UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
ubc_2002-0066.pdf [ 3.71MB ]
Metadata
JSON: 1.0065460.json
JSON-LD: 1.0065460+ld.json
RDF/XML (Pretty): 1.0065460.xml
RDF/JSON: 1.0065460+rdf.json
Turtle: 1.0065460+rdf-turtle.txt
N-Triples: 1.0065460+rdf-ntriples.txt
Original Record: 1.0065460 +original-record.json
Full Text
1.0065460.txt
Citation
1.0065460.ris

Full Text

M A T C H E D F I L T E R BOUNDS F O R FAST FADING 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 this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of E L E C T R I C A L AND C O M P U T E R E N 6 / N E E . R i N < q The University of British Columbia Vancouver, Canada Date OS M A R C H 200Z . DE -6 (2/88) 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 varying filter. The BER 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. 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 non- ideal receive filter is obtained. It was observed that, for the specified model, the system degrada- tion 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 fN^>°°) 7 1 Appendix C: Solving the Integral 74 Appendix D: Calculating the Errors in Amplitude and Phase estimation 77 Bibliography 79 iv L is 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 12 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 fN for the Rician channel, KdB = 0 dB 46 Figure 4.2 BER as a function of the Rician factor and fN 47 Figure 4.3 Effect of the Rician factor for fN = 1.28 47 Figure 4.4 Effect of pulse shape in fast fading Rician channel, KdB = 0 dB 48 Figure 4.5 Channel averaging ability of different pulse shapes 49 Figure 4.6 Effect of the roll-off factor on the system performance, KdB = 0 dB 50 Figure 4.7 Two-beam frequency selective fading, KdB = 0 dB 51 Figure 4.8 Three-beam frequency selective fast fading, fN = 0.64 and a = 0 52 Figure 4.9 Three-beam frequency selective very-slow fading, a = 0 53 Figure 4.10 Three-beam frequency selective fast fading, fN = 0.64 and a > 0 54 Figure 4.11 Three-beam frequency selective very-slow fading, a > 0 55 Figure 4.12 Three-beam fast fading, KdB = 0 dB and fN = 0.64 56 Figure 4.13 Three-beam slow fading, KdB = 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 Figure 4.15 Three-beam fast fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0] 58 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 Glossary Acronyms ASK Amplitude Shift Keying AWGN 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 Drms rms delay spread Eb bit energy fc carrier frequency Afc Coherence bandwidth fd maximum Doppler frequency fN maximum normalized Doppler rate f(t) received pulse shape corresponding to the diffused channel fY(y) pdf of Y h(t) impulse response of the receive filter J0(.) zeroth order Bessel function of the first kind K Rician factor of the channel KdB Rician factor (in dB) N total number of paths in the discrete channel model N noise component at the matched filter output N0 power spectral density of AWGN n(t) AWGN noise Pb probability of bit error p(t) transmitted pulse shape Q(v, w) Marcum's (2-ftmction r(t) received (faded and noisy) signal rcd{t) received signal (for a CD channel) rdp(t) received signal (for a DP channel) Rf(tj, t2) autocorrelation of the process f(t) rfad^ faded signal at the channel output rs(t) received signal without noise (for the DP channel) S(f) power spectrum density of the Rayleigh fading process viii s(t) transmitted signal SfadW received pulse shape T received s ignal durat ion Ar sampl ing interval Af„ Coherence t ime c TM delay spread o f the channel Tp pulse duration v m a x i m u m veloc i ty of M S v(z) Whi t taker funct ion X matched filter output after sampl ing x(t) matched filter output Y s ignal component at the matched filter output z(t, x) impulse response of a l inear channel (general ized notation) zcd(t> x ) impulse response of a C D channel zray(t' x ) impulse response o f the di f fused channel zrice(t> x ) impulse response o f the R i c i an channel zn{t) complex fading process corresponding to the «th beam ix Greek Symbols a2 diffused beam strength a 0 attenuation corresponding to the direct/specular path «„ 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? z (At;(T,,T 2 )) autocorrelation function of z(t, x) a normalized delay parameter 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 x o delay corresponding to the direct/specular path X ma* maximum delay spread X • mm 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 lengthen- ing 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 a fixed symbol/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, fd, and the symbol parameter i.e. pulse duration, T , which are together incorporated in the normalized Doppler rate, fN = fd-T. 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 obtain- ing 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 continu- ous 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 achiev- able 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 perfor- mance 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 receive filter is 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 fad- ing 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 related field are presented in this chapter. Chapter 2 The Wireless Channel 6 Chapter 2 The Wireless Channel 2.1 Introduction T h e e f fect o f a f a d i ng w i r e l e s s channe l on a t ransmi t ted s i gna l is bo th c o m p l e x and interest ing. The ma in purpose o f this chapter is to prov ide a bas ic unders tand ing o f the fad ing phenomenon , its cause and effects; this is necessary background fo r the wo r k presented in this thesis. 2.2 Signal Propagation in a Wireless Channel In a wire less communica t ion system, the transmitted s ignal often reaches its destination v i a mul t ip le paths. The mechanisms that impact s ignal propagation can be c lass i f ied into three types, namely ref lect ion, d i f f ract ion and scattering [1], [11]: • Reflection is caused by the radio wave str ik ing and ref lect ing o f f a smooth surface w i th very large d imens ion compared to the signal wavelength, X. • Diffraction occurs due to the obstruct ion of the radio path between the transmitter and the receiver by a dense body wi th large d imens ion relative to X result ing in the format ion of sec- ondary waves beh ind the obstruct ing body. • Scattering occurs when the signal impinges on either a large rough surface or any surface w i th d imens ion o f the order o f X or less, causing the reflected energy to scatter i n a l l d irect ions. Fading is a cumulat ive effect o f the above mechanisms. 2.2.1 Land Mobile Fading The arrival o f the transmitted s ignal at the receiver over mul t ip le paths gives rise to the Chapter 2 The Wireless Channel 7 phenomenon of fading 1 , in which a signal undergoes random and time varying attenuation and 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 Sta- tion (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 as fixed scatterers 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 varying2 signals from other scatterers and 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 varia- tions in the density of ions in the ionospheric layers, it is referred as Scintillation. This is especially appli- cable 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 = \zcd{t,x)-s{t-x) dx (2.2.1) 0 where TM is the channel's delay spread and zcd(t, x) is the impulse response of the CD channel 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, xn{t), and phase, tp n (0, with respect to the transmitted signal, s(t). The signal, rd (t), at the channel output is rdPW = 5>„C) • e~m° • s(t-xn<t)) . (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 Chapter 2 The Wireless Channel received signal appears as a continuously time-varying signal or a train of pulses respectively. 9 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 transmit- ted 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 (p2> transmitted at time t2, may arrive in con- 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 Spread (TM). Viewed in the frequency domain, signal dispersion is expressed in terms of Coher- ence Bandwidth (Af c ) which implies that the signal components having frequency separation less than Af c are treated alike by the channel. It can be shown [12] that Afc = . Based on M 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. TM = 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 Atc, which is defined as the duration over which the channel's impulse response can be considered as time- invariant. Coherence time is inversely proportional to the Doppler Spread, Atc «*= 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 / „ = / V c o s ( e „ ) (2.2.3) where fn is the Doppler shift [15] for the n t h wave arriving at an angle 0 n with respect to the Chapter 2 The Wireless Channel 11 direction of motion of the MS and / . (fd = ^) 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 indepen- dent 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) • 6(T-T„(0) (2.3.1) n where fc is the carrier frequency of the bandpass signal; oc„(0 and xn(t) are the attenuation and Chapter 2 The Wireless Channel 12 delay of the n t h ray at time t, respectively. 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, zrice{t, x), by taking a constant attenuation, a 0 , and associated delay, x 0 , for the first arriving ray zrice(t, x) = 0 C Q • e - j 2 n f ^ • 8(t - 1 0 ) + Xa„(0 • e-]lKfMt) • 8(t - xn(t)) (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 rs{t), is g iven as •',(*) = 5>«C) • e~J2KfMt) • s(t-xn(t)) . (2.3.4) n The autocorrelat ion funct ion o f z(t, x) is def ined as [12] SR z (Af JCCJ, x 2)) = | • E[z*(t, Tj) • z(f + A/ , x 2)] . (2.3.5) A s s u m i n g uncorre lated scatter ing, i.e. independence o f attenuation and phase shifts assoc iated w i th different path delays, and lett ing At - 0, (2.3.5) can be re-written as SRz(0;(T!, x 2)) = 9t(x,) • 8(T! -x 2) . (2.3.6) The func t i on SR(x), k nown as the delay power spect rum o f the channe l , represents the average power output o f the channel as a funct ion o f the t ime delay x. In urban environments, 9t(x) can be approx imated by the cont inuous one-sided exponential funct ion [13], i.e. SR(x) = ̂ - • e x p f - - M x > 0 (2.3.7) rms ^ rms' where Drms is the rms delay spread. In some previous works [6], the class o f two-s ided cont inu- ous spectra is considered by assuming the Gauss ian spectrum. However, for s impl ic i ty , we conf ine our ana lys i s to one and two beam cases (Doub l e -Sp i ke spec t rum (2.3.8)) w h i c h can eas i ly be extended to a system w i th N , ( N > 2), beams by app ly ing a s imi la r procedure. = ^(8(x) + 8(x -a) ) (2.3.8) A snapshot o f the impu lse response w i th cont inuous one-s ided exponent ia l de lay power spectrum is shown in F igure 2.3. The f igure is plotted assuming Drms = 0.1 sec and a Dopp l e r Chapter 2 The Wireless Channel 14 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 x2, respectively [14]. Thus, on splitting the second term in (2.3.2) into two parts (summation series) each reflect- ing the case of a single Rayleigh faded beam and denoting their respective indices by n{ and n2, Chapter 2 The Wireless Channel 15 with each ray in the group having an associated delay of x„ (0 and ̂ (0, respectively, where xBi(0 = x , + A x B i ( 0 , A T B i ( o « r p (2.3.9) Tp is the transmitted signal block/pulse duration, we obtain zrice{t,i) = a0-e~J2nfcXo-Hi-*0)+zl(t,T) + z2(t,%) (2.3.10) where zx{t,x) = eJ  / c ' X a n , ( 0 ^ 1 • 8 ( T - ( f 1 + A T n | ( 0 ) ) -/27C/-T, -7'2TC/CAT (0 z 2(r, t) = e ; / c 2 X « „ 2 W g • 8 ( T - ( f 2 + AT l l2(0)). (2.3.11) *2~ n2 An upper-bound on Atn(r) can be obtained by realizing [14] that the greatest change in 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 • Tp for a mobile moving at velocity v. Therefore the change in delay (Axn(t)) associated with the change in path-length would be v • T /c, where c is the speed of light. Since (Axn(t))max is much smaller than the pulse duration, we can drop the Axn(t) factor from the delta functions in (2.3.11). However, the same cannot be neglected in the exponential terms due to the large value of fc. 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 X j = 0 and x 2 = a, (2.3.10) simplifies to t r i c e d * ) = OL0-e~i2nfeXo-Hx-x0) + Zl(t)-d(x) + e~i2lifeaz2(t)-^-o) (2.3.12) where zx(t) and z2(t) are complex Gaussian random processes accounting for the attenuation 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 = 0. Normalizing the two random processes such that i^z^r)! 2] = £[|z2(0|2] = 1 and explicitly specifying the strengths of the two Rayleigh faded beams as 0Cj and oc2 respectively, we can re-write (2.3.12) as zrice{t,x) = a0-5(x) + a1z1(0-5(x) + a 2z 2(0-6(x-a) (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 zrice(t,x) = a0-8(x) + a1z1(r)-5(x) (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. Chapter 2 The Wireless Channel 17 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 ) 2 1 (2.3.15) KdB = 10-log10(£) = 10 log 10 where ^fade(^) is the average delay power spectrum of the fading component of the channel, (a 0) 2 is the power of the specular path and the power of the diffused component is obtained by integrating the average delay power spectrum i.e. J'iKfacje(t)dx. 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 Chapter 2 The Wireless Channel 18 a vert ica l ly po lar i zed antenna. The statistical characteristics of the random process zt(t) are based on i so t rop i c scatter ing o f the components/rays r andom ly a r r i v i ng at the rece iver at u n i f o r m l y distr ibuted angles and exper ienc ing s imi lar attenuation over smal l-scale distances. F i g u r e 2.5 dep ic ts the spectra l b roaden ing o f an unmodu l a t ed con t inuous wave ( C W ) carr ier at f requency fc due to Dopp l e r shift caused by the Ray l e i g h channe l . Thus each of the rays arrives at the M S in its own frequency, offset f r om fc w i th in ±fd. The corresponding power spect rum density S(f) and autocorre lat ion funct ion §(t) o f the Ray l e i g h fad ing process z-(r) for the case o f a vert ical wh ip antenna on the M S are given by [15] where b is the average power rece ived by an isotrop ic antenna and (Od = 2nfd, fd be ing the m a x i m u m Dopp l e r frequency, and S(f) = 3 b (2.3.16) <K0 = 1.5-W 0 (ovO (2.3.17) where J0(x) is the zeroth order Besse l funct ion o f the first k ind . Chapter 2 The Wireless Channel F igure 2.5 Power Spectrum of an unmodulated C W carrier Chapter 3 Derivation of the MFB's 20 Chapter 3 Derivation of the MFB'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 time- varying 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 matched filter whose 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 fN = fd- T' where fd 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, fd typically ranges from 50 to 100 Hz for a vehicle speed of 60 km/hr. Therefore a system using a transmission rate of 104 symbols/ 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 Tp. We will refer to a system for which Chapter 3 Derivation of the MFB's 21 fN < 0.01 as slowly fading. For an OFDM system employing hundreds of tones, T would be 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), A = JWb (3.2.1) where p(t) is a unit energy pulse of duration T , so that s(t) has energy Eb. 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 0 • 5(x), and the response of the diffused beam, zray(t, x), i.e. z(t, x) = a 0 • 5(x) + zray(t, x) (3.2.2) where a 0 is the power of the direct/specular component. The real and imaginary parts of zray(t, x) are independent and identically distributed (i.i.d.) zero mean Gaussian random processes. The channel imposes a maximum delay spread of xmax on the transmitted signal s(t). Chapter 3 Derivation of the MFB's 22 T x S igna l R i c i an Channe l F igure 3.1 System M o d e l The received signal r{t) is given by A W G N n(t) Matched r(f) , F i l t e r , x{t)~^ X r(t) = J z(t, X) • s(t - X) dx+ n(t) , (3.2.3) where n(t) is a sample funct ion of the zero mean complex addit ive whi te Gauss ian noise process w i th powe r spectral densi ty NQ. T he receiver 's dec i s ion , as to wh i c h s igna l was transmitted, is based on the real component, Xr, o f the output variable X = x( T), where x(t) = jh(t-x)-r(x)dx, T = Tp + x„ (3.2.4) 3.3 Error Performance Analysis The faded s ignal at the channel output is written as rfad(t) = ±A-sfad(t) (3.3.1) where Chapter 3 Derivation of the MFB's 23 sfadW = j z(t,X)-p{t-X)dX 0 T *max = a0-p(t)+ j zray(t,x)- p(t-x)dx (3.3.2) In (3.3.2) we have assumed that the direct path has zero delay whereas the impulse response of the diffused beam, zray(t, x), extends from xmin to xmax. 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, sfad(t), we can obtain the optimum BER perfor- mance, i.e. h{t) = sfad*(T-t) (3.3.3) 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 = \sfad*(.T-t + *) r(x)dx. (3.3.4) Tx Signal AWGN Channel r ( 0 x{t) t = T Receiver X Figure 3.2 The Transformed System Chapter 3 Derivation of the MFB's 24 Therefore, the decision variable X can be expressed as Xr = ±A • Y + N (3.3.5) where the signal component, Y, is given by y = \ \ S f a d i t ) \ 2 d t 0 - J a0-p(t)+ j zray(t,x) • p(t-x)dx \dt (3.3.6) and the noise component, N, is given by N = Re = Re jsfad*(t)-n(t)dt •o *7 J\a0-p*(t)+ j zray*(t,x)- P*(t-x)dx 0^ T • v "mm n(t)dt (3.3.7) In order to find the probability of error, we need to know the probability density function (pdf) of Xr. For convenience it is assumed that s(t) = A • p(t) is transmitted. Since the noise component, N, is Gaussian, the pdf of Xr conditioned on the signal component, Y, is also Gaussian with mean AY; it's variance can be obtained using Parseval's theorem [18] as a2 - 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 25 The pd f o f Xr condi t ioned on Y is given as fxr(*r\y) = ~ ^ ^ e x P find - ( * - ( A y ) ) 2 ' 2o N (3.3.9) The cond i t iona l probabi l i ty o f error is the probabi l i ty that Xr is less than zero, i.e. Pr(Xr<0\Y) = \fx{xr\y)dxr (3.3.10) The overa l l probabi l i ty o f error Pb is obtained by averaging (3.3.10) over the pd f o f Y Pb = lPr(Xr<0\Y)-fY(y)dy (3.3.11) U s i n g (3.3.9) and (3.3.10) and the fact that Y > 0 f r om (3.3.6), we can rewrite Pb as Pu = oyfj f V .find. exp <xr-(Ay)): 2a N M \ \ dx. fY(y) dy (3.3.12) Equat ion (3.3.12) can be expressed in terms o f the complementary error funct ion, erfc(x), as = \ • \erfc f Ay ^ vfidl) 2 o j \erfc fY(y)dy fY(y)dy (3.3.13) The term Eby/NQ can be interpreted as the received S N R where Eb/NQ is the transmit- ted S N R . The B E R can be calculated us ing (3.3.13) i f the pd f fY{y) is avai lable. The determina- t ion o f fY(y) can be a chal lenging task. The fo l l ow ing quote f r om [14] summar izes the prob lem, Chapter 3 Derivation of the MFB's 26 '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 = Yl + Y2 + Y3 (3.3.14) where Y, = a2- \p\t)dt = (3.3.15) mm mm Y2 = 2 a 0 J Re J zray(t, x) • p(t-t)dz Pit) dt (3.3.16) ^ - J "mm T J zray(t,x) • p(t-x)dx dt = j \f(t)\2dt (3.3.17) an d f(t) = J zray(t,x) • p(t-x)dx . Following [14], we approximate (3.3.17) as a finite summation, i.e. M r 3 = X | / J n = 1 (3.3.18) Chapter 3 Derivation of the MFB's 27 where M = T +T — T • p max min At and / n = f^n - • Atj; the floor function L* J is used to 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, | /„ | 2 , has a central chi- squared degree 2 distribution. However (3.3.18) is a sum of correlated random variables. In order to express Y3 as a sum of independent random variables, we use an orthogonal expansion (Karhunen -Lo'ev e Expansion, [16], [19]), i.e. M fn-jAt= ^Uk-JXk-<?nk k= 1 (3.3.19) where {cpn/t} = j tp^n - ^ • Afj j are orthonormal vectors (3.3.20) and {Uk} are orthogonal complex zero-mean Gaussian random variables with unit variance as shown in (3.3.21) M X Vnl • ^nm* = 5/m = n = 1 1, / = m 0, l± m (3.3.20) E<UrUm*) = 8 (3.3.21) {Xk} and {(p̂ (0} a r e recognized as the eigenvalues and eigen-functions [16] that satisfy the integral equation J Rf(tv t2) • yk(t2) dt2 = Xk • tp^) (3.3.22) where Rf(tp t2) is the autocorrelation function of the process f(t). The function Chapter 3 Derivation of the MFB's 2# Rf(t{, t2) = £[/(*, )-/*(>2)] r T max "max = E (3.3.23) can be expressed as a Hermitian matrix whose (/,m)th element is "max "max ^ 24) = J j Xz[(l-m)At-,(Ta,xb)]p(l'At-Ta)p*(m'At-xb)dTadTb 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 aj ' i - '2;(v xt)] = W'i« X J • W < ' 2 . x i ) ] • ( 3- 3- 2 5) Accommodating A? in (3.3.24) and using (3.3.19), we obtain Rf- At = E[ffH] = cp- A cp" (3.3.26) w here / = TA? • ... f^ T, cp = [cp, ... cpM] with 9* = [<pu ... <Pjif*] a n d A = diag{Xx...XM). 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 3 can now be expressed as a sum of uncorrelated random variables, Uv U2, UM. This follows from (3.3.18) and (3.3.19) from which we can write Chapter 3 Derivation of the MFB's 29 M M X ukfck<Q\k ••• X UkJk~k(?Mk lk = 1 k = 1 M = X V N 2 k = 1 M M X UkfckVxk ••• X UkfckVMk k=\ k=l (3.3.27) where the random variable, U n , can be written as U - V + iW w n rn J n (3.3.28) and \Un\2 = V% + W% has a central chi-squared degree 2 pdf. A l s o , f r om (3.3.19) and (3.3.21), the real component, Vn, and the imaginary component, Wn, o f Un are i.i.d. zero-mean Gauss ian random variables w i th variance a 2 = a 2 = ]r. The characterist ic funct ion o f Y3 is then easi ly * n n It obtained f r om the characterist ic funct ion o f chi-squared degree 2 random variables as M - n ( i ^ ) (3.3.29) Equat ion (3.3.29) gives the characterist ic funct ion o f the s ignal component Y when the fad ing is Ray l e i gh . W r i t i n g (3.3.16) in terms o f f(t) and us ing (3.3.19), we have 2a 0 J Re[f(t)] • p(t)dt = 2 a 0 ( ^ > [ / V ] • Pi • A*) N (• M \ = 2 « o X X ReWk\ • fik • • Pi • (3.3.30) Chapter 3 Derivation of the MFB's 30 where D = At ,T„>% . >0 ' p mm , N = 1 > xm i-„ = 0 p At and i' = i-D+\ Interchanging the order of summations yields M Y2 = 2aQjAt^Re[Uk]- JXk- X cp,'* • k = 1 \i = D M k= 1 N (3.3.31) where B̂  = 2a0*/At • X cp(̂ - pi is a constant. Equation (3.3.31) suggests that Y2 is a weighted sum of independent zero-mean real Gaussian random variables, { Vk}, with variance rj2 . Adding Y2, F 3 and simplifying, we obtain Y3 + Y2 = f M \ / M V« = 1 / Vz = 1 M n = 1 B \ 2 M /B \ 2 = x K A + T ) - x m + s < v w - « = i M n = 1 (3.3.32) From (3.3.14) and (3.3.15), we have Y = M M M « o 2 - X [ y J + X ( t V > 2 + X < * V > : V n = 1 ' n= 1 = 1 (3.3.33) where Vn' = VnJXn + ̂  and W„' = WnJln. 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 Chapter 3 Derivation of the MFB's 31 funct ions o f each term. Rea l i z i ng that the first term, Y}, is a constant, its characterist ic funct ion is written as Y y / T ) ) = exp ( M /R >2Y\ (3.3.34) JJ The second te rm, Yn, has a non-central ch i - squared d i s t r i bu t i on o f degree 1. S i n c e Vn' has K P n variance — and mean — , it's characteristic funct ion is g iven by [12] M (v) = TT / 1 exp 2 A 4 ( 1 - A „ v ) (3.3.35) The th ird term, 7 / / 7 , has a central chi-squared distr ibut ion o f degree 1 w i th Wn' hav ing variance n — . Acco rd ing l y , we have f r om [12] M *Y,„W = n (3.3.36) The characterist ic funct ion o f Y is thus given by Y y ( i » = • «PK | |(D) • ^ y / ; / ( t ) ) (3.3.37) The corresponding pd f o f Y can be obtained numer ica l ly by tak ing the inverse Four i e r t ransform of (3.3.37), i.e. Chapter 3 Derivation of the MFB's 32 oo fy(y) = ^ J VY(v)exv(-jvy) dv . (3.3.38) —oo 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) = oc08(x)+Z-8(x), 0<a 0 <l (3.4.1) where Z is a complex Gaussian random variable; assuming that the channel has unity power gain, £T|Z|2] = 1 - CLQ. However in the general case, when the attenuation may vary within the pulse duration, the channel's impulse response is given by (2.3.14) z(t,z) = a 08(T) + ccjz(08(T) (3.4.2) where a 2 is the power of the diffused beam. The Rician factor of the channel is defined as KdB = 10 • log 1 0 (3.4.3) z(t) is a complex Gaussian random process with £[|z(0l 2] = 1- Thus from (3.2.2) zray(t, x) = aj z(f)8(x) and the corresponding autocorrelation function is given by = ^ ( r , - ^ ) - 8(rj)8(f2). 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 33 T T '"max *max Rf(l,m)= j J $z({l-m)Am%a)h{Tb)p(l'&t-xa)p*(m'At-Tb)dxadxb. (3.4.5) 0 0 Using the channel autocorrelation function for a mobile whip antenna with isotropic scattering [15] <»z(x) = 70(av t) (3.4.6) where J0(x) is a zeroth order Bessel function and (0d = 2nfd is the maximum Doppler angular frequency, (3.4.5) simplifies to RfiUm) = J0(2Kfd(l-m)At)p(l'At)p*(m'At) . (3.4.7) The eigenvalues and eigenvectors corresponding to the matrix RfAt can be easily obtained using mathematical tools [22]. Since xmax = %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 Y3 = af 2 W + W | ) . (3.4.9) Jk= 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 averag- ing depends on the pulse shape. This can also be observed from the expression of the received Chapter 3 Derivation of the MFB's 34 S N R where the signal component Y depends on the pulse p(t). Therefore three pulse shapes are chosen rect pulse: P r e c M ) = I 10, otherwise (3.4.10) hal f s in pulse: PhsW = sin(7Cf/r 0) , o < r < r 0 [0 , otherwise (3.4.11) root raised cos ine pulse: 2T, 1 ( rn(t-aTQy ' l + cosf ^-^-\\,0<t<aT0 aT, o aTQ<t<T0 J2i0(1 +C°< !^))' T0<t*T0«+a) 10, otherwise (3.4.12) where a is the ro l l - o f f factor o f the pu lse and 0 < a < 1. T h e notat ion Tp, i n genera l , refers to the symbol/pulse duration; henceforth Tp = T 0 ( l + a) where a = 0 fo r the rect pulse and the ha l f s in pulse. 3.4.1 Special Cases: 3.4.1.1 Very Slow Fading W h e n the fad ing is very slow, 0<fN< 0.01, us ing (3.4.1), the s igna l componen t Y in Chapter 3 Derivation of the MFB's 35 (3.3.6) reduces to Y = \a0 + Z\2 which implies that E[Y] = 1.0. This shows that the BER does not depend on the pulse shape and Pb is calculated by averaging the error probability for a non-fading AWGN channel over the non-central chi-squared degree 2 pdf, i.e. oo / I 1 r A Eb-y = - • erfc —-— 2 J U N0 1 ( l - a 0 2 ) exp v 1 ~ a o j iJyaQ \iy (3.4.13) Equation (3.4.13) is simplified in [4] and Pb is expressed in terms of the Marcum's ̂ -function Ph = Q(v,w)-h\ + 1 +e exp - v 2 + vf1 I0(vw) (3.4.14) where e = E[\Z\2] • Eb/N0 and v = w K[l +2e-2Ve(l +e)] 2(1+8) K[\ +2e + 2Ve(l +e)] 2(1 +e) (3.4.15) K is the Rician factor, K = al/E[\Z\2], 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. fN^°°, the variance of the signal component Y approaches zero. Since the mean, E[Y] = 1.0, this implies that the pdf, Chapter 3 Derivation of the MFB's 36 lim fY(y) = 8(y- 1). Applying this in (3.3.13), we obtain lim P •8(y-l)dy (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 = 0 and the direct/specular path is absent. Accordingly, the signal component Y in (3.3.14) equals Y3. Using (3.4.2) with ar= £[|z(f)|2] = 1 and cx0 = 0, we have the following results from [8] Eb/N0 + 1 I ' ( ) < / „ < 0.01 ( L L (3.4.18) , otherwise n = 1 m=1 V 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 Pb = \ erM,Ja$Eb/N0) . (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 08(T) + a 1 z 1 ( r )5(x-a , ) . (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 Chapter 3 Derivation of the MFB's 38 Y = a 0 2 + 2a 0 a 1 j Re[zx(t)p(t-Gx)]p(t)dt + ax • j\zx(t)p(t-ax)\2dt (3.5.2) where T = Tp + ax. Using (3.3.30) and (3.3.31), the second term in (3.5.2) can be written as N / M \ Y2 = 2 a 0 a, JXt £ £ • cp,.,* P i , = £)Vjt=l V M £ = 1 (3.5.3) where D = , a , >0 Ar. 1 ,ox = 0 (3.5.4) and TV = Ar , /' = i-D + 1. {^} are the eigenvalues of the matrix Rf At; the (/,m)th element of matrix Rj is defined in (3.4.7). When ax>T,Y2 = 0 which gives Y = oc02 + a 2 - f \zx(t)p(t-ox)\2dt (3.5.5) = CL2 + Ct2 • Y o i r°y where Yray = J |z1(?)jp(^ — cjj )|2<i? is the received signal component from the diffused beam. Chapter 3 Derivation of the MFB's 39 The pd f o f Y, obtained f r om the pd f o f Yray [14], is g iven as fY{y) = 1 2 X r e x p l o , ( y - « o ) N . v > a 2 otherwise (3.5.6) where the constants Bn = and L is the number o f non-zero v = jK eigenvalues. The expression for the B E R is thus given by 2a jerfc 1 a i \Eb-y J No J B X r e xP n = 1 \dy (3.5.7) Per fo rming integration by parts on (3.5.7) we get 1 V B " z u l n = l " e • "a 2^ 2 1  £ £ - [el-y^-y-1/2dy 71 J and us ing (A . 10) (3.5.8) Pb=\ X 5 « « = l erfc ^b aQ ^ 0 ^ erfc(jcijqn)exp ( ct2 ^ V a l 2 ^W. (3.5.9) w h e r e qn = i r + 1 a 2 ^ are constants. F o r a very-s low fad ing case (wi th any degree o f over lapp ing between the d i rect and the d i f fused beams), (3.5.1) and (3.5.2) are given as z(t,x) = a05(x) + Z 1 8 (T -a 1 ) (3.5.10) Chapter 3 Derivation of the MFB's 40 Y = a2 + 2a 0 k • Z r l + (Z 2, +Z 2) = a2(\-k2) + (Zrl + a0k)2 + Z2x (3.5.11) where Zx = Z r l + jZn, E[\ZX\2] = E[Z2rl + Z2X] = 1 -a§; Z r l and Zn are zero mean G a u s s i a n r a n d o m va r i ab l e s and k = j p(t)p(t- o , ) d r is a constant . B y e m p l o y i n g s i m p l e transformations to the non-central chi-squared degree 2 distr ibut ion, the pdf o f Y is wr i t ten as 1-ccg ' 2a2k2-a2 + y} fjy - a2( 1 - k2) • 2a0k I-a2 o J v 1-a 2 (3.5.12) W h e n the two beams are non-over lapping, ol > T and k = 0. Consequent ly (3.3.13) s impl i f ies to Pu = 1 2(1-a 2) jerfc a<5 • exp f y - « 2 > (3.5.13) A p p l y i n g a procedure s imi lar to the one used in obtain ing (3.5.9), the B E R is g iven by Pu = erfc v i ^ 0 j - e x p ( «o2 A Eb/N0 N0 l-oco 2 erfc a 2 & + \-a2,j . (3.5.14) The above express i on can also be d i rec t l y ob ta ined f r o m (3.5.9) by r e a l i z i n g that the ma t r i x Ry • At has on ly a single non-zero eigenvalue, Xx = 1.0, when the fad ing 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 propaga- tion paths of which the first (non-delayed) one is the direct component and the other two arriving at some delays, a, and G 2 , are the diffused components. Accordingly the impulse response is given by z(t, i) = a05(T)+ a1z1(05(i;-a1)+ a2z2(0o(i:-a2), o2>o, . (3.5.15) Comparing (3.2.2) with (3.5.15), zray(t, x) = a,Zj(05(T - a,) + a2z2(f)8(x - a2). Assuming a unity gain channel and ist^,^)!2] = £T|z2(0|2] = 1, we have â  + a 2 + a| = 1 . (3.5.16) The signal component corresponding to (3.5.15) is Y = a20 + 2a0 j Re[f(t)]p(t)dt+ $\f(t)\2dt (3.5.17) o, o, where, from Section 3.3.1, f{t) = a1z1(r)p(r-a,) + a2z2(t)p(t - cs2) • Considering the general case where the two processes Zj(0 and z2(t) have a normalized cross-correlation coeffi- cient p, which is given as 0 (T) p = . ^ r=, lpl<l (3.5.18) where 6 (x) is the cross-correlation function and (L . (T) , i = 1, 2, is the autocorrelation function of the two processes. Assuming both the fading processes have identical statistical characteristics, we can write Chapter 3 Derivation of the MFB's 42 tyZi(T) = <|>z,(x) = JQ((Od • X ) . n2 (3.5.19) The autocorrelation function of zray(t, x) is given as <Rz(tl-t2;(xl,T2)) + a\ • cbZ2(f! - r2)8(x1 - a2)6(x2 - a2) + a^pcb^r, - r2)5(T! - a1)6(x2 -a 2 ) + o^a^cb (tx -?2)8(Xj -G 2 )8(T 2 - a t) which results in the following elements of matrix R^ (3.5.20) Rf{i,m) = • / o ( c o r f ( / _ m ) A 0 t06? p(l'At-al)p*(m'At-ax) + alp(TAt-a2)p*(m'At-ci2) + aja 2pp(l'At-c x) p*(m'At-a2) + a,a2p*/?(/'Ar-a2)p*(m'At- Gj)] . (3.5.21) Using (3.3.31) N / M Y2 = 2a07Ar X X ^ A ' i = D\k=\ M = X vkJhh • k = 1 Pi (3.5.22) When the first two 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, ax = a2 and a2>Tp, the BER is given as (Appendix A) P - I n = 0 n + 1 X k = 1 { k )4 x 'U . (3.5.23) Chapter 3 Derivation of the MFB's 43 Eb{l-a2) where X = 1 + — a n d m e coefficients Cn are defined in (A. 13). In another case when Gx > TQ( 1 + a), i.e. there is no overlap between the direct and the diffused beams, (3.5.17) reduces to Y = a2 + J | / ( 0 I 2 * (3.5.24) - a0 + Y'ray where Yray = J\f(t)\2dt. Comparing (3.5.24) with (3.3.14), we have Yl = a2,, F 2 = 0 and Y3 = Yray. The pdf of Y derived from the pdf of Yray [14] is given as B Im fY(y) = /= i m =i(m-l ) !A,7 LO , otherwise / 1sm - 1 (y-a 0) exp f ( y - c t p V X, > y>ag (3.5.25) where Ld is the number of distinct and non-zero eigenvalues, {Xt} , and rt is the order of Xv The constants Blm are given by (ri-m)\(-jXj)' -(l-jX^V'Wy (v) (3.5.26) ' A / M where *FV ( D ) = TT : »—r is the characteristic function of Yrnv. rrayK -11 (1 _ ll.Vj) 1=1 ' Combining (3.5.25) with (3.3.13) yields Chapter 3 Derivation of the MFB's 44 Pu = a0 (y - ao) dy (3.5.27) Evaluating the above integral (Appendix C) results in the following expression 5 /m 2 2 ^ F a2^ J N o J n + 1 ( M - l ) ! exp \_\\[ 2 J (3.5.28) 2 ' 2 Eb . 1 where | i = m - « and constants Ct = — + r - . T(z) is the Gamma function and v(z) is known as the Whittaker function [23]. rl = 2, V7 applies when the two faded beams are indepen- dent and non-overlapping. However when rl = 1, V/, (3.5.28) simplifies to /= 1 erf v i ^0 j + NQTZ exp ^erfcijc^2) (3.5.29) where erf(x) and erfc(x) are the error functions and complementary error functions respec- tively. Chapter 4 Numerical Results 45 Chapter 4 Numer ica l Results 4.1 Introduction The expressions obtained f r om the analysis in Chapter 3 are used to obtain B E R curves for d i f ferent channe l cond i t i ons . The results for flat f ad ing R i c i a n and f requency se lect ive f ad i ng R i c i an channels are presented in Sect ion 4.2 and Sect ion 4.3 respectively. In the f igures, the B E R plotted against Eb/NQ assuming a unity gain channel. Un less otherwise stated, a rect pulse shape is assumed. In order to ver i fy the results f r om our analysis, s imulat ions us ing M A T L A B ® [22] were per fo rmed for a l l the curves presented. The Ray l e i gh fad ing s imulator used i n the s imu la t i on is based on C la rke ' s mode l , [17], and uses a mod i f i ed vers ion, [24], o f the a lgor i thm in [25]. In a l l cases, the s imu la t i on results were f ound to agree c l o se l y w i t h the ana ly t i ca l curves . The 9 9 % conf idence intervals for the s imulat ion results are w i th in ±5 % of the corresponding mean values. Because o f the c lose agreement, the s imulat ion results are general ly not plotted. The results in Sect ion 4.2 and Sect ion 4.3 assume that the receiver has perfect knowledge o f the channel . In Sect ion 4.4, the effect o f channel est imation errors is studied. 4.2 Flat Rician Fading F i gu r e 4.1 i l lustrates the B E R , Pb, curves for d i f ferent no rma l i z ed D o p p l e r rates, fN , wi th a R i c i a n factor o f O d B . The l im i t i n g case o f very fast fad ing (fN —> °°) w h i c h co inc ides w i th the non-fading A W G N channel case (Sect ion 3.4.1.2) is also shown. The pa i r o f dots marked at interva ls o f 2 d B depict the conf idence intervals. It is observed that the sys tem per fo rmance Chapter 4 Numerical Results 46 improves substantially with the fading rate. 10° 10"'i IO"2 1CT3 ST m Q. -4 10 10"5 IO"" IO"7 0 2 4 6 8 10 12 14 16 18 20 Eb/N0[dB] Figure 4.1 BER as a function of / for the Rician channel, KdB = 0 dB The case 0.0 < fN < 0.01, corresponding to very-slow fading, has the poorest perfor- mance and the curve shifts towards the non-fading case as fN increases. The improvement in performance with fN is more noticeable as we move from the very-slow fading to fN = 0.64 and it tends to diminish with further increase in / . A comparison of the performance for the Rayleigh (KdB —» —<» dB), Rician (KdB = 0 dB) and non-fading (KdB —> °° dB) channels as a function of fN 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 In F i g u r e 4.2, f o r the va lues o f fN chosen, the gap be tween the R a y l e i g h and R i c i a n curves, for a g iven Eb/NQ, tends to increase w i th fN. However , the two curves converge to the A W G N curve as fN —> °°. Thus, the specular component improves the B E R at lower fad ing rates; for very h igh fad ing rates, the imp l i c i t diversity effect becomes dominant in d im in i sh ing the effect of fad ing. The ef fect o f the specu lar component on Pb fo r a range o f R i c i a n factor, K, va lues is i l lustrated in F igure 4.3. A s noted earlier, the B E R decreases w i th K. However, a substantial gain i n per formance is observed for KdB > 0 dB . S ince KdB = 0 d B corresponds to equal powers in the fad ing and non-fading components of the channel, this behavior shows that a greater power i n the non-faded s igna l component relative to the faded component d im in i shes the fad ing effect o f the channel . F igure 4.4 Ef fect o f pulse shape in fast fad ing R i c i an channel , KdB = 0 d B Chapter 4 Numerical Results 49 Unlike in very slow (0.0 < fN < 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 Pb curves for three pulse 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. 1.5 E < 0.5 n r : ! • '• I :• • i. ; i :: I] 7 : i i n —i 7~ i ' i rect half sin RRC (a = 1/4) fading signal _j t i _ : \ \ : : V: :: :: V: \ :: : \ + \ ^ \ \\ \\ Ml 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 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, Tp = 0.5 sec. A sample of a channel fading waveform is also shown. It can be seen that the amplitudes of the half sin and RRC pulse shapes are relatively smaller at the beginning and end of the pulses. Chapter 4 Numerical Results 50 10° 10"' 10" 2 1 0 ° cr CD -4 10 10" 5 lO"' 0 2 4 6 8 10 12 14 16 18 20 Eb/N0[dB] Figure 4.6 Effect of the roll-off factor on the system performance, KdB = 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 normal- ized Doppler rate, fN = fdTp is a function of the total pulse duration, Tp = TQ( 1 + a) where T0 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 for a rect pulse with the same Doppler frequency, fd . 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 Ric ian channel as two and three beam models are presented in 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 the first beam is the direct and the second the diffused beam, are illustrated in Figure 4.7 for both fast (fN = 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 Pb reduces with the 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° io-1 10"2 10"3 cc £ 10 a CL 10-5 10-* 10"7 10 J 0 2 4 6 8 10 12 14 16 18 20 Figure 4.7 Two-beam frequency selective fading, KdB = 0 dB t 100% overlap • 50% overlap 0 No overlap 0.0 <f.,< 0.01 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/non- faded 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 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. 2 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 16 18 20 Figure 4.8 Three-beam frequency selective fast fading, fN = 0.64 and a = 0 Chapter 4 Numerical Results 53 F igures 4.8 and 4.9 refer to fast and s low fad ing condit ions w i th no delay between the two d i f f used beams i.e. a = 0. In both cases it has been f ound that an increase in p results in an improvement in system per formance due to an increase in the rece ived s igna l strength when the beams tend to be correlated and co-phased. A t the other extreme, a cond i t ion o f [0.0;-1.0] wou l d resu l t i n tota l c an ce l l a t i on o f the d i f f u sed beams and recep t i on o f j u s t the non- f aded s i gna l component. A l t hough not shown in the f igures for the sake o f c lar ity, [0.0;-1.0] corresponds to a cons tan t B E R o f 0.5 f o r the R a y l e i g h c hanne l ; f o r the R i c i a n c h a n n e l s i n c e KdB ~-6 d B corresponds to a power o f 0.2 (for a unity gain channel) i n the direct beam, Pb = - erfc '5iVn Compa r i ng F igures 4.8 and 4.9, the gain (obtained w i th an increase in p ) in the latter is seen to be reduced. Th i s is because in fast fad ing there is diversity w i th in a s ingle pulse. 10 12 14 16 18 20 F igure 4.9 Three-beam frequency selective very-s low fad ing, a = 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 b/N 0[dB] F igure 4.11 Three-beam frequency selective very-s low fad ing, a > 0 A n o t h e r m o d e l o f the th ree-beam channe l is c on s i d e r ed i n F i g u r e s 4.12 to 4.16 f o r f requency se lect ive fad ing . It is assumed that the non-faded beam and one of the faded beams coincide. The curves are plotted as a funct ion o f the normal i zed delay (0 < a < 1) and corre lat ion coeff ic ient (-1 < p < 1) between the two faded beams. F igu re 4.12 d isp lays the B E R curves for a fast fad ing R i c i a n channel w i th equa l channel gain for the faded and the non-faded beams. In general, gain in B E R performance is observed w i th an increase i n o. The case [0.0;-1.0] is equivalent to t ransmit t ing the s igna l through a channe l compr i s ing o f on ly the direct path o f hal f the total strength o f the R i c i a n (KdB = 0 d B ) channel . Chapter 4 Numerical Results Figure 4.12 Three-beam fast fading, KdB = 0 dB and/ = 0.64 Figure 4.13 Three-beam slow fading, KdB = 0 dB Chapter 4 Numerical Results 57 Figure 4.13 depicts BER for slow fading Rician channel. The variations of Pb with [a;p] are similar to the fast fading case except for a few cases of which [1.0;0.0] is particularly notewor- thy. 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, we find the former has a lower BER and the gain > 5 dB at Pb = 10~4. 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 lower received s ignal strength in [0.0;0.0] when the fad ing in the two co i n c i - dent d i f fused beams is independent. 10 r 1 1 1 1 1 1 r W d B i F igure 4.15 Three-beam fast fad ing as a funct ion o f K; [o;p] : [0.0; 1.0], [1.0;0.0] Eb/N0(BER) F igure 4.16 Three-beam slow fading as a funct ion 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. Pb increases as the Rician factor is increased from KdB = - o o dB to KdB = 0 dB . This occurs because at low Rician factors, i.e. KdB < 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 non- faded 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 KdB = 3 dB, the non-faded signal dominates and a gain is observed with further increase in K. 10° Eb/N0[dB] 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 Pb with [a;p] for the slow fading channel as observed in Figures 4.16 and 4.17 are found to be similar to the fast fading case. Chapter 4 Numerical Results 60 4.4 Imperfect channel estimation We have analyzed the system behavior when the receiver consists of a correlator demodu- lator 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, o2, proportional to the average power, a 2, of the faded beam. We study the performance for flat fading Rician (KdB = 0 dB) and Rayleigh channels, each with a unit power gain. 10 J 1_ I I I \ - I 1 ^ I 1 2 4 6 8 10 12 14 16 18 Eb/N0[dB] Figure 4.18 Performance degradation in a flat fading Rician channel By modifying a 2 , the BER curves, obtained from simulations with 99% confidence 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 /2, to the fading waveform. In order to study the sensitivity of the (non-ideal) 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 2 is shown in Figures 4.18 and 4.19 respectively. Since a 2 depends on the power of the faded 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 Eb/N0[dB] Figure 4.19 Performance degradation in a flat fading 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 gu res 4.20 and 4.21 co r respond to the B E R fo r the case when the f i l te r is unab le to correct ly trace the phase o f the fad ing wave form. A l t hough , for a l l the values o f a 2 chosen, the B E R for the fast fad ing case is found to be lower than that for the s low fad ing case, the d B spread correspond ing to / = 1.28 is found to be cons iderably greater than the d B spread correspond- i n g to 0<fN < 0 . 0 1 . A s i m i l a r o b s e r v a t i o n can be made in F i g u r e s 4 .22 and 4 .23 w h i c h correspond to the effect o f error in est imating the ampl itude of the fad ing samples. 10 12 14 16 18 F igure 4.22 Ef fect of mis-match in ampl i tude on the B E R in a R i c i an channel C au t i o n shou ld be exerc i sed in de r i v i ng conc lus i ons f r o m a c ompa r i s on o f the f igures s h o w i n g the e f fec t s o f phase and a m p l i t u d e e s t ima t i o n e r ro rs s i n c e the c u r v e s p r e sen ted correspond to the speci f ic mode l assumed in the s imulat ions. The B E R curves wh i c h are plotted Chapter 4 Numerical Results 64 as a f unc t i on o f o 2 reveal a greater degradat ion in rece iver pe r fo rmance caused by imper fec t phase est imat ion than that due to imperfect ampl i tude est imat ion. The plots do not imp l y that in general the receivers are more sensitive to phase errors than to ampl i tude errors. 1 1 1 1 1 1 1 1 : 0 < / < 0 . 0 1 * ^ \ ''.N^/ , fN = \ 1 .W \ - \ \ \ . M F B <T 2 = a 2 / 4 e '• a 2 = a 2 / 2 '. <r2 = a 2 i i i > \ i s \ > S N \ S S \ - H \ ' '• v > \ A V J I I 1 I I 1 J 2 4 6 8 10 12 14 16 18 Eb/N0[dB] F igure 4.23 Ef fect o f mis-match in ampl i tude on the B E R in a Ray l e i gh channel The ca lcu lat ions correspond ing to the mean square error ( M S E ) in ampl i tude est imat ion and mean error in phase est imation are shown in Append i x D. Chapter 5 Conclusions and Suggestions for Future Work 65 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, fN. 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 distribu- tion 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 fN 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 fN 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 na l l y converge w i t h the M F B for the non- fad ing A W G N channe l . The latter ef fect shows the dominant ro le o f the fad ing rate i n l owe r i ng the B E R in fast f ad ing cond i t i ons . In genera l , the B E R per formance improves w i th an increase in K; l i t t le ga in is observed when the rat io o f the powers o f the d i rect and d i f fused beams is less than - 6dB . The sys tem pe r f o rmance was a lso studied for di f ferent transmitted pulse shapes: it was found that the pulse shape affects the B E R s ign i f i cant ly when the channel exhib i ts fast fad ing. Th i s is consistent w i th ear l ier results for the Ray l e i gh case [8], [14]. Results for the frequency selective channel , der ived for the general case of correlated scattering, were also examined. In two-beam frequency selective fad ing, part icu lar ly for s low fad ing, the B E R is very sensitive to the degree o f beam overlap. The B E R performance of a receive fi lter whose impulse response is not perfect ly matched to the channe l f ad ing wave f o rm was stud ied through s imu la t i on . It was observed that, for the spec i f i ed mode l , the sys tem degradat ion was h igher fo r phase m ismatches than fo r amp l i tude mismatches for a g iven error variance. 5.2 Suggestions for Future Work 1. In this thesis, the pdf, fY(y), o f the signal component, Y, is obtained f r om its characterist ic funct ion, ^ ( D ) , through numer ica l integration. To the best o f the author's knowledge, no computa t i ona l l y ef fect ive c l osed f o r m express ion o f the pd f o f unequally weighted non- central chi squared random variables is avai lable. To avo id numer ica l approx imat ion errors and computat ional complexi ty, it wou ld be benef ic ia l to obtain a c losed f o rm express ion for / y ( y ) . 2. In the present work, frequency select iv i ty in the channel is obta ined by assuming a double Chapter 5 Conclusions and Suggestions for Future Work 67 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 both flat and 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. Appendix A : MFB (slow fading ) 68 Appendix A : M F B (slow fading o < fN < 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 08(x)+Z 18(x) + Z 2 8 (T-a) (A.l) = (a 0 + Z 1)8(T) + Z 28(i:-o) where Zx and Z 2 are complex Gaussian random variables with i.i.d. real and imaginary parts Zn = Un + jVn, n = 1, 2 and a2, is the power of the direct/specular path. Using the following expansion, i.e. |Z,+Z2|2 = \Zx\2 + \Z2\2 + 2Re[Zx]Re[Z2] + 2Im[Zx]Im[Z2] (A.2) the corresponding signal component Y can be written as Y = a2 + \ZX\2 + |Z2|2 + 2a0Ul + 2a0U2k + 2UX U2k + 2VX V2k (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>Tp,k = 0 and (A.3) simplifies to Y = (C/j + cc0)2 + U2 + V2 + V2 (AA) Y is thus stated as a sum of non-central chi-squared degree 4 random variable. The cdf of Y is Appendix A : MFB (slow fading) 69 given as [27] oo FY(y) = I C „ P r ( X 4 2 + 2 „ ^ ) (A.5) n = 0 E[\ZA2]E[\Z22] where B = •—• • — is a suitably chosen constant [7]. F r o m [28], we have E[\ZX\2} + E[\Z2\2} M + 1 - ' Nik = 0 Subst i tut ing (A.6) into (A.5) and comput ing the pd f o f Y by dif ferentiat ing FY(y), we obtain fy(y) = ̂ -FY(y) = £ Cn y n = 0 J_ 2(3 1 r ^ - X Jt = 0 *= 1 ( f c - i ) ! 2 p U p ; = y C — e 2 P ^ " 2 p n = 0 n+ 1 (A.7) The B E R is thus g iven by (3.3.13) = \ jerfc X Cn 2 p "2(3 / n = 0 n+l k = 0 k- 1 k\\2$) U P dy (A.8) 1 7 , (F^ys = - ertc 2 'UNQJ F'Y(y)dy Integrating (A.8) by parts and s imp l i f y ing Appendix A : MFB (slow fading ) 70 Pu = erfc V FY(y) \FY{y)d 1 E 2KN, o o erfc (A.9) \ \FY(y)exP(--f)y-^dy 1 3 2 A NR Knowing that [23] Jexp(-4y )y- 1 / 26?y = (A.10) and J e - « y / 2dy = Jn2~nqn 2 ( 2 n - l ) ! ! , n>0 (A. l l ) equation (A.9) further simplifies to oo n = 0 1 - Jfc = 1 (A.12) where (4. = 1 +2(3(£^/iV 0) is a constant. Assuming £[ |Z , | 2 ] = J E , [J^ 2 | 2 ] , the coefficients Cn are given as [27] C 0 = exp(-2ag/(l-ag)) n-l C „ = l / n ; « > 1 7 = 0 , f 2 a 2 / ( l - a 2 ) , n = l Lo, « > I (A. 13) Appendix B : MFB (fast fading) 71 Appendix B : M F B (fast fading fN -> ~) We are interested in finding the limiting case of the variance of the signal component Y, For the flat-fading Rician channel, Y is given by Y = Jjcx0 + <x1z(0|2- \p(t)\2dt (B.l) and its mean E[Y] = \\p(t)\2E[\a0 + alZ(t)\2]dt = j\p(t)\2(a2 + a2)dt (B.2) = a^ + a2 where we have used £[|z(0l2] = 1. Accordingly we have G 2 = E[Y2]-(E[Y])2 (B.3) = E[Y2] - (ag + a 2 ) 2 . Finding the second moment of Y E[Y2] = JJ|p(r1)|2|p(r2)|2JE[|a0 + a 1 z (r 1 ) | 2 |a 0 + a 1z(r 2)| 2] dtxdt2 (B.4) Expanding the expectation term in the integral gives |<x0 + axz(tx)\2 | a 0 + axz(t2)\2 = a$ + a2a2(\z(tx)\2 + \z(t2)\2) + a\ \z(tx)\2\z(t2)\2 + 2aZ a,Re[z(t,)] + 2a3 a,Re[z(t2)] (B.5) + 2a 0 a\Re[z(tx)]\z(t2)\2 + 2a 0 a3xRe[z(t2)]\z(tx)\2 + 4a^a2Re[z(tx)]Re[z(t2)] and taking the expectation of both sides in (B.5) yields Appendix B : MFB (fast fading ) 72 [̂lao + â Cf̂ Plao + ajzC^)! 2] = a 4 + 2aga 2 + £[|z(fj)| 2| z(r 2)| 2] (B.6) Subst i tut ing (B.6) into (B.4) and us ing [14] £[|z('i)| 2 |z(' 2)| 2] = 1 + P 2 (B-7) where p is the normal i zed cross-correlat ion between the two samples o f the real or imag inary part of the process z(t) [16]. Thus, we have E[Y2] = a4 + 2a 2 a 2 + «i JJ|P('I)|2 \P(h)\2 (l + P2) d t \ d t i - (B.8) Since p can be v iewed as the correlat ion of the fading process, f r om [15] p (x ) = J0(2nfDx) (B.9) where J0(x) is the zeroth order Besse l funct ion o f the first k i nd and fD is the m a x i m u m Dopp l e r frequency i n Hertz. A p p l y i n g the transformation tx -12 = i and us ing (B.3), (B.8) and (B.9), the variance o f Y can be written as a 2 = a f JJ|p(r1)|2|p(f,-x)|27g(27c/Dx) <hdtx . (B.10) F o l l o w i n g the ana lys i s i n [14], we subst i tute the autocor re la t i on o f the squared pu l se magnitude wh i ch is g iven by RPW = j"l/>(0l 2 lM '-x)| 2^ ( B . l l ) in (B.10) result ing in a 2 = af J Rp(x) • J$(2nfDT) dx . (B.12) The l im i t i n g var iance o f Y as fN —> <» can be obta ined by tak ing the l im i t as T p ^ o o . 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 ° > " 3 J " ( f t * 1 p 0 7 1 1 = ^ - J l " ( ? ) l 4 ^ (B.14) p o where Q = J |w(g)|4d<7 is a constant. Since an autocorrelation function has it's maximum value o at zero delay, it is convenient to write / y x ) < ^ - (B.15) p which implies 2a\\Q- J&2nfD%) dx a 2 < _ — Q ( B 1 6 ) Taking the limit as Tp —> °° and applying L'Hopital's rule lim a 2 = lim 2af Q J2(2nfDTn)dx = 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. Appendix C : Solving the Integral 74 Appendix C : Solving the Integral To evaluate the following integral encountered in Section 3.5.2: B lm ^ = i X E .rr, i= i m = i ( m - 1)!^ 1 1 i 1 B a 0 2 (C.l) 2 2 X X lm / = l m = l(OT-l)!X / 2 « 0 ( v - a j r - ' e x p f - ^ ^ ^ Substituting the integral expression in the above equation for / - J udv (C.l) 2 C C 0 where u = er/ = — f exp(-t2)dt (C.3) dv = (y-aj)m- 1exp ' ( y - a 2 ^ (C.4) which gives <iu = ——exp - E by\. i/ 2dy (C5) v = - I n = 0 (m- 1)!^ (m-n - 1)! •(y-cc0z)' 2\m-n- 1 exp (C.6) Integrating (C.2) by parts yields Appendix C : Solving the Integral 75 I = -erf (m-n- 1)! • (y -a 0 ^r -" - i exp n+l — y i a n — (m-n-1)! « = o exp ( y - « o ) 2-.ni - rt - 1 (C.7) Ty •exp dy From [23] we obtain the following expression for the integral term in (C.7) -e dy H) a\ = C z z a0^ ^ r C ^ e x p — ^ W j ! (Cjccrf) (C.8) 2 ^ 2 ^ 2 ' 2 . 1 where \x, = m-n, C, = — + r- and T(z) is the Gamma function. WK v ( z ) is known as the ^ 0 h Whittaker function which is defined as [23] ^ , v U ) = , F ( 2 V ) A v ( z ) + , T ( 2 V ) NM , , _ v (z ) rlj-v-x n j + v - x (C.9) where v + MXv(z) = z V z / 2 0 > v-A, + |,2v + l ; z M^_ v (z ) = z ~ V + 2 V z / 2 0 ( - v - ? i + ̂ -2v + l ; z j. 2' (CIO) <E>(a, y; z) is the confluent hypergeometric function [23] _ , . , a z a ( a + l ) z 2 , 0 ( a , y ; z ) = l + — + ^ T T ) % + (C.ll) Equation (C.7) thus simplifies to Appendix C : Solving the Integral 76 I = erf\ 4] (m- 1)!X,' n = 0 n + 1 exp Z-2 a° a 0 ^r(ji)W, • n 5 - (C.12) (C ;a02) which results in the following expression for Pb, L,, r, 73 Zm 2 2,t,„f,(».-i)arL m - l (m- 1)!^ n + l r - o (u -1 ) ! exp H) (Cl?) a 0 2 ; r(i i )W, j ( C ^ ) 2 _ ^ 2 ^ 2 ' 2 Equation (C.l3) can be written in a more simplified form when rt = 1, V/ - 2-2Z, 5 n erfl i= l 2\ II N0n fa2A exp V ^ J (C.H) where erf(x) and erfc(x) are the error and complementary error functions respectively. Appendix D : Calculating the Errors in Amplitude and Phase estimation 77 Appendix D : Calculating the Errors in Amplitude and Phase estimation v actual sample sample with error random error y' y 0 X 0 X 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, se = (x + x') + j(y + y') in the XY-plane. x and y' are both i.i.d. Gaussian random variables with variance, a2* = o2/ = CJ 2/2 where a 2 is the variance of e. The amplitude estimation error is given by er = \se\-\s\ (D.l) where |se| = J(x + x')1 + (y + y ' ) 2 n a s a Rician distribution and l^l = J(x)2 + ( y ) 2 is a 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[e2] = £[|JJ2 + |J|2-2|JJW] = E[\se\2] + \s\2-2\S\E[\se\] ( D 2 ) = a 2 + 2 • \s\2 - [\s\ Jk • fi2 exp(-^) 0>(J, 1 where 0(a, b;z) is the confluent hypergeometric function [23]. 2s[|.sc|2] and £ [ | . s e | ] are the first moments of the non-central chi squared and Rician distributed random variables respectively as shown in [12]. The normalized amplitude estimation error, EN, is written as E[e2] 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, fRQ>(r, (p), of R and 0' which is given by fRv(r,<p) = Z ^ e x p f - 4 1 . (D.5) the mean phase estimation error, E[Qe], is obtained by integrating (D.4) over fRQ'(r, tp), i.e. 2TX OO E[Qe] = — 2 f fratanf, ," S i n ( p l e x p f - ^ W ? . (D.6) E KG2 J J vM+rcoscpJ r v a2) 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, "Matched filter bound for multipath Rician-fading channels," IEEE Trans. Comm., vol. 46, no. 4, pp. 441-445, Apr. 1998. [8] W. Burchill and C. Leung, "Matched filter bound 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, "Matched filter bounds for wireless communication over Ray- leigh 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 in flat, 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: charac- terization," 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. Bibliography 80 [14] W. S. Bu r ch i l l , Performance and Receiver Structures for OFDM on Rayleigh Fading Chan- nels. Ph .D . thesis, Dept. of E lec . & Comp . Eng. , U .B.C. , 1997. [15] W. C. Jakes, Microwave Mobile Communications. N.Y.: Wi ley , 1974. [16] A . Papoul is , Probability, Random Variables, and Stochastic Processes. N.Y.: M c G r a w - H i l l , 1984. [17] R. H . C la rke , " A statistical theory of mobi le-rad io reception," Bell Syst. Tech. J., vo l . 47, pp. 957-1000, 1968. [18] S. Hayk i n , Communication Systems. N.Y.: Wi ley , 3rd edit ion, 1999. [19] W. B. Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise. N.Y.: M c G r a w - H i l l , 1958. [20] E. K r ey s z i g , Advanced Engineering Mathematics. N.Y.: Wi ley , 8th edit ion, 1999. [21] S. S. H a y k i n , Introduction to Adaptive Filters. N . Y : M a c M i l l a n , 1984. [22] The Ma thWorks , Inc., Matlab®, ver. 6.1.0.450, release 12.1, M a y 2001. [23] I. S. Gradshteyn, I. M . Ry zh i k , Table of Integrals, Series and Products. L ondon : A c adem i c Press, 6th edi t ion, 2000. [24] D. J. Young and N . C. Beau l ieu , "The generation o f correlated Ray l e i gh random variates by inverse discrete Four ier transform," IEEE Trans. Comm., vo l . 48, no. 7, pp. 1114-1127, Ju l .2000. [25] J . I. Smi th , " A computer generated mult ipath fading s imulat ion for mob i l e radio," IEEE Trans. Veh. Tech., vo l . VT-24 , no. 3, pp. 39-40, A u g . 1975. [26] J . M . Wozencraft , I. M . Jacobs, Principles of Communication Engineering. N.Y.: Wi ley , 1965. [27] S. Ko t z , N . L. Johnson, D. W. B o y d , "Ser ies representations o f distr ibut ions o f quadratic forms in normal variables II. Non-centra l case," Annals of Mathematical Statistics, vo l . 38, issue 3, pp. 838-848, Jun. 1967. [28] M . Ab ramow i t z and L A . Stegun, Handbook of Mathematical functions. N.Y.: Dove r Pub l i - cations, Inc., 1965.

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
India 6 0
United States 5 1
France 5 0
China 2 13
Republic of Korea 2 0
United Kingdom 1 0
Pakistan 1 0
City Views Downloads
Unknown 6 4
Mumbai 3 0
Beijing 2 0
Chennai 2 0
Washington 2 0
Ashburn 2 0
Roubaix 1 0
Delhi 1 0
Redmond 1 1
Islamabad 1 0
London 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065460/manifest

Comment

Related Items

Admin Tools

To re-ingest this item use button below, on average re-ingesting will take 5 minutes per item.

Reingest

To clear this item from the cache, please use the button below;

Clear Item cache