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Optimal decoding of PSK and QAM signals in frequency nonselective fading channels Bouras, Dimitri 1991

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OPTIMAL DECODING OF PSK AND Q A M SIGNALS IN F R E Q U E N C Y N O N S E L E C T I V E FADING CHANNELS by  DIMITRIOS P. BOURAS Diploma of Electrical Engineering, University of Patras, Patras, Greece, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDEES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard  The UNIVERSITY OF BRITISH COLUMBIA September 1991 © Dimitrios P. Bouras, 1991  In  presenting this  degree at the  thesis  in  University of  partial  fulfilment  of  of  department  this thesis for or  by  his  or  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  representatives.  an advanced  Library shall make it  agree that permission for extensive  scholarly purposes may be her  for  It  is  granted  by the  understood  that  head of copying  my or  publication of this thesis for • financial gain shall hot be allowed without my written permission.  Department of  BecVoCOx,  TE^ne^l  The University of British Columbia Vancouver, Canada  o& u \ jeer A  Date  DE-6 (2/88)  ^  Abstract In this thesis, the maximum likelihood sequential decoder for the reception of coded digitally modulated signals with single or multiamplitude constellations transmitted over a multiplicative, frequency nonselective fading (Rayleigh or Rician) channel and corrupted by additive white Gaussian noise (AWGN), is derived. It is shown that its structure consists of a combination of envelope, multiple differential and coherent detectors. The outputs of each of these detectors are 1  jointly processed by means of an algorithm which is presented in a recursive form. The derivation of this new receiver is general enough to accommodate uncoded as well as trellis coded Phase Shift Keying (PSK) and Quadrature Amplitude Modulated (QAM) schemes. Differentially encoded signals, such as the 7r/4-shift DQPSK scheme which has been selected as the new transmission standard for the emerging all-digital North American and Japanese cellular system, can be also incorporated. Furthermore, there is no assumption made on the power spectral density (or equivalently on the autocorrelation function) of the fading process, although it is considered to be known to the receiver. In order to reduce the overall receiver implementation complexity, several reduced complexity, near optimal versions of the algorithm are presented. These reduced complexity receivers are based on the use of only few multiple differential detectors. Performance evaluation results for reduced complexity trellis coded 7r/4-shift DQPSK and 7r/4-shift 8-DQAM systems have demonstrated that the proposed receivers dramatically reduce the error floor occurring due to fading. For example, for the 7r/4-shift DQPSK scheme operated in a fast (BpT=0.l25) Rayleigh flat-fading channel at a Signal-to-Noise ratio (SNR) of about 25 dB, the new reduced complexity receiver structures result in a bit error rate (BER) reduction of more than three orders of magnitude as compared to a conventional Viterbi receiver. For a slower fading channel, these gains are even higher. What is even more interesting, though, is that this performance was achieved by employing only up to third order multiple 1  As it will be explained later on, a k* order multiple differential detector is a differential detector employing a k* symbol delay element.  ii  differential detectors, thus maintaining reasonable levels of implementation complexity. By further increasing the order of the employed multiple differential detectors, the obtained BER performance improvements were found to be minimal. To verify experimentally the effectiveness of the proposed receivers and to gain additional insight into some of the real-life implementation problems involved, we designed and realized in hardware a prototype 7r/4-shift DQPSK modem. Although its transmitter was designed with off the shelf digital TTL chips, linear analog devices and RF modules, its receiver was implemented by means of a TMS320C30 Digital Signal Processor. Employing this "flexible" receiver, the proposed algorithms could be evaluated very efficiently. The various experimental BER results obtained were in excellent agreement with the equivalent results obtained by means of computer simulation. Motivated by these very encouraging experimental performance results, it is believed that LSI/VLSI implementation of such a 7r/4-shift DQPSK modem would be a very worthwhile task.  iii  Contents Abstract  11  List of Figures  v  "  List of Tables  xiii  Acknowledgments  xiv  Chapter 1  Introduction  1  Section 1.1  The Multipath Fading Environment  1  Section 1.2  PSK and QAM  3  Section 1.3  Digital Transmission over Fading Channels  4  Section 1.4  Research Objectives of this Thesis  5  Section 1.5  Thesis organization  6  Chapter 2  type of Signals  Optimal and Reduced Complexity Detection in Flat Fading Channels  8  Section 2.1  Introduction  8  Section 2.2  System Model Description  8  Section 2.3  Derivation of the Optimal Decoder Structure  13  Section 2.4  Recursive Algorithms  22  Section 2.5  Conclusions  23  Chapter 3  Computer Simulation Performance Evaluation: Software System Description and BER results  25  Section 3.1  Introduction  25  Section 3.2  Software Simulation Model Description  25  The Information Source Block  27  3.2.1  iv  3.2.2  The Transmitter Block  28  3.2.3  The Shaping Filters and the Filtering Operations  29  3.2.3.1  Transmit (Tx-) and Receive (Rx-) filters  29  3.2.3.2  The Filtering Operations  30  3.2.3.3  Examples of Typical Simulation Graphs  32  3.2.4  The Channel Block  34  3.2.5  The Noncoherent Receiver Block  40  3.2.5.1  The FM Limiter/Discriminator Receiver  41  3.2.5.2  The Differential Detector Receiver  42  3.2.5.3  BER Computer Simulation Results  43  3.2.6  47  The Receiver Block  3.2.6.1  Coded Version of the Receiver  48  3.2.6.2  Uncoded Version of the Receiver  53  BER Performance Evaluation Results  53  3.3.1  Coded Schemes  53  3.3.2  Uncoded schemes  58  Conclusions  59  Section 3.3  Section 3.4 Chapter 4  Design and Implementation of a Prototype 7r/4—shift DQPSK System  61  Section 4.1  Introduction  61  Section 4.2  The Synthesized ir/4-shift DQPSK Baseband Generator  62  Section 4.3  RF Modulator/Demodulator (MODEM) and Channel  69  Section 4.4  The Digital Signal Processor (DSP) - based Receiver  72  V  Section 4.5 Chapter 5  Conclusions  .74  Experimental Set-up and Measurements  75  Section 5.1  Introduction  7 5  Section 5.2  Experimental Set-up  76  Section 5.3  Signal Measurements  76  Section 5.4  BER Performance Evaluation Measurements  83  Section 5.5  Conclusions  87  Conclusions and Suggestions for Future Research  89  Section 6.1  Conclusions  89  Section 6.2  Suggestions for Future Research  89  6.2.1  Further Modem Development  89  6.2.2  Symbol Timing Synchronization  90  6.2.3  Frequency Selective Fading  90  6.2.4  CPM Signals .  90  Chapter 6  Bibliography Appendix A  91 Theoretical Derivations  97  Section A. 1  Derivation of Equation 2.18  97  Section A.2  Derivation of Equation 2.27  98  . Section A.3  Derivation of Equation 2.28  102  Section A.4  Derivation of Equation 3.11  104  Section A.5  Derivation of Equation 3.15  105  vi  List of Figures Figure 1.1  The signal space of 8-PSK (a), and 16-QAM (b)  3  Figure 1.2  The signal space of TT/4—shift DPQSK (a), versus the signal space of QPSK (b)  Figure 2.1  4  Block diagram of the transmitter model assumed; C E : Convolutional Encoder, SM: Signal Mapper, DE: Differential Encoder, H (f): T  Transmit filter  8  Figure 2.2  Block diagram of the frequency nonselective fading model. . . . 10  Figure 2.3  Block diagram of the receiver; BPF: Band Pass Filter,  H (f): R  Receiver filter Figure 2.4  Prediction coefficients and MMSPE for land mobile channel with BT  = 0.125; (a): 2  (c): 4  th  F  Figure 2.5  12  nd  order prediction, (b): 3  order prediction,  rd  order prediction, (d): MMSPE  17  Prediction coefficients and MMSPE for land mobile channel with BT  = 0.0125; (a): 2  (c): 4  th  F  nd  order prediction, (b): 3  rd  order prediction,  order prediction, (d): MMSPE  Figure 2.6  Detailed block diagram of the optimal decoder  Figure 3.1  Functional block-diagram of the communication system in software  Figure 3.2  18 20  26  Signal mapping and constellation for (a) 4-state rate-1/2 7r/4-shift DQPSK and (b) 8-state rate-2/3 7r/4-shift 8-DQAM. . 29  Figure 3.3  Example of BPSK signal in the discrete time domain for the sequence 1011. The number of samples/symbol is 4 VII  31  Figure 3.4  Transfer function of cascaded transmit and receive filters for different values of the roll-off factor a  Figure 3.5  33  The simulated 2-symbol eye-diagram for QPSK using roll-off factor a = 0.5  Figure 3.6  33  Baseband signal trajectories of TT/4—shift DQPSK for a = 0.5 and 34  a = 1.0  Figure 3.7  Autocorrelation of 4096 Gaussian white noise samples; variance = 100  Figure 3.8  ;  35  Comparison of computer simulated noise CPDF to the Gaussian theoretical; variance = 100  Figure 3.9  35  The baseband Rayleigh fading and AWGN channel model; s(t) represents the transmitted signal  Figure 3.10  37  Computer simulated amplitude of a faded carrier for the land mobile fading model  Figure 3.11  38  The CPDF for the envelope of the computer simulated complex fading process f(t). For comparison purposes, the theoretical curve has also been included  Figure 3.12  39  Comparison of the simulated faded carrier phase CPDF to the theoretical curve.  . 39  Figure 3.13  The generalized block diagram of the Ricean fading channel.. . 40  Figure 3.14  Integrate-and-dump output for 256 symbols transmitted  Figure 3.15  Theoretical and simulated BER performance of coherently  42  detected QPSK in AWGN. Nyquist filters with a = 0.1,0.4,0.7,1.0 are employed  44 viii  Figure 3.16  Theoretical and simulated B E R performance of 7r/4-shift D Q P S K in AWGN. Nyquist filters with a = 0.1,0.4,0.7,1.0 are employed.  Figure 3.17  BER of differentially detected 7r/4-shift D Q P S K in flat Rayleigh fading. Nyquist filter a = 0.2, BT = B T F  Figure 3.18  45  F  Differential detector in flat Rayleigh fading. Nyquist filter a = 0.35,  46  BT = B T F  Figure 3.20  45  Differential detector in flat Rayleigh fading. Nyquist filter a = 0.2, BT = B T  Figure 3.19  44  BER of differentially detected TT/4—shift D Q P S K in flat Ricean fading. Nyquist filter a = 0.2, BT = B T F  Figure 3.21  Original 4-state trellis with signal numbers used as branch labels  Figure 3.22  50  Super-trellis generation for the 4-state code of Fig. 3.21 and 2  Figure 3.23  47  =1  50  The super-trellis for the 4-state code described in Fig. 3.21, for the case of z = 1. Note that both super-state and super-signal numbers are integers of base 4  Figure 3.24  52  B E R of 4-state Q P S K in flat Rayleigh fading (uniform model). Nyquist filter a = 0.35, B T = 0.125 F  Figure 3.25  B E R of 4-state TT/4—shift D Q P S K in flat Rayleigh fading (uniform model). Nyquist filter a = 0.35, B T = 0.125 F  Figure 3.26  54  B E R of 4-state Q P S K in flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0, B T = 0.125 F  Figure 3.27  54  55  BER of 4-state Q P S K in flat Rayleigh fading (land mobile model). Nyquist filter a = 0.35, B T = 0.125 F  ix  55  Figure 3.28  BER of 8-state 8-QAM in flat Rayleigh fading (land mobile model). Nyquist filter a = 0.35, B T F  Figure 3.29  BER of 8-state TT/4—shift 8-DQAM in flat Rayleigh fading (land mobile model). Nyquist filter a = 0.35, B T F  Figure 3.30  F  = 0.125  F  F  58  = 0.125  58  BER of TT/4—shift DQPSK in flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0, B T F  Figure 4.1  = 0.125  BER of ?r/4—shift DQPSK in flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0, B T  Figure 3.33  .57  BER of 4-state QPSK employing z = 3, in flat Ricean fading (land mobile model). Nyquist filter a = 0.35, B T  Figure 3.32  57  = 0.125  BER of 4-state QPSK in Ricean versus Rayleigh flat fading (land mobile model). Nyquist filter a = 0.35, B T  Figure 3.31  56  = 0.125  = 0.0125  59  Block diagram of the implemented prototype 7r/4—shift DQPSK system  Figure 4.2  61  x/sin(x) equalized raised cosine Nyquist pulse with roll-off factor a = 0.35; the number of samples/(symbol interval) is equal to 16. The sampling instant is at 0  Figure 4.3  Functional block diagram of the 7r/4—shift DQPSK baseband transmitter  Figure 4.4  64  The functional block diagram of the 7i74-shift DQPSK state encoder  Figure 4.5  63  .65  The signal numbering convention and mapping of input information symbols to transmitted differential phases  66  Figure 4.6  Example pulse with J=2, 8 samples/symbol and a = 0.35  67  Figure 4.7  Derivation of the pulse samples for J=2  68  x  Figure 4.8  Functional block diagram of the MODEM and the RF channel. . 70  Figure 4.9  Functional block diagram of the DSP-based digital receiver.... 73  Figure 5.1  Functional block diagram of the experimental layout  75  Figure 5.2  l-channel eye-diagram at modulator input; a = 1.0  76  Figure 5.3  Photograph of the complete experimental set up, in the Communications Research Laboratory of the Department of Electrical Engineering at the University of British Columbia. . . . 77  Figure 5.4  l-channel eye diagram at modulator input; a = 0.5  78  Figure 5.5  l-channel eye diagram at demodulator output; a = 1.0  79  Figure 5.6  l-channel eye diagram at demodulator output; a = 0.5  79  Figure 5.7  Modulated signal space diagram; a = 1.0  80  Figure 5.8  Demodulated signal space diagram; a = 1.0  80  Figure 5.9  Modulated signal space diagram; a = 0.5  81  Figure 5.10  Demodulated signal space diagram; a = 0.5  81  Figure 5.11  The spectrum of the transmitted x/4—shift DQPSK signal; 82  a = 1.0  Figure 5.12  The spectrum of the transmitted 7r/4—shift DQPSK signal; a = 0.5  82  Figure 5.13  The spectrum of 7i74-shift DQPSK in AWGN; a = 1.0  83  Figure 5.14  l-channel eye diagram at demodulator output with AWGN present; a = 1.0  83  Figure 5.15  The spectrum of 7i74-shift DQPSK in AWGN; a = 0.5  84  Figure 5.16  l-channel eye diagram at demodulator output with AWGN present; a = 0.5  84 xi  Figure 5.17  The spectrum of the faded carrier with maximum Doppler frequency set to B  Figure 5.18  F  = 126 Hz  BER of the differential detector receiver in AWGN and Rayleigh fading with B T =0.013, 0.026 and 0.0041 F  Figure 5.19  85  86  BER of the multiple differential detector receiver in AWGN and Rayleigh fading with B T = 0.125 F  xii  87  List of Tables Table 4.2.1  Pulse-sample synthesis implementation specific parameters. . . 64  Table 4.2.2  Next-signal transmitted as a function of the previous-signal and the input  Table 4.2.3  66  The constants used for the generation of the next signal number  67  xiii  Acknowledgments I would like to thank my parents, Panos and Toula, for their continuous moral support and constant encouragement. Dr. Vassilios Makios of the University of Patras, Patras Greece, for his positive influence regarding my post-graduate studies abroad. My supervisor, Dr. P. Takis Mathiopoulos, for providing continuous encouragement, moral support and valuable experience throughout this thesis project. Dimitrios Makrakis of the Communications Research Centre, Department of Communications, Government of Canada, for his valuable help and insightful discussions on algorithmic modelling. Finally, I would like to acknowledge the financial assistance provided by NSERC, in the form of operating grant OGPIN Oil.  xiv  1. Introduction 1.1. The Multipath Fading Environment It is only well known to all involved with mobile communications, that the major disturbance in the transmission of signals over land-mobile and mobile-satellite links, is the presence of multipath fading (see for example [1-5]). Since propagation at VHF/UHF frequencies takes place through scattered and/or reflected waves, for a mobile user the fading effect is produced by a multitude of signals arriving at the receiver's antenna through multiple, random propagation paths which vary according to the user's velocity. As a result, the received signal will be characterized by severe envelope fluctuations as well as randomly changing phase shifts [5], Collectively, the above phenomena create a diffused signal which is referred to as multipath fading (or in short fading). Naturally, the channels in which these phenomena take place arc known as multipath fading channels, or in short, fading channels. Depending upon whether or not there is a line-of-sight (LOS) direct path between the transmitter and the mobile receiver, fading channels can have Rayleigh or Ricean characteristics [6]. For the Rayleigh fading channel, in which such a path does not exist, the probability density function (pdf) of the faded envelope r, follows the Rayleigh distribution, i.e., /(r) = ( £ e - £ , 10  r>0 elsewhere  (1.1)  whereas its phase 0 is uniformly distributed, i.e.,  f(e) = { i>  -*<o<*  t0  (i.2)  elsewhere.  On the other hand, for the Ricean fading channel, in addition to the diffused signal (Rayleigh fading), there is a direct LOS path. The vector sum of the direct signal with a Rayleigh fading 1  Chapter 1.  component results in a composite signal with Ricean statistics. The pdf of the envelope of such a signal is given as [7]  2rJ ±±P-exp[-K -  /(*") =  {  (1 + K)r ]l 2  0  2ry/K{\ + K)  (1.3)  where K denotes the power ratio between the direct and the diffused signal component, S is the power in the diffused component, and Jo(-) the modified Bessel function of order 0. The phase statistics of the Ricean fading channel are described by [8] e~ M - * T  K  \/#cos 0 exp ( - i f sin 0) r , / r— - \ i 27? i[2-erfc(VF^)]  „ , (1.4)  2  +  v  where erfc(-) denotes the complementary error function. For both Rayleigh and Ricean fading channels, the diffused signal component has a two-sided bandwidth of 2Bp = 2vf /c where f c  c  is the carrier frequency, c the propagation velocity and v the velocity of the user. Depending upon the values of v, as well as on the transmitted symbol rate 1/T, the fading could be characterized as either fast or slow. Bp is also known as the fading bandwidth or maximum Doppler shift. The power spectral density of the faded signal, depends upon the fading spectrum assumed. Examples of several such spectra can be found in [6]; two of them, the uniform and the land-mobile, will be used extensively in this thesis (see Chapter 2). Depending upon the values of the delay spread, T^, of the reflected waves, as compared to the symbol duration T, the fading could be frequency selective or frequency nonselective (or flat). For large values of rjT (typically > 0.1) it is probably fair to state that fading is frequency selective. For smaller values of r T, the fading becomes frequency nonselective, i.e., flat. In d  practice, frequency flat and frequency selective fading coexist in mobile radio communication channels (e.g., [9]). However, for the purpose of this thesis, the Rayleigh and Ricean frequency nonselective (flat) fading channel model will be adopted. Section 1.  2  Chapter 1.  1.2. PSK and QAM type of Signals Perhaps the most commonly used bandwidth efficient techniques for transmission of digital signals, are the Phase Shift Keying (PSK) and Quadrature Amplitude Modulation (QAM) techniques [10]. •  M-ary PSK: This is a technique according to which the transmitted symbols are used to produce M distinct carrier phases, which are traditionally, uniformly distributed over [0,2x) [11].  •  Af-ary Q A M : According to this technique, the information symbols are transmitted using a combination of M amplitudes and phases of the transmitted carrier, in such a way that the transmitted signal constellation has a rectangular form [10].  Examples of typical PSK and QAM signal constellations are illustrated in Fig. 1.1. A modulation  •  •  •  (a)8-PSK.  •  (b) 16-QAM.  Fig. 1.1. The signal space of 8-PSK (a), and 16-QAM (b).  format which is unique, is Quadrature PSK (QPSK), since it can be viewed both as 4-PSK and 4-QAM. Furthermore, the recent decision by the EIA/TIA [12] to adopt a QPSK type signal, namely the. 7r/4-shift DQPSK, as the transmission standard for the emerging Northern American digital cellular system, has created aflurryof research activity in systems employing this modulation format [13, 14]. As it can be seen from Fig. 1.2, a pair of input bits are Section 2.  Chapter 1.  0  3  I 2  (a) Ji/4-shift DQPSK. Fig.  (b) QPSK.  1.2. The signal space of W4-shift DPQSK (a), versus the signal space of QPSK (b).  differentially encoding the transmitted carrier phase, so that the carrier phase angle at the z-th symbol interval is given by 6 =  + A0,  {  (1.5)  where A0, is the differential phase, taking values from the alphabet { j> T ' T ' x l * -  T  n  e  principal advantages presented by this modulation scheme, as compared to QPSK, are a) reduced envelopefluctuation,rendering it less prone to distortion imposed by non-linear power amplifier stages, and b) the receiver's ability to recover the transmitted information in a noncoherent (i.e., differential) fashion [15]. This second advantage, also leads to considerable system simplification, as the information bearing carrier signal does not have to be recovered at the receiver end.  1.3. Digital Transmission over Fading Channels The recent trend towards an all digital network (see for example [16]), has led to an increasing need for reliable transmission of digital voice and data in mobile radio applications, i.e., reliable digital transmission over fading channels. However, in such a fading environment, digital transmission schemes, apart from their overall poor performance, exhibit an irreducible errorrate known as "errorfloor"which is caused by the random phase disturbance (otherwise called random-FM) imposed by the fading [5]. In this respect, over the past few years, a large number Section 2.  4  Chapter 1.  of "fading resistive" techniques for communication systems have been proposed, analyzed and evaluated. Examples include interleaving/deinterleaving [17-19], trellis coded modulation [17, 19, 20], prediction/cancellation techniques [21, 22], or combinations of the above. The transmission/detection techniques may be further divided into two broad categories. The first one deals with coherent systems. The majority of these techniques involve the transmission of pilot tones (e.g., [23, 24]), or pilot sequences (e.g., [25-27] which are used to extract the channel impairment. At the receiver, after suitable processing, these pilot tones or sequences can be used as coherent reference to cancel the effects of fading on the digital information signal. The problem of optimal carrier recovery for PSK schemes employing symbol-by-symbol receivers, operated in a Rayleigh faded environment, has been addressed in [28, 29], and more recently in [30]. The second category involves techniques which are based on noncoherent detection (e.g., [31-34, 18, 17, 19]). Noncoherent detection can be further subdivided into differential (e.g., [31-34, 19, 35]) and limiter—FM-discriminator detection (e.g., [15, 36]). For the former one, the phase of the carrier corresponding to the previous transmitted data symbol, is used as demodulation reference. For the latter one, the receiver consists of a limiter/discriminator followed by an integrate-and-dump postdetection filter. Performance comparisons for some of these techniques can be found in [37, 38]. Based on the published literature, it is fair to state that there is no "clear-cut winner" between the various coherent and noncoherent detection techniques. Nevertheless, it seems that in a severe fading environment such as the mobile radio channel, the implementation simplicity and robustness of noncoherent detection make it the preferred detection scheme [34, 38, 39].  1.4. Research Objectives of this Thesis Motivated by the above, this thesis addresses the general problem of optimal detection of both coded and uncoded, differentially encoded or not, PSK and QAM signals in frequency Section 4.  5  Chapter 1.  nonselective Rayleigh or Ricean channels. Its research contributions can be summarized as follows. 1.  Theoretical derivation of the maximum likelihood sequence detector for PSK and QAM signals transmitted over such fading channels. It is shown that the detector's structure can be realized as a combination of envelope, multiple differential and coherent detectors. In addition to the optimal detection algorithm, several reduced complexity, as well as recursive algorithms are also proposed.  2.  Obtain by means of computer simulation, BER performance evaluation results for the reduced complexity algorithms employing various digital modulation schemes, including coded and uncoded 7r/4-shift DPQSK, QPSK, 7r/4-shift DQAM and 8-QAM.  3.  Design, hardware/software implementation of a prototype 7r/4-shift DPQSK modem, employing a Digital Signal Processor (DSP) based receiver, so that the effectiveness of the proposed detector structures could be verified experimentally.  1.5. Thesis organization Including this introductory chapter, the thesis consists of five chapters and three appendixes. Its organization is as follows •  Chapter 2 contains all the theoretical derivations.  The mathematical model for the  communication system under consideration is first presented.  Afterwards, the optimal  detector for channels corrupted by AWGN and flat Rayleigh and Ricean fading is derived. Reduced complexity and recursive forms of the detection algorithm are also presented. •  In Chapter 3, computer simulation software developed for obtaining the BER of a wide variety of digital communication systems is presented. This software is then used to obtain the BER performance of the receiver structures derived in Chapter 2.  Section 4.  6  Chapter 1.  •  Chapter 4 includes the implementation details for the hardware design of the prototype 7r/4-shift DPQSK system.  Firstly, the design and implementation of the DPQSK  baseband generator is described.  Secondly, the hardware implementation of the RF  Modulator/Demodulator (MODEM), as well as the channel, is presented. Thirdly, the DSP-based receiver design and realization is given. •  In Chapter 5, the experimental testbed set up to evaluate the developed prototype 7r/4-shift DPQSK system is presented.  In addition, measurements performed on the  experimental setup, including BER performance results obtained for various experimental MODEM configurations, are given. Finally, comparisons between simulated and measured performances are discussed. •  Chapter 6 contains the conclusions of this thesis, together with suggestions for future work.  •  In Appendix A, some of the detailed theoretical derivations, for the optimal detector derived, are given.  Section 5.  7  2. Optimal and Reduced Complexity Detection in Flat Fading Channels 2.1. Introduction In this chapter, the derivation of the maximum likelihood sequential detector for trelliscoded or uncoded PSK and QAM signals (differentially encoded or not), in channels corrupted by multiplicative frequency nonselective Rayleigh or Ricean fading, as well as additive white Gaussian noise (AWGN), is presented. In addition, various suboptimal, reduced complexity versions of the optimal algorithms are also presented. The organization of the chapter is as follows.  After this introduction, in the second section, a detailed description of the  communications system under consideration, is given. The derivation of the optimal detector for signals transmitted over fading channels is presented in the third section. Some reduced complexity algorithms are also derived in the same section. The structure of two recursive algorithms is presented in the fourth section, while the conclusions of this chapter can be found in the last section.  2.2. System Model Description The block diagram of the transmitter is shown in Fig. 2.1. It consists of a Convolutional  Optional  Optional e  Fig.  J<i«f '+p> t  2.1. Block diagram of the transmitter model assumed; CE: Convolutional  Encoder, SM: Signal Mapper, DE: Differential Encoder, Hr(f)- Transmit filter.  Encoder (CE) which is optional, a Signal Mapper (SM), a Differential Encoder (DE) which 8  Chapter 2. is optional also, a transmit filter with transfer function HTU) and a modulator. The p-bit information word a = [ajt,a|,...,a£] consists of independent and equiprobable bits, taking p  k  values from the alphabet {0,1}. If the CE is used, the output is a q-bit word b\ = [b\,  b\,6|],  resulting in a q/p rate convolution^ code. The SM converts the binary sequences b into 9  k  symbols 6k = 7 * e \ where 7* and  represent the envelope and phase of  j f  respectively.  The mapping of bit combinations b\ to symbols 6k on the signal constellation, can be done according to the set partitioning technique described in [40]. In the special case of PSK signals, 7jt is constant and without loss of generality may be assumed equal to one, while Ojt can be obtained by the following Eq.  L  .=1  If the optional DE is used, the sequence of <5*'s is translated to a new sequence C(a) of cjt's, i.e., C(a) = [ci,C2,...,ci], where c =  =-e>'(**->® *)  (2.2)  0  fc  with $fc representing the phase of c* and © modulo 2ir addition. The signal at the output of the transmitfilterHTU)  c  a  L s(t) = ^c h (t-kT) k  Jb=0  nD  e  represented as (2.3)  T  where hrit) is the impulse response corresponding to Hx(f), T the symbol duration and L the number of symbols to be transmitted. After the modulator, the transmitted signal can be expressed as  x(t) = Re{s(t)e ( * < )} j  2  f  t+p  (2.4)  where Re{-} denotes real part, f is the carrier frequency and p the initial phase of the modulator. c  Section 2.  9  Chapter 2.  It is assumed that x(t) is corrupted by multiplicative,frequencynonselective fading f(t) and AWGN n(r) with one sided power spectral density NQ. Following [5], and as illustrated  Fig.  2.2. Block diagram of the frequency nonselective fading model.  in Fig. 2.2, / ( t ) is modelled as a complex summation of two white and independent Gaussian noise processes, n{(r) and n^(t), filtered by two identical filters, Hp(f).  f'(t) is the fading  signal modulated by f . f(t) can be expressed mathematically as c  (2.5)  /(*) = /'(*) + jf*(t) with both / ( r ) and f®(t) having the same autocorrelation function given by 7  Rf(r) =E{ [f'{t) - J ] 7  [f'(t- )-7 ]} 7  T  =  E{[f (t)-7Q][f^(t-r)-jQ]}  =  Q  oo  N  F  j  (2.6)  \H (f)\ e^dT 2  F  where E{-} denotes the expected value, f = E{f ^)}, f 1  1  Q  = E{f (t)} Q  and iV> is the power  spectral density of n{(t) or nq(t). The cross-correlation of / ( r ) and f®(t) is zero since they 7  are independent. Without any loss of generality, from this point and on, it will be assumed for mathematical convenience that fQ = E{f^(t)} = 0. This by no means compromises Section 2.  10  Chapter 2. the generality of the theoretical analysis, as it amounts to assuming that the transmitted signal carrier has phase equal to 0. In other words, as ~p is varied in order to accommodate the Ricean fading channel, the probability density function (pdf) of the faded carrier phase, will present a maximum at zero degrees. The transfer function of the employed fading filters specifies the spectral characteristics of the fading model considered, e.g., rectangular [41], Gaussian (aeronautical channel) [42-44] and land mobile [5]. In this thesis, two fading models were considered, the rectangular (or uniform) for mathematical convenience in the initial theoretical derivations, and the land mobile. It is well known that the autocorrelation function for the rectangular fading model is given by [6] R f { T )  = ™^B r) F  (  2  J  )  whereas for the land mobile by [5] Rf(r)  = J (2TTB T) 0  (2.8)  F  where JQ(-) is the zero order Bessel function of the first kind and B Normalizing B  to the symbol rate l/T, the higher the B T  F  F  F  is the fading bandwidth.  product, the faster the fading  interference changes with respect to the symbol duration. As well known, the B T product is an F  important parameter when evaluating the performance of digital modulation schemes transmitted over a fading channel. The main reason for this is that the occurrence of "errorfloors"is directly related to the B T, i.e., when B T is increased, the error floor will occur at a higher bit error F  F  rate (BER). For example, for a conventional differentially detected 7r/4-shift DQPSK system, a B T =0.0041 F  will result in an error floor at approximately 2x 10", whereas a B T=0.026 4  F  will  produce an errorfloorat approximately 9x 10". It should be noted that from the open technical 3  literature there exists no clear definition as to which B T product results in fast fading. For the F  purpose of this thesis, however, it will be assumed that for B T > 0.1 we have fast fading. As F  previously mentioned, f(t) will be assumed to have both Rayleigh and Ricean characteristics. Section 2.  11  Chapter 2.  characteristics. For the Ricean fading channel, the K factor, defined as the ratio of the direct signal component to the multipath power [6], will be used in comparing system performance. Following the notation used so far, K can be expressed in deciBells (dB) as ,2  K  1  = 10 log 10  dB  (2.9)  where a ^ and G ^ are the squared variances of / (t) and f®(t), 2  Eq. 2.9, as f  7  1  1  respectively. Clearly, from  —• 0, K —• — oo effectively changing the type of fading from Ricean to Rayleigh.  The block diagram of the receiver is depicted in Fig. 2.3. The received signal x (t),  after  r  •J2nf t t  ^ • 1 Decoder  BPF  y = y(kT) k  Fig. 2.3. Block diagram of the receiver; BPF: Band Pass Filter, HR{/): Receiver filter.  corruption from fading and AWGN, can be expressed as x {t) = x(t)f'(t) T  x (t) T  + n(t) .  (2.10)  is passed through a wideband (roofing) Band Pass Filter (BPF) which limits the Gaussian  noise input to the receiver, without distorting the information signal. After being downconverted , 2  it is passed through a pre-detection low-pass filter HR(/),  which is assumed to be matched to  HTU). From Fig. 2.3, the signal at the output of HR{J) can be expressed mathematically as L  y(t) = f(t)J2 k Ht-kT)-rn (t)-jn (t) c  eJP  a  c  (2.11)  k=0 2  This was assumed for mathematical convenience in order to carry out the theoretical derivations in the complex baseband domain.  Furthermore, this is the worst-case approach, since the fading present will be entirely transformed into the equivalent baseband process, whereas in a conventional differential detector, for example, the fading correlation between adjacent signal samples will result in partial fading interference cancellation.  Section 2.  12  Chapter 2.  where h(t) represents the impulse response corresponding to the product HTU) • HRU)> n (t) 3  and n (t) represent the inphase and quadrature baseband components of the narrowband Gaussian c  noise, respectively. Both n (t) and n (t) have the same variance, equal to C*Q = fy, which is s  c  the received noise power as well, since both have mean equal to 0. In deriving Eq. 2.11, it has been assumed that HRU) is significantly wider than the bandwidth of the fading process, as this is the case for almost any practical application. However, it should be pointed out that for the computer simulations, the performance evaluation results which are presented in Chapter 3, any effects of this filter on the fading process will be included. Furthermore, if HTU)  *  AN<  HRU) are chosen so that h(t) satisfies Nyquist's first criterion, we obtain [2] h(kT )={\ a  i f  <• 0  *= ° otherwise.  (2.12)  Hence, the sampled at t = kT signal y* = y(kT) is given by Vk = (fl + jf?)c e  + n (kT) - jn (kT)  ifi  s  k  where f[ = f*(kT),  f£ = f (kT).  (2.13)  c  The samples y (0 < k < L) are then fed into a decoder  Q  k  for further processing in order to recover the transmitted symbols Q and subsequently the transmitted information sequence a . p  k  2.3. Derivation of the Optimal Decoder Structure The optimal, in the maximum likelihood sense, decoder must choose the data sequence a =  p  p  °li 2' a  p  •••> L a  which maximizes the probability density function (pdf) f [y | C(a),p] with  V = bi>2/2, —>yi]- Using well known probabilistic identities, f [y \ C(a),p] can be expressed as [8] L  f[y\C(a),p]  =Hf[  W  | y.--i,y.--2,...!to,?(3),/>] .  (2.14)  i=l  For mathematical convenience, we will define the variables e{and e  k  e[ = fi + nl - ^ f " )  .  4 = f? + n? =  kit I Section 3.  as ) . |cfc|  13  (2.15)  Chapter 2.  where Re(-) and Im(-) denote real and imaginary part, respectively, and * complex conjugate. From Eq. 2.12 it follows that, n{(k) and n%(k) are given by l  n (k)  =  + n (fcr)Im(c£ -")  n.{kT)Be(cje-»)  e  \c \  ^  e  2  k  and Q  n  {k)  =  n.(kT)hn(cte-iP) - n (fcr)Re(c£ -") e  |c*|  ^  e  2  Since nc(Jtr), n (kT), f£ and fff are Gaussian random variables, from Eq. 2.15 it follows that s  ej. and e® are also Gaussian random variables. Furthermore, as it is shown in Appendix A . l , they have the same autocorrelation function, given by R [C(a),(k-i)T}=E{e{e{}  = E{eQe?}  e  .. = R (k-z)+  o* 6(k-i)  <- > 2  G  f  18  2  where S(-) is the Kronecker 6-function. From Eq. 2.18, it can be seen that for multi-amplitude signal constellations, e.g., QAM, the autocorrelation properties of e{ and e® depend upon the transmitted sequence. For one single amplitude constellations however, e.g. M-PSK and 7r/4—shift DQPSK, where |ct| is constant, ej[ and 2  are independent from the transmitted  sequence C(a). The derivation of the optimal detector, is based on the following theorem from probability theory [8]. Theorem: For the zero mean Gaussian random variables n/, n;_ n / _ , n o with covariance l5  2  Ri-j = E{ninj}, the conditional pdf of n/ given n/_i,n/_2,...,no, is given by C(n/ | n/_!,n/_ ,...,no) =  ,  2  v  exp 7  ^  -{n  - n})  2  x  £  (2.19)  2  In Eq. 2.19, £ j is the minimum mean square prediction error (MMSPE), while n} represents 2  the prediction of n\ based on the previous samples n/_i,n;_2,...,no and is given by  Section 3.  14  Chapter 2.  where pij are the linear prediction coefficients, with / being the order of prediction. The MMSPE can be calculated from the following Eq. (2.21) j=i  Based on the above theorem, the pdf which the optimum detector must maximize, can be rewritten as follows {[y\C(a),p] = fl ^ e x p  1  Re(y c' e-») k  k  f  "2^1  -  * k  "^  jr  N-,f  i=i  1 flm(y cJe->") ^  (2.22)  R*(v*-«S_,e-»)  t  exp  Be(Vk-iCl-i*-»)  T  P M  2a\  In the above equation, a\ = cr| [C(a)] represents the minimum mean square prediction error of (0 < k < L; 0 < * < k) the linear prediction coefficients. The  order k, and p i = Pfc.i [C(a)] k>  values of these coefficients, depend upon both the BpT product, the fading model assumed, the operating Signal-to-Noise ratio (SNR) , the ratio of powers of the fading signal and the Gaussian 3  noise, i.e., Rf(0)/o~G[C(a)], and for multiamplitude signal constellations, on the transmitted sequence. Maximization of the pdf in Eq. 2.22 is equivalent to minimizing the function 2  + Im{y c;e-^) fc  * \  (2.23)  mi I m}y {y _ :_,e- "'l fc  J  iC  The prediction coefficients can be calculated by solving the following set of equations [8] P \C(a)] = D [R \C(a)]] k  k  (2.24)  1  k  where P [C(a)j is a [k x 1] matrix, D is a [fc x 1] matrix and R j [C(a)] is [k x k] matrix with k  k  k  S P ( 3 ) ] = [p ,i[C(a)],p [C(a)],...,p [C(a)]] fc  K  M  = [i2/(l),-R/(2)  Note that SNR=Sj/JV , where E, is the energy per transmitted symbol. 0  Section 3.  15  T  M  i2/(fc)]  T  (2.25)  Chapter 2. with T denoting transpose and  Rf(0) + r^r Rk[C(a)) =  R (i)  Rf(l)  R (0) + r£^ f  R (k)  Rf(k-l)  f  Rf(k)  • • •  f  ••• •••  Rf(k-l) R (0) + f  (2.26)  A  From the above, it is clear that as the SNR increases, the fading autocorrelation terms in the matrix of Eq. 2.26 become dominant over the noise terms a^/lcif, and hence, the values of the prediction coefficients will eventually converge to asymptotic values. These values obtained for SNR —> o o , will depend only upon the fading process. Furthermore, from the matrix shown in Eq. 2.26, it is evident that for single amplitude signals such as M-PSK for which |cfc|  is  constant, the prediction coefficient values will not depend on the transmitted sequence. As numerical examples, for prediction order up to four, the prediction coefficients, as well as the MMSPE, for B T = 0.125 and B T = 0.0125 are plotted in Fig. 2.4 and 2.5. Evident F  F  from bothfiguresare the asymptotic (as SNR —• o o ) values for all prediction coefficients shown, as well as the constant values of the MMSPE for high SNR. Furthermore, as it was expected, the MMSPE values decrease as the order of prediction increases. This clearly indicates that a system employing a higher order of prediction will perform better than the same system utilizing a lower prediction order. One more interesting observation is the quantizing effect on the prediction coefficient values for decreasing B T. This behavior could be important for F  practical implementation of the detection algorithm, as the same coefficient values can be used over a relatively wide SNR range. Returning back to Eq. 2.23, it is shown in Appendix A.2, that maximization of f [y \ C(a), p], with respect to the transmitted information sequence C(a), is shown to be equivalent to Section 3.  16  Fading model: land mobile, B T=0A25 F  Prediction order - 2  Prediction order - 4  11111111111111111 M 11111111111111111111  4 3\ ; 2\ ; 1 — \ ——— 0 \ *<> -1 \ X \ -2 \ \ \ -3 \ \ ^ \ -4 \ -50 5 10 15 20 25 30 35 40 45 50 '•^»  1  n  —  1  0  5  1  1  1  1  1  1  1  1  •  15 20 25 30 35 40 45 50  10 (a)  P22 P21  SNR(dB)  (C)  [J ~ 44  r  43  ;  SNR (dB)  P42 P41  Minimum mean square prediction error (MMSPE)  Prediction order - 3 II 111111111111111111111 ij  A "•A •A  •A\ •A •\  »\ \\ \\ \ ••Si:  0 5 10 15 20253035404550 (b)  P33  0 5 10 15 20 25 30 35 40 45 50 (rt) O r d e r = 4 SNR (dB) ' O Or rd de er r= =2 3  SNR (dB)  v  P32 P31  Fig. 2.4. Prediction coefficients and MMSPE for land mobile channel with BFT = 0.125; (a): 2  3.  nd  order prediction, (b): 3  rd  order prediction, (c): 4* order prediction, (d): MMSPE.  17  Chapter 2. Fading model: land mobile, B r=0.0125 F  Minimum mean square prediction error (MMSPE)  Prediction order- 3 I  '•  •  ll  11  \  —  A -  \  \  1  \  \  \  •  V  \  20 40 60 80 100 120 P33 (b) £ 33 SNR (dB) P32  20 40 60 80 100 120 (*) Order=4 SNR (dB) P31 ' Order=3 Prediction coefficients and MMSPE for land mobile channel with O BFT = 0.0125; rder=2 v  Fig. 2.5. (a): 2  Section 3.  nd  order prediction, (b): 3" order prediction, (c): 4 1  18  th  order prediction, (d): MMSPE.  Chapter 2. minimizing the following function L-kPk+j,j[C(a)]p +j,i+:[C( )]  E j=0  a  k  +  + (2.27)  =1  k*|  *2+,[C(a)]  S  with the terms p^o defined as pjt = - 1 , V 0 < k < L. Several interesting observations can i0  be made with reference to the quantities given in Eq. 2.27 •  In a Rayleigh fading environment, where f  1  = 0, the last two terms become zero, and the  expression to be minimized becomes considerably simpler. Furthermore, the last term is independent of the encoded symbols c* and can be removed from the minimization process. •  In the first term, Re(t/jtj/^_) and I m ^ y ^ ) represent the real and imaginary parts of the t  product between the received sampled signal y^, and the complex conjugated iT seconds delayed version of j/*. Clearly, this operation represents differential detection with a delay element of iT. It can be seen, that differential detectors with delay elements up to kT are involved. Since the detection of y involves all the terms yjt-i, Vk-2, •••> J/o» this corresponds k  to multiple differential detection, where differential detectors with delay elements T, 2T,... up to kT are employed. From now on, we will be referring to a differential detector with delay element kT, a k •  th  order differential detector.  The second term represents, essentially, squared envelope detection  term  contributes to the detection process when multiamplitude signal constellations —such as QAM or multiamplitude PSK— are used. However, for single amplitude signals, where |c,t| is constant, this term can be removed from the decoding process, since it's clearly 2  independent of any transmitted information sequence C(a). Section 3.  19  Chapter 2. •  Finally, the third term represents the coherent detection component and requires an estimate of the initial phase p of the modulator. This requirement is very different from the carrier recovery requirements of a (conventional) coherent receiver for the fading channel. These receivers require an estimate of this initial phase in addition to the phase jitter introduced by fading. This is far from being a trivial task, considering the dynamic behavior of the phase jittering, especially when dealing with fast fading. Due to practical limitations, these estimates are usually very inaccurate and result in poor performance. Note that this coherent detection term, is present for Ricean fading channels, while for Rayleigh fading where f  1  = 0, it is removed from the decoding process.  However, even for Ricean  fading, elimination of the coherent term from the algorithm results in minimal performance degradation. Based on the above observations, a more detailed block diagram for the realization of the detector derived is illustrated in Fig. 2.6, where d,(fc) = ytj/jt_,, i.e., it represents the output of the differential detector with delay element equal to iT. Clearly, the three detection subsystems of this optimal decoder are the Multiple Differential, Envelope and Coherent detectors.  di(k) . Multiple Differential Detector  d (k) k  e WD  _C '35  Envelope Detector  A  a.  8 «-  Cu  Coherent Detector Fig.  Section 3.  ye  jp  k  2.6. Detailed block diagram of the optimal decoder.  20  Chapter 2. The decoding algorithm described by Eq. 2.27, requires the calculation of coefficients involving the terms p*,,-[C(a)] and ^[^(a)]  {I < k < L). In total there are ^{^+1) ^  erms  for pk i, and L terms for al to be computed. For long sequences, i.e., large values of L, the t  processing load (computational complexity) required for the calculation of these parameters, as - well as the memory storage necessary to store all terms, becomes unpractically high. Although the use of more computationally efficient and less memory dependent recursive algorithms, such as the Levinson algorithm [45], can reduce the processing load, the overall complexity will still be high for long sequences . However, this complexity problem can be resolved, by adopting 4  the following approach  •  The maximum order of prediction used can be fixed and equal to a relatively small integer z. If 2 is appropriately chosen, it is expected that this truncation will not result in any significant loss of performance improvement presented by the maximum likelihood form of the algorithm. This expectation stems from the fact that strong correlation usually exists between samples of the fading process with distance of only few symbols. Consequently, terms with greater distance will not contribute significantly to the prediction process, and thus can be dropped. The value chosen for the maximum prediction order z will depend more on the value of the BpT product and less on the type of fading model assumed.  •  The decoding algorithm can be assumed to have been initialized just prior to normal operation, by a known information sequence of length z, e.g., the all zero sequence.  Following the above approach, it is shown in Appendix A.3 that maximization of Eq. 2.27,  Especially with regards to the computational complexity, a sequence of more that about 10 symbols is considered to be long.  Section 3.  21  Chapter 2.  is equivalent to minimizing the following function x  L  OF  - o r r My y*_.)Re(c c* _<) + Im(y ^_,)lm(c c* _J „  (jr.c(a),p) "  2  t  2. fc=l E 1=1  fc  fc  t  t  rT^i,  fc  ^  | || _.|2 2  Cfc  Cfc  C ( a )  J  +  + Er&r,, p7(a)]+  (2.28)  t  . *=i l *l c  ^ ^ - " )  +  fc=i  where  A  t  |  t  [  g  (  5  )  ]  l *l c  ^  Pz,i [C{a)]pz,i+j [C(a)] m  9fl  m  —  z  and  _ f  ™={V ~  k<L-  2,  7. f T  [L — k,  k >  L-  *  ( -30) 2  The quantity to be minimized, as it is given by Eq. 2.28, is general enough to accommodate multiamplitude signal constellations, such as M-ary QAM. Furthermore, for single amplitude signals, such as Af-ary PSK, the second term is constant, so it can be eliminated. Thus, in this case, the expression to be minimized simplifies to F  Q  _  _ A y Refyfci/i^.) Re (c ^_-) + Im(i/ y^_,)lm(q4_.)  (y,C{a),p)  2  t  t  2-^,2-^  \ru\ \ri.  t - i .=i  l *l  2  I  2  \C(a)] +  p 2  c  ( 2  3 1 )  2.4. Recursive Algorithms Section 3.  22  The algorithm can be implemented recursively by keeping an accumulated sum, which  will be called accumulated metric, and updating it as every new signal sample is fed in.  Chapter 2. The incremental quantity added to the accumulated metric will be referred to as metric. Two possible forms for this metric can be derived from Eq. 2.23 and 2.28. Limiting the prediction order in Eq. 2.23, to a maximum equal to z, and removing the outer summation operator, we obtain the following metric expression R  ^ r i  e  i  -  'l[^(a)]  <  - F) - F(I -  ±p.,<m)  +  NI  i ^ m i + j  r^TTTP  " £P^.'Ra)j  5  2  (2.32)  Furthermore, from Eq. 2.28, it is fairly easy to see that a second metric form is the function  ( j  "  \c \ \c -i\  h  c ( 5 ) , p )  , |y*| +  2  2 r  2  k  k  r - ^ M , Re(yt^e-->>)  7-^2^,* R  a  ) J +  l *l  j  -  ^  A  A  3  (  2  3  3  )  ,  *,*L '( )J c  a  |Cfc|  c  which is simplified to yield the following for single amplitude signals AT'  - V Mv>vUM**-d+ 2  ' '  h  {y c(Z) p)  +  lrn(y yUMc cU k  icki'M  ;—T2  l *l  k  +  2 (  2  3  4  )  *,fcl ()J •  A  C7 a  c  The recursive process has to be initialized before normal detection operation is possible. This can be done during the receiver initialization time, by feeding the algorithm with a number of signal samples corresponding to a known sequence, such as the all-zero sequence. For uncoded systems, the number of samples needed for initialization equal to z, the maximum prediction order used, whereas for systems employing convolutional coding, it also depends on the constraint length of the code used.  2.5. Conclusions In this chapter, the optimal decoder was presented, for trellis-coded or uncoded PSK and QAM systems, possibly employing differential phase encoding, operating in channels corrupted by multiplicative frequency nonselective Rayleigh or Ricean fading, as well as AWGN. It was Section 5.  23  Chapter 2.  shown, by using maximum likelihood detection arguments, that its structure can be realized by employing multiple differential detectors, an envelope detector, a coherent detector and a sequential algorithm. Finally various reduced complexity versions of the optimal detection algorithm were derived, in order to facilitate practical implementation, and recursive formulas for metric computation, used in receivers employing sequential decoding, were presented.  Section 5.  24  3. Computer Simulation Performance Evaluation: Software System Description and BER results 3.1. Introduction The purpose of this chapter is twofold. Firstly (in Section 3.2), it presents a somewhat detailed description of computer simulation software which can be used to evaluate the BER performance of a great deal of digital communication systems, under various channel conditions. All programs used, were developed by the author of this thesis, since no communication systems simulation software was provided with BloSim. This library of programs is currently being used by several graduate students in the Department of Electrical Engineering at UBC. Secondly (in Section 3.3), it presents the BER performance evaluation of various modulation schemes, operated in fading channels, employing the receiver structures derived in the previous chapter. The conclusions of this chapter can be found in Section 3.4.  3.2. Software Simulation Model Description The software needed for the system evaluation was implemented using the BloSim general simulation software package. BloSim was written in C by Dr. D. G. Messerschmitt of the Department of Electrical Engineering and Computer Science, at the University of California Berkeley [46]. This software package integrates simulation blocks written in the C programming language, providing only a kernel which supervises the input to and output from such blocks. Use of the C language not only frees the programmer from the constraints imposed by any special simulation language, but also provides great versatility in the implementation of complex algorithms. BloSim version 2.0 and 3.0, released on November 1988, was provided to the Communications Lab with a few examples of simulation blocks, but without any communication systems software. Such software is now available as a product of the research work reported in 25  Chapter 3. Information Source Block  TxFllter Block  Transmitter Block  : real ) • : complex BER counter  Jl Channel Block  RxFllter Block  Receiver Block  Fig. 3.1. Functional block-diagram of the communication system in software.  this thesis, covering a large number of modulation formats, filters, channel models and receiver structures. Using BloSim terminology [46], the simulation blocks are referred to as "stars". They make use of BloSim macro definitions which provide versatile and easy to use input/output functions in the form of FIFO (First In First Out) buffers. Connections between blocks are described in a text file called the "universe file". Through this file and the use of macros for parameter entry, numerical values for specific block-parameters (model parameters of interest) can be entered at run-time, thus considerably minimizing the need for re-compilation between simulation runs. The algorithmic modeling of the digital communication system discussed in the previous chapter is described in the following list of functional blocks, an illustration of which is presented in Fig. 3.1. Sources for the BloSim "stars" implementing these blocks can be found in Appendix C. •  The information source block providing a random bit-stream of O's and 1 's to the transmitter input  •  The transmitter block transforms the incoming bit stream to two outgoing streams of samples/symbol, one for the inphase channel (I-channel) and one for the quadrature channel (Q-channel).  •  The shaping transmit (Tx-) and receive (Rx-) filters perform the filtering of the signal samples according to a desired spectrum shape. Nyquist yfa raised-cosine filters are used  Section 2.  26  Chapter 3.  both at the transmitter output and at the receiver input. •  The channel block simulates the distortion imposed by the type of channel chosen. AWGN, as well as frequency nonselective Rayleigh and Ricean fading channel models were considered.  •  The noncoherent receiver block implements an "integrate and dump" FM-discriminator based or a sampling-differential type of receiver, operating on signal samples from the channel output.  •  The receiver block implements various detection algorithms using the receiver structures presented in Chapter 2.  It should be pointed out that, as is customary in computer simulation, in the system modeling in software, the baseband equivalent was used for all signals. According to this transformation, a modulated signal s(t) = Acos(2irf t + (j>) c  (3.1)  with A denoting the amplitude of the signal, f the carrier frequency and <f> carrier phase, has c  the baseband equivalent signal [2] u(t) = A[cos <f> + j sin <f>) = At?* .  (3.2  The implementation details for the above functional blocks are presented next.  3.2.1. The Information Source Block The source output is used as the input to the transmitter. Desirable features are the highest possible randomness in generating bits 0 and 1, i.e., getting a 0 or a 1 with probability as close as possible to 0.5. To implement this bit stream generator, a pseudo-random number generator existing in the system's C function library was used. This function returns numbers uniformly distributed in the interval [0,1.0). It uses the well known linear congruential formula X +i = (aX + c) mod m , n  Section 2.  n  27  n> 0  (3.3)  Chapter 3. with a and c constants defined in [47]. Using this formula, the generator's period is for all practical purposes "infinite" . The modulus constant m is simply 2 , providing a 48-bit result, 5  48  treated as a double precision floating point number. The bit stream generator maps the two desired alphabet symbols, namely 0 and 1, on the two intervals resulting from the division of the initial output interval [0,1) to two equal subintervals: [0,0.5) and [0.5,1.0). Bit frequency tests performed demonstrated a probability ratio very close to the desired 0.5 even for small samples, e.g., for 4096 bits, 2045 0's and 2051 l's is quite typical.  3.2.2. The Transmitter Block The baseband transmitters implemented fall into two general categories: a) uncoded and b) trellis-encoded, single- and multi-amplitude QPSK, using absolute phases or differential phase shifts (7r/4-shift DQPSK). The signal phase of transmittersfromthefirstcategory always conveys two information bits (one out of four possible phase changes) while the amplitude levels the rest. For the second, encoding is performed using a convolutional encoder with pre-compiled lookup tables. The tables provide a one-to-one mapping of convolutional encoder outputs to points on the two dimensional signal space. Examples of two codes, which are also used in this research, are the 4-state rate-1/2 7r/4-shift DQPSK and the 8-state rate-2/3 7r/4-shift 8-DQAM. Their corresponding signal-space and signal number assignment is illustrated in Fig. 3.2. Both convolutional codes shown were chosen since they are the best, i.e., give the best performance in AWGN, since they provide the maximum free Euclidean distance, for the code complexity considered [40].  5  Strictly speaking, this formula using the constants defined in [47], provides pseudo-random numbers. Nevertheless, for all practical  purposes, the period of the sequence of numbers produced can be regarded to be "infinite". For this reason, from now on, the uniformly distributed samples generated by this equation will be considered to be "random".  Section 2.  28  Chapter 3.  3.2.3. The Shaping Filters and the Filtering Operations 3.2.3.1. Transmit (Tx-) and Receive (Rx-) filters  In the context of software simulation modeling, the following two types of pulse shaping filters were employed [10]  Q  I  uu. r»Aoc»i/ (a) jt/4-shift DQPSK. /  \  ,A  Mapping of transmitted symbols differential phases, t0  K  Mapping of transmitted symbols to differential phases & amplitudes. (b) 7t/4-shift 8-DQAM.  Fig. 3.2. Signal mapping and constellation for (a) 4-state rate-1/2 ir/4-shift DQPSK and (b) 8-state rate-2/3 x/4-shift 8-DQAM. Section 2.  29  Chapter 3. •  The Transmit (Tx-) Filter H (f), T  with 0 < | / | < (1 - a ) / j v  ^-^{fcOfc- )}]  - < w < & elsewhere.  1  •  The Receive (Rx-) Filter H (f), R  ( 1  (l-a)f  N  )  L  N  (3.4)  )f  a  N  <\f\<(l  + ct)f  N  (3.5)  elsewhere.  0  N  a  with 0 < | / | < (1 -  where f  +  is the Nyquist frequency and a the filter roll-off factor taking values  = &  between 0 and 1.0. Both filters are y/a raised cosine filters with the Tx-filter having an amplitude equalizer so that their overall transfer function # T ( / ) . / / R ( / ) satisfies Nyquist's first criterion for pulse transmission. This assumption was also used in Eq. 2.12. In addition, as well known, this split is optimum in the sense that, in an AWGN channel, it provides maximum signal-to-noise (SNR) ratio at the output of the Rx-filter [48].  3.2.3.2. The Filtering Operations When computer simulation is employed, it is easier to realize the signal filtering operation in the frequency domain, rather than in the time domain. Furthermore, all signals and transfer functions must be represented by their discrete time domain and discrete frequency domain equivalents. If such a signal has a period, or equivalendy for our purposes here symbol duration T, the parameter of interest is the number of samples per symbol duration, n . A continuous-time s  signal s(f) which is, say, a series of pulses, will be represented in the discrete-time domain, by the signal s'(kT/n ). Figure 3.3 is an example showing an unfiltered BPSK signal produced by s  the sequence 1011, in the discrete time domain, with n = 4. s  Section 2.  30  Chapter 3.  1  Amplitude  o-  +H f 0  1 2  Discrete Time  5 ?  3  8  9 10 11 12 13 14 15  kT 4  -1-  O-  Fig. 3.3. Example of BPSK signal in the discrete time domain for the sequence 1011. The number of samples/symbol is 4.  The complex Fast Fourier Transform (FFT) [49] was used as the transformation algorithm between the discrete time domain and the discrete frequency domain. The FFT, as it is well known, is a discrete form of the Fourier series expansion and not the Fourier transform. However, if the sample size is sufficiently large, the results obtained are very close to the output from a Discrete Fourier Transform (DFT). The advantage presented by the FFT, as compared to the DFT, is that it is much faster due to its lower computational complexity. The signal s'(kT/n ) is therefore transformed by means of an FFT, to S'(kn /T) 3  s  in the  discrete frequency domain. Then, both real and imaginary parts of S'(kn /T) are multiplied by s  the discrete-frequency equivalent of the desired filter transfer function, e.g., for Hx{f), this is H'(kn /T).  Alternatively, an inverse FFT transformation yields thefiltereddiscrete-time signal  s'(kT/n ).  In this process, the discrete frequency equivalents of the filter transfer functions  3  s  (Eq. 3.4,3.5), i.e., H (kn /T) T  s  and H' (kn /T) R  s  must be constructed. In doing so, the Nyquist  frequency fx must be calculated for the discretefrequencysample available. From Nyquist's sampling theorem it is well known that for a signal having spectral components up to frequency W , the sampling frequency must be at least 2W for no aliasing; note that this factor of 2 also implies 2 samples/(signal period) [2]. However, NRZ type of signals like the BPSK example of Fig. 3.3 do not have a finite bandwidth, making aliasing inevitable. Nevertheless, spectral components of the NRZ signal decrease rapidly in amplitude for higherfrequencies[10]. This Section 2.  31  Chapter 3. in turn implies, that as n increases, the errors caused by aliasing will decrease. In order to s  perform thefilteringoperation in the discrete frequency domain, we have to calculate the index JCN into the array of discrete frequency components, corresponding to the Nyquist frequency fa. For this purpose we define the simulation bandwidth as  B  8  = jr = n3fs  where T = jj is the symbol duration and /  s  (3.6)  denotes the Baudrate (or symbol rate). The  simulation bandwidth corresponds to the entire discrete frequency sample provided by the FFT, so hp? is easily calculated by the following equation  with N denoting the total number of discrete frequency samples. Then, knowing that N = n N , s  3  s  where N is the number of symbols transmitted, substituting in Eq. 3.7 we obtain:  k  N  N = j  •  Using this value which corresponds to the minimum Nyquist frequency  (3-8) the index of any  other frequency desired can be easily calculated. As for a practical value for n , 16 was found s  to be a very satisfactory compromise between desired accuracy and required memory.  3.2.3.3. Examples of Typical Simulation Graphs  This subsection, presents the following three typical examples of plots generated by the developed computer simulation programs: 1) Filter transfer function, 2) eye-diagram and 3) state-space diagram. In Fig. 3.4 the full raised cosinefiltertransfer function H(f) is given for various values of a. Section 2.  32  Chapter 3. x/sin(x) equalized, root of raised-cosine transfer function .1 1 • 1.  i, H < 'r  9l =0.  t=0  0  r' \ \  -  2  -  1  0  f/f  1  2  N  Fig. 3.4. Transfer function of cascaded transmit and receive filters for different values of the roll-off factor a.  In Fig. 3.5, the 2-symbol duration (i.e., 2T) eye-diagram of a QPSK signal simulated by using the discrete-frequency filtering method described, is shown. The value used for the roll-off factor is 0.5. The two sampling instants are revealed by the points through which all signal trajectories pass. Notice that no inter-symbol interference (ISI) exists at the ideal sampling instant.  l-channel eye, alpha=0.5 i 1 • i •i•i •i •i •i •i • i • i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1  •  1  1  1  •  1  1  •  1  •  1  •  1  •  1  •  1  • -  1  •  1  •  1  •  1  •  1  •  1  1  •  1  1  • '  1  •  1  •  1  1  1  1  1  •  1  1  -  1  1  1  • '  • '  1  • '  1  • '  1  • '  1  • '  1  • '  1  • '  1  1  1  - '  Fig. 3.5. The simulated 2-symbol eye-diagram for QPSK using roll-off factor a = 0.5.  Section 2.  33  Chapter 3. Finally, in Fig. 3.6, the baseband signal trajectories (state-space diagram) of a 7r/4-shift DQPSK system employing raised cosinefiltersof a = 0.5 and a = 1.0, are presented.  -1.00  0.00  1.00  -1.00  -0.50  0.00  0.50  1.00  Fig. 3.6. Baseband signal trajectories of jr/4-shift DQPSK for a = 0.5 and a = 1.0.  3.2.4. The Channel Block As previously mentioned, the channel considered in this thesis is assumed to impose two types of interference: a) AWGN and b) multiplicative frequency nonselective Rayleigh or Ricean fading. a) The white, i.e., uncorrelated, and Gaussian noise, which is also needed for simulating the fading interference, is generated by making use of the same C-library "random" number generator as in the case of the "random" information source. The uniformly distributed (i.e. "random") numbers are fed as input to a second routine which transforms their probability density function (PDF) into a Gaussian PDF. A commonly used method of performing this transformation, is by taking advantage of the Central Limit Theorem [8]. According to this theorem, a sufficiendy large number of uniformly distributed samples (e.g., 12), can be added up in order to produce one Gaussian sample. This is quite straight forward to realize, but almost 10 times slower than another method which uses the Box-Muller method for producing Section 2.  34  Chapter 3.  -4000 Fig.  -2000  0  2000  4000 samples  3.7. Autocorrelation of 4096 Gaussian white noise samples; variance = 100.  2 Gaussian samples from 2 uniformly distributed numbers [49]. Both^methods werefirsttested via the autocorrelation of the sequences they produced and yielded statistically identical results. A typical autocorrelation function showing the sufficiently high degree of randomness obtained even for small samples, is presented in Fig. 3.7:  The noise samples were found to follow  faithfully the Gaussian distribution function. This is clearly demonstrated in Fig. 3.8, where the cumulative probability distribution function (CPDF) of the computer simulated Gaussian noise is compared to the theoretical Gaussian CPDF. Verification of the software model for the Gaussian noise generator was also possible from the simulation of modulation schemes like QPSK, the — : Theoretical Gaussian CPDF o : Computer simulated Gaussian noise CPDF  C  o 3 LL  3  JD  a XI co £3 O  JS 3  E  3  o  -20  -10  0  10  20  50  Noise sample value  Fig. 3.8. Comparison of computer simulated noise C P D F to the Gaussian theoretical; variance = 100.  Section 2.  35  Chapter 3. performance of which is well known in the AWGN channel. An excellent match of the simulated results to the theoretical curves expected (e.g., 0.6% away from the theoretical value for QPSK at E / N = 7dB ), demonstrated that the simulated noise model presented the desired accuracy 6  b  0  for all practical purposes. In the complex baseband model, AWGN interference corresponds to having two independent streams of noise samples added to corresponding transmitted inphase and quadrature signal components. b) The frequency nonselective fading was modeled in baseband as a filtered complex Gaussian noise process.  Following [5], a process with Rayleigh distributed envelope and  uniformly distributed phase in  [-7r,7r)  can be produced if we assume Gaussian samples for the  inphase and quadrature signals in the two dimensional signal space. This complex noise process, which can be mathematically expressed as  (3.9)  f(t) = f (t) + jfQ(t) I  is essentially the baseband equivalent of a Rayleigh faded carrier signal. The real and imaginary part are generated byfilteringtwo independent Gaussian noise processes n (f) and n^(t) with 7  identicalfilters.The transfer function chosen for thefiltersdetermines the type of fading assumed. A table of several filter transfer functions together with corresponding fading autocorrelation functions can be found in [6]. As in Chapter 2, in the computer simulation, the uniform and land mobile fading models were also employed. Using the methodology described in [2], it is shown in Appendix A.4 that the baseband model for the faded signal is the complex multiplication of the baseband information signal  x(t) =  Here £  6  x (t)+jx (t) I  Q  (3.10)  denotes the energy per transmitted bit and JVo the' one-sided power spectral density (PSD) of the white Gaussian noise.  Section 2.  36  f (t)  n(t)-jnjt)  s(t)  l  2'  fit)  1  \  x (t) r  Fig. 3.9. The baseband Rayleigh fading and AWGN channel model; s(t) represents the transmitted signal.  with the complex fading process / ( t ) . In other words, the received signal, corrupted by fading and AWGN can be expressed as x (t) = s{t)f(t) + n (t)-jn (t) r  s  (3.11)  c  which is illustrated in block diagram form in Fig. 3.9. For the uniform fading model, the fading filter transfer function is \H (f)\ F  2  = ( 257 * » 0 < | / | < B 10 elsewhere  F  ( 3 1  2  )  whereas for the land mobile fading model o  B  F  f - r - i  for 0 < | / | < B -  ^0  elsewhere .  F  = v/X is the maximum Doppler frequency (or fading bandwidth), with v denoting the  vehicle speed and A = c / / the wavelength corresponding to the carrier frequency f used. c  c  Eq. 3.12 and 3.13 give the general form of the filter transfer function. Thefiltergain, which is not indicated, can be varied according to the desired filter output power, i.e., the desired power of the fading signal. Section 2.  37  Chapter 3. dB  0  500  1000  1500  2000  2500  3000  samples  Fig. 3.10. Computer simulated amplitude of a faded carrier for the land mobile fading model.  The envelope R(t) and phase 6(t) of f(t) are given by  (3.14) 0(t) = arctan A typical faded carrier amplitude, obtained by means of computer simulation is shown in Fig. 3.10. The model used to produce the data for this plot is that of land-mobile fading. The statistical properties of the faded carrier samples produced by the computer simulation were verified by plotting the amplitude and phase CPDF's obtained, together with the theoretically expected. The results are depicted in Figs. 3.11 and 3.12. In Fig. 3.11, the curve plotted indicates the probability of the signal amplitude (envelope) being above the RMS value by as many dB as indicated on the x-axis. In the same figure, the Rayleigh theoretical CPDF is compared to the faded carrier envelope CPDF. The simulated faded carrier phase together with the theoretical CPDF for the uniform CPDF in the interval  [-7r,7r)  are plotted  in Fig. 3.12. The small deviation of the simulated values shown in Figs. 3.11 and 3.12, from the theoretical CPDFs, is due to the small number of signal samples used to generate these two plots. In the actual simulation runs the number of samples were typically 16 to 32 times more than the 8192 symbols x 16 samples/symbol used here. Section 2.  38  Chapter 3.  — : Theoretical Rayleigh CPDF • : Simulated faded carrier amplitude CPDF 10 j 3  3  n  °c b  J5 rt n o > rt 3  1  1  1  l —  T ' T —  1  1  1  1  1  1 ' |  0  10 •1  1  10 -2  10 -3  1  10 •4  10  rn  •j  i , 1 _ 1 _ I._^JL—,.1 i 10-40 -35 -30 -25 -20 -15 -10 -5 05 10 o  E  -8  3'  dB above RMS  Fig. 3.11. The CPDF for the envelope of the computer simulated complex fading process f(t). For comparison purposes, the theoretical curve has also been included. As it is well known, the Ricean fading channel model is realized by simply adding the original undistorted signal to the Rayleigh faded signal [6]. This simulates the direct signal path between base station and mobile. A block diagram of the Ricean fading model is shown in Fig.  O  u  o 3  JD '  b  J5  2  8  0.  >  I3  E 3  o  : Theoretical uniform phase CPDF • : Simulated laded carrier phase CPDF 1.1 T—i—i—i—i—i—i—i—i—i—i—i—r 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 i i i i i i i i i i i -0.1 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1 . 5 2 2.5 3 3.5 J  L  Faded carrier phase (rad)  Fig. 3.12. Comparison of the simulated faded carrier phase CPDF to the theoretical curve. Section 2.  39  Chapter 3. Original Signal  Rayleigh fading channel  Ricean faded signal 11  Direct path  K Fig.  3.13.  3.13. The generalized block diagram of the Ricean fading channel.  The additional parameter for this type of fading is the ratio of the direct signal power to  the power of the Rayleigh faded carrier, which is the previously discussed if-factor. We conclude this subsection by mentioning that in the actual computer simulation, the Tx-filter, the channel and the Rx-filter were all coded into one BloSim star. The reason for this is the fact that a large number of signal samples had to be accumulated in the block implementing the filters before they could be processed. This in turn caused BloSim to grow it's FIFO Input/Output queues between blocks in such a way as to limit the memory available for processing large numbers of transmitted symbols. By lumping the three blocks together, memory space was saved and the overall execution speed was reduced.  3.2.5. The Noncoherent Receiver Block As well known, noncoherent receivers can be classified into two major groups: a) the FM limiter/discriminator type, followed by an integrate-and-dump filter, and b) the differential detector type. The two techniques have been shown to be equivalent for the 7r/4—shift D Q P S K system, in terms of BER performance [50]. For both types of receivers, an amplitude (envelope) limiter is employed before any phase operations so that the computation will not be affected by thefluctuatingenvelope. In the computer simulation, the operation of the limiter was realized in software by dividing the complex sample by it's magnitude, effectively limiting the signal envelope to a constant value, which without loss of generality, was assumed to be equal to 1. Section 2.  40  Chapter 3.  3.2.5.1. The FM Limiter/Discriminator Receiver The FM-discriminator-based noncoherent receiver uses a discriminator to obtain the instantaneous phase change of the carrier. This carrier phase derivative, is the input to the integrate-and-dump filter which computes the carrier phase change between two symbol intervals. Assuming perfect symbol timing recovery , the timing signal will cause a "dump" 7  of the integrator output at the appropriate sampling instant, and will clear it to zero before the integration for the next symbol interval begins. As shown in Appendix A.5, the discriminator is modelled mathematically by the following equation [50] M i )  =  &  X  _ ^ y  (3.15)  where x denotes the inphase signal and y the quadrature signal. However, in order to use it in the discrete-time computer simulation, an equivalent discrete-time model must be obtained. The two time derivatives can be approximated by using any one of the well known formulas from numerical analysis. In our case, the forward reference formula [51] was used  em _ <t>(t + h)- j,® dt h  (316)  The smaller the step h the better the approximation. Using this approximation, Eq. 3.15 becomes A<f>{k) = Ay(k) • x(k - 1) - Ax(ik)  •  y(k - 1)  (3.17)  where k denotes the discrete time variable. Since A<f>(k) = <f>(k) — <f>(k — 1), Ay(k) = y(k) - y(k - 1) and Ax(k) = x(k) - x(k - 1), Eq. 3.17 yields <f>{k) = <f>(k - 1) + [y(k) - y(k - 1)] • x(k - 1) - [x(k) - x{k - 1)] • y{k - 1)  7  (3.18)  As previously mentioned, the problem of obtaining symbol synchronization in a fading channel is not addressed by the research work  reported in this thesis.  Section 2.  41  Chapter 3.  The discrete time step h which determines the approximation accuracy, depends on the number of samples/(symbol interval) n . A typical output, for the 7r/4-shift DQPSK system, from the s  simulated integrate-and-dump receiver block is shown in Fig. 3.14, clearly demonstrating the four possible carrier phase changes.  During the simulation runs it was observed that less than  32 samples per symbol yield great inaccuracy at BER levels below 10" . This relatively large 3  number of samples required limits the maximum number of transmitted symbolsfilteredin an FFT window due to limited computer memory. This is the most serious disadvantage of this model and it is why the second type — the differential detector — was used instead.  3.2.5.2. The Differential Detector Receiver  The differential detector computes the phase change between symbols by operating on samples from the inphase and quadrature channels. Using the same notation as in Eq. 3.17 and 3.18, we can write the output from the envelope limiter as  x(n) = cos [<f>(n) + 6] and y(n) = sin[<£(n) + 6]  T Samples Fig. 3.14. Integrate-and-dump output for 256 symbols transmitted.  Section 2.  42  (3.19)  2T  Chapter 3. where 0 is the carrier initial phase. Then, the phase change between two symbol interval can be calculated by the following two formulas W(n) = x(n)x(n - 1) + y(n)y(n - 1) = cos [<f>(n) - <f>(n - 1)] (3.20) Z(n) = y(n)x(n - 1) - x(n)y(n - 1) = sin [<f>(n) - <t>(n - 1)] . Clearly, W(n) and Z(n) are equivalent to the baseband inphase and quadrature components of a coherendy demodulated QPSK signal. This implementation of noncoherent detection does not present the inaccuracies inherent to the approximation of the phase derivative found in the model for the FM-discriminator, and thus it was employed in our computer simulation model.  3.2.5.3. BER Computer Simulation Results Actual computer simulation data obtained for both receiver models demonstrated their equivalence in terms of BER performance. The computer simulated BER performance of the uncoded QPSK modulation scheme, using a coherent maximum likelihood detector, for different values of the Nyquist filter roll-off factor a, is presented in Fig. 3.15. As it is evident from this figure, the match of the simulation data to the theoretical values is excellent, presenting only minor deviations (up to approx. 5% in the worst case) at an E /No of 10 dB. The reason for this D  is that the number of bits in error counted at these values of E /No, is considerably smaller (e.g., 0  less than 10) than that counted for lower E /No (e.g., more than 100). However, the minimum D  acceptable number of error occurrences during all computer simulation runs was kept at least 100. At a BER equal to 10" this translates into a confidence interval of a factor of about 20% 4  on the BER [52], which results in negligible uncertainty on the E /No scale. D  The simulation results for the noncoherent 7r/4-shift DQPSK scheme, using the differential detector described previously, for the same values of a as in Fig. 3.15, are presented in Fig 3.16. Theoretical values used for comparison were obtained from [53]. As in the case for QPSK, the maximum deviation from the theoretically expected appears at the high E /No end, being again D  in the worst case around 5%. The computer simulation results presented both in Fig. 3.15 and Section 2.  43  Chapter 3. Alpha«1.0 Alpha-0.7 Alpha.0.4 Sim., alpha=0.1 QPSK theoretical  — -B- -  ts - •  • - -B— -  A  Si  03  2  Q. k_  2 (  m  Eb/No (dB) Fig.  3.15. Theoretical and simulated BER performance of coherently detected  QPSK in AWGN. Nyquist filters with a = 0.1,0.4,0.7,1.0 are employed.  Fig 3.16 also prove the accuracy of the software model of the AWGN channel, and especially that of the white Gaussian noise generator block. 0  -5  -3  -1  1  3  5  7  9  11  13  15  Eb/No (dB) Fig.  3.16. Theoretical and simulated BER performance of ^/4—shift DQPSK in AWGN. Nyquist filters with a — 0.1,0.4,0.7,1.0 are employed.  Section 2.  44  Chapter 3. V-125Km/h, BT-0.0041 V-80Km/h, BT=0.0026 V«.40Km/h, BT-0.0013  .Q 03  JD O w — QL  k_  2  <5 m  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  SNR (dB) Fig. 3.17. BER of differentially detected x/4-shift DQPSK in flat Rayleigh fading. Nyquist filter a = 0.2, BT = B T. F  The BER performance of the noncoherent 7r/4-shift DQPSK scheme employing the differential detector receiver, as a function of the signal to noise ratio (SNR), in frequencyflat Rayleigh and Ricean fading channel conditions was also obtained via computer simulation. The obtained BER results with BpT and a as parameters, are summarized in Fig. 3.17 - 3.20.  10  vK  V-100Km/h, BT=0.033 V-80Km/h, BT-0.026 V-40Km/h, BT-0.013  -**I •.  sV :\  JO  \  (TJ -O  o Q.  2  • >  S  10  Qi  \  m  10  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  SNR (dB) Fig. 3.18. Differential detector in flat Rayleigh fading. Nyquist filter a = 0.2, BT = BFT.  Section 2.  45  Chapter 3. In Fig. 3.17, the performance of a 7r/4-shift DQPSK system employing raised cosine niters of a = 0.2 and for B T  products of 0.0013, 0.0026 and 0.0041 is presented. The presence  F  of error floors is evident. Theoretical results in [54] verify the accuracy of our computer simulation results. From Fig.3.18, it can be seen clearly, that higher B T  products will result in higher error  F  floors. On the other hand, different values of a do not seem to have any significant impact on the overall system performance. For example, by comparing Fig. 3.18 (a = 0.2) and 3.19 (a = 0.35), the BER results are almost identical for the same B T product. The explanation for F  this is that the considered fading signal bandwidth B  is much smaller than thefilter'sNyquist  F  frequency and thus the fading characteristic does not change. Finally, in Fig. 3.20, the performance of an a = 0.2 7r/4—shift DQPSK system with differential detector, operated in a Ricean fading channel is illustrated. As expected, the BER performance for K  = 10 dB is significantly better than that for K  dB  dB  — 0 dB.  10 *x— "V  V=100Km/h, BT=0.033 V«=80Km/h, BT-0.026 V-40Km/h, BT-0.013  :\ r V  \ < \  -Q CO -O  \  o  >  10  Q. v_ O k.  -V  k_  CD •»  m  10  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  SNR (dB)  Fig. 3.19. Differential detector in flat Rayleigh fading. Nyquist filter a = 0.35, BT = BFT.  Section 2.  46  Chapter 3.  10  10  CO  JO O i_  Q.  10  i  \\ -Vy-  —\  k_  2  l\ \\ \\ Y\  CD  m  A., A..  10  — \\  01  10  LrrJ  10  15  \  20  25  BT-0.041 BT-0.026 K-10dB, BT-0.013 BT-0.041 BT-0.026 K-OdB, BT-0.013  Fig. 3.20.  30 B—-e  35  40  45  50  SNR (dB)  A — A  B E R of differentially detected 7r/4-shift D Q P S K in flat Ricean fading. Nyquist filter a = 0.2, B T =  B T. F  3.2.6. The Receiver Block The receiver structures derived in Chapter 2 were used in both coded and uncoded digital modulation schemes. The following is a detailed description for the application of the proposed algorithms, implemented in a way that any coded or uncoded PSK or QAM signal constellation can be accommodated. It should be pointed out that the algorithms derived for simulating this particular receiver, were used almost without any alteration for the digital modem implemented in hardware. Section 2.  47  Chapter 3.  3.2.6.1. Coded Version of the Receiver  As presented in Chapter 2, the general form of the quantity to be minimized by the derived optimal algorithm, was given in Eq. 2.23 and for the reader's convenience is repeated here  M  k=l  TF  - ,§ ^  1  k  +  (  ^  T  ^  1  -  - F(.  - E  »,);  +  (3.21)  i=l  where £ represents the number of symbols transmitted, a\ the minimum mean square prediction error of order k, y  k  the complex received signal sample at t = kT, c* the complex signal  being transmitted, p the initial phase of the transmitter carrier and p*,, the prediction coefficients of order k. In a receiver implementing sequential decoding, a decoded symbol is output at every symbol interval . However, a delay of several symbols exists in the decoder which is 8  primarily a function of the constraint length of the convolutional code used. No beginning and end exists for the information stream, since no packetization is performed at this low system level. Consequently, the outer sum from 0 to L in Eq. 3.21 has no practical meaning for the implementation of the algorithm. In practice, the metric can be kept minimum by maintaining an accumulated metric which is updated as every new signal sample is received. This is the same technique used by Viterbi sequential decoders. Furthermore, the prediction order used, typically does not have to be greater that 4, since it turns out that increasing it further, results in negligible performance improvement. Hence, the truncated version of the algorithm, as derived in Eq. 2.32, should be used for calculating the accumulated metric. Again for the reader's convenience, it is repeated here  It should be pointed out that there is a fixed decoding delay of 5-6 times the code constraint length.  Section 2.  48  Chapter 3.  (3.22)  where z is the prediction order used. The metric described by Eq. 3.22 has to be minimized over all possible combinations of the transmitted symbols c*. As it will be explained shortly, this can be accomplished by constructing a trellis structure on the receiver end, different from the one used by the convolutional encoder at the transmitter. From now on, this special trellis decoder will be referred to as super-trellis, it's states as super-states and the signals associated to it's branches super-signals . Clearly, 9  the number of combinations of trellis transitions and associated signals depends on the number of states in the original trellis, as well as it's structure in terms of state connectivity. It is not difficult to recognize from the bracketed portion of Eq. 3.22 that the decoder operates on a sliding window of z past samples plus the current one. This implies that all combinations of possible transmitted symbols Ck,Ck-i, ...,cf.- must be tested. Each combination yields a z  branch (or transition) metric which added to the accumulated metric of every originating superstate, will yield the surviving transition to every merging super-state. The state presenting the minimum accumulated metric will be the origin for the decoding trace-back, as in the well known Viterbi decoding algorithm. Furthermore, it can be seen that the super trellis is generated from the branches of the original trellis code that exist between z+2 trellis nodes. The following example describes the steps involved in the derivation of the super-trellis structure for the simple 4-state trellis code illustrated in Fig. 3.21, for a prediction order of 1, i.e., z = 1. The code and signal phase assignment shown were used to test trellis coded 1/2-rate 7r/4-shift DQPSK using the proposed algorithm. '  For z = 1, the number of nodes required from the original  For simplicity purposes, only trellis codes with no parallel branches are considered here. However, the super-trellis generation procedure  can be extended to accommodate trellis codes with multiple branch transitions.  Section 2.  49  Chapter 3.  4 Fig. 3.21. Original 4-state trellis with signal numbers used as branch labels.  trellis structure is 3. The trellis with signal numbers assigned on each branch is presented in Fig. 3.22.  The transition from state 0 to state 0 and then state 2 shown in Fig. 3.22, will  yield the following on the super-trellis •  an originating super-state described by the combination of states on the first and second node, i.e., state 0 and state 0. A very convenient way to represent this combination is by regarding the two state numbers as the two digits of a base-4 integer (because of the 4 possible states). The convention that will be followed with regards to digit significance is the same for state and signal combinations: least significant digit to most significant digit, as  States  States  Fig. 3.22. Super-trellis generation for the 4—state code of Fig. 3.21 and z = 1.  Section 2.  50  Chapter 3. we go from the left side of the trellis in Fig. 3.22, to the right side. Hence the originating state will have the number (00)4. •  a merging super-state, described by the combination of states on the second and third node of the original trellis, having the number (20)4 and  •  a branch pseudo-signal (super signal) having the number (20)4. The base is 4 because of 4 possible signals (actually signal phases) on the constellation used.  The rest of the super-trellis transitions with accompanying super-signals are constructed by following exactly the same procedure, yielding the trellis structure shown in Fig. 3.23. It should be pointed out here, that the only difference between noncoherent schemes and coherent schemes, is that for the former, the signals assigned to the trellis branches represent phase changes (and possibly amplitudes), while for the latter the signals correspond to absolute phases (and possibly amplitudes). Hence, for example, the code of Fig. 3.21 together with the associated super-trellis of Fig. 3.23, can also be used for rate-1/2 coherent QPSK, in this particular case with the same signal assignment and numbering convention. As demonstrated by this relatively simple example, construction of the super-trellis can be very complicated, especially for higher complexity codes, constellations with large number of signals and/or large numbers for the z parameter (e.g., 3 or 4). For this reason, a recursive algorithm was derived and implemented in software, which constructs the super-trellis by tracing possible paths on the original trellis for z symbol intervals in the past. Connections between super-trellis nodes and super-signals associated to branches, are stored in tables. This recursive algorithm, given an original trellis code and a maximum prediction order, constructs all necessary tables at initialization time. The notation used for the super-signals, i.e., numbering by using as base the number of signals on the constellation, is of particular importance when dealing with multi-amplitude schemes. As pointed out in Chapter 2 (see Eq. 2.26), when the signal mapping uses multiSection 2.  51  Chapter 3. Super-state  Super-state  Fig. 3.23. The super-trellis for the 4-state code described in Fig. 3.21, for the case of z = 1. Note that both super-state and super-signal numbers are integers of base 4. amplitude signals, the prediction coefficients become dependent on the transmitted sequence. In fact, it is not difficult to realize that, the number of transmitted symbol combinations used for this computation is equal to (Number of signals)*. The prediction coefficients for every one of these combinations can be calculated at initialization time and tabulated in such a way that the super-signal number can be used direcdy as a row-index into the table. A column index can then return any one of z required coefficients. The C-language code has been implemented in such a general way, that any signal mapping using any convolutional code can be readily Section 2.  52  Chapter 3. accommodated by simply initializing constellation- and trellis-specific tables.  3.2.6.2. Uncoded Version of the Receiver From the branch metric used for trellis decoding, defined by Eq. 3.22, we can see that the algorithm itself does not necessarily imply the use of a convolutional code. This observation inspired the application of this fading-resistive technique to the uncoded, single-level 7r/4-shift DQPSK. Clearly, without coding and associated trellis, any signal becomes a possibility for the next symbol interval. Thus, all combinations of phase transitions for z symbol intervals in the past, namely cjfc,cjt_i,...,cjt_ , must be tested. The accumulated metrics now are not z  associated to any super-states but to the signal combinations themselves, and the decoding delay is z symbols, the maximum order of prediction used.  3.3. BER Performance Evaluation Results In this section, BER performance evaluation results for both trellis-coded and uncoded modulation schemes using the proposed algorithms will be presented.  All the results are  obtained by means of computer simulation software, which was described in Section 3.2.  3.3.1. Coded Schemes The codes and signal constellations used for both coherent and noncoherent coded schemes are the ones appearing in Fig. 3.2. For the coherent schemes, the absolute phases shown in Fig. 3.2 are transmitted instead of phase differences, as is the case for the noncoherent schemes. Fig. 3.24 shows the BER performance of 4-state trellis-coded QPSK, and Fig. 3.25 the performance of 4-state 7r/4-shift DQPSK, both operated in a Rayleigh fast-fading (BpT  = 0.125)  channel,  with prediction orders 2 = 2,3 and 4. The fading model assumed is the uniform fading model described by Eq. 3.12. The employed filters were raised cosinefilterswith a = 1.0.  The BER  results for both systems, clearly indicate that even for z = 2, the error floor caused by fading Section 3.  53  Chapter 3.  Order-4 Order=3 Order-2 Conventional Viterbi  SNR (dB) Fig. 3.24. BER of 4-state QPSK in flat Rayleigh fading (uniform model). Nyquist filter a = 0.35, BFT = 0.125.  is below 10". Furthermore, it can be seen that, the improvement in performance is very small, 4  beyond some value of the prediction order z (in this case beyond z = 4). Fig. 3.24 also demonstrates the complete failure of the Viterbi decoder using Euclidean distances as a metric. Figs. 3.26 and 3.27 present the BER performance of rate-1/2 4-state  10  Order-4 Order»3 Order-2  ^^^^ >>  10  !5  rc  -O  o Q.  g fi  -2  10 k> -3  10'  10  10  15  20  25  SNR (dB) Fig. 3.25. BER of 4-state x/4-shift DQPSK in flat Rayleigh fading (uniform model). Nyquist filter a = 0.35, B T F  Section 3.  54  = 0.125.  Chapter 3.  Order-4 Order-3 Order-2 Conventional Viterbi  SNR (dB) Fig. 3.26. BER of 4-state QPSK in flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0, B T F  = 0.125.  QPSK employing raised cosinefiltersof a = 1.0 and 0.35. Here, the land mobile fading channel was used, with BpT — 0.125. Comparing the results of Figs. Figs. 3.26 and 3.27, it can be seen that the system with a = 1.0 performs marginally better (at higher SNR) as compared to  0  Order=4 Order=3 Order=2 Conventional Viterbi  SNR (dB) Fig. 3.27. BER of 4-state QPSK in flat Rayleigh fading (land mobile model). Nyquist filter a = 0.35, BFT = 0.125.  Section 3.  55  Chapter 3.  Order-2 Order-1 Conventional Viterbi  0  5  10  15  20  25  30  35  40  45  50  SNR (dB) Fig. 3.28. BER of 8-state 8-QAM in fiat Rayleigh fading (land mobile model). Nyquistfiltera = 0.35, B T F  = 0.125.  the system with a = 0.35. This happens because the higher frequencies of the fading spectrum are attenuated more by the receiver filter. The next two figures present performance evaluation results for 8-QAM type of signals. In particular, Fig. 3.28 illustrates the performance of an 8-state trellis coded 8-QAM system, employing a = 0.35filtersin a land mobile Rayleigh fading channel, with Bf T = 0.125. It is clear that from the figure that with z = 2, the error floor has been reduced to below 10~. 3  In Fig. 3.29, the performance of an equivalent (i.e., same channel conditions andfilters)trellis coded 7r/4—shift 8-DQAM system, is presented. For a fair comparison, we have included the performance of a conventional differentially detected 7r/4-shift DQPSK system. Clearly, the 7r/4-shift 8-DQAM outperforms the conventional system, even for z = 2. In Fig. 3.30, the performance of a 4-state coded QPSK (a = 0.35), operated in a Ricean fading channel, with  = 10 dB, is presented. For comparison purposes, the performance of  the same system in a Rayleigh fading channel is also included. Notice how the performance curves of all these receivers (i.e., 2=2, 3 and 4) almost coincide for the Ricean fading channel. Section 3.  56  Chapter 3. o Order-3 Order-2 Order-1 Uncoded D Q P S K  i  •4  10 I—I—I 0  5  i  i  10  15  ,  1—I—I—I—I—I—I 20  25  1  30  1  35  1—I—L.  40  45  SNR (dB) Fig.  3.29. BER of 8-state x/4-shift 8-DQAM in flat Rayleigh fading  (land mobile model). Nyquist filter a = 0.35, BFT = 0.125.  Finally, Fig. 3.31 presents some BER performance results for a 4-state coded QPSK system, employing a z = 3 receiver and operated in a Ricean fading channel, with K$Q = 0 dB, 5 dB and 10 dB. In the samefigure,the two limiting curves for Rayleigh fading and AWGN channels, 0 Order-4 Order-3 Ricean, O r d e r - 2 Order-4 Order-3 Rayleigh, O r d e r - 2  SNR (dB) Fig.  3.30. BER of 4-state QPSK in Ricean versus Rayleigh flat  fading (land mobile model). Nyquist filter a = 0.35, B T = 0.125. F  Section 3.  57  Chapter 3. 4-state Q P S K ( A W G N ) K-10dB K.5dB K-OdB Rayleigh, O r d e r - 3  SNR (dB) Fig.  3.31. BER of 4-state QPSK employing 2 = 3, in flat Ricean  fading (land mobile model).  Nyquist filter o = 0.35, B T F  = 0.125.  are also included. As expected, with increasing values of the K-factor, the performance of the simulated z = 3 system systems, moves towards the performance in the AWGN channel.  3.3.2. Uncoded schemes  Fig. 3.32. BER of 7r/4-shift DQPSK in flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0, B T F  Section 3.  58  = 0.125.  Chapter 3. The BER performance of 7r/4-shift DQPSK was obtained for the Rayleigh fading channel. The simulation results for BpT = 0.125 and BFT = 0.0125 are shown in Fig. 3.32 and Fig. 3.33 respectively. The Nyquist filter roll-off factor is a = 1.0 and the fading model used is the land mobile. The reduction of the error floors is evident here also, especially for the fast fading (BfT = 0.125) channel. These BER results will be compared later on, in Chapter 5, with experimental BER measurements of an equivalent 7r/4-shift DQPSK system implemented in hardware.  Order=2 Sequential rx, Order-1 Diff. detected DQPSK  Fig. 3.33. B E R of jr/4-shift  DQPSK in  flat Rayleigh fading (land mobile model). Nyquist filter a = 1.0,  BT = F  0.0125.  3.4. Conclusions In this chapter, the development of a computer simulation software library which evaluates the BER performance of a wide variety of digital communication systems was described. Furthermore, by employing the developed software, the BER performance of various PSK and QAM systems operated in fading channels, employing the receivers proposed in Chapter 2, was evaluated. The obtained BER results have indicated that these receivers achieve significant Section 4.  59  Chapter 3. performance improvements, including large error-floor reduction, as compared to equivalent systems employing conventional detectors.  Section 4.  60  4. Design and Implementation of a Prototype 7r/4-shift DQPSK System 4.1. Introduction In this chapter, the design and implementation of a prototype 7r/4-shift DQPSK system will be presented. As illustrated in Fig. 4.1, the overall system consists of a 7r/4-shift DQPSK baseband generator, an RF modulator, the channel, an RF demodulator and digital-signal-processor (DSP) based baseband receiver.  7C/4-shift DQPSK Baseband Generator  Inphase  RF Modulator Quadrature  Channel (AWGN & Fading)  Transmitter  ^  Corrupted Signal  Receiver Fig. 4.1. Block diagram of the implemented prototype w/4—shift DQPSK system.  The main function of the baseband generator is to generate in baseband, a raised cosine filtered 7r/4-shift DQPSK signal. The design was realized using widely available off-the-shelf 61  Chapter 4. digital TTL chips and linear analog devices. Using a prototype RF modulator, the baseband signal is modulated in quadrature, so that it can be transmitted through the communication channel. In the channel, AWGN as well as multiplicative Rayleigh fading are corrupting the transmitted signal. At the receiver, after coherent demodulation which is achieved by an in-house designed 10  and implemented RF demodulator. The received signal is sampled and processed by the DSP which acts as the receiver. Due to availability, we used a TMS320C30 development platform which consists of a PG plug-in card and software development tools for this processor. The big advantage of having a DSP based receiver, is that the algorithms evaluated in Chapter 3, can be directly adapted from the software used for computer simulation. After this introduction, the organization of this chapter is as follows. The second section analyzes the design of the 7r/4-shift DQPSK Baseband Generator, describing the circuitry on a functional block level. The third section describes the RF modulator, demodulator and channel simulator design, while the fourth section of the chapter, deals with the DSP-based implementation of the baseband receiver. Schematics for all hardware blocks analyzed in this chapter, can be found in Appendix B. Finally, the last section contains the conclusions of the chapter.  4.2. The Synthesized 7r/4-shift DQPSK Baseband Generator A transmitter design for a light-weight portable unit cannot incorporate heavy, bulky Nyquist filters for baseband pulse-shaping.  This obvious necessity directed us towards a different  approach to pulse shaping, namely pulse synthesis instead of the more traditionally employed pulse filtering. More specifically, a method similar to the one for M-PSK baseband pulse synthesis presented in [55] was adopted in our design. Pulse synthesis is performed through 1 0  As mentioned previously, this was assumed for mathematical convenience in the complex baseband derivation. It was replicated in  hardware for consistency purposes.  Section 1.  62  Chapter 4. the outputs of which provide the analog waveforms for the inphase (I) and quadrature (Q) modulating channels. It is well known, that in theory a Nyquist-shaped pulse in the time domain, extends from - c o to oo. However, a table of samples cannot have infinite length and furthermore it is well known that the pulse amplitude before and after the sampling instant "dies out" quite fast, e.g., thefirstmaximum after t = 3T is down to 1.6% from the pulse amplitude at t = 0. An example S  illustrating this is the x/sin(x) equalized raised cosine Nyquist pulse shown in Fig. 4.2. From 1.5  0.5  0  -0.5 -90  -80  -70  -60  -50  -40  -30  -20  -10  0  10  20  30  40  50  60  70  80  90  Fig. 4.2. x/sin(x) equalized raised cosine Nyquist pulse with roll-off factor a = 0 . 3 5 ; the number of samples/(symbol interval) is equal to 16. The sampling instant is at 0.  thisfigureit is clear that truncation of the pulse beyond a certain point will not introduce large inaccuracies. The truncation method described in [55] for M-PSK signals, makes use of this observation to limit the number of symbol pulses interfering with each other at any time instant. Once a convenient truncation point has been chosen, the number of overlapping pulses can be determined. The sample tables are then constructed by considering all symbol combinations for the given number of pulses, and generating the samples for the combined output of the pulses in each case. The implementation specific parameters for the 7r/4-shift DQPSK baseband generator implemented, are listed in Table 4.2.1. The two tables for the I and Q channels were stored in Section 2.  63  Chapter 4. Serial Data In (bits)  Fig. 4.3. Functional block diagram of the x/4-shift DQPSK baseband transmitter.  implemented, are listed in Table 4.2.1. The two tables for the I and Q channels were stored in Table 4.2.1. Pulse-sample synthesis implementation specific parameters.  Integral number of symbol periods:  J = 2 (2 past and 2 future symbols used)  Total number of symbol periods:  2xJ+ 1 (the symbol being transmitted) = 5  Number of samples / symbol interval:  8 (or 16)  "Present" signal combinations:  8 (possible 7r/4-shift DQPSK absolute phases)  "Past" and "Future" symbol combinations:  4 = 4 = 16 (4 possibilities/symbol)  Total number of samples in table:  (8 x 16 x 16) x 8 = 16384 (16k samples)  D/A resolution:  256 levels (8-bit words)  Binary storage requirement for I or Q ch.:  16kbytes  J  2  EPROMs. The hardware design also supports 2 selections for the number of samples/symbol, namely 8 and 16. For the latter, the binary storage requirement for each I and Q sample table is 32K bytes. The functional block diagram of the baseband 7r/4—shift DQPSK experimental transmitter is depicted in Fig. 4.3. Section 2.  64  Chapter 4. The following is a fairly detailed description of each individual block.  •  The Serial to Parallel transforms the stream of serial input data bits to parallel form (see Table 4.2.1). Since the integral number of symbol periods used is 2, both the "Future" Bus and the "Past" Bus provide 2x(2 bits/symbol) = 4 bits. The bus feeding the 7i74-shift DQPSK state encoder has two lines providing the 2 bits for the "present" symbol to be sent.  •  The 7i74-shift DQPSK state encoder is the "heart" of the transmitter, providing at every symbol interval the sequence number of the signal to be transmitted from the 7r/4—shift DQPSK signal space. This number depends on the previous signal number and the 2-bit symbol input. A more detailed functional block diagram of the encoder is depicted in Fig. 4.4. The signal numbering convention as well as the phase encoding used are shown in Fig.  o  "5> o  Input Bits +•  _i  Output Signal Number  Latch  Fig. 4.4. The functional block diagram of the jr/4—shift DQPSK state encoder.  4.5. Essentially, this is a differential encoder (see Eq. 2.2) implemented in digital form. The number assignment of the signal states was done in such a way as to render the recursive implementation of the encoder shown in Fig. 4.4 possible. Section 2.  65  Chapter 4.  Fig. 4.5. The signal numbering convention and mapping of input information symbols to transmitted differential phases.  The structure and logic of the recursive 7r/4—shift DQPSK state encoder reveals itself if one observes the previous-signal — next-signal pairs and input symbols listed in Table 4.2.2. The Table 4.2.2 Next-signal transmitted as a function of the previous-signal and the input  Previous signal transmitted (i.e., previous encoder state) - binary  Input  000  001  010  011  100  101  110  111  00  001  010  011  100  101  110  111  000  01  011  100  101  110  111  000  001  010  10  111  000  001  010  011  100  101  110  11  101  110  111  000  001  010  011  100  signal numbers and input dibits are given in binary form. From the bit patterns shown, it can be easily observed that the next signal number can be calculated from the previous signal number and the input dibit through the binary addition of a constant to the previous signal number. This constant depends, as it is expected, on the input dibit, while the addition discards the carry bit. The constants for the four possible input dibits are listed in Table 4.2.3. The two most significant Section 2.  66  Chapter 4. Table 4.2.3 The constants used for the generation of the next signal number.  Additive constant 00  001  Input  01  011  Dibit  10  111  11  101  bits of any constant can be generated easily by a simple logic circuit receiving as input the input symbol (dibit), while the least significant bit is always 1. The previous signal number is saved in a latch and fed into the binary adder together with the corresponding constant. •  The I and Q Memory and accompanying Addressing Logic blocks implement the lookup tables for the shaped baseband pulse samples. Each channel (I or Q) memory is a 16kbyte EPROM when using 8 samples/symbol interval and a 32kbyte EPROM when using 16 samples/symbol interval. An example pulse for J=2, 8 samples/symbol and roll-off factor a = 0.35 is shown in Fig. 4.6 with the sampled values marked by square blocks. The "Present" Symbol interval  -8  -16  M  0  8  16  •  *~  . 2 "Past" symbol intervals Fig.  2 "Future" symbol intervals  4.6. Example pulse with J=2, 8 samples/symbol and a = 0.35.  plot in Fig. 4.6 clearly shows the 5 symbol intervals for J=2, each one holding 8 samples. Section 2.  67  Chapter 4. An example demonstrating the derivation of the pulse samples for all symbol combinations resulting from the parameters in Table 4.2.1, is shown in Fig. 4.7.  In this figure, each  cluster of 5 symbol intervals corresponds to a 2-bit symbol input. This input determines the pulse amplitude and polarity. The result of the sum of the 5 samples at each sampling instant is the number which is tabulated in the EPROM for each channel. The index of this result in the EPROM is determined by the J symbols in the past ("Past Bus"), the J symbols in the future ("Future Bus") and the signal from the 7r/4-shift DQPSK signal space currently being transmitted. Note that the actual sample values saved in the EPROMs are integers from 0 to 255, resulting from re-normalization and truncation (or rounding-up) of thefloatingpoint values. A C-language program used to generate the binaryfilesfor the Iand Q-channel EPROM for any value of the roll-off factor a, the truncation factor J and any number of samples/symbol can be found in Appendix C. It should also be pointed out, that the full a Nyquist raised cosine filter response was used to produce the pulse samples used by the 7r/4-shift DQPSK baseband generator. On the receiver side, a simple Butterworth filter was used, with a high enough cutoff frequency as not to distort the transmitted signal. This method of including the full filter at the transmitter is not at all optimum (see Chapter A d d samples down this column (8 samples / symbol interval shown)  Samples tor the 2 symbols in the past  Samples for the symbol "currently being transmitted symbol interval currently being clocked-out  S a m p l e s for the 2 symbols in the future  Fig.  Section 2.  4.7. Derivation of the pulse samples for J=2.  68  Chapter 4. maximum signal to noise ratio at the output of the receive filter, but greatly simplifies the design with relatively small trade-off in BER performance. •  The Timing Generator provides all the necessary timing signals, including a data rate clock used to clock-out the data bits from the information source and a symbol- and bit-SYNC used at the receiver end. It should be pointed out that the symbol timing recovery circuit needed at the receiver of a real-life modem was omitted, the reason being that symbol synchronization was beyond the research of this thesis. An external clock option was also implemented, so that a variable bit rate system can be implemented. In this case the supplied clock must have a frequency F = ( Number of samples / symbol ) x ( Desired Bit Rate ).  •  The analog output section incorporates two 8-bit D/A converters, one for each channel, and two line drivers. Each line driver includes: a.  a simple 3-pole Butterworthfilterwhich smooths out the "staircase"-like output from the D/A's by removing undesirable high frequencies.  b. a high current driver since the outputs drive 50 Q loads.  The loads are driven  through 50 Q, precision trimmers so that accurate matching, necessary for minimum inter-symbol-interference (ISI), can be performed.  4.3. RF Modulator/Demodulator (MODEM) and Channel Following Fig. 4.1, a more detailed block diagram of the RF Modulator/Demodulator (MODEM) as well as the channel, appears in Fig. 4.8. The carrier frequency employed, as indicated on the block diagram, is 1.5 MHz. The reason for choosing this relatively low carrier frequency is the considerable amount of problems inherent to designs involving carrier frequencies in the VHF/UHF bands. The frequency chosen greatly simplifies the RF design of both the modulator and demodulator, as well as the RF section of the channel model, without compromising in any way the accuracy of the experimental Section 3.  69  Demodulator  Chapter 4. measurements. The choice of carrier frequency was also limited by the frequency coverage of the Gaussian white noise generator used, and the roofing BPFfilteravailable. The generator coverage was specified by the manufacturer to be in the 6 kHz to 25 MHz band. This relatively wideband noise must be limited by using a roofing band-pass filter before it is added onto the faded signal. The BPF employed is centered around the carrier frequency of 1.5 MHz and has a 3 dB bandwidth of 200 kHz. The modulator was implemented using a 90° phase splitter which provides the inphase and quadrature components of the modulated carrier, and two mixers. The mixer outputs are then added by means of a signal combiner, to form the RF modulated signal. A series of amplifier blocks were used to boost the power output from the modulator, since the relatively high overall conversion loss - approximately 20dB down from the 7dBm Local Oscillator (LO) input of the modulating mixer - results to a considerably weak signal. The same layout was used for the design of the RF channel module, the only difference being the baseband I and Q inputs, which in this case are generated from the fading simulator controller presented in [56]. The demodulator was implemented as described in the theoretical derivation (see Fig. 2.3), i.e., the same carrier used for the modulator, is used to coherently downconvert the received signal to baseband. The same type of 90° phase splitter used in the modulator, is also used here to provide inphase and quadrature components of the carrier. In a real-life modem, a noncoherent demodulator would have to be implemented at the RF level and not in baseband, as was the case for our experimental testbed. Two well known noncoherent receiver designs are a) RF differential detector and b) RF limiter/discriminator. •  The RF differential detector implementation uses delayfinesin order to produce a version of the incoming signal, delayed by T seconds, which is then multiplied with the non-delayed signal. The disadvantage of this method is the delay line used, which a) is generally difficult to design and b) will work only for one transmission rate, i.e., only at R =  Section 3.  71  l/T.  Chapter 4. •  The RF limiter/discriminator noncoherent detector, relies on an FM discriminator which provides the first derivative of the carrier phase (i.e., the instantaneous frequency). Using this as input, an integrate-and-dumpfilterprovides the phase change for each symbol period. This design is far more flexible than the RF differential detector, since it can operate on any transmission rate desired, without modification. Furthermore, since it is much easier to implement, it is in general, the preferred design.  Schematic circuit diagrams for the blocks shown in Fig. 4.8 are included in Appendix B.  4.4. The Digital Signal Processor (DSP) - based Receiver The heart of the receiver was implemented using a digital signal processor (DSP) chip. The main advantage presented by a DSP based design over an all-hardware implementation is the great versatility in terms of developing efficient receiving/decoding algorithms. The disadvantage of this approach is the processing time trade-off which prohibits the use of such processors in high speed digital modems running complex and computationally intensive algorithms. However, in a final design, a software/hardware hybrid approach using a few dedicated VLSI integrated circuits can remove this processing time constraint. The DSP development platform used is a Personal Computer (PC) plug-in board, using the Texas Instruments TMS320C30 digital signal processor chip. Software development tools running on the host PC provide assembler, C compiler, linker and utilities for loading and running application software on the DSP board. A somewhat more detailed functional block diagram for the DSP-based digital receiver is depicted in Fig. 4.9. The analog front-end provides for two filtered inputs. The onboard filters are 4  th  order  Butterworth with unity gain, while their cutoff frequency is determined by a plug-in resistor pack [57]. A/D conversions are triggered by using the board's external triggering option. The Section 3.  72  Chapter 4. Bit Clock provided from TX Output Data stream  Parallel to Serial  i LSB  MSB I  TF  DSP Board  Symbol SYNC provided from TX DSP Analog Front - end  Host PC  Fig.  From Demodulator  4.9. Functional block diagram of the DSP-based digital receiver.  The receiver software is divided in two distinct blocks, namely the kernel and the decoding routine. The kernel, written in assembly language, performs hardware-specific initialization and establishes the interrupt-service routine triggered by the Symbol SYNC signal. This interrupt routine, when triggered by the symbol timing signal, reads the input samples and passes them on to the decoding routine. After decoding is done, it writes out to the digital I/O port the symbol returned from the decoding routine (2 bits for 7r/4-shift DQPSK), the bits of which are then clocked-out serially. The decoding routine is an external code module which can be written in assembly or in C. In this case the latter was chosen in order to speed up development time, by making use of the same decoding routines used in the software simulation of the system. However, the C compiler supplied by the chip manufacturer provides assembly output so that the code can be further optimized for speed, if this is desirable.  Section 4.  73  Chapter 4.  4.5. Conclusions This chapter dealt with the design and implementation of a prototype 7r/4-shift DQPSK system. Our discussion included detailed description of the following system modules: 7r/4-shift DQPSK baseband generator, RF modulator and demodulator and a DSP-based receiver.  Section 4.  74  5. Experimental Set-up and Measurements 5.1. Introduction This section describes the experimental set up used to evaluate the prototype 7r/4—shift DQPSK system. It also presents the obtained experimental BER performance evaluation results of the implemented receiver systems which are derived in Chapters 2 and 3. The chapter organization is as follows. After this introduction, in Section 5.2, the experimental set up is described. Section 5.3 presents numerous experimental signal measurements. In Section 5.4, the experimental BER performance evaluation measurements can be found. Finally, Section 5.5 contains the conclusions of the chapter.  Tektronix 115 Pulse Generator  Tt/4-shift DQPSK baseband TX Master Clock  Transmitted Bit Stream  HP1645A Data Error Analyzer Received Bit Stream  TMS320C30 DSP board Symbol SYNC  Host PC  Fading Simulator Controller  RF Modulator  Fading Modulator  W&G Noise Generator  Carrier Noise Summer  HP8656B RF Generator Carrier  RF Demodulator  Distorted Signal Band Pass Filter  Monitor Tektronix 497P Spectrum Analyzer  Fig.  5.1. Functional block diagram of the experimental layout.  75  Chapter 5.  5.2. Experimental Set-up The block diagram of the experimental testbed used for system evaluation is depicted in Fig. 5.1. In addition to the test equipment shown, a Philips PM3055, 50 M H z oscilloscope was used to monitor the baseband signal waveforms. The Pulse Generator shown is also optional, since the 7r/4-shift DQPSK baseband generator has Master Clock generation circuitry onboard. However, the onboard clock was fixed to 200 kBaud, whereas the external timing generator allowed tests at lower Baudrates. This capability proved to be very useful when testing the realtime processing capabilities of the DSP processor used, in conjunction with the computational load presented by the proposed algorithm. Two photographs of the complete experimental set up can be found in Fig. 5.3.  Fig.  5.2.  I-channel eye-diagram at modulator input; a =  1.0.  5.3. Signal Measurements The I-channel eye-diagram  11  at the input of the modulator, using a Nyquist roll-off factor of  a = 1.0, appears in Fig. 5.2. The very small amount of ISI introduced at the sampling instants, is due to pulse truncation used at the 7r/4—shift DQPSK baseband generator. As a decreases, 11  From now on, in this chapter, reference to eye-diagrams will denote the I-channel eye-diagram.  Section 2.  76  Laboratory of the Department of Electrical Engineering at the University of British Columbia.  Chapter 5.  Fig.  5.4. I-channel eye diagram at modulator input; a =  0.5.  the amplitude of the Nyquist pulse tails will increase [2]. As a result, the ISI produced by the implemented pulse truncation synthesizing technique, is expected also to increase. This is evident in Fig. 5.4, where the eye-diagram at the modulator input, with a — 0.5 is given. The eye-diagram at the demodulator output without any channel interference, for a = 1.0, appears in Fig. 5.5. When we compare it with 5.2, the slight degradation introduced by the modulation/demodulation process is evident. The equivalent eye-diagram for a = 0.5 is shown in Fig. 5.6. The phase symmetry of the signal space is also distorted by the non-ideal modulator and demodulator circuitry. Fig. 5.7 shows the signal space at the modulator input while Fig. 5.8 shows the signal space at the demodulator output, both for a = 1.0. The modulated and demodulated signal space for a — 0.5, appear in Figures 5.9 and 5.10, respectively.  This  signal space distortion is mainly a result of the phase unbalance in the 90° splitters used at the modulator and demodulator. For instance, the maximum phase unbalance specification for the components used is 3° [58]. The second factor seriously affecting the signal constellation at the demodulator output, is the power driving the local oscillator (LO) inputs of the mixers used. Section 3.  78  Chapter 5.  Fig. 5.5. I-channel eye diagram at demodulator output; a = 1.0.  A n y deviation greater than 1.5 d B from the specified 7 d B m L O drive, was found to distort the signal space geometry considerably. The spectrum of the transmitted signal gives one more indication of the limitations inherent to the pulse synthesis technique. The spectrum for a = 1.0 is illustrated in F i g . 5.11, and that for a — 0.5 in F i g . 5.12. The Baudrate for both measurements was fixed to 100 kBaud. The spectrum corresponding to a — 0.5 shows considerably more spurious energy around the  Fig. 5.6. I-channel eye diagram at demodulator output; a = 0.5.  Section 3.  79  Chapter 5.  Fig.  5.7. Modulated signal space diagram; or =  1.0.  main "lobe", even when compared to the theoretically wider spectrum for the a = 1.0 case. However, close to the carrier frequency, the bandwidth reduction when using the lower value for a is evident. The spurious energy distorting the spectrum shape, will increase as the value of a decreases and at the same time the truncation limit J (see Table 4.2.1) remains constant. If small values of the Nyquist roll-off factor are to be used efficiently, the number of "past" and "future" symbols considered (see Chapter 4) will have to be increased, as the amplitude of  Fig.  Section 3.  5.8. Demodulated signal space diagram; a =  80  1.0.  Chapter 5.  Fig.  5.9. Modulated signal space diagram; a — 0.5.  the Nyquist pulse tails increases, for decreasing values of a.  The spectrum of the received  signal, for a = 1.0, with A W G N present is illustrated in Fig. 5.13, while the corresponding eye-diagram at the demodulator output is shown in Fig. 5.14. The spectrum and corresponding eye-diagram for the case when a = 0.5 are depicted in Fig. 5.15 and 5.16, respectively. Note the eye closure caused by the noise in both Figures 5.14 and 5.16.  Fig.  Section 3.  5.10. Demodulated signal space diagram; a = 0.5.  81  Chapter 5.  1  i  OOBH  MHZ  j |MI in  j j  n N -nn  Jr  m 1008/  50KHZ ;  M  1  •  -401  )  ml  Ml •••]Ml Ml 30DB  0-1.3  •  IHT 30HZ 10KHZ  Fig. 5.11. The spectrum of the transmitted 7r/4-shift DQPSK signal; a = 1.0.  Finally, the spectrum of the faded carrier with the maximum Doppler frequency Bp set equal to 126 Hz (the maximum B  F  for the fading simulator controller used [56]), is illustrated  in Fig. 5.17.  Section 3.  82  Chapter 5.  Fig. 5.13. The spectrum of ir/4-shift DQPSK in AWGN; a = 1.0.  5.4. BER Performance Evaluation Measurements Measured BER performance results were obtained from the prototype 7r/4-shift DPQSK system, employing two types of receivers, a conventional differential detector receiver (see Eq. 3.20) and a multiple differential receiver based on the detection algorithm of Eq. 2.34.  Fig. 5.14. I-channel eye diagram at demodulator output with AWGN present; a = 1.0.  Section 4.  83  Chapter 5.  Fig.  5.15. The spectrum of jr/4-shift DQPSK in AWGN; a = 0.5.  Measurements performed with the conventional differential detector in an AWGN channel, exhibited a small deviation (approximately 1.5-2 dB at a BER of 10") from the theoretically 4  expected performance. This deviation is due to the following factors. •  The imperfect pulse-synthesizing technique employed by the 7r/4—shift DPQSK baseband generator, which, as it has been demonstrated already (see Fig. 5.5 and 5.6), introduces  Fig. 5.16. I-channel eye diagram at demodulator output with AWGN present; a = 0.5.  Section 4.  84  Chapter 5.  Fig. 5.17. The spectrum of the faded carrier with maximum Doppler frequency set to BF = 126 Hz.  ISI at the sampling instant. •  The non-ideal signal-space at the demodulator output, due to imperfect RF components. A noticeable deviation from ideal quadrature can be observed in Fig. 5.8 and 5.10.  •  The way in which filtering has been realized in hardware. In the computer simulations, y/a Nyquist filters are realized for both the transmitter output and the receiver input. In hardware, however, the full a Nyquist filter is implemented at the 7r/4-shift DPQSK baseband generator, while a simple first order Butterworth filter is used at the mixer outputs in the demodulator.  BER measurements for the conventional differential detector in an AWGN and Rayleigh fading channel, for three values of the BFT product, are presented together with the corresponding computer simulation results, in Fig. 5.18. The close agreement between the experimental and the computer simulated BER results is evident. Finally, in Fig. 5.19, the measured BER performance of the z = 1 and z = 2 reduced complexity receiver, operating in AWGN and Rayleigh fading with BFT = 0.125, are presented. For comparison purposes, the equivalent computer simulated results are also included. It is Section 4.  85  Chapter 5. evident from the BER results, that there is more than an order of magnitude reduction of the error floor. The deviations observed in the experimental measurements presented in Figs. 5.18 and 5.19, are attributed primarily to the DSP analog input subsystem. More specifically, the input voltage range of the A/D input on the DSP board, for a full range operation of the onboard A/D converters, is specified to be +/-3 Volts [57]. However, the faded inphase and quadrature signal at the demodulator output, present considerable amplitudefluctuation,such that frequent signal peeks above +3 Volts or below -3 Volts are clipped. From the measurements in the figures presented, it can be observed that the discrepancies are more severe when the multiple differential detector receiver is used. This is attributed to the memory inherent to this detection algorithm.  -1  10  10 I I I I I I I 10  15  20  25  1  30  1 II II I I II l I II  35  40  45  50  55  60  65  70  M i l l 75  80  SNR (dB)  Fig. 5.18.  Section 4.  BER of the differential  detector receiver in  AWGN  86  and Rayleigh fading with BFT =0.013, 0.026 and 0.0041  Chapter 5. 10 Order-2 Experimental, Order-1 Order-2 Simulation, Order-1  ...  O •  \  .Q  j  !1  10  CO  .Q O  k_ Q. k_  \ \  O o  10"  \  \  s  m 10  5  10  15  20  25  30  35  40  45  50  55  60  SNR (dB) Fig. 5.19. B E R of the multiple differential detector receiver in A W G N and Rayleigh fading with BFT — 0.125.  Finally, some comments with reference to the processing speed of the prototype 7r/4—shift DPQSK system are in order. The limitation of how fast a symbol decision is made, depends upon the speed of the DSP, and the complexity of the algorithm used to make that decision. For the TMS320C30 used in our prototype system, the conventional differential detector would achieve a maximum speed of about 100 kb/s. On the other hand, the receiver with 2 = 1 would achieve approximately 2.4 kb/s, whereas the receiver with 2 = 2, 400 b/s. It should be noted that the above bit rates were achieved without any program code optimization, since the detection routines were written in the C programming language. If assembly language is used instead, it is roughly estimated that the maximum processing speed would increase by at least a factor of three.  5.5. Conclusions The proper operation of the prototype 7r/4-shift DPQSK system designed and implemented in hardware/software, was verified through measurements performed on the baseband inphase and quadrature signals, as well as on the RF spectrum. BER performance data, obtained for Section 5.  87  Chapter 5. both a conventional differential detector and the proposed receivers, in combined A W G N and flat Rayleigh fading channel conditions, demonstrated a very good match of experimental and computer simulation results, verifying the effectiveness of the proposed technique.  Section 5.  88  6. Conclusions and Suggestions for Future Research 6.1. Conclusions In this thesis, the problem of optimal detection of both coded and uncoded, differentially encoded or not, PSK and QAM signals in frequency nonselective Rayleigh or Ricean channels, was treated. A theoretical derivation of the maximum likelihood sequence detector for PSK and QAM signals transmitted over a fading channel, was presented. It was shown that the detector's structure can be realized as a combination of envelope, multiple differential and coherent detectors. In addition to the optimal detection algorithm, several reduced complexity algorithms were also proposed. The BER performance for these reduced complexity algorithms employing various digital modulation schemes, including coded and uncoded 7r/4-shift DPQSK, QPSK, 7r/4-shift DQAM and 8-QAM, was obtained by means of computer simulation. Finally, a hardware/software implementation of a prototype 7r/4-shift DPQSK modem helped verify experimentally the effectiveness of the proposed receivers.  6.2. Suggestions for Future Research 6.2.1. Further Modem Development The prototype 7r/4-shift DPQSK modem designed during the course of this thesis, could be developed further, in several ways such as •  Increasing its processing speed.  •  Incorporating multi-amplitude signals.  •  Reducing the amount of hardware required, by using LSI/VLSI techniques.  An ultimate goal, of course, would be to commercialize the product. 89  Chapter 6.  6.2.2. Symbol Timing Synchronization Throughout this thesis, it was assumed that the receiver had access to a perfect symbol timing signal. However, in practice this does not happen. Consequendy, is will be very useful to investigate methods which can be employed to obtain symbol synchronization.  6.2.3. Frequency Selective Fading Another problem which could be adopted for future research is to obtain optimal sequential receiver structures for a frequency selective fading channel.  6.2.4. CPM Signals Finally, extension of the receiver structures presented in this thesis, in order to include continuous phase modulation (CPM) signals, would be an other interesting research topic.  Section 2.  90  Bibliography [I]  W. C. Y. Lee, Mobile Communications Engineering. McGraw-Hill Book Company, 1982.  [2]  J. G. Proakis, Digital Communications. McGraw-Hill Book Company, second ed., 1989.  [3]  Entire issue, "Portable and mobile communications," IEEE J. Sel. Areas Comm., vol. SAC-7, Jan. 1989.  [4]  Entire issue, "Special issue on fading and multipath channel communications," IEEE J. Sel. Areas Comm., vol. SAC-7, Feb. 1987.  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Suzuki, and T. Nojima, "Performance of pi/4-qpsk transmission for digital mobile radio applications," in the Proceedings of GLOBECOM '89, pp. 443^-47, November 1989. [54] C.-L. Liu and K. Feher, "Performance of non-coherent pi/4-qpsk in a frequency-selective fast rayleigh fading channel," in the Proceedings of SUPERCOM/ICC 90, Atlanta GA, pp. 335.7.1-335.7.5, April 1990. [55] C. A. Siller, W. Debus, and T. L. Osborne, "Spectral shaping and digital synthesis of an M-ary time series," IEEE Comm. Mag., pp. 15-24, Feb. 1989. [56] E. Casas and C. Leung, "A simple digital fading simulator for mobile radio," IEEE Trans. Veh. Tech., pp. 205-212, Aug. 1990. [57] SPECTRUM Signal Processing Inc., The TMS320C30 System Board Technical Reference Manual, Issue 1.0, May 1990. [58] Mini-Circuits, RFIIF signal processing guide. SF-89/90.  96  Appendix A. Theoretical Derivations A.1. Derivation of Equation 2.18. For the reader's convenience, we rewrite Eq. 2.15 /...  rl • -/..  k — Jk'  MykC* e-JP)  Q  k  —  l Jt|  "  '  L  Q  Q  — ''-  n  Im(y cle-»)  (A.l)  k  —  u  s  c  The autocorrelation matrix terms are then given by R [C(S), (* - i)T] = E{e{ei) =  (A.2)  E^e?}  e  Since fading and white Gaussian noise are assumed to be independent, substituting Eq. A . l into the above equation, we obtain: (A.3)  R [C{a), {k - i)T] = R {k -i) + E{nl(k)nl{i)} e  f  By using Eq. 2.16 and 2.17, i.e., n {kT)Re(cle-">) + n ( t r ) I m ( c ; e ^ )  7 n  e\k)  s  (A.4)  c  —  ; ~$ \ k\ c  and Q  _ n (kT)lm(cle-">) s  - n (*r)Re( £e->') c  C  (A.5)  Ck  and substituting in the £{n^(fc)nf(i)} term of Eq. A.3, we get n,(kT)Re(c' e- )  n (kT)lm{cle-»>)  lp  E{ni(k)n' (i)} = E  k  c  .o  n (iT)Re(e*e--' )  "T"  n (iT)lra(cte-  p  c  c  5  .  • . i  ! To Ci  ~f~  } (A.6)  Since n (kT) and n (kT) are uncorrected, the expected value of all cross-product terms is c  s  zero. Hence, from the remaining terms £  f / n  ( j b ) n  /  ( 0  i  _ fnAkT)Re(cle-»)n,(iT)Re(c;e-»)  |  L  \  |c I |c | 2  fc  n (fer)Im( ;e-^)n (»T)Im( r --'>) j e  e  C  |cjt| ki|  2  2  x  C  2  e  J (A.7)  97  Since the power of the Gaussian noise at the output of the receive filter HR(f) is equal to 2 _  No  (A.8)  Equation A . 7 becomes Re(c* Ci)  (A.9)  k  a 8{k-i)E2  ]c  G  |2| c  l fc| l t|  |2"  where 6(-) is the Kronecker delta function, nonzero only for k = i. Combining Eq. A.3 and A.9 we get E{e{e }=E{e e<?}=R (k-t) I  +  Q  l  k  f  a 8(k — i) G  c  (A.10)  fc  which is Eq. 2.18.  A.2. Derivation of Equation 2.27. The general form of the metric as presented in Eq. 2.23, is given by the function  +  i=l  \m{y1c' t-»}  —  k  k  "  L  *  - E  t=l  Im{»*_,-c;_,.e->"} Pk,i " 1  n  2  >  (A.11)  L  Cfc-  For simplification of the symbolism, the dependence of the prediction coefficients and the MMSPE on the transmitted sequence, has been dropped. By defining p^o = — 1  0 < k < L,  the above equation becomes it  I.  i f  2 ^ a\  k=\  +  |Cfc_;  «=0 fc  k  i=0 T 2 Im{y -.- ;-.-e~ }  (A.12)  J>  -EM,  t  C  t=0  The first squared term in Eq. A.12 can be expanded as '  -E  Re{y c^} \ k\  i=o  77V"„ f  ^  t  c  2  2  ^ j.olT\^  1^ 1=0 Pk.i  t'=0  fc fc n  + 2 / 2/=0^  V*  R e  Pk.l 2^ Pk.i t'=0  {^-« fc-« " } c  e  |Cfc| —To  98  J P  , /y-*  Re{yk- 4_ e-"' t  2_> P .i  + .i=0  k  t  Ck\  (A.13)  The third term of Eq. A. 13 can be combined with the second squared term within the brackets of Eq. A.12 to yield the following Re{y _,c^_, -^} fc  e  2^Pk,i  PJFC ,  2  ,=o  k  + 2^''  —-  l*-«'l c  ip  |Cfc_,|  t=0  lcfc-,1  i=0  A  Z  + 2_^  r~Ji  —~ |Cfc-  t=0  2 (Re{yfc-.-Cfc-,-e" })2 , fc ^  l_,Pk,x  :  Re{yfc-,c*_ e-^}Re{y _ 4_ e-^}  Z_> Pk,*Pk,l  1  :  fc  —  |cfc-.'| |cfc-;|  ,=0  i  ¥  |cjfc_;| |c _/|  /=0  /  /  2  2  4  /  y>  2  +  2  fc  (A.H) The first and third term of Eq. A. 14, can be combined to yield  E  -  1 0  |2  fc  * k-i .0-3P \yk-i . .4  2 Pk,i  c  e  = £Pfc,« t=0  |yfc-tl I iI  (A.15)  Re(AB*)  while the addition of the second and fourth term, by using the identity Re(A)Re(B) + lm(A)lm(B), leads to the expression ^  ^  2^  Re{yfc-,4_-e Pfc,«Pfc,r  J  2  2  Re{y _,y£_ }Re{c _,4_ } +  ^  2^  '"yJ_,cjfc_ie '"}  |cjt_,| |cit_/|  i=o ... 1=0  J^  J  fc  /  Jt  Pfc.iP*,'  k  |cfc_,-| |cjt_/| 2  _,, 1=0  t'=0  (A.16)  lm{y -iy* _ }lm{c - c* _ }  /  k  l  k  i  k  l  2  By combining the intermediate results, Eq. A.12 can be rewritten as  1  L 1  1  1  —• >  '  ( (/'!>*,•) JV  —7T <  t=o  / fcfc  +V E  Pfc, E P f c /  1  t=o  /=o  r  6  1  ^ / " "  |Cfc  " i r  i  -'  1  +  E  P h t i + |c  «=o  1  f  i  r  *' -  1  -i  V V Re{y -iy _,}Re-{cfc-ic;_|}-HnT{yfc-iy;_, jlm{c -ic;_|} t=0 L, . . . /=0Pk,iPk,l , >• > fc  fc  I  /  •  ,  fc  Equation A.17 can be simplified further and for this purpose it is split up to four terms 99  A n  >  v^-l'J  rp T l  Y^ Y^  1  z^i2Z^  =  Jfc = l  i=0 .  1 Jt=l  rp T  f  3 =  fc  2 bfc-ti  ki o  Jb=l  * \  /  fc  lm{y _ y _,}lm{c _ c _ }  /  k  i  k  k  l  k  l  ;—[2]—12 l fc-t'l \ k-l\ c  c  2  c  1 , 7 / ^ 2  fc  l fc-t|  =  2_> 3  *=1  Re{yfc-ty -jRe{c _,c _ } +  P * M /=0  R {^-. fc-, " } e  C  e  2^  2_>  1  >=0  »=0  l fc-tl  t=0  J >  j2  c  /  (A.18) The 1/2 factor was dropped since it is only a scaling factor. Observing term T i , it is clear that (ij) combinations appear in pairs that can be lumped together if the following identity is used  Re(A)Re(B) + Im(A)Im(B) + Re{A*)Re{B*) + lm(A*)lm(B*) = (A. 19) = 2{Re(A)Re(£) + Im(A)Im(5)} .  This transforms T i to  jfc=l  k i=Q j=i+l  \ k-i\ \ k-j\ c  c  (A.20) By writing out in more detail the sum in the expression for T i , it is fairly clear to see that the individual terms can be grouped in a different way, that is, with respect to combinations of 100  the form {y y* _ )- By rewriting T i 12 as k  ^ *  T  k  t  2 ^  ^  Re{y - yt_ }Re{c - cl_ } k  i  j  k  = Z . ? r Z . Z . P*..-P*.»— k = l °* i = lj=i+l 2 = —{PioPn^Pio} \  i  —  +  j  '  lm{y - yl_ }lm{c - cl_ } k  • ,2, \ k-i\ \Ck-j\  i  j  k  i  - — -  2  j  =  L  C  a  2 + —{P20P21^P21 + P20P21^20 + P2lP22Vlo}  +  .  %  — { P 3 0 P 3 1 ^ 3 2 + P 3 0 P 3 2 ^ 3 1 + P 3 0 P 3 3 ^ 3 0 + P 3 l P 3 2 ^ 2 1 + P 3 l P 3 3 ^ 2 0 + P32P33^Plo}  (PLQPLIVL.L-I + (PL1PL2 PL-1,L-2+ —  A  21  )  h PLOPLLVLO)  H  V  <  +  ( -  PL\PLL*#L-2fi)  <  +(PL,L-IPL,LVIO)  with R e ( y y ; ) R e ( ^ ) + I (y,y*)lm(c,c*) t  C l  m  -Ptj -  —7- ^ | C , | \Cj\  (A.22)  and grouping with respect to the ^3,^ terms, the expression for T i becomes T  i  =  2  J2  ^  Re  + My*g*-i} {*fc-i} y> Im c c  {^^-»} { ^4-J Re  c  |c*| |c*-i|  k=i .=1  2  Pk+j,jPk+j,i  +J  2 J  =  0  (A.23) The same type of term re-grouping can be applied to the expressions for T and T . Expanding 2  3  the expression for T , we obtain 2  T 2 =  k  1  Eil> M1  2  |2  I  _L/ 2 k i l ! , 2 Jyopl d  ^r  D  1 0  ^! ^ 2  1 1  |co| / 2  2 + T l i P20 7 - 7 2 " + P 2 l 7 - T T + P22^  cr  + +  j.  [2  |c r J  M_ + ML . kiL . M_ 2  L{r? _ 2 1 P30 .  ^3  I  ^  C  2  ^  0  D  |  2 3  2  + P31 , . |C |  ^r,  2  „2  2 2  „2  + ^32 . + P33 7 ^ | Ci C | |  2  0  I  .  ,  „2  M  From this point and on, commas between subscripts will be omitted whenever appropriate, for clarity purposes.  101  (A.24)  which becomes the following, when grouping with respect to the |y;| /|c,| terms, is applied:  k=l 1^1  .=0 k+, a  For T3, expanding yields  T  3 = £  =  £  P*.i L  ft.."  T - ^  ~2 {(PIO^I +PllQo)(piO +Pll)}  +  (P20Q2 + P21&1 + +P2oQo)(P20 + P21 + P22)}  1. V  + ~|{ ( P  where  =  3 0  /  ^  + ^"{(PLO&L  3  >  (A.26)  + P3l02 + P32&1 + P33Qo)(P30 + P31 + P32 + P33)}  +PL1&L-1  h PLL&o)(PLO  +  Qi = — i i i - i -  + PLl +  >-  r PLL)}  (A.27)  and grouping with respect to the Q,- terms, results to the final expression for T 3 , which is the following Re(  Ts = 2  F  f  fc=l  7t" E^X>^,P)  l*:l C  <> A 2 8  »=0 *+* J=0  The fourth term, namely T , does not depend on the information sequence transmitted and 4  therefore it can be eliminated from the metric calculation process. Combining terms T i , T and T , we arrive at Eq. 2.27. 3  A.3. Derivation of Equation 2.28. The incremental metrics for the recursive implementation of the algorithm, can be obtained by inspection, from the expanded forms of T i through T . From every line in Eq. A.21, A.24 3  and A.26, only the terms for which, the second digit of the prediction coefficient subscript is 102  2  not greater than the maximum prediction order, are kept. Thefirstdigit is changed for all to the maximum prediction order. This essentially corresponds to having a sliding prediction window of size z + 1. Furthermore, the MMSPE used is always cr .. For example, for a maximum 2  prediction order of z = 2, the expanded form of the sum for T 2 is modified as  " 2 SP20 7—2 J + ^22 -~2 \ 1 1 2 + P21 2 [ \C |C | |Ci| J M\ i2 1 2 2 A  0  11  2  2  |ml y r i 2 b1  2  4  12/21  o + P 2 1 ^ f + P22/T 2 1^20/T 2 I |C | |c | |c 2 | N i2 1 , J_J 2 JysJl 2 m1 , 2 + ~2 1 P20 I 71 + P21 T + P22 71 2 I ||cs] C5| |C | |C |  +  CT  4  3  tof1  D  (A.29)  2  1  A  I  2  \yi 1 CT  2 (  .  |2 | c|i |  4  3  2 |yx-il  . 2  2  |Ci_i|  1  2  \CL-2\  Notice that each term in Eq. A.29 includes the z + 1 most recent signal samples only, and no terms for samples before 2/3, since the algorithm is assumed to have been initialized by z samples prior to normal operation. By inspection, it is not difficult to realize that the incremental metric for T , i.e., A T , is 2  2  1  I  I  2  m  where m  { L — k,  k > L—z .  ^  ^  Similar observations lead to the incremental forms for T i and T 3 , i.e., A T i and A T 3 respectively: A  _  T  ^  Re(y y*_.)Re(c c^_.) + I m ( y t / £ _ ) I m ( c f c 4 _ . ) ^ I |21 |2 / fc  '  J  fc  fc  Mk-.l  AT  - oTT (^ fc R e  c  e  )  3P  Pz.i  n  103  t  J=  p ,ip ,i+j „2 t  o  z  °*  (  A  3  2  )  The summation limit m given by Eq. A.31, applies only to transmission of information blocks having afinitelength L. However, for sequential decoders performing metric computation after every new sample input, m is always equal to z. Finally, by combining the expressions for all incremental terms, we get the complete incremental metric  My*yfc-J ( * I--i) Re  K  K  ' '  c  c  i=l  + Im(y y*_ )lm(c c*_,) fc  |cjfc|  t  fc  \Ck-i\  (A.33)  |2  +  {SSl r,4p7(j)] + 5 s J ! 2 ^ ! l A . . » P ( S ) ] l Jfc|  Cfc  c  with B  .  -±± [°( ^  v =  %  a  ^  P z  '  i + j  771  r-^VfpijjE^.PP)] oT^  and m =  £, L — k,  (A34) *  m  k ^ L—z k > L —z .  A.4. Derivation of Equation 3.11. Following [5], it can be assumed that the diffusive behavior of the land-mobile radio channel is adequately modelled by the following function: s'(t) = R(t)Ai cos {u t - fa - 6(t))  (A.35)  c  where s'(t) represents the received signal, R(t) the Rayleigh faded carrier envelope, A , an information amplitude signal, u> the carrier radian frequency, <pi the information bearing phase c  and 6(t) the random phase due to fading. This phase 6(t) is assumed to be uniformly distributed in  [-7r,7r).  Using trigonometric identities, s'(t) can be manipulated as follows s'(t) = R(t)A [cos 6(t) cos (u t - 4>i) + sin0(f) sin (u t - fa)] t  c  c  (A.36) f (t) cos (u; t -<l>i) + f (t) J  Q  c  104  sin [u> t - <f>i) c  where / (*) = R(t)cos9{t) and fQ(t) = R(t)sinfl(r). Furthermore 7  c o s + / ( t ) s i n w <sin0i) + (f (t) sinu tcos<f>i - f (t)  s'(t) = ^4,{ (/ (<) cos 7  7  Q  cosu t sin(j>i)}  Q  c  c  e  (A.37) = At {(f  1  (t) cos 4>i -  (t) sin <£,•) cos  + ( / ' ( i ) sin 0,- + f  Q  (t) cos <£,) sin u> t} . c  The baseband equivalent of s'(t) is [2]  '(*) = A,- [(/ (0 cos c6, - / « ( t ) sin 7  + j (/'(*) sin & +  cos (A.38)  = (/ (<) + ;7 (<))^,-(cos^-rjsin^) 7  Q  .  But the term A,-(cos 0,- + j sin^,-) is the baseband equivalent of the modulated carrier while / ( t ) + jfQ(t) is the complex fading process generated byfilteringwith identical fading filters 7  two independent Gaussian noise sources (see Fig. 3.9).  A.5. Derivation of Equation 3.15. Assuming that an envelope (amplitude) limiter normalizes the signal amplitude to one, we would like to derive an expression for the derivative of the carrier phase <p(t), as a function of the inphase (x(t)) and quadrature (y(t)) signals. Since tan [<p(t)] = y(t)/x(t), taking the first derivative we get  dt  dt  (A.39)  From the above equation, and since 1  Eq. 3.15 follows.  105  d<t>(t)  (A.40)  

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