - C D M A is extended to slowly fading downlink Rayleigh multipath channels known as type I detectors. Computer simulation results show that type I detectors do not work well in slowly fading multipath channels even though such a scheme provides very- good performance in A W G N with M A I . In slowly fading channels, the orthogonality of the time domain signature sequences is preserved. We propose a type II detector which uses a cascade implementation with the time domain detection output acts as the input of the frequency domain detection. Computer simulations show that type II detectors provide much better performance than type I detectors. Blind multiuser detection for T F - M C - C D M A is further extended for slowly fading Rayleigh multipath channels with Doppler shift induced ICI. In mobile channels. Doppler shifts combined with multipath effects create random subcarrier frequency shifts which in turn cause subcarrier frequency mismatch. Such mismatch leads to the loss of orthogonality among subcarriers thus creating ICI. Our analysis shows that Doppler shifts induced ICI has the effect of destroying the common-channel property in downlink channels. The downlink T F - M C - C D M A signal becomes a quasi-uplink signal because of user dependent subchannel gains. The type II detector for slowly fading Rayleigh multipath channels is further extended to apply in such channels with Doppler shift induced ICI. It is shown through analysis and computer simulations that the proposed type II detectors implemented using C M O E RLS algorithm, blind subspace SVD algorithm, and the blind subspace PASTcl adaptive subspace tracking algorithm, without modifications, provide robust performance in multipath channels with Doppler shift induced ICI. in Contents Abst rac t i i Contents v i i Lis t of Tables ix Lis t of Figures x i Lis t of Symbols x i i Lis t of Abbrevia t ions x i i i Acknowledgements x v i Dedica t ion x v i i 1 In t roduct ion 1 1.1 O v e r v i e w o f C u r r e n t W i r e l e s s C o m m u n i c a t i o n S y s t e m s 1 1.1.1 L i m i t a t i o n s o f D S - C D J V I A i n B r o a d b a n d W i r e l e s s C h a n n e l s . . 2 1.2 M u l t i c a r r i e r - b a s e c l T r a n s m i s s i o n 3 1.2.1 M u l t i c a r r i e r C D M A 4 1.2.2 T i m e - F r e q u e n c y S p r e a d M u l t i c a r r i e r C D M A 4 1.3 C h a l l e n g e s a n d M o t i v a t i o n 5 1.3.1 M u l t i u s e r D e t e c t i o n i n T F - M C - C D M A S y s t e m s 6 i v 1.3.2 Blind Signal Processing Techniques 6 1.4 Review of Related Works 7 1.5 Thesis Contributions 9 1.6 Thesis Outline 12 2 T i m e Frequency Spread Mul t i c a r r i e r Spread Spec t rum System M o d e l 13 2.1 Introduction 13 2.2 Transmitter Model 14 2.2.1 Equivalent Complex Baseband Signal 16 2.2.2 Allocation of Spreading Sequences 18 2.3 Channel Models 21 2.3.1 A W G N Channel 21 2.3.2 Downlink Multipath Rayleigh Fading Channel 22 2.3.3 Cyclic Prefix Insertion and Removal of Block ISI 26 2.4 T F - M C - C D M A Demodulation 28 2.4.1 Passbancl Model Demodulation 28 2.4.2 Equivalent Complex Baseband Model Demodulation 30 2.5 Summary 31 3 B l i n d Mul t i u se r Detect ion for a T F - M C - C D M A Signal in A W G N 33 3.1 Introduction . 33 3.2 Linear Multiuser Detection for T F - M C - C D M A 34 3.3 Adaptive Blind Multiuser Detection for T F - M C - C D M A 35 3.3.1 Direct Matrix Inversion (DMI) Method 38 3.3.2 C M O E Recursive Least Squares (RLS) Method 40 3.3.3 Blind Subspace-Based Method 43 v 3.4 Performance Measure . . ' 47 3.4.1 Receiver Complexity : 49 3.5 Simulation Results 50 3.6 Summary 53 4 T F - M C - C D M A B l i n d Mul t iu se r Detect ion in Slowly Fading Rayle igh M u l t i p a t h Channel , 60 4.1 Introduction 60 4.2 Linear Multiuser Detection for T F - M C - C D M A in Frequency Selective Fading Channels 61 4.3 Blind Multiuser Detection in Slowly Fading Channels 63 4.3.1 Type I Receiver 64 4.3.2 Type II Receiver 65 4.3.3 C M O E RLS Blind Detection 68 4.3.4 Blind Subspace Multiuser Detection 74 4.4 Performance Evaluation Measure 79 4.4.1 Receiver Complexity 79 4.5 Simulation Results 83 4.6 Summary 87 5 T F - M C - C D M A B l i n d Mul t iu se r Detect ion in Downl ink M o b i l e Ray le igh Fading Channe l w i t h Doppler Induced I C I 92 5.1 Introduction 92 5.2 Modeling Intercarrier Interference 93 5.3 Blind Multiuser Detection 96 5.3.1 Type II Detection Formulation 97 5.3.2 C M O E RLS Blind Detector 98 v i 5.3.3 Blind Subspace Multiuser Detection 99 5.4 Simulation Results 103 5.5 Summary 105 6 Conclus ion and Future Research I l l 6.1 Conclusion I l l 6.2 Future Research 113 6.2.1 MIMO T F - M C - C D M A Multiuser Detection 113 6.2.2 Bayesian Monte-Carlo Blind Multiuser Detection for T F - M C - C D M A 114 6.2.3 T F - M C - C D M A Blind Multiuser Detection in Fast Fading Chan- nels 115 Bib l iography 116 vii List of Tables 2.1 Summary of channel fading models in mobile wireless channels . . . . 26 3.1 B l i n d T F - M C - C D M A D M I Receiver Algor i thm 40 3.2 B l ind C M O E R L S Receiver Algor i thm 42 3.3 B l i n d Subspace S V D Receiver Algor i thm • 45 3.4 B l i n d Subspace P A S T c l Receiver Algor i thm 48 3.5 Summary of T F - M C - C D M A B l i n d Multiuser detection complexity for A W G N channel 50 3.6 Simulation parameters 50 4.1 Type I C M O E R L S B l i n d Receiver Algor i thm 72 4.2 Type II C M O E R L S Bl ind Receiver Algor i thm 73 4.3 Type I B l i n d Subspace (SVD) Receiver Algor i thm 77 4.4 Type II B l i n d Subspace (SVD) Receiver Algor i thm 78 4.5 Type I B l i n d Subspace P A S T c l Receiver Algor i thm 80 4.6 Type II B l ind Subspace P A S T c l Receiver Algor i thm 81 4.7 Summary of T F - M C - C D M A B l i n d Multiuser detection complexity for slowly fading Rayleigh multipath channels 83 4.8 I T U - A Pedestrian Rayleigh Fading Channel Power delay profile . . . 84 4.9 Simulation parameters 84 5.1 Type II C M O E R L S Bl ind Receiver Algor i thm for multipath fading channels with Doppler induced ICI 99 vi i i 5.2 Type II Blind Subspace (SVD) Receiver Algorithm 101 5.3 Type II Blind Subspace PASTd Receiver Algorithm 102 5.4 Summary of simulation parameters 103 ix List of Figures 2.1 T F - M C - C D M A transmitter model 14 2.2 T F - M C - C D M A baseband transmitter implementation using I D F T / I F F T 17 2.3 T F spreading sequence matrix 19 2.4 Cyclic Prefix Insertion -. . . 27 2.5 Passband Demodulator 28 2.6 Complex Baseband Modulator and Demodulator 31 3.1 T F - M C - C D M A Multiuser Detector Architecture 37 3.2 B E R Performance with 49 Users 52 3.3 B E R Performance with 77 Users 53 3.4 B E R Performance with 105 Users 54 3.5 Output SINR with 49 Users at SNR lOdB 55 3.6 Output SINR with 77 Users at SNR lOdB 56 3.7 Output SINR with 105 Users at SNR lOdB 57 3.8 BER, performance plotted against the number of users at SNR lOdB . 58 3.9 Output SINR plotted against the number of users at SNR lOdB . . . 59 4.1 Type I detector architecture 65 4.2 Type II detector architecture 66 4.3 BER, performance for 35 users case 85 4.4 B E R performance for 55 users case 86 x 4.5 B E R performance for 75 users case 87 4.6 Output SINR for 35 users case at SNR 12clB 88 4.7 Output SINR for 55 users case at SNR 12clB 89 4.8 Output SINR for 75 users case at SNR 12dB 90 4.9 Mean square channel estimation error for 35 users at SNR 16clB . . . 91 5.1 B E R performance for 35 users case 104 5.2 B E R performance for 55 users case 105 5.3 B E R performance for 75 users case 106 5.4 Output SINR for 35 users case at SNR 12clB 107 5.5 Output SINR for 55 users case at SNR 12clB 108 5.6 Output SINR for 75 users case at SNR 12dB 109 5.7 Mean square channel estimation error for 35 users at SNR 16clB . . . 110 xi List of Symbols argmaxj-} Argument maximizing the expression in the bracket argminj-} Argument minimizing the expression in the bracket | • | Absolute value of a complex number * Convolution hire Natural logarithm of x Re{-} Real part of a, complex number <5(-) Dirac delta function || • || Frobenius norm sgn{-} Sign of a real number E{-} Expectation [•]* Complex conjugate [•]r Matrix or vector transposition [•]H Matrix or vector Hermitian transposition Im Identity matrix with dimension m x m xii List of Abbreviations 1G First Generation 2G Second Generation 3G Third Generation 4G Fourth Generation A M P S Advanced Mobile Phone System A W G N Additive White Gaussian Noise B E R Bit Error Rate B P S K Binary Phase Shift Keying C D M A Code Division Multiple Access C M O E Constrained Minimum Output Energy D F T Discrete Fourier Transform DMI Direct Matrix Inversion D S - C D M A Direct Sequence C D M A E D G E Enhanced Data Rates for G S M Evoluti E V D Eigenvalue Decomposition xiii F D M A Frequency Division Multiple Access F F T Fast Fourier Transform FIR Finite Impulse Response GPRS General Packet Radio Service G S M Global Systems for Mobile Communications HSDPA High Speed Downlink Packet Access IDFT Inverse Discrete Fourier Transform IEEE Institute of Electrical and Electronic Engineers IFFT Inverse Fast Fourier Transform ISI Inter-symbol Interference ITU International Telecommunication Union L M M S E Linear Minimum Mean Square Error L T V Linear Time Varying M A I Multiple Access Interference M C - C D M A Multicarrier C D M A M C - D S - C D M A Multicarrier Direct Sequence C D M A M M S Multimedia Messaging Service M M S E Minimum Mean Square Error M O E Minimum Output Energy xiv M U D Multiuser Detection/Detector O F D M Orthogonal Frequency Divison Multiplexing PASTcl Projection Approximation Subspace Tracking with deflation RLS Recursive Least Squares RMS Root Mean Square SINR Signal to Interference and Noise Ratio SMS Short Messaging Service SNR Signal to Noise Ratio SVD Singular Value Decomposition T D L Tapped Delay Line T D M A Time Division Multiple Access T F Time Frequency T F - M C - C D M A Time Frequency spread Multicarrier C D M A U M T S Universal Mobile Telecommunication System W G N White Gaussian Noise W L A N Wireless Local Area Network WSSUS Wide Sense Stationary Uncorrected Scattering xv Acknowledgements First and foremost, I thank my parents for everything, without their support, this work would not have been possible. I would like to thank my supervisor Dr. Cyril Leung for his valuable guidance and criticisms. I would also like to thank my col- leagues Jingjun L i and Derrick Wan who always extend their helping hands without hesitation. Finally, I would like to thank my thesis examination committee members Dr. Lutz Lampe and Dr. Shahriar Mirabbasi for their time and efforts. W I L S O N T A M THE UNIVERSITY OF BRITISH COLUMBIA October 2007 xvi my parents. xv i i Chapter 1 Introduction 1.1 Overview of Current Wireless Communication Systems During the past two decades, wireless communications have been one of the fastest growing industries in the world. Much of the growth in the past 10 years can be attributed to the ever increasing popularity of the mobile cellular market. According to the G S M Association (GSMA), the 2 billionth mobile phone user was subscribed in June 2006 and it was estimated that new users are subscribing at a rate of 1000 per minute world wide. The demand for ubiquitous connectivity anywhere, anytime in the near future is certainly unquestioned. The first wave of wireless revolution came with the introduction of the first generation (1G) analog systems such as A M P S which was built upon frequency di- vision multiple access (FDMA) . In 1G systems, voice service was the only service provided to subscribers. With the emergence of digital communication technology, evolution towards second generation digital systems such as G S M (2G), and later EDGE/GPRS(2.5G) provided subscribers with a wider variety of data communica- tion services on top of the still dominant voice service which include limited internet 1 access and the popular short messaging service (SMS). Current third generation (3G) mobile communication systems are built upon code division multiple access ( C D M A ) using direct sequence spreading codes in the time domain and allow users to enjoy much higher data rates than ever before. C D M A systems are robust to narrow band interference and provide an efficient util ization of both time and frequency and thus are able to deliver enhanced voice services and sufficiently high data rates to support a broader range of services such as multimedia, streaming applications, multimedia messaging service ( M M S ) and internet surfing capability. Another distinct advantage of C D M A over traditional F D M A and T D M A is a cellular re-use factor of one [1]. Combined with soft capacity limits in C D M A based cellular networks (which offer a graceful degradation when the multiple access interference exceeds a tolerable level as new users are admitted into the system), this creates the possibility of intelligent soft hand-off procedures when mobile users roam across cell boundaries. Al though the in- ternational standardization body for fourth generation ( 4 G ) has not yet released the official specifications for the next generation system requirements; it is anticipated that 4 G systems wi l l accommodate broadband services at data rates comparable to those in contemporary W L A N systems. 1.1.1 Limitations of D S - C D M A in Broadband Wireless Chan- nels Exist ing 3 G standards using D S - C D M A as the access air interface are adequate in achieving high data rates as long as the channel is not severely frequency selective. When a channel is frequency selective, a R A K E receiver must be used. For R A K E receivers to perform well., they require very accurate channel information and channel t iming information which are usually acquired through training sequences and/or the use of pilot symbols. Ut i l iz ing the R A K E receiver, however, does not eliminate the 2 problem of intersymbol interference (LSI) caused by the frequency selectivity of the channel. As a result, equalization techniques may be required to mitigate the ISI. The combination of the R A K E receiver, channel estimation and equalization significantly increases the complexity in the design of the receivers. 4G systems will very likely operate in broadband wireless channels and the severely frequency selective nature of the channel will pose difficult challenges in the design of D S - C D M A receivers of practical complexity. Due to these limitations of DS-CDMA, multicarrier based transmission multiple access schemes have emerge as potential candidates for next generation access air interfaces [2]. 1.2 M u l t i c a r r i e r - b a s e d T r a n s m i s s i o n The concept of transmitting information over multiple subcarriers in parallel sub- streams was first proposed by [3] in 1966. In 1971. it was shown in [4] that such transmission can be realized using the discrete Fourier transform (DFT). However, the implementation of such a transmission scheme at the time it was first proposed was beyond existing technology. A major advancement in signal processing tech- nology came with the invention of the fast Fourier transform (FFT) which enabled efficient computation of the discrete Fourier transform (DFT). This made possible the practical implementation of multicarrier transmission. The first popular standardized multicarrier based system was based on orthogonal frequency division multiplexing (OFDM) used in current W L A N standards IEEE 802.11a and 802.llg. The concept of O F D M is to transmit at high data rates over parallel narrowband subchannels in which the corresponding subcarriers are orthogonal. The advantage of O F D M based multicarrier transmission is inherent in broadband wireless communication systems in which the channel is highly frequency selective. By dividing the available spectrum into narrowband subcarriers and appropriately choosing the subcarrier bandwidth. 3 each na r r owband subchanne l can be t reated as f requency non-select ive fad ing. T h e success of O F D M , pa r t i cu l a r l y i n W L A N systems, has evoked its poss ib le app l i c a t i on i n b r oadband mob i l e commun i ca t i on s [5] and so lv ing the need for comp lex equa l i za- t i on i n 3 G D S - C D M A systems. 1.2.1 Multicarrier C D M A Mu l t i c a r r i e r C D M A ( M C - C D M A ) . f irst p roposed i n [6.7] is a mu l t i p l e access scheme tha t is based on a comb ina t i on of O F D M and D S - C D M A . T h e m a i n idea of M C - C D M A is to use O F D M t ransmiss ion whi le separa t ing mu l t i p l e access users us ing C D M A . T w o popu l a r imp lementa t i ons of M C - C D M A exist: the first one reta ins the name M C - C D M A , the other is k nown as M C - D S - C D M A . M C - C D M A spreads the user's s ignature sequence i n the f requency d oma i n by p r e -mu l t i p l y i ng a ch ip at each O F D M subcar r ie r b ranch . Therefore, the sequence of ch ips across a l l the subcarr iers const i tu tes the user's f requency d oma i n s ignature sequence. M C - D S - C D M A , on the other hand , spreads the o r ig ina l d a t a sequence i n t ime f irst us ing D S - C D M A and then t r an sm i t t i n g the t ime doma i n spreadecl d a t a s t ream us ing O F D M t ransmiss ion . A deta i l ed compar i son of the two mu l t i ca r r i e r mu l t i p l e access schemes is g iven i n [8] and [9]. B o t h M C - C D M A and M C - D S - C D M A re ta in the robustness to f requency select ive fad ing wh i le un comp rom i s i ng the benef i ts of us ing C D M A . 1.2.2 Time-Frequency Spread Multicarrier C D M A Recent ly , a new mu l t i ca r r i e r based mu l t i p l e access scheme known as t ime f requency spread M C - C D M A ( T F - M C - C D M A ) was proposed by [10,11] wh i c h combines M C - C D M A and M C - D S - C D M A . T h e ma i n idea beh i nd T F - M C - C D M A is to spread the da t a i n b o t h the t ime and f requency doma ins . The r e are many advantages to T F - M C - C D M A . F i r s t l y , l ike M C - C D M A and M C - D S - C D M A , T F - M C - C D M A inher i t s 4 the benefits of O F D M based transmission and is robust to frequency selectivity in wireless channels. Secondly, the maximum user capacity for T F - M C - C D M A is given by the product of the spreading gain of the time domain spreading and the spreading gain of the frequency domain spreading. This allows the use of short codes in both the time and frequency domain to achieve the same maximum user capacity in M C - C D M A and D S - C D M A employing long codes and high chip rates. Through both time and frequency domain spreading, T F - M C - C D M A is able to benefit from time and frequency diversity gains. Thirdly, T F - M C - C D M A allows a very flexible means for resource allocation. Because T F - M C - C D M A systems are not fixed with solely time domain spreading or solely frequency domain spreading, its adaptability is very- attractive in supporting a much wider range of services and data rates in broadband wireless systems [5,12]. 1.3 Challenges and Motivation Multiple access interference (MAI) is a common problem in all C D M A based multiple access systems. Whether it is M C - C D M A , M C - D S - C D M A , DS-CDMA, or T F - M C - C D M A . the performance of the system depends on the severity of the M A I among the users. M A I results from the non-orthogonality of the user signals at the receiver. This lack of orthogonality can be due to either the use of non-orthogonal codes (i.e. M-sequences or Gold sequences), or the orthogonality of the codes is corrupted by the channel, or both. It was shown in [13] that multiuser detection techniques are quite effective in combating MAI ; however, it is also known that multiuser detection methods almost always significantly increase the complexity of the receiver. Further- more, [13] has shown that the complexity of the optimal multiuser detection method grows exponentially with the number of users, making it unattractive for practical implementation. Suboptimal. low complexity multiuser detection methods will be a 5 key aspect of receiver design in C D M A based systems. 1.3.1 Multiuser Detection in T F - M C - C D M A Systems Since T F - M C - C D M A systems employ both time and frequency spreading, multiuser detection will very likely be used in both the time and the frequency domain because M A I can be present in both domains for reasons discussed previously. An optimal multiuser detection method (which minimizes the probability of symbol error) would be overly complex. Linear multiuser detection methods (a class of suboptimal mul- tiuser detection methods) are attractive due to their simple implementation. The idea of linear multiuser detection is to compute a set of weights such that using these weights, the linearly weighted chip samples at the receiver produce an output that minimizes a certain criterion such as the M A I (the linear decorrelating detector) or the mean square error (the linear M M S E detector). For linear multiuser detection, the challenge is to devise methods that can efficiently compute those weights and at the same time gives acceptable performance. 1.3.2 Blind Signal Processing Techniques Non-blind multiuser detection requires the knowledge of the signature waveforms of all users in the system as well as accurate channel information. In many applications, the receiver only knows its own signature waveform. This is particularly true in receivers used in mobile handsets. Furthermore, estimation of channel parameters usually requires training sequences and/or pilot symbols both df which represent overhead. These difficulties motivate the use of blind signal processing techniques in which [14] first applied to D S - C D M A multiuser detection problems. In T F - M C - C D M A , the blind multiuser detection problem is. based on only the received signal, the desired user's own time domain spreading signature and its frequency domain 6 signature, to mitigate M A I in the received signal and perform detection when the channel is unknown to the receiver. 1.4 Review of Related Works The original multicarrier C D M A system was first proposed in [6,7]. T F - M C - C D M A has been proposed more recently, as a means of achieving diversity gains in both the time and the frequency domain and also greater flexibility and efficiency in resource utilization. It was shown in [10] that T F - M C - C D M A outperformed single carrier systems using the R A K E receiver in frequency selective Rayleigh fading channels when system parameter values are properly matched to channel attributes such, as coherence bandwidth and the delay spread. The uplink and downlink performances of T F - M C - C D M A systems in indoor and outdoor wireless channels were investigated in [15] and it was found that the system performance benefitted from both time and frequency diversity gains. The presence of M A I is a fundamental limitation in any C D M A based system and T F - M C - C D M A , unfortunately, suffers the same limitation. Multiuser detection methods enhance the performance of C D M A based systems by mitigating M A I . The complexity of multiuser detection methods ranges from the most complex optimal multiuser detectors to non-linear and linear detectors. The optimal multiuser detector and a few selection of non-linear and linear multiuser detectors for D S - C D M A systems were extensively investigated in [13]. Non-blind, linear multiuser receivers, namely the decorrelation detector and the M M S E detector, were studied for T F - M C - C D M A in A W G N channels in [16] and [17]; in particular, it was shown in [17] that the total supported users in the system can be grouped using the time domain spreading sequences with users in the same group sharing the same time domain signature sequence but have different frequency domain signature sequence. Such a scheme reduces the potentially large joint cross-correlation matrix 7 into two smaller cross-correlation, matrix, one for the time domain and one for the frequency domain. Complexity reduction is achieved by applying separate time and frequency domain detection and the authors in [17] have shown that such method gives comparable performance to the joint time and frequency domain detection. However, the drawback of methods proposed in [17] is that it requires knowledge of all users' time and frequency domain signature sequences as well as accurate knowledge of the noise power. When the desired user only knows its own time and frequency domain signature sequences, blind multiuser detectors must be used. This assumption is generally ap- propriate for receivers used in mobile handsets. Linear blind multiuser receivers are especially attractive due to their low complexity compared to the optimal multiuser detector. The constrained minimum output energy detector (CMOE) for D S - C D M A was investigated in [14] and [18] which demonstrated all linear blind detectors can be represented in a canonical form and such form facilitated adaptive implementations which was investigated in [19]. It was shown in [20] that the linear blind multiuser detector for D S - C D M A can be equivalently formulated using the subspace decompo- sition by decomposing the data correlation matrix into a signal eigen-subspace and a noise eigen-subspace. It was further shown in [20] that both the blind clecorrelating detector and the blind M M S E detector can be computed through subspace decom- position. Many recent works, such as that of [21] and [22], have applied the blind subspace methods to M C - C D M A and M C - D S - C D M A systems. Subspace methods have shown considerable promise because the framework allows for adaptive imple- mentation using existing subspace tracking algorithms [20, 23, 24] and blind joint channel estimation [22,25]. Subcarrier frequency offsets in mobile channels clue to Doppler shifts causes inter- carrier interference (ICI) in systems using O F D M based transmission [26]. It is shown 8 in [27] that using a time domain signal analysis, the ICI in M C - C D M A can be mod- eled as weighted frequency domain sampling. The effect of Doppler induced ICI in mobile fading channels has been investigated for pure O F D M systems in [28] and [26]. The main result is that ICI creates leakage of energy from one subcarrier to another when subcarriers are no longer orthogonal due to the random offsets caused by the Doppler shifts in the channel. In multipath fading channels, each path arriving at the receiver experiences a different Doppler frequency shift as a result of the random angle of arrival. Therefore for T F - M C - C D M A , it is in general that clue to multipath propagation and user mobility, the orthogonality of the subcarriers are corrupted as a result of ICI and at the same time, the orthogonality (if orthongonal codes are used) of the frequency domain spreading sequences are lost clue to the combination of both multipath fading and ICI. Past research focus has been on mitigating solely the ICI problem in O F D M systems: this is achieved through ICI cancellation and suppres- sion techniques [29-31] all of which adds to the complexity of the receiver. In most downlink applications, the orthogonality of the spreading sequences is critical for high speed data transfer such as HSDPA for 3G mobile systems. For T F - M C - C D M A to perform well, the receiver must be able to overcome the two major impairments, M A I and ICI. 1.5 T h e s i s C o n t r i b u t i o n s The objective of this thesis is to propose blind multiuser detection methods for T F - M C - C D M A in three different propagation environments: (1) the A W G N channel, (2) the downlink multipath Rayleigh fading channel and (3) the mobile Rayleigh fading channel with Doppler shift induced ICI. The main contributions of the thesis are summarized as follows: • An improved time and frequency domain signature sequence allocation scheme 9 is proposed. This scheme clarifies the time signature grouping scheme in [17] by minimizing the number of users within a time domain signature group (and also the number of users sharing the same frequency domain signature sequence). Our proposed scheme ensures that all time domain groups have a minimal num- ber of users sharing the same time domain signature sequences and thereby keeping the number of interfering signals in the time and frequency domain to a minimal. • A new suboptimal decoupling approach to blind linear multiuser detection for T F - M C - C D M A is proposed. The original linear blind multiuser detection prob- lem for T F - M C - C D M A can be suboptimally decoupled into two D S - C D M A multiuser detection problems. This allows for separate blind detection for the time and frequency domain multiuser detection with each of them being equiv- alent to a blind multiuser D S - C D M A detection problem. Using the decoupling strategy, the linear blind multiuser detector for each of the time and frequency domain problem can be solved using existing linear blind multiuser detection algorithms for DS-CDMA: - Direct matrix inversion (DMI) method - Constrained minimum output energy (CMOE) recursive least squares (RLS) method - Blind subspace method using singular value decomposition (SVD) - Blind subspace method using PASTd adaptive subspace tracking The performance of each method in A W G N channel is investigated where non- orthogonal spreading sequences are used. Performance in moderate, moderate heavy, and full system capacity loads were investigated for each implementation method. 10 The decoupling method for T F - M C - C D M A blind linear multiuser detection is extended to detection in slowly fading Rayleigh multipath channel on the downlink. This is known as the type I blind detector. It is shown that the C M O E RLS. blind subspace multiuser detection using SVD and adaptive sub- space tracking using PASTcl are all applicable to type I implementation. The only added complexity comes from the joint blind channel estimation. Us- ing computer simulations, the performance for type I detector using C M O E RLS. blind subspace detection using SVD and adaptive subspace tracking us- ing PASTcl algorithm are investigated. A T F - M C - C D M A type II blind detector is also proposed for the same channel model. In contrast to type I detector which maintains the same parallel architecture as for the A W G N case, the type II de- tector uses a cascade implementation in which the output of the time domain detector is fed into the frequency domain detector. Since the time domain sig- natures remain orthogonal in slowly fading channels, the type II detector is able to reduce the number of interferers for the frequency domain detection stage by matched filtering out users that does not share the same time domain signature. The performance of the blind C M O E RLS algorithm, blind subspace detection using SVD, and adaptive subspace tracking using PASTd are investigated us- ing computer simulations for the type II implementation. Simulation results showed that type II blind detectors provide better performance than type I blind detectors. We further extend TF-MC-CDMA blind multiuser detection to slowly fading mobile Rayleigh multipath downlink channel with Doppler shift induced ICI. Our work contributes to the modeling and analysis of Doppler shift induced ICI in T F - M C - C D M A and showed that the random Doppler induced ICI in a mobile channel has the effect of destroying the common multiuser-channel property 11 of the downlink channel. As a result, the original T F - M C - C D M A downlink blind multiuser detection problem becomes a quasi-uplink multiuser detection problem where the channel now becomes user dependent. We extended the type II blind multiuser detection methods for slowly fading Rayleigh multipath downlink channel (without ICI) to the one with Doppler induced ICI. It is shown that the original method developed for the downlink channel without ICI still provides very good performance. Computer simulation results confirm that the blind type II receiver, implemented using the blind C M O E RLS, blind subspace using SVD and blind subspace using adaptive subspace tracking with PASTcl give robust performance in mobile channels with Doppler induced ICI. 1.6 Thesis Outline The roaclmap for the rest of the thesis is as follows: Chapter 2 presents the T F - M C - C D M A transmission system model as well as the channel models used in the thesis. Chapter 3 discusses blind multiuser detection algorithms for T F - M C - C D M A signal in A W G N channels when non-orthogonal spreading sequences are used. Chapter 4 extends the T F - M C - C D M A blind multiuser detection methods proposed in Chapter 3 to slowly fading Rayleigh multipath channels. Chapter 5 further extends the work of Chapter 4 to blind multiuser detection in mobile Rayleigh multipath channels with Doppler shift induced ICI. Finally, conclusions and future research directions are discussed in Chapter 6. 12 Chapter 2 Time Frequency Spread Mult icarr ier Spread Spectrum System Mode l 2.1 I n t r o d u c t i o n In this chapter, the T F - M C - C D M A system and channel models are described. The chapter is divided into three sections; in Section 2.2, the transmitter model and mod- ulation for the T F - M C - C D M A system are described and the motivation for the time frequency spread spectrum system is discussed. The model is the basis for the dis- cussions of receiver designs in the remaining chapters. Section 2.3 describes the addi- tive white Gaussian noise (AWGN) channel with multiple access interference (MAI) and the synchronous downlink multipath Rayleigh fading channel that is used to assess T F - M C - C D M A system performance in the remaining chapters. For the down- link Rayleigh multipath fading channel model, four sub-models, namely, frequency non-selective slowly fading, frequency non-selective fast fading, frequency-selective slow fading, and frequency-selective fast fading channels are discussed. Section 2.4. 13 F i g u r e 2.1: T F - M C - C D M A t r ansmi t t e r m o d e l discusses the d e m o d u l a t i o n process for T F - M C - C D M A signals . A s u m m a r y of the chapter is p r o v i d e d i n Sec t ion 2.5. T h e t r ansmis s ion m o d e l cons idered for the T F spread M C - C D M A sys tem follows tha t i n [17]. T h e t r ansmi t t e r c a n be thought of as K users t r a n s m i t t i n g s imul t aneous ly and synchronous ly . T h e b lock d i a g r a m for the kih user is shown i n F i g . 2.1. It is assumed tha t a b i t s t ream. c/fc[i] £ { 0 ; 1}, f rom a b i n a r y s y m m e t r i c source ( B S S ) is to be sent for the kth user, k G {1, 2 . . . . . A " } . T h e b i t s t ream is t hen m o d u l a t e d us ing b i n a r y phase shift k e y i n g ( B P S K ) in to bk[i] G { — 1, + 1}. T h e B P S K m o d u l a t e d sequence is passed t h r o u g h a l inear t ime invar ian t ( L T I ) filter w i t h a r ec tangu la r pulse shape impu l se response of d u r a t i o n Tb r e su l t ing i n an ou tpu t waveform g iven by 2.1. 2.2 T r a n s m i t t e r M o d e l oo (2.1) 14 The signal bk(t) is then mixed with a direct sequence (DS) time domain (TD) code ak(t) given by 2.2 with spreading gain N and chip duration Tc, N-1 a f c(t) = J ] a A ; [ j ] F f c ( t - . ? T c ) .7=0 (2.2) where NTC = TJ, and i r = ak is the k user time domain ak[0) . . . ak[N-l] chip sequence. The DS time domain spread signal is replicated over M branches onto M sub carriers of freciuency um, {u.m: m = 1. 2... . . A4}. In branch m, ak(t)bk(t) is pre-multiplied by a frequency domain chip sequence ck[m] of duration Tc. We further assume that the set of subcarrier frequencies satisfy the orthogonality condition over the interval Tc given by exp (j (umt + <pm)) exp (-j(u>it + <bi))dt = 0, m^l (2.3) JO and exp (j (umt + 0m)) exp ( - j (cu i, t + fa)) dt ̂  0, m = I (2.4) J o Hence, together with the carrier frequency uc, the transmitted signal s f e(t) for the A;'-'1 user is given by the sum of the outputs of all M subcarrier branches, i.e. A'J (2.5) Sk{t) = \ JTYI bk(t)ak(t)ck[m] cos {(uc + um) t} 171=1 where P is the energy of sk(t). i.e. P= / sk(t)sl(t)dt •Jo (2.6) The overall transmitted passband signal is *(*) = ] > > ( * ) A:=l /\ A'/ (2.7) ^2 ^ ^fc(OaA:(0CfcH COS {(CJ C + Um) t} k = l 771=1 15 2.2.1 Equivalent Complex Baseband Signal It is k n o w n f rom [32] a n d [33] tha t a t r a n s m i t t e d passbancl s igna l can be w r i t t e n i n the form s(t) = v /2Re {sLP(t) exp (juct)} (2.8) where sr,p(t) is the equivalent lowpass c o m p l e x baseband s igna l of s(t). E q u a t i o n 2.7 can be r e -wr i t t en as / \ M M s(t) = \ / 2 R e \\/^Y,zZ h(t)a>k(t)ck[,n) exp {jumt} exp {juct} \ (2.9) k=\ m.= l A c o m p a r i s o n of 2.8 a n d 2.9 shows that rjj '< M SLp(t) = yjrfJ2Yl h(t)ak(t)ck[m] exp (jwmt) IT K \ M 1 A:=l l?n=l J T h e t e r m i n the c u r l y braces i n 2,10 can be rea l ized a n d i m p l e m e n t e d us ing the I D F T [4] w h e n the subcar r ie r frequency set { o ; m } ^ = 1 is chosen to have m i n i m u m subcar r ie r frequency separa t ion and satisfy the o r thogona l i t y c o n d i t i o n g iven by 2.3 a n d 2.4. T o find the m i n i m u m possible subcar r ie r frequency separa t ion , the o r thogona l i t y c o n d i t i o n g iven i n 2.3 can be re -wr i t t en as ex-p(j(um - (jj{)t) e x p ( j ( 0 m - 4>i))dt = 0, m ^ I (2.11) E q u a t i o n 2,11 holds i f a n d on ly i f (um — ui)~^ = n7l> n is a n Y nonzero integer (2.12) T h e m i n i m u m subcar r ie r frequency separa t ion is o b t a i n e d for n — 1 a n d is equal to f r . If m i n i m u m subcar r ie r frequency spac ing is used, the subcar r ie r frequencies are 16 Figure 2.2: T F - M C - C D M A baseband transmitter implementation using I D F T / I F F T then given by ujm=LO0 + ^ , m = 0 , l , . . . ; M - l (2.13) c where UQ is an arbitrary frequency offset. For u0 = 0 , the complex baseband signal SLp{t) can be written as rfj K (M~l f 2nm \ 1 siAt) = V^X>(*K(*) E CfcNexp U-y^t) (2.14) A:=l ( m . = 0 ^ c ' ) It is shown in [34] that the signal Sr,p(t) can be reconstructed from samples taken at a rate of ^ samples per second, i.e. J c sLp[n] = sLP = y / ^ E 6*a*M | E C f c H exp | (2-15) The term inside the curly braces in 2.15 is an M-point I D F T . Using this fact, the T F - M C - C D M A modulation can be efficiently implemented using I F F T if M is chosen to be a power of 2. This is shown in Fig. 2.2. 17 2.2.2 A l l o c a t i o n of Spreading Sequences In T F - M C - C D M A . each user is allocated a pair of signature sequences, one for the time domain spreading (TDS) and the other for the frequency domain spreading (FDS). As long as no two users share the same TDS and FDS signature sequence pair, the system is able to support up to Nx M users, where N and M are the TDS and FDS spreading gains respectively. In [17], a scheme is proposed that allocates TDS and FDS sequence pairs by grouping the K users in the system into N groups with each group being represented by one of the Ar TDS sequences in the set {a 1 ; a 2 ; a ^ } . where ak ak[l] ••• ak[N] r , and with each group supporting at most x = 77 users. The users within a group are then separated by the x FDS signatures chosen T . is the vector that Cfe[l] . . . ck[M] from the set {ci, Co.,c,w} where ck = contains the FDS of the kth user. However, [17] did not specify how to choose the TDS and FDS signature sequence pairs to minimize M A I in the system. To achieve minimization of M A I , it is desirable to minimize the number of users sharing the same TDS signature sequence and the number of users sharing the same FDS signature sequence. This motivates us to propose a simple, systematic way of allocating the time and frequency signature sequences that achieves this result. The TDS and FDS signature sequence pairs can be viewed in an arranged two dimensional array, the T F sequence matrix, as shown in Fig. 2.3. Using the T F sequence matrix, each user is assigned a pair of T D S / F D S sequences according to the following rule. In the first case, suppose the number of available TDS sequence AT is equal to the number of available FDS sequence M. Therefore, the T F sequence matrix is a square matrix of size N by N (or M by M). Then, 1. The pairs along the main diagonal of the TF-sequence matrix are assigned first sequentially going from left to right. 2. Once the main diagonal pairs are used up. this means that the last main diagonal 18 T - d o m a i i i sequences; c i r— 1 — z — r ± : — I 1 1"" r — Cv Figure 2.3: T F spreading sequence matrix pair belonging to the last column has just been assigned. The next pair is selected from the first column. From the first column, let R[i — 1] be the row number of the last signature sequence pair assigned from the first column. Then, the row number of the next pair assigned is computed as ((R[i — 1] + N) mod M) + 1. 3. The following pair to be assigned will come from the next column to the right of the column number of the last assigned pair. If the last column is reached, the next column will be again the first column. We again use the row number of the last pair assigned from the current column of consideration to compute the row number of the next pair assigned using the same method as the previous step. 19 4. This process is continued until all pairs have been used up For the case when the number of available TDS sequence N is not equal to the number of available FDS sequence M, without loss of generality, suppose that N < A4. and hence a long rectangular T F sequence matrix with VY columns and M rows(this is because we can always switch TDS and FDS labels), then 1. The pairs along the main diagonal of the TF-sequence matrix are assigned first sequentially going from left to right. 2. Once the main diagonal pairs are used up, this means that the last main diagonal pair belonging to the last column has just been assigned. The next pair is selected from the first column. From the first column, let R[i — 1] be the row number of the last signature sequence pair assigned from the first column. Then, the row number of the next pair assigned is computed as (R[i — 1] + N) mod M. Therefore, the next assigned pair is located on column 1 and row (R[i — 1] + N) mod M. Note that the modulo operation gives row numbers going from 0 , M — 1. Since the row number in the-TF-sequence matrix goes from 1..... Ad. this means that a computed 0th row actually corresponds to the A4th row. 3. The following pair to be assigned will come from the next column to the right of the column number of the last assigned pair. If the last column is reached, the next column will be again the first column. We again use the row number of ' the last pair assigned from the current column of consideration to compute the row number of the next pair assigned using the same method as the previous step. 4. This is continued until all pairs in the TF-sequence matrix are assigned. 20 It was shown in [16,17] that if the nth time group (i.e. the group of users with the same time domain signature sequence) contains Xn users, then, the passband transmitted signal in 2.7 can be equivalently re-written as rrjp N x<> M s ^ = V Jd zZ E E M*K(*)<Um] cos {(uc + um) t} (2.16) 71 = 1 K= 1 771=1 The equivalent complex baseband signal can also be re-written as M scp(t) = sJlZ YZ zZ zZ bvAt)an(t)cK[m] exp (jumt) (2.17) 77=1 f t = l 777=1 2.3 Channel Models In this section, the channel models to be used for the performance analysis of TF- M C - C D M A are reviewed. In particular, two channel models are presented: (1) the A W G N channel and (2) the downlink multipath Rayleigh fading channel. For the latter, four types of fading are considered (based on different channel characteristics) from the wide sense stationary uncorrelated scattering (WSSUS) channel model. 2.3.1 A W G N Channel The non-fading channel for T F - M C - C D M A is modeled as r(t) = s(t) + n(t) (2.18) where n(t) is a W G N process with double sided power spectral density equal to ^ and r(t) is the received signal. When non-orthogonal spreading sequences axe used for the T D and FD, the severity of the M A I is determined by the pairwise cross correlations of the TDS sequences {pij'.i.j = 1,2..... K}, and also the pairwise cross correlation of the frequency-domain spreading sequences {PVj] i.j = 1, 2..... K}, where M i l (2-19) faj = ii E Ci[m]cj[m] 777 = 0 21 In [16] and [17]. the authors investigated the performance of non-blind T F - M C - C D M A multiuser detection using this channel model and M-sequences was used for both the TDS and FDS signature sequences. 2.3.2 Downlink Mul t ipa th Rayleigh Fading Channel The multipath Rayleigh fading channel is modeled as a tapped-delay line linear time varying (LTV) filter [35;36]. This model is motivated by the fact that copies of the original signal arrive at the receiver with different delays and complex attenua- tions. Following the analysis given in [33] and [1], the statistical properties of the multipath Rayleigh fading channel can be described by the autocorrelation function B,hit,(t,t''.T.T') of the L T V channel impulse response h(t,r), where t is the time vari- able and r is the delay variable. Rhh(t, t', T, T') = E {h*(t, r)h(t\ r')} (2.20) The wide sense stationary (WSS) assumption states that the autocorrelation function depends only on the difference of the variables t and t' . i.e. Rhh(t, t + At , T, T') = Rhh(At, r, r') (2.21) where At = \t-t'\. In the frequency domain (i.e. the Fourier transform of the t variable), WSS means that contributions from different Doppler frequencies of the channel are uncor- related. For Gaussian process, WSS directly implies strict sense stationarity which is equivalent to stating that the statistics of the channel does not change with time. The uncorrelated. scattering (US) assumption states that the statistics of the different delays are uncorrelated, i.e. R,hk(t.,t'.T.T>) = PMl(t,t',r)5(r - r') (2.22) 22 where Phh{t.t',r) is called the delay cross power spectral density function. Equiv- alently. in a multipath fading channel, the US assumption states that paths with different delays are uncorrelatecl and that any path does not provide any information on another path. The tapped delay line (TDL) model is a popular model because it satisfies the assumptions of the WSSUS channel [1], Using the T D L channel model, the continuous time L T V channel impulse response of a multipath fading channel with L resolvable propagation paths is given by L - l h{t,T) = YJ9i(t)5{r-Tl) (2.23) 1=0 where gi(t), 77 represent the complex path gains and the delay of the Ith path. The time correlation of the complex path gains of the ltk path is given by their corre- sponding Doppler spectrum. To use the T D L model in computer simulation when doing analysis, the path delays must be converted or rounded to an integer multiple of the sampling time. Hence, in discrete computer simulation with sampling time Ts. the tapped delay line model channel response is L ' - l h(t,r) = ^2gi{t)5(T-lTs) (2.24) 1-0 where L' is the length of the channel impulse response in units of Ts. The Doppler spectrum in Rayleigh fading is obtained as following the derivation given in [1]. Suppose the mobile station is moving at speed v and the Ith path arrives in an incident angle 6. then it experiences a Doppler shift given by v=—cos(e) = fdcos{6) (2.25) c where fci denotes the maximum Doppler shift and is equal to It is further assumed that the arrival incident angle is uniformly distributed, i.e. Pe(e) = ̂ , # e ( - 7 T ; + 7 r ] (2.26) 23 Using 2.25 and 2.26 and doing a transformation of variables yields the probability density function for the Doppler shift u Pv(v) = - , , 1 m M = , 1 (2.27) Assuming that an ideal isotropic antenna is used and received power is uniformly distributed over all incident angles..hence the Doppler power spectral density, or the Doppler spectrum is given by * D ( V ) = ,A'1 (2.28). where o\ is the mean received power. The autocorrelation function is obtained by taking the inverse Fourier transform of the Doppler power spectral density, <p/(/),(At.) = olJ0(2irfdAt) (2.29) In Rayleigh fading, the gain on the path I, cji, is assumed to be zero mean complex Gaussian with variance afv Using this assumption, the envelope or the amplitude magnitude has Rayleigh distribution given by the P D F »('•) = 5 e x p (-4) (2.30) From the above discussion on Rayleigh multipath fading channel, four types of fading can be distinguished. The root mean square (RMS) delay spread measures the time dispersive nature of the channel. This is can be computed directly from the power delay profile. In practice, the delay of the first arrival path is set to zero and the delay of all other paths that arrives at the receiver later are all measured relative to the first path. The RMS delay spread is given by / = 1 (2.31) X X I 24 where al[l] is the mean received power of path / and 77 is the delay of path /. One can also define the coherence bandwidth Be being proportional to the inverse of the RMS delay spread Bc « — (2.32) Trm,s Then, with respect to the symbol period Ts (and system bandwidth Bs = ^-), two types of fading are differentiated. If Ts » r r m s or equivalently Bs << Be, then the channel is a flat fading channel or a frequency non-selective channel. If Ts « r r m s or equivalently Bs >> Bc, then the channel is a frequency selective channel. A flat fading channel is non-dispersive and only scales the transmitted signal by a complex gain. A frequency selective channel is dispersive and this dispersion induces intersymbol interference (ISI) The Doppler spread or Doppler bandwidth Bo is defined as the range of frequen- cies over which the Doppler spectrum is non-zero. In Rayleigh fading, the Doppler bandwidth is approximately equal to the maximum Doppler shift BD « fd = ^ (2.33) The coherence time Tc can be defined as being proportional to the inverse of the Doppler bandwidth Tc « (2.34) D With respect to the symbol period, one can differentiate two other types of fading. If Ts << Tc or equivalently Bs >> Bp. then the channel is a slowly fading channel. If Ts >> Tc or equivalently Bs << Bp, then the channel is a fast fading channel. In a slowly fading channel, the channel does not change much or can be assumed static over a symbol period or several symbol periods. In fast fading, the channel changes within a symbol period. Table 2.1 summarizes the four types of fading model. C H A N N E L T Y P E CONDITIONS frequency non-selective, slowly fading T s » Trms, Bs « Bc Ts « Tc, Bs » BD frequency selective, slowly fading Ts « rrms. Bs » Bc Ts « Tc, Ba » BD frequency non-selective, fast fading T s » Trms. Bs « Bc Ts » Tc, Bs « BD frequency selective, fast fading T s « Trms, Bs » Bc Ts » Tc, Bs « BD Table 2.1: Summary of channel fading models in mobile wireless channels 2.3.3 Cyclic Prefix Insertion and Removal of Block ISI In using O F D M based transmission, implementation using the IDFT and D F T mod- ulation and demodulation requires the insertion of cyclic prefix. This is crucial to the performance in frequency selective channels for any system that uses O F D M based transmission including T F - M C - C D M A . Without loss of generality, let us assume our channel is frequency selective and that the equivalent discrete time channel (after sampling our signal) has L taps, then it can be written as a discrete time varying finite impulse response (FIR) filter L-\ • h[k,n] = E # [ f c M n - l\ ( 2- 3 5) /.=o Then, the received O F D M block consisting of M transmitted symbols then will experience block ISI with the previous O F D M block with L overlapping symbols. The problem with block ISI is that the output, which is obtained by linear convolution with the channel impulse response, has a different value than the circular convolution between the signal and the channel impulse response. In O F D M implementation using IDFT and DFT, the circular convolution operation in the time domain is equivalent to multiplication operation in the frequency domain [37,38]. Ideally, we would like to have the circular convolution operation have the same output as the true linear convolution. To avoid block ISI and have circular convolution and linear convolution 26 [L ~ 1] - 4/- Figure 2.4: Cyclic Prefix Insertion to produce the same output, a prefix is inserted consisting of the L — 1 cyclically rotated symbols to our original O F D M data block as shown in Fig. 2.4. Note that we can also achieve the same result by zero padding instead of adding cyclic prefix. By using the cyclic prefix, we can write the output as the circular convolution of the input signal and the channel impulse response. Denoting d as our T original O F D M data block, h = then we have fjo 9L-1 as the channel impulse response. (2.36) y = d * h DFTM {y} = DFTM {d * h} = DFTM {d} o DFTM {h} where o denotes element by element multiplication and DFTM denotes the M-point DFT. From the above, one can define an equivalent frequency domain transfer func- tion H of the channel impulse response h M H = DFTM {h} = i-o 9i exp -j2iTml L-l 1=0 , M > L (2.37) 27 Figure 2.5: Passband Demodulator 2.4 T F - M C - C D M A Demodulation In this section, the demodulator used for demodulating TF-spread M C - C D M A sig- nal is discussed. The purpose of the demodulator is to extract the set of sufficient statistics from the observation data with no loss of information. The set of sufficient statistics at the output will be used for detection processing. In particular, we will discuss two equivalent demodulation process: (1) Bandpass model demodulation and (2) complex baseband model demodulation. 2.4.1 Passband Model Demodulation The system block diagram of the demodulator is shown in Fig. 2,5. From the demod- ulator system block diagram, the received signal (signal plus noise) coming from the demodulator output at the nUl time domain chip of the subcarrier is given by = I ° r{tyPTc(t - nTc) cos{uit)dt (2.38) J{n-l)Tc The noise component, specifically, of the output of the nth time domain chip epoch of the Ith subcarrier is given by piTc = n{t)PTc{t-nTc)cos{u)it)dt (2.39) J(n-1)TC The statistical properties of the noise term NJP are straight forward to derive using the A W G N properties of the noise process n(t). E {/v*W} = E \ n(t)PTc(t - nTc) cos(uj,t)d,i U(n-1)TC -nTc E{n(t)}PTc(t - nTc) cos(uj,t)d,t (2-40) ' ( n - l ) T c = 0 Furthermore, we have • n T c fjTc '(n-l)Tc J(j-1)TC x cos(oj/t) cos(u)mr)dtdT E {N®N}m)} = I"10 lJlC E (n(t)n(r)} PTe(t - nTc)PTc(r - jTc)x 1 J(n-l)Tr. J(i-1)TC 5(t-T)PTc(t-nTc)Prc(r-jTc)x (n-\)TcJU-l)Tc 1 x cos(o;/i) cos(umT)dtdT nTc rjTc AT / -rr&(t ~ T)8ni8mlPTc(t - nTc)PTc{r - jTc)x ( n - l ) T c J(j-1)TC Z x cos(cu/t) cos(umr)dtd,T = '-^SnjSml. / COS2 (UJ it) dt 1 J(n-l)Tc N0TC - y ^ ^ j ^ i i (2.41) Hence, at the output of the demodulator, we accumulate N time domain chips from each of M carriers for a total of MN buffered chips per transmitted symbol. 29 2.4.2 Equivalent C o m p l e x Baseband M o d e l Demodu la t i on The demodulation for the equivalent baseband model implemented using IDFT mod- ulation is accomplished using the inverse operation - the -DFT. Applying the M point IDFT to the sampled received signal given in 2.15, DFTM {sLP[n}} = - = £ SuM exp {^^) (2-42) n = 0 Expanding and applying the orthogonality principle yields M-\ / nr K CM-i r , / 'znrnn \ l \ x DFTM {sLP[n}} = b ^ E C * H exp [j^) x exp 7i.=0 \ k.=l {m=0 P sr^ 7 r i (l2n m - m ) n 7 7 z L E } ^ exp I .A:=| 77?.=0 n = 0 V -p /\ M-l -J^~ E E fefcafc.?cfcHM5mm/ fc=| 771=0 = v / P ^ E 6 f c « f c j C f c [???/] fc=i The sampled complex lowpass noise process is given by W[m] = DFTM { w M DFTM{w[n}}- M - l (2.43) (2.44) 1 N T ^ r n (-j2irmn\ -= } w [n exp — s/M ^ V M J 71=0 The statistical properties of the complex baseband sampled noise process are £ {VF[m]} = E E W " ] } e x P ,1=0 V 7 W / (2.45) = 0 30 Figure 2.6: Complex Baseband Modulator and Demodulator E {W[m]W[l]} = ± £ £ £ { W [ n ] « ; [ n ' ] } exp (̂ fp) exp ( - ^ ) 71=0 7 1 / = o ^ / \ - / = N05mI (2.46) Fig. 2.6 summarizes the complex baseband modulation and demodulation. 2.5 Summary In this chapter, the main principles of T F - M C - C D M A transmission and modulation are presented. The unique feature of T F - M C - C D M A is the realization of using two signature sequences for multiple access: one in the time domain (DS-CDMA), and the other in the frequency domain ( M C - C D M A ) . We have also provided an efficient TDS and FDS sequence pair allocation scheme that minimizes the number of inter- ferers in the each time domain group (and also for each frequency domain group). Furthermore, the characteristics and assumptions of the commonly used multipath fading channel model are reviewed. Finally, the demodulation for T F - M C - C D M A is 31 presented. 32 Chapter 3 B l i n d Mult iuser Detection for a T F - M C - C D M A Signal in A W G N 3.1 Introduction In this chapter, blind multiuser detection methods for T F - M C - C D M A systems in A W G N are described. The results of this chapter build on the work in [16,17] which uses separate time and frequency linear multiuser detection by using the clecorrela- tion detector and the M M S E detector when the receiver has perfect knowledge of the signature sequences of all users in the system as well as the noise power of the channel. Furthermore, the decorrelation detector and the M M S E detector considered are implemented using matrix inversion. It is shown in [17] that by using a two stage detection scheme, one for the time-domain spreading and the other for the frequency domain spreading, the size of the matrix to be inverted in order to compute the mul- tiuser detector weights can be reduced. The motivation for the work in this chapter is to extend the two stage non-blind linear multiuser detection proposed by [17] to a fully blind approach. To make a valid comparison between the our results and those in [16. 17], the work presented in this chapter retains the same transmission 33 and channel model (AWGN) as those in [16.17], but our work focuses on blind signal processing techniques which do not require the receiver knowledge of the signature sequences of other users or the noise power. Furthermore, we also investigate into more computationally appealing adaptive multiuser detection methods. The roadmap for this chapter is as follows: Section 3.2 describes the linear mul- tiuser detection problem formulation for T F - M C - C D M A . Section 3.3 discusses the algorithms for T F - M C - C D M A blind multiuser detection. Section 3.4 provides a dis- cussion of the performance measures and evaluations for blind multiuser detection methods for T F - M C - C D M A . Section 3.5 presents computer simulation results of the blind multiuser detection algorithms presented in this chapter. Finally, a summary of the chapter is provided in Section 3.6. 3.2 Linear Multiuser Detection for T F - M C - C D M A The purpose of using multiuser detection in T F - M C - C D M A is to mitigate the M A I in both the T D and FD and hence provide a better performance than that of con- ventional single user matched filter detection. It was shown in [17] that the exact bit error rate performance of the T F - M C - C D M A single user matched filter detector in A W G N channel is given by K p r { ^ M = j = T y - y q bne{-i.+\} 6A-e{-i,+i} 1 - E bkPlkfilk (3-1) k=2 where, without loss of generality,. it is assumed that the desired user is user 1. It is shown by computer simulation in [16] and [17] that the non-zero cross correlation {Pifc}jt=2 a n c l {Pik}'k=2 °f tb16 time and frequency domain signature sequences of the desired user and the other users cause the matched filter receiver performance to have a relatively high bit error rate (BER) floor. Proceeding from the passband demodulated signal given by 2.38, the discrete 34 signal output of the nlh sampled time chip from the mtk subcarrier stream of the T F - M C - C D M A passband demodulator in A W G N channel is given by = 5.,(7m) + N™, n = 1 , N : m = 1, ...,lVI (3.2) where I? P T h S™ = J~^hak[n]ck[m] (3.3) fc=i and it was shown in 2.40 and 2.41 that Nn'^ is a zero-mean Gaussian random vari- able with covariance ^-^fSnjdmi. After demodulation, the observation data for one information symbol is collected in the N by M matrix Z where the elements of Z are given by zmn = Z-m\ n=l,...,N;m = l,...,M (3.4) Based only on the observation data and the knowledge of user l's T D and FD signature sequence pair, our goal is to compute two sets of multiuser detection coef- ficients (one for the TD, one for the FD) such that the mean square error (MSE) is minimized. Mathematically, this is stated as ( \V ( .w* f ) = arg min E j\\Ab\ — ( w ' v Z ) * w y | | " | (3.5) where A = yj^r^ i g l n e energy of user one (other users also have the same energy). The optimal linear M M S E detector for T F - M C - C D M A is given by the solution to the optimization problem given in 3.5. 3.3 Adaptive Bl ind Multiuser Detection for T F - M C - C D M A In order to find a solution for 3.5, one must simultaneously solve for the time domain and frequency domain detector coefficients. This is generally a computationally com- 35 plex problem. This motivates us to propose an intuitive albeit suboptimal approach by decoupling the joint optimization problem into two separate optimization prob- lems, one for the blind T D M U D and one for the blind FD M U D . To do this, we first note that if one considers one particular subcarrier stream rn. then this corresponds to mLh column of the observation matrix Z. TSV^ " N[m) + N{m) _ N s(m) + n (m) (3.6) Substituting 3.3 into 3.6 yields A' A:=.l 2PTr M 2 (bkck[m])ak + n ^ (3.7) where ak ak[l] ak[N]] T is the time domain spreading chip sequence of user k and n''' is a zero mean Gaussian random vector with covariance ^f^flNxN- It might be noted that 3.7 is equivalent to a multiuser signal model in D S - C D M A in A W G N channel. The synchronous D S - C D M A signal model has the form [13,39] A" r[i] = Yl AkBk[i]sk + W W where Ak, Bk[i], sk are the received amplitude, the ith k=l transmitted symbol and signature sequence of the kth user respectively and w[z] is a zero mean complex Gaussian random vector. If one denotes bkck[m\ as Bk, then 3.7 can be interpreted as a D S - C D M A multiuser signal. Similarly, considering the nUl time chip epoch, one would get the nth row of the matrix Z and get the equations of the same form as 3.6 zj: = s ( l ) " o(2) On + MM) T T (3.8) 36 Time Domain M U D W Demodulator ourout RehvfZ ] w , •r requeney Domain MUD Figure 3.1: T F - M C - C D M A Multiuser Detector Architecture Again, combining with 3.3 we obtain ^ = E y i 7 T ( 6 f c a f c [ n ] ) C f c + n^ ( 3 - 9 ) where is a zero mean Gaussian random vector with covariance ^ ^ T v / x M - Denot- ing bkak[n] as Bk- then 3.9 can also be interpreted as a D S - C D M A multiuser signal. This result is significant because the joint T F - M C - C D M A blind multiuser detection problem is now split into two separate D S - C D M A blind M M S E multiuser detection problem and the}' can be simultaneously processed in parallel. Furthermore, because blind linear M M S E multisuer detection in D S - C D M A has been extensively studied, many techniques developed in past literatures can be applied. From 3.5 and 3.7. the optimization problem in 3.5 is decoupled into two D S - C D M A blind L M M S E multiuser detection problems, namely w * = argmin E j || A (bkck[m]) — w'fz^ ||2 } w ' e C J V X l f 2 1 (3.10) w * = argmin E \ \\A (bkak,[n]) - w'/zT\\ \ W / e C U x 1 1 ' > The block diagram for a detector for the decoupled problem is shown in Fig. 3.1. In Fig. 3.1. the sheer (the right most block in the diagram) threshold is set to zero. 37 It should be noted here that the decoupled D S - C D M A optimization problems in 3.10 can be modified, by considering both the received signal and its complex conjugate, to the Widely Linear' (WL) formulation which was shown in [40] to improve contemporary linear M M S E processing. However, in this thesis, although we only limit the analysis to linear M M S E processing, the application of the W L M M S E processing to the decoupling technique presented here is straight forward. Three different methods are now considered for solving the decoupled blind linear M M S E multiuser detection problem: direct matrix inversion (DMI), recursive least squares (RLS) and the blind subspace detection. 3.3.1 Direct Matrix Inversion (DMI) Method The direct matrix inversion method (DMI) for blind detection of D S - C D M A multiuser signals was discussed in [39]. The direct matrix inversion method involves computing the inverse of the data correlation matrix of the received signal. Using the analysis of the DMI method for D S - C D M A blind detection in [39], 3.10 can be re-written as 3.11 by expanding w t = argmm w t e C A ' x l wj — argmm w f E | Z M ( Z M ) W j w , _ 2 w f Re {A*E {z^ {bkck[m])}} w ? £ {z£ ( V / ) " } w , - - 2wf Re [A*E {zTn (bkak[n})}} vfeCMxl Setting the derivatives with respect to w t and w/ in 3.11 to zero yields 2C t w* - 2 | A | 2 a f c = 0 2 C / W } - 2 | A | 2 c f c = 0 (3.H) (3.12) since E{zW (bkck[m])} = Aak E{zZ(bkak[n])} = Ack (3.13) 38 Defining Ct, EE E ^ < » ) ( z ( m ) ^ > (3.14) the detector coefficients for user one axe then given by (3.15) w * = \A\-C7xan w*f = \A\" C f 1Cfc The matrices Ct and Cf can be estimated through their corresponding sample cor- relations from the received signal either through block sampling averaging or aclap- tively/recursively updated using a forgetting factor as in Ct\p,m] = XCt\p,m - 1] + (zW\p])(zW\p])H (3.16) Cf[p,n] = XCr[p,n - 1] + (zn\p]T)(zn\p]T)H where A is the forgetting factor. Ct[p,m] is the update of the time domain sample correlation matrix for the rnu> subcarrier stream, {rn : m = 1,2. .... M}. during the pih transmitted bit, {p : p = 0,1, 2,...}, and C/[p, ri] is the update of the frequency domain sample correlation matrix for the nth time chip epoch, {??, : n = 1,2, ...,N} during the pih transmitted bit. It can be noted that Ct[p, 0] is equivalent to Ct[p — 1, M] and similarly C/•[?;, 0] is equivalent to Cf[p — 1,N]. Note that the received power A only affects w* and through positive scaling. The decision of the estimated information bit is unaffected because using BPSK, the decision is computed as bi\p] — sgn (Re { ( w f ' Z ) * w / } ) -and is unaffected by positive scaling of the parameters w* and w^. Therefore, estimation of the received power is not needed for detection. Furthermore, the detector coefficients need not be computed every symbol time. In practice, for non-fading channels, the sample correlation matrices for both the time and frequency domain reaches convergence after a relatively few data symbols and re-computing the detector coefficients from the 39 / / Blind T F - M C - C D M A DMI Receiver in A W G N In the pUl symbol, input matrix Z[p] for(m = 1,2,...,M) update Ct[p,m] = XCt[p,m - 1] + (z^n){p})(z^[p})H end for for(n = 1,2,..., AO update Cf{p./n] = XCf p. n - 1] + (z n [p] T)(z n [p] T ) / / end for Compute Detectors w* = C^ax w*f = Cjlci Perform detection 61[p] = sgn(Re{(w t* t fZ)*w* f}) Table 3.1: Blind T F - M C - C D M A DMI Receiver Algorithm updates of the sample correlation matrices further beyond that point does little change to the performance of the detector. For fading channels, periodic re-computation of the detector coefficients are necessary clue to the time varying nature of the channel. The period between update would depend on the coherence time of the fading channel. The DMI algorithm is summarized in Table 3.1. 3.3.2 C M O E Recursive Least Squares (RLS) Method The main drawback of the DMI method is the need for inverting the sample correlation matrices C t and Cf. To avoid computing matrix inversions, the method of recursive least squares (RLS) can be used. The RLS method is based on solving constrained minimum output energy ( M O E ) / C M O E detector for blind multiuser detection in DS- C D M A first proposed in [14]. It is shown in [13] that the D S - C D M A C M O E detector and the blind linear M M S E detector optimization problems give solutions that differ only by a positive scaling constant. Therefore, minimizing the M O E is equivalent to minimizing the MSE. It is later shown in [18] that the C M O E detector can be generalized to problems that have more than one user signature code constraint. 40 Using the analysis given in [39]. the C M O E alternative representation of the T F - M C - C D M A M M S E decoupled optimization problem in 3.10 is given by w* = argmm E { | | w f z<"l> | | 2 j , w f al = 1 w t e c - < v x i ' ( 3 1 7 ) ŵ  = argmin E j ||w^z^||2 I , wf C i = 1 Based on the C M O E constrained optimization problem above, the RLS algorithm computes the detector weights according to the minimization of the sum of exponen- tially weighted mean-square output with a forgetting factor [39] wt* = argmin £ A ' ^ M ||wf z(m)|| , wf ai = 1 M . . . . . . | | 2 ( 3 - 1 8 ) w*f = argmin ]C ^ % | | w f z n | | > w f c i = 1 W / e c A ' x 1 11=0 The first summation in 3.18 states that the sum is over all subcarrier stream going from the zeroth subcarrier stream from the zeroth transmitted symbol to the Ith subcarrier stream of the pth symbol. Similarly, the second summation in 4.15 states that the sum is over all time chips going from the zeroth time chip from the zeroth transmitted symbol to the jUl chip of the pth symbol. The solution to the optimization problem in 3.18 (up to a real positive scaling constant) has a similar form to the DMI method w f = cr̂ i 1 (3.19) W*f = C T ^ l where again. CAp. ml = AC, [p. m - 1} + z<m)z(m>" 1 J L (3.20) Cf\p1n]=XCf\p,n-l]+zl(zl)ff To compute the detector, the RLS algorithm avoids computing the inverse of the correlation directly by using the matrix inversion lemma [39] to recursively estimate 41 / / Blind T F - M C - C D M A C M O E RLS Receiver in A W G N Initialization: $,;[0,0] = W , */[0,0] ^IMxM In the pth symbol, input matrix Z[p] for (m= 1.2 V ) Compute Kalman Gain: ki\p,m\ — — '• — l+A-' lz(m)[p]H* t[p,jn-l]z(m)[jrj] Compute Update: <&t[p,m] = A " 1 ($>f\j),m - 1] - k/ (z ( m ) [p]) 7 / <&t[p. m - 1] end for for'/; = 1,2,..., AT) Compute Kalman Gain: k f r „ „ 1 = A - ^ / [ p , n - l ] z n [ p ] r • / U i ' J H - A - | ( z , 1 . [ p ] T ) H * / [ p , „ - l ] z „ . [ p ] ^ Compute Update: $/[p,n] = A " 1 ($f\p,n - 1] - k/ (z^ p ] 7 ) " $/[p,n - 1] end for Compute Detectors w* = ^tax = $ f C i Perform detection 6:l[?;] = sgn(Re{(w; t f Z)*w*}) : 3.2: Blind C M O E RLS Receiver Algorithm the matrix inverse. Letting ~ Ct[j] 1 and <&/[.?'] ~ Cf[j] 1 •, the matrix inversion lemma gives d, r„-i _ \-i<r> r„- _ A - 2 $ t [ 7 - i ] z < " O z < " > " $ t [ j - i ] ^ t U J - A q - M J 1J i + A - i z ( » ' ) « $ t [ ; i - i ] z ( " ' ) , 3 2 1 x where j is the iteration index. The RLS algorithm has a much faster convergence than the least mean squares (LMS) method using gradient descent method and is therefore less sensitive to the choice of initial values [39]. The T F - M C - C D M A blind M O E RLS multiuser detection algorithm is summarized in Table 3.2. 42 3.3.3 B l i n d Subspace-Based M e t h o d The method of subspace blind multiuser detection for D S - C D M A was first proposed in [20]. The idea is to decompose the autocorrelation matrix into a signal subspace and a noise subspace through eigenclecomposition. To see this, we consider the decoupled multiuser detection problem for the time domain (the exact analysis holds for the frequency domain multiuser detection problem) C , = E { z « (z<'>)"} = St | A | 2 S f + a*INxN (3.22) where [ai|a2|...|aD-f r 2 A'o TC (3.23) | A | 2 = diag(A\,.... A2Ut) and Ui, represents the number of linearly independent TDS signature sequence used in the system, and Ai represents the superposition of the amplitudes of all users that uses the TDS signature i. i = Ut. It is assumed that the spreading signature sequences in the time and frequency domain form linearly independent sets. This is usually not a problem in practical applications because the transmitter and receiver can be coordinated to use sequence sets that are linearly independent. On average, the effect of the channel does not affect the linear independence of the sequence sets. Assuming this condition is satisfied, then the eigendecomposition of the autocorrelation matrix can be written as (3.24) where JM̂ }. a r e t r i e Ut largest eigenvalues of C t in descending order, the N by Ut matrix contains the corresponding Ut eigenvectors as its columns and the N by N — Ut UJ? contains the Ar — Ut eigenvectors corresponding to the eigenvalue of. 43 The blind L M M S E detector is obtained by substituting the subspace decompo- sition of Ct into 3.19. i.e. w* = C ^ a , = ( U ^ u F ' + ^ I f ' u f l A = U ^ C A W j ^ U ^ a x + ^ U ^ a , (3-25) = u i i ) ( A i t ) ) U i t ) W a 1 where the last equality holds since the eigenvectors contained in the matrix span the signal subspace which means that the vector a x can be expressed as a linearly combination of the eigenvectors in the matrix ui*') and therefore (U<<>)"ai = 0 (3.26) Using a similar analysis for the FD case in the decoupled multiuser detection problem yields w} = U ( / ) ( A . ( / ) ) - 1 U i / ) / / c 1 (3.27) Hence, the blind L M M S E detector for both the FD and T D detection can be computed once Tj£°, A i 4 ) , and A . l 7 ) are estimated. Two methods are shown in [20] and [39] to estimate the subspace parameters, namely, singular value decomposition and adaptive subspace tracking using PASTd algorithm. S i n g u l a r V a l u e D e c o m p o s i t i o n The subspace parameters can be tracked by applying the singular value decomposi- tion of the sample data matrix, or equivalently. the eigenvalue decomposition of the sample correlation matrix. The sample correlation matrix can be obtained through block sampling averaging or recursively updating using an exponentially weighted window with a forgetting factor as in 3.20. As in the DMI method, the subspace parameters need not be computed every symbol duration. In practice, for non-fading channels, the sample correlation matrices for both the time and frequency domain 44 / / Blind T F - M C - C D M A Subspace (SVD) Receiver in A W G N In the pth symbol., input matrix Z[p] for(m = 1,2, . . . ,M) Compute Update: C t[p,m] = XCt\p,m - 1] + {z{m)\p)){z^\p])H end for for: „ = 1. 2,.... N) Compute Update: Cf[p,n] = XCf\p,n- 1] + (Zn[p]T)(zn[p}T)H end for Apply Eigenvalue decomposition to C t and C / to get eigenvectors and eigenvalues: Compute Detectors w* = U ^ W V W ) / f c : l Perform detection ^[p] = sgn (Re { ( w ^ Z ) * w * f } ) Table 3.3: Blind Subspace SVD Receiver Algorithm reaches convergence after a relatively few data symbols and re-computing the sub- space matrices and detector coefficients from the updated sample correlation matrices further beyond that point does little improvement to the performance of the detector. For fading channels, periodic re-computation of the detector coefficients are necessary clue to the time varying nature of the channel. The period between update would de- pend on the coherence time of the fading channel. The T F - M C - C D M A blind subspace algorithm using SVD is summarized in Table 3.3. Adap t ive Subspace Tracking U s i n g P A S T d A computationally more attractive algorithm for estimating the subspace parame- ters is adaptive subspace tracking. In addition to its computational advantage, it is adaptable to changes in channel conditions even when the channel is time varying and thus more robust in practical applications. The PASTd algorithm was first pro- 45 posed by [22] and first applied to D S - C D M A systems by Wang [30]. The concept of adaptive subspace tracking can be summarized as follows [22] [24]. We consider the optimization problem of the scalar function J(W) = E{\\r[i] - WW"r[i] || 2} (3.28) where v[i] is a noisy observation vector with correlation matrix given by C r = E {r[z]r['i]w} and W £ CNxq,q < N. There are two important properties of this optimization problem: 1. W is a stationary point (the point where the gradient is zero) of J(W) if and only if W can be factored as W = U r Q where U r is a N by q matrix that contains any q distinct eigenvectors of Cr, and Q is any q by q unitary matrix. 2. Al l stationary points of J(W) are saddle points only except the case when U r contains the q dominant eigenvectors (corresponding to the q largest eigenval- ues) of C r . In that case. J(W) attains the global minimum. Here, we are interested in the case when q = 1. Then the solution to the op- timization problem is simply given by the eigenvector corresponding to the largest eigenvalue. However, in practice, we only have access to the observation samples r[i] and not G r : hence, we modify the original optimization problem into the exponential weighted sum with a forgetting factor 7 i J(W) = Yi'~n \\r[n] - W[z]W[i]"r[n]||2 (3.29) The PASTcl algorithm has two main components: (1) Projection approximation and (2) deflation. The projection approximation is W[i]wr[n] ~ W[n - l]Hr[n] = y[n], n = 1, 2, ...i (3.30) Then, applying the approximation, (3-29) becomes i J(W) « J(W) = ||r[n] - W[*]y[n]||2 (3.31) n-0 46 The RLS algorithm can once again be applied to solve for W[z]. which, by property (2) above, is an approximation to the eigenvector of C 7 . corresponding to the largest eigenvalue. The deflation technique can now be applied. Once an estimate of the dominant eigenvector is obtained, the projection of the current observation vector r[i] onto it can be removed using a Gram-Schmidt like procedure and the problem is now "deflated" or dimensionally reduced. Thus, if we repeat the same procedure (i.e. applying the RLS algorithm again and solving for W[z]), we would get an estimate of the next dominant eigenvector. Once we have the estimates of the eigenvectors and eigenvalues, the blind subspace LMiVISE detector can be computed. The algorithm is summarized in Table 3.4. 3.4 Performance Measure The most commonly used performance measure for a multiuser detector is the bit error rate (BER). In T F - M C - C D M A , parameters such as the TDS factor and FDS factor also play a significant role in determining the overall system performance and efficiency in resource utilization. Another measure for blind multiuser detectors is the output signal to interference plus noise ratio (SINR). This is computed as the ratio of the desired signal to the sum of the total interference plus ambient noise. For T F - M C - C D M A linear multiuser detectors, the output SINR is defined as SINE,(wt, w f = — - — H r , —irnvx 3-32 E{Var ((w,"Z[2;])*w/|6][p]) } From the output SINR information, we can evaluate the convergence speed for mul- tiuser detection algorithms. Also important measures are how B E R performance and the output SINR, changes with the number of users in the system. To summarize, bit error rate performance evaluated with multiuser capacity, output SINR,, resource utilization efficiency and rate of convergence are important in designing multiuser ' 47 / / Blind T F - M C - C D M A Subspace (PASTcl) Receiver in A W G N In the pth symbol, input matrix Z[p] and previous estimates for(m = 1 , 2 , M ) Input z ( m ) [p] Set x[° = z('n)[p] for(fc = 1,2,...,[/,,) Projection Approximation: yf* = v^\p,m — l j ^ x ^ Eigenvalue Update: AJ^ [p, m] = ̂ \k^[p.m, — 1] Eigenvector Update: Vk vSkl)\p,m] = u^ ;[p,m - 1] + r,(*)r Deflation: x (0 y,, (0* 4° - UfcV,™]?/? 4 ° - u ^ b v m - l]:(/{0 e n d for e n d for for(n = 1 . 2 . . V ) Input zn[p\r Set 4f) = zn\p]T for (A; - 1.2 I\ Projection Approximation: yyk ;(/) „ ( / ) u ^ n - l ] ^ Eigenvalue Update: A^fp , n] = 7AJ^ [p,n — 1] + Eigenvector Update: u p. ?7,] = uh [p, n — 1] + Deflation: x).;+ 1 (/) _ „ ( / ) xk u[P\p,n]yl vifh X[f){p,n] ,t„(/)' x ^ - u ^ n - l ] ^ ,(/) e n d for e n d for Compute Detectors wr = tjW(Ai t ) )- 1 tj i t ) H a 1 w* f = U ( / ) / A ( / ) ^ l T j ( / ) f f C l W ) Perform detection bi\p] = sgn (Re {(w: w i l l Table 3.4: Blind Subspace PASTcl Receiver Algorithm 48 receivers. 3.4.1 Receiver Complexity In this section, we briefly discuss the complexity of the blind receivers for T F - M C - C D M A in A W G N proposed in this chapter. The DMI receiver requires two matrix inversions: one for the time domain and one for the frequency domain. The size of the sample correlation matrices are N by N and M by M for the time and frequency domain respectively. Using normal Gaussian elimination to find the matrix inverse, the complexity is 0{N3 + M3). The C M O E RLS algorithm has a complexity of 0(N2) per update for the time domain detector and 0(M2) per update for the frequency domain detector. However, in our algorithm, there are M updates per symbol for the T D detector and N updates per symbol for the FD detector. Therefore, the total complexity for the C M O E RLS algorithm is 0(NM2 + MN2). The subspace SVD involves computing the singular value decomposition on the received data, or equivalently the eigendecomposition of the sample correlation matrices for both the time and frequency domain. The worst case complexity for eigendecomposition for the time domain is 0(N3) and for the frequency domain is 0(Mi). Therefore, the total complexity for the blind subspace SVD method is 0{M:i + A^ 3). The adaptive subspace tracking algorithm using PASTcl has a complexity of 0{NN) per update for the time domain where N denotes the number of time domain signature/groups used. Therefore, the maximum complexity occurs when N = N (when all available time domain sequences are used) and the complexity will be 0(A[2) per update for the time domain. For the frequency domain, the complexity is 0(xnM) where Xn denotes the number of users within the nth time domain group or using the nth time domain signature sequence. Under a fully loaded system, the adaptive subspace tracking multiuser detection using PASTd is then 0(N2 + M2) per update. Suppose 49 Multiuser Detector Complexity DMI 0(N'A + M:i) C M O E RLS • 0(NM2 + MN2) Subspace SVD 0{N:i + M 3 ) Subspace PASTcl 0{MUtN + NXnM) Table 3.5: Summary of T F - M C - C D M A Blind Multiuser detection complexity for A W G N channel T D spreading gain 7 FD spreading gain 15 Number of subcarriers 15 TDS sequence type M-sequences FDS sequence type M-sequences Table 3.6: Simulation parameters the number of TDS sequence used is Ut and Xn denotes the number of users within the nth T D group, then the total complexity for the blind subspace PASTcl algorithm is 0(MUtN + NxnM) per symbol. The complexity of the algorithms are summarized in Table 3.5. 3.5 Simulation Results Computer simulations were carried out to illustrate the performances of the four types of blind detectors for T F - M C - C D M A discussed in this chapter: the DMI method, the M O E RLS method, the subspace SVD method, and the adaptive subspace tracking PASTcl method. The parameter values chosen are listed in Table 3.6. The number of users used in the simulation reflect three levels of loading in a typical T F - M C - C D M A system: • Case 1: 49 users - moderate load • Case 2: 77 users - moderate heavy load 50 • Case 3: 105 users - full load Figures 3.2, 3.3 and 3.4 show the B E R performance for each case. Not surprisingly, the DMI method gives the best B E R performance. The subspace method using SVD can achieve almost the same BER, as the DMI method. The adaptive subspace tracking al- gorithm using PASTcl has a slightly degraded performance: the degradation increases slightly with loading and at a target BER, of 10~3, the penalty is approximately l dB for 105 users. It can be seen that the adaptive subspace tracking outperforms the C M O E RLS receiver with comparable complexity. The difference or gap between the performance of the C M O E R.LS receiver and other blind multiuser receiver consid- ered, namely the DMI, blind subspace SVD, and the blind subspace PASTcl detector, increases with system loading. In a fully loaded system with 105 users, the differ- ence in performance between the adaptive subspace tracking receiver and the C M O E RLS is more than 6clB at a target BER, of 10~4. Nonetheless, comparing with the matched filter detector, the C M O E RLS blind multiuser receiver remains a much better alternative. The output SINR, curves as a function of the number of data smaples are shown in Fig. 3.5, 3.6 and 3.7 and agree with the B E R performance results. The DMI and the subspace SVD methods have almost identical output SINR curves. The output SINR, of the adaptive subspace tracking receiver using PASTcl is approximately l d B less than that of the DMI and subspace SVD receiver for the 49 and 77 user case and slightly greater than ldB less for the 105 (full capacity) users case. The output SINR curves also show the convergence rate of the proposed receivers. It can be seen that the C M O E RLS converges the fastest, the output SINR reaches stabilizes after approximately 60 data samples. The convergence of the DMI and subspace SVD are similar: they reach convergence after approximately 200 samples. The adaptive subspace tracking using PASTcl has the slowest convergence. The result shown for 51 N = 7, M = 15, 49 Users (moderate load), A W G N Channel 10 - B — DMI - 0 — C M O E RLS - © — subspace (SVD) V subspace(PASTd) matched filter 10" 1 0 - 4 | i J \ \ \ I 0 5 10 15 20 SNR (dB) Figure 3.2: B E R Performance with 49 Users the PASTcl tracking has been accelerated by using doing an SVD decomposition on the first 50 data samples. Even with the acceleration, the PASTcl tracking algorithm has the slowest rate of convergence. Finally, Fig. 3.8 and 3.9 shows the B E R and output SINR as a function of the number of users in the system. In Fig.,3.8, it can be seen that the B E R performance deteriorates fairly quickly going from 5 users to about 45 users. Beyond 50 users, the B E R degradation stabilizes to a steady increase. Fig. 3.9 shows similar results that the system output SINR degrades fairly quickly from 5 users to 45 users and then stabilizes to a steady degradation as the number of users in the system increases. 52 10u N = 7, M = 15, 77 Users (moderate heavy load), AWGN Channel DMI CMOE RLS subspace(svd) V subspace(PASTd) matched filter 4 -e CC _ 2 LU 10 CO 10" 1 c p l 1 1 1¥—* ' ^ 1 0 5 10 15 20 SNR (dB) Figure 3.3: B E R Performance with 77 Users 3.6 Summary In this chapter, the blind multiuser detection problem for T F - M C - C D M A in A W G N was investigated. It was shown that the blind L M M S E detector for T F - M C - C D M A can be suboptimally decoupled into two D S - C D M A blind multiuser detection prob- lems, one for the frequency domain, and one for the time domain. Three methods for solving the decoupled multiuser detection problem, namely the DMI method, C M O E RLS method, and the subspace method were studied. For the subspace multiuser detection method, two implementations were considered: the singular value decom- position method and the adaptive subspace tracking method using the PASTcl algo- rithm. Finally, simulation B E R curves for the different multiuser detectors studied were presented and discussed. 53 N = 7, M = 15, 105 Users (Full Load), AWGN Channel S — DM ^ — C M O E RLS G— subspace(SVD) V - subspace(PASTd) 10 SNR(dB) F i g u r e 3.4: B E R Per formance w i t h 105 Users 54 N = 7, M = 15, 49 Users (moderate load), AWGN Channel, Output SINR 500 1000 1500 2000 Number of Data Samples F i g u r e 3.5: O u t p u t S I N R w i t h 49 Users at S N R l O d B 55 N = 7, M = 15, 77 Users (moderate heavy load), A W G N Channel, Output SINR I I DMI • _i ~~ ^ 1 1 subs pace (PASTd) : / / / :CMOE RLS /subspace(svd) l i i 0 500 1000 1500 2000 Number of data samples F i g u r e 3.6: O u t p u t S I N R w i t h 77 Users at S N R lOc lB 56 N = 7, M = 15, 105 Users (full load), AWGN Channel, Output SINR DMF~-L___^ ; / / subspace(PASTd) '• I / C M O E F =iLS 1 subspace(svd) / i i 0 500 1000 1500 2000 Number of Data Samples F i g u r e 3.7: O u t p u t S I N R w i t h 105 Users at S N R l O d B 57 Figure 3.8: B E R performance plotted against the number of users at SNR lOdB •58 N = 7, M = 15, A W G N Channel, Output SINR vs. Number of Users 10 CO C/0 13 el-'s O -0— C M O E RLS -0— subspace (SVD) V subspace (PASTd) 10 20 30 40 50 60 70 number of users 80 90 100 110 Figure 3.9: Output SINR, plotted against the number of users at SNR lOdB 59 Chapter 4 T F - M C - C D M A B l i n d Mult iuser Detection in Slowly Fading Rayleigh Mul t ipa th Channel 4.1 Introduction The distinct advantage of using multicarrier based transmission is to combat fre- quency selective fading by transmitting in multiple subcarrier streams such that each stream experiences frequency non-selective fading. In this chapter, the channel is assumed to be varying slowly with respect to overall transmission rate. Rayleigh fading multipath channels are common in urban mobile communications where line of sight signals are often not available and all received energies are from reflected and scattered signals. In this chapter, blind linear multiuser detection algorithms are proposed for T F - M C - C D M A system in downlink multipath fading channels. In par- ticular, the focus is on the design of low complexity receiver algorithms suitable for implementation in mobile handsets. In most realistic situations, the mobile receiver may only know its own signature sequence and does not know the channel without 60 using training sequences or pilot symbols. In this chapter, we discuss blind multiuser detectors for T F - M C - C D M A where the receiver has no knowledge of the channel information and has only knowledge of its own T D and FD signature sequences. This chapter is organized as follows: Section 4.2 reviews the signal model in down- link slowly fading Rayleigh multipath channels in a T F - M C - C D M A system. Section 4.3 presents the T F - M C - C D M A blind multiuser detection problem in slowly fading Rayleigh multipath channels. Section 4.4 discusses the blind multiuser detection al- gorithms. Section 4.5 reviews performance measures. Section 4.6 provides computer simulation results. Finally, section 4.7 concludes with a summary of the chapter. 4.2 Linear Multiuser Detection for T F - M C - C D M A in Frequency Selective Fading Channels An important motivation for using multicarrier based transmission is to maintain a robust system performance in frequency selective channels. This is achieved through transmission in multiple narrowband subcarriers such that each subcarrier stream experiences frequency non-selective or flat fading. T F - M C - C D M A systems inherit this characteristic that is also present in M C - C D M A , O F D M and M C - D S - C D M A systems. A T F - M C - C D M A signal in frequency selective downlink fading channel model with L paths represented by a tapped-delay line time varying channel impulse response h(t, r) is considered where gi(t) is the time varying complex gain of path I and Tc is the chip duration. Referring to the analysis in Section 2.3.3, the sampled received signal can be written in terms of the discrete time convolution of the transmitted signal and the discrete (4.1) /=i 61 time channel response given by 'i'Lp[n-; k] — SLp{n] * h[k, n] + w[n (4.2) where s^pfn] is given by 2.15. w[n] is a complex W G N random variable with variance N0, and h[k, n] is the equivalent discrete time channel impulse response of h(t. r) given by 2.35. The discrete time demodulated equivalent baseband signal is given by the D F T of 4.2 DFTM {/-,/->• k]} = DFTM {sLP[n\ * h[n, k}} + DFTM {w[n]} (4.3) Without using multicarrier modulation, the received signal would experience ISI as a result of the frequency selectivity of the channel. However, in multicarrier modulation, the use of I F F T / F F T implementation and the insertion of a cyclic prefix (as discussed in Section 2.3.3) eliminates the ISI between O F D M blocks. The insertion of a cyclic prefix (longer than the delay spread of the channel) enables the F F T demodulation process to be equivalent to a circular convolution between the signal and the channel in the time domain [37, 38]. Hence in the frequency domain, under the circular convolution theorem, the D F T of the transmitted signal and the D F T of the discrete time channel response multiplies. At the F F T demodulator output, each subcarrier stream is a flat faded signal. The complex channel gain for each subcarrier stream is obtained by computing the D F T of the discrete channel impulse response. Using this fact, along with 2.37, 2.42, and 2.43, the demodulated downlink T F - M C - C D M A signal in a frequency selective channel can be expressed as K Zn[m] = \ / p y ^ bkak[n]ck[m]H[m] + VF„.[m], m = 1, 2,.... M (4.4) A;=l or equivalently Zn[m] = H[m]Sn[m] + Wn[m], rn 1.2 M (4.5) L-l 62 Based on only the knowledge of the desired user's T D and FD signature se- quences, we propose blind linear M M S E multiuser detector that performs blind de- tection for the T F - M C - C D M A signal and jointly estimate the channel. 4.3 Bl ind Multiuser Detection in Slowly Fading Channels The decoupling method proposed in Chapter 3 for blind linear multiuser detection for T F - M C - C D M A signal in A W G N can also be applied in the case for the slowly fading Rayleigh multipath channel. Similarly to the blind multiuser detection problem for the A W G N channel with M A I case, the full discrete time T F - M C - C D M A demodu- lated signal naturally decouples into one which involves only the T D spreading and one that only involves the FD spreading. The m t h column of Z gives the T D multiuser detection problem z[m] = \/PH[m} E {bkCk[m}) ak + W[m], m = 1, 2 , M The nth row of Z gives the FD multiuser detection problem (4.6) where / ' E (bkak[n}) ck + W,'/. n = 1, 2 , N k=l ck = diag(ck)B. H = H[l] H[2] Ck[l] H[M] d,iag(ck) ck[M] (4.7) (4.8) It can be noted here that, more compactly written, the channel impulse response g and its D F T H is related by H = FMxLg where FMXL is the M by L Fourier matrix. 63 Matched filtering can be applied for the T D multiuser detection if orthogonal codes are used. Examining 4.6, the orthogonality of the T D signature sequences is preserved when the channel is fading slowly with respect to the symbol duration. On the downlink, the channel is synchronous in nature and thus warrants the use of orthogonal signature sequences. Two types of detection, type I and type II detection, are proposed and their performance are investigated. Our proposed type I detector uses separate "parallel" time and frequency domain processing through decoupling where the output of the T D M U D does not feed into the FD M U D or vice versa. The proposed type II detector uses a cascade implementation where the output of the T D processing is input into the FD processing. In the next two subsections, type I and type II detection are discussed. 4.3.1 Type I Receiver The implementation of our proposed type I receiver uses the decoupled T D and FD processing introduced in Chapter 3 with the exception that the matched filter can be applied to the T D detection since orthogonality is preserved in the time domain for slowly fading channels. Therefore, for the T D detection, matched filtering is applied with the desired user's TDS sequence. Fig. 4.1 shows the system block diagram of the type I receiver. The frequency domain M U D is computed through the observation data given by the rows of the observation matrix Z in 4.7. The blind L M M S E for type I FD M U D solves the optimization problem J (4-9) z I = \ ^ E (MfcH) ck + n£ A:=l Re-examining 4.9, if one let Bk = bkak[n] and realizing that ck = diag(ck)H is the effective FD signature, then 4.9 is equivalent to the blind L M M S E multiuser W j — argmin E P(bkak[n]) - w'/xl 64 W, - IK xrequency domain M U D RefafZjW,}- Channel estimation Figure 4.1: Type I detector architecture optimization problem for M C - C D M A in a slowly fading frequency selective channel. The M C - C D M A multiuser signal model was investigated in [22,25,41,42]. 4.3.2 Type II Receiver The implementation of the type II receiver uses a cascade architecture in which the T D matched filtering is first applied and then the output is input into the FD multiuser detector. Matched filtering of the desired user's time domain signature is applied at the first stage. The motivation for processing the T D detector using the matched filter first is to filter out interferers from other time groups (interferers that does not have the same T D signature as the desired user). Fig. 4.2 shows the system block diagram of the type II receiver. The type II receiver implementation takes advantage of the preserved orthogonality of the TD signature sequences to reduce the number of interferers for the FD detection. Reversing the process where the FD detection is computed first does not achieve the same result. The reason is that matched filtering can not be applied to the FD detection because orthogonality of the FDS is lost clue 65 * z z a f Z • frequency domain M U D r , ,f ..H A • R e ^ z j _J Channel Estimation Figure 4.2: Type II detector architecture to the effect of the channel. As a result, it is not possible to filter out interferers that lmve different FDS than the desired user. The equivalent multiuser signal model using T D signature grouping given in Section 2.2.3 is useful in the analysis of type II receivers. At the output of the demodulator, the elements of the N by M observation matrix Z using the T D signature grouping can be written as A' Xj Zn [m] = ^PJ2Y1 b^a» [J] ̂  H H H + Wn [m], a - L . . . ; . V m = 1,..., M ; , = 1 K=l In vector notation. 4.10 is rewritten as A' Xj z[rn] = \/~P bjKcK\m\H[rn]sL.j + W[m .7 = 1 « = The inner product of 4.11 with user l's T D signature sequence yields afz[m] = z'[m N Xj ^Y, E bjn cK [m] H im] a f a ? + a f WH i=:i K=I XI = \ / P //jm (6 1 k C ( C [m]) + w' [m], m = I....., M K . = l Equation 4.12 is simplified further in vector form to get z'[l] z = z'llVI] XI K=l (4.10) (4.11) (4.12) (4.13) 66 where biK denotes the transmitted symbol of the user with T D signature sequence 1 and FD signature sequence K. We assume our desired user has T D signature se- quence number 1 and FD signature sequence number 1. The blind L M M S E detector optimization problem for the type II FD stage detection is thus given by Comparing with 4.9. the FD blind L M M S E multiuser detection problem given by 4.14 has the same form and is equivalent to blind multiuser detection for M C - C D M A mul- tiuser signal in slowly fading frequency selective channel. Applying matched filtering for the FD M U D is inappropriate in this case because the orthogonality of the fre- quency domain spreading sequences is lost due to the effect of the channel. Therefore, blind multiuser detection techniques must be used for the FD processing. However, the C M O E RLS and blind subspace methods used for solving the decoupled T F - M C - C D M A multiuser detection problem in A W G N in Chapter 3 can be applied to the FD processing for both type I and type II detectors but with added complexity as a result of the need for joint blind channel estimation that must be performed in order to obtain the blind L M M S E detector for the FD multiuser detector. The type I and type II FD multiuser detection problem have identical forms comparing 4.9, 4.13 and 4.14. If the system has K active users and K < PI, then it is possible that all users have different FD signature sequence because for spreading sequences of length PI, there can be at most PI linearly independent sequences. Therefore, for K < PI, the type I FD detector is required to suppress K — l interfering FD signature sequences. However, for the type II FD detector, it is only required to suppress Xi ~ 1 interfering FD signature sequences where Xi 1S the number of users sharing the same T D signature sequence as the desired user. By construction and definition of xi- Xv < K- If K > PI, then there will be some users that share the same FD signature se- (4.14) 67 quence and this means that all M linearly independent FD signature sequences are used in the system. Therefore, for the. type I FD detector, it is required to suppress M — 1 interfering FD signature sequences. However, for the type II FD detector, it is again only required to suppress xi ~ 1 interfering FD signature sequences. Further- more, it is easy to see that xi < M. To summarize, since the type I and type II detector FD M U D problem differ only in the number of users, we can then solve both problems using the same methods assuming the FD M U D problem has [/-users (for type I, U = M: and for type II U = Xi if K > M and U = K for type I and U = Xi for type II if K < M). To solve the FD M U D problem, two methods used in Chapter 3 are revisited and applied to the T D M U D problem, namely, the C M O E RLS method and the subspace blind detection method. 4.3.3 C M O E R L S Bl ind Detection From our analysis in the previous section, we have shown that the T F - M C - C D M A FD blind linear M M S E M U D problem for both type I and type II (differing only in the number of interfering FD signature sequences) is equivalent to blind linear M M S E M U D for M C - C D M A signal. It was shown in [22] that the C M O E RLS algorithm can be applied to M C - C D M A signal. We apply the C M O E RLS algorithm to the FD M U D of our proposed type I and type II detectors for T F - M C - C D M A signal and investigate its performance in slowly fading Rayleigh multipath channel. Using the C M O E RLS blind detection algorithm, we seek a solution for the blind L M M S E multiuser detection problem given by 4.14. Using the C M O E criterion formulation, the same C M O E RLS optimization problem is arrived as presented in Section 3.3.2 with a slight modification in the constraint where the constraint is now on the effective FD signature sequence of the desired user instead of the original FD 68 signature sequence. The type I detector frequency domain C M O E RLS M U D problem is given by M w} = argmin V A ^ 1 - " ||w ẑr||2, w f c j = 1 (4.15) where C i = diag(ci)H is the effective signature waveform and j[p] denotes the jth time chip of the pth transmitted symbol. The summation in 4.15 states that the sum is over all time chips going from the zeroth time chip from the zeroth transmitted symbol to the jth chip of the pUl symbol. The solution to the C M O E RLS optimization problem is given by 3.19. i.e. w*f=(c?Cfciy1C]'c1 (4.16) where the data correlation matrix is given by the recursive updates from the columns of Z. namely Cf[p, n] = AC/[p. n - 1] + z'M)11. n= 1..... 7Y (4.17) Recall that Cr[p,0] is equivalent to C / [p — 1, N}. The type II C M O E RLS M U D problem is given by p w* = argmin V A h | | w f z'[i]||2, wj'c;, = 1 (4.18) w / e C M x ' . = 0 The solution of 4.18 is also given by 4.16 where now the data correlation matrix is given by the recursive updates from the output of the T D matched filter. C / W = E A"-V[i]z'[i]" =XCj[P - 1] + z'\p]z'\p]H (4-!9) • Applying the C M O E RLS algorithm presented in Chapter 3 for the T F - M C - C D M A signal in A W G N . the inverse of the correlation matrix is approximated using the matrix inversion lemma and is given by 69 where j denotes the iteration index arid q = zJn for type I detection and q = z ' for type II detection. To complete the'solution, it is necessary to solve for the effective FD signature C i = G?zag(ci)H. Since the user only knows its own FDS signature and not the channel, the channel must be jointly estimated in order to perform the detection. Once the channel parameters are estimated, the FD blind multiuser detector is given 4.16 for both type I and type II detectors. B l i n d Channe l Es t ima t ion For the blind channel estimation for the FD M U D detector using the C M O E RLS algorithm, we make use the analysis for M C - C D M A blind channel estimation given in [22]. From the C M O E solution given in equation (4-16), the type I detector minimum output energy (MOE) is given by MOE,{w}) = E{ j v/}Hzl f } (4.21) Similarly, the M O E for type II is given by MOEn{w}) = E{ | W * / V | 2 } (4.22) Expanding either 4.21 or 4.22 yields the same result, i.e. MO£(wJ) = E{W<zl\2} • /-{(w^zD ( w * " ^ = £ { w ; " z M " w ; } = w f / 5 K ( z ^ } w } = W y C f\V I Substituting the C M O E solution given by 4.16 into 4.23 yields H r i (4.23) MOE(w}) = [ ( c f C y ^ i ) C y ' c r J C / ( ( c f C j ' c i ) ' Cj% 70 (4.24) It was shown in [22] that the M O E expression given in 4.24 can be maximized with respect to the effective FD signature sequence of the desired user C i to estimate the effective FD signature, and in the process, the channel. Using this approach, 4.25 gives the optimization problem for the channel estimation, namely argmax {MOE{w))} = argmax [ ( c f C ^ C i ) _ 1 } (4.25) Ci Cl -1 Equivalently, one can minimize the reciprocal of 4.25 and therefore, the estimated effective signature is given by ai = a igmin{ (c fC7 1 c 1 ) } (4.26) C] Using the fact that C j = C i H where C i = diag(ci), H = FMxLg and the RLS esti- mate <&j ~ Cj1, the channel response can be estimated by solving the optimization problem in equation (4-26), applying the substitutions expanding (4-26) yields g = argmin { g ^ F ^ C f ^ C i F ^ g } (4.27) g For a normalized channel impulse response g (i.e. | | g | | = 1), the solution, unique up to a multiplicative complex constant, to 4.27 is the normalized eigenvector correspond- ing to the smallest eigenvalue of the matrix F f / x L C f <J>/-CIFA,/XL- The uniqueness of the solution is guaranteed if two conditions are met: (1) the composite matrix [ci|c2|...|c/<] has rank equal to the number of linearly independent FD sequence used in the system U\ (2) the channel estimation problem is solvable only if the number of linearly independent FD sequence used in the system U satisfy L < M — U for a channel with a discrete time impulse response of length L [42] (this condition will be discussed again in more detail in Section 4.3.4). The second condition states that the maximum system capacity is reduced for multipath channels since rewriting condition (2) yields U < M — L. This is because in a multipath fading channel, only U out of a possible M linearly independent FD sequences can be used in the system. Therefore, 71 / / Type I Blind T F - M C - C D M A C M O E RLS Receiver / / in Slowly Fading Rayleigh Multipath Channel Initialization: * / [ 0 ! 0 ] = I M X M In the pth symbol, input matrix Z[p] Set w* = ai for(n = 1.2. ....AT) Compute Kalman Gain: , r i _ \-1$f\p,n-l]z„.[p]T f[P;nl - i + A - i ( z „ b n " * / b . » - l ] a n b F Compute Update: *f\p, n] = A " 1 (&f\p, n - 1 } - kf {zn[p)r)H n - 1] end for Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: C i = F^v /xtg Compute FD Detector w; <T> / c! Perform detection &1[p] = sgn(Re{(w t * w Z)*w}}) Table 4.1: Type I CA'IOE RLS Blind Receiver Algorithm the maximum user capacity for T F - M C - C D M A system in a slowly fading multipath channel is now given by the product N x ( M — L). The channel estimation solution is unique up to a multiplicative complex constant which means that only the magnitude of the channel impulse response can be uniquely estimated and there is an unknown phase. However, this can be resolved by using differential detection or phase tracking by periodical'insertion of pilot symbols [22]. The algorithms we presented assume that this complex constant has been perfectly estimated: however, the algorithms can be easily modified using differential BPSK. Finally, the type I and type II C M O E RLS blind detection algorithm for T F - M C - C D M A are summarized in Table 4.1 and 4.2 respectively. 72 / / Type II Blind T F - M C - C D M A C M O E RLS Receiver / / in Slowly Fading Rayleigh Multipath Channel Initialization: * / [ ° ] = ! M X M In the pth symbol, input matrix Z[p] Set w*. = ai Compute: z' = afZ[p] Compute Kalman Gain: k f y - x-1*I\p-iWr R / [ P J ~ l + A - i ^ ) " * ^ , - ! ] ^ Compute Update: */LP] = f*/[P - 1] - k ; (z'T)H - 1] Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: C i = F M x L g Compute FD Detector w*f = 3?/Ci Perforin detection bi [p] = sgn (Re { [ Table 4.2: Type II C M O E RLS Blind Receiver Algorithm 73 4.3.4 B l i n d Subspace M u l t i u s e r Detec t ion It was shown in [22,25,41,42] that the blind subspace multiuser detection can be applied to M C - C D M A signals. In this section, we apply the blind subspace multiuser, detection method for our proposed T F - M C - C D M A type I and type II FD multiuser detection in 4.9 and 4.14. The correlation matrices for the type I and type II detectors can be re-written as Cf E {zi {£)"} = S , | A | 2 S f + a ? W ( 4 2 g ) Cf = E {z'z' f /} = S , ! A | 2 S f + a}IMxM where S / = [ c 1 | c 2 | . . . | c £ / ] (4.29) contains the effective FDS signature waveforms of each user. Recall that the FD blind M U D for the T F - M C - C D M A type I and type II detector only differs in the number of interfering FD sequences. Therefore for type I detector, U = K if K < M and U = M if K > M and for type II detector, U = Xi- | A | 2 denotes diag(A\, A\,...,Al,) where At is the total energy of all users using the same FD signature sequence, i = 1,2,...,[/. The eigendecomposition of the correlation matrix then yields the subspace components [39] ; (4.30) Ain = diag(\l»t...,W) Then, based on the discussion of subspace detection in Section 3.3.3, the blind L M M S E detector for the' type I and type II FD M U D is given by w * = U ^ ( A ^ ) " 1 U ( / ^ c 1 (4.31) Estimating the effective FDS signature waveform is the last step to complete the blind subspace solution given by 4.31. 74 B l i n d Channel Es t ima t ion It was shown in [42] that the effective'signature waveform and the channel response can be jointly estimated by exploiting the orthogonality nature between the noise subspace and the signal subspace for M C - C D M A signals. For our proposed type I and type II detector joint channel estimation using the blind subspace method, we follow similar analysis as given in [42]. The orthogonal set of eigenvectors Us from the eigendecomposition of the cor- relation matrix Cj spans the signal subspace of Sf = [ci |c-21 . . . |cV], then the orthog- onality between the signal and noise subspace gives the result U ^ ' c ; , . = 0 (4.32) Using this fact, the least squares method can be used to estimate the effective signa- ture waveform and the unknown channel response Ci = argmin C l VlPHcA2\ (4.33) Using the definition of the effective signature waveform and expanding 4.33 yields an expression almost identical to 4.27. g = argmin \gHFf/xLCf U ^ U ^ C x F ^ g } (4.34) Upon restricting the solution to normalized g, the solution for g is given once again by the eigenvector corresponding to the smallest eigenvalue of the composite matrix F ^ x t C f fjiP~(jlPnC\FMxL. Such estimation is unique up to a complex constant as in the case with the channel estimation using the C M O E RLS estimates described in section 4.3.3. Using the subspace decomposition, the approach to channel esti- mation for both the C M O E RLS method and the subspace'method relies on 4.32 being solvable. Recall that by definition, the matrix U , 7 contains all eigenvectors corresponding to the full eigenvalue decomposition of the sample data correlation 75 matrix that does not belong to the signal space collection. Suppose the signal space has rank U. then the matrix U; , contains M — U eigenvectors. Also recall that the length of the unknown channel impulse response is L and hence the number of unknowns to solve for is also L . Therefore 4.32 contains M — U equations and we are seeking for the solution for L unknowns. For the system of equations to be solvable, the number of equations must be greater than or equal to the number of unknowns and therefore, we arrive at the condition that M — U > L which was stated without justification in Section 4.3.3- Hence, the maximum users that can be supported is limited by the channel impulse response length and the number of subcarriers used. Therefore, with M being fixed, the maximum number of users that can be supported in a slowly fading Rayleigh multipath channel is N x (iVI — L). For our proposed T F - M C - C D M A type I and type II detector, we present here two implementations for the blind subspace multiuser detection in slowly Rayleigh fading channel, namely, the singular value decomposition of the data matrix and adaptive subspace tracking using the PASTcl algorithm introduced in Chapter 3, and investigate their performance in slowly fading Rayleigh multipath channel. S i n g u l a r V a l u e D e c o m p o s i t i o n The singular value decomposition implementation applies the exact eigendecomposi- tion to the sample correlation matrix. The sample correlation matrix can be obtained either through batch sample averaging or an exponentially weighted recursive update with a forgetting factor which is more suitable for dynamic channels. The compu- tation of subspace parameters using SVD need not be re-computed for every data sample. However, the interval between re-computation of the blind detector should accommodate to the changing dynamics of the channel such as the channel coherence time. The type I and type II detector implementation using the SVD method is 76 / / Type I Blind T F - M C - C D M A Subspace (SVD) Receiver / / in Slowly Fading Rayleigh Multipath Channel In the pth symbol, input matrix Z[p] Set w. = a fori// ••- 1.2 V) Compute Update: Cf\p,n] = XCf[p,n - 1] + (z,Mr)K[p]T)H end for Apply Eigendecomposition to C / to get Subspace Eigenvectors: Subspace Eigenvalues: Form Subspace matrices: ui^, Ai^.U.^ Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: c, = F M x / j g Compute FD Detector Perform detection b1[p] = s g n ( R e { ( w ^ Z ) * w } } ) Table 4.3: Type I Blind Subspace (SVD) Receiver Algorithm summarized in Table 4.3 and Table 4.4 respectively. P A S T d Adap t ive Subspace Tracking Alternatively, the subspace parameters can also be estimated using the PASTd adap- tive subspace tracking algorithm used in Chapter 3. The adaptive subspa.ce tracking using the PASTd algorithm offers a computation advantage over the singular value implementation method as discussed in .Chapter 3. The adaptive algorithm need not be modified for the slowly fading channels. Again, the only added complexity comes from the joint channel estimation step which the channel coefficients are obtained once the subspace parameters are computed. However, we note that the PASTd al- gorithm only tracks the subspace eigenvectors belonging to the signal subspace but 77 / / Type II Blind T F - M C - C D M A Subspace (SVD) Receiver / / in Slowly Fading Rayleigh Multipath Channel In the pth symbol, input matrix Z[p] Set w,* = a:i Compute: z' = afZ[p], Compute Update: Cf[p} = XCf[p-l) + (zUJ\(Ap]T)H Apply Eigendecomposition to Cf to get Subspace Eigenvectors: j u ^ j Subspace Eigenvalues: j ^ ^ j Form Subspace matrices: uiJ), Al/0, Vnf) Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: 6] = FMxLg Compute FD Detector w} = U ^ ) ( A i / ) ) - 1 U ^ f f c 1 Perform detection hip] = sgn (Re{(w} ,Hz'T Table 4.4: Type II Blind Subspace (SVD) Receiver Algorithm 78 in fact we also need the estimate of the noise subspace eigenvectors as well. To do that, we invoke a orthogonality relationship relating the noise subspace eigenvectors and the signal subspace eigenvectors [23] UiPlJiP" = I - U ^ U ^ " (4.35) Using 4.35. we do not have to explicitly update the noise subspace eigenvectors using the PASTcl algorithm. It can be computed through the signal subspace. The algo- rithms for type I and type II detectors using PASTcl adaptive subspace tracking are summarized in Table 4.4 and 4.5 respectively. 4.4 Performance Evaluation Measure The performance evaluation criteria for blind multiuser receivers presented in Section 3.4 applies to the slowly fading channel as well. The output SINR is defined as in section 3.4 except that the expectation is now over all channel realizations for a given Doppler rate and delay profile. Q T M T 7 ( N g{wfzMW /|b : l[p]}2 . M A ' / t l W / . W f ) = — — — - — - — T . — — (4.3o 1 / ; E{Var ( W ^ Z ^ W / I M P ] ) } V 1 Furthermore, because our blind multiuser receivers perform joint channel estimation, we also evaluate the mean square error in the channel estimation. This is defined as the variance of the channel estimation error as a function of the number of samples processed. This is defined as, £(g) = £ { | | g - g | | 2 } (4.37) 4.4.1 Receiver C o m p l e x i t y The receiver computational complexity in the downlink slowly fading Rayleigh multi- path channel is slightly different from .the A W G N channel case discussed in Chapter 79 / / Type I Blind T F - M C - C D M A Subspace (PASTcl) Receiver / / in Slowly Fading Rayleigh Multipath Channel In the pth symbol, input matrix Z[p] and previous estimates Set W ( = ai •for(n'= 1,2,...,N) Input zn[p]r Set = zn[p]T for (A; = l,2,...,U) Projection Approximation: yk = uk [p.n — l}Hx.k Eigenvalue Update: \\p[p.n} = jX[P\p.n — 1] Eigenvector Update: uk%,n] = v4P\p,n - 1] + (x<'> - uk»\p,n - l]y^ Deflation: x j ^ = x [ J ) - u[f)\2J,n]y[f) end for end for Form Signal Subspace Matrices: U . ^ . A^ Compute: t f t ' W = I - U ^ U ^ " Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: 61 = F M x L g Compute FD Detector v} = iji'\AP)-1WHcl Perform detection S1|p] = s g n ( R e { ( w r w Z ) * w * f } ) Table 4.5: Type I Blind Subspace PASTcl Receiver Algorithm 80 II Type II Blind T F - M C - C D M A Subspace (PASTcl) Receiver / / in Slowly Fading Rayleigh Multipath Channel In the pth symbol, input matrix Z[p] and previous estimates /\(/) f l ( / ) \ t / Set = a ;i Compute: z ' = a f Z[p] Input z'[p]r Set 4f) = z ' [p] r for(A; = 1,2,...,17) Projection Approximation: y[P — u[P\p — l j ^ x j ^ Eigenvalue Update: \[f)\p] = jX{kf)[p - 1] + Eigenvector Update: 4P* - « f b - 1 ) + ( = 4 ° - - %F> Deflation: = x<;n - u ^ W e n d for Form Signal Subspace Matrices: \jiJ\ A.^ Compute: UPW" = I - tjif)\Jl/)H Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: C i = F / V / X ^ g Compute FD Detector w;; = u<'W ))-1u$ / )"c1 Perform detection S1[p] = s g n ( R e { ( w ; ) t f z / r } Table 4.6: Type II Blind Subspace PASTcl Receiver Algorithm 81 3 even though the methods used in both cases are inherently the same (CMOE RLS, blind subspace using SVD. and blind subspace using PASTd subspace tracking). The difference stems from the fact that in slowly fading downlink channels, matched filter- ing is applied for the time domain detection because orthogonality of the T D signature sequence are preserved in slowly fading channels. Furthermore, there is added com- plexity for receivers in slowly fading downlink channel coming from the joint channel estimation computations. The C M O E RLS channel estimation involves computing an eigendecomposition of a L by L matrix to find the eigenvector corresponding to the smallest eigenvector; this has complexity 0 ( L 3 ) . Both the blind subspace methods using SVD and adaptive subspace tracking PASTd also involve computing an E V D of a L by L matrix to find the eigenvector corresponding to the smallest eigenvector to estimate the channel. The FD detector computation for both C M O E RLS type I and type II detectors have the same complexity as in the A W G N channel case which is 0(M2) per update. Because there are N updates per symbol for type I. the total complexity for type I is 0(NM2). For type II, only one update is computed per symbol, hence the total complexity of type II C M O E RLS is 0(lVI2). Hence in this case, there is a computational advantage going from type I to type II for the C M O E RLS implementation. The FD detector computation using blind subspace method through SVD is based on the data autocorrelation matrix rank. For type I, this is min {K. M}. In most realistic situations, M will be less than K since one of the main motivations for using T F - M C - C D M A is to use short signature codes [17]. For type II. this is xi a n d as discussed before, x.i < M- Using SVD, the complexity for the type I blind subspace detector computation is 0(Mi) and for the type II detector O(xi)- In this case, when the system is below full capacity, there is computation advantage going from type I to type II. The frequency domain detector computation using blind subspace method using PASTd adaptive subspace tracking is also based 82 Multiuser Detector Type Complexity C M O E RLS Type I 0(NM2 + L 3 ) C M O E RLS Type II 0{1VI2 + L 3 ) Subspace SVD Type I 0(M3 + L 3 ) Subspace SVD Type II oixi + L:i) Subspace PASTcl Type I 0(NM'z + L 3 ) Subspace PASTcl Type II 0{xiM + L 3 ) Table 4.7: Summary of T F - M C - C D M A Blind Multiuser detection complexity for slowly fading Rayleigh multipath channels on the sample correlation matrix rank. Using the PASTcl adaptive subspace tracking, the complexity for the type I blind subspace detector computation is 0(lVI2) under normal system capacity and for the type II detector is 0(xiM) per update. For type I using PASTcl, there are N updates per symbol, therefore, the total complexity for type I receiver is 0(NM2). However, for type II, there is only one update per symbol, therefore the total complexity for type II PASTd detector is O(xiM). We also see that using the PASTd subspace tracking algorithm, there is computation advantage going from type I to type II. Together with the joint channel estimation estimates, the detector complexities for the multiuser detection methods considered are summarized in Table 4.7. 4.5 Simulation Results Computer simulations were carried out to illustrate the performance of the following multiuser receivers in downlink slowly fading Rayleigh multipath channels: C M O E RLS, subspace SVD, and the adaptive subspace tracking using PASTD implemented in both type I and type II architectures. The channel model used in our simulation is the ITU-A model for evaluating outdoor/indoor pedestrian environment. The power delay profile for the ITU-A model is given in Table 4.8. The T F - M C - C D M A chip period is chosen to be 100ns. For simplicity, we round the relative delays of the delay 83 Tap Relative Delay (ns) Average Power (dB) Doppler Spectrum 1 0 0 < ^\A-(t)2 2 110 -9.7 < 3 190 -19.2 < -W:i-(t)2 4 410 -22.8 < ^\A-(t)2 Table 4.8: ITU-A Pedestrian Rayleigh Fading Channel Power delay profile T D Spreading Gain 8 FD Spreading Gain 16 Number of Subcarriers 16 TDS Sequence Walsh-Haclamarcl FDS Sequence Walsh-Hadamarcl Number of FDS Sequence used 11 Table 4.9: Simulation parameters profile to the nearest 100ns. Furthermore; we assume the mobile user is moving at speed of 3km/hr which corresponds to normal walking pace in urban areas. The system parameters chosen are shown in Table 4.9. We consider three levels of system loading, light moderate (35 users), moderate (55 users) and high (75 users). The B E R performance curves for 35 users. 55 users and 75 users are presented in Figures 4.3. 4.4. and 4.5. It can be seen from the B E R figures that the T F - M C - C D M A type I detectors are far inferior in performance to the type II detectors especially in the 35 and 55 users case. This result is not surprising given that the type II cascade processing eliminates interferers from other time-groups by matched filtering while type I detectors do not enjoy this property. There is a large difference in performance between type I and type II blind subspace PASTcl detectors and it can be seen that type I blind subspace PASTcl detector has the poorest performance among all receivers considered. For light to moderate loads (i.e. 84 downlink slowly fading Rayleigh channel, 3km/hr, 35 Users 10 Avg SNR (dB) Figure 4.3: B E R performance for 35 users case the 35 users and 55 users case), it can be seen that the type II blind subspace PASTd detector's performance matches closely to that of the type II blind subspace SVD although there is a slight (<ldB) penalty for using an adaptive tracking algorithm instead of computing the SVD. However, at higher system loading, the blind subspace SVD detector has a better good performance. For the type II detectors (in the 35 users and 55 users case), we see similar trends as in Chapter 3 for the TF-VIC-CD M A blind multiuser detection problem in A W G N . We see that the subspace detection using SVD provides the best performance and the subspace detection using PASTd subspace tracking but provides fairly good performance. We also present output SINR plots to confirm our B E R performance results and to look into the convergence of the proposed detectors. Figures 4.6, 4.7 and 4.8 provide 85 downlink slowly fading Rayleigh channel, 3km/hr, 55 Users 10 AVG SNR (dB) Figure 4.4: B E R performance for 55 users case the output SINR versus data samples plot for 35 users, 55 users, and the 75 users system load. The convergence of the blind subspace PASTd algorithm is relatively slow compared to the other two. This is expected since we have already observed this relatively slow convergence in Chapter 3. In our simulations, the PASTcl algorithm has been accelerated by applying a batch SVD decomposition to the first 50 data samples. The SINR plots confirm that type II detectors outperform type I detectors. We also studied the channel estimation capabilities for our proposed detectors. The sampled mean square channel estimation error is plotted against the data samples in Fig. 4.9 for type I and type II C M O E RLS. blind subspace SVD. and blind subspace PASTcl adaptive subspace tracking detectors. Again, the channel estimation error plots confirm the type II detectors estimates the channel much better than type II detectors. The cascade implementation of type II detectors remove M A I using the 86 downlink slowly fading Rayleigh channel, 3km/hr, 75 Users 10 Avg SNR (ciB) Figure 4.5: B E R performance for 75 users case orthogonality of the T D spreading codes thus giving better joint channel estimation during the FD detection stage. 4.6 Summary In this chapter, the T F - M C - C D M A blind multiuser detection problem in a down- link slowly fading Rayleigh multipath channel was investigated. Two types of blind multiuser detectors were proposed and investigated. The first type, known as type I detector, retains the same parallel processing form as the blind multiuser detector for the A W G N channel with M A I by using the decoupling method discussed in Chapter 3. The second type, known as type II detector, uses cascade processing where the output of the T D processing is used as the input to the frequency domain processing. 87 SINR Comparisons, 35 Users subspace PASTcl (type II) 2000 3000 Data Samples 5000 Figure 4.6: Output SINR for 35 users case at SNR 12clB It was found that both methods involve solving an equivalent M C - C D M A multiuser detection problem (that differ only in the number of FD interfering sequences in slowly fading channel for the FD multiuser detector coefficients. The type I and type II detectors were then applied with three multiuser detection methods, namely, the C M O E RLS method, blind subspace method using SVD and blind subspace method using PASTcl adaptive subspace tracking. Computer simulation results show that type II detectors outperform type I detectors. The cascade implementation of the type II detector reduces the amount of interference going into the FD M U D stage by applying matched filtering in the T D detection. 88 Output SINR Comparison, 55 Users Data Samples Figure 4.7: Output SINR for 55 users case at SNR 12clB 89 Output SINR Comparison, 75 Users 4.5 I ! ! ! Data Samples Figure 4.8: Output SINR for 75 users case at SNR 12dB 90 LU <2- 0.5 I LU <£ 0.5 LU w 0.5 Blind subspace SVD, 35 Users ^type 1 -type II — 500 1000 1500 Blind subspace PASTd, 35 Users 500 1000 1500 CMOE RLS, 35 Users 500 1000 Data Samples 1500 2000 i i ^ ^ • t y p e II type I L. i i 2000 ^Jype I 2000 Figure 4.9: Mean square channel estimation error for 35 users at SNR 16dB 91 Chapter 5 T F - M C - C D M A B l i n d Mult iuser Detection in Downlink Mobi le Rayleigh Fading Channel with Doppler Induced ICI 5.1 Introduction In this chapter, the random frequency offset caused by mobile Doppler shifts in down- link multipath fading channels is investigated. Previous discussions in Chapters 3 and 4 have implicitly assumed that any frequency offset is compensated exactly at the re- ceiver. Frequency offset is a result of the mismatch in frequency between the detected signal and the demodulator's oscillators. Perfect frequency matching is important in multicarrier based modulation because subcarrier frequency offset or mismatch results in leakage from one subcarrier to other subcarrier as a result of loss of orthogonal- ity creating intercarrier interference (ICI). Random frequency mismatch is caused by the Doppler shift in mobile communication multipath channels and the effect of in- 92 tercarrier interference is most severe when subchannels or subcarriers have minimal separation. The result is a loss of orthogonality in the FDS codes. In this chapter, we investigate the performance of the T F - M C - C D M A blind multiuser detectors proposed in Chapter 4 in multipath fading channels with Doppler induced ICI. The roadrnap of this chapter is as follows: section 5.2 discusses the mathemat- ical model of ICI in T F - M C - C D M A and its analysis in T F - M C - C D M A downlink transmission. Section 5.3 discusses the blind multiuser detection algorithms used in combating ICI in downlink multipath fading channels. Section 5.4 presents computer simulation results. A Summary of the chapter is provided in Section 5.5. 5.2 Modeling Intercarrier Interference In this section, a mathematical model for ICI in T F - M C - C D M A caused by mobile Doppler shifts as a result of user mobility is discussed. It is shown in [26] that intercarrier interference (ICI) in O F D M systems caused by mobile Doppler shifts can be modeled as a frequency shift within the channel impulse response. Because of multipath propagation, each path arrives at the mobile receiver isotropic antenna with incident angle 6>,; and experiences a Doppler shift ;y,;(Hz). In [43], it was shown that if the Doppler shift is considered into the tapped delay line channel model, then the continuous time varying channel impulse response for a L path multipath channel is given by If the channel is assumed to be changing slowly with respect to the transmission rate so that the complex channel gains, path delays and Doppler frequency shifts are relatively constant over at least an O F D M data block, then we can rewrite 5.1 as L - l 5.1) L - 1 (5.2) 93 The Fourier transform with respect to the delay variable r gives the function L - l h(f,t) = j2^eJ2nvi(t~Tl)e~j2nfv (5-3) A n important result in [27] showed that using a time domain analysis from the received signal, the effect of Doppler induced ICI in M C - C D M A systems can be viewed as a form of weighted sampling in the frequency domain. Using this result and sampling the channel frequency response given in 5.3 with integer multiples of the frequency separation between subcarriers / s . the Doppler induced ICI weight coefficients can be found. Assuming that M subcarriers are used, this yields L-l h(rn0fs, t) = J2 gie>2ltv,it-r0e-i2wmof'T> (5.4) / . = o where fs = TTJJJ is the subcarrier frequency spacing and Ts is the duration of an O F D M symbol transmission. Samples of the channel impulse response in the t variable at multiples of Ts is yields the discrete channel impulse response L-l h{m0fs, iT,) = ]T g t e i 2 n v ' i i T ' - T < h - j 2 n m o f ' T ' (5.5) If we approximate and discretize the path delays to integer multiples of Ts, and let Ei = T~ be the normalized Doppler offset of the I path, then we can rewrite 5.5 as is L~l /J2me,\ f-]2ne,l\ f-j2nmQl h(mo,i) = }^giexp ^—— j exp ^ M j exp M ' -° (5.6) L ' / . 7 ' 27 re i ( * -0^ (-j2~mj gi e^p y 1=0 9i exp 77 exp M 7 1 V M The frequency domain channel transfer function is then the M point D F T of h(n, i) with respect to the variable i. I (j2irei(i - l)\ f-j2irm.ol\ (-j2wmi H(rn0, m) = - ^ ^ 9 l exp ^ J exp ^ — — j exp ^ — — Jul \ ' ( 5 - 7 ) 1 '-^ v - (j2ir((ei - m)i - etl)\ f-j2nm0V = M E E 9i exp [ ¥ f J exp ^ — — i = o / = o v 7 v 94 It is shown in [28] that using O F D M based transmission, the D F T demodulated re- ceived signal is weighted by the frequency domain channel transfer function H(mo, m— m0). Applying this result to the T F - M C - C D M A received signal yields Hp K rLp[m0] = J— bkak[n] ]P ck[m}H(m0, m - m0) + W[m0] (5.8) A.-1 771=0 The term H(mo, rn — ni0) weighted by the frequency domain spreading chip sequence ck[m] is the ISI induced by the channel and represents the leakage of energy from the mth subcarrier to the mo"1 subcarrier. Expanding 5.8 yields / K A'I-1 rLp[mo] = J£[ E hak[n] E ck[m] k=l m=0 , > y 7=0 /.=0 V 7 J Equation (5-10) is the received T F - M C - C D M A signal model that includes both the effects of both multipath fading and Doppler induced ICI. This result can be used to model T F - M C - C D M A signals in downlink mobile fading channels. If the mobile receiver is stationary relative to the transmitting basestation and as a result, no Doppler frequency shifts, then we can set ei — 0 for / = 1.2.....L. By doing this, then we have K h'I-l rLp[m0} = E bkak[r>] E c fc [ m ] k = l 777=0 x U E E' 9i exP ( ^ - ^ ) exp { = ^ ) ) + W[m0] (5.10) V i=o i=o v J J = E hak[n] E L ck[m]5mmo E fji exp (=&P*) + W[m0] k=l -777=0 ( = 0 which is the received signal model for the slowly fading Rayleigh multipath channel discussed in Chapter 2 and Chapter 4. We now define the effective frequency domain chip sequence (as a result of ICI) as Ad-l Gk[rn0] = Y ck[m]H(m0, m - m0) (5.11) 777=0 95 Equation (5-8) can bc rewritten as K '1'LP m°] = E t>k(-i'k[n}Gk[mo\ + W[m0] (5.12) fc=i or in a more familiar form rip (5.13) Defining the effective frequency domain complex subchannel gain as G'k[mQ] Gk[mo] Ck[m0] (5.14) Using (5-14) and (5-15). the received signal can be written as i< •I'LP m0] = Y2bkak[n}Gk[mo}ck[m0] + W[m0] (5.15) which is an equation identical in form to 4.4 with the exception that 5.15 is a quasi- uplink synchronous model (even though we are analyzing a downlink channel) since each user experiences a user dependent channel gain GJ..[mo] instead of a common channel gain. As a result, we can also define an equivalent channel impulse response Based on the above discussion of the modeling of ICI due to Doppler shifts in a mobile communication channel, we can summarize two important effects of ICI in T F - M C - C D M A : (1) it destroys the orthogonality of the F D S codes and (2) it transforms a downlink synchronous model into a quasi-uplink synchronous model. 5.3 Bl ind Multiuser Detection This section discusses using the blind multiuser detection methods proposed in Chap- ter 4 to the T F - M C - C D M A signal given by 5.15 without knowledge of the ICI co- efficients, the Doppler frequencies, and other interfering users' time and frequency g' = IDFT(G'k). 96 domain signature sequences. Since it was shown in Chapter 4 that type II detectors out-performs type I detectors, we only discuss type II detectors. In particular, we revisit the C M O E RLS method and the subspace blind multiuser detection method using SVD and adaptive subspace tracking through the PASTcl algorithm to show- that all these algorithms gives robust performance in mobile multipath fading chan- nels with Doppler induced ICI. 5.3.1 Type II Detection Formulation Based on the previous discussion of the type II detectors in Chapter 4, we begin by rewriting the T F - M C - C D M A received signal given by 5.15 using the groupings of TDS sequences, with all users sharing the same time domain signature sequence forming a group. This gives A' Xj rLp[nio} = Zn[mo] = \fP &n*anb1 c«MG' r e[m0] + ^ « K ] (5.16) j = l K - l Using vector notation, 5.16 can be re-written as N Xi z[m0] = N / P ^ ^ ^ [ m o l G ' J m o l a j + W[m0] (5.17) J = l K.= l The orthogonality of the TDS sequences is preserved when the channel is varying slowly with respect to the transmission rate and thus we can simplify 5.17 by applying matched filtering for the time domain detection N Xj afz[m 0] = z'[mo] = \fP bj«c«[m0]C7'K[mo]af a,- + af W[m0] 3 = 1 * = 1 (5.18) X i • \ / = ^YblK,cK\m0]G'K[mo\ + w' h-,=i Re-writing 5.18 using vectors yields X l = y/P ^ & i « c K + w' (5.19) K , = l 97 where now the effective FDS waveform is given by c f c = diag(ck)G':k G', = G'k[l] G'k[2] ok[l] • G'k[M] T d,tag(ck) = ck[M] (5.20) Then, the same optimization problem for the type II blind L M M S E detector as in Chapter 4 (see equation 4.14) is obtained with the slight difference in the effective FDS signature since the complex channel gain vector is now user dependent. However, the FD blind L M M S E optimization problem remains the same as before, i.e. W | = argmin E W f e C M x l P{bu w f z' ^5.21) 5.3.2 C M O E R L S Bl ind Detector In this section, we show the C M O E RLS blind detector for blind multiuser detection can be applied in mobile Rayleigh fading channel with Doppler shift induced ICI. Since the optimization problem in 5.21 is similar to that in 4.14 except the effective FDS waveform has a quasi-uplink interpretation. Using this fact, the C M O E RLS algorithm remains the same as that in chapter four, namely w* = ( c f Cfcx) (5.22) where C ^ 1 is approximated recursively using the matrix inversion lemma, i.e. A - 2 * / [ 7 - - l ] z ' z ' w * / [ j - - l ] (5.23) Using the results in Section 4.3.3, the effective FDS sequence, and hence the effective frequency domain channel gain for the desired user can be estimated using the M O E expression and then maximizing it with respect to the desired user's effective FD / / Type II Blind T F - M C - C D M A C M O E RLS Receiver / / in Slowly Fading Rayleigh Multipath Channel / / with Doppler induced ICI Initialization: $ / [ 0 ] = I A 7 XJW In the pUl symbol, input matrix Z[p] Set w* = a:1 Compute: z' = a f Z[p] Compute Kalman Gain: K F [ P L ~ I + A - I ( Z ^ ) h * / [ P - I ] Z " - Compute Update: * / W = A " 1 ( * / [ P - 1] - k / ( z ' T ) 7 / $ / [ P " 1]) Perform Channel Estimation: Find smallest eigenvector g' corresponding to the smallest eigenvalue of Compute Estimated Effective FD Signature: ci = Faxes' Compute FD Detector Wj = $ yCl Perform detection S1[p] = s g n ( R e { ( w } ) t f 2 / r } ) Table 5.1: Type II C M O E RLS Blind Receiver Algorithm for multipath fading chan- nels with Doppler induced ICf signature. This again ends up being a familiar problem to find the normalized eigen- vector corresponding to the smallest eigenvalue of the matrix F f / x / C f $ ^ C I F M X L - The algorithm is summarized in Table 5.1. 5.3.3 B l i nd Subspace Multiuser Detection The blind subspace multiuser detector for T F - M C - C D M A signal in mobile fading channel with Doppler induced fCI can be formulated in the same way as presented in Section 4.3.4. The data correlation matrix from the T F - M C - C D M A multiuser detection problem in 5.15 can be expanded iiito C ; = E { z ' z ' " } = S ; I A | 2 S f + ajIMxM (5.24) 99 where S / = [£ 1 |c 2 | . . . |c x l ] (5.25) contains the effective FD signatures of the users within the same time domain signa- ture group as the desired user (user 1). Applying the eigendecomposition to Cf gives the signal and noise subspace decomposition The solution to the type II FD blind L M M S E detector is given by (5.26) W} = U ^ ( A ^ ) " 1 U ^ C ] (5.27) The effective FD signature can be estimated using the exact same procedure as in Section 4.3.4 by exploiting the orthogonality nature between the noise subspace and the signal subspace. Combined with the least squares estimation, we arrive again at the familiar optimization problem for the channel estimation g' = argmin { g ^ F ^ C f U ^ U ^ C i F ^ g } (5.28) Solving the optimization problem in 5.28 again requires finding the normalized eigen- vector corresponding to the smallest (least dominant) eigenvalue of the composite matrix F1MxLC{{\J^Pt](;Pfr C\F M X L - As in Chapter 4 where it was implicitly assumed that the channel ICI has been perfectly compensated, the channel estimation in this case is also unique up to a multiplicative complex constant for the same reasoning discussed in Section 4.3.4. Therefore, the maximum user capacity for the T F - M C - C D M A downlink system in a multipath fading channel with Doppler induced ICI is also N x (iVI — L) as long as the two conditions stated in Section 4.3.3 are satisfied. Therefore, assuming the two conditions are satisfied, the Doppler induced ICI effect 100 / / Type II Blind T F - M C - C D M A Subspace (SVD) Receiver / / in Slowly Fading Rayleigh Multipath Channel / / with Doppler induced ICI In the pth symbol, input matrix Z[p] Set w* = a-i Compute: z' = a{FZ[p] Compute Update: C / b ] = A C / [ p - l ] + (z 'b]^)(z 'b] r ) / f Apply Eigendecomposition to Cy to get Subspace Eigenvectors: < uA > Subspace Eigenvalues: f . Form Subspace matrices: U ^ , , tjlP Perform Channel Estimation: Find smallest eigenvector g' corresponding to the smallest eigenvalue of -pH f~<Hf-[(f)j-!(f)Hri ir Compute Estimated Effective FD Signature: cq = F^/x/^g' Compute FD Detector Perform detection b1[p] = s g n ( R e { ( w p / / z - r } ) Table 5.2: Type II Blind Subspace (SVD) Receiver Algorithm does not change the user capacity of the system as the system is still capable of sup- pressing N x ( M — L) — 1 interferers. However, there is a penalty to pay in terms of B E R performance because ICI reduces the average output SINR as a result of the leakage of subcarrier energy. The subspace parameters can be obtained using the SVD method or the PASTcl adaptive subspace tracking. Both methods require no modification for the Doppler induced ICI channel. We summarize the both the type II blind subspace detector algorithm using SVD and using PASTcl adaptive subspace tracking in Table 5.2 and 5.3 respectively. 101 / / Type II Blind T F - M C - C D M A Subspace (PASTcl) Receiver / / in Slowly Fading Rayleigh Multipath Channel / / with Doppler induced ICI In the pth symbol, input matrix Z[p] and previous estimates l A f c ; U f c /*=, Set = a] Compute: z' = af'Z[p] Input z'[p]L Set x(/} = z'\p]r ibr: /• 1,2, ...,xi.) Projection Approximation: = u ^ [ p — l]'"'x^ Eigenvalue Update: X[f) [p] = -yX[f) [p - 1] + ;y[7) Eigenvector Update: u ^ b ] = - 1] + (x<" - u i ^ " 1 ] ^ ) Deflation: x^ : 1 = x[ J ) - u{kf) \p]y[f) end for Form Signal Subspace Matrices: U s . A s Compute: U ^ U ^ " = I - U ^ U . ^ Perform Channel Estimation: Find smallest eigenvector g ' corresponding to the smallest eigenvalue of T?H (-iHjTiDfiif)1-!(~< P Compute Estimated Effective FD Signature: C i = F ^ / x L g ' Compute FD Detector w} = u ^ ( A i / ) ) - 1 u i / ) H e . Perform detection b 1b] = s g n ( R e { ( w } ) / / z ' r Table 5.3: Type II Blind Subspace PASTcl Receiver Algorithm 102 T D spreading gain , 8 FD spreading gain 16 Number of subcarriers 16 TDS sequence type Walsh-Haclamard FDS sequence type Walsh-Haclamarcl Number of FDS sequence used 11 Carrier frequency 1.9 GHz Table 5.4: Summary of simulation parameters 5.4 Simulation Results Computer simulations were carried out to illustrate the performance of the proposed T F - M C - C D M A type II C M O E RLS receiver, blind subspace receiver using SVD, and the blind subspace receiver using PASTcl adaptive subspace tracking for downlink slowly fading Rayleigh multipath channels with Doppler induced ICI. The channel model used is again the ITU-A 4 path model for evaluating outdoor/indoor pedestrian test environments provided in Table 4.5. The T F - M C - C D M A chip period is 100ns. Again, for simplicity, the relative delays of the multipath arrivals are rounded to the nearest 100ns and we do not modify the average receiving powers. The mobile speed considered here is pedestrian speed which corresponds to 3km/hr. The simulation parameters are summarized in Table 5.4. Three types of system loading are considered: 35 users (light moderate load), 55 users (moderate load), and 75 users (high load). Figures 5.1, 5.2 and 5.3 show the B E R performance curves for the 35 users, 55 users, and 75 users load respectively. Comparing the 35 users case and 55 users case with the simulation results in Chapter 4 (perfect frequency offset correction), it can be seen that the performance is still very good for all three type II detection methods. In particular, we once again see the closeness in performance between the type II blind subspace SVD and the blind subspace PASTcl detectors for the 35 users and 55 users case. However, as the system 103 Slowly Fading Rayleigh w/ Doppler ICI, 35 Users, 3km/hr 10° L j J Avg SNR (dB) Figure 5.1: B E R performance for 35 users case reaches a high user loading, the type II blind subspace PASTcl detector degrades sig- nificantly. This observation is also consistent with the simulation results from Chapter 4. From the B E R performance simulation results, we can see that without using any frequency offset corrections, the type II blind subspace SVD, PASTcl detectors and the C M O E RLS detector are robust to Doppler induced ICI in the channel. We next present the output SINR results. Figures 5.4, 5.5 and 5.6 show the output SINR against the number of data samples processed for the 35 users, 55 users and 75 users case respectively. The output SINR results show that in all three system load cases, the slowest to converge is the blind subspace PASTcl algorithm. The convergence speeds of tire C M O E RLS detector and the blind subspace SVD algorithm are similar. In all user load cases, the blind subspace SVD performs the best which confirms the result of the B E R performance results above. At high user 104 Slowly fading Rayleigh w/ Doppler ICI, 55 Users, 3km/hr L I J J 0 5 10 15 20 Avg SNR (dB) Figure 5.2: B E R performance for 55 users case loading, the PASTd does not perform well. Finally, we present the mean square channel estimation error plots in Fig. 5.7 for all three type II detector implementation methods. This channel estimation is different from the channel estimation in Chapter 4 where we are truly estimating the channel's path gains. The channel estimation here estimates the effective frequency domain complex subchannel gains given in 5.14. It can be seen from Fig. 5.7 that all three detector schemes give similar channel estimation performance. 5.5 Summary In this chapter, the mobile Doppler shift induced ICI is analyzed and derived for the T F - M C - C D M A slowly fading downlink model. This was shown to have two important 105 Slowly fading Rayleigh w/ Doppler ICI, 75 Users, 3km/hr 10° L ! J J 0 5 10 15 20 Avg SNR (dB) Figure 5.3: B E R performance for 75 users case effects: (1) the loss of orthogonality of the FDS sequences and (2) clue to ICI, the pure downlink model becomes a quasi-uplink synchronous blind multiuser detection problem. We also show that we need not modify the T F - M C - C D M A type II C M O E RLS method and the blind subspace method proposed in Chapter 4. It is shown that they can be directly applied as long as the rank criteria is fulfilled. However, computer simulation results confirms that due to Doppler induced ICI, there is a performance degradation as compared to the performance without ICI. 106 Output SINR Comparison, 35 Users, Doppler ICI Channel 1 1 1 subspace SVD (type — ^^•subspace PASTd (type II) / / ' - / CMOE RLS (type II) / 0 1000 2000 3000 4000 5000 Data Samples Figure 5.4: Output SINR for 35 users case at SNR 12dB 107 Figure 5.5: Output SINR for 55 users case at SNR 12dB 108 F i g u r e 5.6: O u t p u t S I N R for 75 users case at S N R 12dB 109 Blind subspace SVD, 35 Users LU £ 0.2 h 500 1000 1500 Blind subspace PASTd, 35 users 500 1000 1500 CMOE RLS, 35 Users 500 1000 Data Samples 1500 2000 2000 V . 2000 Figure 5.7: Mean square channel estimation error for 35 users at SNR 16clB 110 Chapter 6 Conclusion and Future Research 6.1 Conclusion T F - M C - C D M A has shown significant promise to accommodate the demands of 4G broadband mobile communication systems. It combines the benefits of D S - C D M A and M C - C D M A that are high user capacity, robust to frequency selective fading, and flexibility in providing a framework for adaptive resource allocation. Any system using C D M A based multiple access is interference limited, and this is no different for T F - M C - C D M A . In fact, T F - M C - C D M A must contend with interference in both the spreading in time and frequency. The method of multiuser detection mitigates the near far problem and suppresses the multiple access interference thus providing much better performance than single user matched filter detection. Non-blind multiuser detection requires the knowledge of the spreading sequences of all users and accurate channel information. Blind multiuser detection methods avoid the use of pilot symbols and training sequences to acquire this information and provide much superior band- width utilization than non-blind multiuser detection. This is especially important in receivers operating in next generation mobile handsets. This thesis is focused on extending the applicability of blind multiuser detection methods to T F - M C - C D M A 111 systems in three different channels: (1) the A W G N . (2) the downlink slowly fading multipath Rayleigh channel, and (3) the downlink slowly fading multipath mobile Rayleigh channel with Doppler shift induced ICI. In Chapter 2, the modeling of T F - M C - C D M A transmission and demodulation is considered. The WSSUS channel model is also discussed and the equivalent frequency domain channel transfer function for a frequency selective channel is obtained for system using multicarrier based transmission. One of the key results developed in Chapter 2 is that a systematic allocation of the time and frequency domain signature sequences for T F - M C - C D M A so as to reduce the total multiple access interference by minimizing the number of users sharing the same T D signature (and the same FD signature). This allocation scheme is not specified in [17]. In Chapter 3, the blind linear multiuser detection problem for T F - M C - C D M A in A W G N with M A I is discussed. The key result of Chapter 3 is that one can subop- timally decouple the T F - M C - C D M A multiuser detection problem into two separate blind multiuser D S - C D M A problems, one for the T D spreading and one for the FD spreading. This has the advantage of utilizing parallel processing. The blind DMI al- gorithm, blind C M O E RLS algorithm, the blind subspace algorithm using SVD, and the blind adaptive subspace algorithm using PASTcl were applied to the proposed decouple detector and their performances were investigated. In Chapter 4, the blind multiuser detection problem for TF-A4C-CDMA in a slowly fading downlink multipath Rayleigh channel is discussed. The key result is that two types of T F - M C - C D M A blind detectors are proposed based on.the results developed in Chapter 3. The first type, known as the type I detector, follows the same decoupled processing method as in Chapter 3 for the T F - M C - C D M A blind detection in the A W G N . The second type, known as the type II detector, uses a cascade processing with the output of the T D processing serving as input to the FD 112 processing. Computer simulations have shown that type II detectors outperform type I detectors. The type II detector takes advantage of the orthogonality of the time domain spreading sequences to eliminate interference from users belonging to other time domain signature groups. Therefore, the signal coming from the time domain detector only contains interferers from users within the same time domain user groups which is guaranteed to be less than or equal to the frequency domain spreading gain. In Chapter 5 ; the T F - M C - C D M A blind multiuser detection problem in slowly- fading downlink multipath Rayleigh channel with Doppler shift induced TCI is dis- cussed. We first presented the modeling of Doppler shift induced ICI in a WSSUS multipath fading channel. There are several important results that came from the analysis: firstly, the ICI creates leakage of energy from one subcarrier to another that results in a loss of orthogonality in the frequency domain spreading codes: and secondly, the downlink multiuser detection problem becomes a quasi-uplink problem. 6.2 Future Research The results presented this thesis on blind multiuser detection for T F - M C - C D M A provide a glimpse of the immense possibilities and potential applications of T F - M C - C D M A systems in future mobile communication systems. There are many interesting research areas that can be branched off into various directions in conjunction with other currently new research results in the areas of M I M O . Bayesian Monte Carlo methods, and signal processing techniques for fast fading channels. 6.2.1 M I M O T F - M C - C D M A Multiuser Detection M I M O using multiple transmit and receive antennas can further increase the capac- ity of T F - M C - C D M A systems. Such a system can fully utilize time, frequency, and spatial diversity. Some ground work has been laid in [21] investigating the perfor- 113 mance of un-coded T F - M C - C D M A subspace detection using multiple antennas. The integration of transmit diversity with space-time coding methods such as space-time trellis coding and space-time block coding can further improve the performance of T F - M C - C D M A systems. The addition of M I M O transmission and reception will in- crease the complexity receiver, hence the challenge remains to design receivers that incorporates spatial diversity in T F - M C - C D M A systems while maintaining a practical level of complexity. 6.2.2 Bayesian Monte-Carlo Blind Multiuser Detection for T F - M C - C D M A The Bayesian Monte-Carlo technique is a new signal processing technique that can be used in any type of statistical inference problems. In the communications area, blind multiuser detection, blind equalization are all special types of statistical inference problem. The main idea of Bayesian Monte-Carlo signal processing is to approxi- mate the solution by simulating samples from the posterior distribution. There are two popular Bayesian Monte-Carlo techniques used for such applications: (1) Markov chain Monte Carlo (MCMC) which is a batch processing method and (2) Sequential Monte Carlo (SMC) which is a, recursive processing method. It was shown in [44] that near optimal results can be obtained by applying the SMC technique to blind detec- tion in O F D M systems. Such detectors have the potential to achieve near optimal performance because this type of detection method approximates the optimal M A P estimate that truly minimizes the probability of error. Another advantage of such methods is that they can be incorporated into any model with a fully blind approach with all unknown parameters jointly estimated. 114 6.2.3 T F - M C - C D M A Bl ind Multiuser Detection in Fast Fad- ing Channels The blind multiuser detection techniques discussed in this thesis applies to slowly fading channels. That is. the channel is approximately constant for at least the duration of an O F D M symbol. However, if the channel changes rapidly within an information symbol, then other techniques must be used. Several challenging issues arise in fast fading: (1) The fast fading channel gain need to be tracked for each subcarrier stream in order to perform multiuser detection. (2) One can not rely on the orthogonality of the time domain signature grouping of the users since in fast fading channel, the orthogonality of the time domain signature will in almost all cases be lost. 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