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Performance of differentially detected GMSK in a mobile radio channel Mok, William Chiu Hung 1998

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PERFORMANCE OF DIFFERENTIALLY DETECTED GMSK IN A MOBILE RADIO CHANNEL by WILL IAM CHI H U N G M O K B.ENG. A .C.G.I. (Electrical and Electronics Engineering), Imperial College, University of London, United Kingdom, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDffiS DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A April 1998 © William C. H. Mok, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract As the demand for current cellular voice-oriented networks to include other services such as data and video applications is increasing rapidly, the reliability of information transmission over the mobile radio environment is important. Performance evaluation in terms of the bit error rate (BER) and block (packet) error rate (BKER) of modulation schemes in such an environment is of considerable research interest. In this thesis, we study the BER as well as the B K E R performance of Gaussian-filtered Minimum Shift Keying (GMSK) modulation, currently used in the Global System for Mobile Communication (GSM) standard, in additive white Gaussian noise (AWGN) as well as flat and frequency selective Rayleigh fading. 1-bit and 2-bit differential detectors are used for demodula-tion. A detailed analysis is carried out to examine the IF filtering effect on the BER performance over the fading channels. The BKER performance is mainly studied through simulations. Since the premodulation Gaussian filter at the G M S K transmitter introduces significant Intersymbol Interference (ISI), we also examine the performance of a decision feedback (DF) scheme which partially removes the ISI, over such channels. The BER results in both flat and frequency selective (FS) Rayleigh fading show that 2-bit conventional (with no decision feedback) G M S K performs slightly better than 1-bit conventional GMSK when Eb/NQ < 25 dB. Above this value, the latter performs better and has a lower irreducible error. Results also show that 2-bit conventional G M S K is more sensitive to the IF filtering distortion than 1-bit conven-tional G M S K in Rayleigh fading. The percentage improvement in BKER of 1-bit DF G M S K over 1-bit conventional GMSK is almost equal to that of the corresponding BERs. In the case of 2-bit ii Ill DF GMSK, the B K E R performance shows an increased improvement over 2-bit conventional G M S K when compared to the corresponding B E R performance, owing to increased error propagation. Since fading gives rise to bursty bit errors, the actual B K E R is lower than the predicted assuming independent bit errors. In all fading cases, 1-bit conventional G M S K has a better B K E R performance over 2-bit conventional G M S K when Eb/NQ >25 dB. In the case of decision feedback, similar performance trends are observed. Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement x Chapter 1 Introduction 1 1.1 Motivation and Objectives 1 1.2 Outline of the Thesis 2 Chapter 2 Background 4 2.1 Continuous Phase Modulation 7 2.2 Rayleigh Fading 14 2.3 Coherent Detection 17 2.4 Non-coherent Detection 18 2.5 Intersymbol Interference 19 Chapter 3 B E R Performance of G M S K in Rayleigh Fading 20 3.1 BER Performance of 1-bit Detection in Flat Rayleigh fading channel 21 3.1.1 Unfiltered case 23 3.1.2 Unfiltered Case with Decision Feedback 35 3.1.3 Filtered Case 37 3.2 BER Performance of 1-bit Detection in FS Rayleigh fading channel 46 3.2.1 Unfiltered case 46 3.2.2 Filtered case 52 iv V 3.3 BER Performance of 2-bit Detection in flat Rayleigh fading channel 58 3.3.1 Unfiltered case 59 3.3.2 Unfiltered Case with Decision Feedback 66 3.3.3 Filtered case 68 3.4 BER Performance of 2-bit Detection in FS Rayleigh fading channel 71 3.4.1 Unfiltered case 71 3.4.2 Filtered case 75 Chapter 4 Results and Discussions 81 4.1 Simulation Model 81 4.2 Error Performance in a Non Fading Environment 85 4.3 Comparison of BER Performances between 'filtered' and 'unfiltered' cases in Rayleigh Fading 89 4.4 BER Performance in Slow Rayleigh Fading 94 4.5 BKER Performance in Rayleigh Fading 103 Chapter 5 Conclusions 118 Glossary 121 Bibliography 126 Appendix A 130 Appendix B 132 Appendix C 135 Appendix D 141 List of Tables Table D . l Phase shifts 6;_;- (in degrees) for BtT= 0.25 in 1-bit detection 141 Table D.2 Delayed phase shifts 0/_y(in degrees) for BtT= 0.25 with x = 0.2T in 1-bit detection 141 Table D.3 Differential phase angles Att5 ; and Act^  t (in degrees) of the one-bit detector corresponding to various input data combinations for BtT= 0.25. The contributions of bj_2 and bj+2 are ignored 141 Table D.4 Phase shifts y[ _ • (in degrees) for BtT = 0.25 in 2-bit detection 142 Table D.5 Delayed phase shift <jlri_ - (in degrees) for BtT= 0.25 with x = 0.2T in 2-bit detection 142 Table D.6 Differential phase angles Afys l and Atyd t (in degrees) of the one-bit detector corresponding to various input data combinations for BtT= 0.25. The contributions of Cj_3 and Cj+2 are ignored 142 vi List of Figures Figure 2.1 Block diagram of a digital communication system 4 Figure 2.2 General block diagram of continuous phase modulation 8 Figure 2.3 Information carrying phase pattern of (a) non-differentially encoded GMSK (b) differentially encoded GMSK 13 Figure 2.4 Response of Gaussian lowpass filter to a rectangular pulse of duration T. 13 Figure 2.5 A diagram to illustrate the differences between flat and frequency selective fading 16 Figure 2.6 Two Ray Model: the block diagram of a frequency selective Rayleigh fading channel 17 Figure 3.1 Block diagram of a 1-bit GMSK differential receiver 22 Figure 3.2 Block diagram of a 1-bit DF GMSK receiver 35 Figure 3.3 Phase-state diagram of 1-bit differential detection: (a) before feedback (b) after feedback , 36 Figure 3.4 Block diagram of 2-bit GMSK transmitter (for differential detection) 58 Figure 3.5 Block diagram of a 2-bit GMSK differential receiver 59 Figure 3.6 Block diagram of a 2-bit DF GMSK receiver 66 Figure 3.7 Phase-state diagram of 2-bit differential detection: (a) before feedback (b) after feedback 67 Figure 4.1 BER of 1 and 2-bit conventional GMSK in a non fading environment with AWGN. For 2-bit 'detection', 'thres = 0' denotes the case where a decision threshold of zero is used 86 Figure 4.2 BKER of 1-bit conventional GMSK in a non fading environment with AWGN...87 Figure 4.3 BKER of 2-bit conventional GMSK in a non fading environment with AWGN. Note that the threshold is optimized for every Eb/NQ 88 Figure 4.4 CDF of the number of bit errors in a block at = 16 dB in a non fading environment for (a) 1-bit conventional GMSK (b) 2-bit conventional G M S K with optimum threshold 89 vii Figure 4.5 Average BER as a function of fDT for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in (a) flat fading (b) FS fading with CDR = 20 dB, x = 0.27 91 Figure 4.6 Average BER as a function of fDT for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in FS fading with (a) CDR = 20 dB, x = 0.47 (b) CDR = 10 dB, x = 0.27" (c) CDR = 10 dB, x = OAT 92 Figure 4.7 Average BER as a function of CDR for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in FS fading for fDT = 0.002 with (a) x = 0.2T (b) x = 0.47" 93 Figure 4.8 Average BER as a function of Eb/N0 in flat fading 95 Figure 4.9 Average BER as a function of Eb/Na in FS fading with CDR = 20 dB, x = 0.27" 98 Figure 4.10 Average BER as a function of Eb/N0 in FS fading with CDR = 20 dB, x = 0.47" 99 Figure 4.11 Average BER as a function of Eb/NQ in FS fading with CDR = 10 dB, x = 0.27 100 Figure 4.12 Average BER as a function of Eb/Na in FS fading with CDR = 10 dB, x = 0.27. Note that dx is chosen to be 0.5, instead of 1 for the simulated 1-bit DF GMSK curve and d2 is chosen to be 1.5, instead of 2 for the simulated 2-bit DF GMSK curve 101 Figure 4.13 Average BER as a function of Eb/N0 in FS fading with CDR = 10 dB, x = 0.47 102 Figure 4.14 Average BKER as a function of Eb/N0 for (a) 1-bit conventional G M S K and 1-bit DF GMSK in flat fading (b) 2-bit conventional G M S K and 2-bit DF G M S K in flat fading (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.27 (d) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB x = 0.27 104 Figure 4.15 Average BKER as a function of Eb/NQ for (a) 1-bit conventional G M S K and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.47 (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.47 (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.27 (d) 2-bit conventional GMSK and 2-bit DF G M S K in FS fading with C D R = lOdB, x = 0.27 106 viii Figure 4.16 Average BKER as a function of Eb/Na for (a) 1-bit conventional G M S K and 1-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.47/ (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB,x = OAT 108 Figure 4.17 Average BKER as a function of Eb/N0 for 1-bit conventional G M S K and 2-bit conventional GMSK in (a) flat fading (b) FS fading with CDR = 20 dB, x = 0.27/ (c) FS fading with CDR = 20 dB, x = OAT (d) FS fading with CDR = 10 dB, x = 0.27/ 110 Figure 4.18 Average BKER as a function of Eb/NQ for 1-bit conventional G M S K and 2-bit conventional GMSK in FS fading with CDR = 10 dB, x = OAT I l l Figure 4.19 CDF of the number of bit errors in a block at Eb/NQ = 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in flat fading (b) 2-bit conventional GMSK and 2-bit DF GMSK in flat fading 112 Figure 4.20 CDF of the number of bit errors in a block at Eb/N0 = 20 dB for 2-bit conventional GMSK and 2-bit DF GMSK in flat fading 113 Figure 4.21 CDF of the number of bits errors in a block at Eb/Na = 40 dB for 1-bit and 2-bit conventional GMSK in flat fading 113 Figure 4.22 CDF of the number of bit errors in a block at Eb/N0 = 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.27 (b) 2-bit conventional GMSK and 2-bit DF G M S K in FS fading with CDR = 20 dB, x = 0.27/ (c) 1-bit conventional G M S K and 1-bit DF G M S K in FS fading with CDR = 20 dB, x = 0.4T (d) 2-bit conventional G M S K and 2-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.47/ 114 Figure 4.23 CDF of the number of bit errors in a block at Eb/N0 = 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.27/ (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.27/ (c) 1-bit conventional G M S K and 1-bit DF G M S K in FS fading with CDR = 10 dB, x = 0.47/ (d) 2-bit conventional G M S K and 2-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.47/ 115 Figure 4.24 CDF of the number of bit errors in a block at Eb/N0 = 40 dB for (a) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 20 dB, x = 0.2T (b) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 20 dB, x = 0.47/ (c) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 10 dB, x = 0.27/ (d) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 10 dB, x = 0.47/ 116 ix Acknowledgment I would like to express my gratitude to my research supervisor, Dr. Cyril Leung, for his continuous guidance and valuable advice throughout the course of the research work. Also, the sincere assistance from my colleagues is much appreciated. This work was partially supported by NSERC grant OGP0001731. Finally, I would like to thank my parents and my sister for their constant encouragement and support throughout my graduate study at UBC. Chapter 1 Introduction Today, the area of wireless communications is experiencing unprecedented growth, with the development of a cellular integrated services network, known as the Universal Mobile Telecommunication System (UMTS) [1] being under way. Currently, the Global System for Mobile Communication (GSM) forms the standard for most mobile voice networks worldwide [2]. The introduction of General Packet Radio Service (GPRS)1 [3], intended to support frequent non continuous transmission of bursty packet data up to data rate of 200 kbits/s in the G S M networks, will provide a proving ground for the integrated services to be supported in future UMTS networks. Therefore, G S M is now evolving from a voice-oriented network that generally supports circuit switching, to include a packet switching platform which improves bandwidth utilization for packet data transmission. Gaussian-filtered Minimum Shift Keying (GMSK) is the modulation scheme currently used in the G S M standard. It belongs to a modulation class, known as Continuous Phase Modula-tion (CPM) which has been closely examined due to its improved spectral characteristics and power efficiency relative to that of Phase shift keying (PSK) and Frequency modulation (FM). 1.1 Motivation and Objectives In connection with the current development trend in the G S M standard, it is worthwhile to study the bit error rate (BER) as well as the block (packet) error rate (BKER) performances of G M S K in a mobile radio channel. Differential detection is to be used, because it eliminates the need for precise regeneration of the reference carrier, difficult to achieve in fading conditions, 1 Before the introduction of GPRS, G S M is already supporting packet switching using the Short Message Service (SMS) intended to provide infrequent, non continuous transmission of very short data bursts of length up to 160 char-acters (seven bit data). 1 Chapter 1 Introduction otherwise. 2 The BER performance of GMSK in various channels has been investigated in [5], [9]-[20], [22],[23]. However, the BKER performance has rarely been studied. In this thesis, a simulation model of G M S K employing 1-bit and 2-bit differential detection is developed to study the BER and B K E R performance in non fading additive white Gaussian noise (AWGN) channel, flat and frequency selective (FS) Rayleigh fading channels. A number of papers which analyse differential detection of G M S K in flat and FS Rayleigh fading channels, assume that the intermediate-frequency (IF) filter at the receiver has a negligible effect on the signal spectrum but little justifi-cation has been given [9],[10],[13],[19],[21]. Therefore, a detailed analysis is also presented to compare the BER performance of G M S K when the IF filtering effect is taken into account and when it is not. The numerical results from the analysis are verified by comparison with the simula-tion results. Since the premodulation Gaussian filter at the transmitter for G M S K introduces significant ISI, a decision feedback scheme was proposed in [22] to partially remove the effect of destructive ISI. The performance of this scheme was simulated in [22] for a non fading environ-ment and it was analyzed in [13], [23] and simulated in [16] for flat fading channels; but not in FS fading channels. In this thesis, the BER and BKER performances of this scheme in flat and FS Rayleigh fading channels are investigated. In particular, the B E R performances over such channels will be analysed and compared with the simulation results. The B K E R performance is used to look at the bit error distribution under this scheme. 1.2 Outline of the Thesis This thesis is organized as follows. In Chapter 2, some background material including the mathematical modelling of mobile radio Chapter 1 Introduction channels is reviewed. 3 In Chapter 3, we analyze the BER performance of differentially detected G M S K with and without decision feedback in flat and frequency selective Rayleigh fading channels. In Chapter 4, the BER and BKER performances in AWGN, flat and FS Rayleigh fading channels are presented. In particular, the numerical BER results from chapter 3 are compared with the simulation results. In Chapter 5, the conclusion and some recommendations for future work are given. Chapter 2 Background Figure 2.1 shows the functional blocks of a digital communication system [6]. The input source may be either an analog or a digital signal. The source encoder maps the analog signal into a sequence of binary digits with as little redundancy as possible. The channel encoder encodes the binary information sequence so as to introduce some redundancy into the information. The added redundancy can assist the receiver to reconstruct the transmitted data from the received signal which is corrupted by noise and distortions. The output of the channel encoder is modulated onto a carrier. The focus in this thesis is on the modulator/demodulator section of the system. digital signal source transmitted signal Source Encoder Channel Modulator Encoder Transmitter earner Transmission Channel (^Demodulator received signal Noise, interference and distortion Receiver Clock (synch.) carrier reference (coherent) Channel Source Decoder Decoder output digital signal Figure 2.1 Block diagram of a digital communication system 4 Chapter 2 Background 5 There are a number of considerations that enter into the choice of a modulation technique for use in a wireless application. In a communication medium or channel with additive white Gaussian noise, Shannon derived a formula for the capacity, C (bits/s), as a function of the bandwidth of the channel, W (Hz), and the average received signal to noise power ratio, SNR [6] where Ps is the transmitted signal power, Eb is the transmitted energy per bit, Gn is the received (postdetection) noise power, Rb is the transmission rate in bits/s and N0 is the one-sided power spectral density (PSD) of AWGN. The significance of the channel capacity formula given above is that it puts an upper limit on Rb for reliable communication over a noisy channel. Specifically, one can transmit informa-tion, with as small an error probability as desired, as long as Rb is smaller than C. Using (2.1), we can plot Rb/W versus Eb/N0, giving the bandwidth efficiency plane [7]. From the plane, we can see that there is a trade-off to be made between the transmitted power Ps and the channel bandwidth W in maximizing the channel capacity and achieving a specified level of probability of bit error (BER). In designing a digital communication system, the objectives are to minimize the C = Wlog 2(l +SNR) (2.1) 2 2 P is denned as the desired (average) signal power at the point before AWGN is added. In this case, it is equal to the transmitted signal power. Chapter 2 Background 6 BER and the parameters Ps and W; but also to maximize Rb. However, these objectives are in conflict with each other. For example, if we reduce Ps, with Rb and W remaining constant in the system, then the BER is degraded. The ratio Rj/W is a measure of the bandwidth efficiency. For example, m-ary phase shift keying (MPSK) requires a bandwidth at IF (double-sided bandwidth) equal to the symbol rate, Rs [7]: W = Rs (2.2) Rb where Rv = . Therefore, the bandwidth efficiency of MPSK modulated signals can be ex-1 log2m pressed as R — = log 2m. (2.3) On the other hand, non-coherent m-ary orthogonal frequency shift keying (MFSK) modulation requires a bandwidth equal to mRs corresponding to the m different orthogonal waveforms. Hence, the bandwidth efficiency of MFSK modulation is equal to Rh \og0m w = — <2-4> Vv m At a bit rate of 9600 bits/s and a symbol rate of 2400 symbols/s, M P S K has a bandwidth efficiency of 2 (hence m=4) and requires an Eb/Na of 17.5 dB to achieve a B E R of 10~5, whereas 4-FSK requires an Eb/Na of only 8.1 dB but can only achieve a bandwidth efficiency of 0.5 [7]. Chapter 2 Background 7 2.1 Continuous Phase Modulation Most mobile radio products are designed with Class-C power amplifiers, which offer the highest power efficiency among the common types of power amplifiers. However, this class of amplifiers are highly nonlinear, which requires the signal to be amplified to have a constant envelope. Any amplitude fluctuations in the input signal will result in a spectral widening of the output signal, which gives rise to adjacent channel interference (ACI) as explained in the next section [2]. Continuous phase modulation (CPM) [8] is a class of modulation schemes that can achieve a good compromise between power and bandwidth efficiency while maintaining a specified level of BER at the expense of only a reasonable increase in system design complexity. In CPM, the phase is constrained to be continuous, making memory being present in the input to the phase or frequency modulator. The RF signal after modulation has a constant envelope which contributes to a couple of advantages. Firstly, constant envelope signals are less sensitive to amplifier nonlinearities due to the lack of fluctuations in the signal amplitude, resulting in a more compact power spectrum. Sidelobe spreading, as compared with that of non-constant envelope signals is reduced hence improving the bandwidth efficiency3. Secondly, they are more resistant to adjacent channel interference (ACI), interference due to signals of adjacent channels. This is due to their compact power spectrum so that more of these energy is contained in the channel bandwidth [6],[8]. In CPM, the transmitted signal s(t) is given as If we define bandwidth as the frequency band that contains 99.9% of the signal power, then the smaller the normal-ized bandwidth (bandwidth-bit duration product) of a modulation scheme, the higher the bandwidth efficiency it can achieve. Chapter 2 Background 2£„ s(t) = l^Cos(2nfct + Q>(t,b)) (2.1.1) where ' f $(t;b) = 2nh\ ]T bjgiy-jT) \dv h = 2fdT (2.1.2) (2.1.3) where Es is the transmitted energy per symbol, T is the symbol duration, h is the modulation index,/^ is the peak frequency deviation, and g(t) will be described later. Premodulation Filter NRZ Sequence HJf) d(t) s(t) FM Modulator 2nh Figure 2.2 General block diagram of continuous phase modulation Assuming the transmitted power Ps = E /T = 0.5 , (2.1.1) can be written as s(t) = cos(2nfct + §(t;b)) (2.1.4) The transmitted information is contained in the phase which is used to modulate a carrier Chapter 2 Background 9 in such a way that its phase is continuous. In Figure 2.2, {bj} denotes the non-return-to-zero (NRZ) sequence of amplitudes obtained by mapping fc-bit blocks of binary digits from the information sequences {a.j} into the amplitude levels ± 1 , ±3 , .. . ± (ra - 1) where ra = 2k. After the premodulation filter with frequency response Ht(f), the output signal d(t) can be expressed as oo d(t) = X bjg(t-jT) (2.1.5) j = -°° with g(t) = ht{t) ® p(t), where ® denotes convolution, p(t) is a rectangular pulse of duration T and unity amplitude, ht(t) is the impulse response of the premodulation filter and g(t) is the response of the transmitted rectangular pulse to the pre-modulation filter. If g(t) = 0 for t > T, the CPM signal is called full response CPM. If g(t) ^ 0 for t >T, then the CPM signal is called partial response CPM [8]. From the definition of F M modulation, one can observe that the phase response described in (2.1.2) is a function of the integral of the pulse g(t). The shape of g(t) determines the smooth-ness of the transmitted information carrying phase. The rate of change of the phase is proportional to the parameter h, which is the modulation index. The pulse g(t) is normalized such that oo J g(t)dt = 0.5 [8]. —oo A variety of CPM signals can be obtained by varying the three parameters, h, g(t) and w. A few popular schemes include Tamed Frequency Modulation (TFM), Raised Cosine (L-RC), Chapter 2 Background 10 Rectangular (L-REC) and Gaussian Minimum Shift Keying (GMSK). M S K is a member of the L - R E C scheme which has a rectangular pulse as its pulse response: g(t) = 1 0<t<LT 2LT . (2.1.6) 0 otherwise In MSK, L is 1, m = 2 and h is 0.5. It should also be pointed out that the premodulation filter is not required in the implementation of MSK. Following from (2.1.2), the phase of the carrier in the interval nT < t < (n + 1)7/ [6] is where n- 1 W;b) = \% £ bj+ Kbnq(t-nT) ; = — (2.1.7) „ 1 , ft-nT n- 1 e» = f X *y (2-1-8) ; = - o o ^(r) = \g(%)dx. (2.1.9) Substituting (2.1.7) into (2.1.4) gives Chapter 2 Background 11 1 , ft-nT s(t) = cos(2nfct + Qn + ^nbn^ T " (2.1.10) = cos {**{fc + ^fK} ~ \™bn + G„) nT < t < (» + 1)7" The equation above indicates that MSK signal is a sinusoid having one of the two possible frequencies in the interval nT <t<(n + 1)T .If we define these frequencies as fi=fc-jf (2.1.H) fl = fc + 4T then equation (2.1.10) can be written as st(t) = cos(27u/.r + e n + i m t ( - l ) ' _ 1 j , nT<t<(n+\)T, i = 1,2. (2.1.12) The frequency separation A / = f2-f\ = is the minimum frequency separation that is necessary to ensure orthogonality of the signals s^t) and s2(t) over a signalling interval of length T and explains the name Minimum Shift Keying. One of the advantages of M S K is that it does not introduce any intersymbol interference (ISI). The transmitted pulse corresponding to each symbol is confined within the bit duration, so that there is no adjacent interference between symbols. MSK allows a simple implementation as in the F M form depicted in Figure 2.2. It can also be realized in the form of 4-phase PSK as it is the only linear modulation in the CPM family [2]. Gaussian-filtered Minimum Shift Keying (GMSK) was invented in an attempt to improve the M S K power spectrum [27],[28]. In GMSK, the power spectrum is made more compact, with Chapter 2 Background 12 spectral sidelobes further reduced by applying a lowpass filter (i.e. the premodulation filter as shown in Figure 2.2) to the data stream prior to modulation in its F M form. The constant envelope property of C P M is preserved as filtering is performed before modulation. The premodulation filter chosen is a Gaussian lowpass filter because it satisfies certain properties suitable for re-shaping the output power spectrum, hence the name GMSK. The impulse response of the premod-ulation Gaussian filter, ht(t) is given by where K is an arbitrary constant (the choice of K is elaborated in Chapter 3) and Bt is the 3-dB bandwidth of the filter. Figure 2.3 shows the information carrying phase §(t;b) of MSK and G M S K for a particu-Figure 2.2, is a differentially encoded version of the actual transmitted information sequence. Note that the phase changes for GMSK are smooth due to pulse shaping by premodulation filter-ing, whereas M S K has linear, sharp phase changes during data transitions. To illustrate the bandwidth efficiency of CPM signals in general, we can define the bandwidth as the frequency band that contains 99.9% of the signal power. The 3-dB bandwidth of the premodulation Gaussian lowpass filter is denoted by Bt. For MSK where BtT = °°, the normalized bandwidth is 2.76 and for G M S K with BtT= 0.25, it is approximately 1 [27]. Hence, G M S K (with BtT < o o ) achieves a higher bandwidth efficiency than MSK at the expense of the introduction of ISI. In general, as BtT decreases, the bandwidth efficiency of the signal increases but the spreading of the pulse also (2.1.13) lar data sequence. Note that in differentially encoded GMSK, the data sequence {bj} shown in Chapter 2 Background 13 becomes more severe, causing more ISI. (a) (b) Figure 2.3 Information carrying phase pattern of (a) non-differentially encoded GMSK (b) differentially encoded GMSK Figure 2.4 illustrates the response of the Gaussian filter to a rectangular pulse of period T and the effect of variation of BtT on pulse spreading. Both MSK and GMSK can be coherently or non-coherently detected, as further described in Sections 2.3 and 2.4. 1 1 1 1 1 i 1 1 1 B(T = ~(MSK) -A BtT = 0.5 h B(T = 0.25 jX y B t T = 0 1 0 1 2 3 4 5 6 7 8 9 Figure 2.4 Response of Gaussian lowpass filter to a rectangular pulse of duration T. Chapter 2 Background 14 2.2 Rayleigh Fading In most mobile radio channels, the transmitted wave travels to the receiver via multiple paths, reflected from objects such as buildings or scattered from objects close to the receiver. The channel in which the transmitted signal can propagate via multiple paths is called a multipath channel. Associated with each path, there exists an attenuation factor and a propagation delay; the former accounts for the amplitude variations in each multipath component and the latter accounts for the phase variations. Due to the constructive and destructive interferences of these components in the vincinity of the receiver, there will be amplitude and phase variations in the received signal. Fading occurs when the receiver moves through an area of destructive interfer-ence [2], [6]. A set of channel correlation functions and power spectra that define the characteristics of a fading multipath channel can be developed. The characterization involves the following parame-ters: the time variations of the channel can be measured by the Doppler spread and the coherence time. The Doppler spread fD is the width of the Doppler spectrum, which represents the strength of the Doppler shift at different frequencies, caused by the movement of the mobile. The coherence time tc is approximately equal to the reciprocal offD and is a measure of the time interval over which the channel conditions remain approximately constant. The frequency variations of the channel can be measured by the coherence bandwidth and the multipath delay spread. The coherence bandwidth Afc is a measure of the bandwidth over which the signals spaced in frequency, will be similarly affected by the channel. The reciprocal of Afc is approxi-mately equal to the multipath delay spread4, xd, which is a measure of the time dispersion of a 4 Another measurement of delay spread is the root mean squared (rms) delay spread [42]. Chapter! Background 15 transmitted signal caused by the channel [6], [29], [30]. Fading channels are characterized as flat if all frequencies of the signalling bandwidth are similarly affected by the channel; otherwise they are frequency selective, where different frequencies are subjected to different fading. Therefore from the above definition, if the transmitted signal bandwidth is small in comparison with A / c , the channel is said to be flat or frequency non-selective; otherwise it is said to be frequency selective (FS). The sum of these multipath components or phasors, is a complex Gaussian process. The magnitude of the process has a Rayleigh distribution. It will change into a Ricean distribution in the presence of a line-of-sight component [2]. In a Rayleigh fading channel, the transmitted signal is randomly modulated with a complex Gaussian process denoted as Zs(t), where and xs(t), ys(t) are independent, zero mean Gaussian lowpass processes. A model for mobile radio channels has been proposed by Clarke [31]. The Doppler power spectrum derived from this model is given by Zs(t) = xs(t) + jys(t) (2.2.1) \f\<f. D (2.2.2) 0 1/1 >/ D where fD is the maximum Doppler frequency (the Doppler spread) given by • - Z - Vll D ~ X - c (2.2.3) Chapter 2 Background 16 In Equation (2.2.3), v denotes the speed of the mobile unit in metres/second (m/s),/c is the carrier frequency in Hz and c is the speed of light which is equal to 3 x 1 0 8 m/s. The implementation of the Rayleigh fading channel in the software simulation will be further described in Chapter 4. Since the multipath delay spread is approximately equal to the reciprocal of A / c , we can also categorize a channel as flat or frequency selective according to the delay spread. Figure 2.5 illustrates the differences between flat and FS fading. In general, multipath signal components of which their path delays are closely distributed about a mean delay would be regarded as one Rayleigh faded signal (via the Central Limit Theorem). The term 'closely distributed' means the difference in path delays are small relative to a symbol duration. The mobile channel in which only one such path exists is known as a flat Rayleigh fading channel (path 1). Signal components with path delays closely distributed about a mean delay different from that of path 1, would be regarded as a second delayed Rayleigh faded path, represented by path 2 in Figure 2.5. If the \ Receiver Figure 2.5 A diagram to illustrate the differences between fiat and frequency selective fading Chapter 2 Background 17 difference in the mean path delays, T , is significant relative to a symbol duration, then there exists frequency selectivity. A frequency selective channel model contains multiple paths of which their mean path delay differences are significant with respect to a symbol duration. In this thesis, we will use the two ray model (represented by path 1 and path 2 in Figure 2.5) which is simple and adequate enough to simulate frequency selectivity in mobile radio channels. The model can be implemented as shown in Figure 2.6. It has been reported in [38] that channels with rms delay spread less than about 0.1 of a symbol duration are considered flat; otherwise they are considered as frequency selective. PATH 1 AWGN signal at output of transmitter Rayleigh faded signal Zs(t) and Zj(t) are independent complex Gaussian processes PATH 2 Figure 2.6 Two Ray Model: the block diagram of a frequency selective Rayleigh fading channel 2.3 Coherent Detection In coherent detection systems, the receiver requires an estimate of the carrier frequency and phase in order to recover the transmitted data in the received signal. A majority of methods in combating fading in coherent systems involve the transmission of pilot tones [33],[34] or pilot sequences [35],[36]. The pilot scheme involves the transmission of a pilot tone at a convenient frequency in the data spectrum. The pilot tone is extracted at the receiver and the channel impair-ments can then be deduced from the tone. The information from the channel can assist in the Chapter 2 Background 18 reconstruction of a coherent reference signal. The disadvantage of this scheme is that it resembles a double sideband modulation scheme which is bandwidth inefficient because the sidebands need to have zero response around dc, to allow bandwidth for pilot transmission. An improvement to this scheme is proposed in [34]. 2.4 Non-coherent Detection In contrast to coherent systems, non-coherent detection does not need to generate a carrier reference at the receiver, hence allowing simple implementation. Non-coherent detection can be subdivided into limiter/discriminator [4],[11],[15],[17] and differential detection [9],[10],[13],[16],[18]. Limiter/discriminator consists of a hard limiter to remove the amplitude variations of the received wave, followed by a bandpass filter [26]. The resulting wave is a constant-envelope sinusoidal and is demodulated by a discriminator. However, this detection scheme suffers from the F M threshold effect when the signal carrier to noise ratio falls below a certain level. In differential detection, information, instead of being transmitted using absolute carrier phases, is encoded into carrier phase differences [24]. Information is therefore recovered at the receiver by obtaining the difference in phase of received signal at current time and at some time instant in the past, usually at multiples of the symbol period. Since information is transmitted and detected through phase differences of the carrier, the absolute carrier phase is not important and so the need for carrier recovery is eliminated. Differential detection does not suffer from the threshold effect and is affected little by severe channel delay distortions [21]. Its performance in AWGN is worse than that of coherent detection, but it has an improved performance in fading channels [9],[10],[27]. This is because it is difficult to precisely regenerate the reference carrier in coherent detection under fading conditions. Moreover, differential detection improves perform-ance by cancelling the phase distortion between adjacent symbols during phase subtraction. In the Chapter 2 Background 19 case of GMSK, the inherent ISI due to pulse shaping can be partially eliminated using decision feedback [22]. The differential detection of GMSK is described in Chapter 3. 2.5 Intersymbol Interference Intersymbol interference (ISI) introduced by the premodulation filter in the transmitted G M S K signals was described in Section 2.1 and in this section, we will describe the ISI induced by the channel. In any communication systems, it is desirable to transmit information at a rate as high as possible while maintaining the reliability of transmission. Due to non ideal channel frequency response characteristics, a succession of pulses transmitted at rates on the order of the channel bandwidth W will be distorted and they overlap each other so that each of the pulses can no longer be distinguished independently. The overlapping affects decision making at sampling instants since there is interference contributed from other adjacent symbols, giving rise to ISI. A signal pulse can be designed that allows us to transmit at symbol rates equal to or smaller than 2W the Nyquist rate, with zero or a controlled amount of ISI. Non-constant envelope digital modula-tion schemes make use of filters to satisfy Nyquist's criteria for zero ISI or duobinary coding which introduces ISI in a controlled manner in order to achieve a transmission rate equal to 2W [6],[25], assuming that the channel's response characteristics are known a priori. If the channel conditions are not known a priori, we have to employ equalization methods to eliminate or reduce ISI [6]. In the case of GMSK, it is not possible to use filters for pulse shaping to avoid ISI. This is because the pulse shaping in G M S K is performed using the premodulation Gaussian lowpass filter which can improve on the bandwidth efficiency as well as maintaining the constant envelope property of GMSK signals. In this thesis, we will only look at the effect of ISI in G M S K signals caused by the premodulation filter and consider the use of decision feedback [22] to combat ISI. Chapter 3 BER Performance of GMSK in Rayleigh Fading A careful analysis of BER performance of a digital communication system generally takes into account the following factors which cause signal distortions during transmission: additive white Gaussian noise, the statistical characteristics of the flat fading channel, the frequency selectivity in the channel, Doppler spread in the carrier frequency due to the motion of the mobile unit, intersymbol interference due to the premodulation filter, intersymbol interference due to bandlimiting by the IF filter, the co-channel interference from a user in another cell and the adjacent channel interference which arises between channels whose frequency bands are in proximity. Some of these factors have been described in the previous chapter. In this chapter, we will analyse the IF filtering effect on the BER performances of 1 and 2-bit differentially detected G M S K (hereafter called 1 and 2-bit GMSK) in flat and FS Rayleigh fading channels. We will also look at the BER performances when a decision feedback scheme is employed on such channels. As mentioned in Chapter 1, the BER performances of 1 and 2-bit G M S K have been analysed in [9], [10], [13] under the assumption that the IF filtering effect is negligible (the 'unfiltered' case) in flat Rayleigh fading channels but the IF filtering effect on the BER perform-ances has not been analysed. There are also papers on the BER performances of 1 and 2-bit G M S K in FS Rayleigh fading channels [14], [18], [20] where the 'unfiltered' case is assumed in [14], and the filtering effect has been taken into account (the 'filtered' case) in [18] and [20]. However, there has not been a direct comparison between the B E R performances for the 'unfiltered' and the 'filtered' case. 20 Chapter 3 BER Performance of GMSK in Rayleigh Fading 21 In [9] and [10], the modulation index, h, of the GMSK signal is chosen so as to minimize the probability of error. In this thesis, we will use h = 0.5 which allows G M S K to be demodu-lated using coherent detection as that employed in MSK. Hence, choosing such a value of h will allow us to compare the performance of differential detection with results from other papers in which coherent detection is looked at. We will use the approach developed in [10] which deals with the performance of 2-bit G M S K in flat Rayleigh fading channel to evaluate the B E R performances of 1 and 2-bit GMSK in both flat and FS Rayleigh fading channels. This approach eliminates the need for computing a double integral which appears in [20],[21]. In the case of a non-fading channel, AWGN with two-sided power spectral density (PSD) N(/2, is simply added to the transmitted GMSK signal before the input of the receiver. The BER performance of 1 and 2-bit differentially detected G M S K in AWGN channel is analysed in [5]. The 2-bit detection performs better than the 1-bit detection and the performance advantage is greater for lower values of the bandwidth-time product, BtT of the Gaussian premodulation filter. The improvement in the BER performance for the 2-bit case is due to the asymmetric eye opening of the 2-bit detector eye patterns, enabling one to bias the detection threshold towards the larger opening of the eye, as compared to that of the 1-bit case. This makes 2-bit detection less sensitive to errors caused by AWGN. 3.1 B E R Performance of 1-bit detection in Flat Rayleigh fading channel Figure 2.2 in Chapter 2 illustrates a G M S K transmitter, consisting of a premodulation Gaussian lowpass filter and a F M modulator with modulation index, h = 0.5 giving a peak frequency deviation/^ equal to 1/(47/). The output of the premodulation filter, d(t) and the Chapter 3 BER Performance of GMSK in Rayleigh Fading 22 transmitted signal at the output of the modulator, s(t) are given by (2.1.5) and (2.1.4) respectively. Figure 3.1 shows the receiver for the 1-bit detection case. The output of the 1-bit detector V'(t) is o obtained from multiplying the output of the IF filter, r(t) by a 1-bit period delayed, 90 phase shifted version of itself. The product is then passed through a lowpass filter to produce V(t), which is sampled at t = IT to give V(IT), where / is the Ith sampling interval: le {...,-1,0, 1,2, ...} . A hard decision is then made on V(IT) to decide whether a "1" or a "0" was sent. In an ideal channel (no fading and no noise), the receiver output is given by (3.1.23). The operation of the 1-bit differential detector will be described mathematically in Section 3.1.1. n(t) Ht) Detector 90° Delay T phase shift V(IT) Sample Hard Decision "0" or "V Sj( t): signal after fading with average power o 2 n(t): AWGN which produces narrowband noise with 2 average noise power <s at the IF filter output Figure 3.1 Block diagram of a 1-bit GMSK differential receiver Chapter 3 BER Performance of GMSK in Rayleigh Fading 23 3.1.1 Unfiltered case After propagation through the flat fading mobile radio channel, the signal after fading, 2 sj(t), with average power <JS, is given by sf(t) = Re[(xs(t)+jys(t))-&xpMs(t) + (Oct)] (3.1.1) = /J«[{cos(co c r + (|),(0) + j s in ( (D c r + <|>,(0)} • {xs(t) + jys(t)}] = x,(f)cos(cocf + ^(r)) - ys(t) sin(cocr + 0,(0) where xs(t)and ys(t) are independent stationary zero mean random Gaussian lowpass processes that define the Rayleigh fading channel and §s(t) is the information carrying phase, as described in Chapter 2. We first consider the 'unfiltered' case in which the filter bandlimits the noise but does not distort the signal. Assuming the noise is Gaussian and broadband prior to IF filtering, the filtered noise, n(t) can be written as n(t) = xn(t)cos®ct-yn(t)sm(£>ct (3.1.2) where xn(t) and yn(t) are independent stationary zero mean Gaussian lowpass processes. We denote r(t) as the sum of Sj(t) and n(t): r(t) = sf(t) + n(t). (3.1.3) The received signal, denoted by r(i) at the output of the IF filter is the sum of the filtered noise and the filtered signal after fading5, which should have been distorted by the filter. In the The signal after fading will be referred to as signal, hereafter. Chapter 3 BER Performance of GMSK in Rayleigh Fading 24 'unfiltered' case, we assume that the IF filter does not distort the signal Sj(t) and hence the received signal is equal to r(t). r(r) = r(0 = xs(t)cos((act + ^s(t))-ys(t)sm((i)ct + ^s(t)) + xn(r)coscoc? - yn(t)smo)ct = [xs(t)costys(t) -ys(t)smtys(t)]cos(i)ct - [xs(t)sm§s(t) + ys(t)cos$s(t)]sin(Oct + xn(t)cos(dct - yn{t) sincocr = X(r)cosco cr- Y(t)s'm(Oct = Re{Z(t)exp(ja>ct)} where Z(0 = X(t) + jY(t) , (3.1.5) X(t) = xs(t) cos^CO-y,(0sin<|),(0 + xn(t) , (3.1.6) Y(t) = xs(t)sm<$>s(t) + ys(t)cos<bs(t) + yn(t) . (3.1.7) The processes xs(t), ys(t), xn(t) and yn(t) are independent and of zero mean, hence the follow-ing relations can be obtained: xs(t) = ys(t) = = yB(r) = 0 Xi(t)yi(t) = yjijxjf) = Xi(t)Xj(t) = yjfjyjf) = 0 *,-(0y/0 = y,.(f)*;(0 = 0 xi(t)yj(t + ^3) = Xiit + fyyjit) = yi{t)Xj{t + t) = y,(r+ £)*,•( f) = 0 xJiTfyy~(7) = y.(t + Z>)xi(t) = 0 *,•(* +$)*,•(') = yi{t + Z)yi(t) = a?p.(cj where i, j = s,n j which will be used later in this section. In (3.1.8), the 'bar' denotes statistical average. Chapter 3 BER Performance of GMSK in Rayleigh Fading 25 Using (3.1.5), (3.1.6) and (3.1.7), Z(f) can be rewritten as Z(0 = xs(t)cos(<bs(t))-ys(t)sm($s(t)) + j[xs(t) sin(<|>,(0) + y5(f)cos(<MO)] + xnW + 7>„(0 = Z^OexpEy^COl+Z^O (3.1.9) where Zs(t) and Zn(t) are independent complex zero mean Gaussian processes given by Z(t) = x.(t)+jy.(t), z n(0 = xn(t)+jyn(t). (3.1.10) (3.1.11) The output of the detector V'(t) (see Figure 3.1) is given by V'(t) = r(t)-r(t-T)\n positive phase shift (3.1.12) = Re{Z(t)exp(j«>ct)} • Re{Z(t-T)exp(j(dc(t-T))exV(jn/2)} where Z(r) is defined in (3.1.5) and r(t-T)\n = Re{Z(t-T)exp(jac(t-T))} - positive phase shift positive phase shift = /?e{Z(r-7/)}cosco c(f-7/)| Im{Z(t-T)}smm(t-T) I - positive phase shift positive phase shift = Re{Z(t-T)cos(oJt-T + n/2)}-Im{Z(t-T)sin(oJt-T + n/2)} (3.1.13) = Re{Z(t-T)expU®c(t-T))txp(jn/2)} Representing Z(t) by its real and imaginary parts as in (3.1.5) and multiplying out (3.1.12) Chapter 3 BER Performance of GMSK in Rayleigh Fading 26 yields, V\t) = Re{Z(t)expU®ct)} • Re{Z(t-T)exv)(j(£>c(t-T))exip(jn/2)} = (X(r)coscoc?-r(r)sincocO . (-X(t - T) sin®c(t - T) - Y(t - T)cos(&c(t - T)) = -l-X(t)Y(t-T)[cos(4nfct-2nfcT) + cos(2nfcT)] 1 (3.1.14) + ^Y{t)X(t- T)[-cos{AKfct-2nfcT) + cos(2nfcT)] - l-X(t)X(t- T)[sm(4nfct- 2nfcT) - sin(2nfcT)] + l-Y(t)Y(t-T)[sin(4llfct-2KfcT) + sm(2KfcT)] In (3.1.14), the product contains signal components at around 2fc, which are rejected by the lowpass filter to yield V(r) (see Figure 3.1). Assuming that fc is chosen such that 2nfcT = (OcT = 2%m where m is a positive integer, the output of the lowpass filter is V(t) = - \x(t)Y(t-T)+l-Y{t)X(t-T) = ±Re[-j(X(t) + jY(t))(X(t-T)- jY(t- T))} = l-Re[-jZ{t)Z*(t-T)] (3.1.15) Chapter 3 BER Performance of GMSK in Rayleigh Fading Expanding (3.1.15) and substituting (3.1.9) for Z(t) gives 27 V(t) = ±Re{-j[Zs(t)exp(Jt(t))+Zn(t)]-[Z*(t-T)exp(-j<$>s(t-T)) + Zn*(t-T)]} = X-Re{-jZs(t)Z*{t-T)zxVj($s{t)-$s{t-T)) + ZN{t)} \ (3.1.16) = l-Re{ [- jxs{t)xs{t -T)- jys(t)ys(t -T)- xs(t)ys(t -T) + ys(t)xs(t- T)] • [cos A(|>5(r) + jsin A^(0] + ZN(t)} = ±[xs(t)xs(t -T) + ys(t)ys(t- T)] sinA$s(t) + \[ys(t)xs(t- T)-xs(t)ys(t- T)]cosA^s(t) +n\t) where ZN(t) represents all terms associated with the noise processes Zn(t) and Zn*(t - T), and n'(t) is the real part of ZN(t). The sampled output of the receiver V(IT) (see Figure 3.1), is obtained by sampling V(t) at time t=lT. The phase change over the 1-bit period is A(t>J;/ = 4>,(zr)-4>a(zr-r) (3.1.17) oo IT oo IT-jT = 2nfd X bj J g(v-jT)dv = 2izfd £ bj J g(v)dv j = -oc (l-l)T j = -oo a-\)T-jT where g(t) is the response of the premodulation Gaussian filter with impulse response ht(t) given by (2.1.13) to a transmitted rectangular pulse and bj is the jth transmitted symbol, i.e. 8(t) = (3.1.18) Chapter 3 BER Performance of GMSK in Rayleigh Fading 28 Equation (3.1.17) can be written as oo Ws,i= 1 bjQi-j (3- 1- 1 9) j = -~ where 0/_ ;is equal to Qi-j = ^rf 8(u-JT)du. (3.1.20) J ZlJ(t-l)T When 0.2 < BfT < 0.3, 9,_ -is negligible for | / - j\ > 3 [22] (Table D . l in Appendix D shows the values of 0, • for BtT= 0.25), therefore A$s l can be simplified to *Ki = b^Q.i + b i ^ + b f i o + b ^ + b , ^ (3.1.21) Table D.3 in Appendix D shows the differential phase angles Atys l for all possible input sequence combinations. The transmitted signal s(t) after propagation through an ideal channel where there is no fading and no noise, is just the signal at the output of the modulator, defined in equation (2.1.4) in Chapter 2 and repeated here for convenience: s(t) = cos(cocr + (j)5(0)- (3.1.22) Comparing (3.1.1) with (3.1.22), it is clear that the lowpass Gaussian processes xs(t) and ys(t) in (3.1.1) is equal to 1 and 0, respectively for all values of t, when the channel is not fading. Chapter 3 BER Performance of GMSK in Rayleigh Fading 29 Hence, in an ideal channel, the receiver output V will be6 V = ^ s i n A ^ . (3.1.23) In Table D.3 (Appendix D), it is noted that Atys t (or equivalently sinAc^ t) is positive when the transmitted bit bt is T and negative when bl is '0'. Consequently, the decision rule at the receiver is: decide the transmitted bit is a T if V > 0 and vice versa. In an ideal channel, correct decisions are always made and therefore the decision b\ made at time t = IT is always equal to the corresponding transmitted bit, denoted as bt. After propagation through the mobile channel, the transmitted bandpass signal spectrum is being shaped by the Doppler spectrum defined in (2.2.2) in Chapter 2 and the bandpass noise spectrum is being shaped by the Gaussian IF filter with 3-dB bandwidth Bt, and frequency response H^{f) given by Hif(f) = H(f-fc)+H(-f-fc) (3.1.24) where H(f) is the equivalent baseband frequency response of the filter: tf(/) = K e x p { - ( ^ J . f } . (3.1.25) In (3.1.25), K is an arbitrary constant. Assuming that the receiver has an omnidirectional antenna [9], the equivalent baseband spectra of the unmodulated carrier fading signal7 and noise, Hereafter, we omit the time dependence of V on T, assuming it to be understood. Chapter 3 BER Performance of GMSK in Rayleigh Fading 30 denoted as Ws(f) and Wn(f), respectively, can be obtained by performing a frequency transla-tion of / (the carrier frequency) towards the baseband, on their bandpass spectra: W , ( / ) = 2 ^ \f\<fD i— \J \ — J u 0 \f\>fD Wln2 2 Wn(f) = —^=exp\ -[ ) ln2 \ (3.1.27) where//) is the maximum Doppler frequency. In (3.1.27), Wn(f) is obtained by multiplying the one-sided noise spectral density Na by H(f) defined in (3.1.25) and choosing K, where a2n = N0Bn (3.1.28) and Bn is the equivalent noise bandwidth given by B„ = (3.1.29) The probability density function (p.d.f.) of the sampled output of the receiver V can be derived by taking the inverse Fourier Transform of the characteristic function F(v) of Vas follows [10]. Define a column matrix Z 7 The unmodulated carrier fading signal is equal to Re{Zs(t)exp(ja>ct)} , where Z^(f) is defined in (3.1.10). Chapter 3 BER Performance of GMSK in Rayleigh Fading 31 z = Z{IT) Z((l-\)T)_ z2 (3.1.30) Its covariance matrix C z is denned to be C 2 = i ( Z - Z ) * ( Z - Z ) (3.1.31) where the bar denotes taking the statistical average. Using the relations defined in (3.1.8), it is found that Zj = Z 2 = 0 and hence C z can be written as C = -^ 2 7 * ^ 1 7 * Zj z 2 z , * z 2 Z 2 * Z j z 2 (3.1.32) = a 1 p* IP 1. where p is the correlation coefficient between Z 2 and Zj [9],[21], and is given by a2p = a52p5(r)exp(;A(l)^) + ^p„(r) (3.1.33) In (3.1.32), a represents the average power of the received signal, which is given by: 2 2 2 (3.1.34) 2 2 In (3.1.34) and (3.1.33), as, G„ are the average powers of the signal, Sj-(t), and noise, respec-Chapter 3 BER Performance of GMSK in Rayleigh Fading 32 tively; ps(t) and p„(r) are the normalized autocorrelation functions of the unmodulated carrier fading signal and noise, respectively. We start by obtaining ps(T) and pn(T) , which we find by taking the inverse Fourier Transform of the spectra defined in (3.1.26) and (3.1.27), respectively. This yields [9] ps(T) = J0(2nfDT) , (3.1.35) p„(T) = exp ln2 (3.1.36) 2 2 Defining the average signal-to-noise ratio (SNR) to be S = <5s/<5n , p is evaluated as p = ^ ^ y o ( 2 ^ D ^ e x P O ' A ( l > , ) / ) + ( 5 ^ ) e x P { - ( ^ „ : r ) 2 / l n 2 } • C 3 - 1 - 3 7 ) The characteristic function F(v) of the sampled output V is defined as a Fourier Transform on the p.d.f. of V and can be shown to be equal to [39] F(v) = [det(I-2jvC * Q ) ] _ 1 (3.1.38) where I is a 2 x 2 identity matrix and Q is the 2 x 2 Hermitian matrix of the quadratic form of V. Using (3.1.38), F(v) is evaluated in Appendix A as F(v) = [ l-2va 2 ( p - p * ) + 4vV ( l - | p | 2 ) ] 1 . (3.1.39) Solving the quadratic equation in the denominator of F(v), the two poles of the character-Chapter 3 BER Performance of GMSK in Rayleigh Fading 33 istic function can be found to be Kpj±Ji^pr) ^1.2 2 , ,2 2a ( l - |p | 2 ) (3.1.40) where pr and py are the real and imaginary parts of p , respectively. The p.d.f. of the receiver output V is equal to 1 P(V) = ^JF(v)e-JvVdv (3.1.41) Using (3.1.41), it is shown in Appendix B that the conditional p.d.f.'s of V are given by P{V\bl = ±\) = 1 exp ( 1 2 ! " P r exp 2a 2 ( l - |p | 2 ) 2cr 2(l-|p| 2) y<o y>o (3.1.42) In (3.1.42), pr and py are the real and imaginary parts of p (see (3.1.37)), respectively, with p being a function of Atys 1 8 which depends on the input bit, bt, as described earlier. According to the decision rule described earlier, error occurs if V < 0 when b\ = 1 or vice versa. The probability of errors for each of the cases are given as follows: In the FS fading case, p is also a function of A<() . , which depends on b,, as defined in Section 3.2. Chapter 3 BER Performance of GMSK in Rayleigh Fading 34 Pel = P(V<0 |6 Z = 1) = \P(V\b,= \)dv = \ 1 - (3.1.43) ( Pe2 = P(V>0|A, = - 1 ) = J/>(V|i, = - l ) A = I 1 + — ^ (3.1.44) The unconditional probability of error is (3.1.45) Substituting (3.1.37) into (3.1.43) and (3.1.44), (3.1.45) is equal to P = -e 2 X7 o (27t / Dr)sin(0 o + A(p5 t) f 1 + 2 1 + A / l - [ X / o ( 2 7 t / D r ) c o s ( 0 o + A(p,>/) + Fexp{-(7i5 iT) 2/ln2}] : X7 o (27c / Dr)sin(-0 O + A(p5 ,) V 1 - [XJ0(2nfDT)cos(- 6 0 + Acp, {) + yexp{-(7Cfi IT)Vln2}] (3.1.46) S 1 where X = -—-, Y = -—-, 0O represents the signal and A(p5 t represents the ISI terms: o -f* 1 o "H 1 ' A ( P, , / = & / + 2e-2 + ^ +10-l+Vl0l+V2e2- (3.1.47) Chapter 3 BER Performance of GMSK in Rayleigh Fading 3.1.2 Unfiltered Case with Decision Feedback 35 HO * 1 Delay T 90° F F phase shift F Lowpass Filter Sample Hard Decision F F V(t) V(t) V(IT) or V Delay T Figure 3.2 Block diagram of a 1-bit DF GMSK receiver Decision feedback receiver improves the system performance by partially removing the effect of ISI. At the sampling instant t = IT, when bt is to be decided, the receiver has already made a decision, bi_\on the transmitted bit A phase shift X equal to t>i_\Qx can be introduced to eliminate the ISI contribution due to bl_ x§x. The effect of ISI is always destructive in states 4 and 5 since bl^b[_l and therefore these two states are located closest to the thresh-old, as shown in Figure 3.3(a). With decision feedback, we move these two states away from the threshold. The new differential angles for these two states are as follows: D F _ JA (rV/ + 0 l for state 3 A<^s'1 ~ \A$S z - 9 J for state 4 (3.1.48) Chapter 3 BER Performance of GMSK in Rayleigh Fading 36 In doing so, some of the states are moved closer to the threshold (see Figure 3.3(b) and Table D.3 in Appendix D). However, this only causes minimal BER performance degradation because states 3 and 4 are the critical states in determining the performance. Therefore, the unconditional probability of error is identical to (3.1.46) with A(j)5 t DF replaced by A$s l as follows: = bfi. + A ^ (3.1.49) where A%j = bl + 2Q_2 + bl+lQ_1+bl^Ql + bl_2Q2-b[-lQl (3.1.50) Chapter 3 BER Performance of GMSK in Rayleigh Fading 37 3.1.3 Filtered Case In the previous section, we considered the 'unfiltered' case where the IF filter's effect is only to bandlimit the noise and not to distort the signal. In reality, the IF filter at the receiver distorts the signal's phase and amplitude. Its effect can be taken into account by convolving the received signal with the filter's impulse response h^(t) given by hif(t) = 2Re{h(t)exv}(j(i)ct)} (3.1.51) where h(t) is the equivalent lowpass impulse response of the IF filter, given by the inverse Fourier Transform of the filter's baseband frequency response H(f) defined in (3.1.25) 2n2(B/2)2t2~ hT2 (3.1.52) where Bi has been defined in Section 3.1.1. The 'filtered' received signal from the output of the IF filter is denoted as r(t) which we can express in a similar form as r(t) defined by equation (3.1.4) for the "unfiltered" case (Section 3.1.1): r{t) = X(r)cosco cr-Y(t)sm(£> ct , (3.1.53) = Re{Zif(t)exp(j(Dct)} where Z^(r) is the complex envelope of r(t). The 'filtered' signal r(t) is related to the 'unfiltered' signal r(t) by the convolution integral: Chapter 3 BER Performance of GMSK in Rayleigh Fading 38 ~r(t) = r(t)®hif(t) oo = J r(u)hif{t - u)du • (3.1.54) —oo o o = j Re{Z(u) exp(y'cocw)} • Re{h(t - u)exp(j(dc(t - u))}du —oo Assuming that the center frequency of the IF filter is the carrier frequency, it can be shown that the filtering effect is the same regardless of whether the signal is at baseband or at IF. That is, the bandpass system can be equivalently represented by the complex envelope Z(f) given in (3.1.9) and the lowpass function h(t) [26]. The complex envelope Z^(t) of r(t) is related to the complex envelope Z(t) of r(t) by the following convolution: Zif(t) = Z{t)®h{t) (3.1.55) oo = J Z(u)h(t - u)du —oo where <8> denotes convolution. It is only required to perform the convolution integral defined in (3.1.55) over the interval, denoted as Tt, for which the value of h(t) is not negligible. Assuming that the fading rate is chosen such that it is much less than the bit rate, i.e. | / D | « \/Th, then the complex amplitude Zs(t) defined in (3.1.10) is essentially constant over the interval Ti. In this case, Z^(t) in the absence of noise, denoted by Z^Jt) is given by Chapter 3 BER Performance of GMSK in Rayleigh Fading 39 Z'if(t) = \Zs{u)exV[j{$s{u))]h{t-u)du = [xs(t) + jys(t)] j exp[j(^s(u))]h(t - u)du = xs(t) j coss(u))h(t - u)du Ti - ys(t) J s'm($s(u))h(t - u)du Ti + jxs(t) j sin(§s(u))h(t - u)du Ti + jys(t) j cos(§s(u))h(t - u)du (3.1.56) Consequently using (3.1.56), the received signal r(t) in (3.1.54) can be expressed as ~r{t) = Re{Zif{t)&xxy(j(act)} = Re{(Z'if{t) + Zn{t))exV(jact)} n 1 = Re{[{xs(t)Ac{t)-ys(t)As{t) + xn(t))+ j(xs(t)As(t) + ys(t)Ac(t)+yn(t))]expU(Oct)} = Re{(X(t) + jY(t))exVUact)} where Zn[t) is the zero mean complex Gaussian process for noise. From above, it can be seen that Zif(t) = X(t)+jY(t) = Zs(t)AC0(t)+Zn(t) where X(t) = xs(t)Ac(t)-ys(t)As(t)+xn(t), (3.1.59) Y(t) = xs(t)As(t)+ys(t)Ac(t) + yn(t), (3.1.60) A(t) = A(t) + jA(t), (3.1.61) Chapter 3 BER Performance of GMSK in Rayleigh Fading 40 Ac(t) = cos<|>5(0®MO = j" coss(u))h(t - u)du T, As(t) = sin<^(r) ® h(t) = j sin((j)5(M))/i(r - u)du The term can be written as (3.1.62) (3.1.63) Evaluating (3.1.57) yields: r(0 = Re{(X(t)+jY(t))exV(j<»ct)} = X ( r ) c o s c o c r - T ( f ) s i n a > c r = x M) A c(t) cos a ct-xM) AM) sin a ct . (3.1.64) - [ys(t)As(t)cos(£>ct + ys(t)Ac(t)sin(£>ct] + xn(t) cos a>ct - yn(t) sin (£>ct xM)AM)cos(OJ-xM)A(t)sm(o t (3.1.65) xs(t)A(t)cos§s(t)cos(dj-xs(t)A(t)sin$s(t)sin(xict ^ \ 66) = xM)A(t)cos((i) t + is(*)) where A M) and A (t) can be expressed in terms of A(t) and §s(t); AM) = A(0cos^(r), AM) = A(r)sin<t>,(r); (3.1.67) (3.1.68) Chapter 3 BER Performance of GMSK in Rayleigh Fading 41 and tys(t) = tan" = tan • s i n < | > 5 ( 0 ® V O -A c(r)J Lcos^r)® (3.1.69) A2(t) = A2(t)+A2(t) [cos^(t)®hif(t)]2 + [sin^(t)®hif(t)]2 . (3.1.70) Similarly, the term y„(f)A (f)coscocr+ y_(r)A (r)sina> f (3.1.71) can be written as yJ(f)A(r)sin(^i(r)cos cocf + yJ(f)A(r)cos0 iS(Osincocr = y s(0A(0sin(co cf + <j),(0) (3.1.72) Hence using (3.1.66) and (3.1.72), (3.1.64) can be written as ~r{t) = xs(t)A(t)cos(<dct + tys(t))-ys(t)A(t)sm(G)ct + <ks(t)) + xn(t) cos a ct - yn(t) sin on ct (3.1.73) The output of the 1-bit differential detector is given by (3.1.15), where Z(f) is now replaced with Zif(t) defined in (3.1.58): Chapter 3 BER Performance of GMSK in Rayleigh Fading 42 V(t) = ±Re{-jZif(t)Zif*(t-T)} = l-Re{-j(Zs(t)Aco(t) + Zn(t))(Z*(t- T)Aco*(t-T)+ Z*(t- T))} = l-Re{- j(Zs(t)Z*(t - T)AC0(t)AC0*(t- T)) + ZN(t)} 1 (3 1 74) = -{-AM)Ap-T)xs{t)xs(t-T) + As{t)Ac(t-T)xs(t)xs(t-T) - Ac(t)Ac(t- T)xs(t)ys(t- T) - As(t)As(t - T)xs(t)ys(t - T) + Ac(t)Ac{t-T)ys(t)xs{t-T) + As(t)As(t - T)ys(t)xs(t - T) - Ac(t)As(t- T)ys(t)ys(t -T) + As(t)Ac(t- T)ys(t)ys(t -T) + ri{t)} where Aco{t), Ac(t) and As(t) are defined in (3.1.61), (3.1.67) and (3.1.68), respectively. It is noted that and As(t)Ac(t-T)-Ac(t)As{t-T) ( 3 1 7 5 ) = A(t)A(t-r)sin(A<i),(0) Ac(t)Ac(t-T) + As(t)As(t-T) = A(t)A(t-T)cos(A$s(t)) (3.1.76) where A M O = is(t)-Ut-T). (3.1.77) Hence using (3.1.75) and (3.1.76), V(r)from (3.1.74) can be expressed as V(t) = l{A(t)A(t-T)[xs(t)xs(t-T)+ys(t)ys(t-T)]sm(A^s) 2 . (3.1.78) + A(t)A(t-T)[y(t)x(t-T)-x(t)y(t-T)]cos(A^s) + n'(t)} Chapter 3 BER Performance of GMSK in Rayleigh Fading 43 The phase change over the 1-bit period from t = (/- 1)7" to t = IT at the Ith sampling interval, denoted as A§s t is now given by A ^ , z = MIT) - MIT ~T) = tan J sin(§s(u))h(lT-u)du J) J coss(u))h(lT-u)du tan J sin(<))s(u))h(lT - T -u)du Ti J cos((j)s(u))h(lT - T - u)du (3.1.79) where the integration is taken over the interval Ti for which h(.) is not negligible. As in Section (3.1.1), the probability of error is given by equations (3.1.43), (3.1.44) and (3.1.45); where pr and p ; are the real and imaginary parts of p , which is now given by Gjfjjp = A 1 A 2a 2 p J ( r ) exp( ;A(^ / ) + c 2 p„(7 ' ) (3.1.80) a, = .2 2 2 (3.1.81) Co = .2 2 2 A2°s+°n> (3.1.82) A] = A2(IT-T) A22 = A2(IT) (3.1.83) The parameters ps(T) and pn(T) are given in (3.1.35) and (3.1.36). The average powers of the 2 2 signal and noise are denoted as os and cn, respectively. Equations (3.1.80), (3.1.81)and (3.1.82) are derived as shown in Appendix C. It is noted that in the unfdtered case, the IF fdter can be Chapter 3 BER Performance of GMSK in Rayleigh Fading 44 considered to be an ideal bandpass filter with equivalent noise bandwidth Bn. In this situation, A(t) in (3.1.70) will be equal to 1 for all values of t and the signal's phase is undistorted i.e. A<j)5; / = ;. This brings (3.1.80) back to (3.1.33) and (3.1.81) will be equal to (3.1.82), which is identical to (3.1.34). Therefore, the unconditional probability of error can be obtained from (3.1.45) as P = -e 2 r f \_ 2 U 1 b,= \ V f 1 + 2 1 + V )2-(Wbi= , + exp{-(7c5Mr)2/ln2})2; A / (a 1 a 2 /a 2 ) 2 - (W f c i = _1 + exp{-(7t5n7')2/ln2})2; (3.1.84) where (3.1.85) wb, = ±i = 5A 1A 2y 0(2K/Dr)cosA(|), ; Z, (3.1.86) 2 , 2 and S = Gs/on. The subscripts in U and W refer to the corresponding input bit bl for the parameter A<|)5> /, which is a function of bl. With decision feedback, the new differential angles are given as ;DF (3.1.87) where bi-\ is the decision made by the receiver on the transmitted bit bt_ x . The unconditional Chapter 3 BER Performance of GMSK in Rayleigh Fading 45 probability of error in the case of decision feedback is identical to (3.1.84) with A^ s > / replaced by In the sections which deal with the performance of GMSK with DF in a FS channel only, the BER analysis for 'filtered' GMSK with DF will be considered. This is because the 'filtered' analysis is more realistic as it takes into account the filtering effect. Moreover, both analytical and simulation results show that the differences in BER between the 'filtered' and 'unfiltered' cases with DF is very small. Chapter 3 BER Performance of GMSK in Rayleigh Fading 46 3.2 B E R Performance of 1-bit detection in FS Rayleigh fading channel 3.2.1 Unfiltered case In a commonly used frequency selective fading channel [42], the received signal is the sum of two incoming signals, each with independent Rayleigh fading characteristics: the desired signal Sjr(t) given in (3.1.1) and a delayed signal J^(f) which also undergoes Rayleigh fading that is independent of that for s^(t). This is known as the two-ray model as described in the previous chapter. The delayed signal d^(t) is given by df(t) = xd(t-x)cos(a>c(t-x) + ^s(t-T))-yd(t-x)cos(oic(t-x) + ^s(t-%)) . (3.2.1) In (3.2.1), xjt) and yd(t) are independent stationary zero mean Gaussian lowpass processes. The time delay difference between the two signals is denoted as t. In the 'unfiltered' case, the received signal denoted by r(t) at the output of the IF filter is equal to r(t), which is given by the sum of Sf(t), df(t) and the noise n(t) defined in (3.1.2). r(t) = sf(t) + df(t) + n(t) = x5(0cos(oy + <|>5(0) ~ y4(0sin(cBcr + <|>5(0) + xd(t - x)cos(foc(r - T) + §s(t - x)) - yd(t - %) cos(coc(r - T) + §s(t - x)) • (3-2.2) + xn(t) cos coct -yn(t) sin ®ct In (3.2.2), if the fixed phase shift of cocx is included in the random process xd(t-%) and yd(t -1), then r(t) can be expressed as r(f) = X(0coscocr-y(f)sincoc? = /?«{Z(0exp(y(0c0} (3.2.3) Chapter 3 BER Performance of GMSK in Rayleigh Fading 47 where Z(r) = X(t)+jY(t) , (3.2.4) X(t) = xs(t)cos§s(t)-ys(t)sm<bs(t) + xd(t - x)cos())J(r - x) - yd(t - x) sin - x) + xn(t) Y{t) = ^(Osin^CO+y^Ocos^CO + xd(t - x) sin <!>,(* - x) + yd(t - x) cos05(? - x) + yn(t) We can re-write (3.2.4) using complex envelope notation as Z(r) = ^(Ocos^CO-y^Osin^CO + xd(t - x)cos(j)J(r - x) - yd(t - x) sin^f - x) + j[xs(t) sin 0,(0 + ^ (Ocos^CO + xd(t - x) sin^Cr - x) + yd(t - x) cos - x)] + xn(t) + jyn(t) = Zs(t) exp [y<|>5(0] + Zd(t - T)exp[./<t>5(f - *)] + Zn(t) where Zs(t), Zd{t) and Zn(t) are independent zero mean complex Gaussian processes. The processes Zs(t) and Zn(r)are given by (3.1.10) and (3.1.11), respectively and Z^(0is given by The output of the lowpass filter V(t) in the 1-bit differential receiver is given in (3.1.15). Substi-Zd(t) = xd(t) + jyd(t). (3.2.8) tuting Z(r) from (3.2.7) yields Chapter 3 BER Performance of GMSK in Rayleigh Fading 48 V(t) = ^Re{-j[Zs(t)cxV(j^s(t)) + Zd(t-x)exp(j^{t-x)) + Zn(t)]-[Z*(t-T)exp(-j^s(t-T)) + Zd*(t-x-T)cxp(-j^(t-x-T))+Zn*(t-T)]} = l-Re{- j[Zs(t)Z*(t - T)exVj(tys(t) -6)s(t- T)) + Zs(t)Zd*(t- x - T) e x p y X K O -tys(t-x-T)) + Zd(t - x)Z*(t - 7>xpy(<|>,(' -1) - W ~ T)) + Zd(t-x)Zd*(t- x - T)expMs(t-x) - 4>,(f - x - T))] + ZN{t)} (3.2.9) where ZN(t) represents all terms associated with the noise processes Zn(t) and Zn*(t- T). Substituting (3.1.10) and (3.2.8) for Zs(t) and Zd(t) into (3.2.9) gives V(t) = l-{(xs(t)xs(t-T) + ys(t)ys(t-T))SmA$s(t) + (xd(t- x)xd(t- x - T) + yd(t-x)yd{t-x - T))s\nA$d(t) + (-xs(t)ys(t-T)+ys(t)xs(t-T))coSA^s(t) + (- xd(t - x)yd(t -x-T)+ yd(t- x)xd(t - x - 7j)cosA(^(r) (3.2.10) + (- xs(t)yd(t- x-T)+ ys(t)xd(t- x - r))cosA(|)'(0 + {xs(t)xd{t -x-T)+ ys(t)yd(t- x - T)) sin Af(r) + (-xd(t - x)ys(t -T) + yd{t - x)xs{t- T))cosAc>"(0 + (xd(t- x)xs(t -T) + yd(t - x)ys{t - T)) sinA<|>"(f) + n\t)} where A ^ O = <$>s(t)-<bs(t-T), (3.2.11) A<M0 = $s(t-T)-$s(t-i-T), (3.2.12) Chapter 3 BER Performance of GMSK in Rayleigh Fading 49 A<t>'(0 = <bs(t)-<$>s(t-T-T), (3.2.13) A<|)"(0 = ( ^ - t ) - ^ - ! ' ) , (3.2.14) and n'(t) is the real part of ZN(t). The sampled output of the receiver V is obtained by sampling V(t) at time t = IT. The 2 signal power cs and the correlation coefficient p can be inferred from (3.1.34) and (3.1.33) evaluated for the flat fading 'unfiltered' case (Section 3.1.1), but with the effect of second delayed signal dj-(t) also taken into account, as described earlier in this section. This yields: a 2 = a 2 + o^ + a 2 , (3.2.15) a 2p = a2p,(r)exp(;A(t), ) /) + a ^ ( r ) e x p ( 7 A ^ / ) + a 2 p„, (3.2.16) 2 2 2 where os, od and cn are the average powers of the desired signal, delayed signal and noise, respectively and p s(r), pd(r) and pn(r) are the normalized autocorrelation functions of the unmodu-lated carrier of the desired fading signal, the unmodulated carrier of the delayed fading signal 9 and noise, respectively. Assuming that the desired signal and the delayed signal experience the same Doppler spreading, then p s(T) is equal to pd(T) = jQ(2nfDT). We can observe that the probability of error defined in (3.1.45) in the FS case is related with the desired signal phase change A ^ l given in (3.1.17) and the delayed signal phase change The unmodulated carrier of the delayed fading signal is equal to Re{Zd(t)exp(j(Oct)}, where Z d(t) is denned in (3.2.8). Chapter 3 BER Performance of GMSK in Rayleigh Fading 50 A<j)^  ; , which is the phase difference over the 1-bit period from (l-l)T-T to IT-i. A§d t is given by A ^ , = ^(IT-T)-MIT-T-T) oo IT-X oo IT-x-jT . (3.2.17) = 2nfd X bj J g(v-jT)dv = 2nfd £ bj J g(v)dv j = - o o ( / _ i ) 7 - _ t j = -°° (l-i)T-x-jT Equation (3.2.17) can be written as oo Wd,i = 2 bfii-j (3-2-18) where 0* is equal to 6/_7- = g(u-jT)du (3.2.19) ^ 2IJ(l-\)T-x and is the/ f c transmitted symbol. As with Atys t , A$d l can be simplified to A$d, l = bl + 26-2 + + 10-1 + M o + bl _ j 0! + bl _ 2 0 2 . (3.2.20) 2 2 We define the average signal-to-noise ratio (SNR) to be S = os/on and the average 2 2 desired (carrier) signal-to-delayed signal ratio (CDR) to be D = cs/cjd. By direct substitution of S and D into (3.2.16), p is given as Chapter 3 BER Performance of GMSK in Rayleigh Fading 51 P = [DS + S + D)Jo(^fDT)^V(ms, /) + [DS + S S + D)j0(2KfDT)exp(jAh, /) + [DS fs + ^ Vi-^Bjf/iXnl)} (3-2.21) The probability of error is given in (3.1.45). By substituting (3.2.21) for p (hence p r and p •), (3.1.21) for Atys l and (3.2.17) for A$d we can derive the unconditional probability of error in FS Rayleigh fading channel for the unfiltered case as P„ = (Pel + Pel) (3.2.22) r f 1 1 -X} sin(0o + A(ps z) + X 2 s in(0 o + Aqd ,) Jl-f 1 + 2 1 + V [X 1 cos(0 o + A(pi;,) + X 2 cos(0 o + Acp ,^) + Fexp{-(7tfl,7/) Vln2}] Xj sin(-0 o + A(ps>,) + X 2 s in ( - 0O + Acp^,) . / l - t Z j C o s ^ O o + Acp^  ,) + X 2 cos ( - 0O + A(prf> z) + 7exp{-(7t5 (.r) 2/ln2}] 2; y = D = D S + s + D ' J ° i 2 n f D T ) ' A ( ? s > 1 i s § i v e n m (3-1-47) and A(P</,/ = b[ + 2 0_ 2 + &z + ,0_i + fcM§i + ^/_202 • (3.2.23) Chapter 3 BER Performance of GMSK in Rayleigh Fading 52 3.2.2 Filtered case We can apply a similar analysis described in the 'filtered' case for flat fading channel (section 3.1.3) by including the IF filtering effect on the signal and the delayed signal components in a FS channel. The 'filtered' received signal from the output of the IF filter is denoted as r(r) which is given by r(t) = Re{Zif(t)expUG>ct)} (3.2.24) where Z^(t) is the complex envelope of r(t). As described in the 'filtered' case for flat fading, the complex envelope Z(y(r) is related to the complex envelope Z(f) of the 'unfiltered' received signal r(t) by equation (3.1.55), where Z(t) is given by (3.2.4). The received signal r{t) will have a similar form as that derived for the flat 'filtered' case in Section 3.1.3, with the terms associated with the delayed signal also taken into account. Consequently, the received signal given in (3.2.24) can be expressed as ~r(t) = Re{Zif(t)exp(j®ct)} = Re{[xs(t)Ac(t)-ys(t)As(t) + xd(t - x)Ac(t -x)-yd(t- x)As(t — x)+ xn(t)] (3-2-25) + j[xs(t)As(t)+ys(t)Ac(t) + xd(t - x)As(t — x) + yd(t - x)Ac(t — x)+ yn(t)] exp(;0)cr)} = Re{(X(t) + jY(t))exp(j(Oct)} where Zn(t) is the zero mean complex Gaussian lowpass process for noise. From above, it can be seen that Chapter 3 BER Performance of GMSK in Rayleigh Fading 53 Zif(t) = X(t) + jY(t) (3.2.26) = Z,(t)Aco(t) + Zd(t - T)Aco(t- X ) + Z„(0 where X(t) = xs(t)Ac(t)-ys(t)As(t) , (3.2.27) + xd(t- x)Ac{t- x) -yd(t- x)As(t - x) + xM) •7(0 = xs(t)As(t) + ys(t)Ac(t) , (3.2.28) + xd(t - x)As(t - x) + yd(t - x)Ac{t- x) + yn{t) and Aco(t) is defined in (3.1.61), Ac(t) in (3.1.62) and As(t) in (3.1.63). Alternatively, r(t) can be expressed as ~r{t) = xs(t)A(t) cos ((act + ^ (0)-y 5(r)A(r)sin(co cf + ^ (r)) + xd(t - x)A(t- x)cos(cocr + §s(t - x)) - yd(t - x)A(t - x)cos(oocf + <j>5(f - x)) + xn(t)cos(x)ct - yn(t) sincocr (3.2.29) where the filtered signal's phase (j>s(f) and the receive filtered signal amplitude A(t) are given by (3.1.69) and (3.1.70), respectively. The terms Ac(t) and As(t) can be expressed in terms of A{t) and ^(f), as given in (3.1.67) and (3.1.68). The output of the 1-bit differential detector is given by (3.1.15), where Z(t) is now replaced with Z(y(r) defined in (3.2.26): Chapter 3 BER Performance of GMSK in Rayleigh Fading 54 V(t) = ±Re{-jZif't)Zif*(t-T)} = l-Re{-j(Zs(t)Aco(t) + Zd(t-x)Aco(t-x) + Z „ ( f ) ) • (Z*(t - T)AC0*(t -T) + Z/(t-x- T)Aco*(t-x-T)+Z*{t-T))} 1 (3.2.30) = ^Re{-j(Zs(t)Z*(t-T)Aco(t)Aco*(t-T) + Zd(t - x)Z*(t - T)Aco(t - x)Aco*(t- T) + Zs(t)Zd*(t - x - T)AC0(t)AC0*(t- x - T) + Zd(t-x)Z/(t-x- T)Aco(t-x)Aco*(t-x-T))+ ZN(t)} where ZN(t) represents all terms associated with noise processes Zn(t) and Zn*(t- T). Substituting (3.1.10) for Zs(t), (3.2.8) for Zd(t), (3.1.11) for Z „ ( f ) , (3.1.61) for Aco(t) and multiplying out (3.2.30) gives V(t) = |[^(0^(f-r) + y J (0y 5 (r-7 ' )]A(0A(r-7 ' )s in(A^(0) + [xd(t - x)xd(t -x-T) + yd(t- x)yd{t -x-T)]A(t-x)A(t-x- T) sin(A^(0) + [ys(t)xs(t-T)-xs(t)ys(t-T)]A(t)A(t-T)cos(AUt)) + [xd(t- x)xs(t- T) + yd(t- x)ys(t - T)]A(t- x)A{t- T)sin(A^"(0) (3.2.31) + [yd(t- T)xs(t -T)- xd{t - x)ys(t- T)]A(t- x)A(t - 7)cos(A^"(0) + [xs(t)xd(t-x- T) + ys(t)yd(t-x- T)]A(t)A(t-x- r)sin(A^'(0) + [ys(t)xd(t-x-T)-xs(t)yd(t-x- T)]A(t)A(t — x — T)cos(A<j>;(0) + [yd(t-x)xd(t-x-T)-xd(t-x)yd(t-x-T)]A(t-x)A(t-x-T)cos(A$d(t)) + n'(t) where A(j)s(r) is given by (3.1.77) and Chapter 3 BER Performance of GMSK in Rayleigh Fading 55 (3.2.32) A<t>'(0 = 6)s(t)-$s(t-i--T), (3.2.33) A<|>"(f) = faft-T)-$s(t-T). (3.2.34) At the Ith sampling interval, the desired differential phase change over the 1-bit period, Atys I is given by (3.1.79) and the delayed differential phase angle, A ^ ; / is given as = tan - l - tan J sin(())s(u))h(lT -x - u)du J cos(§ s{u))h(lT - x - u)du | sm(§s(u))h(lT - x - T - u)du Jj J cos(§ s(u))h(lT - x - T - u)du T, (3.2.35) where T- denotes the time interval over which h(,) is not negligible. The probability of error is given in (3.1.45) where p is given by a , o 2 p = A 1 A 2 a 2 p , ( r )expOAt>, J )+A T l A t 2 a^(7;)exp(jA^, / ) + a2p„(r) (3.2.36) and Chapter 3 BER Performance of GMSK in Rayleigh Fading 56 2 .2 2 .2 2 2 (3.2.37) 2 .2 2 .2 2 2 a 2 = A2as+Ax2ad + an. , (3.2.38) A\ = A2{IT-T) A\ = A2(IT) A2xX = A2(lT-T + x) A\2 = A2(lT + x) (3.2.39) ps(T) and pn(T) are given in (3.1.35) and (3.1.36). The average powers of the signal, Sjr(t), and 2 2 noise are denoted as os and G n , respectively. In Appendix C, the derivations of (3.2.37), (3.2.38) and (3.2.39) are shown. Therefore, the total probability of error can be obtained from (3.1.45) as P . = x r f 1 2 L V f 1 - — Ui,bl=i + U2ibi=1 ) - ( W 1 ) f c ( = 1 + W 2 ; , ( = 1 + exp{-(7u5.T)/ln2}) J ( 3 2 4 Q ) 1 + 2 1+ , )2 " (Wlt b i = _x + W2> b i = _x + exp{-(7t5 . r) 2 /ln2}) 2 ; where f/j fc _ ± 1 and fc _ ± 1 are given in (3.1.85) and (3.1.86), respectively and U2,b, = ±l = DNA\A2Jo(2KfDT)sinA^d,h (3.2.41) W 2,b, = ±i = DNAlA2J0(2KfDT)cosAt^dii. (3.2.42) In (3.2.41) and (3.2.42), the parameter A§d> / depends on the corresponding input bit bl and DN Chapter 3 BER Performance of GMSK in Rayleigh Fading 57 2 2 is the average delayed signal to noise ratio, defined as cd/an. With decision feedback, the new differential angles are given as ~ DF ~ ~ A ^ / = A ^ z - d , ^ . ^ ! , (3.2.43) = A i ^ - d ^ e p (3.2.44) where dx is a variable that can be adjusted to provide the optimum feedback angle so that the BER is minimized. This is because in FS fading, the phase shift £>/_ i0] is not always optimum owed to the presence of the delayed signal. The total probability of error in this case is identical to ~ ~DF ~ ~DF (3.2.40) with A(j)iS> / replaced by A ^ ; and A(j)^ / by Aty^ /. Chapter 3 BER Performance of GMSK in Rayleigh Fading 58 3.3 BER Performance of 2-bit detection in flat Rayleigh fading Figure 3.4 and Figure 3.5 depict the transmitter and receiver for the 2-bit detection case. The transmitter is similar to that for the 1-bit detection case except that it includes a differential encoder before the premodulation filter. This is because if the differential encoder is absent, the decisions made by the detector will be a differentially encoded version of the input data sequence and not the original input. This will become clearer in the subsequent section, where the mathematical analysis is shown. From Figure 3.4, we have bj = Cj • C y _ j since bj is either T or r ' — • 1 T Cj-l Gaussian F M lowpass filter modulator s(t) Figure 3.4 Block diagram of 2-bit GMSK transmitter (for differential detection) The output of the 2-bit detector V'(t) is obtained from multiplying the output of the IF filter, r(t) by a 2-bit period delayed version of itself. The product is then passed through a lowpass filter to produce V(t), which is assumed to be sampled at time t = IT to give V(IT), where / denotes the Ith sampling interval: le {...,-1,0, 1,2, ...}. A hard decision is then made on V(IT) to decide whether a '1' or a '0' was sent. In an ideal channel (no fading and no noise), the receiver output is given by (3.3.9). The operation of the 2-bit differential detector will be described mathematically in section 3.3.1. Chapter 3 BER Performance of GMSK in Rayleigh Fading 59 Detector Delay 2T V(0 V(IT) Hard Decision "0" or "!' Sj(t): signal after fading with average power ^ s n(t): AWGN which produces narrowband noise with 2 average power °„ at the IF filter output Figure 3.5 Block diagram of a 2-bit GMSK differential receiver 3.3.1 Unfiltered case In the 'unfiltered' case, the received signal is equal to r{t) which is given by (3.1.4) and the complex envelope Z(t) of r(t) is given by (3.1.5) or equivalently (3.1.9). Since we assume the IF Gaussian filter has no effect on the received signal, except for bandlimiting the noise, therefore the signal's phase is undistorted at the output of the filter. The output of the detector V'(t) (see Figure 3.5) is given by Chapter 3 BER Performance of GMSK in Rayleigh Fading 60 V(t) = [r(t)-r(t-2T)] = Re{Z(t)exp(j(tict)} • Re{Z{t-2T)exV{j(Sic{t-2T))} • ( 3 3 > 1 ) Multiplying out (3.3.1) yields V"(0 = Re{Z(t)exp(j(£>ct)} • Re{Z(t-2T)exp(ja>c(t-2T))} = (X(t) cos a y - 7(0 sin a y ) • (X(t - 2T) coscoc(f -2T)-Y(t-2T) sincoc(f - 2T)) = ±X(t)X(t-2T)[cos(4nfct - 2nfcT) + cos(2nfcT)] (3 3 2^  -±Y(t)X(t-2T)[sm(4%fct-2nfcT) + sm(2nfcT)] -±X(t)Y(t-2T)[sm(4nfct-2KfcT)-sm(2nfcT)] + X-Y{t)Y(t-2T)[- cos(4nfct-2KfcT) + cos(2nfcT)] The lowpass filter rejects the signal components shown in (3.3.2) at around 2fc. Assuming that fc is chosen such that 2nfcT = (£>CT = 2%m where ra is a positive integer, the output of the lowpass filter is V(0 = ^[X(t)X(t-2T) + Y(t)Y(t-2T)] = l-Re[(X(t) + jY(t))(X(t-2T)-jY(t-2T))] • (3.3.3) = ±Re{Z(t)Z*(t-2T)} Substituting (3.1.9) for Z(t) into (3.3.3) gives Chapter 3 BER Performance of GMSK in Rayleigh Fading 61 V(t) = l-Re{[Zs(t)cxp(JW)+Zn(t)]-[Z*(t - 2T)exv(-j<$>s(t- IT)) + Z*(t - 2T)]} = ±Re{Zs(t)Zs*(t - 27>xpy(<M0 " <M'" 2 r » + Z ^)> (3.3.4) = l-[xs(t)xs(t-2T) + ys(t)ys(t-2T)]cosA<bs(t) + \ [xs(t)ys(t- 2T) - ys(t)x(t- 2T)] sin A^(f) + n'(t) where ZN(t) represents all terms associated with the noise processes Zn(t) and Zn*(t- 2T), and n'{t) is the real part of ZN(t). The sampled output of the receiver V(IT) (denoted as V hereafter) is obtained by sampling V(t) at t = IT. The phase change over the 2-bit period from t = (l-2)T to t = IT, denoted as A(j)5 l where A(|> l = <bs(lT)-tys(lT-2T) ' (3.3.5) o o IT-jT = 2Kfd X cj J S(v)dv ; = -•» 0-2)T-jT and g(r) is the response of the premodulation Gaussian filter to a transmitted rectangular pulse, given in (3.1.18). Equation (3.3.5) can be written as o o A t , i = X CJVI-J (3.3.6) Chapter 3 BER Performance of GMSK in Rayleigh Fading 62 where V(/,_ ; is equal to IT Vl.j = 2^ 1 8(u-jT)du. (3.3.7) ( / -2)7 When 0.2 < 5-7 < 0.3 , \|/,__. is negligible for |/ - ; | > 3 [22] (Table D.4 in Appendix D shows the values of for 5 ( T = 0.25), therefore Atys l can be simplified to A4>,,/ = c / + 2 ¥ - 2 + c / + i ¥ - i + c / ¥ 0 + c z _ 1 \ l / 1 + c /_ 2\)/ 2 + c / _3\| /3 . (3.3.8) As in the 1-bit case, the transmitted signal s(t) after propagation through an ideal channel is defined by (3.1.22). Comparing (3.1.22) with (3.3.4), the lowpass Gaussian processes xs(t) and ys(t) in (3.3.4) is equal to 1 and 0, respectively for all values of t, when the channel is not fading. Hence, in an ideal channel, the receiver output V at time IT will be V = \cosA$Stl. (3.3.9) It is noted from Table D.4 that when 0 < |A<t>^ . z| < 7 i /2 , the product cl_ lcl is -1 (cl_l and cz are of opposite signs), and when 7t/2 < |A(t)5 [\<n, cl_ xcl is 1 (ct_ j and ct are of equal signs). Since we have bl = cl_lcl , the decision rule at the receiver is: decide the transmitted bit b, is a ' 1' when V < 0 and vice versa. Therefore, Chapter 3 BER Performance of GMSK in Rayleigh Fading 63 A$s,l = ±(Vo + Vi) + A ( r V / = ± 2 \ | / 0 + A ( p 5 , / A<bs,l = ± ( V 0 - V l ) + A ( P 5 , / = 0 + A%,l where \\f0 is equal to \\fx (from Table D.4) and given bt = 1 given bl - -1 (3.3.10) (3.3.11) Following a similar procedure as in the 1-bit detection case, we define a column matrix Z: z= Z(IT) 2," Z( ( / -2 )T) z 2 (3.3.12) Its covariance matrix C is given by (3.1.32), namely 1 p* IP 1. (3.3.13) where p is the correlation coefficient between Z 2 and Z j , and is given by: a2p = G 2 P j (27>xpaA^ ,) + a 2p„(27;). (3.3.14) The average power of the received signal denoted as a given by (3.1.34). The average 2 2 powers of the signal, Sjr(t) and noise are denoted as Gs, Gn, respectively; and the normalised autocorrelation functions of the unmodulated carrier fading signal and noise are denoted as ps(t) and p„(r) , respectively. The parameters ps(2T) and p„(2T) are evaluated as [10]: ps(2T) = JQ(4nfDT) , (3.3.15) Chapter 3 BER Performance of GMSK in Rayleigh Fading 64 p„(27/) = exp {-(InBiT) Vln2} (3.3.16) 2 , 2 Defining the average signal-to-noise ratio (SNR) to be S = Gs/Gn, p is evaluated to be P = [^ ] / 0 (47t / D r )exp ( jA( t ) , ; / ) + ( ^ T ) e x p { - ( 2 7 U f i „ r ) 2 / ( l n 2 ) } . (3.3.17) The characteristic function F(v) of the sampled output V is given in (3.1.38). For the 2-bit detection case, it is evaluated to be: F(v) = [ l - 2 > G 2 ( p + p*) + 4 v 2 o - 4 ( l - | p | 2 ) f 1 . (3.3.18) Solving the quadratic equation in the denominator of F(v), the two poles of the character-istic function can be found to be: _ 7(Pr±7T-p]) P l < 2 ~ 2 , ,2 2a 2 ( l - | p | 2 ) (3.3.19) where p r and py- are the real and imaginary parts of p , respectively. The p.d.f. of the receiver output V, P(V), is given by (3.1.41). Following the analysis shown in Appendix B for 1-bit detection, P(V) for 2-bit detection can be evaluated to be P{V\bl = ±\) = exp 4a 4 a 2 ^ 2 exp 2a 2 ( l - | p | 2 ) . , 2a 2 ( l - | p | 2 ) •V V<0 V>0 (3.3.20) Chapter 3 BER Performance of GMSK in Rayleigh Fading 65 In (3.3.20), p r and pj are the real and imaginary parts of p (see (3.3.17)), respectively with p being a function of A$s 1 1 0 which depends on the input bit, bt, as shown in (3.3.10). According to the decision rule described earlier, error occurs if V > 0 when bl = 1 and vice versa, the probability of errors for each of the cases are given as follows: Pel = P(V<0|fc, = - l ) = \P{V\b^-l)dv = \ f P A 1- P r V (3.3.21) oo Pel = P(V>0\b[=l) = jP(V\b^l)dv = \ V (3.3.22) The unconditional probability of error is given by P„ = (Pel + Pel) = \Pe\+\- \{(Pel\^s, l = 2 V 0 + A(P*, /) + \(Pel\A^, / = ~2Vo + A<P,, /) (3.3.23) since for the case when bl = T , A§s t can have either of the two values, as shown in (3.3.10). Substituting (3.3.14) into (3.3.21) and (3.3.22), (3.3.23) yields In the FS fading case, p is also a function of A<(> . , which depends on b, , defined in Section 3.4. Chapter 3 BER Performance of GMSK in Rayleigh Fading 66 X7 0(47t/Dr)cos(A(p 5) + yexp{-(2nB I.r) V ln2} Jl-[XJ0(4nfDT)sm(A<?s)]2 1 + 2 1 + XJ0(4nfDT)cos(2\\fQ + Aq>5) + rexp{-(27ifl lT) Vln2} Jl - [XJ0(4nfDT)sin(2i|/0 + A(p5)]2 1 + 2 1 + V X7 0(47t/Dr)cos(-2v|/ 0 + Aq>5) + 7exp{-(2jcB,-r) V ln2} Jl - [XJ0(4nfDT)sm(-2y0 + Acp5)]2 (3.3.24) where X = 5+1 ,Y = 5+1 3.3.2 Unfiltered Case with Decision Feedback 1 < Delay 2T A, F F N Lowpass Filter Sample F F Hard V(0 V(0 "t-l V(IT) Delay T "1-2 Delay T Figure 3.6 Block diagram of a 2-bit DF GMSK receiver Decision feedback can also be applied to 2-bit detection. Observe from Figure 3.7(a) that states 17 and 24 are subjected to the greatest amount of ISI, therefore the objective is to move these two states and also other states away as far as possible from the threshold. The phase shift X, can be implemented as follows [22]: Chapter 3 BER Performance of GMSK in Rayleigh Fading 67 0 i f c / _ 2 = C / _ 1 It is noted that the biased threshold is only employed in a non-fading channel with AWGN, as described in the beginning of this chapter. In the case of fading, we will only consider the zero-level threshold because there is little performance improvement in a fading channel with a biased threshold. Decision Threshold Decision Threshold 22,23 21 20 (a) 24 9,10 biased threshold 22,24 ! ! \ k 21,23 r ol 1 1 1 1 1 T W j 11,12, / 13,14 / 15,16 18,20 18,19 ^ I 1 17 1 (b) 10,11, 14,15 12,16 Figure 3.7 Phase-state diagram of 2-bit differential detection: (a) before feedback (b) after feedback Therefore, the new probability of error is identical to (3.3.24) with A$s t replaced by DF A$s l as follows: A C z = c / V o + c / - i ¥ i + A C / DF (3.3.26) where Chapter 3 BER Performance of GMSK in Rayleigh Fading 68 A ( P 5 , / = cl + 2^-2 + Cl-2V2  +  Cl-3^3-^ • (3.3.27) 3.3.3 Filtered case The input to the receiver, r(t) is given by (3.1.64) in section 3.1.3. It is also given by (3.1.57) when expressed in complex envelope notation where Z^(t) is given by (3.1.58). The output of the 2-bit differential detector is given by (3.3.3), where Z(r) is replaced with Z^(r): V(t) = ±Re{Zif(t)Zif*(t-2T)} = ±Re{(Zs(t)AC0(t) + Zn(t))(Z*(t - 2T)Aco*(t -2T) + Z „ * ( f ) ) } = ±Re{Zs(t)Z*(t- 2T)Aco(t)Aco*(t- 2T) + ZN(t)} = l-Re{Ac{t)Ac(t- 2T)xs(t)xs(t - 2T) + As(t)As(t - 2T)xs(t)xs(t - 2T) { 3 3 2 S ) - Ac(t)As(t- 2T)xs(t)ys(t - 2T) + As(t)Ac(t - 2T)xs(t)ys{t - 2T) + Ac(t)As(t - 2T)ys(t)xs(t- 2T) - As(t)Ac(t - 2T)ys(t)xs(t -2T) + Ac(t)Ac(t - 2T)ys(t)ys(t- 2 7 ) + As(t)As(t- 2T)ys(t)ys(t- 2T) + n'(t)} where Aco(t), Ac(t) and As(t) are defined in (3.1.61), (3.1.62) and (3.1.63), respectively. It is noted that Ac(t)Ac(t - 2T) + As(t)As(t- 2T) (3.3.29) = A(t)A(t-2T)cos(A$s(t)) and As{t)Ac{t-2T)-Ac{t)As{t-2T) = A(t)A(t-27/)sin(A(j>i(r)) (3.3.30) Chapter 3 BER Performance of GMSK in Rayleigh Fading 69 where A ^ ( 0 = Ut)-Ut-2T). (3.3.31) Hence using (3.3.29) and (3.3.30), V(t) from (3.3.28) can be expressed as V(t) = i { A ( 0 A ( r - 2 r ) [ x J ( 0 ^ ( f - 2 r ) + y , ( 0 y s ( r - 2 r ) ] c o s ( A i J ( 0 ) ( 3 3 3 2 ) + A(t)A(t-2T)[xs (t)y(t-2T)-ys (t)x(t-2T)]sin(AUt)) + n'(t)} The phase change over the 2-bit period from t = (I - 2)T to t = IT at the Ith sampling interval, denoted as A(j) 5 l , is now given by A<|>.z = UIT)-UIT-2T) = tan J sm(§s(u))h(lT -u)du T J cos(§s(u))h(lT-u)du L.T, (3.3.33) - tan J sin(§ s(u))h(lT - 2T - u)du h J cos (0 s{u))h(lT -2T - u)du where the integration is taken over the interval Ti for which h(.) is not negligible. The analogue of (3.1.80) for the 2-bit detection case is given as a,a2p = A 1 A 2 a 2p i ( 2 r)expO , A ^ > / ) + a 2p„(27') (3.3.34) Chapter 3 BER Performance of GMSK in Rayleigh Fading 70 where 2 .2 2 2 a, = Axas + <5n, (3.3.35) 2 .2 2 2 °~2 = A20s+°n> (3.3.36) A\ = A2(IT-2T) A22 = A2(IT) (3.3.37) The parameters ps(2T) and pn(2T) are given in (3.3.15) and (3.3.16). Equations (3.3.34), (3.3.35) and (3.3.36) can be derived in a similar manner as that for 1-bit detection, shown in Appendix C. As in Section 3.3.1, the probability of error is given by equations (3.3.21), (3.3.22) and (3.3.23), where pr and py are the real and imaginary parts of p, which is given by (3.3.34). Therefore, the unconditional probability of error can be obtained as 1 + 2 f Wb = 1 + exp{-(27i5 nr) 2/ln2}A 1 + — — (<*l°2/CTn) -Ub,= l2 J 2 \ w 1 -V fc/ = _ 1 + exp{-(27Cfi l ir)Vln2} (G^ / a 2 ) 2 ' - U b i = _2 (3.3.38) where ub, = ±i = SA^jQiAnf^sinAtysj, W b, = ±i = 5A 1 A 2 7 0 (4 j c / o r ) cosA^ i / , (3.3.39) (3.3.40) Chapter 3 BER Performance of GMSK in Rayleigh Fading 71 2 2 and S = as/an. The subscripts in U and W refer to the corresponding input bit bl for the parameter A§Si /, which is a function of c{ and bt = ct • cl _ x . With decision feedback, the new differential angle is given as A ^ F / = A^j-l (3.3.41) where A. is given in (3.3.25). The total probability of error in the case of decision feedback is identical to (3.3.38) with Acj)^  / replaced by A^j. 3.4 BER Performance of 2-bit Detection in FS Rayleigh fading channel 3.4.1 Unfiltered case In a commonly used frequency selective fading channel model, the received signal is the sum of two incoming signals, each with independent Rayleigh fading characteristics. It is given by (3.2.2) or equivalently by (3.2.3) if expressed in complex envelope form. The complex envelope Z(r) of r(r) is given by (3.2.7). The output of the 2-bit differential detector is given by (3.3.3). Substituting (3.2.7) for Z(t) into (3.3.3) yields Chapter 3 BER Performance of GMSK in Rayleigh Fading 72 V(t) = l-Re{Z{t)Z*{t-2T)} = ±Re{(Zs(t)expj($s(t))+Zd(t-x)expMs(t-x)) + Zn(t)) •(ZsM-2T)txp(-j$s(t-2T)) + Zd*(t-x-2T)exp{-j§M-x-2T)) + Z*{t-2T))} = l-Re{Zs{t)Z*{t- 2r)exp;((t),(0 - 2 T ) ) 2 (3.4.1) + Zs(t)Zd*(t- x - 2T)expMs(t) - $s(t - T - 27)) + Zd(t - x)Z*(t- 2T)cxpMs(t -x)- ^(f - 2D) + Zd(t- x)Zd*(t - x - 2T)expj($s(t -x)- <$>s(t -x- 27)) + ZN(t)} Substituting (3.1.10) and (3.2.8) for Zs(t) and Zd(t) into (3.4.1) gives V(t) = ±{(xs(t)xs(t-2T) + ys(t)ys(t-2T))cosA<$>s(t) + (xs(t)ys(t - 2T) - ys(t)xs(t- 2D) sinA^CO + (xs(t)xs(t -x-2T)+ ys(t)ys(t- x - 2T))cosA^'d(t) + (xs(t)ys(t-x - 2D -y s(t)x s(t-x- 2D)sinA$ d(t) (3.4.2) + (xd(t- x)xs(t - 2D + yd(t - x)ys(t - 2T)) cos A^"d(t) + (xd(t - x)ys(t - 2D - yd{t - x)xs(t- 2D) sinA$'d(t) + (xd(t - x)xd(t - X-2T) + yd(t - x)yd{t -x- 2D)cos A<^(r) + (xd(t- x)yd{t- x — 2T) — yd{t - x)xd{t -x- 2D) sin A ^ O + n\t)} where A<|>5(0 = $s(t)-$s{t-2T), (3.4.3) A0d(O = <|>,(f — x) — <|>5(f - X - 2D , (3-4.4) A$\t) = <|>5(0-<Mf-T-2D, (3.4.5) Chapter 3 BER Performance of GMSK in Rayleigh Fading 73 A<|>"(0 = t(t-x)-Mt-2T), (3.4.6) and ri(t) is the real part of ZN(t). As in the 1-bit FS case (section 3.2.1), the probability of error depends on both the desired signal phase change A<j)5 l given in (3.3.8) and the delayed signal phase change A(|)rf t , where where I S equal to (3.4.7) A 0 d i / = §s{lT-x)-$s{lT-x-2T) o o IT-x oo IT-x-jT = 2*fd X cj J S(v-jT)dv = 2nfd ]T ^ J g(v)dv y = -°° U-2)T-t j = -°° (l-2)T-x-jT Equation (3.4.7) can be written as A$d,i = X c $ l - j (3-4-8) , - = ^ r ~ T g{u-jT)du (3.4.9) and cy is the jth transmitted differentially encoded symbol. As with A<|>s i , A(|)rf l can be simplified to A<^d,l = cl + lV-2 + cl+\V-\ + c$0 + Cl-\V\ + cl-lVl + cl-lVl- (3.4.10) In (3.4.10), unlike the case of A§s , , \ j / 0 is not equal to . The values of • for Chapter 3 BER Performance of GMSK in Rayleigh Fading 74 \l - j\ < 3 with BtT = 0.25 and x = 0.2 T is given in Table D.5 in Appendix D. 2 2 We define the average signal-to-noise ratio (SNR) to be 5 = cs/on and the average 2 2 desired (carrier) signal-to-delayed signal ratio (CDR) to be D = as/(5d. The analogue of (3.2.16) for the 2-bit case is given by a2p = a,2p5(2r)expO-A^ /) + a 2p r f(2T)expaA^ /) + a 2p„. (3.4.11) By direct substitution of 5 and D into (3.4.11), p is given as P = [DS +SS + D ) j Q ( 4 n f D T ) e X 1 ? U A ^ > l ) + [DS + S + j J 7 o ( 4 7 I / D ? > x p ( 7 A ( ^ , ) D \ 2 ( 3 A 1 2 > 1 exp{-(27t5„T') V(ln2)} DS + S + D The probability of error is given in (3.3.23) where Pel and Pe2 are given by (3.3.21) and (3.3.22), respectively. By substituting (3.4.12) for p (hence pr and p •), (3.3.8) for A<j>5>/ and (3.4.10) for A(j)rf ; . In this instance, the unconditional probability of error is derived as Chapter 3 BER Performance of GMSK in Rayleigh Fading 75 P = r L 2V 1 + 1 + 2 f V 1 + 2 1 + 2 1 + 1_ 2 1-f V X,cos(2\)/0 + Acp^j) +X 2cos(\j/ 0 + vj/! + Acp^ ,) + Y Jl -[A r 1sin(2\(/ 0 + A(p j ] /) + A:2sin(\j/0 + vi/1 +Aq>d>/)]2. X,cos(-2\|/ 0 + A(p 5 /) +X 2cos(-\j/ 0-\j/ 1 + A(prf i#) + F Jl-[Xl sin(-2\|/0 + Aq>5>,) + X 2 sin(-\j / 0 - ijjr, + A(prf>,)]' ' X 1 cosA(p 5 / + Z 2cos(\i/ 0-\}/ 1 +AqcU) + Y ^ -[XjSinCAcp^;) +X 2 sin(\j / 0 - \j / 1 + A<pd>,)]' Z,cosA(p5 j + XjCost-ij/o + iij/, + A(p^ /) + T Jl-[Xl sin(A(pS),) + X 2 s i n ( - y 0 + vj/, + A<pd>,)]' (3.4.13) where (3.4.14) (3.4.15) v D J ( 2 * W DS+S+D In 2 (3.4.16) A ( P</ , / = cZ + 2 V - 2 + c Z + l ¥ - l + C / _ 2 V 2 + c/-3 l ir/3' (3.4.17) and 5 = c]/a2n,D = a)/a]. 3.4.2 Filtered case The input of the receiver, r(t) is given by (3.2.29) in section 3.2.2. It is also given by (3.2.25) when expressed in complex envelope notation where Z^(t) is given by (3.2.26). The Chapter 3 BER Performance of GMSK in Rayleigh Fading 76 output of the 2-bit differential detector is given by (3.3.3), where Z(r) is replaced with ZM): V(t) = ±Re{Zif(t)Zif*(t-2T)} = l-Re{(Zs(t)AC0(t) +Zd(t-x)Aco{t- x) + ZM)) • (Z*(t- 2T)Aco*(t-2T) + Z/(t-x- 2T)Aco*(t-x-2T) + Zn*(t- 2T))} = l-Re{ZM)Zs*{t-2T)AC0{t)Ac*(t-2T) ( 3 A 1 8 ) + ZM)Zd*(t - x - 2T)Aco(t)Aco*(t -x-2T) + Zd(t- x)Z*(t - 2T)Aco(t - x)Aco*(t -2T) + Zd(t- x)Zd*(t - x - 2T)Aco(t - x)AC0*(t - x - 2T) + ZN(t)} where ACM), AM) and As(t) are defined in (3.1.61), (3.1.62) and (3.1.63), respectively, and ZN(t) represents all terms associated with the noise processes ZM) and Zn*(t - T). Substituting (3.1.10) for ZM), (3.2.8) for Zd(t), (3.1.11) for Zn(t) into (3.4.18) and using (3.3.29) and (3.3.30), gives Chapter 3 BER Performance of GMSK in Rayleigh Fading 77 V(t) = -A(t)A(t-2T)[xs(t)xs(t-2T) + ys(t)ys(t-2T)]cos(AUt)) + A(t)A(t-2T)[xM)ys(t-2T)-ys(t)xs(t-2T)]sin{AUt)) + A(t)A(t- x - 2T)[xs(t)xd(t- x — 2T) + ys(t)yd(t - x - 27/)] cos(A(j>' (r)) + A(t)A(t - x - 2T)[xs(t)yd(t — x — 2T) — ys(t)xd(t - x - 2T)] sin(A0'(O) + A(t - x)A(t- 2T)[xd{t - x)xs{t - 2T) + yd(t - x)ys(t - 27/)]cos(A<K'(0) ( 3 A 1 9) + A(t-x)A(t-2T)[xd(t-x)ys(t-2T)-yd(t-x)xs(t-2T))sm(A^(t)) + A(t - x)A(t -x- 2T)[xd(t- x)xd(t- x-2T) + yd(t- x)yd(t- x - 27/)]cos(A<jv(r)) + A(t - x)A{t - x - 2T)[xd(t- x)yd(t -X-2T)- yd(t- x)xd(t — x — 2T)) sin(A^(0) where A<()5(£) is given in (3.3.31) and AWO = Mt-x)-Mt-x-2T) (3.4.20) (3.4.21) Af(0 = Ut-D-Ut-2T) (3.4.22) At the Ith sampling interval, the desired differential phase over the 2-bit period, A ^ / is given by (3.3.33) and the delayed differential phase angle, Atj)^  / is given as Chapter 3 BER Performance of GMSK in Rayleigh Fading 78 A(k/ = <$>S(IT-T)-UIT-T-2T) = tan -l - tan J sin (§s(u))h(lT - x - u)du Is j cos(§s(u))h(lT— x — u)du J sin(§ s(u))h(lT - x- 2T - u)du Ii | cos (§s(u))h(lT -x-2T - u)du (3.4.23) where Tt is defined earlier. The probability of error is given in (3.3.23) where p is given by a,a2p = A 1 A 2 a 2 p 5 ( 2 r ) e x p ( y A ^ ( / ) + A T l A x 2 a ^ ( 2 r ) e x p ( ; A ^ ) / ) + a 2 p „ (2 r ) (3.4.24) and 2 .2 2 .2 2 2 ° 1 = A \ ^ s + A ^ d + (Sn > (3.4.25) 2 .2 2 .2 2 2 ° 2 = A2°s+Ai2ad + °n » (3.4.26) A 2 = A2(IT-2T) A\ = A2(IT) A2xl = A2(lT-2T + x) A22 = A2(lT + x) (3.4.27) p s(2T) and pw(27) are given in (3.3.15) and (3.3.16). The average powers of the signal, Sj(t) Chapter 3 BER Performance of GMSK in Rayleigh Fading 79 2 2 and noise are denoted as a5 and cn, respectively. Equations (3.4.24), (3.4.25) and (3.4.26) are derived in a similar manner, as in 1-bit detection (Section 3.2.2). Similar to the analysis done for the 1-bit detection and using (3.3.21), (3.3.22), (3.3.23), the unconditional probability of error in the 2-bit case is obtained as P . = r 1 + 2 Wi bl- i + W2 b ! + exp{-(27ufi„r) 2/ln2} 1 + — Wi, b, = -i + W2, b, = - l + exp{-(27tZ?„r) 2/ln2} 1 2~2 2 (cxa2/an) — ( C / i , f c / = _ i -i-(3.4.28) where ^ i , * , = ±i = SAXA2J Q{4nf DT)smA$sJ, (3.4.29) U2b, = ±\ = DNAlA2Jo(4nfDT)S'mA$d,h (3.4.30) W i,b, = ±i = 5 A 1 A 2 7 0 ( 4 7 t / D r ) c o s A ^ / , (3.4.31) W 2,b, = ±l = DNAlA2Jo(4nfDT)COsA<bd,h (3.4.32) 2 2 2 2 and S = os/Gn, DN = od/an (the average delayed signal to noise power ratio). In (3.4.29) to (3.4.32), the parameters A$Sj / and A(j)^; depend on the corresponding input bit . Chapter 3 BER Performance of GMSK in Rayleigh Fading 80 With decision feedback, the new differential angles are given as A<j)^  = AtySi i - d2X, (3.4.33) A$7,i = A4d,i-d2X, (3.4.34) where d2 is a variable that can be adjusted to provide the optimum feedback angle so that the BER is minimized, as in 1-bit detection. The probability of error in the case of decision feedback ~ ~ DF ~ ~DF is identical to (3.4.28) with A(j)S) / replaced by A ^ , / and A<t>^  / by A ^ /. Chapter 4 Results and Discussions Chapter 4 Results and Discussions In this chapter, the BER formulae derived in Chapter 3 are illustrated with numerical examples. In order to validate the formulae, comparisons with simulation results are also presented. 4.1 Simulation Model The transmitter consists of a random binary source and a G M S K modulator. The source output data bits are generated using a pseudo-random number generator. It should be pointed out that in 2-bit detection, the data sequence requires differential encoding. The simulation written in C, is implemented as an equivalent baseband model, which has been described in Chapter 3. The baseband transmitted signal is given in (3.1.9), with Zs(t) and Zn(t) set to 1 and 0, respectively. The received signal, distorted by fading and noise, is then passed through the equivalent baseband Gaussian IF filter and then to the receiver implemented as a 1-bit differential detector for 1-bit detection or as a 2-bit differential detector for 2-bit detection. The zero mean random Gaussian noise process is generated using the method described in [44] and the generation of the random Gaussian processes (the in-phase and quadrature components) that characterize the Rayleigh fading waveform is described below. The Rayleigh fading waveform that characterizes multipath propagation is generated using the method described by Jakes [32]. The Jakes model is a discrete, time averaged approxi-mation to the desired Doppler spectrum defined in (2.2.2) in the form of Nos low-frequency oscillators. As described in the previous chapters, the in-phase and quadrature components, denoted as xs(t) and ys(t) respectively, are given by: 81 82 xs{t) = 2 ^ cosPncos(conO + j2cos((aDt) (4.1.1) n = 1 ys(t) = 2 2 s i n P „ c o s ( a y ) (4.1.2) n = 1 where B„ = is to provide the appropriate gains for the sinusoids, and n (Nos+l) (0„ = G)dcos(2nn/Ns). (4.1.3) In (4.1.3), Ns = 2(2N0S + 1) and a>D is the maximum Doppler frequency in radians/sec. The above technique can be extended to provide multiple independent fading processes [32]. Two independent Rayleigh fading waveforms can be generated and used to implement the FS two-ray model as described in Chapter 2. The following is a list of parameters being used in obtaining the numerical or simulation results: (1) sampling frequency (fs) The 'sampling frequency' is the number of samples per second. It is set to 8 • 105 samples per second in the simulations. (2) bit rate (Rb) The 'bit rate' is the transmission rate in number of bits per second. In the simulations, the bit rate is chosen to be 16000 bits per second. Hence, this gives 50 samples per bit using the value of fs 83 stated above. (3) signal power (Ps) The 'signal power' is the desired signal power at the point before A W G N is added. When the channel is non fading with only AWGN, the signal power is given by the transmitted power of the signal, s(t), defined in (2.1.4) and is set to 0.5; whereas in the case of Rayleigh fading, it is given by the average power of the signal, Sy(r), defined in (3.1.1) and is equal to 0.5(xs + ys). 2 (4) postdetection noise power (o"n) In this thesis, error rates are usually expressed as a function of Eb/NQ, where Eb is equal to P/Rb in a non fading AWGN channel and it is an average value in the case of fading because the signal power (after fading) varies; and N0 is the one-sided PSD of AWGN. The received signal is passed through the IF filter and the noise is therefore being bandlimited. The postdetection noise 2 . . , power <5n is given by where Bn is the equivalent noise bandwidth of the IF filter and is equal to 2 A / l n ^ ^ 1 ' ^ ' k e * n § ^ e 3-dB bandwidth of the IF filter. Throughout the simulations, Bi is chosen to be 0.9Rb because it is optimum for a varieties of channel parameters (in both fading and non fading channels) and BER performances [5],[18],[20], hence Bn is equal to 15328.8 Hz. n (4.1.4) 84 (5) normalised 3-dB bandwidth of the pre-modulation filter (BtT) The parameter BtT is chosen to be 0.25. The reasons for choosing this value is three-fold: (1) this value is assumed in a number of papers and thus allows a direct comparison between our results and previous works. (2) GMSK is made more spectrally compact by having a smaller BtT (in the G S M standard, BtT = 0.3), at the cost of significant ISI. (3) the effectiveness of the decision feedback scheme can be more clearly demonstrated, given the amount of inherent ISI. (6) number of low-frequency oscillators in the Jakes model in simulating Rayleigh fading (Nos): The number Nos is set to 8. By choosing p n = — — in (4.1.1) and (4.1.2), the average ' os ) powers of xs(t) and ys(t), denoted as xs and ys, are given by Nos and Nos + 1, respectively. (7) doppler frequency (fD) The Doppler frequency has been defined in (2.2.3). The Doppler frequency-time product fDT (also known as the normalized Doppler spread) is used as a parameter in the simulations, where T = l/Rb is the bit duration. The values that are chosen for fDT are given in later sections. (8) time delay difference in a FS channel (x) The parameter x is the time delay difference between the main (undelayed) and the secondary (delayed) signal in a FS channel, as described in Chapter 3. Two values, 0.2Tand 0.47; are used in the simulations. 85 (9) decision feedback coefficients (dx and d2) The decision feedback coefficient, dx, for 1-bit DF is chosen to be 1 [22] as it minimizes the BER for most of the cases of fading considered in this thesis. The corresponding parameter d2, for 2-bit DF is chosen to be 2 in all the curves presented, except that in Figure 4.12. The simulation results presented in this thesis have 99% confidence intervals of approxi-mately ±1 % in a non fading AWGN channel, 95% confidence intervals of approximately ±5 % in flat Rayleigh fading and 95% confidence intervals of approximately ±10 % in FS Rayleigh fading. We will denote conventional (no decision feedback) G M S K as 'conventional' and G M S K with decision feedback as DF. In the figure legends, 't' stands for theoretical results, 's' for simulation results, 'c' for conventional detection and 'DF' for detection with decision feedback. 4.2 Error Performance in a Non Fading Environment In this section, the simulated BER and BKER results of 1 and 2-bit 'conventional' in a non fading channel with AWGN are shown. A numerical procedure for calculating the BER is given in [5]. For our purposes, it is more convenient to obtain the BER, which is used later to calculate the BKER, from simulations. The simulation values were verified to agree with the values in [5]. Figure 4.1 shows the BER of 1 and 2-bit 'conventional' GMSK in a non fading environment with AWGN. In the 2-bit detection case, two curves are shown: the first curve results if the detection threshold is optimized for every Eb/Na and the second curve is for a fixed detection threshold of zero [5]. As mentioned in Chapter 3, the BER performance of 2-bit detection can be much improved when the threshold is optimized. The advantage is apparent when Eb/ND is greater 8 6 than about 10 dB, as shown in Figure 4.1. A fixed zero threshold 2-bit 'conventional' shows a better performance than 1-bit 'conventional' when Eb/N0 is less than 17 dB. However, for Eb/N0 greater than 17 dB, 1-bit 'conventional' has a better BER performance. Eb/N0 (dB) Figure 4.1 BER of 1 and 2-bit conventional GMSK in a non fading environment with AWGN. For 2-bit detection, 'thres = 0' denotes the case where a decision threshold of zero is used. The block error rate (BKER) is the probability that there is at least one bit error in a packet of N bits. Assuming independent bit errors, the BKER is given by l - ( \ - P e y (4.2.1) where Pe is the BER. The probability of more than M bit errors in a block of length N bits, denoted as P(M, N), is important in the performance evaluation when using block coding in an error control scheme. The CDF is equal to the probability that a block has less than or equal to M errors, given by 87 M Pr{number of errors < M} = ^ Prinumber of errors = n} n = Q M - x {y.w n = 0 (4.2.2) and Pr{number of errors <M} = 1 - P(M, N). (4.2.3) Figure 4.2 and Figure 4.3 show the B K E R for 1-bit and 2-bit 'conventional' in a non fading environment with AWGN, respectively. Both of the figures show that the BKER for /V = 255 bits obtained from simulation is just slightly lower than that obtained from (4.2.1), using the 16 18 20 22 E b / N 0 (dB) Figure 4.2 BKER of 1-bit conventional GMSK in a non fading environment with AWGN corresponding simulated BER. This is because for differential detection, the bit errors are correlated (rather than independent) as each decision depends on the current phase and the phase at 1-bit period or 2-bit periods earlier (depending on whether it is 1-bit or 2-bit differential detection). It is also observed that as N increases, the difference between the B K E R from (4.2.1) 88 and the simulated BKER decreases. This observation agrees with that in [45] which investigates the block (packet) error rates of differentially encoded coherent BPSK, in which the problem of correlation between bit errors also arises. Eb/N0 (dB) Figure 4.3 BKER of 2-bit conventional GMSK in a non fading environment with AWGN. Note that the threshold is optimized for every E^/N' . Figure 4.4 shows the CDF of the number of bit errors in a block at Eb/N0 = 16 dB in a non fading environment for 1-bit and 2-bit 'conventional', with N = 255, 511 and 1023 bits. In both cases, the CDFs obtained from simulations agree very well with the corresponding theoreti-cal CDFs obtained from (4.2.2). This indicates that (4.2.2) (and (4.2.1)) is valid for approximating the block error statistics in a non fading environment, despite the fact that differential detection introduces some correlation between the bit errors. In summary, the performance of 2-bit detection can be much improved when the threshold is optimized. However, if a fixed zero threshold is used in 2-bit detection, it only shows a better performance than 1-bit detection for Eb/N0 < 17 dB. In both cases of detection, small correlation 89 (a) (b) Figure 4.4 CDF of the number of bit errors in a block at E^/Ng - 16 dB in a non fading environment for (a) 1-bit conventional GMSK (b) 2-bit conventional GMSK with optimum threshold. exists between the bit errors, introduced by differential detection. 4.3 Comparison of BER Performances between 'filtered' and 'unfiltered' cases in Rayleigh Fading In a fading environment, the received signal power varies and therefore the parameter Eb/N0 refers to the ratio of average energy (after fading) per bit to the noise PSD. In this section, the average BER performances are obtained from the numerical formulae derived in Chapter 3. For 1-bit detection, the 'unfiltered' and 'filtered' BERs in flat fading are obtained by (3.1.46) and (3.1.84), respectively. In FS fading, the corresponding BERs are given by (3.2.22) and (3.2.40). For 2-bit detection, the 'unfiltered' and 'filtered' BERs in flat fading are obtained by (3.3.24) and (3.3.38), respectively. In FS fading, they are given by (3.4.13) and (3.4.28). 90 Figure 4.5(a) compares the BER of 1-bit and 2-bit 'conventional' for both 'filtered' and 'unfiltered' cases in flat fading, with the Doppler frequency-time product fDT as a variable parameter for Eb/NQ at 20 dB and at 60 dB. For both cases, we can see that at Eb/Na = 20 dB, the BER is almost insensitive to fDT. However, at Eb/NQ = 60 dB, the BER increases with fDT. This is because at lower values of Eb/N0, the BER is noise-limited. Conversely, at high values of Eb/N0 (>40 dB) the noise has a negligible effect on the BERs, which are now dominated by the effect of the Doppler spread. In subsequent sections, results that show irreduci-ble error floors being present at Eb/No>40 dB with fDT > 0.001, are displayed. Error floors occur when the BER cannot be further lowered by increasing Eb/N0, due to the effect of Doppler spread. When the 'filtered' and the 'unfiltered' cases are compared: at Eb/NQ = 60 dB, the 'filtered' BER is on average about 25% higher than the 'unfiltered' BER of 1-bit detection for all values of fDT shown; and this difference is about 35% for 2-bit detection. This suggests that the BER for 2-bit detection is more sensitive to the IF filtering effect than that for the 1-bit case and can be explained by the fact that the BER expressions depends on the parameter J0(2nfDnT) where n = 1, 2 , for 1-bit and 2-bit detection, respectively. Since J0(.) is a decreasing function of its argument and a smaller / 0 (.) means an increased B E R (see equations (3.1.84) and (3.3.38)), the BER degradation through IF filtering is more apparent in the 2-bit case. Figure 4.5(b) shows the BER of 1 and 2-bit 'conventional' for both 'unfiltered' and 91 'filtered' cases as a function of fDT, with average carrier to delayed signal power ratio (CDR) = 20 dB and time delay difference x = 0.27 in a FS channel for Eb/Na at 20 dB and at 60 dB. In both cases, we see that at Eb/N0 = 20 dB, the BER is almost insensitive to fDT, as in the flat fading case. At Eb/N0 = 60 dB, the BER increases much more gradually, especially for small values of fDT (fDT < 0.001) when compared with the flat fading case. This is because in a FS channel, the BER is controlled by the combined effect of fDT, CDR and x. cr L U C O 0.000 1-bit, f i l t e red r& 2 - b i t , f i l t e red • 1-bit, un f i l t e red A 2 - b i t , un f i l t e red 10'' 10 ' L U C O 10 - ' 0.001 0.002 0.003 10"' 0.000 (a) - D 1 - b i t filtered - A 2-bit , f i l t e red • 1 -bit, un f i l t e red | A - A Z - b i t , un f i l t e r ed E./N = 20dB b o E./N = 60dB b o 0.001 0.002 (b) 0.003 Figure 4.5 Average BER as a function of fDT for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in (a) flat fading (b) FS fading with CDR = 20 dB, x = 0.2T . When Eb/Na = 60 dB and fDT < 0.001, the effects of Doppler and noise are negligible and the parameters CDR and x, which are fixed for the results shown in Figure 4.5(b), overwhelms the 92 Doppler effect on the BER performance. This explains why the BER is almost insensitive at low values of fDT. A comparison of the 'filtered' and the 'unfiltered' BER shows that the BER 10"' 10-' cr LU CB 10"J 10"' 0.000 -H 1 -bit, filtered - A 2-bit, filtered JQ - EH-bit, unfiltered IA - A2-bit, unfiltered E./N = 20dB b o E./N = 60dB b o _L 0.001 0.002 fnT 10"1 UJ IO"2 H CO 10"; 0.003 0.000 -€H-biL filtered -A 2-bit, filtered In - EI1 4ML unfiltered |A - A2 -b iL unfiltered E./N = 20dB b o E./N = 60dB b o _L 0.001 0.002 0.003 10"1 cr o iu 10-2 IO"-0.000 E./N = 20dB b o -Ol-bitfWtered -A2-bit,Wtered • 1 -bit, unfiltered IA - A 2-bit unfiltered E./N = 60dB b o 0.001 0.002 0.003 Figure 4.6 Average BER as a function of /DT for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in FS fading with (a) CDR = 20 dB, i = OAT (b) CDR = 10 dB, T = 0.2T (c) CDR = 10 dB, T = 0.4T . 93 degradation due to IF filtering is negligible when fDT is small and it increases with fDT for Eb/N0 = 60 dB, whereas it remains almost constant independent of fDT when Eb/N0 = 20 dB. 2-bit detection is also more sensitive to IF filtering. Similar reasons as stated earlier are accounted for the observations. Similar observations can also be deduced from Figure 4.6. In the case of flat fading (Figure 4.5(a)), it is observed that at both Eb/N0 = 20 dB and 60 dB, the percentage differ-ence between the 'filtered' and the 'unfiltered' BER is almost constant for the values of fDT. CDR CDR (a) (b) Figure 4.7 Average BER as a function of CDR for 'filtered' and 'unfiltered' cases of 1 and 2-bit conventional GMSK in FS fading for fDT = 0.002 with (a) T = 0.2T (b) x = 0.4T . From Figure 4.7, it can be seen that at both Eb/Na = 20 dB and 60 dB, the percentage difference between the BER in the 'unfiltered' and 'filtered' cases increases with CDR. This is because for a small CDR, the relatively high delayed signal power tends to mask the effect of the IF filtering; but as C D R increases, the IF filtering effect becomes more apparent. At 94 Eb/N0 = 60dB, with CDR = 60dB for both x = 0.2T and x = OAT, the BERs for the 'unfiltered' and 'filtered' cases approach the corresponding BERs of the flat fading case. This is expected because at such a high CDR, the delayed signal power is much weaker than that of the carrier signal power and can be considered to be negligible regardless of x, resembling the case of flat fading. In the case when Eb/No = 20 dB, the BERs stay almost constant as the CDR increases. This section investigated the effect of IF filtering on the BER performances of 1-bit and 2-bit conventional GMSK as a function of fDT and CDR. For both 'filtered' and 'unfiltered' cases, the BER is almost insensitive to fDT at low Eb/NQ but at high Eb/N0, the BER is dominated by the effect of the Doppler spread, thus it increases with fDT.A comparison of the 'filtered' and the 'unfiltered' BER in FS fading shows that the BER degradation due to IF filtering is negligible when fDT is small and it increases with fDT for high Eb/N0, whereas the BER degradation is almost the same for all values of fDT when Eb/N0 is small. In the case of flat fading, it is observed that for a given Eb/N0, the percentage difference between the 'filtered' and the 'unfiltered' BER is almost constant, independent of the values of fDT considered. In all the fading cases that were considered, 2-bit detection is more sensitive to IF filtering than 1-bit detection. 4.4 BER Performance in Slow Rayleigh Fading With slow fading, the signal is assumed to be constant for the duration of over at least one bit period, i.e. the product fDT should be much less than 1. In this case, fD is chosen to be 32 95 Hz, giving fDT= 0.002 where T is given by the reciprocal of the bit rate Rb. Figure 4.8 shows the average BER in a flat Rayleigh fading channel. Irreducible error floors start to appear at Eb/N0 ~ 45 dB . It can be seen from the figure that for 1 and 2-bit 'conventional' and 1-bit DF, the simulation results are in good agreement with the theoretical 30 Eb/N0 (dB) Figure 4.8 Average BER as a function of Eb/Ng in flat fading. results as given by (3.1.84), (3.3.38) and (3.1.84)*, respectively. The '*' indicates that for the decision feedback schemes, the BER formulae are based on those of the corresponding 'conven-tional' detection, but with the pertinent values of differential angles substituted, as described in Chapter 3. In the case of 2-bit DF, the simulations BER values are substantially higher than the theoretical results from (3.3.38)*. This is because (3.3.38)* assumes that all decisions being fed back are correct in order to simplify the analysis. However in the simulation, true decisions, some of which incorrect, are being fed back. In the 2-bit DF case, the phase shifting rule looks at two consecutive decisions which are then differentially encoded in order to determine the feedback 96 angle. Therefore, if one decision is incorrect, after differential encoding error propagation occurs to give double-bit errors. This explains why the simulated BER is higher than the theoretical BER from (3.3.38)*. To verify this explanation, a simulation was run in which all of the decisions being fed back are correct. In this case, it was found that the average BER for 2-bit DF is in close agreement with the analytical result from (3.3.38)*. The improvement in BER for 1-bit DF over 1-bit 'conventional' are about 55% and 60% at Eb/N0 = 30 dB and 60 dB, respectively. For 2-bit DF, the corresponding percentages are about 50% and 40%, respectively. It is observed that 2-bit 'conventional' performs slightly better than 1-bit 'conventional' when Eb/N0 <25 dB. For example, 2-bit 'conventional' provides an approxi-mate 1 dB gain over 1-bit 'conventional' at a BER = 10 _ 1 . When Eb/Na ~ 25 dB, there is a switch over between the two performances: 1-bit 'conventional' now performs better than 2-bit 'conventional' and has a lower irreducible error floor. In the cases of FS fading, there also exist similar trends, as shown in subsequent figures. This is expected because the BER performances of 1-bit and 2-bit 'conventional' (with zero-level decision threshold) in a non fading AWGN channel shows a similar trend (see Figure 4.1). In the case of decision feedback, there is almost no differ-ence between 1-bit DF and 2-bit DF when Eb/Na <30 dB. However, beyond Eb/Na = 30 dB, 1-bit DF has a better performance and has a lower error floor. We can compare the BER performances of 1 and 2-bit conventional G M S K with that of coherent detection obtained in [27] 1 1 in flat Rayleigh fading (known as the dynamic B E R 97 performance). When fDT = 0.0025 1 2 (fD = 40 Hz), the BER obtained in [27] is about l x l O - 3 at Eb/N0 = 60 dB (the error floor), whereas they are 1.68 x 10"4(from (3.1.84)) and 5.72 x 10"4 (from (3.3.38)) for 1 and 2-bit conventional G M S K , respectively. When fDT = 2.5 x 10" 4(/ D = 4 Hz), the BER from [27] is 1 x 10~5 and the corresponding BERs for 1 and 2-bit 'conventional' are 4.25 x 10 6 and 8.15 x 10 6 , respectively. Hence, in both cases, differen-tial detection achieves a lower BER over coherent detection in fiat Rayleigh fading. The improve-ment comes from the fact that differential detection cancels the phase distortion between adjacent symbols during phase comparison. In FS fading, the theoretical BERs for 1 and 2-bit 'conventional'; 1 and 2-bit DF G M S K are given by (3.2.40), (3.4.28), (3.2.40)* and (3.4.28)*, respectively. Figure 4.9 shows the average BER in FS Rayleigh fading channel with x = 0.2T and CDR = 20 dB. Irreducible error floors start to appear at around Eb/Na = 40 dB. As in the flat fading case, the simulation results for 1 and 2-bit conventional G M S K and 1-bit DF G M S K are in good agreement with the theoretical results. In the case of 2-bit DF GMSK, the average BER from simulation is about 50% higher than the theoretical results from (3.4.28)* for Eb/N0 > 30 dB. The degree of disagreement is less than that in the flat fading case. The improvements in BER of 1-bit DF G M S K over 1-bit conven-tional G M S K are about 50% and 15% at Eb/NQ = 30 dB and 60 dB, respectively. In 2-bit DF GMSK, the corresponding percentages are about 45% and 20%, respectively. 1 1 The experimental results in [27] were reproduced in [12] by software simulation. 1 2 The bit duration Tis chosen to be 6.25 10 5s , which is also assumed in this thesis. 98 E b / N 0 (dB) Figure 4.9 Average BER as a function of Eb/N' in FS fading with CDR = 20 dB, x = 0.2T. Figure 4.10 shows the average B E R in FS Rayleigh fading with x = OAT and CDR = 20 dB . Irreducible error floors occur at around Eb/Na = 40 dB. There is good agreement between the simulation results and the theoretical results for 1 and 2-bit 'conventional' and 1-bit DF. In the case of 2-bit DF, the average BER from simulation is about 30% higher than the corresponding theoretical result for Eb/N0 > 30 dB. The improvement in BER with 1-bit DF over 1-bit 'conventional' is about 40% at Eb/N0 = 30 dB. However at Eb/N0 = 60 dB, decision feedback does not improve the BER at all. The improvement in BER obtained with 2-bit DF are about 40% and 30% at Eb/N0 = 30 dB and 60 dB, respectively. Figure 4.11 shows the average B E R in FS Rayleigh fading with x = 0.2T and CDR = 10 dB . It shows that irreducible error floors start to occur at about Eb/No=A0 dB. Similar to the previous fading cases described earlier, the simulation results agree with the 99 theoretical results for 1 and 2-bit 'conventional' and 1-bit DF. With 2-bit DF, the average BER from simulations are about 30% and 25% higher than the corresponding theoretical results for Eb/N0 at 30 dB and 60 dB, respectively. 10° I i i i i i i i i i i i I 0 10 20 30 40 50 60 Eb/N0 (dB) Figure 4.10 Average BER as a function of Eb/N in FS fading with CDR = 20 dB, x = 0.47* The improvement in B E R with 1-bit DF over 1-bit 'conventional' is about 30% at Eb/Na = 30 dB . At Eb/N0 = 60 dB, decision feedback even degrades the BER by about 20%. This is because the feedback coefficient, d,, that we assume throughout the simulations (dx = 1) is not optimal in this case. For 2-bit DF, the improvement in BER over 2-bit 'conventional' is about 20% at Eb/Na = 30 dB. At Eb/Na = 60 dB, the BER obtained with decision feedback is about 5% higher than that of 'conventional' though theoretical result shows a 25% improvement in BER. This is because the analysis does not take into account of error propagation, as explained earlier. 100 Figure 4.12 shows the average BER in FS fading at Eb/N0 from 40 dB to 60 dB for the same CDR and T as in Figure 4.11, but with new optimal feedback coefficients. In 1-bit DF, both theoretical and simulation results show that the optimal feedback coefficient, dx is equal to 0.5. In this case, 1-bit DF exhibits a slight improvement in BER when compared to 1-bit 'conven-tional'. With 2-bit DF, the optimal feedback coefficient13, d2, from theory is 2 whereas it is 1.5 from simulation. The difference is again due to the fact that 'error propagation' is not considered in the theoretical analysis. The simulated BER from 2-bit DF is approximately the same as that of 2-bit 'conventional' when the new coefficient is used. Figure 4.13 shows the average B E R in FS Rayleigh fading with x = OAT and 1 3 For both 1-bit DF and 2-bit DF, the optimum coefficient is chosen in steps of 0.25 from the original value; the step size being fine enough to give an accurate optimum value. 101 • 1-bit conventional, s A 1-bit DF, s DC 111 00 • 2-bit conventional, s * 2-bit DF, s 1-bit conventional, t 1-bit DF, t ro 2-bit conventional, t 2-bit DF, t > < chosen to be 0.5, instead of 1 for the simulated 1-bit DF GMSK curve and d2 is chosen to be 1.5, instead of 2 for the simulated 2-bit DF GMSK curve. CDR = 10 dB . In this case, irreducible error floors occur at around Eb/Na = 30 dB, which is lower than the corresponding values for the other cases of fading previously described. There is good agreement between the simulation results and the theoretical results for 1 and 2-bit 'conven-tional' and 1-bit DF. In the case of 2-bit DF, the average BER from simulation is about 30% higher than the theoretical results for both Eb/N0 >30 dB. The improvement in BER with 1-bit DF over 1-bit 'conventional' is about 20% at Eb/N0 = 30 dB. However, the BER is not improved at all when Eb/N0 = 60 dB. The improvement in BER obtained with 2-bit DF over 2-bit 'conven-tional' is about 25% at both Eb/N0 = 30 dB and 60 dB. This section presented BER results of differentially detected G M S K in Rayleigh fading. For 1 and 2-bit conventional G M S K and 1-bit DF GMSK, the simulation results are in good agreement with the theoretical results. In the case of 2-bit DF GMSK, the simulation BER values are higher than the theoretical results due to the fact that error propagation is not taken into account in the theoretical results. The improvement in BER using decision feedback is greatest in 102 20 30 40 E b / N 0 (dB) Figure 4.13 Average BER as a function of Eb/N in FS fading with CDR = 10 dB, T = 0.47. flat fading. There is almost no BER improvement with decision feedback for the FS channel when CDR = 10 dB and x = 0.2T. In the different cases of fading that have been described, 2-bit conventional G M S K performs slightly better than 1-bit conventional G M S K when Eb/N0 < 25 dB. However, at a higher Eb/N0, the latter performs better and has a lower irreduc-ible error floor. In the case of decision feedback, there is almost no difference in performance between 1-bit D F G M S K and 2-bit D F G M S K when Eb/N0 <30 dB. However, when Eb/NQ > 30 dB, 1-bit DF G M S K has a better performance and a lower error floor. It is also important to note that the corresponding BER performances for 1-bit and 2-bit conventional G M S K and their DF schemes, degrade as we go from the case of flat fading shown in Figure 4.8 to FS fading with CDR = 10 dB and x = 0.4T shown in Figure 4.13. 103 4.5 BKER Performance in Rayleigh Fading The block lengths used in the simulation are N=255, 511 and 1023 bits. As in Section 4.2, fD is chosen to be 32 Hz, the corresponding fDTB are 0.51, 1.022 and 2.046, respectively; and TB is the duration of a block. Since fDTB is not much less than 1 in all of the cases, the signal cannot be considered to be constant over a block and therefore the environment is not slow fading with respect to BKER performance in this circumstance. Figure 4.14(a), Figure 4.14(c), Figure 4.15(a), Figure 4.15(c) and Figure 4.16(a) show the BKER performances of 1-bit 'conventional' and 1-bit DF for block lengths N=255, 511 and 1023 bits, in flat and FS Rayleigh fading with CDR = 20 dB, x = 0.27/; CDR = 20 dB, x = OAT, CDR = 10 dB, x = 0.2T and CDR = 10 dB, x = OAT, respectively. Figure 4.14(b), Figure 4.14(d), Figure 4.15(b), Figure 4.15(d) and Figure 4.16(b) show the corresponding B K E R performances of 2-bit 'conventional' and 2-bit DF for the same block lengths and fading parame-ters, respectively. In general, we see that irreducible error floors appear at Eb/NQ in the range of 40 dB to 50 dB. With 1-bit detection: in Figure 4.14(a) for flat fading, Figure 4.14(c) and Figure 4.15(a) for FS fading where the CDR = 20 dB, the B K E R approximately doubles when N doubles at Eb/NQ > 50 dB. In Figure 4.15(c) and Figure 4.16(a) for FS fading where the CDR = 10 dB, the B K E R increases with N as expected but the increase is less than double when TV doubles. In the case of 2-bit detection, the BKER performances show similar trends, as in 1-bit detection. 104 Figure 4.14 Average BKER as a function of Eb/No for (a) 1-bit conventional GMSK and 1-bit DF GMSK in flat fading (b) 2-bit conventional GMSK and 2-bit DF GMSK in flat fading (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, % = 0.2T (d) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB , T = 0.2T. 105 Figure 4.14(a) shows the BKER performance of 1-bit 'conventional' and 1-bit DF in flat fading. With DF, at Eb/NQ= 60 dB, the improvement in B K E R over 1-bit 'conventional' is approximately 55% for the values of N considered. The percentage improvement is almost equal to that of the corresponding BER. Figure 4.14(c) shows the BKER performance of 1-bit 'conventional' and 1-bit DF in FS Rayleigh fading with x = 0.2T and C D R = 20dB . The BKER obtained from 'conventional' is improved by about 15% when DF is applied, for the values of N considered at Eb/Na = 60 dB. The percentage improvement is roughly equal to that of the corresponding BER performance. Figure 4.15(a) shows the BKER performance of 1-bit 'conventional' and 1-bit DF in FS Rayleigh fading with x = 0.4T and CDR = 20 dB. With decision feedback, the improvement in BKER is about 40% at Eb/N0 = 30 dB but there is almost no improvement at Eb/NQ = 60 dB, for the values of N considered. The corresponding BER performance shows a similar percentage in improvement. Figure 4.15(c) shows the BKER performance of 1-bit 'conventional' and 1-bit DF in FS Rayleigh fading with x = 0.2T and CDR = 10 dB. With decision feedback, the improvement in BKER is about 30% at Eb/N0 = 30 dB for N = 255 bits and it is about 25% when N= 1023 bits. However, the BKER is slightly degraded at Eb/Na = 60 dB, for all three values of N. As with the BER performance, the degradation is due to the fact that the feedback coefficient has not been chosen appropriately at Eb/N0 = 60 dB. CU > < 10"' — • 1-bit c, block length N=255bits — A 1-bit c, block length N=511bits —•1-bit c, block length N=1023bits E - - • 1-bit DF, block length N=255brts - -A1-bitDF,block length N=511bits - -*1-bitDF, block length N=1023Ms S> 10-to CD > < J l j I i L 10 20 30 40 50 60 Eb/N 0 (dB) (a) 10'' — • 2-bit c, block length N=255bits —4 2-bit c, block length N=511 bits — • 2-bit c, block length N=1023bhs • - - B 2-bit DF, block length N=255bits | A - -A2-bitDF, block length N=511bits * - - *2-bit DF, block length N=1023bits • B * 10 20 30 40 Eb/N 0 (dB) (b) 50 60 rr LU CO CD > < 10"' 10 u 1 -bit c, block length N=255bils 1 -bit c, block length N=511 bits «1-bit c, block length N=1023bits B 1-bit DF, block length N=255bits lA - - A1-bit DF, block length N=511bits | * - - • 1-bit DF, block length N=1023bits cr LU CO 0) CD CO i CD > < 10" — • 2-bit c. block length N=255bHs — i 2-bit c, block length N=511btts — • 2-bit c. block length N=1023Ms • - - • 2-bit DF, block length N=255bits |A - - A2-M DF, block length N=511bils \ » - - * 2-bit DF, block length N=1023bits 10 20 30 40 Eb/N 0 (dB) ( C ) 50 60 10 20 30 40 Eb/N 0 (dB) (d) 50 60 Figure 4.15 Average BKER as a function of EB/NQ for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.42" (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB , T = 0.4F (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB, x = Q.2T (d) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.27/. 107 Figure 4.14(b) shows the BKER performance of 2-bit 'conventional' and 2-bit DF in flat fading. When DF is applied, the improvement in BKER over 2-bit 'conventional' is approxi-mately 70% for the values of N considered. In the case of BER performance, the percentage improvement is only about 40%. Better improvement with DF shown by the BKER performance is due to the fact that with 2-bit DF, the bit errors tend to occur in pairs owing to error propagation whereas the number of blocks with an odd number of errors is reduced. Figure 4.14(d) shows the BKER performance of 2-bit 'conventional' and 2-bit DF in FS Rayleigh fading with x = 0.27/ and CDR = 20dB . At Eb/NQ= 60 dB, the BKER obtained with DF is approximately a 50% improvement over that of 'conventional', for the values of N considered. In the case of BER performance, the improvement is only about 20%. Figure 4.15(b) shows the BKER performance of 2-bit 'conventional' and 2-bit DF in FS Rayleigh fading with x = OAT and CDR = 20 dB. With decision feedback, the percentage improvement in BKER is about 45% at Eb/NQ = 60 dB, for the values of N considered. The corresponding percentage improvement in the BER performance is about 30%. Figure 4.15(d) shows the BKER performance of 2-bit 'conventional' and 2-bit DF in FS Rayleigh fading with x = 0.2T and CDR = 10 dB. When decision feedback is applied, the improvement in BKER is about 25% at Eb/N0 = 60 dB for N = 255 bits but it decreases to about 18% when N = 1023 bits. However, we have previously mentioned that the BER is slightly degraded (see Figure 4.11) because the decision feedback coefficient that is assumed in other fading cases, is not optimal in this case. The reason that there exists an improvement in BKER is due to error propagation, as described earlier. 108 Figure 4.16(a) shows the BKER performances of 1-bit 'conventional' and 1-bit DF in FS Rayleigh fading with x = OAT and CDR = 10 dB. At EB/N0 = 60 dB, decision feedback does not improve the BKER, for all three values of N. This is similar to the trend shown by the BER performance. Figure 4.16(b) shows the corresponding BKER performance of 2-bit 'conventional' and 2-bit DF. When decision feedback is applied, the improvement in BKER is about 35% at EB/NA = 60 dB for N = 255 bits and it is about 15% when N = 1023 bits. rr L U DO CD O) CO 3 10" — a 1-bit c, block length N=255bils — * 1-bit c. block length N=511bils — • 1-bit c, block length N=1023bits |Q - - • 1-bit DF, block length N=255bils |A - - Al-bit DF, block length N=511 bits I* - - *1-brt DF, block length N=1023bits 10 20 30 40 Eb/N 0 (dB) (a) 50 60 10 u rr L U m CD D> 2 cu 3 10" — • 2-bit c, block length N=255bits — A 2-bit c, block length N=511bits — • 2-bit c, block length N=1023bils • - - •2-bit DF, block length N=255bits | A - - A2-bH DF, block length N=511bits - - * 2 - M DF, block length N=1023bits _1_ _1_ 10 20 30 40 50 Eb/N 0 (dB) (b) 60 Figure 4.16 Average BKER as a function of Eb/N for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB, T = OAT (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB , t = 0.47. Figure 4.17 and Figure 4.18 show the BKER performances of 1-bit and 2-bit 'conven-tional' for blocks lengths N= 255, 511 and 1023 bits in fiat and FS fading with the same parame-ters (CDR and x) as for Figure 4.14 to Figure 4.16. In all cases, 1-bit 'conventional' has a better 109 performance over 2-bit 'conventional' when Eb/N0 >25 dB because it has a lower BER and a lower irreducible error floor for high Eb/N0, For flat and FS fading with CDR = 20 dB at Eb/Na = 60 dB, the improvement is about the same for all values of N. For FS fading with CDR = 10 dB, the improvement drops as N increases. When the channel goes from flat fading to FS fading, the improvement also decreases for all values of N. For example, there is around 70% improvement with flat fading but it is down to about 50% for FS fading with x = OAT and CDR = 10 dB, for N = 255 bits. The cumulative density function (CDF) of the number of bit errors in a block for 1-bit and 2-bit detection in flat fading at Eb/N0 = 40 dB for N = 255, 511 and 1023 bits are shown in Figure 4.19 and Figure 4.21, respectively. Figure 4.19(a) compares the CDFs between 1-bit 'conventional' and 1-bit DF. It can be seen that the Pr{errors <M} for 1-bit 'conventional' is always lower than that for 1-bit DF. This is because the BER for 1-bit 'conventional' is always higher than that for 1-bit DF as shown in Figure 4.8. Figure 4.19(b) compares the CDFs between 2-bit 'conventional' and 2-bit DF. The CDF for 2-bit DF shows a stair-case-like shape: for example, the Prjerrors < 1} is just slightly higher (about 0.2% higher) than 7V{errors <0} , but the Pr{errors <2} is about 3.5% higher than Pr{ errors < 1} . This implies that the number of blocks that contain only one bit error is much less than those that have double-bit errors. The predomination of double-bit errors was discussed in Section 4.4. rr LU m 0 CD « i CD > < 20 30 40 Eb/N 0 (dB) ( C ) — • 1-bit c, block length N=255bits — * 1-bit c, block length N=S11 bits —•1-bit c, block length N=1023bits !• - - B 2-bit c, block length N=255bits |A - - A2-bit c, block length N=511bits | » - - » 2-bit c block length N=1023bits J I I l _ 10 20 30 40 Eb/N 0 (dB) (d) 50 60 Figure 4.17 Average BKER as a function of E^/N' for 1-bit conventional GMSK and 2-bit conventional GMSK in (a) flat fading (b) FS fading with CDR = 20 dB, x = 0.27 (c) FS fading with CDR = 20 dB, x = 0.47 (d) FS fading with CDR = 10 dB, x = 0.2T. I l l tr LU CO CD cn co * CD 3 20 30 40 Eb/N 0 (dB) Figure 4.18 Average BKER as a function of E^/N' for 1-bit conventional GMSK and 2-bit conventional GMSK in FS fading with CDR = 10 dB, t = 0.41. Generally, the probability that a block contains an odd number of errors is less than that of an even number of errors. This is clearly illustrated in Figure 4.20 which shows the CDFs at Eb/N0 = 20 dB . We see that in the CDF of 2-bit DF, 'flat tops' appear when M goes from even to odd and they are joined by 'slopes' when M changes from odd to even. Figure 4.21 shows the CDFs of 1-bit and 2-bit 'conventional' in flat fading for N = 255, 511 and 1023 bits at Eb/Na = 40 dB. It can be seen that for M < 2, 1-bit 'conventional' shows a better performance than 2-bit 'conventional', for all three values of N. This is indicated by its higher 7Jr{errors < M} over 2-bit 'conventional'. However, when M > 2, the performance of 1-bit 'conventional' is equivalent to or slightly worse than that of 2-bit 'conventional', although 1-bit 'conventional' has a lower BER. If we compare the simulated B K E R which is equal to 112 l-Pr{errors = 0} in Figure 4.21 and that obtained with (4.2.1) using the corresponding average BER in flat fading, we found that the simulated BKER 1 4is lower than that obtained in (4.2.1) for both 1-bit and 2-bit 'conventional' with all three values of N. This is because the assumption of independent bit errors made in (4.2.1) is not realistic in the case of fading, where error bursts occur during fades and hence correlation exists between bit errors. (a) (b) Figure 4.19 CDF of the number of bit errors in a block at E^/No - 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in flat fading (b) 2-bit conventional GMSK and 2-bit DF GMSK in flat fading. Both Figure 4.22 and Figure 4.23 compare the CDFs of 1-bit and 2-bit 'conventional' with their corresponding decision feedback schemes in FS fading, for the appropriate values of CDR and x, when N = 255, 511 and 1023 bits. Similar trends as in Figure 4.21 can be observed. As an example, for N= 255 bits, the BKER obtained from (4.2.1) is 0.08653 and 0.138468, for 1-bit and 2-bit 'con-ventional', respectively. Figure 4.21 CDF of the number of bits errors in a block at E^/N' GMSK in flat fading = 40 dB for 1-bit and 2-bit conventional 114 0.95 f 0.9 £ 0.85 N 0.75 x block length N=255bits o block length N=511 bits * block length N=1023bits 1-bit c 1-bitDF x block length M=255bi1s o block length N=511bits * block length N=1023bits 2-bit c 2-bit DF (a) (b) 0.65 x block length N=255bits o block length N=511bits * block length N=1023bits 1-bit c 1-bitDF 15 M 0.3' x block length N=255bits o block length N=511bits x block length N=1023bits 2-bit c 2-bit DF 15 M 20 25 20 25 30 0 5 10 (c) (d) Figure 4.22 CDF of the number of bit errors in a block at E^/N = 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.2T (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB, x = 0.2T (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 20 dB, x = OAT (d) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 20 dB, x = OAT 30 115 (c) (d) Figure 4.23 CDF of the number of bit errors in a block at E^/N o = 40 dB for (a) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB, X = 0.2T (b) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB, x = 0.2T (c) 1-bit conventional GMSK and 1-bit DF GMSK in FS fading with CDR = 10 dB , X = 0.47 (d) 2-bit conventional GMSK and 2-bit DF GMSK in FS fading with CDR = 10 dB, x = OAT. 116 Figure 4.24 compares the CDFs between 1-bit and 2-bit 'conventional' in FS fading (for C D R = 20 dB, x = 0.27/; C D R = 20 dB, x = 0.47/; C D R = 10 d B , x = 0.27 and 0.31 1 1 1 1 1 1 ' 1 1 1 01 1 1 1 1 1 1 1 0 2 4 6 8 10 12 14 16 18 20 0 10 20 30 40 50 60 70 M M (c) (d) Figure 4.24 CDF of the number of bit errors in a block at Eb/NQ = 40 dB for (a) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 20 dB, x = 0.27 (b) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 20 dB, x = 0.47 (c) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 10 dB, x = 0.27 (d) 1-bit and 2-bit conventional GMSK in FS fading with CDR = 10 dB, x = 0.47. 117 CDR = 10 dB , x = 0.47") with N = 255, 511 and 1023 bits at Eb/Na = 40 dB. In all of the cases, 1-bit 'conventional' performs better than 2-bit 'conventional' when M is small and the advantage disappears when M increases. This section examined the BKER performances of 1-bit and 2-bit differential detection of GMSK in flat and FS Rayleigh fading. In general, we see that irreducible error floors appear at Eb/N0 in the range of 40 dB to 50 dB. The percentage improvement in BKER of 1-bit DF GMSK over 1-bit conventional GMSK is almost equal to that of the corresponding BERs presented in Section 4.4. In the case of 2-bit DF GMSK, the BKER performance shows an increased improvement over 2-bit conventional GMSK when compared to the corresponding BER performance. This is due to the presence of increased error propagation in 2-bit DF GMSK. In all cases, 1-bit conventional GMSK has a better BKER performance over 2-bit conventional GMSK when Eb/Na >25 dB. This is because 1-bit conventional GMSK has a lower BER and a lower irreducible error floor in this Eb/N0 range. In the case of 2-bit DF GMSK, for a given block, its probability of containing an odd number of errors is less than that of an even number of errors, due to error propagation. When examining the CDFs of 1-bit and 2-bit conventional GMSK especially at Eb/N0 >20 dB, the simulated BKER is always lower than the corresponding BKER obtained with (4.2.1) which assumes independent bit errors. This is because fading causes the bit errors to occur in bursts, thus reducing the BKER. Chapter 5 Conclusions In this thesis, we examined the BER and BKER performances of GMSK employing 1-bit and 2-bit differential detection in AWGN, flat and FS Rayleigh fading channels through analysis and computer simulations. A detailed analysis was carried out to look at the IF filtering effect on the BER performance over fading channels. The BKER performance, which gives information on the bit error distributions, was mainly studied through simulations. Since the premodulation Gaussian filter at the GMSK transmitter introduces significant ISI, we also examined the perform-ance of a decision feedback scheme which partially removes the effect of destructive ISI, over such channels. The effect of IF filtering on the BER performance over Rayleigh fading channels was examined. For both 'filtered' and 'unfiltered' cases, the BER is almost insensitive to fDT at low Eb/Na but at high Eb/NQ, the BER is dominated by the effect of the Doppler spread, and it increases with fDT. A comparison of the 'filtered' and the 'unfiltered' BER in FS fading, shows that the BER degradation due to IF filtering is negligible when fDT is small but it increases with fDT for high Eb/NQ, whereas the BER degradation is almost the same for all values of fDT when Eb/N0 is small. In the case of flat fading, it is observed that for the same Eb/NQ, the percentage difference between the 'filtered' and the 'unfiltered' B E R is almost constant, independent of fDT. The BER results of differentially detected G M S K as a function of Eb/N0 in Rayleigh fading were then presented. For 1 and 2-bit conventional GMSK and 1-bit DF GMSK, the simula-118 Chapter 5 Conclusions 119 tion results are in good agreement with the theoretical results. In the case of 2-bit DF GMSK, the simulation BER values are higher than the theoretical results. This is because error propagation is not taken into account in the analysis. The improvement in BER using decision feedback is greatest in flat fading. In the different cases of fading that were considered, 2-bit conventional G M S K performs slightly better than 1-bit conventional GMSK for Eb/Na < 25 dB. For higher Eb/NQ, the latter has a lower irreducible error floor than the former. This is expected because the BER performances of 1-bit and 2-bit conventional G M S K (with a fixed zero decision threshold) in a non fading A W G N channel also shows a similar trend. In the case of decision feedback, similar performance trends are observed. As expected, the corresponding BER performances for 1-bit and 2-bit conventional GMSK and their DF counterparts, degrade as the CDR decreases. In FS fading for the same CDR, the BER performance also degrades when the time delay difference, x, is increased. For the purpose of computing the BKER in a non fading AWGN channel, it was found that the bit errors can be assumed to be independent, though differential detection introduces some correlation between them. The BKER results of 1-bit and 2-bit differential detection of GMSK in flat and FS Rayleigh fading show that irreducible error floors generally appear at Eb/N0 in the range of 40 dB to 50 dB. The percentage improvement in B K E R of 1-bit DF G M S K over 1-bit conventional G M S K is almost equal to that of the corresponding BERs at high Eb/Na. In the case of 2-bit DF GMSK, the B K E R performance shows an increased improvement over 2-bit conventional GMSK when compared to the corresponding BER performance, due to the increased presence of error propagation in 2-bit DF GMSK. Moreover, due to error propagation, there is a higher probability that a given block contains an even number of errors than that of an odd Chapter 5 Conclusions 120 number of errors. Since fading gives rise to bursty bit errors, the actual BKER is better than the predicted assuming independent bit errors. Some suggestions for related future research work are given as follows. • To derive formulae for BKER performance in very slow fading, taking into account bit error correlation introduced by differential detection, and attempt to extend the analy-sis to the slow fading case. • Since GMSK and CPM in general have memory such that their information carrying phase follows a certain trellis path, Viterbi decoding can be applied for error correction after differential detection [6]. The BER and BKER performances of differentially de-tected CPM can be analyzed with the application of Viterbi algorithm over the mobile radio channel. • To investigate the BER and BKER performances in Nakagami fading used to model fading in microcellular systems [38]. Glossary ACI - Adjacent channel interference {dj} - Input information sequence of logical ' 1' and '0' AWGN - Additive White Gaussian Noise Bt - 3-dB bandwidth of the premodulation Gaussian filter Bi - 3-dB bandwidth of the IF Gaussian filter Bn - Equivalent noise bandwidth of the IF Gaussian filter {bj} - Non-return-to-zero (NRZ) signalling representation of {a^ } Pn - Gain of the nth oscillator in the Jake's fading model BER - Bit Error Rate BKER - Block (Packet) Error Rate g c - Speed of light equal to 3 x 10 m/s C - Channel capacity {c .} - Differentially encoded sequence of {bj} CDF - Cumulative density function CDR - Average carrier (desired signal) to delayed signal power ratio CPM - Continuous Phase Modulation dx and d2 - Decision feedback coefficients for 1-bit and 2-bit DF GMSK, respectively dj-(t) - Delayed (secondary) signal after fading D(f) - Doppler spectrum 121 122 DF - Decision feedback A / c - Coherence bandwidth A0 J f l and A<|>d> ,; A<j>4i / and A<jv, / - 'Unfiltered' differential phase angle (phase change) for the desired signal and the delayed signal, respectively. The 'tilde' denotes the 'filtered' case. U W , A.DF K~,DF ~DF Ac^ z and A ^ , ; A ^ / and A ^ / - 'Unfiltered' differential phase angle (phase change) for the desired signal and the delayed signal after decision feedback, respectively. The 'tilde' denotes the 'filtered' case. - (Average) energy per bit or (average) energy per symbol, as specified fc a n d °>c - Carrier frequency in Hz and radians/s, respectively fd - Peak frequency deviation f D a n d fflD - Doppler spread or maximum Doppler frequency in Hz and radians/s, respectively - Sampling frequency F(v) - Characteristic function of the sampled output V F M - Frequency Modulation FS - Frequency selective FSK - Frequency Shift Keying g(t) - Response of the premodulation filter to a transmitted rectangular pulse G M S K - Gaussian-filtered Minimum Shift Keying GPRS - General Packet Radio Service G S M - Global System for Mobile Communication 123 h - Modulation index n,y(r) - Impulse response of IF filter h(t) - Equivalent lowpass impulse response of IF filter ht(t) - Impulse response of the premodulation filter Hi/if) - Frequency response of IF filter H(f) - Equivalent baseband frequency response of IF filter Ht(f) - Frequency response of the premodulation filter IF - Intermediate frequency ISI - Intersymbol Interference Ja(.) - Zeroth order Bessel function of the first kind A, - Decision feedback angle m - Number of bits per symbol in modulation MFSK - m-ary FSK MPSK - m-ary PSK MSK - Minimum Shift Keying n(t) - B andlimited noise N - Block length in bits iV 0 - One-sided power spectral density of Gaussian noise Nos - Number of low-frequency oscillators in the Jakes Model con - Frequency of the nth oscillator in the Jakes Model p.d.f. - Probability density function 124 Pe - BER (for a non fading AWGN channel) or average BER (for fading) P(M, N) - Probability of more than M bit errors in a block of length N bits $(t;b) or Q>s(t) - Transmitted information carrying phase §s(t) - Filtered information carrying phase Ps - Desired (average) signal power at the point before AWGN is added PSD - Power spectral density or power spectrum PSK - Phase Shift Keying r(0 - 'Unfiltered' received signal r(t) - 'Filtered' received signal Rb - Bit rate Rs - Symbol rate RF - Radio frequency p,(.), p d ( . ) andp„( . ) - Normalized autocorrelation functions of the unmodulated carrier of the desired fading signal, the delayed fading signal and noise, respectively s(t) - Transmitted signal Sjr(t) - Desired (main or undelayed) signal after fading 2 Gd - Average power of dj(t) 2 Gn - Postdetection noise power, i.e. power of n(t) 125 Cs - Average power of sy(r) SNR - (Average) signal-to-noise ratio, it is an average value in fading tc - Coherence time x - Time delay difference between the two signals in a FS channel %d - Multipath delay spread T - Bit or symbol duration TB - Block duration UMTS - Universal Mobile Telecommunication Systems v - Velocity of the mobile unit V - Sampled output of lowpass filter in the receiver W - Channel bandwidth Wn(t) - Equivalent baseband spectrum of noise Ws(t) - Equivalent baseband spectrum of the unmodulated carrier fading signal xs(t), ys(t), xd(t), yd{t), xn(t) and yn(t) - Independent zero mean, lowpass Gaussian processes Z(t) - Complex envelope of r(t) Ztf(t) - Complex envelope of r(t) Zd(t) - Equal to xd(t) + jyd(t) ZM) -Equal to xn(t)+jyn(t) Zs(t) - Equal to xs(t) + jys(t) Bibliography [I] S. 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Appendix A Evaluation of the Characteristic function of the Detector output The problem to be considered is to find the p.d.f. of the detector output V(IT) (denoted as V hereafter) through its characteristic function F(v) [39] F(v) = [^f ( I -2yvC *Q) ] _ 1 (A.l) where Q is the 2x2 Hermitian matrix of the quadratic form and it satisfies V = \zT*QZ. (A.2) 1-Bit Detection For 1-bit detection, Vis given by V = \Re{-jzxz2*} (A.3) where z\ = Z(IT) and z 2 = Z((l-1)T). V can be readily shown to be Hermitian since Q is Hermitian. Equating (A.2) with (A.3) and ignoring the scalar factor, Q can be obtained as Q = 0 j . (A.4) -j q 130 Appendix A 131 The covariance matrix C z is given in (3.1.31). Therefore F(v) is obtained as F(v) = det 2 2 1 + 2VG p -2va 2 2 2VG 1 - 2VG p -l (A.5) = [ l -2va 2 (p -p*)+4vV( l - |p | 2 ) ] 1 The derivation of the covariance matrix C z for filtered 1-bit detection in flat and FS Rayleigh fading channels is given in Appendix C. 2-Bit Detection Using the above procedure, Q can be obtained as Q = 0 1 1 0 (A.6) for the 2-bit case. The corresponding characteristic function F(v) is found to be F ( v ) = det 2 2 1-2jvG p -2jvo 2 2 -2JVG 1-2JVG p (A.7) = [ 1 - 2;va2(p + p* ) + 4vV( 1 - |p| 2)f 1 Appendix B 132 Appendix B Evaluation of the p.d.f. of the 1-bit detector output from its Characteristic function The residue theorem can be applied to evaluate the following improper integral, as a contour integral. The contour integral is being taken in the anti-clockwise sense around a closed path [41]. Therefore, the characteristic function F(v) can be taken to be | F ( v ) e - > ^ v = j 2 ' I - ' ' ( ' ' " [ F ( z ) e " J f t ] ) V < ° <B.l) l-2nj(res[F(z)e jVz]) V>0 where the residue of a simple pole at z =PQ for a function f(z), is defined to be r « / (z ) = lim (z-p0)f(z) (B.2) The characteristic function F(v) for one-bit detection is derived in Appendix A and is given by (A.5). The two poles of F(v) are found to be P*>2 ~ 2 , ,2 2 a ( l - | p | 2 ) (B.3) Appendix B Consider the case when V < 0, res[F(z)e jVz] lim (z-p1)F(z)e j V z 2 ^ P i lim 1 z^Pl(z-pl){z-p2) 4 G 4 ( 1 _ | p | 2 ) 1 e~jVz = lim *-^>4o - 4 ( l - |p | 2 ) (z-p2) j f Jl~P2r + Pj J = ; exp<{ J ! — - — - i-V^ 4 a 2 ^ T ^ l 2 a 2 ( l - | p | 2 ) J Therefore the conditional p.d.f. of V when V < 0, is given by P(V\bl = ±1) = • J F(v)e ; v V J v where is the /r/z input bit. ^•27U7-rE5[F(z)e" 7'V z] 1 [ 71-Pr + P j 4a 2 / /rrp~ 2 e X P l2a 2 (l- |p| 2 ) Appendix B 134 For the case V> 0, the analysis is similar. res[F(z)e jVz] = lim (z- p2)F(z)e = lim -jVz -jVz ^ ^ 4 a 4 ( l - | p | 2 ) (z-P\) 2 a 2 ( l - | p | 2 ) V (B.6) Therefore, the conditional p.d.f. of Vwhen V > 0, is given by P(V\bl = ±l) = ^z\F(v) dv = ~ • 2nj • res[F(z)e-jVz] 4 a 2 J T p " 2 exp-! A/1 - P r - P j ' ^ 2 o 2 ( i - i p i 2 ) ; V V>0 (B.7) Appendix C 135 Appendix C Derivation of the Covariance matrix C z for Filtered 1-Bit Detection in Flat and FS Rayleigh Fading Channels Flat Rayleigh Fading Channel The filtered received waveform at the receiver output r(t) is given as r(t) = X(r)cosco cr-Y(t) sin® ct (C.l) where X(t) = xs(t)Ac(t) - ys(t)As(t) + xn{t), (C.2) Y(t) = xs(t)As(t) + ys(t)Ac(t) + yn(t), (C.3) and Ac(t) = cos(|)s(0® hif(t), , (CA) As(t) = sm$s(t)®hif(t). (C.5) T We define a vector a = [xs(t), ys(t), xn(t), yn(t)\, where xs(t), ys(t), xn(t) and yn(t) are mutually independent zero mean Gaussian processes. Thus, the following relations are obtained: Appendix C 136 xs(t) = ys(t) = xn(t) = yn{t) = 0 x (Oy,-(0 = y,-(0*,-(0 = *,-(0*,-(0 = y,-(0y,-(0 = o (Oy/0 = y,-(0*/0 = o (C.6) (f + $)y,(0 = y,-(f + ^ ,.(0 = 0 where i , j = s, n j where the bar denotes taking the ensemble average. Hence, the covariance matrix of a is given by _ — T C a = [a-a][a-a] a! 0 0 0 0 a; 0 0 0 0 a„ 0 0 0 0 a; (C.7) 2 2 where as and aM are the average received powers of the signal after fading and noise. Define a vector e = [Xv Yx, X2, Y2], where Xj(0 = X(0 , r,(0 = Y(t),X2(t) = X(t-T),Y2(t) = Y(t-T) (C.8) and X- = Xt(lT) , F- = y,.(/r) ,(i= 1,2;/is an integer). (C.9) Appendix C 137 Its covariance matrix H e is defined as H e = [ e - e ] [ e - e f (CIO) where the bar denotes taking the statistical average. H e is computed to be where H e = aT 0 0"lPl ° 2 P 2 0 a 2 -c22p2 a 2 p j o"iPi " a 2 p 2 a 2 0 a 2 p 2 a l P l 0 a 2 (C.l l) 2 . 2 2 2 2 . 2 2 2 ° " 2 = A2°,+<V a i P i = A i A 2 G ? P , ( 7 , ) c o s ( A ^ ; / ) + a 2p„(r), °1>P2 = - ^ 1 ^ 2 a ? P , ( r ) s i n ( A ^ , / ) ' A2(t) = A2(t) + A2(t) , (C.12) (C.13) (C.14) (C.15) (C.l 6) A<j>5(/ = tan" 1(A 5(/r)/A c(/r))-tan" 1(A ,(/r-r)/A c(/r-7')) ) (C.17) and Appendix C 138 A] = A2(IT-T) A\ = A2(IT) (C.18) Define the vector Z to be Z = [Z(IT) Z(IT-T)] (C.19) where Z(t) = Xi(t) + yYi(t) and its covariance matrix is defined as C z = X-[Z-Zf[Z-Z]T (C.20) The elements of C z can be computed from H e as G21pl+jO2202 CTlPl -7"a2p2 O ^ P 2 a ,a 2 p a 2 (C.21) where 0"jO2p is given as a , a 2 p = o 2 p 1 - ; a 2 p 2 = A j A ^ p ^ e x p C / A ^ / ) + CT2pn(T) (C.22) F S Rayleigh Fading Channel In the case of FS fading, the filtered received waveform at the receiver output is given in (C.l), where X(t) = xs(t)Ac(t)-ys(t)As(t) + xd(t- x)Ac(t -x)-yd(t- x)As(t -x) + xn(t) (C.23) Appendix C 139 Y(t) = xs(t)As(t) + ys(t)Ac(t) . (C.24) + xd(t- x)As(t - x) + yd(t- x)Ac(t -x) + yn(t) T We define a vector a = [xs, ys, xd, yd, xn, yn]. Following the same procedure as in the filtered FS fading case and using the relations given in (C.6) in which i,j now denote the subscripts s,d or n and i ^  j, we can obtain the corresponding covariance matrix H e as given in (C.l 1) where Oj = A1os+AxlGd + cn , (C.25) <h = AWs + k\i^d + a«. (c-26) a 2 P l = A1A2a2p,(7')cos(Ac>J; /) + A , 1 A, 2 aJ P r f ( r ) cos (A^ ; ) + a 2p„(r), (C.27) a2p2 = - [A 1 A 2 CT 2 p 5 ( r )s in(A^ i / ) + A T l A T 2 a ^ ( r ) s i n ( A ^ ) / ) ] , (C.28) with Acj)^  / being given in (C.17) and A<j>4/ = tan X(AS(IT-x)/AC{IT-x)) -tan 1{AS(IT- T-x)/Ac{lT- T-x)). (C.29) Equation (C.l 8) defines the variables A j and A 2 and A 2 ! = A2(lT-T + x) . (C30) A 2 2 = A 2 ( / 7 + T ) The corresponding covariance matrix C z can be similarly obtained as that in the filtered flat fading 2 2 case. C z is given in (C.20) with ax, G 2 given by (C.25) and (C.26), respectively and Appendix C 140 2 2 o,a 2p = o , p 1 - 7 a 2 P 2 _ (C.3i) = A l A 2 a 5 2 p * ( r ) e X P O ' A < k /) + Ax\Az2°dPd(T)exV(JA$d, l) + C2npn(T) Appendix D Appendix D Differential Phase Angles of 1-bit and 2-bit Detection 141 BtT 9-3 0-2 0-1 0 O 01 02 0. 0.25 0.0009 0.6 18.2 52.4 18.2 0.6 0.0009 Phase shifts 6^  _ . (in degrees) for B,T = 0.25 in 1-bit detection. BtT 0-3 0-2 0-1 00 01 02 03 0.25 0.00017 0.2 11.3 50.3 26.7 1.5 0.0042 Table D.2 Delayed phase shifts (in degrees) for BtT= 0.25 with x - 0.27 in 1-bit detection. Bit Combinations State ''/-I  h l  /;/+/ 1 1 1 1 88.8 88.3 1 1 -1 2 52.4 34.9 -1 1 1 3 52.4 65.7 -1 1 -1 4 16.0 12.3 1 -1 1 5 -16.0 -12.3 1 -1 -1 6 -52.4 -65.7 -1 -1 1 7 -52.4 -34.9 -1 -1 -1 8 -88.8 -88.3 Table D.3 Differential phase angles A§s ^ and Aty^ ^ (in degrees) of the one-bit detector corresponding to various input data combinations for BtT- 0.25. The contributions of and b[+2 are ignored. Appendix D 142 ¥-3 ¥-2 ¥-1 ¥ 0 ¥l ¥ 2 • ¥ 3 " 0.25 0.00095 0.6 18.8 70.6 70.6 18.8 0.6 Table D.4 Phase shifts \|/ ; _ • (in degrees) for B,T - 0.25 in two-bit detection. B,T ¥ - 3 - ' ¥-2 _ ¥-i ¥o '. ¥i ¥2 ¥ 3 0.25 0.0002 0.2 11.5 61.5 77.0 28.2 1.5 Table D.5 Delayed phase shift _ • (: degrees) for B,T - 0.25 with x = 0.2 T, in two-bit detection. Bit Combinations Suic ' cl-2 cl-l cl cl+l 1 1 - 1 1 9 37.6 24.2 1 - 1 1 1 10 37.6 55.2 1 i . i - . 11 0.0 -32.2 . - 1 1 - 1 12 0.0 -1.2 - 1 1 - 1 1 13 0.0 1.2 - 1 - 1 1 1 14 0.0 32.2 .1 1 -1 -1 15 -37.6 -55.2 -1 -1 1 -1 16 -37.6 -24.2 1 - . - 1 1 17 -103.6 -98.8 . - . -1 -1 18 -141.2 -155.2 . . -1 -1 1 19 -141.2 -121.8 -1 -1 -1 -1 20 -178.8 -178.2 1 1 1 1 21 178.8 178.2 . 1 1 - 1 22 141.2 121.8 - 1 1 1 1 23 141.2 155.2 - 1 1 1 . 1 24 103.6 98.8 Table D.6 Differential phase angles A^^ ; and A())^  ; (in degrees) of the two-bit detector corresponding to various input data combinations for BtT- 0.25. The contributions of C/_3 and c;+2 are ignored. 

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