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Matched filter bounds for fast fading Rician channels Dinesh, Vaibhav 2002

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M A T C H E D F I L T E R BOUNDS F O R F A S T F A D I N G RICIAN C H A N N E L S by VAIBHAV DINESH B. E. (Electronics and Communications), Guru Jambheshwar University, India, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March 2002 © Vaibhav Dinesh, 2002  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  of  the  University  of  British  Columbia,  I  agree  for  this or  thesis  reference  thesis by  this  for  his thesis  and  study.  scholarly  or for  her  I further  purposes  gain  that  agree  may  be  It  is  representatives.  financial  requirements  shall  not  that  the  Library  permission  granted  by  understood  be  for  allowed  the  The University of British Columbia Vancouver, Canada  Date  DE-6  (2/88)  O S M A R C H 200Z .  COMPUTER  for  make  it  extensive  head  without  EN6/NEE.RiN<q  advanced  shall  that  permission.  Department of E L E C T R I C A L AND  an  of  copying my  my or  written  Abstract A new matched filter bound (MFB) for a dispersive Rician fading channel with unrestricted normalized Doppler rate is presented. Analytical expressions are derived for BPSK modulation. The channel model is based on the standard linear time varyingfilter.The BER performance, in general, is found to improve with an increase in the fading rate due to the inherent diversity present within a single pulse. The shape of the transmitted pulse is shown to affect the BER in fast fading conditions. It has been found that the specular component improves the BER at lower fading rates; for very high fading rates, the implicit diversity effect becomes dominant in diminishing the effect of fading. Discrete (two and three beam) models are assumed for the dispersive channel. The error bounds for the three-beam model are derived as a function of the inter-beam delay and correlation parameters. In general, the delay spread was found to result in lowering the BER for both the slow and fast fading cases when the fading in the beams is uncorrelated. In the two-beam frequency selective case, the BER is very sensitive to the degree of beam overlap. Finally, assuming errors in estimating the channel fading waveform, the BER for a nonideal receive filter is obtained. It was observed that, for the specified model, the system degradation is higher for phase mismatches than for amplitude mismatches for a given error variance.  ii  T a b l e  o f  C o n t e n t s  Abstract  ii  List of Figures  v  Glossary  vii  Acknowledgment  xi  Chapter 1 Introduction  1  1.1  Motivation  1  1.2  History of Related Work  3  1.3  Thesis Contribution  4  1.4  Thesis Organization  5  Chapter 2 The Wireless Channel  6  2.1  Introduction  6  2.2  Signal Propagation in a Wireless Channel  6  2.2.1 Land Mobile Fading  6  2.2.2 Fading Channel  8  2.3  Channel Model and it's statistical Characterization  11  2.3.1 The Linear Channel  11  2.3.2 RF Spectrum and Autocorrelation of the Fading Process  17  Chapter 3 Derivation of the MFB's  20  3.1  Introduction  20  3.2  System Model  21  3.3  Error Performance Analysis  22  3.3.1 Deriving the pdf of Y  26  iii  3.4 Flat Fading  32  3.4.1 Special Cases:  34  3.4.1.1 Very Slow Fading  34  3.4.1.2 Very Fast Fading  35  3.4.1.3 Rician Factor, K  36  3.5 Frequency-Selective Fading  37  3.5.1 Two-Beam Case  37  3.5.2 Three-Beam Case  41  Chapter 4 Numerical Results  45  4.1 Introduction  45  4.2 Flat Rician Fading  45  4.3 Frequency-Selective Rician Fading  50  4.3.1 Two-Beam Case  51  4.3.2 Three-Beam Case  52  4.4 Imperfect channel estimation  60  Chapter 5 Conclusions and Suggestions for Future Work  65  5.1 Conclusions  65  5.2 Suggestions for Future Work  66  Appendix A : MFB (slow fading 0.0 < /  < 0.01)  68  Appendix B : MFB (fast fading f ^>°°)  7 1  Appendix C : Solving the Integral  74  Appendix D: Calculating the Errors in Amplitude and Phase estimation  77  Bibliography  79  N  iv  L i s t of Figures Figure 1.1  Received signal corresponding to slow and fast fading channels  2  Figure 2.1  Continuously Dispersive and Discrete Path channel outputs  9  Figure 2.2  Linear System model  Figure 2.3  3-D view of the continuous-time impulse response, \z(t, x)|  14  Figure 2.4  3-D view of the discrete-time impulse response, \z(t, x)\  17  Figure 2.5  Power Spectrum of an unmodulated CW carrier  19  Figure 3.1  System Model  22  Figure 3.2  The Transformed System  23  Figure 4.1  BER as a function of f  Figure 4.2  BER as a function of the Rician factor and f  47  Figure 4.3  Effect of the Rician factor for f  47  Figure 4.4  Effect of pulse shape in fast fading Rician channel, K  Figure 4.5  Channel averaging ability of different pulse shapes  Figure 4.6  Effect of the roll-off factor on the system performance, K  Figure 4.7  Two-beam frequency selective fading, K  Figure 4.8  Three-beam frequency selective fast fading, f  Figure 4.9  Three-beam frequency selective very-slow fading, a = 0  53  Figure 4.10  Three-beam frequency selective fast fading, f  54  Figure 4.11  Three-beam frequency selective very-slow fading, a > 0  55  Figure 4.12  Three-beam fast fading, K  56  Figure 4.13  Three-beam slow fading, K  = 0 dB  56  Figure 4.14  Three-beam fast fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0]  57  12  N  for the Rician channel, K  = 0 dB  dB  N  N  = 1.28 dB  = 0 dB  N  dB  dB  = 0 dB  = 0 dB N  = 0 dB and f  N  48 49  dB  dB  46  = 0.64 and a = 0  = 0.64 and a > 0  = 0.64  50 51 52  Figure 4.15  Three-beam fast fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0]  Figure 4.16  Three-beam slow fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0] ....58  Figure 4.17  Three-beam slow fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0] ....59  Figure 4.18  Performance degradation in a flat fading Rician channel  60  Figure 4.19  Performance degradation in a flat fading Rayleigh channel  61  Figure 4.20  Effect of mis-match in phase on the BER in a Rician channel  62  Figure 4.21  Effect of mis-match in phase on the BER in a Rayleigh channel  62  Figure 4.22'  Effect of mis-match in amplitude on the BER in a Rician channel  63  Figure 4.23  Effect of mis-match in amplitude on the BER in a Rayleigh channel  64  Figure D. 1  Resolution of the fading sample with/without error  77  vi  58  Glossary Acronyms  ASK AWGN  Amplitude Shift Keying Additive White Gaussian Noise  BER  Bit Error Rate  BFSK  Binary Frequency Shift Keying  BPSK  Binary Phase Shift Keying  BS  Base Station  CD  Continuously Dispersive  CW  Continuous Wave  DP  Discrete Path  EFD  Intrinsic Frequency Diversity  ISI  Inter-Symbol Interference  LOS  Line Of Sight  MFB  Matched Filter Bound  MS  Mobile Station  MSE  Mean Square Error  OFDM  Orthogonal Frequency Division Multiplexing  PAM  Pulse Amplitude Modulation  PSK  Phase Shift Keying  RRC  Root Raised Cosine  SNR  Signal to Noise Ratio  vii  Symbols  a  roll-off factor of the root raised cosine pulse  A  transmitted signal amplitude  c  speed of light  D  rms  rms delay spread  E  b  bit energy  f  carrier frequency  c  Af  Coherence bandwidth  f  d  maximum Doppler frequency  f  N  maximum normalized Doppler rate  f(t)  received pulse shape corresponding to the diffused channel  f (y)  pdf of Y  h(t)  impulse response of the receive filter  J (.)  zeroth order Bessel function of the first kind  K  Rician factor of the channel  c  Y  0  K  Rician factor (in dB)  N  total number of paths in the discrete channel model  N  noise component at the matched filter output  dB  N  power spectral density of AWGN  n(t)  AWGN noise  P  probability of bit error  p(t)  transmitted pulse shape  Q(v, w)  Marcum's (2-ftmction  r(t)  received (faded and noisy) signal  r {t)  received signal (for a CD channel)  r (t)  received signal (for a DP channel)  0  b  cd  dp  Rf(tj, t )  autocorrelation of the process f(t)  fad^  faded signal at the channel output  r (t)  received signal without noise (for the DP channel)  S(f)  power spectrum density of the Rayleigh fading process  2  r  s  viii  s(t)  transmitted signal  SfadW  r e c e i v e d pulse shape  T  r e c e i v e d s i g n a l duration  Ar  s a m p l i n g interval  Af„  Coherence time  T  delay spread o f the c h a n n e l  T  pulse duration  v  m a x i m u m velocity of M S  c  M  p  v  (z)  Whittaker function  X  m a t c h e d filter output after s a m p l i n g  x(t)  m a t c h e d filter output  Y  s i g n a l c o m p o n e n t at the m a t c h e d filter output  z(t, x)  i m p u l s e response o f a linear c h a n n e l ( g e n e r a l i z e d notation)  cd(t> ) x  z  i m p u l s e response o f a C D channel  ray(t' )  i m p u l s e response o f the d i f f u s e d c h a n n e l  rice(t> )  i m p u l s e response o f the R i c i a n c h a n n e l  z {t)  c o m p l e x f a d i n g process c o r r e s p o n d i n g to the «th b e a m  x  z  x  z  n  ix  Greek Symbols  diffused beam strength  a  2  a  attenuation corresponding to the direct/specular path  0  strength of the nth faded beam  «„  nth path attenuation characteristic function 8(r)  delta function  r(z)  Gamma function  <p„(0  nth path phase/ eigen-function  <K0  autocorrelation function of the Rayleigh fading process cross-correlation function of the fading processes  A.  signal wavelength/ eigenvalue  p  normalized correlation coefficient of the fading processes  <K(x)  delay power spectrum of the channel delay power spectrum of the diffused channel  9? (At;(T,,T ))  autocorrelation function of z(t, x)  a  normalized delay parameter  z  2  variance of the additive error (complex Gaussian random variable) °n  delay corresponding to the nth faded beam variance of the noise component  e»  angle of arrival of the nth wave  o  delay corresponding to the direct/specular path  x  X ma*  Xmm •  maximum delay spread delay with respect to the first faded component nth path delay maximum Doppler angular frequency  Acknowledgment This thesis is a tribute to my parents, Shri Dinesh C. Jain and Smt. Trishla Jain for their infinite support and encouragement. I am equally grateful to my brother, Dr. Gaurav Dinesh and other near and dear ones for their well wishes and moral support throughout. Of special mention is my thesis advisor, Dr. Cyril Leung, whose guidance and critical thinking helped me reach the goal. I would also like to thank my friends especially Cyril Iskander, Xinrong Wang and Farshid Agharebparast for providing with useful suggestions and help. Thanks also go to the members of my thesis examination committee for their time and efforts. This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) grant #OGP0001731.  Chapter 1 Introduction Mankind has been on an unending quest to improve his communication ability. It began with the use of sign language and gradually developed into well defined speech. Communication at a distance was a challenge. The practice of sending handwritten messages/letters through messengers is a very old one and still in prevalence. However, the twentieth century has seen dramatic changes in the field of telecommunications: the telegraph, telephone, internet and cellular phones are but the most visible examples. For many people, the need to be able to communicate anytime, anywhere is satisfied using cellular systems. However the design of mobile communication systems presents challenging problems because of the random time-varying nature of the land mobile radio channels.  1.1 Motivation One of the main problems with transmission over land mobile radio channels is the multipath fading phenomenon which results in amplitude and phase fluctuations of the received signal, at times causing significant dispersion leading to interference.  Various diversity schemes [1] have been proposed to combat fading. Most of the commonly used schemes involve multiple transmission and/or reception of the signal in space, time, frequency and polarization planes. Although these methods can lower the error rates, they often achieve this at the cost of different system parameters like the transmission rates, bandwidth and system complexity. Orthogonal Frequency Division Multiplexing (OFDM) [2] is a concept that works on a simple principle of inducing time diversity inherently in the system by lengthening the symbol period so that the system is more robust to impulse noise and channel fades.  1  Chapter! Introduction  2  Since the coherence time of a channel is inversely proportional to the carrier frequency, the application of higher operating frequencies (in the GHz range), for fixed (or relatively less increase in the) transmission rates, would increase the fading rate in the channel resulting in the channel to be treated as time variant over a single symbol. Figure 1.1 illustrates the difference  between a fast and a slow flat-fading channel response assuming afixedsymbol/pulse duration. It is important to realize that a system is referred to as slow or fast fading with respect to both, the channel parameter i.e. Doppler rate, f , and the symbol parameter i.e. pulse duration, T , which d  are together incorporated in the normalized Doppler rate, f  N  =  f -T. d  t [sec]  Figure 1.1 Received signal corresponding to slow and fast fading channels In addition, as shown in [3], microcellular systems i.e. mobile radio systems configured with many small cells having low-powered base stations, often have Rician rather than Rayleigh envelope fading characteristics. The deployment of such systems in urban mobile radio channels  Chapter 1 Introduction  3  and the use of multi-carrier modulation schemes like OFDM, provides the motivation for the study of the effect of frequency dispersion along with time dispersion in Rician fading channels. 1.2 History of Related Work Communication over wireless channels has been studied since the early 1960's. In [4], the error probabilities for coherent and non-coherent multichannel (or multibranch) communication system involving Rician fading are obtained. The paper is among the pioneering works in obtaining the performance bounds for the coherent and non-coherent multichannel systems subjected to Rician and Rayleigh fading. It shows that the effectiveness of the multichannel reception is more pronounced in the case of channels with small Rician factors. However, it does not consider time dispersion within each of the channels. Reference [5] discusses signal propagation through a Rayleigh fading channel without any explicit diversity (multichannel reception). The paper presents more general results than those in [4] by considering cases of a single-beam (flat-fading) and two-beam (frequency-selective fading) signal reception taking into account correlation between the beams. An extension of the above work appears in [6] which provides the performance analysis of a Rayleigh channel with continuous and discrete delay profiles for a space-diversity mobile radio receiver. This work compares the gain in bit error rate performance, obtained with space diversity over the gain obtained with frequency selectivity in the channel and the effectiveness of frequency selectivity, in lowering the error rate, in a single branch system over a multiple branch system. The Rayleigh fading channel is a commonly used model when there is no line of sight (LOS) path between the transmitter and the receiver. When LOS path is present, a Rician fading model can be adopted. Matched filter bounds over a multipath Rician fading channel of discrete  Chapter 1 Introduction  4  and continuous dispersion are derived in [7]. The paper emphasizes that the diversity gain achievable with intrinsic frequency diversity(IFD) (or frequency selectivity) diminishes with the increase in the Rician factor of the channel. It is worth noting that [4] and [7] refer to the diversity gain from different aspects. The former defines diversity gain with respect to multiple branch system when the diversity is induced in the system whereas the latter relates it to the IFD of the channel. A more recent and useful contribution to the system presented in [6] appears in [8], [14] and [9] where the channel is no longer assumed to be time-invariant. The effect of fading rate in the Rayleigh fading channel is studied. In [8] a general algorithm for obtaining matched filter bounds for an uncoded BPSK modulation scheme is devised and the effect of pulse shape on receiver's performance is analyzed. Similar approach is applied in [9], although in the frequency domain and the assumption of uncorrelated scattering in the channel is made in deriving the results for a continuously dispersive channel. The exploitation of Doppler diversity (frequency dispersion) in fast fading channels is also discussed in [10]. 1.3 Thesis Contribution The research presented in the thesis involves deriving the optimum bit error rate performance of a system, comprising of a linear time-varying Rician fading channel, with uncoded BPSK modulation. In compliance with the previous results, [8] and [14], the probability of error is found to be significantly affected by the channel's fading rate and unlike in slow fading case, the shape of the transmitted pulse poses an issue. The system behavior as a function of different parameters like the Rician factor, normalized Doppler rate etc. for frequency flat and in addition, the delay spread and beam correlation for frequency selective channels is studied. Finally, bit  Chapter 1 Introduction  5  error rate performance of a system employing a non-optimum receivefilteris presented.  1.4 Thesis Organization The research work, background information and conclusions are, in all, presented in five chapters, a brief description of which follows: •  Chapter 1: The chapter introduces the challenge undertaken in the present research and chronologically provides the background of the problem. It also specifies, in brief, the contributions of the thesis.  •  Chapter 2: The objective is to provide a necessary and basic understanding of the fading phenomenon, the communication system, the wireless channel and other related topics.  •  Chapter 3: This chapter forms the core of the thesis as it presents the solution to the problem stated in the introductory chapter. A detailed analysis of the matched filter bounds is presented.  •  Chapter 4: All the analytical and simulation results are graphically presented in this chapter. It includes physical interpretation of the results and comparisons between some of the plots.  •  Chapter 5: The conclusions derived from this research and some useful suggestions for future work in relatedfieldare presented in this chapter.  Chapter 2 The Wireless Channel  6  C h a p t e r 2 T h e Wireless C h a n n e l 2.1 Introduction T h e effect o f a fading wireless channel o n a transmitted signal is both c o m p l e x a n d i n t e r e s t i n g . T h e m a i n p u r p o s e o f this c h a p t e r i s t o p r o v i d e a b a s i c u n d e r s t a n d i n g o f the f a d i n g p h e n o m e n o n , its cause a n d effects; this i s necessary b a c k g r o u n d f o r the w o r k p r e s e n t e d i n this thesis.  2.2 Signal Propagation in a Wireless Channel In a w i r e l e s s c o m m u n i c a t i o n system, the transmitted s i g n a l often reaches its destination v i a m u l t i p l e paths. T h e m e c h a n i s m s that i m p a c t signal propagation c a n be c l a s s i f i e d into three types, n a m e l y reflection, d i f f r a c t i o n and scattering [1], [11]:  • Reflection i s c a u s e d b y the r a d i o w a v e s t r i k i n g a n d r e f l e c t i n g o f f a s m o o t h surface w i t h v e r y large d i m e n s i o n c o m p a r e d to the signal wavelength, X.  • Diffraction occurs due to the obstruction o f the r a d i o path between the transmitter a n d the receiver b y a dense b o d y w i t h large d i m e n s i o n relative to X r e s u l t i n g i n the f o r m a t i o n o f seco n d a r y waves b e h i n d the obstructing body.  • Scattering occurs w h e n the signal i m p i n g e s o n either a large r o u g h surface o r any surface w i t h d i m e n s i o n o f the order o f X o r less, c a u s i n g the reflected energy t o scatter i n a l l d i r e c t i o n s .  Fading is a c u m u l a t i v e effect o f the above m e c h a n i s m s .  2.2.1 Land Mobile Fading T h e arrival o f the transmitted s i g n a l at the receiver over m u l t i p l e paths gives rise to the  Chapter 2 The Wireless Channel  7  phenomenon of f a d i n g , in which a signal undergoes random and time varying attenuation and 1  delay. It is useful to distinguish between two types of fading effects [1]: Large-scale fading which represents average signal power attenuation or path loss due to  motion over large areas and small-scale fading which refers to the changes in signal amplitude and phase owing to small changes in the location of the receiver with respect to the transmitter. Although both fading aspects are significant from different perspectives, the present work focuses on small-scale fading as encountered in land mobile fading channels. It has been often observed that, due to the dense population of sky-scrapers and other tall structures within a city, there seldom exists a direct line of sight (LOS) path between a Base Station (BS) and a Mobile Station (MS) whereas in sub-urban areas this is usually not the case. Rather, there are generally some prominent structures which may act asfixedscatterers or signal reflectors. Thus, inspite of the lack of a LOS path, the MS may continue to receive a signal of a fixed, appreciable strength in addition to the randomly varying signals from other scatterers and 2  reflectors which are in relative motion to the mobile. Accordingly, the channel is modeled as Rician fading if it is characterized by the presence of LOS, fixed scatterers or signal reflectors in the medium or Rayleigh fading if the specular path (i.e. the non-fading component of the channel) is absent. The Rician fading channel is considered  When the multipath effect is caused by physical changes in the propagating medium for example variations in the density of ions in the ionospheric layers, it is referred as Scintillation. This is especially applicable in radio astronomy; however both terms, fading and scintillation, refer to a signal's random fluctuations and differ mainly in their causes of occurrence [11]. Here the random variations in the signal are being treated as variations with respect to different symbols or variations within a symbol of a finite duration.  Chapter 2 The Wireless Channel  8  as a more general model in which the absence of the specular path results in the Rayleigh fading channel.  2.2.2 Fading Channel In a wireless communication system, the received signal is often a random transformation of the transmitted one. Two types of channel models have been proposed in [7]: continuously dispersive (CD) and discrete path (DP) channel models. A CD channel has a continuum of diffused components. For a transmitted signal, s(t), the received signal can be written as [12]  'cdM  = \z {t,x)-s{t-x) cd  dx  (2.2.1)  0  where T  is the channel's delay spread and z (t, x) is the impulse response of the CD channel  M  cd  which is defined as the response of the channel at time t to an impulse applied at time  t-x.  On the other hand, we model the DP channel as consisting of a large number of discrete paths (multipath); each path n has independent time varying attenuation, oc„(0, delay, x {t), and n  phase, tp (0, with respect to the transmitted signal, s(t). The signal, r (t), at the channel n  d  output is dW  r  P  =  5>„C) • e~ ° m  • s(t-x <t)) n  .  (2.2.2)  n  The difference between the above two channel models can be clearly observed in Figure 2.1 which displays the response of a CD and DP channel to a short pulse (ideally an impulse). The  9  Chapter 2 The Wireless Channel  received signal appears as a continuously time-varying signal or a train of pulses respectively.  Figure 2.1 Continuously Dispersive and Discrete Path channel outputs Small variations in the spatial separation between the transmitter and the receiver and/or random changes in the surroundings have two major manifestations: • Time-spreading and • Frequency-spreading of the signal Time-spreading, commonly known as signal dispersion, is the elongation of the transmitted signal as it undergoes multipath propagation. It is caused by the fact that the different signal paths vary in length causing copies of the signal to reach the receiver at different times. As a result, not only is the signal spread in time but the signal components arriving at the receiver also differ in phase. Thus, a signal component with phase (p> transmitted at time t , may arrive in con2  2  junction with another component of phase (pi transmitted at time tj, resulting in constructive or  Chapter 2 The Wireless Channel  10  destructive interference between the two. We can characterize signal dispersion in terms of Delay Viewed in the frequency domain, signal dispersion is expressed in terms of Coher-  Spread (T ). M  ence Bandwidth (Af  less than Af  c  c  ) which implies that the signal components having frequency separation  are treated alike by the channel. It can be shown [12] that Af = c  M  . Based on  this, we can further classify fading as: • Flat or frequency non-selective fading: When A f  c  is large relative to the signal bandwidth,  the channel is said to be flat. Viewed in the time domain, the channel ideally causes no delay spread in the transmitted signal, i.e. T  M  = 0.  • Frequency Selective fading: If the coherence bandwidth is small compared to the signal bandwidth, the channel is said to be frequency selective. The attendant delay spreading causes inter-symbol interference (ISI). Frequency-spreading of the signal is a direct consequence of the spatial changes of the receiver relative to its surroundings, accounting for propagation path changes. As a result, the channel offers a time-variant response. This is characterized by the Coherence Time At , which is c  defined as the duration over which the channel's impulse response can be considered as timeinvariant. Coherence time is inversely proportional to the Doppler Spread, At  c  «*= 4- [12]. The id  Doppler spread is a function of the MS velocity, v, and signal wavelength, X, and is a measure of the spectral broadening of the signal / „ = /Vcos(e„)  where f  (2.2.3)  is the Doppler shift [15] for the n wave arriving at an angle 0 with respect to the th  n  n  Chapter 2 The Wireless Channel  11  direction of motion of the MS and / . (f  d  = ^) is the maximum Doppler frequency.  Depending on the channel's coherence time with respect to the signal duration, the Doppler spreading results in: • Fast Fading: The channel exhibits a rapidly time-varying impulse response relative to the symbol duration and the signal experiences spectral broadening. Fast fading often causes the occurrence of frequent deep fades in the received signal. • Slow Fading: When the channel's coherence time is long compared to the symbol duration, the channel is considered as slowly fading. Equivalently, the Doppler spread is small. It is important to understand that the time-spreading and frequency-spreading are independent phenomenons. A wireless channel may exhibit any combinations of Flat/Frequency Selective and Fast/Slow fading. 2.3 Channel Model and its Statistical Characterization 2.3.1 The Linear Channel In terms of the complex lowpass equivalent form of a bandpass system [12], we can express the impulse response of the channel as z(t, x), which represents the response of the channel at time t to an impulse applied at time t-x. In this section an expression for z(t, x) applicable for both flat and frequency selective fading channels is obtained. In doing so, we follow an approach similar to that of [12]. The general expression of the impulse response is given as z(t, x) = X « „ ( 0 • e~  J2nfMt)  (2.3.1)  • 6(T-T„(0)  n  where f is the carrier frequency of the bandpass signal; oc„(0 and x (t) are the attenuation and c  n  12  Chapter 2 The Wireless Channel  delay of the n ray at time t, respectively. th  Equation (2.3.1) is a general expression from the point of view that it accounts for both Rician and Rayleigh fading channels and accommodates for all types of fading discussed in Section 2.2.2. Since the specular path is a non-fading component of the channel, we can implement the impulse response of a Rician fading channel, z {t, x), by taking a constant attenuation, a , and rice  0  associated delay, x , for thefirstarriving ray 0  z (t, x) = 0 C Q • erice  j 2 n f  ^ • 8(t - 1 ) 0  + Xa„(0 • e-  ]lKfMt)  • 8(t - x (t)) n  (2.3.2)  Figure 2.2 shows the block diagram representation of the response r(f) of a fading channel z(t, x), corrupted with noise n(t), to an input s{t).  Channel  n{t)  AWGN Figure 2.2 Linear System model The signal r(t) can be written as  r(t) = J z(t,x)s(t-x)dx  + n(t)  (2.3.3)  Substituting (2.3.1) in (2.3.3) and neglecting the noise term for the moment, the received signal,  Chapter 2 The Wireless Channel  13  r {t), is given as s  •',(*) = 5>«C) • e~  J2KfMt)  .  • s(t-x (t)) n  (2.3.4)  n  T h e a u t o c o r r e l a t i o n f u n c t i o n o f z(t, x) i s defined as [12]  SR (Af z  JCCJ, x )) = | • E[z*(t, Tj) • z(f + A / , x ) ] . 2  (2.3.5)  2  A s s u m i n g u n c o r r e l a t e d s c a t t e r i n g , i.e. i n d e p e n d e n c e o f attenuation a n d phase s h i f t s a s s o c i a t e d w i t h different path delays, a n d letting  At - 0, (2.3.5) c a n be re-written as  SR (0;(T!, x )) = 9t(x,) • 8(T! - x ) . z  2  (2.3.6)  2  T h e f u n c t i o n SR(x), k n o w n as the d e l a y p o w e r s p e c t r u m o f the c h a n n e l , represents the average p o w e r output o f the c h a n n e l as a f u n c t i o n o f the t i m e d e l a y x . I n u r b a n e n v i r o n m e n t s , 9t(x) c a n be a p p r o x i m a t e d b y the c o n t i n u o u s one-sided e x p o n e n t i a l f u n c t i o n [13], i.e.  SR(x) = ^ - • e x p f - - M rms  where  D  rms  ^ rms'  x >0  is the r m s d e l a y spread. In some p r e v i o u s w o r k s  (2.3.7)  [6], the class o f t w o - s i d e d c o n t i n u -  ous spectra i s c o n s i d e r e d b y a s s u m i n g the G a u s s i a n s p e c t r u m . H o w e v e r , f o r s i m p l i c i t y , w e c o n f i n e o u r a n a l y s i s t o o n e a n d t w o b e a m cases ( D o u b l e - S p i k e s p e c t r u m  (2.3.8)) w h i c h  can easily be  extended t o a s y s t e m w i t h N , ( N > 2), beams b y a p p l y i n g a s i m i l a r p r o c e d u r e .  = ^(8(x) + 8 ( x - a ) )  (2.3.8)  A snapshot o f the i m p u l s e response w i t h c o n t i n u o u s o n e - s i d e d e x p o n e n t i a l d e l a y p o w e r s p e c t r u m is s h o w n i n F i g u r e  2.3.  T h e f i g u r e is p l o t t e d a s s u m i n g  D  rms  = 0.1 sec a n d  a Doppler  14  Chapter 2 The Wireless Channel  frequency of 10 Hz. One needs to be careful while comparing Figure 2.1, pertaining to the response of the fading channel to an impulse, with the three-dimensional plot of the impulse response, \z(t, x)|; since z(t, x) is a function of two time variables, t and x, whereas the response of the channel is a function of t only. It is, however, possible to trace the channel's response to an impulse from the three-dimensional plot of it's impulse response.  0.2 x [sec]  0.3 0.4  0  Figure 2.3 3-D view of the continuous-time impulse response, \z(t, x)| It should be noted that the terms 'one beam' and 'two beam' models do not simply refer to the arrival of one and two rays at the mobile; rather each beam may be interpreted as a collection of several rays with varying path-delays closely distributed about the mean delays x, and x , 2  respectively [14]. Thus, on splitting the second term in (2.3.2) into two parts (summation series) each reflecting the case of a single Rayleigh faded beam and denoting their respective indices by n and n , {  2  Chapter 2 The Wireless Channel  15  with each ray in the group having an associated delay of x„ (0 and ^(0, respectively, where x (0 Bi  =  x,+Ax (0,  AT  Bi  B i  (o«r  p  (2.3.9)  T is the transmitted signal block/pulse duration, we obtain p  z {t,i) = a -e~  -Hi-* )+z (t,T)  + z (t,%)  J2nfcXo  rice  0  0  l  (2.3.10)  2  where z {t,x) = e  J  /c  x  'X n,(0^ a  -/27C/-T,  z (r,t) = e  ;  / c 2  1  1  -7'2TC/ AT C  X « „ *2W ~  g  n  n |  (2.3.11)  (0  •8(T-(f +  2  2  •8(T-(f +AT (0))  2  AT (0)). ll2  2  An upper-bound on At (r) can be obtained by realizing [14] that the greatest change in n  the path length occurs when the mobile is directly in line with the direction of arrival of the signal. Over the duration T , this would correspond to a change in path-length of v • T for a mobile p  moving at velocity v. Therefore the change in delay (Ax (t)) n  associated with the change in  path-length would be v • T /c, where c is the speed of light. Since (Ax (t)) n  max  is much smaller  than the pulse duration, we can drop the Ax (t) factor from the delta functions in (2.3.11). n  However, the same cannot be neglected in the exponential terms due to the large value of f . c  Assuming that each of the two beams consists of a large number of independent rays and  Chapter 2 The Wireless Channel  16  the phases of the rays are uniformly distributed from 0 to 2n [15], the Central Limit Theorem [16] can be applied to (2.3.11). Taking Xj = 0 and x = a, (2.3.10) simplifies to 2  t r i c e d * )  =  OL -e~  -Hx-x )  + (t)-d(x) + e~  i2nfeXo  0  z (t)-^-o)  i2lifea  0  Zl  2  (2.3.12)  where z (t) and z (t) are complex Gaussian random processes accounting for the attenuation x  2  and phase of the two independent beams. Since the direct/specular path is assumed to have the shortest path-length and the first Rayleigh faded component arrives with negligible delay, we have x = 0. Normalizing the two 0  random processes such that i^z^r)! ] = £[|z (0| ] = 1 and explicitly specifying the 2  2  2  strengths of the two Rayleigh faded beams as 0Cj and oc respectively, we can re-write (2.3.12) as 2  z {t,x) rice  =  a -5(x) + a z (0-5(x) + a z (0-6(x-a) 0  1  1  2  2  (2.3.13)  Equation (2.3.13) is a general expression for the impulse response of a time-varying, frequency selective channel. The impulse response corresponding to a flat fading channel is z (t,x) rice  = a -8(x) + a z (r)-5(x) 0  1  1  (2.3.14)  Figure 2.4 illustrates the impulse response corresponding to (2.3.13) assuming a unity gain channel with a Rician factor of 1. The figure is plotted with a Doppler frequency of 20 Hz and a delay(a) of 0.2 sec.  17  Chapter 2 The Wireless Channel  0.20 0  Figure 2.4 3-D view of the discrete-time impulse response, \z{t, %)\ The Rician factor K is defined as the power of the specular component relative to the power in the Rayleigh faded or diffused component and is often expressed in dB. Thus («o)  (  K  dB  where ^f d (^) a  e  = 10-log (£) = 10 log 10  2  1  10  (2.3.15)  is the average delay power spectrum of the fading component of the channel,  ( a ) is the power of the specular path and the power of the diffused component is obtained by 2  0  integrating the average delay power spectrum i.e. J'iKf j (t)dx. ac e  2.3.2 RF Spectrum and Autocorrelation of the Fading Process The present work is based on Clarke's model [17] which assumes a fixed transmitter with  18  Chapter 2 The Wireless Channel  a v e r t i c a l l y p o l a r i z e d antenna. T h e statistical characteristics o f the r a n d o m process z (t) t  are based  o n i s o t r o p i c s c a t t e r i n g o f the c o m p o n e n t s / r a y s r a n d o m l y a r r i v i n g at the r e c e i v e r at u n i f o r m l y distributed angles a n d e x p e r i e n c i n g s i m i l a r attenuation over s m a l l - s c a l e distances.  F i g u r e 2.5 d e p i c t s the s p e c t r a l b r o a d e n i n g o f an u n m o d u l a t e d c o n t i n u o u s w a v e ( C W ) c a r r i e r at f r e q u e n c y f  c  due to D o p p l e r shift c a u s e d b y the R a y l e i g h c h a n n e l . T h u s e a c h o f the  rays arrives at the M S i n its o w n frequency, offset f r o m f  c  s p e c t r u m d e n s i t y S(f)  w i t h i n ±f .  a n d a u t o c o r r e l a t i o n f u n c t i o n §(t)  d  The corresponding power  o f the R a y l e i g h f a d i n g p r o c e s s z-(r)  f o r the case o f a v e r t i c a l w h i p antenna o n the M S are g i v e n b y [15]  S(f)  =  3b  (2.3.16)  w h e r e b is the average p o w e r r e c e i v e d b y an i s o t r o p i c antenna a n d (O  d  = 2nf , d  f  d  b e i n g the  m a x i m u m D o p p l e r frequency, and  <K0 = 1 . 5 - W ( o v O 0  w h e r e J (x) 0  is the zeroth order B e s s e l f u n c t i o n o f the first k i n d .  (2.3.17)  Chapter 2 The Wireless Channel  F i g u r e 2.5 P o w e r S p e c t r u m o f an u n m o d u l a t e d C W carrier  Chapter 3 Derivation of the MFB's  20  Chapter 3 Derivation of the M F B ' s 3.1 Introduction Matched Filter Bounds (MFB's) are useful to system designers in setting a goal for the practical realization of the system [6]. Assuming perfect equalization, the channel is considered free from Inter-symbol Interference (ISI). However the transmitted signal gets subjected to timevarying channel fades which, in most of the previous analyses, are assumed constant during a symbol duration. This chapter presents the MFB analysis for an uncoded BPSK system on a Rician fading channel with an arbitrary normalized Doppler rate. We assume an adaptive  matchedfilterwhose impulse response varies in accordance with the received (faded) signal. By varying the Rician factor, the derived expressions can be used to obtain the bit error rate (BER) performance in strongly (Rayleigh) and weakly fading channels and compare their behavior under fast and slow fading conditions. Although developed for the cases of one and two beam (in addition to the non-fading component) channels, the analysis can be extended to include multiple (>2) beams with arbitrary arrival times and inter-beam correlations. We define a channel as slowly or fast fading with respect to the normalized Doppler rate f  N  = f - T' where f d  d  is the Doppler frequency and T is the symbol duration. In land mobile  radio systems operating in the range of 900 MHz to 1.8 GHz, f  d  typically ranges from 50 to 100  Hz for a vehicle speed of 60 km/hr. Therefore a system using a transmission rate of 10 symbols/ 4  sec would have a maximum normalized Doppler rate of 0.01. This means that there are negligible variations in the channel characteristics over the duration T . We will refer to a system for which p  Chapter 3 Derivation of the MFB's  21  f < 0.01 as slowly fading. For an OFDM system employing hundreds of tones, T would be N  hundreds of times longer and in this case an analysis applicable to fast fading is required. 3.2 System Model We assume BPSK modulation for simplicity, although the analysis presented in the chapter can be applied to any PAM scheme with minor modifications. Figure 3.1 shows the complex baseband representation of the communication system. A signal, s(t), is sent by the modulator over the Rician channel. It is convenient to write s(t) = ±A-p(t),  (3.2.1)  A = JW  b  where p(t) is a unit energy pulse of duration T , so that s(t) has energy E . b  From Section 2.3.1, the complex baseband equivalent form of the impulse response of the time-varying channel is denoted by z(t, x) which is defined as the response of the channel at time t to an impulse applied at time t-x [12]. The response z(t, x) can be written as the sum of the impulse response of the direct path, a • 5(x), and the response of the diffused beam, z (t, x), 0  ray  i.e. z(t, x) = a • 5(x) + z (t, x) 0  ray  (3.2.2)  where a is the power of the direct/specular component. The real and imaginary parts of 0  z (t, x) are independent and identically distributed (i.i.d.) zero mean Gaussian random ray  processes. The channel imposes a maximum delay spread of x  max  on the transmitted signal s(t).  22  Chapter 3 Derivation of the MFB's  AWGN  n(t) Matched  r(f) ,  F i l t e r  , x{t)~^ X  T x Signal  Rician Channel F i g u r e 3.1  System M o d e l  T h e r e c e i v e d signal r{t)  is given b y  r(t)  w h e r e n(t)  = J z(t, X) • s(t - X) dx+ n(t)  ,  (3.2.3)  i s a s a m p l e f u n c t i o n o f the zero m e a n c o m p l e x additive w h i t e G a u s s i a n n o i s e process  w i t h p o w e r spectral density N .  T h e r e c e i v e r ' s d e c i s i o n , as t o w h i c h s i g n a l w a s t r a n s m i t t e d , i s  based o n the real c o m p o n e n t , X ,  o f the output variable X = x( T), w h e r e  Q  r  x(t) = jh(t-x)-r(x)dx,  T = T + x„ p  (3.2.4)  3.3 Error Performance Analysis T h e f a d e d s i g n a l at the channel output is written as  r (t) fad  where  =  ±A-s (t) fad  (3.3.1)  23  Chapter 3 Derivation of the MFB's  fadW  s  z(t,X)-p{t-X)dX  = j 0  (3.3.2)  T  *max  = a -p(t)+  j z (t,x)-  0  ray  p(t-x)dx  In (3.3.2) we have assumed that the direct path has zero delay whereas the impulse response of the diffused beam, z (t, x), extends from x ray  to x .  min  max  It is well known that a matched filter (MF) is the optimal receiver for a signal transmitted over a non-fading AWGN channel [18]. Therefore, by having the impulse response, h(t), of the receive filter matched to the complex signal, s (t), we can obtain the optimum BER perforfad  mance, i.e. h{t) =  (3.3.3)  s *(T-t) fad  Thus the entire system in Figure 3.1 can be interpreted in the form shown in Figure 3.2. Using (3.3.3) in (3.2.4), we can write the filter output as  *(0 = \ fad*(. s  T  t  +  (3.3.4)  *) r(x)dx.  r(0  x{t)  X  t = T  Tx Signal Figure 3.2  AWGN Channel The Transformed System  Receiver  24  Chapter 3 Derivation of the MFB's  Therefore, the decision variable X can be expressed as (3.3.5)  X = ±A • Y + N r  where the signal component, Y, is given by  y  =  \\ fadit)\ dt S  2  0  (3.3.6) a -p(t)+  -J  j z (t,x)  0  ray  • p(t-x)dx  \dt  and the noise component, N, is given by  N = Re js *(t)-n(t)dt fad  •o  *7 = Re J\a -p*(t)+ 0  0^  (3.3.7) j z *(t,x)ray  *(t-x)dx  P  n(t)dt  T •  "mm  v  In order to find the probability of error, we need to know the probability density function (pdf) of X . For convenience it is assumed that s(t) = A • p(t) is transmitted. Since the noise r  component, N, is Gaussian, the pdf of X conditioned on the signal component, Y, is also r  Gaussian with mean AY; it's variance can be obtained using Parseval's theorem [18] as a  2  -  2  •Wfad'iT-tfdt  (3.3.8)  Equation (3.3.8) shows that the noise variance, being a function of Y, is itself a random variable.  Chapter 3 Derivation of the MFB's  The pdf of X  r  25  c o n d i t i o n e d o n Y i s g i v e n as  -(*-(Ay)) ' 2  fx (*r\y)  =  r  ~ ^ ^  e  x  find  P  T h e c o n d i t i o n a l p r o b a b i l i t y o f error is the p r o b a b i l i t y that X  is less than zero, i.e.  r  P (X <0\Y) r  T h e o v e r a l l p r o b a b i l i t y o f error  P  b  Using  =  r  Equation  (3.3.12) can  V  that  Y>  exp  from  (3.3.11)  (3.3.6), w e can \  r  2a N  .find.  dx.  rewrite  P as b  \ f (y)  (3.3.12)  y  d  Y  M  be expressed i n terms o f the c o m p l e m e n t a r y error f u n c t i o n ,  E y/N  Q  Ay  ^  vfidl)  \erfc o  erfc(x), as  f (y) y d  Y  (3.3.13) f (y) y d  j  Y  can be interpreted as the received S N R w h e r e  ted S N R . T h e B E R can be c a l c u l a t e d u s i n g  Y  0  :  2  t i o n o f f (y)  (3.3.10) o v e r the p d f o f Y  Y  <x -(Ay))  f  b  r  r  r  = \ • \erfc  The term  r  lP (X <0\Y)-f (y)dy  oyfj f =  x  b  (3.3.9) a n d (3.3.10) and the fact  Pu  (3.3.10)  \f {x \y)dx  P is obtained b y averaging  =  (3.3.9)  2o N  (3.3.13) i f the  pdf  f {y) Y  E /N b  Q  i s the  transmit-  i s available. T h e d e t e r m i n a -  c a n be a c h a l l e n g i n g task. T h e f o l l o w i n g quote f r o m [14] s u m m a r i z e s the p r o b l e m ,  26  Chapter 3 Derivation of the MFB's  'The effects of fast fading, correlations and diversity are reflected solely in how they affect the pdf of Y\ 3.3.1 Deriving the pdf of Y Expanding (3.3.6), Y can be written as Y = Y  l +  Y  2 +  3  (3.3.14)  =  (3.3.15)  Y  where Y, = a - \p\t)dt 2  Y  2  = 2 a J Re J z (t, x) • p(t-t)dz 0  ray  mm  • p(t-x)dx dt  J z (t,x) ray  (3.3.17)  "mm  T  = j  ray  (3.3.16)  mm  ^ - J  and f(t) = J z (t,x)  Pit) dt  \f(t)\ dt 2  . Following [14], we approximate (3.3.17) as a finite  • p(t-x)dx  summation, i.e. M  r = 3  X|/J n= 1  (3.3.18)  27  Chapter 3 Derivation of the MFB's  T +T  where M =  p  —T • max  and /  min  = f^n  n  -  • Atj; the floor function L* J is used to  At  denote the largest integer <x. Since f(t) is a zero mean complex Gaussian random process comprising of i.i.d. real and imaginary components, each of the M terms, |/„| , has a central chi2  squared degree 2 distribution. However (3.3.18) is a sum of correlated random variables. In order to express Y as a sum of independent random variables, we use an orthogonal expansion 3  (Karhunen -Lo'ev e Expansion, [16], [19]), i.e. M  f -jAt=  (3.3.19)  ^U -JX -<?  n  k  k  nk  k= 1  where {cp } = j t p ^ n - ^ • Afj j are orthonormal vectors (3.3.20) and {U } are orthogonal n/t  k  complex zero-mean Gaussian random variables with unit variance as shown in (3.3.21) M  X  Vnl  n=  = /m  • ^nm*  5  1  E<U U *) r  {X } and {(p^(0} k  a r e  =  1, / = m 0, l± m  (3.3.21)  =8  m  (3.3.20)  recognized as the eigenvalues and eigen-functions [16] that satisfy the  integral equation J R (t f  v  t ) • y (t ) dt = X • tp^) 2  k  2  2  k  where Rf(t t ) is the autocorrelation function of the process f(t). The function p  2  (3.3.22)  Chapter 3 Derivation of the MFB's  R (t , t ) = £[/(*, f  {  )-/*(> )]  2  r  2#  2  T  (3.3.23)  max "max  =E  can be expressed as a Hermitian matrix  whose (/,m)th element is  ^ 24)  "max "max  = J j  X [(l-m)At-,(T ,x )]p(l'At-T )p*(m'At-x )dT dT z  a  b  a  b  a  b  1 1 where /' = / - r , m' = m-- and 1 </,ra< M . From (2.3.5), the autocorrelation of i s  s  i v e n  °y a j ' i - ' ;(v  x  2  t)] =  W'i«  X  J •W < ' 2 . i)] •  ( - - )  x  3  3  25  Accommodating A? in (3.3.24) and using (3.3.19), we obtain R - At = E[ff ] H  f  / = TA? •  w here  A = diag{X ...X ). x  M  ... f^ , T  = cp- A cp"  cp = [cp, ... cp ] with M  (3.3.26)  9* = [<p ... <Pjif*]  a  n  d  u  The eigenvectors are real and the eigenvalues are real and non-negative  [20]. This also follows from [21] which shows that the eigenvalues of a positive definite matrix, Ry, are both real and positive. F can now be expressed as a sum of uncorrelated random variables, U U , 3  This follows from (3.3.18) and (3.3.19) from which we can write  v  2  U . M  29  Chapter 3 Derivation of the MFB's  M  M  M X kfck<Q\k lk = 1 u  k  M  X UkfckVxk k=\  ••• X kJ ~k ?Mk k= 1 U  (  ••• X k=l  kfckVMk  U  (3.3.27)  M  = X V N k=  2  1  w h e r e the r a n d o m variable, U , can be written as n  U w  and  (3.3.28)  - V + iW n J n r  n  \U \ = V% + W% has a central c h i - s q u a r e d degree 2 pdf. A l s o , f r o m (3.3.19) a n d (3.3.21), 2  n  the real c o m p o n e n t ,  V , and the i m a g i n a r y c o m p o n e n t , W , o f U are i.i.d. z e r o - m e a n G a u s s i a n n  n  r a n d o m variables w i t h variance a  2  = a  n  = ]r. T h e characteristic f u n c t i o n o f Y  2  3  *n  i s then e a s i l y  It  n  obtained f r o m the characteristic f u n c t i o n o f chi-squared degree 2 r a n d o m variables as  M  -n( Equation  (3.3.29)  i ^ )  (3.3.29) gives the characteristic f u n c t i o n o f the s i g n a l c o m p o n e n t Y w h e n the f a d i n g is  Rayleigh. Writing  (3.3.16) i n terms o f f(t) and u s i n g (3.3.19), w e have  2 a J Re[f(t)] • p(t)dt 0  (3.3.30)  = 2 a ( ^ > [ / V ] • Pi • A*) 0  N (• M  = «o X 2  \  X Wk\ • fik • Re  • Pi •  30  Chapter 3 Derivation of the MFB's  . >0  ,T„>%  At  where D =  '  p  mm  and i' =  p  ,N =  At  1 > x -„ = 0  i-D+\  mi  Interchanging the order of summations yields M  Y  = 2a jAt^Re[U ]-  2  Q  X cp,'* •  JX -  k  k  \i = D  k = 1  (3.3.31)  M k=  1  N  where B^ = 2a */At • X cp ^- p is a constant. Equation (3.3.31) suggests that Y is a 0  (  i  2  weighted sum of independent zero-mean real Gaussian random variables, { V }, with variance k  rj . Adding Y , F and simplifying, we obtain 2  2  3  f M  \  / M  /  Vz =  Y +Y = 3  2  V« =  1  M  B\  2  /B \  M  (3.3.32)  1  = x K A + T ) - « x= i m  M  2  +  n =1  s  n =1  <vw-  From (3.3.14) and (3.3.15), we have M  Y =  «o -X[yJ V  where V ' = V JX n  n  n  n=  M  M  2  +X(tV> +X<*V> 2  '  1  n=  + ^ and W„' = W Jl . n  n  1  = 1  :  (3.3.33)  In (3.3.33) 7 is represented as a sum of three  independent terms. The characteristic function of Y is therefore the product of the characteristic  31  Chapter 3 Derivation of the MFB's  f u n c t i o n s o f each t e r m . R e a l i z i n g that the first term, Y , is a constant, its characteristic f u n c t i o n is }  written as (  /R  M  Y y / T ) ) = exp  >2Y\  (3.3.34)  JJ  The second term, Y , n  K  h a s a non-central c h i - s q u a r e d d i s t r i b u t i o n o f d e g r e e 1. S i n c e V ' has n  P n  variance — a n d m e a n — , it's characteristic f u n c t i o n is g i v e n b y [12]  M  (v) =  T h e t h i r d term, 7  / / 7  2  TT /  exp  1  4(1-A„v)  A (3.3.35)  , has a central chi-squared distribution o f degree 1 w i t h Wn' h a v i n g variance  n  — . A c c o r d i n g l y , w e have f r o m [12]  M *Y,„W  =  (3.3.36)  n  T h e characteristic f u n c t i o n o f Y is thus g i v e n b y Y (i» = y  • «P (D) • ^ K||  y / ; /  (t)) (3.3.37)  T h e c o r r e s p o n d i n g p d f o f Y c a n be obtained n u m e r i c a l l y b y t a k i n g the inverse F o u r i e r t r a n s f o r m o f (3.3.37), i.e.  Chapter 3 Derivation of the MFB's  32  oo fy(y) = ^ J V (v)exv(-jvy) —oo  dv .  Y  (3.3.38)  3.4 Flat Fading In flat fading, all the frequency components suffer similar attenuation and phase-shift by the channel and the delay power spectrum of the channel SR(x) is ideally an impulse. Therefore, for very-slow fading, we may write the channel's impulse response as z(?,x) = oc 8(x)+Z-8(x), 0  0<a <l 0  (3.4.1)  where Z is a complex Gaussian random variable; assuming that the channel has unity power gain, £T|Z| ] = 1 - CLQ. However in the general case, when the attenuation may vary within the pulse 2  duration, the channel's impulse response is given by (2.3.14) (3.4.2)  z(t,z) = a 8(T) + ccjz(08(T) 0  where a is the power of the diffused beam. The Rician factor of the channel is defined as 2  K  dB  = 10 • log  (3.4.3)  10  z(t) is a complex Gaussian random process with £[|z(0l ] = 1- Thus from (3.2.2) 2  z (t, x) = aj z(f)8(x) and the corresponding autocorrelation function is given by ray  = ^ ( r , - ^ ) - 8(rj)8(f ). 2  From (3.3.24) the (l,m)th element of the corresponding Hermitian matrix Rj is given by  Chapter 3 Derivation of the MFB's  T  33  T  '"max *max  f(l,m)=  R  j  J $ ({l-m)Am% )h{T )p(l'&t-x )p*(m'At-T )dx dx .  0  0  z  a  b  a  b  a  b  (3.4.5)  Using the channel autocorrelation function for a mobile whip antenna with isotropic scattering [15] <»(x) = 7 (av t) z  (3.4.6)  0  where J (x) is a zeroth order Bessel function and (0 = 2nf is the maximum Doppler angular 0  d  d  frequency, (3.4.5) simplifies to RfiUm) = J (2Kf (l-m)At)p(l'At)p*(m'At) 0  d  .  (3.4.7)  The eigenvalues and eigenvectors corresponding to the matrix RfAt can be easily obtained using mathematical tools [22]. Since x  max  =%  min  = 0, from (3.3.30), N = M and  i' = / which gives M f M  \  i = 1 VJfc = 1 M  /  (3.4.8)  i  and from (3.3.27) we have M  W + W|).  Y = af 2 3  Jk=  (3.4.9)  1  Since the channel is assumed to be time-varying within the pulse duration, we expect the pulse shape to have a significant effect on the system performance as the effective channel averaging depends on the pulse shape. This can also be observed from the expression of the received  34  Chapter 3 Derivation of the MFB's  S N R where the signal c o m p o n e n t Y depends o n the pulse p(t). T h e r e f o r e three pulse shapes are chosen rect pulse:  P r e c M )  =  (3.4.10)  I  otherwise  10, h a l f sin pulse:  sin(7Cf/r ) , o < r < r 0  PhsW = [0 ,  (3.4.11)  0  otherwise  root raised c o s i n e pulse:  1 ( rn(t-aT y ' l + cosf 2T, aT,o ^-^-\\,0<t<aT Q  0  aT <t<T Q  J i ( °< ^))' +C1  2  !  (3.4.12)  0  T < *T « a)  0  0  10,  t  0  +  otherwise  w h e r e a i s the r o l l - o f f f a c t o r o f the p u l s e a n d 0 < a < 1. T h e n o t a t i o n T , p  refers to the s y m b o l / p u l s e duration; henceforth T  p  i n general,  = T ( l + a) w h e r e a = 0 f o r the rect p u l s e 0  and the h a l f s i n pulse.  3.4.1 Special Cases: 3.4.1.1 Very Slow Fading W h e n the f a d i n g is v e r y slow,  0<f < N  0.01, u s i n g (3.4.1), the s i g n a l c o m p o n e n t Y i n  35  Chapter 3 Derivation of the MFB's  (3.3.6) reduces to Y = \a + Z\ which implies that E[Y] = 1.0. This shows that the BER 2  0  does not depend on the pulse shape and P is calculated by averaging the error probability for a b  non-fading AWGN channel over the non-central chi-squared degree 2 pdf, i.e.  oo  /  1 exp (l-a ) v  I  1 r A b-y = - • erfc —-— 2 J U N E  iJyaQ  2  0  1  ~  a  \iy  (3.4.13)  o j  0  Equation (3.4.13) is simplified in [4] and P is expressed in terms of the Marcum's ^-function b  v + vf I (vw) exp 2  P = Q(v,w)-h\  +  h  where e = E[\Z\ ] • E /N 2  b  0  1 +e  1  0  (3.4.14)  and  v=  K[l + 2 e - 2 V e ( l +e)] 2(1+8)  w  K[\ +2e + 2Ve(l +e)] 2(1 +e)  K is the Rician factor, K = al/E[\Z\ ], 2  (3.4.15)  I^(x) is the Bth order modified Bessel function of the  first kind and the Marcum's Q-function is defined as [12]  G(v, w) = e x p ^ - ^ ^ j X g J/„(vw),  w>v>0  (3.4.16)  n=0  3.4.1.2 Very Fast Fading It has been shown in Appendix B that in the extreme condition when the normalized Doppler rate increases without limit, i.e. f ^°°, N  the variance of the signal component Y  approaches zero. Since the mean, E[Y] = 1.0, this implies that the pdf,  36  Chapter 3 Derivation of the MFB's  lim f (y) Y  = 8(y- 1). Applying this in (3.3.13), we obtain  •8(y-l)dy  lim P  (3.4.17)  which gives the BER for a non-fading AWGN channel. Thus using an extremely long signaling pulse relative to the channel's coherence time or having a very high Doppler frequency has essentially made the fading channel look like a non-fading one [14]. 3.4.1.3 Rician Factor  Equation (3.3.13) gives the general expression of the BER for a Rician fading channel. Varying the Rician factor, K, corresponds to varying the power of the non-fading component with respect to that of the fading component. We consider two extreme cases corresponding to a Rayleigh channel (K = 0) and a non-fading channel (K —> oo). When K = 0, a = 0 and the direct/specular path is absent. Accordingly, the signal 0  component Y in (3.3.14) equals Y . Using (3.4.2) with a = £[|z(f)| ] = 1 and cx = 0, we 2  3  r  0  have the following results from [8]  E /N b  (  L  0  ( ) < / „ < 0.01  +1I'  (3.4.18)  L  , otherwise V  n = 1 m=1 m*n  J  Chapter 3 Derivation of the MFB's  37  where L is the number of non-zero eigenvalues of the matrix Rj-At. On the other extreme when K -> oo, the channel's impulse response (3.4.2) reduces to that of a non-fading AWGN channel and the BER is given by lim  P = \ erM,Ja$E /N ) b  b  0  .  (3.4.19)  3.5 Frequency-Selective Fading When there is a significant amount of delay spread relative to the pulse duration, the channel is said to be frequency selective. This implies that the channel is selective in it's response to the signal at different frequencies and the fading at different frequencies is not necessarily correlated. We present the general cases of two and three beam signal reception when the direct (non-faded) and diffused versions of the transmitted pulse arrive at different time instants. 3.5.1 Two-Beam Case In our two-beam case, the channel is modeled with the following impulse response z(t,x) = a 8(T) + a z ( r ) 5 ( x - a , ) . 0  1  1  (3.5.1)  Assuming that the non-faded beam has the shortest path length, the first and the second terms correspond to the non-delayed (direct component) and delayed (faded component) responses of the channel respectively. Considering a unity gain channel, ocj = By varying the delay parameter a , , we can study the influence of signal dispersion on the system performance. The signal component Y for the present case is written as  38  Chapter 3 Derivation of the MFB's  Y = a + 2a a j 2  0  0  1  Re[z (t)p(t-G )]p(t)dt + a • j\z (t)p(t-a )\ dt (3.5.2) 2  x  x  x  x  x  where T = T + a . Using (3.3.30) and (3.3.31), the second term in (3.5.2) can be writtenas p  x  N  /  Y = 2 a a, JXt £  \  £  0  2  M  • cp,.,*  , = £)Vjt=l  P i  V  (3.5.3)  M  £= 1 where  Ar.  D = 1  and TV =  >0  ,a,  ,o = x  (3.5.4) 0  , /' = i-D + 1. {^} are the eigenvalues of the matrix Rf At; the (/,m)th  Ar  element of matrix Rj is defined in (3.4.7). When a >T,Y x  2  = 0 which gives  Y = oc + a - f 2  2  0  =  CL + Ct 2  o i  where Y  2  x  x  (3.5.5)  • Y °y  r  = J |z (?) p(^ — cjj )|<i? is the received signal component from the diffused beam. 2  ray  2  \z (t)p(t-o )\ dt  1  j  39  Chapter 3 Derivation of the MFB's  The pdf of  Y, obtained f r o m the p d f o f Y  ray  1  (y-«o)  Xr  2  f {y) =  [14], i s g i v e n as N  e x p  .  v>a  2  (3.5.6)  Y  lo,  w h e r e the constants B  otherwise  a n d L i s the n u m b e r o f n o n - z e r o  =  n  v = jK  eigenvalues. T h e e x p r e s s i o n f o r the B E R is thus g i v e n b y  2a 1  P e r f o r m i n g integration b y parts o n  V  1  z  u  B  e x  o J  N  n= 1  (3.5.7)  \dy  P  (3.5.7) w e get  "  l n = l  X r  b  J  a i  B  \E -y  jerfc  e  2  • "a ^ 2  1  "  £  £[e -y^-y- dy 71 J l  1/2  (3.5.8) and u s i n g ( A . 10)  P =\ b  X «  erfc  w h e r e  q =i r n  +  1 a ^ 2  erfc(jcijq )exp n  ^0^  « =l  ( ct ^ 2  ^b Q a  5  (3.5.9)  V l ^W. a  2  are constants.  F o r a v e r y - s l o w f a d i n g case ( w i t h any degree o f o v e r l a p p i n g b e t w e e n the d i r e c t a n d the diffused beams),  (3.5.1)  and  (3.5.2)  are g i v e n as  z(t,x) = a05(x) + Z 8 ( T - a ) 1  1  (3.5.10)  40  Chapter 3 Derivation of the MFB's  Y = a +2a k • Z 2  0  = a (\-k ) 2  where  Z =Z x  2  rl  + jZ , E[\Z \ ] = E[Z n  2  + (Z + a k) + Z  2  2  rl  + (Z , + Z ) 2  rl  0  x  + Z ] = 1 -a§; Z  2  X  (3.5.11)  2  and  2  rl  X  rl  Z  n  are  z e r om e a n  k = j p(t)p(t- o , ) d r i s a c o n s t a n t . B y e m p l o y i n g s i m p l e  Gaussian random variables and  transformations to the non-central chi-squared degree 2 distribution, the p d f o f Y is w r i t t e n as  ' 2a k -a 2  2  + y}  2  2  2  W h e n the t w o beams are n o n - o v e r l a p p i n g ,  2  0  o  I-a  1-ccg  fjy - a ( 1 - k ) • 2a k 1-a  v  J  2  (3.5.12)  o > T a n d k = 0. C o n s e q u e n t l y (3.3.13) s i m p l i f i e s l  to  Pu =  1 jerfc 2(1-a )  f  • exp  2  y - «  2  >  (3.5.13)  a<5  A p p l y i n g a p r o c e d u r e s i m i l a r to the one used i n o b t a i n i n g  (  Pu =  erfc vi  ^0 j  -exp  «o  (3.5.9), the B E R is g i v e n b y  E /N  2 A  b  N  0  l-oco  0  erfc  x  2  &  +  \-a ,j 2  . (3.5.14)  2  T h e above expression c a n also be directly obtained f r o m Ry • At has o n l y a s i n g l e non-zero eigenvalue, X  a  (3.5.9) b y r e a l i z i n g that the m a t r i x  = 1.0, w h e n the f a d i n g is very slow.  Chapter 3 Derivation of the MFB's  41  3.5.2 Three-Beam Case In the three-beam channel model, the transmitted signal arrives following three propagation paths of which thefirst(non-delayed) one is the direct component and the other two arriving at some delays, a, and G , are the diffused components. Accordingly the impulse response is 2  given by z(t, i)  o >o, .  = a 5(T)+ a z (05(i;-a )+ a z (0o(i:-a ), 1  0  1  1  2  2  2  2  (3.5.15)  Comparing (3.2.2) with (3.5.15), z (t, x) = a,Zj(05(T - a,) + a z (f)8(x - a ). Assuming a 2  ray  2  2  unity gain channel and ist^,^)! ] = £T|z (0| ] = 1, we have 2  2  2  a^ + a + a| = 1 .  (3.5.16)  2  The signal component corresponding to (3.5.15) is Y = a + 2a j Re[f(t)]p(t)dt+ $\f(t)\ dt 2  (3.5.17)  2  0  0  o,  o,  where, from Section 3.3.1, f{t) = a z (r)p(r-a,) + a z (t)p(t - cs ) • Considering the 1  1  2  2  2  general case where the two processes Zj(0 and z (t) have a normalized cross-correlation coeffi2  cient p, which is given as 0  p = .  where 6  ^  (T)  r=,  (x) is the cross-correlation function and  lpl<l  (L.(T), i  (3.5.18)  = 1, 2, is the autocorrelation  function of the two processes. Assuming both the fading processes have identical statistical characteristics, we can write  42  Chapter 3 Derivation of the MFB's  = <| >,(x) = J ((O • n  ty (T)  z  Zi  Q  d  (3.5.19)  X) .  2  The autocorrelation function of z (t, x) is given as ray  R (t -t ;(x ,T ))  <  z  l  2  l  2  + a\ • cb (f! - r )8(x - a )6(x - a ) 2  Z2  1  2  2  2  + a ^ p c b ^ r , - r )5(T! - a )6(x - a ) 2  1  + o^a^cb (t -? )8(Xj 2  x  2  2  -G )8(T - a ) 2  2  t  (3.5.20) which results in the following elements of matrix R^ f{i,m) = • o ( r f ( ) 0 t ? p(l'At-a )p*(m'At-a ) + alp(TAt-a )p*(m'At-ci ) + aja pp(l'At-c ) p*(m'At-a ) + a,a p*/?(/'Ar-a )p*(m'At- Gj)] .  R  /  c o  / _ m  A  06  l  2  x  x  2  2  2  2  2  (3.5.21) Using (3.3.31) M Y = 2a 7Ar X X ^ A i = D\k=\ M N  2  /  '  0  = X kJhh v  Pi  (3.5.22)  •  k=1  When thefirsttwo beams arrive simultaneously i.e. Gj = 0, the three-beam case may be interpreted as a two-beam signal reception of which the first one is Rician faded and the second one is Rayleigh faded. Under these circumstances, for the special case of slow fading Rician channel when the fading in the two diffused components is uncorrelated, a = a and a >T , x  2  2  p  the BER is given as (Appendix A) P  n+1  - I n=0  X  k=1  {  k  )4  x  'U.  (3.5.23)  Chapter 3 Derivation of the MFB's  43  E {l-a ) 2  b  where X = 1 +  —  a n  d  m  e  coefficients C are defined in (A. 13). n  In another case when G > T ( 1 + a), i.e. there is no overlap between the direct and the x  Q  diffused beams, (3.5.17) reduces to Y = a + J|/(0I * 2  2  (3.5.24)  - 0 Y' a  +  ray  where Y  = J\f(t)\ dt. Comparing (3.5.24) with (3.3.14), we have Y = a ,, F = 0 and 2  ray  2  l  Y = Y . The pdf of Y derived from the pdf of Y 3  ray  ray  BIm f (y) = Y  /  [14] is given as  1m - 1 s  /=i =i(m-l)!A,7 ( y - a ) 0  2  f  (y-ctpV  exp  X,  > y>ag  (3.5.25)  m  LO , otherwise  where L is the number of distinct and non-zero eigenvalues, {X } , and r is the order of X d  t  t  v  The constants B are given by lm  ( -m)\(-jXj)'  -(l-jX^V'Wy  'A/  ri  M  where *F  (D) =  V  r ay  K  r  TT : »—r is the characteristic function of Y . (1 _ ll.Vj)  -11  1=1  rnv  '  Combining (3.5.25) with (3.3.13) yields  (3.5.26)  (v)  44  Chapter 3 Derivation of the MFB's  (y - ao)  Pu =  dy  (3.5.27)  a  0  Evaluating the above integral (Appendix C) results in the following expression ^ F  5 /m  2  2  J  N  a^ 2  o  J  (3.5.28) n+ 1  exp \_\\[  (M-l)!  2J 2 ' 2  E .1 where | i = m - « and constants C = — + r - . T(z) is the Gamma function and b  t  v  (z) is  known as the Whittaker function [23]. r = 2, V7 applies when the two faded beams are indepenl  dent and non-overlapping. However when r = 1, V/, (3.5.28) simplifies to l  erf /= 1  + vi  ^0  j  N TZ Q  exp  ^erfcijc^ ) 2  (3.5.29)  where erf(x) and erfc(x) are the error functions and complementary error functions respectively.  Chapter 4 Numerical Results  45  C h a p t e r 4 N u m e r i c a l Results 4.1 Introduction T h e expressions obtained f r o m the analysis i n C h a p t e r 3 are u s e d to obtain B E R c u r v e s f o r d i f f e r e n t c h a n n e l c o n d i t i o n s . T h e results f o r flat f a d i n g R i c i a n a n d f r e q u e n c y s e l e c t i v e f a d i n g R i c i a n channels are presented i n S e c t i o n 4.2 and S e c t i o n 4.3 respectively. In the figures, the B E R p l o t t e d against  E /N b  Q  a s s u m i n g a u n i t y g a i n c h a n n e l . U n l e s s otherwise stated, a rect p u l s e shape  is a s s u m e d .  In order to v e r i f y the results f r o m o u r analysis, s i m u l a t i o n s u s i n g M A T L A B ®  [22] w e r e  p e r f o r m e d f o r a l l the c u r v e s presented. T h e R a y l e i g h f a d i n g s i m u l a t o r u s e d i n the s i m u l a t i o n is b a s e d o n C l a r k e ' s m o d e l , [17], and uses a m o d i f i e d v e r s i o n , [24], o f the a l g o r i t h m i n [25]. In a l l cases, the s i m u l a t i o n results w e r e f o u n d to agree c l o s e l y w i t h the a n a l y t i c a l c u r v e s . T h e 9 9 % c o n f i d e n c e intervals f o r the s i m u l a t i o n results are w i t h i n ±5 % o f the c o r r e s p o n d i n g m e a n values. B e c a u s e o f the c l o s e agreement, the s i m u l a t i o n results are g e n e r a l l y not plotted.  T h e results i n S e c t i o n 4.2 and S e c t i o n 4.3 assume that the receiver has perfect k n o w l e d g e o f the c h a n n e l . In S e c t i o n 4.4, the effect o f c h a n n e l estimation errors is studied.  4.2 Flat Rician Fading F i g u r e 4.1 i l l u s t r a t e s the B E R , P , b  c u r v e s f o r d i f f e r e n t n o r m a l i z e d D o p p l e r rates, f  w i t h a R i c i a n f a c t o r o f O d B . T h e l i m i t i n g case o f v e r y fast f a d i n g (f  N  N  ,  —> °°) w h i c h c o i n c i d e s  w i t h the n o n - f a d i n g A W G N channel case ( S e c t i o n 3.4.1.2) is also s h o w n . T h e p a i r o f dots m a r k e d at i n t e r v a l s o f 2 d B d e p i c t the c o n f i d e n c e i n t e r v a l s . It is o b s e r v e d that the s y s t e m p e r f o r m a n c e  Chapter 4 Numerical Results  46  improves substantially with the fading rate.  10°  10"'i IO"  2  1CT  3  ST m Q.  -4  10  10"  5  IO""  IO"  7  0  2  4  6  8  10  12  14  16  18  20  E /N [dB] b  Figure 4.1 BER as a function of / The case 0.0 < f  N  0  for the Rician channel, K  < 0.01, corresponding to very-slow fading, has the poorest perfor-  mance and the curve shifts towards the non-fading case as f  N  performance with f  N  = 0 dB  dB  increases. The improvement in  is more noticeable as we move from the very-slow fading to f  N  = 0.64  and it tends to diminish with further increase in / . A comparison of the performance for the Rayleigh (K  dB  (K  dB  = 0 dB) and non-fading (K  dB  —» —<» dB), Rician  —> °° dB) channels as a function of f  N  is shown in Figure  4.2. As expected, the Rician channel has a lower BER than the Rayleigh channel.  Chapter 4 Numerical Results  Chapter 4 Numerical Results  48  I n F i g u r e 4 . 2 , f o r the v a l u e s o f f  N  c u r v e s , f o r a g i v e n E /N , b  A W G N c u r v e as f  N  Q  c h o s e n , the g a p b e t w e e n the R a y l e i g h a n d R i c i a n  tends to increase w i t h f .  H o w e v e r , the t w o c u r v e s c o n v e r g e to the  N  —> °°. T h u s , the specular c o m p o n e n t i m p r o v e s the B E R at l o w e r f a d i n g rates;  f o r v e r y h i g h f a d i n g rates, the i m p l i c i t diversity effect b e c o m e s d o m i n a n t i n d i m i n i s h i n g the effect of fading. T h e e f f e c t o f the s p e c u l a r c o m p o n e n t o n P  f o r a r a n g e o f R i c i a n f a c t o r , K,  b  illustrated i n F i g u r e 4.3. A s noted earlier, the B E R decreases w i t h K. i n p e r f o r m a n c e is o b s e r v e d f o r K  dB  > 0 dB . Since K  dB  v a l u e s is  H o w e v e r , a substantial g a i n  = 0 d B c o r r e s p o n d s to e q u a l p o w e r s i n  the f a d i n g a n d n o n - f a d i n g c o m p o n e n t s o f the c h a n n e l , this b e h a v i o r s h o w s that a greater p o w e r i n the n o n - f a d e d s i g n a l c o m p o n e n t relative to the f a d e d c o m p o n e n t d i m i n i s h e s the f a d i n g effect o f the c h a n n e l .  F i g u r e 4.4 E f f e c t o f pulse shape i n fast f a d i n g R i c i a n c h a n n e l , K  dB  = 0 dB  Chapter 4 Numerical Results  49  Unlike in very slow (0.0 < f  N  < 0.01) and nonfading conditions where the BER does not  depend on the shape of the transmitted pulse, p(t), the pulse shape does influence the BER in fast fading environments [8], [14]. This is illustrated in Figure 4.4 where the P curves for three pulse b  shapes: rect, half sin and root raised cosine (RRC), are shown. The parameter 'a' refers to the roll-off factor for the RRC pulse. Due to less effective channel averaging for the half sin and RRC (a > 0) pulses, the BER, for the pulse shapes chosen, is found to be the lowest for the rect pulse.  n  r  rect half sin RRC (a = 1/4) fading signal  1.5  —i  E <  7~  i  ' i  !  :  : \  • '• I  \  0.5 :: I]  :i  n  \+\ ^ \  7  i  \\ \\  _j  0  : : V: :: ::  V: \ :: :  :• • i. ; i  0.05  0.1  0.15  0.2  0.25  0.3  0.35  t  0.4  i_  0.45  Ml 0.E  f [sec]  Figure 4.5 Channel averaging ability of different pulse shapes Figure 4.5 shows the rect, half sin and RRC (a = 0.25) pulse shapes, all of the same pulse duration, T = 0.5 sec. A sample of a channel fading waveform is also shown. It can be p  seen that the amplitudes of the half sin and RRC pulse shapes are relatively smaller at the beginning and end of the pulses.  50  Chapter 4 Numerical Results  10°  10"'  10"  2  10°  cr  CD  -4  10  10"  5  lO"'  0  2  4  6  8  10  12  14  16  18  20  E /N [dB] b  0  Figure 4.6 Effect of the roll-off factor on the system performance, K  dB  = 0 dB  It is important to understand the basis on which comparison among different pulse shapes is made, especially as it applies to the R R C pulse with varying roll-off factors. Since the normalized Doppler rate, f  = fT  N  T  0  d  p  is a function of the total pulse duration, T  p  = T ( 1 + a) where Q  is the duration of a rect or half sin pulse, a roll-off factor of 1.0 (for instance) for the R R C  pulse would correspond to twice the normalized Doppler rate f o r a rect pulse with the same Doppler frequency, f  d  . Figure 4.6 shows how an increase in ' a ' adversely affects the B E R due to  degradation in channel averaging ability of the pulse.  4.3 Frequency-Selective Rician Fading The results for fast and slow fading R i c i a n channel as two and three beam models are presented i n this section. The curves are plotted for varying amount of delay (or overlap) and  Chapter 4 Numerical Results  51  correlation between the beams and comparison is also made with respect to the Rician factor.  4.3.1 Two-Beam Case MFB for the two-beam Rician channel, of which thefirstbeam is the direct and the second the diffused beam, are illustrated in Figure 4.7 for both fast (f  N  = 0.64) and very slow fading  conditions. The curves are plotted for normalized delays of 0.0, 0.5 and 1.0 for the diffused beam which account for 100% overlap, 50% overlap and no overlap between the two beams. As expected the results for zero delay match with the flat fading BER and P reduces with the b  decrease in the amount of overlapping (interference) between the two beams. Also the dB spread is found to be more in the slow fading case because in slow fading, diversity is achieved only from the extent of beam independence whereas in fast fading there is an additional advantage from the implicit diversity in each symbol.  10° t 100% overlap • 50% overlap 0 No overlap  io-1 10"  2  10"  3  cc £  a  10  CL  10-  5  10-* 0.0 <f.,< 0.01 10"  7  10  J  0  2  4  6  8  10  12  Figure 4.7 Two-beam frequency selective fading, K  14  dB  16  18  = 0 dB  20  Chapter 4 Numerical Results  52  4.3.2 Three-Beam Case Figures 4.8 to 4.11 are plotted for the three-beam channel comprising of a direct/nonfaded beam and two diffused/faded beams. The curves are presented for varying normalized delays, rj, and correlation coefficients, p, between the two faded components assuming no  overlapping with the non-faded component. The correlation [26] between the fading processes in the two beams is implemented as follows: two (independent) sets of fading samples are generated 2  using the fading simulator. Samples from one of the sets are then scaled by the factor v 1 - p and a fraction (p) of the corresponding samples from the other set is added to the scaled samples. The two correlated sets of samples are then normalized independently.  10° + [0.0;-0.5] • [0.0; 0.0] * [0.0; 0.5]  rr UJ  m  a.  10"  0  2  4  6  8  10  12  14  Figure 4.8 Three-beam frequency selective fast fading, f  16  N  18  20  = 0.64 and a = 0  53  Chapter 4 Numerical Results  F i g u r e s 4.8  a n d 4.9  d i f f u s e d b e a m s i.e. a  refer to fast and s l o w f a d i n g c o n d i t i o n s w i t h no delay b e t w e e n the two  = 0. In b o t h cases it has b e e n f o u n d that a n i n c r e a s e i n p r e s u l t s i n an  i m p r o v e m e n t i n s y s t e m p e r f o r m a n c e due to an increase i n the r e c e i v e d s i g n a l strength w h e n the b e a m s tend to be correlated and co-phased. A t the other extreme, a c o n d i t i o n o f  [0.0;-1.0]  would  r e s u l t i n t o t a l c a n c e l l a t i o n o f the d i f f u s e d b e a m s a n d r e c e p t i o n o f j u s t the n o n - f a d e d s i g n a l c o m p o n e n t . A l t h o u g h not s h o w n i n the figures f o r the sake o f clarity, c o n s t a n t B E R o f 0.5  f o r the R a y l e i g h c h a n n e l ; f o r the R i c i a n c h a n n e l s i n c e K  dB  c o r r e s p o n d s to a p o w e r o f  C o m p a r i n g F i g u r e s 4.8  [0.0;-1.0] c o r r e s p o n d s  0.2  = - erfc  (for a unity g a i n channel) i n the direct b e a m , P  a n d 4.9,  b  ~-6  '5iV  dB  n  the g a i n (obtained w i t h an increase i n p ) i n the latter is seen to be  reduced. T h i s is because i n fast f a d i n g there is diversity w i t h i n a s i n g l e pulse.  10  F i g u r e 4.9  to a  12  14  16  T h r e e - b e a m frequency selective v e r y - s l o w f a d i n g , a  18  =  20  0  Chapter 4 Numerical Results  54  Figures 4.10 and 4.11 illustrate the BER curves assuming a delay between the diffused beams. The curves corresponding to the cases [1.0;-1.0] and [1.0; 1.0] (not shown in figures) coincide since the filter can resolve the two beams independently. In contrast to a = 0 case, a gain in performance is observed (especially for a —> 1) with lesser correlation (|p| —> 0) between the two beams. This occurs due to the diversity effect achieved when the fading in the two beams is uncorrelated. Also the gain is less for the fast fading case due to the inherent diversity present within a single pulse.  Figure 4.10 Three-beam frequency selective fast fading, /  = 0.64 and a > 0  Chapter 4 Numerical Results  55  E /N [dB] b  0  F i g u r e 4.11 T h r e e - b e a m frequency selective v e r y - s l o w f a d i n g , a > 0 A n o t h e r m o d e l o f the t h r e e - b e a m c h a n n e l i s c o n s i d e r e d i n F i g u r e s 4.12 t o 4.16 f o r f r e q u e n c y s e l e c t i v e f a d i n g . It i s a s s u m e d that  the non-faded beam and one of the faded beams  coincide. T h e curves are plotted as a f u n c t i o n o f the n o r m a l i z e d d e l a y (0 < a < 1) a n d c o r r e l a t i o n c o e f f i c i e n t (-1 < p < 1) between the t w o faded beams.  F i g u r e 4.12 d i s p l a y s the B E R curves f o r a fast f a d i n g R i c i a n c h a n n e l w i t h e q u a l c h a n n e l g a i n f o r the faded and the non-faded beams. In general, g a i n i n B E R p e r f o r m a n c e i s o b s e r v e d w i t h an i n c r e a s e i n  o . T h e case [0.0;-1.0] is e q u i v a l e n t to t r a n s m i t t i n g the s i g n a l t h r o u g h a c h a n n e l  c o m p r i s i n g o f o n l y the direct path o f h a l f the total strength o f the R i c i a n (K  dB  = 0 d B ) channel.  Chapter 4 Numerical Results  Figure 4.12 Three-beam fast fading, K  dB  Figure 4.13 Three-beam slow fading, K  dB  = 0 dB and/  = 0 dB  = 0.64  Chapter 4 Numerical Results  57  Figure 4.13 depicts BER for slow fading Rician channel. The variations of P with [a;p] b  are similar to the fast fading case except for a few cases of which [1.0;0.0] is particularly noteworthy. Since in a slow fading channel, the signal amplitude does not vary within the symbol duration, interference between the direct beam and the coinciding diffused beam results in a greater loss in the received signal energy when they tend to be out of phase. On comparing [1.0; 1.0] with [1.0;0.0], in contrast to the fast fading case, wefindthe former has a lower BER and the gain > 5 dB at P = 10~ . 4  b  To facilitate comparison and in particular to observe the trends in system performance as a function of the Rician factor in fast and slow fading conditions, curves are plotted for four combinations of [o;p].  W  d B  i  Figure 4.14 Three-beam fast fading as a function of K; [a;p] : [0.0;0.0], [1.0; 1.0] In Figure 4.14, the gain is seen to be more for [1.0; 1.0] than for [0.0;0.0] for small K  Chapter 4 Numerical Results  58  values because o f the l o w e r r e c e i v e d s i g n a l strength i n  [0.0;0.0] w h e n  the f a d i n g i n the t w o c o i n c i -  dent d i f f u s e d b e a m s is independent.  10 r  1  1  1  1  1  W  Figure  4.15  d B  1  r  i  T h r e e - b e a m fast f a d i n g as a f u n c t i o n o f  K; [o;p] : [0.0; 1.0], [1.0;0.0]  E /N (BER) b  Figure  4.16  0  T h r e e - b e a m s l o w f a d i n g as a f u n c t i o n o f  K; [a;p] : [0.0;0.0], [1.0; 1.0]  Chapter 4 Numerical Results  59  In Figure 4.15, the case [0.0; 1.0] seems to follow an opposite trend i.e. P increases as the b  Rician factor is increased from K  dB  Rician factors, i.e. K  dB  = - o o dB to K  dB  = 0 dB . This occurs because at low  < 0 dB, when the strength of the direct beam is lower than that of the  diffused beams, the faded component influences the received signal amplitude more than the nonfaded component; a decrease in the power of the diffused beam, due to an increase in K, tends to adversely affect the system performance. However for K  dB  = 3 dB, the non-faded signal  dominates and a gain is observed with further increase in K.  10°  E /N [dB] b  0  Figure 4.17 Three-beam slow fading as a function of K; [rj;p] : [0.0; 1.0], [1.0;0.0] The variations of P with [a;p] for the slow fading channel as observed in Figures 4.16 and 4.17 b  are found to be similar to the fast fading case.  60  Chapter 4 Numerical Results  4.4 Imperfect channel estimation We have analyzed the system behavior when the receiver consists of a correlator demodulator which is perfectly matched to the fading channel. The case when the receive filter makes some error in estimating the channel variations is considered in the present section.  It is assumed that in estimating the channel fading samples, the receiver is liable to make random errors which are Gaussian distributed with zero mean and variance, o , proportional to 2  the average power, a , of the faded beam. We study the performance for flat fading Rician 2  (K  dB  = 0 dB) and Rayleigh channels, each with a unit power gain.  10  J  2  1_  4  I 6  I \ 10  I  8  - I  12  1  14  ^  I  16  1 18  E /N [dB] b  0  Figure 4.18 Performance degradation in a flat fading Rician channel By modifying a , the BER curves, obtained from simulations with 99% confidence 2  intervals within ± 7 % of the mean value, are plotted for varying degree of error in estimation. A  Chapter 4 Numerical Results  61  rect pulse shape is used in the simulations. The mis-matched signal in the receiver is simulated by adding a complex Gaussian random process, with i.i.d. real and imaginary parts each with power spectral density a / 2 , to the fading waveform. In order to study the sensitivity of the (non-ideal) 2  filter under different fading conditions, results for both fast and very-slow fading channels are shown. The degradation in system performance for the Rician and Rayleigh channels as a function of a is shown in Figures 4.18 and 4.19 respectively. Since a depends on the power of the faded 2  2  beam, the degradation in performance tends to be more for the Rayleigh channel. ..o 10  t  i  i  i  i  i  1  1  i  3  E /N [dB] b  0  Figure 4.19 Performance degradation in aflatfading Rayleigh channel The following plots (Figures 4.20 to 4.23) illustrate the influence of error in estimating the amplitude and phase, individually, on the BER.  Chapter 4 Numerical Results  Chapter 4 Numerical Results  63  F i g u r e s 4 . 2 0 a n d 4.21 c o r r e s p o n d to the B E R f o r the c a s e w h e n the f i l t e r is u n a b l e to c o r r e c t l y trace the phase o f the f a d i n g w a v e f o r m . A l t h o u g h , f o r a l l the values o f a  2  c h o s e n , the  B E R f o r the fast f a d i n g case is f o u n d to be l o w e r than that f o r the s l o w f a d i n g case, the d B spread c o r r e s p o n d i n g to / i n g to  0<f  N  = 1.28 is f o u n d to be c o n s i d e r a b l y greater than the d B spread c o r r e s p o n d -  < 0 . 0 1 . A s i m i l a r o b s e r v a t i o n c a n be m a d e i n F i g u r e s 4 . 2 2 a n d 4 . 2 3 w h i c h  c o r r e s p o n d to the effect o f error i n estimating the a m p l i t u d e o f the f a d i n g samples.  10  12  14  16  18  F i g u r e 4.22 E f f e c t o f m i s - m a t c h i n a m p l i t u d e o n the B E R i n a R i c i a n c h a n n e l C a u t i o n s h o u l d be e x e r c i s e d i n d e r i v i n g c o n c l u s i o n s f r o m a c o m p a r i s o n o f the f i g u r e s s h o w i n g the effects o f phase and a m p l i t u d e e s t i m a t i o n errors s i n c e the c u r v e s p r e s e n t e d c o r r e s p o n d to the s p e c i f i c m o d e l a s s u m e d i n the s i m u l a t i o n s . T h e B E R c u r v e s w h i c h are p l o t t e d  Chapter 4 Numerical Results  as a f u n c t i o n o f o  2  64  r e v e a l a greater d e g r a d a t i o n i n r e c e i v e r p e r f o r m a n c e c a u s e d b y i m p e r f e c t  phase e s t i m a t i o n than that due to i m p e r f e c t a m p l i t u d e e s t i m a t i o n . T h e plots d o not i m p l y that i n general the receivers are m o r e sensitive to phase errors than to a m p l i t u d e errors.  1  1  1  1  1  1  1  1  :  0</<0.01  *  ^\  ''. ^/ N  ,  =  N  f  1.W  \  -\\  \ \  .  \ <T  2  e  •'  a  '.  <r  2  > S  N  \  =  a /4  =  a /2  =  a  2  is \  MFB  S  S  2  2  \ \  2  Ji  iI  iI  I  I  2  >1  4  6  8  10  12  -H ''•  \  A V  J  1  14  v>  16  18  E/N[dB] b  0  F i g u r e 4.23 E f f e c t o f m i s - m a t c h i n a m p l i t u d e o n the B E R i n a R a y l e i g h c h a n n e l T h e c a l c u l a t i o n s c o r r e s p o n d i n g to the m e a n square error ( M S E ) i n a m p l i t u d e e s t i m a t i o n and m e a n error i n phase estimation are s h o w n i n A p p e n d i x D .  65  Chapter 5 Conclusions and Suggestions for Future Work  C h a p t e r  5  C o n c l u s i o n s  a n d  S u g g e s t i o n s  f o r  F u t u r e  W o r k  5.1 Conclusions Matched filter bounds for fast fading Rician channels have been derived to establish lower bounds on the B E R for uncoded B P S K modulation over flat and frequency-selective fading channels. The analysis holds for any value of the normalized Doppler rate, f . N  The B E R perfor-  mance, in general, is found to improve with an increase in the fading rate due to the inherent diversity present within a single pulse. A l l the analytical results were verified to be in close agreement with simulation results.  Unlike the Rayleigh fading case where the statistical distribution of the (received) random signal component, after diagonalizing the covariance matrix of the fading samples, is given by the central chi squared distribution, the Rician case is more complex. The new random signal component is a sum of three terms: a constant term, a scaled version of the signal component encountered in the Rayleigh case and a term which is dependent on the second term. The distribution was derived by separating the i.i.d. real and imaginary parts of the uncorrelated complex Gaussian random variables encountered in the second term and grouping all the real components and imaginary components separately; the random signal component can then be expressed as a sum of a constant term and a random variable with a non-central chi squared distribution.  A comparison between the B E R curves for the Rayleigh and the Rician channels, as a function of f  N  in flat fading suggests that under slow fading conditions, the performance gain is  attributed mainly to the presence of the direct/specular component in the Rician channel whereas when f  N  increases without limit, the gap between the two tends to reduce and both the curves  Chapters Conclusions and Suggestions for Future Work  66  f i n a l l y c o n v e r g e w i t h the M F B f o r the n o n - f a d i n g A W G N c h a n n e l . T h e latter e f f e c t s h o w s the d o m i n a n t r o l e o f the f a d i n g rate i n l o w e r i n g the B E R i n fast f a d i n g c o n d i t i o n s . I n g e n e r a l , the B E R p e r f o r m a n c e i m p r o v e s w i t h an i n c r e a s e i n K; l i t t l e g a i n i s o b s e r v e d w h e n the ratio o f the p o w e r s o f the d i r e c t a n d d i f f u s e d b e a m s i s less t h a n - 6 d B . T h e s y s t e m p e r f o r m a n c e w a s a l s o s t u d i e d f o r different t r a n s m i t t e d p u l s e shapes: i t was f o u n d that the p u l s e shape affects the B E R s i g n i f i c a n t l y w h e n the c h a n n e l e x h i b i t s fast f a d i n g . T h i s is c o n s i s t e n t w i t h e a r l i e r results f o r the R a y l e i g h case [8], [14]. R e s u l t s f o r the f r e q u e n c y selective c h a n n e l , d e r i v e d f o r the g e n e r a l case o f c o r r e l a t e d scattering, were also e x a m i n e d . In t w o - b e a m frequency selective f a d i n g , p a r t i c u l a r l y f o r s l o w f a d i n g , the B E R is very sensitive to the degree o f b e a m o v e r l a p .  T h e B E R p e r f o r m a n c e o f a receive filter w h o s e i m p u l s e response is not p e r f e c t l y m a t c h e d to the c h a n n e l f a d i n g w a v e f o r m w a s s t u d i e d t h r o u g h s i m u l a t i o n . It w a s o b s e r v e d that, f o r the s p e c i f i e d m o d e l , the s y s t e m d e g r a d a t i o n w a s h i g h e r f o r p h a s e m i s m a t c h e s t h a n f o r a m p l i t u d e m i s m a t c h e s f o r a g i v e n error variance.  5.2 Suggestions for Future Work 1.  In this thesis, the pdf,  f (y), Y  o f the signal c o m p o n e n t ,  Y, is o b t a i n e d f r o m its characteristic  f u n c t i o n , ^ ( D ) , through n u m e r i c a l integration. To the best o f the author's k n o w l e d g e , n o c o m p u t a t i o n a l l y e f f e c t i v e c l o s e d f o r m e x p r e s s i o n o f the p d f o f  unequally weighted non-  central chi squared random variables is available. To a v o i d n u m e r i c a l a p p r o x i m a t i o n errors and c o m p u t a t i o n a l c o m p l e x i t y , it w o u l d be b e n e f i c i a l to obtain a c l o s e d f o r m e x p r e s s i o n f o r  /y(y).  2.  In the present w o r k , f r e q u e n c y s e l e c t i v i t y i n the c h a n n e l i s o b t a i n e d b y a s s u m i n g a d o u b l e  Chapter 5 Conclusions and Suggestions for Future Work  spike model for the delay power spectrum of the fading channel with arbitrary correlation between the fading waveforms. The analysis is limited to a single branch system i.e. without explicit diversity. Deriving the MFB's for a multibranch system would be a useful extension to this work.  3.  MFB's for bothflatand frequency selective fading in a rapidly time-varying Rician channel have been derived assuming BPSK modulation. The same for any PAM(ASK) scheme can also be obtained with some minor modifications; the new (variable) threshold boundaries of the decision maker would be a function of the received signal component. Deriving the error bounds other modulation schemes can prove to be a little challenging especially when the transmitted pulse, p(t), is a complex entity.  4.  The BER corresponding to imperfect channel estimation were obtained by simulations and presented in Section 4.4. The theoretical analysis for the same is complicated since the additive Gaussian error in the matched signal, i.e. the impulse response of the receive filter, does not correspond exactly to the assumption that the noise component in the decision variable is Gaussian.  67  Appendix A : MFB (slow fading )  Appendix A  68  : M F B (slow fading o < f  N  < o.oi)  The impulse response of a slow fading frequency-selective Rician channel assuming reception of two faded (Rayleigh) components in addition to the direct one, i.e. three-beam channel, when one of the faded beams and the non-faded beam coincide, is given as z(t,%) = a 8(x)+Z 8(x) + Z 8 ( T - a ) 0  1  (A.l)  2  = ( a + Z )8(T) + Z 8(i:-o) 1  0  where Z  x  and Z are complex Gaussian random variables with i.i.d. real and imaginary parts 2  Z = U + jV , n  2  n  n  n = 1, 2 and a , is the power of the direct/specular path. Using the following 2  expansion, i.e. |Z,+Z | = \Z \ + \Z \ + 2Re[Z ]Re[Z ] 2  2  2  x  2  2  x  2  + 2Im[Z ]Im[Z ] x  (A.2)  2  the corresponding signal component Y can be written as Y = a + \Z \ + |Z | + 2a U 2  2  X  + 2a U k  2  0  2  l  0  + 2U U k + 2V V k  2  X  2  X  2  (A.3)  where k = J p{t)p(t - <3)dt is a constant symbolizing the amount of pulse overlap (between the a two diffused beams) during signal reception.  For the special case when the fading in the delayed and non-delayed beams is uncorrelated and there is no overlapping i.e. <5>T ,k p  = 0 and (A.3) simplifies to  Y = (C/j + cc ) + U + V + V 2  0  2  2  2  (AA)  Y is thus stated as a sum of non-central chi-squared degree 4 random variable. The cdf of Y is  69  Appendix A : MFB (slow fading)  g i v e n as [27] oo  F (y)  IC„P ( 4  =  Y  r  n  E[\ZA ]E[\Z ] 2  where  B =  X  2 + 2  „ ^ )  (A.5)  =0  2  2  •—• 2  •  — is a suitably chosen constant [7]. F r o m [28], w e have 2  E[\Z \ } + E[\Z \ } X  2  M  +  1  -  '  Nik  =0 S u b s t i t u t i n g ( A . 6 ) into ( A . 5 ) a n d c o m p u t i n g the p d f o f Y b y differentiating F (y), Y  fy(y)  = ^-F (y) Y  = £ C  n  n= 0  y  = ^y C" — e P 2p n =  2(3  n+  2  1 r  J_ Jt =  ^  w e obtain  - X  *= 1( f c - i ) ! 2 p U p ;  0  1  (A.7)  0  T h e B E R i s thus g i v e n b y (3.3.13) n+l  = \  X  jerfc /  17 = -  2  ,  ertc  (F^y  s  n  C  2  n=0  F' (y)dy Y  'UN  QJ  Integrating ( A . 8 ) b y parts a n d s i m p l i f y i n g  p  k-  "2(3  k\\2$)  k= 0  1 U P  dy (A.8)  70  Appendix A : MFB (slow fading )  erfc  Pu =  Y  Y  o o  V  1 E \ 2KN, 1 2 A  (A.9)  erfc  \F {y)d  F (y)  \F (y)ex (--f)y-^dy Y  P  3 N  R  Knowing that [23] (A.10)  Jexp(-4y )y- 6?y = 1/2  and  Je-«y/  dy = Jn2~ q  2  n  n  2  (2n-l)!!,  (A.ll)  n>0  equation (A.9) further simplifies to oo  (A.12)  1n= 0 where  (4.  Jfc = 1  = 1 +2(3(£^/iV ) is a constant. Assuming £ [ | Z , | ] = E [J^ | ], the coefficients C 2  0  ,  J  2  2  n  are given as [27] C  0  = exp(-2ag/(l-ag)) n-l  C„=l/n;  «>1 7= 0  ,  f2a /(l-a ),  n=l  Lo,  «> I  2  2  (A. 13)  71  Appendix B : MFB (fast fading)  Appendix B  : M F B (fast fading f  -> ~)  N  We are interested in finding the limiting case of the variance of the signal component Y,  For the flat-fading Rician channel, Y is given by Y =  Jjcx + <xz(0|- \p(t)\ dt 2  0  (B.l)  2  1  and its mean E[Y] = \\p(t)\ E[\a  +  2  0  = j\p(t)\ (a 2  a (t)\ ]dt 2  lZ  + a )dt  2  (B.2)  2  = a^ + a  2  where we have used £[|z(0l ] = 1. Accordingly we have 2  G  2  =  E[Y ]-(E[Y]) 2  2  (B.3)  = E[Y ] - (ag + a ) . 2  2  2  Finding the second moment of Y E[Y ] 2  = JJ|p(r )| |p(r )| E[|a + a z ( r ) | | a + a z(r )| ] dt dt 2  1  2  2  2  J  1  0  (B.4)  2  1  1  0  2  x  2  Expanding the expectation term in the integral gives < | x + a z(t )\  2  0  x  x  | a + a z(t )\  2  0  x  2  = a$ + a a (\z(t )\ 2  2  + \z(t )\ ) + a\  2  \z(t )\ \z(t )\  2  x  2  2  + 2aZ a,Re[z(t,)]  + 2a3  2  x  2  a,Re[z(t )] 2  (B.5) + 2a a\Re[z(t )]\z(t )\  2  x  0  +  2  + 2a  4a^a Re[z(t )]Re[z(t )] 2  and taking the expectation of both sides in (B.5) yields  x  2  a Re[z(t )]\z(t )\ 3  0  2  x  2  x  Appendix B : MFB (fast fading )  72  ^[lao + a^Cf^Plao + ajzC^)! ] 2  = a 4 + 2aga +  (B.6)  £[|z(fj)| | (r )| ]  2  2  2  z  2  S u b s t i t u t i n g (B.6) into (B.4) and u s i n g [14]  £[|z('i)| |z(' )| ] = 2  2  1 +  2  P  (B-7)  2  w h e r e p is the n o r m a l i z e d cross-correlation between the t w o samples o f the real o r i m a g i n a r y part o f the process z(t)  [16]. T h u s , w e have  E[Y ] = a4 + 2 a 2  2  a  2+  «i JJ|P('I)| \P(h)\ ( + P ) 2  2  l  2  d  t  \  d t  i-  (B.8)  S i n c e p c a n b e v i e w e d as the correlation o f the f a d i n g process, f r o m [15]  p ( x ) = J (2nf x) 0  (B.9)  D  w h e r e J (x) is the z e r o t h order B e s s e l f u n c t i o n o f the first k i n d a n d f 0  D  f r e q u e n c y i n H e r t z . A p p l y i n g the transformation  t -1  2  x  is the m a x i m u m D o p p l e r  = i and u s i n g (B.3), (B.8) a n d (B.9), the  variance o f Y c a n b e written as  a  2  (B.10)  = a f JJ|p(r )| |p(f,-x)| 7g(27c/ x) <hdt . 2  2  1  D  x  F o l l o w i n g the a n a l y s i s i n [14], w e s u b s t i t u t e the a u t o c o r r e l a t i o n o f the s q u a r e d p u l s e m a g n i t u d e w h i c h is g i v e n b y RW P  in  = j"l/>(0l lM'-x)| ^ 2  2  (B.ll)  (B.10) r e s u l t i n g i n  a  2  = af  J R (x) • J$(2nf T) dx .  T h e l i m i t i n g v a r i a n c e o f Y as f  p  N  D  (B.12)  —> <» c a n b e o b t a i n e d b y t a k i n g the l i m i t as T ^ o o . p  Appendix B : MFB (fast fading )  73  Writing p(t) in terms of u(t), a unit energy and unit duration pulse, i.e. p(t) = -j=u(±).  (B.13)  Substituting (B.13) in (B.l 1) and evaluating at T = 0 T  V°> "  J  3  "(ft*  0  p  1  7  1  1  =  ^ - J l " ( ? ) l p  4  (B.14)  ^  o  where Q = J |w(g)| d<7 is a constant. Since an autocorrelation function has it's maximum value 4  o  at zero delay, it is convenient to write /yx)<^-  (B.15)  p  which implies 2a\\Q-  J&2nf %) dx D  2<_—Q  a  (  B  1  6  )  Taking the limit as T —> °° and applying L'Hopital's rule p  lim a  2  = lim 2af Q J (2nf T )dx 2  D  n  =0  (B.17)  Thus the variance of the signal component Y approaches zero for any unit energy pulse as the normalized Doppler rate increases without limit.  74  Appendix C : Solving the Integral  Appendix C  : Solving the Integral  To evaluate the following integral encountered in Section 3.5.2:  ^  =  i  B lm  XE  .rr,  a  2  (C.l)  0  i= i = i ( m - 1)!^ m  lm X X 1 i l 1 l(OT-l)!X B  (v-ajr-'expf-^^^  2 2 1  /  =  m  =  /  2  «  0  Substituting the integral expression in the above equation for /  - J udv  (C.l)  2  CC  0  where  u = er/  = —  f  exp(-t )dt 2  ' (y-a ^  (C.3)  2  dv = (y-aj) - exp m  1  (C.4)  which gives y\.  E b  <iu = ——exp -  v =- I  (m- 1)!^  n=0  (m-n  Integrating (C.2) by parts yields  i/ dy  2\m-n- 1  - 1)! •(y-cc0 )' z  2  exp  (C5)  (C.6)  75  Appendix C : Solving the Integral  I =  -erf  •(y-a ^r-"- exp i  1)!  (m-n-  0  2-.ni -rt-  n+l  — y i n — a  (m-n-1)!  «=o  (y-«o)  exp  (C.7) 1  dy  •exp  Ty  From [23] we obtain the following expression for the integral term in (C.7)  H) -e  dy  = C  a ^ ^rC^exp—^Wj  z z  0  where \x, = m-n,  !  (Cjccrf)  2 ^ 2 ^ 2 ' 2  a\  C, = — +. 1 r- and T(z) is the Gamma function. W ^0 h  K  v  (C.8)  ( z ) is known as the  Whittaker function which is defined as [23] ^,vU)  =  ,  F (  2  V  )  A  rlj-v-x  v  ( z )  ,  +  T ( 2 V ) N  M,,_ (z) v  (C.9)  nj v-x +  where M (z) Xv  = z  v+  V  z / 2  0 > v-A, + |,2v + l ; z (CIO)  M^_ (z) = z~ v  V+ 2  V  z / 2  j. 2'  0 ( - v - ? i + ^-2v + l ; z  <E>(a, y; z) is the confluent hypergeometric function [23] _, . , 0(a,y;z) = l  Equation (C.7) thus simplifies to  az a(a + l)z , — +^ T ) % 2  +  +  T  (C.ll)  76  Appendix C : Solving the Integral  4](m- 1)!X,'  I = erf\  (C.12) n+  1  exp Z-2  n =0  a  °  a  ^r(ji)W,  0  (C a ) 2  •n -  ;  0  5  which results in the following expression for P , b  L,,  r,  73Zm  2 ,t,„f,(».-i)arL 2  m-l  r  -  o  H)  n+l  (m- 1)!^  (u-1)!  (Cl?) a  exp  2; 0  r(ii)W,  j  (C^)  2 _ ^ 2 ^ 2 ' 2  Equation (C.l3) can be written in a more simplified form when r = 1, V/ t  fa A  2\  - 2-2Z, n erfl 5  i= l  2  II exp Nn V^J 0  where erf(x) and erfc(x) are the error and complementary error functions respectively.  (C.H)  77  Appendix D : Calculating the Errors in Amplitude and Phase estimation  Appendix D  : Calculating the Errors in Amplitude and Phase estimation  actual sample  v  sample with error random error  y'  y 0  X X  0  x'  Figure D.l Resolution of the fading sample with/without error Figure D.l illustrates the resolution of the fading sample, s = x + jy, the error variable, e = x + jy' and the fading sample with error, s = (x + x') + j(y + y') in the XY-plane. x and e  y' are both i.i.d. Gaussian random variables with variance, a * = o / = CJ /2 where a is the 2  2  2  2  variance of e.  The amplitude estimation error is given by  e = \s \-\s\ r  where |s | = J(x + x') + (y + y ' ) 1  e  2n  a  sa  (D.l)  e  Rician distribution and l^l = J(x)  2  + ( y ) is a 2  constant. The mean square error (MSE) is therefore a function of the amplitude of the fading sample and is given by  Appendix D : Calculating the Errors in Amplitude and Phase estimation  78  E[e ] = £[|JJ2 + |J| -2|JJW] 2  2  = E[\s \ ] 2  e  \s\ -2\ \E[\s \] 2  +  S  e  ( D 2 )  = a + 2 • \s\ - [\s\ Jk • fi e x p ( - ^ ) 0>(J, 1 2  2  2  where 0(a, b;z) is the confluent hypergeometric function [23]. 2s[|.s | ] and £ [ | . s | ] are the first 2  c  e  moments of the non-central chi squared and Rician distributed random variables respectively as shown in [12]. The normalized amplitude estimation error, E , is written as N  E[e ] 2  Since the phase of the fading sample is uniform from 0 to 2n, the phase estimation error, 0 , is independent of 0 and can be obtained by assuming 0 = 0 in Figure D.l. It thus follows  Letting R = \e\ and knowing from [19] the joint pdf, f >(r, (p), of R and 0' which is RQ  given by  f v(r,<p) = Z ^ p f - 4 1 .  (D.5)  e x  R  the mean phase estimation error, E[Q ], is obtained by integrating (D.4) over f '(r, e  2TX  E[QE] = — e  KG  2 2J  RQ  tp), i.e.  OO  fJ fratanf, ," S i n ( p  vM+rcoscpJ  l e xvp fa-)^ W ? r  2  .  (D.6)  Bibliography  79  Bibliography [I]  T. S. Rappaport, Wireless Communications Principles & Practice. N.J.: Prentice Hall PTR, 1996.  [2] L. Litwin, "An introduction to multicarrier modulation," IEEE Potentials, pp. 36-38, Apr/ May 2000. [3] R. J. C. Bultitude and G. K. Bedal, "Propagation characteristics on microcellular urban mobile radio channels at 910 MHz," IEEE Journal on Selected Areas in Comm., vol. 7, no. 1, pp. 31-39, Jan. 1989. [4] W. C. Lindsey, "Error probabilities for Rician fading multichannel reception of binary and N-ary signals," IEEE Trans. Inform. Theory, pp. 339-350, Oct. 1964. [5] J. E. Mazo, "Exact matched filter bound for two-beam Rayleigh fading," IEEE Trans. Comm., vol. 39, no. 7, pp. 1027-1030, Jul. 1991. [6] M.V. Clark, L. J. Greenstein, W. K. Kennedy, and M. Shafi, "Matched filter performance bounds for diversity combining receivers in digital mobile radio," IEEE Trans. Veh. Tech., vol. 41, no. 4, pp. 356-362, Nov. 1992. [7] K. W. Yip and T. S. Ng, "Matchedfilterbound for multipath Rician-fading channels," IEEE Trans. Comm., vol. 46, no. 4, pp. 441-445, Apr. 1998.  [8] W. Burchill and C. Leung, "Matchedfilterbound for OFDM on Rayleigh fading channels," Electronics Letters, vol. 31, no. 20, pp. 1716-1717, Sep. 1995. [9] N. J. Baas and D. P. Taylor, "Matchedfilterbounds for wireless communication over Rayleigh fading dispersive channels," IEEE Trans. Comm., vol. 49, no. 9, pp. 1525-1528, Sept. 2001. [10] R. Boudreau, J. Y. Chouinard and A. Yongacoglu, "Exploiting Doppler-diversity inflat,fast fading channels," 2000 Canadian Conference on Electrical and Computer Engineering, vol.1, pp. 270-274, 2000. [II] B. Sklar, "Rayleigh fading channels in mobile digital communication systems part I: characterization," IEEE Comm. Mag., pp. 90-100, Jul. 1997. [12] J. G. Proakis, Digital Communications. N.Y.: McGraw-Hill, 3rd Edition, 1995. [13] D. C. Cox and R. P. Leek, "Distributions of multipath delay spread and average excess delay for 910 MHz urban mobile radio paths," IEEE Trans. Antennas Propagat., vol. AP-23, pp. 206-213, Mar. 1975.  80  Bibliography  [14]  W . S. B u r c h i l l ,  Performance and Receiver Structures for OFDM on Rayleigh Fading Chan-  nels. P h . D . thesis, D e p t . o f E l e c . & C o m p . E n g . , U . B . C . , 1997.  [15] W. C . Jakes, Microwave Mobile Communications. N.Y.: W i l e y , 1974. [16]  A . Papoulis,  Probability, Random Variables, and Stochastic Processes. N.Y.: M c G r a w - H i l l ,  1984. [17]  R. H . C l a r k e , " A statistical theory o f m o b i l e - r a d i o reception,"  Bell Syst. Tech. J., v o l . 4 7 , pp.  9 5 7 - 1 0 0 0 , 1968. [18] [19]  S. H a y k i n ,  Communication Systems. N.Y.: W i l e y , 3 r d e d i t i o n , 1999.  W . B . D a v e n p o r t , Jr. and W . L. R o o t ,  An Introduction to the Theory of Random Signals and  Noise. N.Y.: M c G r a w - H i l l , 1958.  Advanced Engineering Mathematics. N.Y.: W i l e y , 8th e d i t i o n , 1999.  [20]  E. K r e y s z i g ,  [21]  S. S. H a y k i n ,  [22]  T h e M a t h W o r k s , Inc., Matlab®, ver. 6.1.0.450, release 12.1, M a y 2 0 0 1 .  [23]  Introduction to Adaptive Filters. N . Y : M a c M i l l a n , 1984.  I. S. G r a d s h t e y n , I. M . R y z h i k ,  Table of Integrals, Series and Products. L o n d o n : A c a d e m i c  Press, 6th e d i t i o n , 2 0 0 0 . [24]  D . J . Y o u n g a n d N . C . 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