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Blind multiuser detection for time-frequency spread multicarrier CDMA Tam, Wilson 2007

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B l i n d 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 O F THE REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF APPLIED  SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES (Electrical and Computer Engineering)  T H E U N I V E R S I T Y 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. Multicarrier 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 ) system 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 multipath 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 multiuser 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 R L S 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 verygood 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  Abstract  ii  Contents  vii  L i s t of Tables  ix  L i s t of F i g u r e s  xi  L i s t of S y m b o l s  xii  L i s t of A b b r e v i a t i o n s  xiii  Acknowledgements  xvi  Dedication  xvii  1  Introduction 1.1  Overview 1.1.1  1.2  1.3  1  of C u r r e n t W i r e l e s s  Communication  Systems  L i m i t a t i o n s of 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  1 Channels  .  .  2  Multicarrier-basecl Transmission  3  1.2.1  Multicarrier C D M A  4  1.2.2  Time-Frequency  4  Challenges 1.3.1  Spread Multicarrier C D M A  and Motivation  5  M u l t i u s e r Detection in T F - M C - C D M A  iv  Systems  6  1.3.2  2  6  1.4  Review of Related Works  7  1.5  Thesis Contributions  9  1.6  Thesis Outline  12  T i m e Frequency Spread M u l t i c a r r i e r Spread S p e c t r u m  System  Model  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  2.4  2.5 3  Blind Signal Processing Techniques  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  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  Summary  31  B l i n d M u l t i u s e r D e t e c t i o n for a T F - M C - C D M A Signal i n 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  4  Performance Measure . . '  47  3.4.1  49  Receiver Complexity :  3.5  Simulation Results  50  3.6  Summary  53  T F - M C - C D M A B l i n d M u l t i u s e r D e t e c t i o n i n Slowly F a d i n g R a y l e i g h Multipath Channel 4.1  Introduction  4.2  Linear Multiuser Detection for T F - M C - C D M A in Frequency Selective  4.3  4.4  5  ,  60 60  Fading Channels  61  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 R L S Blind Detection  68  4.3.4  Blind Subspace Multiuser Detection  74  Performance Evaluation Measure  79  4.4.1  79  Receiver Complexity  4.5  Simulation Results  83  4.6  Summary  87  T F - M C - C D M A B l i n d Multiuser Detection in Downlink Mobile R a y l e i g h F a d i n g C h a n n e l w i t h D o p p l e r 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 R L S Blind Detector  98  vi  5.3.3  6  Blind Subspace Multiuser Detection  99  5.4  Simulation Results  103  5.5  Summary  105  C o n c l u s i o n and F u t u r e Research  Ill  6.1  Conclusion  Ill  6.2  Future Research  113  6.2.1  M I M O 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 CDMA  6.2.3  114  T F - M C - C D M A Blind Multiuser Detection in Fast Fading Channels  115  Bibliography  116  vii  List of Tables 2.1  Summary of channel fading models i n mobile wireless channels . . . .  26  3.1  B l i n d T F - M C - C D M A D M I Receiver A l g o r i t h m  40  3.2  B l i n d C M O E R L S Receiver A l g o r i t h m  42  3.3  B l i n d Subspace S V D Receiver A l g o r i t h m  3.4  B l i n d Subspace P A S T c l Receiver A l g o r i t h m  3.5  Summary of T F - M C - C D M A B l i n d Multiuser detection complexity for  •  45 48  A W G N channel  50  3.6  Simulation parameters  50  4.1  T y p e I C M O E R L S B l i n d Receiver A l g o r i t h m  72  4.2  T y p e II C M O E R L S B l i n d Receiver A l g o r i t h m  73  4.3  T y p e I B l i n d Subspace ( S V D ) Receiver A l g o r i t h m  77  4.4  T y p e II B l i n d Subspace ( S V D ) Receiver A l g o r i t h m  78  4.5  T y p e I B l i n d Subspace P A S T c l Receiver A l g o r i t h m  80  4.6  T y p e II B l i n d Subspace P A S T c l Receiver A l g o r i t h m  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  . . .  4.9  Simulation parameters  5.1  T y p e II C M O E R L S B l i n d Receiver A l g o r i t h m for multipath fading  84 84  channels w i t h Doppler induced ICI viii  99  5.2  Type II Blind Subspace (SVD) Receiver Algorithm  101  5.3  Type II Blind Subspace P A S T d 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  2.4  Cyclic Prefix Insertion  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 S N R 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  19 -. . .  x  27  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 S N R 16clB . . .  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  91  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  I  Identity matrix with dimension m x m  m  xii  List of Abbreviations 1G  First Generation  2G  Second Generation  3G  Third Generation  4G  Fourth Generation  AMPS  Advanced Mobile Phone System  AWGN  Additive White Gaussian Noise  BER  Bit Error Rate  BPSK  Binary Phase Shift Keying  CDMA  Code Division Multiple Access  CMOE  Constrained Minimum Output Energy  DFT  Discrete Fourier Transform  DMI  Direct Matrix Inversion  DS-CDMA  Direct Sequence C D M A  EDGE  Enhanced Data Rates for G S M Evoluti  EVD  Eigenvalue Decomposition xiii  FDMA  Frequency Division Multiple Access  FFT  Fast Fourier Transform  FIR  Finite Impulse Response  GPRS  General Packet Radio Service  GSM  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  LMMSE  Linear Minimum Mean Square Error  LTV  Linear Time Varying  MAI  Multiple Access Interference  MC-CDMA  Multicarrier C D M A  M C - D S - C D M A Multicarrier Direct Sequence C D M A MMS  Multimedia Messaging Service  MMSE  Minimum Mean Square Error  MOE  Minimum Output Energy  xiv  MUD  Multiuser Detection/Detector  OFDM  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  TDL  Tapped Delay Line  TDMA  Time Division Multiple Access  TF  Time Frequency  T F - M C - C D M A Time Frequency spread Multicarrier C D M A UMTS  Universal Mobile Telecommunication System  WGN  White Gaussian Noise  WLAN  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 colleagues 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  THE UNIVERSITY  OF BRITISH  COLUMBIA  October 2007  xvi  T A M  my parents.  xvii  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 division multiple access ( F D M A ) . 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 E D G E / G P R S ( 2 . 5 G ) provided subscribers with a wider variety of data communication services on top of the still dominant voice service which include limited internet 1  access and the popular short messaging service ( S M S ) . Current t h i r d generation ( 3 G ) 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 utilization of b o t h 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 w i t h 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. A l t h o u g h the international 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 w i 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 Channels  Existing 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. W h e n 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 timing information which are usually acquired through training sequences and/or the use of pilot symbols. U t i l i z i n g 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 D S - C D M A , multicarrier based transmission multiple access schemes have emerge as potential candidates for next generation access air interfaces [2].  1.2  Multicarrier-based Transmission  The concept of transmitting information over multiple subcarriers in parallel substreams 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 technology 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 I E E E 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 n a r r o w b a n d s u b c h a n n e l c a n b e t r e a t e d as f r e q u e n c y n o n - s e l e c t i v e f a d i n g . T h e success of O F D M , p a r t i c u l a r l y i n W L A N s y s t e m s , h a s e v o k e d i t s p o s s i b l e a p p l i c a t i o n i n b r o a d b a n d m o b i l e c o m m u n i c a t i o n s [5] a n d s o l v i n g t h e n e e d for c o m p l e x e q u a l i z a tion in 3 G D S - C D M A  1.2.1  systems.  Multicarrier C D M A  M u l t i c a r r i e r C D M A ( M C - C D M A ) . first p r o p o s e d i n [ 6 . 7 ] is a m u l t i p l e access scheme t h a t is b a s e d o n a c o m b i n a t i o n of O F D M a n d D S - C D M A . T h e m a i n i d e a of M C CDMA  is t o use O F D M t r a n s m i s s i o n w h i l e s e p a r a t i n g m u l t i p l e access users u s i n g  C D M A . T w o p o p u l a r i m p l e m e n t a t i o n s of M C - C D M A exist: t h e first one r e t a i n s t h e name M C - C D M A ,  t h e o t h e r is k n o w n as M C - D S - C D M A .  MC-CDMA  spreads the  user's s i g n a t u r e sequence i n t h e f r e q u e n c y d o m a i n b y p r e - m u l t i p l y i n g a c h i p a t e a c h O F D M s u b c a r r i e r b r a n c h . T h e r e f o r e , t h e sequence of c h i p s across a l l t h e s u b c a r r i e r s c o n s t i t u t e s t h e user's f r e q u e n c y d o m a i n s i g n a t u r e sequence. M C - D S - C D M A , o n t h e o t h e r h a n d , s p r e a d s t h e o r i g i n a l d a t a sequence i n t i m e first u s i n g D S - C D M A a n d t h e n t r a n s m i t t i n g t h e t i m e d o m a i n spreadecl d a t a s t r e a m u s i n g O F D M t r a n s m i s s i o n . A d e t a i l e d c o m p a r i s o n of t h e t w o m u l t i c a r r i e r m u l t i p l e access schemes is g i v e n i n [8] a n d [9]. B o t h M C - C D M A  and M C - D S - C D M A  retain the robustness to frequency  selective f a d i n g w h i l e u n c o m p r o m i s i n g t h e b e n e f i t s of u s i n g C D M A .  1.2.2  Time-Frequency Spread Multicarrier C D M A  R e c e n t l y , a n e w m u l t i c a r r i e r b a s e d m u l t i p l e access s c h e m e k n o w n as t i m e f r e q u e n c y spread M C - C D M A  ( T F - M C - C D M A ) w a s p r o p o s e d b y [10,11] w h i c h c o m b i n e s M C -  C D M A a n d M C - D S - C D M A . T h e m a i n i d e a b e h i n d T F - M C - C D M A is t o s p r e a d t h e d a t a i n b o t h the time a n d frequency domains. MC-CDMA.  Firstly, like M C - C D M A  T h e r e are m a n y a d v a n t a g e s t o T F -  and M C - D S - C D M A , T F - M C - C D M A  4  inherits  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 veryattractive 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 , D S - C D M A , 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 M A I ; however, it is also known that multiuser detection methods almost always significantly increase the complexity of the receiver. Furthermore, [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. A n optimal multiuser detection method (which minimizes the probability of symbol error) would be overly complex. Linear multiuser detection methods (a class of suboptimal multiuser 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 appropriate 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 decomposition 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 decomposition. 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 implementation 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 intercarrier 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 modeled 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 suppression 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  Thesis Contributions  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: • A n 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 number 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 problem 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 equivalent 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 D S - C D M A : - 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 nonorthogonal 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 subspace tracking using PASTcl are all applicable to type I implementation. The only added complexity comes from the joint blind channel estimation. Using computer simulations, the performance for type I detector using C M O E RLS. blind subspace detection using S V D and adaptive subspace tracking using 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 detector uses a cascade implementation in which the output of the time domain detector is fed into the frequency domain detector. Since the time domain signatures 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 R L S algorithm, blind subspace detection using SVD, and adaptive subspace tracking using PASTd are investigated using 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 T F - M C - C D M A 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 S V D 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 T i m e Frequency Spread M u l t i c a r r i e r Spread Spectrum System M o d e l  2.1  Introduction  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 modulation 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 discussions of receiver designs in the remaining chapters. Section 2.3 describes the additive 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 downlink 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 a n s m i 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  c h a p t e r is p r o v i d e d i n S e c t i o n 2.5.  2.2  Transmitter  Model  T h e t r a n s m i s s i o n m o d e l c o n s i d e r e d for the T F s p r e a d M C - C D M A s y s t e m follows t h a t i n [17]. T h e t r a n s m i t t e r c a n b e t h o u g h t of as K users t r a n s m i t t i n g s i m u l t a n e o u s l y a n d s y n c h r o n o u s l y . T h e b l o c k d i a g r a m for the k  ih  user is s h o w n i n F i g . 2.1. It is a s s u m e d  t h a t a b i t s t r e a m . c/ [i] £ { 0 1}, f r o m a b i n a r y s y m m e t r i c source ( B S S ) is to b e sent fc  for t h e k  th  ;  user, k G { 1 , 2 . . . . . A " } .  T h e b i t s t r e a m is t h e n m o d u l a t e d u s i n g b i n a r y  p h a s e shift k e y i n g ( B P S K ) i n t o b [i] k  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 i n e a r t i m e i n v a r i a n t ( L T I ) filter w i t h a r e c t a n g u l a r p u l s e s h a p e i m p u l s e response of d u r a t i o n T  b  r e s u l t i n g i n a n o u t p u t w a v e f o r m g i v e n b y 2.1. oo (2.1)  14  T h e signal b (t) is then mixed w i t h a direct sequence (DS) time domain ( T D ) code k  a (t) given by 2.2 w i t h spreading gain N and chip duration T , k  c  N-1 a (t) = J ] a fc  A ;  (2.2)  [j]F (t-. T ) f c  ?  c  .7=0 i  where NT  C  = TJ, and  chip sequence.  a [0)  ...  k  r = a is the k  a [N-l]  user time domain  k  k  T h e D S time domain spread signal is replicated over M branches  onto M sub carriers of freciuency u , {u. : m = 1. 2..... A4}. In branch m, a (t)b (t) is m  m  k  k  pre-multiplied by a frequency domain chip sequence c [m] of duration T . We further k  c  assume that the set of subcarrier frequencies satisfy the orthogonality condition over the interval T given by c  exp (j (u t + <p )) exp (-j(u>it + <bi))dt = 0, m  m^l  m  (2.3)  JO  and  exp (j (u t m  +0  m  )) exp ( - j (cu i, t +fa))dt ^ 0, m = I  (2.4)  J o  Hence, together w i t h the carrier frequency u , the transmitted signal s (t) for the A;' '  - 1  c  fe  user is given by the sum of the outputs of all M subcarrier branches, i.e.  A'J Sk{t) = \  JTYI k(t)a (t)c [m]  cos {(u + u ) t}  b  k  k  c  m  (2.5)  171=1  where P is the energy of s (t). i.e. k  P=  / •Jo  s (t)sl(t)dt  (2.6)  k  T h e overall transmitted passband signal is  *(*) = ] > > ( * ) A:=l /\  (2.7)  A'/  ^2 ^ ^fc(O A:(0 fcH COS {(CJ + U ) t} k = l 771=1 a  C  C  15  m  2.2.1  Equivalent Complex Baseband Signal  It is k n o w n f r o m [32] a n d [33] t h a t a t r a n s m i t t e d passbancl s i g n a l c a n b e w r i t t e n i n the form  s(t) = v / 2 R e {s (t) e x p (ju t)} LP  (2.8)  c  w h e r e sr,p(t) is t h e e q u i v a l e n t l o w p a s s c o m p l e x b a s e b a n d s i g n a l of s(t).  E q u a t i o n 2.7  c a n b e r e - w r i t t e n as /\  s(t) = \ / 2 R e  M  \\/^Y,zZ M  h(t)a (t)c [,n) e x p {ju t} e x p {ju t} \ >k  k  m  c  (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 t h a t  '<  rjj SLp(t) =  M  yjr J2Yl  h(t)a (t)c [m]  f  IT  k  k  \  K  A:=l  e x p (jwmt)  1  M  l?n=l  J  T h e t e r m i n t h e c u r l y b r a c e s i n 2,10 c a n b e r e a l i z e d a n d i m p l e m e n t e d u s i n g t h e IDFT  [4] w h e n t h e s u b c a r r i e r frequency set { o ; } ^ m  = 1  is c h o s e n t o h a v e m i n i m u m  s u b c a r r i e r f r e q u e n c y s e p a r a t i o n a n d satisfy t h e o r t h o g o n a l i t y c o n d i t i o n g i v e n b y 2.3 a n d 2.4. T o find t h e m i n i m u m p o s s i b l e s u b c a r r i e r f r e q u e n c y s e p a r a t i o n , t h e o r t h o g o n a l i t y c o n d i t i o n g i v e n i n 2.3 c a n b e r e - w r i t t e n as  ex-p(j(u  m  - (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 h o l d s i f a n d o n l y i f  (u  — i)~^ u  m  =  >  n7l  n  is  a  n  Y n o n z e r o integer  (2.12)  T h e m i n i m u m s u b c a r r i e r frequency s e p a r a t i o n is o b t a i n e d for n — 1 a n d is e q u a l t o fr.  If m i n i m u m s u b c a r r i e r frequency s p a c i n g is used, t h e s u b c a r r i e 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  uj =LO m  0  + ^  ,  m = 0,l,... M-l  (2.13)  ;  c where UQ is an arbitrary frequency offset. For u  0  SLp{t)  = 0 , the complex baseband signal  can be written as  rfj  siAt)  K  M  = V^X>(*K(*)  l  E  A:=l  1  f m \  (~  2n  CfcNexp  U-y^t) ^  (m.=0  c  '  (2.14)  )  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. Jc  s p[n] = s L  The  LP  =y/^E  6  * *M | E a  Cfc  H  exp  |  (2-15)  term inside the curly braces in 2.15 is an M - p o i n t 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. T h i s is shown in F i g . 2.2.  17  2.2.2  A l l o c a t i o n of S p r e a d i n g 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 T D S 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 T D S and FDS sequence pairs by grouping the K users in the system into N groups with each group being represented by one of the A T D S sequences in the set { a r  1;  a  2 ;  a^}.  where a  r k  a [l] k  •••  a [N] k  , 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  from the set {ci, Co.,c,w}  where c =  Cfe[l] . . .  contains the FDS of the k  k  th  c [M] k  . is the vector that  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 T D S 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 T D S sequence A is equal to T  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 T D S 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 T D S 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. selected from the first column.  The next pair is  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 0  th  A4  th  row actually corresponds to the  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 n  th  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 s  rrjp  N  x<> M  ^ = V Jd zZ E E M*K(*)<Um] cos {(u  c  71 =  1 K= 1  + u ) t} m  (2.16)  771=1  The equivalent complex baseband signal can also be re-written as scp(t)  =  sJlZ YZ zZ zZ bvAt)a (t)c [m] M n  K  exp (ju t)  (2.17)  m  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 T F 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 F D , the severity of the M A I is determined by the pairwise cross correlations of the T D S sequences {pij'.i.j  = 1,2..... K}, and also the pairwise cross correlation  of the frequency-domain spreading sequences {P j] i.j = 1, 2..... K}, where V  Mil faj  = ii  (2-19) Ci[m]cj[m]  E 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 M u l t i p a t h 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 attenuations. 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,hi ,(t,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. R (t, t', T, T') = E {h*(t, r)h(t\ r')} hh  (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. R (t, t + A t , T, T') = R (At, hh  hh  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 uncorrelated. 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, (t.,t'.T.T>) hk  = P (t,t',r)5(r Ml  22  - r')  (2.22)  where P {t.t',r) hh  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  (2.23)  h{t,T) = Y 9i(t)5{r-T ) J  l  1=0  where gi(t), 77 represent the complex path gains and the delay of the I  th  time correlation of the complex path gains of the l  tk  path. The  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 T . s  the tapped delay line model channel response is L'-l  (2.24)  h(t,r) = ^2 {t)5(T-lT ) gi  s  1-0  where L' is the length of the channel impulse response in units of T . s  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 I  th  path arrives  in an incident angle 6. then it experiences a Doppler shift given by v=—cos(e) c  = f cos{6) d  where f i denotes the maximum Doppler shift and is equal to c  (2.25) It is further assumed  that the arrival incident angle is uniformly distributed, i.e.  (e) = ^,  Pe  #e(-7T +7r] ;  23  (2.26)  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) =  ,'  (2.28).  A 1  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.) = olJ (2irf At) /(/)  0  (2.29)  d  In Rayleigh fading, the gain on the path I, cji, is assumed to be zero mean complex Gaussian with variance af  v  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 R M S delay spread is given by  (2.31)  / = 1  XXI 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 R M S delay spread B  « —  c  (2.32)  Trm,s  Then, with respect to the symbol period T (and system bandwidth B = ^-), two s  types of fading are differentiated. If T » s  r  rms  s  or equivalently B << Be, then the s  channel is a flat fading channel or a frequency non-selective channel. If T « s  r  rms  or  equivalently B >> B , then the channel is a frequency selective channel. A flat fading s  c  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 frequencies over which the Doppler spectrum is non-zero. In Rayleigh fading, the Doppler bandwidth is approximately equal to the maximum Doppler shift B  D  « f = ^ d  (2.33)  The coherence time Tc can be defined as being proportional to the inverse of the Doppler bandwidth T « c  D  (2.34)  With respect to the symbol period, one can differentiate two other types of fading. If T << Tc or equivalently B >> Bp. then the channel is a slowly fading channel. s  s  If T >> Tc or equivalently B << Bp, then the channel is a fast fading channel. In s  s  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.  CHANNEL TYPE frequency non-selective, slowly fading  CONDITIONS T  »  s  T « s  frequency selective, slowly fading  T « s  T « s  frequency non-selective, fast fading  T  »  s  T » s  frequency selective, fast fading  T  «  s  T » s  T , B « rms  s  T, B » c  s  r . B » rms  s  T, B » c  a  T . B « rms  s  T, B « c  s  T , B » rms  s  Tc, B « s  B  c  B  D  B  c  B  D  B  c  B  D  B  c  B  D  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 I D F T and D F T modulation 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#[ M - \ f c  n  l  (- ) 2  35  /.=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 I D F T and D F T , 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 =  fjo  as the channel impulse response.  9L-1  then we have y  = d *h  DFT  M  (2.36)  {y}  = DFT  M  {d * h} = DFT  M  {d} o DFT  where o denotes element by element multiplication and  M  DFTM  {h}  denotes the M-point  D F T . From the above, one can define an equivalent frequency domain transfer function H of the channel impulse response h  M H  = DFT  M  {h}  =  i-o  9i exp  L-l  -j2iTml  , M > L (2.37) 1=0  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 signal 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 M o d e l Demodulation  The system block diagram of the demodulator is shown in Fig. 2,5. From the demodulator system block diagram, the received signal (signal plus noise) coming from the  demodulator output at the n  time domain chip of the  Ul  = I  °  r{tyP (t  subcarrier is given by  - nT ) cos{uit)dt  Tc  (2.38)  c  J{n-l)T  c  The noise component, specifically, of the output of the n  time domain chip epoch  th  of the I  th  subcarrier is given by piT  c  =  n{t)P {t-nT )cos{u)it)dt Tc  (2.39)  c  J(n-1)T  C  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} =  n(t)P (t - nT ) cos(uj,t)d,i  E \  Tc  c  U(n-1)T  C  -nT  c  '(n-l)T  E{n(t)}P (t  - nT ) cos(uj,t)d,t  Tc  (- ) 2  c  40  c  = 0 Furthermore, we have E  {N®N} } m)  =  I" •nT  J(n-l)T '(n-l)T.  1  l  10 fjTc  c  r  JlC  E  (n(t)n(r)} P (t - nT )P (r c  Te  Tc  -  jT )x c  J(i-1)T J(j-1)T  C C  c  x cos(oj/t)  cos(u) r)dtdT m  5(t-T)P (t-nT )P (r-jT )x Tc  (n-\)T JU-l)T c  x cos(o;/i)  rc  c  cos(u T)dtdT m  c  AT  rjT  nT  c  / (n-l)T  c  1 c  c  x cos(cu/t)  -rr&(t ~ T)8 8 P (t ni  J(j-1)T  ml  Tc  - nT )P {r c  Tc  -  jT )x c  Z C  cos(u r)dtd,T m  = '-^SnjSml. /  COS (UJ it) dt 2  J(n-l)T  1  c  NT 0  -  C  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 Complex Baseband M o d e l Demodulation  The demodulation for the equivalent baseband model implemented using I D F T modulation is accomplished using the inverse operation - the -DFT. Applying the M point I D F T to the sampled received signal given in 2.15, DFT  SuM exp {^^)  {s [n}} = - = £  M  LP  (2-42)  n=0  Expanding and applying the orthogonality principle yields DFT  M  nr  /  M-\  K  CM-i  {s [n}} =  ^  b  LP  \  7i.=0  E  k.=l  * H ,exp / 'znrnn [j^) \ l \  C  r  {m=0  x  x exp  7 7P zsr^ L  7  E  i  r  /\  -J^~ E  2n  n=0  .A:=| 77?.=0  -p  (l  -  m  } ^ exp I  m  )  n  V  M-l  E  fc fc.? fcHM5 /  fe  a  c  mm  fc=| 771=0  =  v/P^E  6 «fcjC f c  f c  [???/]  fc=i  (2.43) The sampled complex lowpass noise process is given by W[m] = DFT  { w  M  DFT {w[n}}M  M  (2.44)  M-l  1 NT^ n (-j2irmn\ -= } w [n exp — s/M ^ V J r  M  71=0  The statistical properties of the complex baseband sampled noise process are  £ {VF[m]} = E ,1=0  E  W"]}  e x  P V  = 0  30  7 W  /  (2.45)  Figure 2.6: Complex Baseband Modulator and Demodulator  E W[m]W[l]} {  =  ±  £  £  71=0  =  £ { [n]«;[n']} W  71/=o  exp  (^fp) ^  exp /  ( - ^ ) \  -  /  N5 0  mI  (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 interferers 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 Multiuser 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 clecorrelation 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 multiuser 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 multiuser 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 discussion 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 F D and hence provide a better performance than that of conventional 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 bne{-i.+\} 6 -e{-i,+i} A  q  1 - E  kPlkfilk  b  (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  tb  16  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 n  sampled time chip from the m  lh  tk  subcarrier stream of the  T F - M C - C D M A passband demodulator in A W G N channel is given by = 5.,  ( m) 7  + N™,  n = 1,N:  m = 1, ...,lVI  (3.2)  where  I? P T S™  h  = J~^ha [n]c [m] k  (3.3)  k  fc=i  and it was shown in 2.40 and 2.41 that Nn'^ is a zero-mean Gaussian random variable with covariance ^-^fS jd i. n  m  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 z  = Z- \ m  mn  n=l,...,N;m  = l,...,M  (3.4)  Based only on the observation data and the knowledge of user l's T D and F D signature sequence pair, our goal is to compute two sets of multiuser detection coefficients (one for the T D , one for the FD) such that the mean square error (MSE) is minimized. Mathematically, this is stated as arg min  (\V(.w* ) = f  where A = yj^r^  i l g  n  e  E  j\\Ab\  — (w' Z)* wy||"| v  (3.5)  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 B l i n d Multiuser Detection for T F MC-CDMA  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 com35  plex problem. This motivates us to propose an intuitive albeit suboptimal approach by decoupling the joint optimization problem into two separate optimization problems, one for the blind T D M U D and one for the blind F D M U D . To do this, we first note that if one considers one particular subcarrier stream rn. then this corresponds to m  Lh  column of the observation matrix Z. TS ^ V  "  N[  m)  {m)  N  +  s  _  (m)  +  n  (m)  (3.6)  N  Substituting 3.3 into 3.6 yields A'  2PT  r  M 2  A:=.l  where a user  k  a [N]]  a [l]  k  k  (3.7)  n^  +  k  is the time domain spreading chip sequence of  T  k  (b c [m])a  k  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 where A , B [i], s k=l W  k  k  k  are the received amplitude, the  i  th  user respectively and w[z] is a  transmitted symbol and signature sequence of the k  th  zero mean complex Gaussian random vector. If one denotes b c [m\ k  k  as B , then 3.7 k  can be interpreted as a D S - C D M A multiuser signal. Similarly, considering the n  Ul  time chip epoch, one would get the n  th  row of the  matrix Z and get the equations of the same form as 3.6 s  (l)  "  o(2)  On  z: = j  T  +  MM)  36  T  (3.8)  Time D o m a i n MUD 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  ^ = Eyi7T where  ( 6 f c a f c [ n ] ) C f c+  n  ^  (3  -  9)  is a zero mean Gaussian random vector with covariance ^ ^ T v / x M - Denot-  ing b a [n] as B - then 3.9 can also be interpreted as a D S - C D M A multiuser signal. k  k  k  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 (b c [m]) — w'fz^ k  k  ' w * = argmin E \ \\A (b a ,[n]) - w'/z \\ w  || } 2  e C J V X l  f  2 1  \  T  k  W  /  e C  U  x  1  k  '  1  (3.10)  >  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 D M I 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 f E | w eC ' A  M ( M) j , _ W  Z  Z  w  2  w  f Re {A*E {z^  {b c [m])}} k  k  x l  (3.H)  t  wj — argmm w ? £ {z£ ( V / ) " } w , - - 2 w f Re [A*E {z  T n  (b a [n})}} k  k  v eC  Mxl  f  Setting the derivatives with respect to w and w / in 3.11 to zero yields t  2C w* - 2 | A | a  f c  2C  f c  2  t  }-2|A| c 2  / W  = 0  (3.12)  = 0  since E{zW  (b c [m])} = Aa k  k  k  E{zZ(b a [n])} k  = Ac  k  k  38  (3.13)  Defining C,  EE E ^ < » )  t  ( (m)^ z  >  (3.14)  the detector coefficients for user one axe then given by w* =  \A\-C7 an x  (3.15)  w* = \A\" C f Cfc 1  f  The matrices C and Cf can be estimated through their corresponding sample cort  relations from the received signal either through block sampling averaging or aclaptively/recursively updated using a forgetting factor as in C \p,m]  = XC \p,m  - 1] + (zW\p])(zW\p])  C [p,n]  = XC [p,n  - 1] +  t  t  H  (3.16) f  r  where A is the forgetting factor. C [p,m] t  correlation matrix for the rn  u>  p  ih  (z \p] )(z \p] ) n  T  T  n  H  is the update of the time domain sample  subcarrier stream, {rn : m = 1,2. .... M}. during the  transmitted bit, {p : p = 0,1, 2,...}, and C/[p, ri] is the update of the frequency  domain sample correlation matrix for the n  th  during the p  ih  time chip epoch, {??, : n = 1,2, ...,N}  transmitted bit. It can be noted that C [p, 0] is equivalent to C [p — t  t  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 B P S K , the decision is computed as bi\p] — sgn (Re { ( w f ' Z ) * / } ) -and is unaffected by w  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 D M I Receiver in A W G N In the p symbol, input matrix Z[p] for(m = 1,2,...,M) update C [p,m] = XC [p,m - 1] + (z^ {p})(z^[p}) end for for(n = 1,2,..., AO update C {p./n] = XC p. n - 1] + (z [p] )(z [p] ) end for Compute Detectors w* = C^ax Ul  n)  t  H  t  T  T  f  w* = f  f  n  //  n  Cj ci l  Perform detection 6 [p] = sgn(Re{(w * Z)*w* }) tf  1  t  f  Table 3.1: Blind T F - M C - C D M A D M I 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 D M I algorithm is summarized in Table 3.1.  3.3.2  C M O E Recursive Least Squares ( R L S ) M e t h o d  The main drawback of the DMI method is the need for inverting the sample correlation matrices C and Cf. To avoid computing matrix inversions, the method of recursive t  least squares (RLS) can be used. The R L S method is based on solving constrained minimum output energy ( M O E ) / C M O E detector for blind multiuser detection in DSC 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 M S E . 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*  w c-<vxi  l  j , wf a = 1  2  l  '  te  w^  z<"> ||  = argmm E { | | w f  (  j ||w^z^|| I , wf C i 2  = argmin E  3  1  7  )  1  =  Based on the C M O E constrained optimization problem above, the R L S algorithm computes the detector weights according to the minimization of the sum of exponentially weighted mean-square output with a forgetting factor [39]  w* t  = argmin £ M  w* = argmin ]C ^ e c ' 11=0 x  m  w  ...... %  f  A  || f z( )|| , wf ai  A'^M  = 1 ( 3  | | 2  ||  f n||  w  z  >  = 1  wfci  -  1 8 )  1  W/  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 subcarrier stream of the p  I  th  symbol. Similarly, the second summation in 4.15 states  th  that the sum is over all time chips going from the zeroth time chip from the zeroth transmitted symbol to the j  chip of the p  Ul  th  symbol.  The solution to the optimization problem in 3.18 (up to a real positive scaling constant) has a similar form to the D M I method  = cr^i  wf  (3.19)  1  W* = C T ^ l f  where again.  CAp. ml = AC, [p. m - 1} + z< )z( >" m  1  J  L  m  (3.20)  C \p n]=XC \p,n-l]+zl(zl)  ff  f  1  f  To compute the detector, the R L S 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 R L S Receiver in A W G N Initialization: $, [0,0] = W , */[0,0] ^I In the p symbol, input matrix Z[p] f o r ( m = 1.2 V) Compute Kalman Gain: ;  MxM  th  ki\p,m\  —  —  '•  —  l+A-' z( )[p]H* [p,jn-l]z( )[jrj] l  m  m  t  Compute Update: <& [p,m] = A " ($>f\j),m - 1] - k/ (z [p]) <&[p. m - 1] 1  (m)  7/  t  t  end for for'/; = 1,2,..., AT)  Compute Kalman Gain: k  f  r „ •  /  U  „1 '  i  A-^ [p,n-l]z [p] /  =  n  r  H-A- (z, .[p]T) * [p,„-l]z„.[p]^  J  |  H  1  /  Compute Update: $/[p,n] = A " ($ \p,n  - 1] - k/ ( z ^ p ] ) " $/[p,n - 1]  1  7  f  end for Compute Detectors w* = ^ a = $fCi Perform detection 6 [ ;] = s g n ( R e { ( w ; Z ) * w * } ) t  x  tf  :l  ?  :  3.2: Blind C M O E R L S Receiver Algorithm  the matrix inverse. Letting  ~ C [j] t  1  and <&/[.?'] ~ Cf[j] •, the matrix inversion 1  lemma gives d,  r„-i  ^tUJ  _ \-i<r> -  A  q  r„-  -MJ  _ 1J  A- $ [7-i]z<"Oz<">"$ [j-i] 2  t  t  i A-i (»')«$ [ -i] ("') +  z  t  ; i  z  ,  3  2  1  x  where j is the iteration index. The R L S 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<'>)"} = S | A | S f + a*I 2  t  NxN  (3.22)  where  [ai|a |...|a 2  r  2  D f  (3.23)  A'o T  C  |A|  = diag(A\,....  2  A ) 2  Ut  and Ui, represents the number of linearly independent T D S signature sequence used in the system, and Ai represents the superposition of the amplitudes of all users that uses the TDS signature i. i =  U . It is assumed that the spreading signature sequences t  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 matrix  JM^}.  a r etrie  U largest eigenvalues of C in descending order, the N by U t  t  t  contains the corresponding Ut eigenvectors as its columns and the N by  N — Ut UJ? contains the A — U eigenvectors corresponding to the eigenvalue of. r  t  43  The blind L M M S E detector is obtained by substituting the subspace decomposition of C into 3.19. i.e. t  w* = C ^ a , = ( U ^ u F ' + ^ I f ' u f l A = U^CAWj^U^ax + ^ U ^ a , = ui (Ai )Ui i )  t )  t ) W  a  (3-25)  1  where the last equality holds since the eigenvectors contained in the matrix  span  the signal subspace which means that the vector a can be expressed as a linearly x  combination of the eigenvectors in the matrix ui*' and therefore )  (U<<>)"ai = 0  (3.26)  Using a similar analysis for the F D case in the decoupled multiuser detection problem yields w} = U ( ( A . ( ) ) - U i / )  /  1  / ) / /  c  1  (3.27)  Hence, the blind L M M S E detector for both the F D 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 P A S T d algorithm.  Singular Value Decomposition  The subspace parameters can be tracked by applying the singular value decomposition 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 D M I 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 [p,m] = XC \p,m - 1] + {z \p)){z^\p]) end for for: „ = 1. 2,.... N) Compute Update: {m)  t  H  t  C [p,n] = XC \p,n1] + ( [p] )(z [p} ) end for Apply Eigenvalue decomposition to C and C / to get eigenvectors and eigenvalues: T  f  f  Zn  T  H  n  t  Compute Detectors  w* = U ^ W V  W  ) / f  c  : l  Perform detection ^[p] = sgn (Re { ( w ^ Z ) * w * } ) f  Table 3.3: Blind Subspace S V D Receiver Algorithm reaches convergence after a relatively few data symbols and re-computing the subspace 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 depend on the coherence time of the fading channel. The T F - M C - C D M A blind subspace algorithm using S V D is summarized in Table 3.3.  A d a p t i v e Subspace T r a c k i n g U s i n g P A S T d A computationally more attractive algorithm for estimating the subspace parameters 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 P A S T d algorithm was first pro45  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] || }  (3.28)  2  where v[i] is a noisy observation vector with correlation matrix given by C E {r[z]r['i] } and W £ C ,q w  < N.  Nxq  r  =  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 Q where U r  r  is a N by q matrix that  contains any q distinct eigenvectors of C , and Q is any q by q unitary matrix. r  2. A l 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 eigenvalues) of C . In that case. J(W) attains the global minimum. r  Here, we are interested in the case when q = 1. Then the solution to the optimization 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 : hence, we modify the original optimization problem into the exponential r  weighted sum with a forgetting factor 7 i  \\r[n] - W[z]W[i]"r[n]||  J(W) = Yi'~  n  (3.29)  2  The PASTcl algorithm has two main components: (1) Projection approximation and (2) deflation. The projection approximation is W[i] r[n] ~ W[n - l] r[n] w  H  = 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]|| n-0  46  2  (3.31)  The R L S algorithm can once again be applied to solve for W[z]. which, by property (2) above, is an approximation to the eigenvector of C . corresponding to the largest 7  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 T D S 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,(w , t  w  f  = — - — , —irnvx H E{Var ((w,"Z[2;])*w |6 [p]) } r  /  -  3  32  ]  From the output SINR information, we can evaluate the convergence speed for multiuser 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 p symbol, input matrix Z[p] and previous estimates th  for(m = 1 , 2 , M ) Input z [p] Set x[° = z(' )[p] ( m )  n  for(fc = 1,2,...,[/,,)  Projection Approximation: yf* = v^\p,m — l j ^ x ^ Eigenvalue Update: AJ^ [p, m] = ^\ ^[p.m, — 1] Vk Eigenvector Update: k  (0* y,,  r,(*)r[p,m - 1] + vS \p,m] = u^ l)  ;  k  Deflation: x (0  4 ° - u ^ b v m - l]:(/{  0  4° - UfcV,™]?/?  end for end for  for(n = 1 . 2 . . V ) Input z [p\ r  n  Set 4  = z \p]  f)  T  n  I\ „(/) u ^ n - l ] ^ Projection Approximation: y Eigenvalue Update: A;(/) ^ f p , n] = 7AJ^ [p,n — 1] +  f o r (A;  -  1.2  y  k  Eigenvector Update: p. ?7,]  u  =  [p, n — 1] +  u  h  (/) Deflation: x).;  k  x  end for end for  fh  f)  x ^ - u,(/)^ n - l ] ^  ,t„(/)' u[P\p,n]yl  _ „ ( / ) +1  vi  X[ {p,n]  Compute Detectors wr = t j W ( A i ) - t j i t)  1  t)H  )/A(/)^lTj(/) w* = U ( / W ) f  Perform detection bi\p] = sgn (Re {(w:  a  1  f f C l  w  ill  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 D M I 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{N  3  + M ). The C M O E RLS algorithm has a complexity of  0(N )  3  per update for the time domain detector and 0(M )  2  per update for the frequency  2  domain detector. However, in our algorithm, there are M updates per symbol for the T D detector and N updates per symbol for the F D detector. Therefore, the total complexity for the C M O E R L S algorithm is 0(NM  2  + MN ).  The subspace  2  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(N ) 3  and for the frequency domain is 0(M ).  Therefore, the  i  total complexity for the blind subspace S V D method is 0{M  :i  + A^ ). The adaptive 3  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  the time domain. For the frequency domain, the complexity is denotes the number of users within the n  th  domain signature sequence.  per update for  0(x M) n  where Xn  time domain group or using the n  th  time  Under a fully loaded system, the adaptive subspace  tracking multiuser detection using P A S T d is then 0(N  2  49  + M ) per update. Suppose 2  Multiuser Detector DMI C M O E RLS • Subspace S V D Subspace PASTcl  Complexity 0(N'  +  0(NM 0{N  +  A  2  :i  0{MU N t  M) :i  MN ) 2  + M ) 3  +  N M) Xn  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 F D spreading gain Number of subcarriers TDS sequence type FDS sequence type  7 15 15 M-sequences M-sequences  Table 3.6: Simulation parameters  the number of T D S sequence used is U and Xn denotes the number of users within t  the n  T D group, then the total complexity for the blind subspace PASTcl algorithm  th  is 0(MU N t  + Nx M) n  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 D M I method, the M O E R L S method, the subspace S V D 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 D M I 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 algorithm using PASTcl has a slightly degraded performance: the degradation increases slightly with loading and at a target BER, of 10~ , the penalty is approximately l d B 3  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 considered, 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 difference 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~ . Nonetheless, comparing with the 4  matched filter detector, the C M O E R L S 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 D M I and the subspace S V D 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 D M I and subspace S V D receiver for the 49 and 77 user case and slightly greater than l d B 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 R L S converges the fastest, the output SINR reaches stabilizes after approximately 60 data samples. The convergence of the D M I and subspace S V D 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  J  i  - 4 |  0  5  \  10 S N R (dB)  \  \ 15  I 20  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  N = 7, M = 15, 77 Users (moderate heavy load), AWGN Channel 10  u  4  -e V  CC LU CO  10  _  DMI CMOE RLS subspace(svd) subspace(PASTd) matched filter  2  10"  1  c  p l  0  1  1  5  10  1¥—*  '  15  ^  1  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 problems, 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 decomposition method and the adaptive subspace tracking method using the PASTcl algorithm. 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 P e r f o r m a n c e w i t h 105 U s e r s  54  N = 7, M = 15, 49 Users (moderate load), AWGN Channel, Output SINR  500  F i g u r e 3.5:  1000 Number of Data Samples  1500  O u t p u t S I N R w i t h 49 U s e r s at S N R l O d B  55  2000  N = 7, M = 15, 77 Users (moderate heavy load), A W G N Channel, Output SINR I  I  DMI •  1  _i  1 subs pace (PASTd) / / /subspace(svd)  l  0  ~~  ^  :  /  :CMOE RLS  i  i  500  1000 Number of data samples  1500  F i g u r e 3.6: O u t p u t S I N R w i t h 77 U s e r s at S N R l O c l B  56  2000  N = 7, M = 15, 105 Users (full load), AWGN Channel, Output SINR  DMF~-L___^ ;  / I  / subspace(PASTd) •' C M O E F=iLS  /  1 subspace(svd)  /  0  i  i  500  1000 Number of Data Samples  1500  F i g u r e 3.7: O u t p u t S I N R w i t h 105 U s e r s at S N R l O d B  57  2000  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—  CMOE RLS  -0—  subspace (SVD)  V  10  20  subspace (PASTd)  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 Blind Multiuser Detection in Slowly Fading Rayleigh M u l t i p a t h Channel 4.1  Introduction  The distinct advantage of using multicarrier based transmission is to combat frequency 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 particular, 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 F D signature sequences. This chapter is organized as follows: Section 4.2 reviews the signal model in downlink 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 algorithms. 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  A n 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 (4.1)  /=i where gi(t) is the time varying complex gain of path I and T is the chip duration. c  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  61  time channel response given by 'i'Lp[ -;  SLp{ ]  k] —  n  * h[k,  n  n]  +  (4.2)  w[n  where s^pfn] is given by 2.15. w[n] is a complex W G N random variable with variance N , and h[k, n] is the equivalent discrete time channel impulse response of h(t. r) 0  given by 2.35. The discrete time demodulated equivalent baseband signal is given by the D F T of 4.2 DFT  M  {/-,/->• k]} = DFT  M  {s [n\ LP  * h[n, k}} + DFT  M  {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  Z [m] n  = \ / p y ^ b a [n]ck[m]H[m] k  k  + VF„.[m], m = 1, 2,.... M  (4.4)  A;=l  or equivalently Z [m] = H[m]S [m] n  n  + W [m], n  L-l  62  rn  1.2  M  (4.5)  Based on only the knowledge of the desired user's T D and F D signature sequences, we propose blind linear M M S E multiuser detector that performs blind detection for the T F - M C - C D M A signal and jointly estimate the channel.  4.3  B l i n d 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 demodulated signal naturally decouples into one which involves only the T D spreading and one that only involves the F D spreading. The m  t h  column of Z gives the T D multiuser detection problem z[m]  The n  th  =  \/PH[m}  {bkC [m})  E  a  k  k  +  m  W[m],  =  1,  2 , M  (4.6)  row of Z gives the F D multiuser detection problem (b a [n}) c + W,'/.  /' E  k  k  n = 1,  k  2 , N  (4.7)  k=l  where c  k  =  diag(c )B. k  H =  H[l]  H[M]  H[2]  (4.8)  Ck[l] d,iag(c ) k  c [M] k  It can be noted here that, more compactly written, the channel impulse response g and its D F T H is related by H = F g MxL  where 63  F XL M  is the M by L Fourier matrix.  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 F D 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 F D 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 F D 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 T D S 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 F D M U D solves the optimization problem W j — argmin E  P(b a [n]) k  k  -  w'/xl  (4-9)  J  (MfcH) c + n £  z I = \ ^ E  k  A:=l  Re-examining 4.9, if one let B  k  = b a [n] and realizing that c k  k  k  = diag(c )H k  is  the effective F D signature, then 4.9 is equivalent to the blind L M M S E multiuser  64  W, - IK  RefafZjW,}-  xrequency domain M U D  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 F D 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 T D signature sequences to reduce the number of interferers for the F D detection. Reversing the process where the F D detection is computed first does not achieve the same result. The reason is that matched filtering can not be applied to the F D detection because orthogonality of the FDS is lost clue  65  * z  af  z  Z  frequency domain MUD  •  r , ,f ..H  A  •Re^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 F D S 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'  Z [m] = n  Xj  ^PJ2Y1 ^ » b  ;, = 1  [J] ^ H H H  a  + W [m], a - L . . . . V n  ;  m = 1,..., M  =l  K  (4.10) In vector notation. 4.10 is rewritten as A'  Xj  z[rn] = \/~P  (4.11)  bj c \m\H[rn]sL.j + W[m K  K  .7 = 1 « =  The inner product of 4.11 with user l's T D signature sequence yields Xj  N  afz[m] =  ^Y, E  z'[m  i=:i  bjn cK  [ ] m  H  i]f m  a  a  ?  + af  WH (4.12)  K=I  XI  = \/P  //jm (6 K.=  1kC(C  [m]) + w' [m], m = I....., M  l  Equation 4.12 is simplified further in vector form to get z'[l] XI  z  (4.13)  = K=l  z'llVI] 66  where bi denotes the transmitted symbol of the user with T D signature sequence K  1 and F D signature sequence K. We assume our desired user has T D signature sequence number 1 and F D signature sequence number 1. The blind L M M S E detector optimization problem for the type II F D stage detection is thus given by (4.14) Comparing with 4.9. the F D 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 multiuser signal in slowly fading frequency selective channel. Applying matched filtering for the F D M U D is inappropriate in this case because the orthogonality of the frequency domain spreading sequences is lost due to the effect of the channel. Therefore, blind multiuser detection techniques must be used for the F D 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 F D multiuser detector. The type I and type II F D 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 F D 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 F D detector is required to suppress K — l interfering F D signature sequences. However, for the type II F D detector, it is only required to suppress Xi ~ 1 interfering F D 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 < KIf K > PI, then there will be some users that share the same F D signature se67  quence and this means that all M linearly independent F D signature sequences are used in the system. Therefore, for the. type I F D detector, it is required to suppress M — 1 interfering F D signature sequences. However, for the type II F D detector, it is again only required to suppress xi ~ 1 interfering F D signature sequences. Furthermore, it is easy to see that xi < M. To summarize, since the type I and type II detector F D M U D problem differ only in the number of users, we can then solve both problems using the same methods assuming the F D 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 F D 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 R L S method and the subspace blind detection method.  4.3.3  C M O E R L S B l i n d 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 F D 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 R L S algorithm can be applied to M C - C D M A signal. We apply the C M O E R L S 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 R L S 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 R L S 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 F D signature sequence of the desired user instead of the original F D  68  signature sequence. The type I detector frequency domain C M O E RLS M U D problem is given by M  |w^z| , r2  w} = argmin V where C i = diag(ci)H time chip of the p  A^  1 -  "  wfcj = 1  (4.15)  is the effective signature waveform and j[p] denotes the  j  th  transmitted symbol. The summation in 4.15 states that the sum  th  is over all time chips going from the zeroth time chip from the zeroth transmitted symbol to the j  th  chip of the p  symbol.  Ul  The solution to the C M O E R L S optimization problem is given by 3.19. i.e. w* =(c?Cfc y C]'c  (4.16)  1  f  i  1  where the data correlation matrix is given by the recursive updates from the columns of Z. namely C [p, n] = AC/[p. n - 1] + z'M) .  n= 1..... 7Y  11  f  (4.17)  Recall that Cr[p,0] is equivalent to C / [ p — 1, N}. The type II C M O E R L S M U D problem is given by p  w* = argmin V A w/eCMx'  .  =  h  | | w f z'[i]|| , 2  wj'c;, = 1  (4.18)  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[  - 1] + z'\p]z'\p]  H  P  ( -!9) • 4  Applying the C M O E R L S 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 = z  J n  for type I detection and q = z ' for  type II detection. To complete the'solution, it is necessary to solve for the effective F D 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 F D blind multiuser detector is given 4.16 for both type I and type II detectors. Blind Channel Estimation For the blind channel estimation for the F D M U D detector using the C M O E R L S 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/} zl H  f }  (4.21)  Similarly, the M O E for type II is given by MOE {w})  = E{ | W * / V | }  (4.22)  2  n  Expanding either 4.21 or 4.22 yields the same result, i.e.  MO£(wJ)  =  E{W<zl\ } • /-{(w^zD 2  ( w * " ^  = £{w;"zM"w;} = =  (4.23)  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  MOE(w})  = [(cfCy^i)  Cy'crJ  70  r  C / ((cfCj'ci)  i  '  Cj%  (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 F D signature sequence of the desired user C i to estimate the effective F D signature, and in the process, the channel. Using this approach, 4.25 gives the optimization problem for the channel estimation, namely argmax { M O E { w ) ) } = argmax [ ( c f Ci  C ^ C i )  Cl  _  1  (4.25)  } -  1  Equivalently, one can minimize the reciprocal of 4.25 and therefore, the estimated effective signature is given by a = aigmin{(cfC7 c )}  (4.26)  1  i  1  C]  Using the fact that C j = C i H where C i = diag(ci), mate <&j ~ Cj , 1  H = F g  and the R L S esti-  MxL  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 } g  (4.27)  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 corresponding to the smallest eigenvalue of the matrix F f  /  x  L  C f <J>/-CIFA,/ LX  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 F D sequence used in the system U\ (2) the channel estimation problem is solvable only if the number of linearly independent F D 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 F D sequences can be used in the system. Therefore, 71  / / Type I Blind T F - M C - C D M A C M O E R L S Receiver / / in Slowly Fading Rayleigh Multipath Channel Initialization: */[0 0]=IMXM In the p symbol, input matrix Z[p] Set w* = ai for(n = 1.2. ....AT) Compute Kalman Gain: !  th  ,  r  i _  f[P; l n  -  \- $ \p,n-l]z„.[p] 1  T  f  i+A-i( „bn"*/b.»-l]anbF z  Compute Update: *f\p, n] = A " (& \p, 1  f  n-1}-  k  {z [p) ) r  f  n - 1]  H  n  end for Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective F D Signature: Compute F D Detector w; <T> / c! Perform detection  Ci  =  F^v/xtg  & [p] = s g n ( R e { ( w * Z ) * w } } ) w  1  t  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 B P S K . 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: */[°]  = !MXM  In the p symbol, input matrix Z[p] Set w*. = ai Compute: z' = afZ[p] Compute Kalman Gain: th  k R  f /  [  y  -  P  ~  J  x- * \p-iW 1  r  I  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 F D Signature: Compute F D Detector w*  f  Ci =  F  M  x L g  = 3?/Ci  Perforin detection bi [p] = s g n (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 D e t e c t i o n  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 F D multiuser detection in 4.9 and 4.14. The correlation matrices for the type I and type II detectors can be re-written as E {zi  C  f  C  {£)"}  = S, | A | S f + a ? W 2  = E {z'z' } = S , ! A | f/  f  2  Sf +  (  4  2  g  )  a}I  MxM  where S  /  (4.29)  = [c |c |...|c ] 1  2  £ /  contains the effective FDS signature waveforms of each user. Recall that the F D 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 F D 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 | denotes diag(A\, 2  A\,...,Al,)  where A is the total energy of all users using the same F D signature sequence, t  i = 1,2,...,[/.  The eigendecomposition of the correlation matrix then yields the  subspace components [39]  (4.30)  ;  Ain = diag(\l» ...,W) t  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 F D 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  Blind Channel Estimation 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 orthogonality 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 signature waveform and the unknown channel response  VlP cA \ H  Ci = argmin  (4.33)  2  Cl  Using the definition of the effective signature waveform and expanding 4.33 yields an expression almost identical to 4.27. g = argmin \g Ff Cf  U ^ U ^ C x F ^ g }  H  /xL  (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  Cf  fjiP~(jlP C\F . n  MxL  Such estimation is unique up to a complex constant as  in the case with the channel estimation using the C M O E R L S estimates described in section 4.3.3. Using the subspace decomposition, the approach to channel estimation for both the C M O E R L S 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.  Singular Value Decomposition  The singular value decomposition implementation applies the exact eigendecomposition 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 computation of subspace parameters using S V D 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 S V D 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 p symbol, input matrix Z[p] Set w. = a fori// ••- 1.2 V) Compute Update: th  C \p,n] f  = XC [p,n f  - 1] +  (z,M )K[p] ) r  T  H  end for Apply Eigendecomposition to C / to get Subspace Eigenvectors: Subspace Eigenvalues: Form Subspace matrices: ui^, A i ^ . U . ^ Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of Compute Estimated Effective F D Signature: c, = F Compute F D Detector  M x / j  g  Perform detection b [p] = s g n ( R e { ( w ^ Z ) * w } } ) 1  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 A d a p t i v e Subspace T r a c k i n g Alternatively, the subspace parameters can also be estimated using the P A S T d adaptive subspace tracking algorithm used in Chapter 3. The adaptive subspa.ce tracking using the P A S T d 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 P A S T d algorithm 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 p symbol, input matrix Z[p] Set w,* = a i Compute: z' = afZ[p], Compute Update: th  :  C [p} f  = XC [p-l)  +  f  (zU \(Ap] ) J  T  H  Apply Eigendecomposition to Cf to get Subspace Eigenvectors: j u ^ j Subspace Eigenvalues: j ^ ^ j Form Subspace matrices: ui , Al , V Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of J)  /0  f)  n  Compute Estimated Effective F D Signature: 6] = Compute F D Detector w} = U ^ ( A i ) - U ^ c Perform detection )  hip]  / )  1  F g MxL  f f  1  = sgn ( R e { ( w } , z' H  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 -  (4.35)  U ^ U ^ "  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 algorithms 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{wfzM |b [p]} W/  2  :l  M A ' / t l W / . W f ) = ———-—-— .—— E{Var (W^Z^W/IMP])}  . (4.3o  T  1  /  ;  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  £  4.4.1  (4.37)  Receiver Complexity  The receiver computational complexity in the downlink slowly fading Rayleigh multipath 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 p symbol, input matrix Z[p] and previous estimates th  Set W ( = a i •for(n'= 1,2,...,N) Input z [p] r  n  Set = z [p] for (A; = l,2,...,U) T  n  Projection Approximation: y  k  =u  [p.n — l} x. H  k  k  Eigenvalue Update: \\p[p.n} = jX[P\p.n — 1] Eigenvector Update: u %,n] k  =  v4P\p,n - 1] +  Deflation: x j ^ = x [ end for end for  (x<'> - u »\p,n k  -  l]y^  u[ \2J,n]y[  J )  f)  f)  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 F D Signature: Compute F D Detector  v} =  61 =  F  M  x  L g  iji'\AP)- W c 1  H  l  Perform detection S |p] 1  =  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 p symbol, input matrix Z[p] and previous estimates /\(/) (/)\ th  t /  f l  Set =ai Compute: z ' = a f Z[p] Input z'[p] ;  r  Set 4 = z'[p] for(A; = 1,2,...,17) r  f)  Projection Approximation: y[P — u[P\p — l j ^ x j ^ Eigenvalue Update: \[ \p] = jX [p f)  { f) k  - 1] +  Eigenvector Update:  4P* - « f  b -1) +  Deflation: end for  (=4°  = x<; - u n  ^  -  - %F>  W  Form Signal Subspace Matrices: \ji \  A.^  J  Compute: UPW" =Itji \J / Perform Channel Estimation: Find smallest eigenvector g corresponding to the smallest eigenvalue of f)  l  )H  Compute Estimated Effective F D Signature: Compute F D Detector  w;; = u<'W )- u$ "c )  1  Ci = F  / V / X  ^g  /)  1  Perform detection S[p] = s g n ( R e { ( w ; ) z } t f  / r  1  Table 4.6: Type II Blind Subspace PASTcl Receiver Algorithm  81  3 even though the methods used in both cases are inherently the same ( C M O E 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 filtering 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 complexity 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 ) . Both the blind subspace methods 3  using S V D 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 F D detector computation for both C M O E R L S type I and type II detectors have the same complexity as in the A W G N channel case which is 0(M ) 2  per update. Because there are N updates per symbol for type I. the total  complexity for type I is 0(NM ). 2  For type II, only one update is computed per  symbol, hence the total complexity of type II C M O E R L S is 0(lVI ). 2  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 F D detector computation using blind subspace method through S V D 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(M ) i  O(xi)-  and for the type II detector  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 C M O E RLS C M O E RLS Subspace S V D Subspace S V D Subspace PASTcl Subspace PASTcl  Type Type I Type II Type I Type II Type I Type II  Complexity 0(NM + L ) 0{1VI + L ) 0(M + L ) 2  3  2  3  3  3  oixi + L ) :i  0(NM'  + L )  0{xiM  + L )  z  3  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(lVI ) 2  normal system capacity and for the type II detector is 0(xiM)  under  per update. For type  I using PASTcl, there are N updates per symbol, therefore, the total complexity for type I receiver is 0(NM ). 2  However, for type II, there is only one update per symbol,  therefore the total complexity for type II P A S T d detector is O(xiM).  We also see  that using the P A S T d 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 P A S T D implemented in both type I and type II architectures. The channel model used in our simulation is the I T U - 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  <  2  110  -9.7  3  190  -19.2  ^\A-(t) <  <  -W -(t) < ^\A-(t) :i  4  410  2  -22.8  2  2  Table 4.8: ITU-A Pedestrian Rayleigh Fading Channel Power delay profile T D Spreading Gain F D Spreading Gain Number of Subcarriers TDS Sequence FDS Sequence Number of FDS Sequence used  8 16 16 Walsh-Haclamarcl Walsh-Hadamarcl 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 P A S T d detector's performance matches closely to that of the type II blind subspace S V D 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 P A S T d 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 P A S T d 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 S V D 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 S N R (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 F D detection stage.  4.6  Summary  In this chapter, the T F - M C - C D M A blind multiuser detection problem in a downlink 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 S N R 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 F D interfering sequences in slowly fading channel for the F D 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 F D 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 S N R 12clB  89  Output SINR Comparison, 75 Users 4.5  I  !  !  !  Data Samples  Figure 4.8: Output SINR for 75 users case at S N R 12dB  90  Blind subspace SVD, 35 Users  ^type 1  LU  <2- 0.5 I -type II —  500  1000  1500  2000  Blind subspace PASTd, 35 Users i  i  LU  <£ 0.5 ^ ^ • t y p e II  type I i  i  1000  1500  2000  1500  2000  L.  500  CMOE RLS, 35 Users  LU  w 0.5  ^Jype  500  I  1000 Data Samples  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 Blind Multiuser Detection in Downlink M o b i l e Rayleigh Fading Channel w i t h Doppler Induced I C I 5.1  Introduction  In this chapter, the random frequency offset caused by mobile Doppler shifts in downlink multipath fading channels is investigated. Previous discussions in Chapters 3 and 4 have implicitly assumed that any frequency offset is compensated exactly at the receiver. 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 orthogonality 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 mathematical 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 L-l  5.1)  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-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 / . the Doppler induced ICI weight coefficients can be s  found. Assuming that M subcarriers are used, this yields L-l  h(rn f , 0  s  t) = J2 gie>  - 0e-  2ltv,it  r  '>  (5.4)  i2wmof T  /.=o  where f = TTJJJ is the subcarrier frequency spacing and T is the duration of an O F D M s  s  symbol transmission. Samples of the channel impulse response in the t variable at multiples of T is yields the discrete channel impulse response s  L-l  iT,) = ]T g e  h{m fs, 0  i 2 n v  t  '  i i T  '- <hT  j 2 n m o f  ' '  (5.5)  T  If we approximate and discretize the path delays to integer multiples of T , and let s  Ei = T~ be the normalized Doppler offset of the I path, then we can rewrite 5.5 as is  ~  L  h(mo,i)  l  J2me,\  f-]2ne,l\  /  = }^giexp  ^——  j exp ^  j  M  f-j2nm l Q  exp  M  '-° L  (5.6) '  .7'27re (*-0^ (-j2~mj 9i exp 77 M 7exp V M gi e^p y /  i  1  1=0  The frequency domain channel transfer function is then the M point D F T of h(n, i) with respect to the variable i. H(rn , 0  m) = -  I  ^  ^  (j2irei(i  9 l  exp ^  - l)\  f-j2irm.ol\  J exp ^  —  — j exp ^  Jul  \  EE  (-j2wmi  1 '-^ v (j2ir((ei - m)i - e l)\ f-j2nm V = M 9i exp [ J exp ^ — — i = o  /=o  t  ¥  0  f  v  7  94  v  —  —  '(5-7)  It is shown in [28] that using O F D M based transmission, the D F T demodulated received signal is weighted by the frequency domain channel transfer function H(mo, m— m ). Applying this result to the T F - M C - C D M A received signal yields 0  Hp rLp[m ]  K  b a [n] ]P c [m}H(m ,  = J—  0  k  k  k  A.-1  0  m - m ) + W[m ] 0  0  (5.8)  771=0  The term H(mo, rn — ni ) weighted by the frequency domain spreading chip sequence 0  c [m] is the ISI induced by the channel and represents the leakage of energy from the k  m  th  subcarrier to the mo" subcarrier. Expanding 5.8 yields 1  /  = J£[  r p[mo] L  A'I-1  K  E ha [n] E c [m] k  k  k=l  y  m=0  7 = 0 /.=0  ,  V  >  J  7  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 rLp[m } 0  x U V  h'I-l  E b a [r>] E  =  k  k= l  E  c  k  fc[ ] m  777=0  E ' 9i exP ( ^ - ^ ) exp { = ^ ) )  i=o i=o  = E ha [n] E c [m]5 k  k=l  L  J  J  v  k  + W[m ] 0  (5.10)  E fji exp (=&P*) + W[m ]  mmo  0  (= 0  -777=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[rn ] = Y 0  c [m]H(m , k  0  777=0  95  m - m) 0  (5.11)  Equation (5-8) can bc rewritten as K '1'LP  °]  m  =  E  t>k(-i'k[n}G [mo\ + W[m ] k  0  (5.12)  fc=i  or in a more familiar form  (5.13)  rip Defining the effective frequency domain complex subchannel gain as  G' [m ] k  G [mo] k  Q  (5.14)  Ck[m ] 0  Using (5-14) and (5-15). the received signal can be written as  •I'LP  i< m ] = Y2 kak[n}G [mo}c [m ] + W[m ] b  0  k  k  0  0  (5.15)  which is an equation identical in form to 4.4 with the exception that 5.15 is a quasiuplink 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. A s a result, we can also define an equivalent channel impulse response g' =  IDFT(G' ). k  Based on the above discussion of the modeling of I C I due to Doppler shifts in a mobile communication channel, we can summarize two important effects of I C I i n 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  B l i n d Multiuser Detection  T h i s section discusses using the blind multiuser detection methods proposed i n Chapter 4 to the T F - M C - C D M A signal given by 5.15 without knowledge of the I C I coefficients, the Doppler frequencies, and other interfering users' time and frequency  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 R L S method and the subspace blind multiuser detection method using S V D and adaptive subspace tracking through the PASTcl algorithm to showthat all these algorithms gives robust performance in mobile multipath fading channels 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 T D S sequences, with all users sharing the same time domain signature sequence forming a group. This gives A'  Xj  &n*anb1 «MG' [m ] + ^ « K ]  r p[nio} = Z [mo] = \fP L  c  n  re  j = l  (5.16)  0  K - l  Using vector notation, 5.16 can be re-written as Xi  N  z[m ] = 0  N/P  ^ ^ ^ [ m o l G ' J m o l a j + W[m ]  (5.17)  0  J= l  K.=  l  The orthogonality of the T D S 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 Xj  N  afz[m ] = z'[mo] = \fP  bj«c«[m ]C7' [mo]af a,- + af W[m ]  0  0  3 = 1  *  K  0  (5.18)  = 1  Xi  •  \  /  = ^Yb ,c \m ]G' [mo\ + w' lK  K  0  K  h-,=i  Re-writing 5.18 using vectors yields  Xl  =  y/P ^ & i « c K,=  97  l  K  + w'  (5.19)  where now the effective FDS waveform is given by c  fc  =  diag(c )G': k  k  T  G',  =  G' [l]  •  G' [2]  k  k  G' [M] k  (5.20)  o [l] k  d,tag(c ) = k  c [M] k  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 F D blind L M M S E optimization problem remains the same as before, i.e. W | = argmin E W f  5.3.2  eC  P{b  ^5.21)  w f z'  u  M x l  C M O E R L S Blind Detector  In this section, we show the C M O E R L S 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 R L S  algorithm remains the same as that in chapter four, namely w* = ( c f  (5.22)  Cfc ) x  where C ^ is approximated recursively using the matrix inversion lemma, i.e. 1  A- * [ --l]z'z' * [j--l] 2  w  /  7  /  (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 F D  / / Type II Blind T F - M C - C D M A C M O E R L S Receiver / / in Slowly Fading Rayleigh Multipath Channel / / with Doppler induced ICI Initialization: $/[0]=IA7XJW  In the p symbol, input matrix Z[p] Set w* = a Compute: z' = a f Z[p] Compute Kalman Gain: Ul  :1  K F [ P L  ~  I A-I(Z^) * [ -I]Z"h  +  /  P  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 F D Signature: ci = Faxes' Compute F D Detector Wj = $ yCl  Perform detection S [p] = s g n ( R e { ( w } ) 2 } ) t f  / r  1  Table 5.1: Type II C M O E R L S Blind Receiver Algorithm for multipath fading channels with Doppler induced ICf signature. This again ends up being a familiar problem to find the normalized eigenvector corresponding to the smallest eigenvalue of the matrix F f  /  x  /  C f $^CIFMXL-  The algorithm is summarized in Table 5.1.  5.3.3  B l i n d 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;  IA|  99  2  S f + ajI  MxM  (5.24)  where  S = [£ |c |...|c ] /  1  2  (5.25)  xl  contains the effective F D signatures of the users within the same time domain signature group as the desired user (user 1). Applying the eigendecomposition to Cf gives the signal and noise subspace decomposition  (5.26)  The solution to the type II F D blind L M M S E detector is given by W} = U ^ ( A ^ ) "  1  U ^  C  ]  (5.27)  The effective F D 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 eigenvector corresponding to the smallest (least dominant) eigenvalue of the composite matrix F C{ \J^Pt] ;P 1  {  MxL  (  fr  C\F X L - As in Chapter 4 where it was implicitly assumed M  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 p symbol, input matrix Z[p] Set w* = a-i Compute: z' = a{ Z[p] Compute Update: th  F  C b ] = A C [ p - l ] + (z'b]^)(z'b] ) Apply Eigendecomposition to Cy to get r  /  /f  /  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) r H  i  ir  Compute Estimated Effective F D Signature: cq = F^/x/^g' Compute F D Detector Perform detection b [p] = s g n ( R e { ( w p z - } ) / /  r  1  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 suppressing 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 S V D 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 S V D and using PASTcl adaptive subspace tracking in Table 5.2 and 5.3 respectively.  101  // // // In  Type II Blind T F - M C - C D M A Subspace (PASTcl) Receiver in Slowly Fading Rayleigh Multipath Channel with Doppler induced ICI the p symbol, input matrix Z[p] and previous estimates th  l /*=, Set = a] Compute: z' = af'Z[p] Input z'[p] A f c  ; U f c  L  Set  x/ (  }  =  z'\p]  r  ibr: /• 1,2, ...,xi.) Projection Approximation:  = u ^ [ p — l]'"'x^  Eigenvalue Update: X[ [p] = -yX[ [p - 1] + ;y[ f)  f)  7)  Eigenvector Update: - 1] +  u^b] =  (x<" - u i ^ " 1 ] ^ )  Deflation: x^ = x[ - u \p]y[ end for Form Signal Subspace Matrices: U . A { f)  J)  f)  k  :1  s  s  Compute: U ^ U ^ " = I - U ^ U . ^ Perform Channel Estimation: Find smallest eigenvector g ' corresponding to the smallest eigenvalue of T?H  (-iHjTiDfiif) -!(~< P 1  Compute Estimated Effective F D Signature: Compute F D Detector  w} = u ^ ( A i ) - u i / )  1  / ) H  Ci  =  F^/xLg'  e.  Perform detection b b] = s g n ( R e { ( w } ) z ' //  r  1  Table 5.3: Type II Blind Subspace PASTcl Receiver Algorithm  102  T D spreading gain F D spreading gain Number of subcarriers TDS sequence type FDS sequence type Number of FDS sequence used Carrier frequency  ,8 16 16 Walsh-Haclamard Walsh-Haclamarcl 11 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 S V D and the blind subspace PASTcl detectors for the 35 users and 55 users case. However, as the system  103  10°  L  Slowly Fading Rayleigh w/ Doppler ICI, 35 Users, 3km/hr 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 significantly. 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 R L S detector and the blind subspace S V D 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  0  5  10 Avg SNR (dB)  J  15  20  Figure 5.2: B E R performance for 55 users case loading, the P A S T d 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  0  !  5  J  10 Avg SNR (dB)  J  15  20  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 Data Samples  4000  Figure 5.4: Output SINR for 35 users case at S N R 12dB  107  5000  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  500  1000  1500  2000  Blind subspace PASTd, 35 users  LU  £ 0.2 h  500  1000  1500  2000  1500  2000  CMOE RLS, 35 Users  V. 500  1000 Data Samples  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 bandwidth 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 F D 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 suboptimally 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 F D spreading. This has the advantage of utilizing parallel processing. The blind DMI algorithm, 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 F D  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 discussed. 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 capacity 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 perfor113  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 increase 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 TF-MC-CDMA  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 approximate 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 ( M C M C ) 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 S M C technique to blind detection 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 B l i n d Multiuser Detection in Fast Fading 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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