Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Chu, Haobing
Master of Applied Science - MASc
University of British Columbia
The performance analysis of modulation techniques has been studied extensively. In this thesis, we investigate the performance of a new modulation technique called spiral modulation. Spiral modulation uses complex spirals in the complex plane to modulate data and form complex signals. Spiral modulation was proposed by Astrapi Corporation and the company claimed that spiral modulation is better than the conventional modulations and can exceed the Shannon limit. In this thesis, we explore the potential of spiral modulation and verify the claims by using MATLAB simulation according to the Astrapi's spiral modulation patents. We first present the system structure and the concept of spiral modulation. Then, we focus on the bit-error rate (BER) performance analysis and the spectral efficiency analysis by using MATLAB simulation. In the BER performance analysis, our simulation results reveal that the BER performance of spiral modulation can be better than some conventional modulations when the constellation size is equal or larger than eight if the bandwidth is not taken into consideration. In the spectral efficiency analysis, we estimate the bandwidth of spiral modulation signals by using the periodogram method. The results show that the bandwidth of spiral modulation signal is not strictly limited. A spectral efficiency plane is plotted, and it shows that the spectral efficiency of spiral modulation is worse than the conventional modulation techniques. We conclude that, contrary to the claim made by Astrapi, the spiral modulation can not exceed Shannon capacity boundary on the spectral efficiency plane.
IS SPIRAL MODULATION REALLYUSEFUL?byHaobing ChuB.Eng., Birmingham City University, UK, 2012M.Sc., The University of Warwick, UK, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2016c© Haobing Chu, 2016 The undersigned certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled: IS SPIRAL MODULATION REALLY USEFUL? Submitted by Haobing Chu in partial fulfillment of the requirements of The degree of Master of Applied Science . Julian Cheng, School of Engineering Supervisor, Professor (please print name and faculty/school above the line) Jahangir Hossain, School of Engineering Supervisory Committee Member, Professor (please print name and faculty/school in the line above) Chen Feng, School of Engineering Supervisory Committee Member, Professor (please print name and faculty/school in the line above) Yang Cao, School of Engineering University Examiner, Professor (please print name and faculty/school in the line above) External Examiner, Professor (please print name and university in the line above) 16/Dec/2016 (Date submitted to Grad Studies) Additional Committee Members include: Please print name and faculty/school in the line above Please print name and faculty/school in the line above AbstractThe performance analysis of modulation techniques has been studied extensively. Inthis thesis, we investigate the performance of a new modulation technique called spiralmodulation. Spiral modulation uses complex spirals in the complex plane to modulatedata and form complex signals. Spiral modulation was proposed by Astrapi Corporationand the company claimed that spiral modulation is better than the conventional modula-tions and can exceed the Shannon limit. In this thesis, we explore the potential of spiralmodulation and verify the claims by using MATLAB simulation according to the Astrapi’sspiral modulation patents. We first present the system structure and the concept of spiralmodulation. Then, we focus on the bit-error rate (BER) performance analysis and thespectral efficiency analysis by using MATLAB simulation. In the BER performance anal-ysis, our simulation results reveal that the BER performance of spiral modulation can bebetter than some conventional modulations when the constellation size is equal or largerthan eight if the bandwidth is not taken into consideration. In the spectral efficiency anal-ysis, we estimate the bandwidth of spiral modulation signals by using the periodogrammethod. The results show that the bandwidth of spiral modulation signal is not strictlylimited. A spectral efficiency plane is plotted, and it shows that the spectral efficiencyof spiral modulation is worse than the conventional modulation techniques. We concludethat, contrary to the claim made by Astrapi, the spiral modulation can not exceed Shannoncapacity boundary on the spectral efficiency plane.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Organization and Contributions . . . . . . . . . . . . . . . . . . . . . 4Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Generalization of the Euler’s Formula . . . . . . . . . . . . . . . . . . . . . 62.2 The General Spiral Modulation Formula . . . . . . . . . . . . . . . . . . . . 72.3 Gray Coding for Spiral Modulation . . . . . . . . . . . . . . . . . . . . . . . 162.4 Shannon-Hartley Theorem and Astrapi’s Claims . . . . . . . . . . . . . . . 192.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Chapter 3: BER Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 21iiiTABLE OF CONTENTS3.1 System Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.1 Transmitter Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 Receiver Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 BER Comparison Between Conventional Modulations and Spiral Modulation 253.2.1 Phase Shift Keying and Quadrature Amplitude Modulation . . . . . 263.2.2 Spiral Modulation With Time Reversal . . . . . . . . . . . . . . . . 273.2.3 Spiral Modulation With Rotation Reversal . . . . . . . . . . . . . . 393.2.4 Spiral Modulation With Both Reversals . . . . . . . . . . . . . . . . 443.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Chapter 4: Spectral Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . 514.1 Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Spectrum-Efficiency Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Summary of Accomplished Work . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68ivList of TablesTable 2.1 Conventional modulation sets for an alphabet of eight [10] . . . . . . 15Table 2.2 Spiral modulation sets for an alphabet of eight [10] . . . . . . . . . . 16Table 3.1 Parameter values of spiral modulation with time reversal . . . . . . . 28Table 3.2 Symbol waveform equations of spiral modulation with time reversal . 29Table 3.3 Parameter values of spiral modulation with rotation reversal . . . . . 39Table 3.4 Symbol waveform equations of spiral modulation with rotation reversal 40Table 3.5 Parameter values of spiral modulation with both reversals . . . . . . 45Table 3.6 Symbol waveform equations of spiral modulation with both reversals 46vList of FiguresFigure 2.1 Complex spirals of the generalized Euler’s formula . . . . . . . . . . 7Figure 2.2 Time reversal representation . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.3 (a) Real part waveform when K1 = 1 or ω1 = 0 (b) Imaginary partwaveform when K1 = 1 or ω1 = 0 . . . . . . . . . . . . . . . . . . . 9Figure 2.4 (a) Real part waveform when K1 = −1 or ω1 = pi (b) Imaginarypart waveform when K1 = −1 or ω1 = pi . . . . . . . . . . . . . . . 9Figure 2.5 Rotation reversal representation . . . . . . . . . . . . . . . . . . . . 10Figure 2.6 (a) Real part waveform when K2 = 1 or ω2 = 0 (b) Imaginary partwaveform when K2 = 1 or ω2 = 0 . . . . . . . . . . . . . . . . . . . 11Figure 2.7 (a) Real part waveform when K2 = −1 or ω2 = pi (b) Imaginarypart waveform when K2 = −1 or ω2 = pi . . . . . . . . . . . . . . . 11Figure 2.8 (a) Real part waveform for different m values (b) Imaginary partwaveform for different m values . . . . . . . . . . . . . . . . . . . . 12Figure 2.9 Generalized Euler’s formula when m = 3 and T = 1, 2 and 3 . . . . 13Figure 2.10 Generalized Euler’s formula when m = 3 and T = 1, 2 and 3 (a)Real part waveform (b) Imaginary part waveform . . . . . . . . . . 14Figure 2.11 Complex plane plot of 8-ary Spiral modulation with K1 = ±1 for0 < t < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 3.1 The transmitter structure of the spiral modulation [2] . . . . . . . . 23Figure 3.2 Receiver of spiral modulation . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.3 Complex plane plot of spiral modulation with time reversal . . . . 30Figure 3.4 BER for BPSK and binary spiral modulation with time reversal . . 32Figure 3.5 BER for 4PSK and 4-ary spiral modulation with time reversal . . . 32viLIST OF FIGURESFigure 3.6 BER for BPSK and binary spiral modulation with time reversalwhen T = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.7 BER for 4PSK and 4-ary spiral modulation with time reversal whenT = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.8 Waveforms of binary spiral modulation with time reversal when T =1 (a) Real part waveform (b) Imaginary part waveform . . . . . . . 35Figure 3.9 Waveforms of binary spiral modulation with time reversal when T =6 (a) Real part waveform (b) Imaginary part waveform . . . . . . . 35Figure 3.10 BER for 8PSK and 8-ary spiral modulation with time reversal . . . 36Figure 3.11 BER for 16QAM and 16-ary spiral modulation with time reversal . 36Figure 3.12 BER for binary spiral modulation with time reversal for different mvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.13 Complex plane plot of spiral modulation with rotation reversal . . 41Figure 3.14 BER for BPSK and binary spiral modulation with rotation reversal 42Figure 3.15 BER for 4PSK and 4-ary spiral modulation with rotation reversal . 42Figure 3.16 BER for 8PSK and 8-ary spiral modulation with rotation reversal . 43Figure 3.17 BER for 16QAM and 16-ary spiral modulation with rotation reversal 43Figure 3.18 Complex plane plot of spiral modulation with both reversals . . . . 47Figure 3.19 BER for BPSK and binary spiral modulation with both reversals . 48Figure 3.20 BER for 4PSK and 4-ary spiral modulation with both reversals whenT = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.21 BER for 8PSK and 8-ary spiral modulation with both reversals . . 49Figure 3.22 BER for 16QAM and 16-ary spiral modulation with both reversals . 49Figure 4.1 Comparison between theoretical PSD and simulated PSD of rectan-gular pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.2 PSD of spiral modulation signals when T = 1 . . . . . . . . . . . . . 54Figure 4.3 PSD of spiral modulation signals when T = 2 . . . . . . . . . . . . . 54Figure 4.4 PSD of spiral modulation signals when T = 3 . . . . . . . . . . . . . 55Figure 4.5 Waveform of real part of spiral modulation when T = 1 . . . . . . . 56viiLIST OF FIGURESFigure 4.6 Waveform of imaginary part of spiral modulation when T = 1 . . . 56Figure 4.7 91.98% bandwidth of spiral modulation signals when T = 1 . . . . . 57Figure 4.8 91.98% bandwidth of spiral modulation signals when T = 2 . . . . . 58Figure 4.9 91.98% bandwidth of spiral modulation signals when T = 3 . . . . . 58Figure 4.10 The effect of parameter T on spiral modulation signal under differentbandwidth criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.11 PSD of sinc pulse and rectangular pulse . . . . . . . . . . . . . . . . 60Figure 4.12 Spiral modulation with time reversal: . . . . . . . . . . . . . . . . . 61Figure 4.13 Spiral modulation with rotation reversal: . . . . . . . . . . . . . . . 62Figure 4.14 Spiral modulation with both reversals: . . . . . . . . . . . . . . . . 63viiiList of AcronymsAcronyms Definitions4PSK 4-ary Phase-Shift Keying8PSK 8-ary Phase-Shift Keying16PSK 16-ary Phase-Shift Keying64PSK 64-ary Phase-Shift KeyingAWGN Additive White Gaussian NoiseBER Bit-Error RateBPSK Binary Phase-Shift KeyingPSD Power Spectral DensityQAM Quadrature Amplitude ModulationQPSK Quadrature Phase-Shift KeyingSDR Software Defined RadioSNR Signal-to-Noise RatioixAcknowledgementsI would like to appreciate many people for their help. First and foremost of all, I wouldlike to express my deepest gratitude to my supervisor Dr. Julian Cheng for his enthusiasm,guidance, advice, encouragement and support. It is an honor to study and work with Dr.Julian Cheng.I am deeply grateful to Dr. Jahangir Hossain and Dr. Chen Feng for their great effortand significant amount of time to serve on my M.A.Sc. committee. I would also liketo express my thanks to Dr. Yang Cao for serving as my university examiner. I reallyappreciate their valuable time and constructive comments on my thesis.My thanks to all my friends at UBC Okanagan for their help and friendship. Specialthanks to my dear colleagues Changle Zhu, Chunpu Wang, Fang Fang, Guanshan Ye, HaoLiu, Hui Ma, and Yanjie Dong for their help on study and life at UBC. I would also liketo thank all my friends in Kelowna, Vancouver and Hefei.Finally, I would like to special thank to my dear parents for their support during theseyears at UBC. But more importantly, thanks for their greatest love and support.xTo My ParentsxiChapter 1Introduction1.1 Background and MotivationDigital modulation techniques are essential in many digital communication systems,whether it is a satellite communication system or a mobile cellular communication sys-tem. In the past decades, modulation techniques have been well studied and researchedthoroughly, and many promising results have been developed. In digital modulation, infor-mation bits are transformed into waveforms that are compatible with the characteristicsof the channel. The information bits are binary digits taking on values of either 1 or 0.In digital modulation techniques, the modulation process consists of mapping the infor-mation bits into an analog signal for transmission over the channel, and the demodulationprocess consists of determining bit sequence based on the signal received over the channel.Usually, baseband modulation is for short distance transmission in which a sequence ofdigital symbols are used to create a square pulse waveform with certain features such aspulse amplitude, pulse width, and pulse position. The square pulse waveform then canbe recovered at the receiver. For long distance transmission and wireless transmission,bandpass modulation is usually used. A sequence of digital symbols are used to alter theparameters of a high-frequency sinusoidal signal. Sinusoidal waveform is used as carrierfor transmission propose. Therefore, according to the three features of sinusoid, bandpassmodulation includes amplitude modulation, frequency modulation, phase modulation or acombination of them. A variety of modulation schemes can be derived from these com-binations. Quadrature amplitude modulation (QAM) and phase-shift keying (PSK) arestandard bandpass modulation techniques, and they are widely used [1–3].In nowadays communications, people are demanding rapid data transmission. The use11.2. Literature Reviewof digital modulation provides more capacity, higher data security, better quality communi-cations, and quicker system availability. But due to the fact of maximum channel capacitythat was introduced by Shannon [4], none of these traditional modulation methods canexceed this limitation. Recently, a new modulation technique called spiral modulationhas been developed by a company called Astrapi Corporation [5]. On its website, thecompany claims itself the pioneer of a revolutionary method of spiral-based modulation, agame-changing technology in communications technology [5]. Astrapi Corporation holdsseveral key patents on spiral modulation. The company claims that spiral modulation pro-vides better performance than the conventional modulation techniques. More suspiciously,it claims that spiral modulation can exceed the Shannon limit [6]. Moreover, the com-pany doesn’t provide any performance study of the spiral modulation. In order to verifythese claims on spiral modulation, we study in this thesis the performance of the spiralmodulation in additive white Gaussian noise (AWGN) channels.1.2 Literature ReviewThe telecommunication industry faces a fundalmental problem with the exponentialgrowth of data transmission demand. Shannon limit provides a sharp limit on the maximumrate in an AWGN channel for a specified amount of bandwidth and signal-to-noise ratio(SNR) [4]. There is no existing modulation technique that can exceed this limit. However,Astrapi believes that spiral modulation provides a fundamental solution to this problem.Astrapi Corporation is a U.S. based company that was founded in 2013 by Jerrold Prothero.Dr. Jerrold Prothero received his Ph.D. in mechanical engineering from University ofWashington in 1998 and is currently the CEO of Astrapi Corporation. He developed ageneralization of Euler’s formula which is described in Chapter 2. Building on this formula,Dr. Prothero invented the general spiral modulation formula and it can be used for spiralmodulation. Astrapi claims that this spiral modulation can increase spectral efficiencydramatically and can even exceed the Shannon limit. Astrapi owns several patents onspiral modulation. In this section, we provide a literature review on these patents.In [7], Prothero provided a generalization of Euler’s formula. Different from the Euler’s21.2. Literature Reviewformula that corresponds to a rotating circle in the complex plane, the generalization ofEuler’s formula can be expressed as a rotating spiral in the complex plane. In addition,reference [7] describes a transmitter structure of spiral modulation. This transmitter struc-ture generates a complex waveform for transmission purpose. The details of this structureis illustrated in Chapter 3. In Astrapi’s patents and white papers, no receiver structureis given. Therefore, we will apply the traditional correlator receiver to analyze the perfor-mance of spiral modulation in Chapter 3.Reference [8] provides the general spiral modulation formula. This formula can beused to map information bits into a complex waveform. In the traditional modulationtechniques, only amplitude, frequency and phase are used for signal modulation. But inthe general spiral modulation formula, in addition to amplitude, frequency and phase,spiral modulation also uses time reversal, rotation reversal and a parameter m to representdifferent symbols. This modulation method will be illustrated in Chapter 2 in great detail.We will compare the spiral modulation with QAM to show that the spiral modulation canallow more modulating waves than the classical modulations.In [9], a Gray coding method for spiral modulation is presented. In the spiral modula-tion, the concept of signal constellation doesn’t apply. Therefore we can not code symbolsaccording to a constellation diagram using a Gray mapping rule. In [9], the author pro-vides an algorithm to show which symbols are most likely to be confused with each other.By using this algorithm, we can apply a Gray code to the symbols. This method will beillustrated in Chapter 2.In [10], a software defined radio (SDR) implementation for spiral modulation is pre-sented. This SDR consists of:− A memory system which stores a plurality of spiral modulation sets,− A microprocessor which selects one of the spiral modulation sets from different mod-ulation combinations to accommodate channel conditions and receives analog signalsfrom an analog-to-digital convertor,− A digital-to-analog convertor that transforms digital signals to analog signals,31.3. Thesis Organization and Contributions− A power amplifier which boosts the signal power,− An antenna that receives the signals from channels,− An analog-to-digital convertor that transforms analog signals to digital signals andfeeds the signal back to the microprocessor.1.3 Thesis Organization and ContributionsThis thesis consists of five chapters. A summary of each chapter and its contributionsare presented as follows.Chapter 1 reviews some background on modulation techniques. Traditional modula-tions have been studied thoroughly and the well-known Shannon-Hartley theorem has beenestablished for over sixty years. Recently, however, a new modulation called spiral modula-tion has been developed by Astrapi and the company claims spiral modulation can exceedthe Shannon capacity boundary. The motivation of this thesis is to verify the claims madeby Astrapi and explore the potentials of spiral modulation. A literature review of the spiralmodulation is provided. Finally, a summary of contributions of this thesis is presented.Chapter 2 presents the background on spiral modulation, including generalization ofEuler’s formula, the general spiral modulation formula, and Gray coding for spiral modu-lation. In addition to amplitude modulation, frequency modulation and phase modulation,spiral modulation also uses time reversal, rotation reversal and the parameter m to modu-late signals. At last, we review the Shannon-Hartley theorem. Several key claims of Astrapiare stated and they will be verified in the following chapters.Chapter 3 first describes the transmitter and receiver structures of the spiral modula-tion. Then a bit error rate (BER) comparison between the conventional modulations andthe spiral modulation is presented. In this comparison, we compare spiral modulation withtwo conventional modulations: PSK and QAM. Chapter 3 shows how a data stream can bemodulated to a complex waveform by using spiral modulation and its symbol waveforms.It also shows the complex plane plot of each spiral modulation with different modulationorders. Finally, a summary of the BER performance of spiral modulation is presented. The41.3. Thesis Organization and Contributionsresults show that the BER performance of spiral modulation is better than the traditionalmodulations when the modulation order is equal to or larger than eight, assuming thatbandwidth is not taken into consideration.In Chapter 4, we study the spectral efficiency of spiral modulation. A comprehensivespectrum analysis is presented. In this analysis, we compute the spectrum of spiral mod-ulation by using the periodogram method and define the bandwidth of spiral modulation.A spectral efficiency plane is plotted to show the spectral efficiency of spiral modulation,and we compare it with the traditional modulations. The result shows that the spectralefficiency of the spiral modulation is worse than the traditional modulations. We concludethat, unlike the claim made by Astrapi, the spiral modulation cannot exceed Shannon’scapacity boundary.Finally, we summarize the contents and the thesis contributions in Chapter 5, andsuggest some further research topics related to the spiral modulation.5Chapter 2BackgroundIn this chapter, we first present a detailed background review of spiral modulation,including modulation equation, modulation methods and Gray coding of spiral modulation.Then a brief description of Shannon-Hartley theorem is presented. At last, we summarisesome Astrapi’s key claims.2.1 Generalization of the Euler’s FormulaThe Euler’s formula [11]eit = cos(t) + i sin(t) (2.1)where i2 = −1, is the most remarkable equation in mathematics and it has wide applicationsin communication theory. This formula is the basis for a variety of digital modulations.In (2.1), it can be seen that cos(t) and sin(t) are two periodic functions, and the Euler’sformula corresponds to a unit circle in the complex plane as t ranges through the realnumbers. In the unit circle, t is the angle between a line connecting the origin with a pointon the unit circle and the positive real axis. Different from the famous Euler’s formula,Dr. Jerrold Prothero cleverly developed a generalization of the Euler’s formula as [6]eti(2(2−m))(2.2)where m is a positive real number that is greater than 2. Apparently, when m = 2, eq.(2.2) specializes to the traditional Euler term that corresponds to a circle in the complexplane. But when m is larger than 2, eq. (2.2) corresponds to a spiral in the complex plane.Figure 2.1 shows the graphical representation of (2.2) for m = 2.1, 2.5 and 3.62.2. The General Spiral Modulation Formula−35 −30 −25 −20 −15 −10 −5 0 5−15−10−50510Real axisImag axism = 3m = 2.5t = 0m = 2.1t → ∞Figure 2.1: Complex spirals of the generalized Euler’s formulaIn Figure 2.1, there are three spirals and their trajectories for different values of mbecome longer when the time variable t takes lager values. When m becomes larger, thespiral becomes more and more outwards. In [8], the author uses this generalization of theEuler’s formula to design signal modulation.2.2 The General Spiral Modulation FormulaThe general spiral modulation formula is given as [7]fm(t) = K0eiω0e[K1eiω1 ](t+t0)i[K2eiω2 ](2(2−m))0 < t < T. (2.3)In (2.3), T is the symbol duration and the first level is specified by K0 and ω0 such thatK0 can be used for amplitude modulation and ω0 can be used for phase modulation.The second level is specified by K1, ω1 and t0 such that K1 can be used for frequencymodulation, and K1 and ω1 can be used for time reversal. The parameter t0 can be used72.2. The General Spiral Modulation Formulafor time shift when time reversal is applied. The term time reversal refers to that the spiraldirection can be either outwards or inwards. For example, time reversal can be achievedby assuming K1 = ±1 with ω1 = 0, or equivalently ω1 = 0 and ω1 = pi with K1 = 1. Anexample of a complex spiral signal with time reversal is illustrated in Figure 2.2.−35 −30 −25 −20 −15 −10 −5 0 5−15−10−50510Real axisImag axis Outward spiral when K1 = 1 or ω1 = 0Inward spiral when K1 = −1 or ω1 = piInwardOutwardStarting location for inward spiral (circle)Starting location for outward spiral (solid line)Figure 2.2: Time reversal representationIn Figure 2.2, the solid line represents an outward spiral and the circle represents aninward spiral. In the time domain, it indicates whether the waveform is starting from highamplitude to low amplitude or vise versa. The real part and the imaginary part are plottedversus t in Figures 2.3-2.4. It can be seen that the real part waveform in Figure 2.3 are timereversed version of the real part waveform in Figure 2.4. The imaginary part waveform inFigure 2.3 are time reversed version of the imaginary part waveform in Figure 2.4. Timereversal in the spiral modulation is different from the concept of time reversal in wirelesscommunications. In wireless communications, the main purpose of time reversal in suchapplications is to focus energy in space and time on the point of interest by filtering thesignal through the complex conjugate and time inverted channel impulse response [12].82.2. The General Spiral Modulation Formula0 1 2 3 4 5−40−20020Time variable tAmplitude(a)0 1 2 3 4 5−20−10010Time variable tAmplitude(b)Figure 2.3: (a) Real part waveform when K1 = 1 or ω1 = 0 (b) Imaginary part waveformwhen K1 = 1 or ω1 = 00 1 2 3 4 5−40−20020Time variable tAmplitude(a)0 1 2 3 4 5−20−10010Time variable tAmplitude(b)Figure 2.4: (a) Real part waveform when K1 = −1 or ω1 = pi (b) Imaginary part waveformwhen K1 = −1 or ω1 = pi92.2. The General Spiral Modulation FormulaThe third level is specified by K2, ω2 and m such that K2 and ω2 can be used for rotationreversal. The term of rotation reversal indicates whether the spiral form is clockwise oranticlockwise. For example, rotation reversal can be achieved by using K2 = ±1 withω2 = 0, or equivalently ω2 = 0 and ω2 = pi with K2 = 1. An examples is shown in Figure2.5.−35 −30 −25 −20 −15 −10 −5 0 5−15−10−5051015Real axisImag axis Anti−clockwise spiral when K2 = 1 or ω2 = 0Clockwise spiral when K2 = −1 or ω2 = piAnti−clockwiseClockwiseFigure 2.5: Rotation reversal representationFrom Figure 2.5 one observes that the two curves are both outwards, but the solid oneis anti-clockwise and the dotted one is clockwise. The real part and the imaginary part areplotted versus t in Figures 2.6-2.7.102.2. The General Spiral Modulation Formula0 1 2 3 4 5−40−20020Time variable tAmplitude(a)0 1 2 3 4 5−20−10010Time variable tAmplitude(b)Figure 2.6: (a) Real part waveform when K2 = 1 or ω2 = 0 (b) Imaginary part waveformwhen K2 = 1 or ω2 = 00 1 2 3 4 5−40−20020Time variable tAmplitude(a)0 1 2 3 4 5−1001020Time variable tAmplitude(b)Figure 2.7: (a) Real part waveform when K2 = −1 or ω2 = pi (b) Imaginary part waveformwhen K2 = −1 or ω2 = piFrom Figures 2.6-2.7, it can be seen that the two real parts are the same and the twoimaginary parts are reflections to each other with respect to the horizontal axis. But not112.2. The General Spiral Modulation Formulaevery rotation reversal paired signals have the same real parts and different imaginaryparts. Some rotation reversal paired signals have the same imaginary parts but differentimaginary parts. Some rotation reversal paired signals have different real parts and differentimaginary parts.Furthermore, m is a parameter to establish a spiral. A higher value of m, e.g. m > 2,corresponds to more rapid growth, but low frequency spiral trajectory. For example, thethree spirals in Figure 2.1 are complex spirals. Then the real part and the imaginary partcan be plotted versus t separately to have a better understanding of parameter m.0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40−30−20−10010Parameter tAmplitude(a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−15−10−50510Parameter tAmplitude(b) m = 2.1m = 2.5m = 3m = 2.1m = 2.5m = 3Figure 2.8: (a) Real part waveform for different m values (b) Imaginary part waveform fordifferent m valuesFigures 2.8 shows that both real part and imaginary part are not periodic waveformsand the amplitude increases when t is increased. When m becomes larger, the amplitudechanges rapidly versus t. Therefore, this kind of waveforms have been named as non-122.2. The General Spiral Modulation Formulaperiodic amplitude varying waveforms by Astrapi Corporation [7].At last, t represents the time variable which lasts from 0 < t < T where T is the symbolduration which also indicates the length of a spiral trajectory in the complex plane [7]. Inspiral modulation, when m > 2, eq. (2.2) corresponds to a spiral in the complex plane andAstrapi uses this spiral to modulate signal, resulting the amplitude of spiral modulatedsignal changes with time. For example, setting m = 3 and T = 1, 2, and 3 in (2.2), thecorresponding spiral signal can be plotted in Figure 2.9.−5 −4 −3 −2 −1 0 1 2012345678Real axisImag axis T = 1T = 2T = 3Figure 2.9: Generalized Euler’s formula when m = 3 and T = 1, 2 and 3The real part and the imaginary part are plotted versus t in Figure 2.10.132.2. The General Spiral Modulation Formula0 0.5 1 1.5 2 2.5 3−6−4−202Parameter tAmplitude(a) 0 0.5 1 1.5 2 2.5 302468Parameter tAmplitude(b) T = 1T = 2T = 3T = 1T = 2T = 3Figure 2.10: Generalized Euler’s formula when m = 3 and T = 1, 2 and 3 (a) Real partwaveform (b) Imaginary part waveformFigure 2.10 shows when t increases, the maximum amplitude also increases. Afterunderstanding these parameters, we know that (2.3) can be used for signal modulation.Different from conventional modulation formats that use amplitude, frequency and phaseto modulate the signal, spiral modulation has three new parameters that the conventionalmodulation techniques do not have: m, time reversal and rotation reversal. We can use(2.3) to modulate different symbol waveforms. For example, for binary modulation, bit 0can be modulated to a complex waveform with K2 = 1 (anti-clockwise spiral) and bit 1 canbe modulated to a complex waveform with K2 = −1 (clockwise spiral). Because of timereversal and rotation reversal, spiral modulation has more options to modulate signals. Inone of Astrapi’s patents [10], all possible modulation sets of an alphabet size of 8 betweenconventional modulation and spiral modulation are listed.142.2. The General Spiral Modulation FormulaTable 2.1: Conventional modulation sets for an alphabet of eight [10]Phases Amplitudes Frequencies Symbols8 1 1 84 2 1 84 1 2 82 2 2 82 4 1 82 1 4 81 8 1 81 4 2 81 2 4 81 1 8 8152.3. Gray Coding for Spiral ModulationTable 2.2: Spiral modulation sets for an alphabet of eight [10]Phases Amplitudes Frequencies Rotation Time Symbols4 1 1 1 0 84 1 1 0 1 82 2 1 1 0 82 2 1 0 1 82 1 2 1 0 82 1 2 0 1 82 1 1 1 1 81 4 1 1 0 81 4 1 0 1 81 2 1 1 1 81 1 4 1 0 81 1 4 0 1 81 1 2 1 1 81 2 2 1 0 81 2 2 0 1 8Table 2.1 shows the possible modulation sets of conventional modulation techniques.In (2.3), when m > 2, it generates a spiral in the complex plane and one may utilize timereversal and rotation reversal to modulate the signal. In Table 2.2, it can be seen that foran alphabet of 8, we can use time reversal, rotation reversal or both reversals to modulatethe signal to form 8 different symbols.2.3 Gray Coding for Spiral ModulationSimilar to conventional modulation methods, spiral modulation also uses Gray cod-ing. A method for performing Gray coding of spiral modulation has been introduced in162.3. Gray Coding for Spiral Modulation[9]. For spiral modulation, the conventional concept of signal constellation doesn’t apply.Therefore, we cannot apply Gray coding symbols according to the constellation in thespiral modulation. Reference [9] provides an algorithm to determine which waveforms aremost likely to be confused with each other, and applies a Gray encoding technique to thesymbols. In this section, this algorithm is described.Algorithm 1: Compute the information on the likelihood of inter-symbol confusionof spiral modulationinput : Totally M waveforms for a modulation order of Moutput: A matrix with the information on the likelihood of inter-symbol confusionStep 1: For each waveform si(t), n points are sampled such that each waveform isdiscrete in time and denoted as si(n) ;Step 2: Create normalized version of si(n) such that sˆi(n) = si(n)/‖si(n)‖, where‖ · ‖ denotes the norm operation;Step 3: Form a matrix Ai,j such that Ai,j = sˆi(n) ◦ sj(n), where i denotes the row ina matrix, j denotes the column in a matrix and ◦ denotes the inner productoperation;Step 4: Take the diagonal elements of Ai,j such that d = diag(Ai,j), where diag(·)denotes diagonal elements of a diagonal matrix ;Step 5: Form a matrix Di,j such that each row is the vector d and there are totallyM rows;Step 6: Form a matrix Bi,j such that Bi,j = |Ai,j −Di,j |;In Algorithm 1, step 2 provides correlation for each waveform. In step 3, each elementof Ai,j represents the strength of the match for the ith waveform with the jth waveform, andtherefore provides information on the likelihood of inter-symbol confusion. In step 6, thematrix Bi,j has zero along the diagonal and all positive values elsewhere. Smaller non-zerovalues in Bi 6=j,j imply a greater risk that the receiver will incorrectly identify a transmittedsymbol i as the symbol j. Therefore, we can apply Gray coding to the waveforms accordingto this matrix Bi,j . The method for doing so is described in Algorithm 2.An example of Gray coding of 8-ary spiral modulation with rotation reversal and phase172.3. Gray Coding for Spiral ModulationAlgorithm 2: Apply Gray code to spiral modulationinput : The matrix Bi,joutput: Gray coded waveformsStep 1: Create an empty ordered list L ;Step 2: Find the column j with the smallest non-zero value in the matrix Bi,j ;Step 3: Find the row i with this smallest non-zero value in the column j;Step 4: Add this value to the list L, and this value indicates that the ith waveform ismost likely to be confused with the jth waveform;Step 5: Remove column j from the matrix Bi,j ;Step 6: Repeat the previous 4 steps until all symbol indices appear exactly once in L;Step 6: Apply a Gray code to L such that successive symbols in L are assigned bitsequences that differ by one bit ;shift is shown in Figure 2.11.−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52real axisimag axis 000001011010110111100101Figure 2.11: Complex plane plot of 8-ary Spiral modulation with K1 = ±1 for 0 < t < 1In Figure 2.11, the time variable t is set as 0 < t < 1 and the parameter m is equal182.4. Shannon-Hartley Theorem and Astrapi’s Claimsto 3. The solid line represents rotation reversed spiral and the dashed line represents nonrotation reversed spiral. Because there is no constellation for spiral modulation, we canonly show Gray coding in a complex plane with all 8 complex waveforms plotted. Afterapplied Gray code method as described above, each spiral is assigned to a symbol. Twosuccessive values differ in only one bit are mapped to two more likely confused spirals.2.4 Shannon-Hartley Theorem and Astrapi’s ClaimsIn radio communication systems, communication channels play an important role instudying and designing modulation schemes. Different modulation techniques are chosenaccording to different channel characteristics in order to optimize their performance. Indigital communications, AWGN channel model is widely used for simulation analysis [2].In this model, channel adds a white Gaussian noise to the signal and we assume that fadingdoes not exist. Therefore, the received signal can be expressed asr(t) = s(t) + n(t) (2.4)where s(t) is the transmitted signal and n(t) is the additive white Gaussian noise [2]. Thepower spectral density (PSD) of AWGN is [13]SN (f) = N0/2,−∞ < f <∞ (2.5)where N0 is the noise power per unit of bandwidth and the term 1/2 indicates that power issymmetric across both positive and negative frequencies. Eq. (2.5) implies that AWGN isa random process with a flat PSD over all frequencies. In this thesis, AWGN channel modelis adopted for simulation purpose. In information theory, for such an AWGN channel witha finite bandwidth W , the capacity C of this AWGN channel can be expressed in terms ofavailable bandwidth and the signal-to-noise ratio. The capacity relationship can be statedas [13]C = W log2(1 +SN0W)(2.6)= W log2(1 +SN)(2.7)192.5. Summarywhere S is the average received signal power and N = N0W is the average noise power.Eq. (2.7) is the well-known Shannon-Hartley theorem [4]. This theorem shows that themaximum rate at which information can be reliably transmitted over an AWGN channelwith a specified bandwidth. It is the most fundamental theorem in information theory andhas been standing for over sixty years. However Astrapi claims that spiral modulation canexceed the Shannon limit [6]. Astrapi also has several other claims about spiral modulationin their patents. In this thesis, some key claims are singled out to be verified. These claimsare listed below.− Claim 1: The performance of spiral modulation is better than conventional modula-tions [6].− Claim 2: Spiral modulation has a better spectral efficiency than conventional modu-lations [6].− Claim 3: Spiral modulation can exceed the Shannon limit [6].2.5 SummaryIn this chapter, we presented the background knowledge of spiral modulation, includinggeneralization of the Euler’s formula, the general spiral modulation formula, Gray codingfor spiral modulation. A detailed modulation method of spiral modulation was presented.We also reviewed the Shannon-Hartley theorem. At last, some key claims of Astrapi werestated and they are to be verified in the next two chapters.20Chapter 3BER Performance AnalysisIn this chapter, we first describe the transmitter and receiver structure of spiral mod-ulation. Then we use the general spiral modulation formula to analyze the BER of spiralmodulation by using MATLAB simulation. We also present the symbol waveform equa-tions and complex plane plot of spiral modulating signals with different modulation orders.All performance results are compared with the conventional modulation techniques.3.1 System StructureThe transceiver of spiral modulation is more complicated than the conventional modu-lation schemes. In this section, the transmitter and receiver structures of spiral modulationare described. In Astrapi’s patents, no receiver structure is provided. Therefore, we applythe traditional receiver structure to analyze spiral modulation.3.1.1 Transmitter StructureAccording to (2.3), every spiral modulated signal is a complex signal, i.e. every mod-ulated signal has a real part and an imaginary part. In [7], Astrapi claimes that a two-component transmission is used for spiral modulation. Here, the two-component trans-mission means that the transmitter generates the real part signal and the imaginary partsignal separately. From (2.3), by setting K0 = K1 = K2 = 1 and ω0 = ω1 = ω2 = 0, eq(2.3) becomesfm(t) = eti(2(2−m))(3.1)= et(eipi/2)(2(2−m))(3.2)= etei(2(1−m)pi)(3.3)213.1. System Structure= et[cos(2(1−m)pi)+i sin(2(1−m)pi)] (3.4)= et cos(2(1−m)pi)[cos(t sin(2(1−m)pi)) + i sin(t sin(2(1−m)pi)]. (3.5)From (3.5), it can be seen that both the real part and the imaginary part of a spiral mod-ulation can be distinguished from each other. In [7], a block diagram of the transmitterstructure is given to describe how the spiral waveforms are generated and are multipliedwith carrier signals with a carrier frequency at least twice times the baseband signal fre-quency. The structure is shown in Figure 3.1.223.1. System Structurem-factorπ*2(1-m)sin & cosexp oft*cos(π2(1-m))sin cosClock t-valuesin & cos oft*sin(π2(1-m))ImaginarypartRealpartIm{f(t)}Re{f(t)}mDataX XX Xsinexp expcosXsin(2πf ct)Xcos(2πf ct)+s(t)Figure 3.1: The transmitter structure of the spiral modulation [2]3.1.2 Receiver StructureIn Astrapi’s patents and white papers, no receiver structure is given. Therefore, we willapply the traditional correlator receiver to analyze the performance of spiral modulation. InMATLAB simulation, correlation detection is applied [13]. The receiver structure diagramis shown in Figure 3.2.233.1. System StructureXXLPFLPFr(t) = Re{f(t)} cos(2πfct) + Im{f(t)} sin(2πfct)cos(2πfct)sin(2πfct)Find the correlation between f(t) and each reference signals. Then make decisionRe{f(t)}Im{f(t)}Correlator DetectionFigure 3.2: Receiver of spiral modulationIn the transmitter structure, the information signal f(t) can be treated as an envelopeof a carrier signal. Therefore, we first need to perform the envelope detection. The envelopedetection process is given in the following description. A received signal r(t), which has areal part and an imaginary part, goes through each multiplication operator by multiplyingeach carrier waveform. The mathematical process is shown below. For the upper branch,we haveUpperbranch = r(t) cos(2pifct) (3.6)= Re {f(t)} cos2(2pifct) + Im {f(t)} sin(2pifct) cos(2pifct) (3.7)=12Re {f(t)}+ 12Re {f(t)} cos(4pifct) + 12Im {g(t)} sin(4pifct). (3.8)For the lower branch, we haveLowerbranch = r(t) sin(2pifct) (3.9)= Im {f(t)} sin2(2pifct) +Re {f(t)} cos(2pifct) sin(2pifct) (3.10)=12Im {f(t)} − 12Im {f(t)} cos(4pifct) + 12Re {g(t)} sin(4pifct). (3.11)The output is passed through a lowpass filter, and the real part and imaginary partscan be recovered. At the last block, the correlator detection comsists of M correlators,each correlator performs a correlation between the inputs and the reference signal, fi(t)(i = 1, . . . ,M). The correlation detection includes two steps. In the first step, the detectedenvelope f(t) is correlated with each reference signals to form a set of random variable,243.2. BER Comparison Between Conventional Modulations and Spiral Modulationzi(T ) (i = 1, . . . ,M) at the output of the correlators at time t = T , where T is the symbolduration. The mathematical process is shown belowzi(T ) =∫ T0f(t)fi(t)dt i = 1, . . . ,M. (3.12)In the second step, a symbol decision is made. A reasonable decision rule is to choose thewaveform, fi(t) that matches best or has the largest correlation with f(t). In other words,the decision rule isChoose the fi(t) whose index corresponds to the max zi(T ) (3.13)3.2 BER Comparison Between Conventional Modulationsand Spiral ModulationIn this section, the performance of spiral modulation is investigated by using MATLABsimulation and it is compared with several conventional modulations. Table 2.2 shows thatthe spiral modulation has more modulation combinations than the conventional modula-tions. According to Table 2.2, we can divide the spiral modulation into three differentschemes as follows− Spiral modulation with rotation reversal,− Spiral modulation with time reversal,− Spiral modulation with both reversals.For example, for a modulation order of 8, 8PSK simply applies 8 different phases to form 8symbol waveforms. But for spiral modulation, spiral modulation with rotation reversal (ortime reversal) uses 4 different phases to form 4 symbol waveforms and then apply rotationreversal (time reversal) to these waveforms to form another 4 rotation-reversed (time-reversed) symbol waveforms. Also, spiral modulation with both reversals uses 2 differentphases to form 2 symbol waveforms and then apply rotation reversal to form another 2waveforms and apply time reversal to these 4 waveforms to form another 4 waveforms. Inthis section, we will study the BER performance of each spiral modulation scheme andcompare it with that of the conventional modulations.253.2. BER Comparison Between Conventional Modulations and Spiral Modulation3.2.1 Phase Shift Keying and Quadrature Amplitude ModulationIn this section, we describe two of the most commonly used conventional modulations:phase shift keying and quadrature amplitude modulation. In the following sections, thesetwo modulations are compared with different spiral modulation schemes.For PSK, all of the information is encoded in the phase of the transmitted signal. Thus,the PSK transmitted signal is given by [2]SPSKi (t) = Ag(t) cos[2pi(i− 1)M]cos 2pifct−Ag(t) sin[2pi(i− 1)M]sin 2pifct 0 ≤ t ≤ T(3.14)where A is typically a function of the signal energy; g(t) is the pulse shaping function;fc is the carrier frequency; T is the symbol duration; M is the modulation order andi = 1; . . . ,M . In MATLAB simulation, for the convenience purpose, we only simulate thebaseband transmission. All PSK transmitted signals SPSKi (t) have equal energy [2]ESPSKi=∫ T0(SPSKi (t))2dt (3.15)= A2. (3.16)For QAM, the information is encoded in both the amplitude and phase of the trans-mitted signal. Thus, the QAM transmitted signal is given by [2]SQAMi (t) = Aig(t) cos(θi) cos 2pifct−Aig(t) sin(θi) sin 2pifct 0 ≤ t ≤ T (3.17)where Ai is typically a function of the signal energy; g(t) is the pulse shaping function andmust maintain the orthonormal properties of basis functions; fc is the carrier frequency;T is the symbol duration; i = 1, . . . ,M and M is the modulation order. The signal energybetween symbols with different amplitude are not the same. The energy of each symbol is[2]ESQAMi=∫ T0(SQAMi (t))2dt (3.18)= A2i . (3.19)263.2. BER Comparison Between Conventional Modulations and Spiral ModulationThen the average signal energy isE¯QAMS =∑Mi=1EPSKSiM. (3.20)Once we have the signal energy, we can define the SNR in the simulation. In order tomake conventional modulations have the same SNR definition with spiral modulation, theSNR is defined asSNR[db] =EbNo[db] (3.21)=EsNo[db] + 10 log(log2(M)) (3.22)where Eb is the energy per bit; Es is the energy per symbol; No is the noise power andEsNo=ESPSKiNofor PSK (EsNo =E¯SQAMiNofor QAM).3.2.2 Spiral Modulation With Time ReversalThe reason why spiral modulation can have more modulation combination than con-ventional modulations is that it introduces two additional parameters to modulate thesignal. One is time reversal and the other one is rotation reversal. In spiral modulationwith time reversal, only time reversal is applied. Therefore, according to the general spiralmodulation formula in (2.3), when only time reversal is applied, eq. (2.3) becomesfm(t) = K0eiω0eK1eiω1 (t+t0)i(22−m). (3.23)In Astrapi’s patent [7], they use m = 3 to illustrate spiral modulation. Therefore, wechoose m = 3 to form a spiral. Then, eq. (3.23) becomesf3(t) = K0eiω0eK1eiω1 (t+t0)i(22−3). (3.24)Then, we can use eq. (3.24) to modulate signals. Table 3.1 shows the parameter values ofspiral modulation with rotation reversal for different modulation orders.273.2. BER Comparison Between Conventional Modulations and Spiral ModulationTable 3.1: Parameter values of spiral modulation with time reversalModulationOrderTime Reversal Phases AmplitudeCorrespondingconventionalmodulations2K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1n/a n/a BPSK4K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1ω0 = 0, pi n/a 4PSK8K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1ω0 = 0,pi2 , pi,3pi2n/a 8PSK16K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1ω0 = 0,pi4 ,pi2 ,3pi4 , pi,5pi4 ,3pi2 ,7pi4K0 = 1,√2 16QAMAccording to Table 3.1, we can generate waveforms for different modulation orders.Then we apply the Gray coding algorithms which are introduced in Section 2.3 to eachmodulation order. After computing the information on the likelihood of inter-symbolconfusion and applying Gray coding, the symbol waveform equations are shown in Table3.2.283.2. BER Comparison Between Conventional Modulations and Spiral ModulationTable 3.2: Symbol waveform equations of spiral modulation with time reversalModulationOrderSymbol Waveform Equations2 S0(t) = eti(2−1), S1(t) = e−(t+t0)i(2−1)4S00(t) = eti(2−1), S01(t) = e−(t+t0)i(2−1) , S11(t) = eipieti(2−1),S10(t) = eipie−(t+t0)i(2−1)8S000(t) = eti(2−1), S001(t) = e−(t+t0)i(2−1) ,S011(t) = eipi2 eti(2−1), S010(t) = eipi2 e−(t+t0)i(2−1),S110(t) = eipieti(2−1), S111(t) = eipie−(t+t0)i(2−1),S101(t) = ei 3pi2 eti(2−1), S100(t) = ei 3pi2 e−(t+t0)i(2−1)16S0000(t) = eti−(2−1), S0001(t) = e−(t+t0)i(2−1) ,S0011(t) =√2eipi4 e−(t+t0)i(2−1), S0010(t) =√2eipi4 eti(2−1),S0110(t) = eipi2 e−(t+t0)i(2−1), S0111(t) = eipi2 eti(2−1),S0101(t) =√2ei3pi4 e−(t+t0)i(2−1), S0100(t) =√2ei3pi4 eti(2−1),S1100(t) = eipie−(t+t0)i(2−1), S1101(t) = eipieti(2−1),S1111(t) =√2ei5pi4 e−(t+t0)i(2−1), S1110(t) =√2ei5pi4 eti(2−1),S1010(t) = ei 3pi2 e−(t+t0)i(2−1), S1011(t) = ei 3pi2 eti(2−1),S1001(t) =√2ei7pi4 e−(t+t0)i(2−1), S1000(t) =√2ei7pi4 eti(2−1)The resulting complex plane plot diagram of spiral modulation with time reversal for0 < t < 1 is shown in Figure 3.3. Note that the complex plane plot is not the same asa conventional constellation diagram in which symbols are represented by points. In thecomplex plane plot of spiral modulation, symbols are represented by complex spirals. InFigure 3.3, the solid line represents non time reversed spiral and the circle represents timereversed spiral. It can be seen that the signals which are represented by circles are timereversed version of the signals which are represented by solid line.293.2. BER Comparison Between Conventional Modulations and Spiral Modulation1 1.1 1.2 1.3 1.4 1.5 1.6 1.700.20.40.60.811.21.4 Real axisImag axis OutwardInwardLine−0Circle−1(a) Binary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5−1−0.500.511.5Real axisImag axis line−11Circle−01Circle−10line−00(b) 4-ary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52Real axisImag axis Line−011Circle−010Line−110Circle−111Line−000Line−101Circle−001Circle−100(c) 8-ary−3 −2 −1 0 1 2 3−3−2−10123Real axisImag axis Line−0000Circle−0001Circle−0011Circle−0111Line−0110Line−0100Circle−1101Line−1100Line−1110Circle−1111 Circle−1011Line−1010Line−1000Line−0010Circle−0101Circle−1001(d) 16-aryFigure 3.3: Complex plane plot of spiral modulation with time reversal303.2. BER Comparison Between Conventional Modulations and Spiral ModulationIn order to simulate the BER performance, we first need to define the SNR of spiralmodulation. The energy of a spiral modulation signal isESi =∫ T0(Si(t))2dt (3.25)=∫ T0(Re{Si(t)})2dt+∫ T0(Im{Si(t)})2dt. (3.26)Thus, the average signal energy for a modulation order of M isE¯S =∑Mi=1ESiM. (3.27)Once we have the signal energy, we can define the SNR according to 3.21. Then, we canhave the same SNR definition in conventional modulations and spiral modulation. In thesimulation for a certain SNR range (a certain EsNo range), we haveEsNo=ESPSKiNPSKoorE¯SQAMiNQAMo=E¯SN spiralo(3.28)where NPSKo / NQAMo / Nspiralo is the noise power of AWGN which is added on PSK/QAM/ spiral modulation signals. Therefore, for a certain SNR range, we can generate thecorresponding AWGN in MATLAB with certain noise power.The BER comparison of conventional modulations and spiral modulation with rotationreversal in AWGN channel is shown in Figures 3.4 - 3.11.313.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−2Spiral−T=1simulation−2Spiral−T=2simulation−2SpiralT=3simulation−BPSKtheoretical−BPSKFigure 3.4: BER for BPSK and binary spiral modulation with time reversal−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−4Spiral−T=1simulation−4Spiral−T=2simulation−4Spiral−T=3simulation−4PSKtheoretical−4PSKFigure 3.5: BER for 4PSK and 4-ary spiral modulation with time reversalFigures 3.4 and 3.5 show that the BER performance of spiral modulation with timereversal is worse than BPSK and 4PSK under different value of T . But when T is increased323.2. BER Comparison Between Conventional Modulations and Spiral Modulationthe BER performance improves. When T is equal to 6, it can be seen from Figures 3.6and 3.7 that spiral modulation can have the same performance as BPSK and 4PSK. Itcan be shown numerically that further increasing T will not contribute to the performanceimprovement.−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−2Spiral−T=6simulation−BPSKtheoretical−BPSKFigure 3.6: BER for BPSK and binary spiral modulation with time reversal when T = 6333.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−4Spiral−T=6simulation−4PSKtheoretical−4PSKFigure 3.7: BER for 4PSK and 4-ary spiral modulation with time reversal when T = 6One may ask why binary (or 4-ary) spiral modulation with time reversal is worse thanBPSK (or 4PSK) when T is less than 6 is shown in Figure 3.8-3.9. In Figure 3.8, whenT = 1, both real and imaginary parts of two symbol are similar to each other. In otherwords, the receiver is difficult to distinguish these two symbol when T = 1. In Figure 3.9,when T = 6, two symbols can be distinguished easily from each other. Therefore, whenT is increased, the BER performance improves. But eventually, the BER performance ofbinary (or 4-ary) spiral modulation with time reversal can not exceed the performance ofBPSK (or 4PSK).343.2. BER Comparison Between Conventional Modulations and Spiral Modulation0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 111.21.41.61.8The parameter TAmplitude(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.511.5The parameter TAmplitude(b) Imag part of symbol 0Imag part of symbol 1Real part of symbol 0Real part of symbol 1Figure 3.8: Waveforms of binary spiral modulation with time reversal when T = 1 (a) Realpart waveform (b) Imaginary part waveform0 1 2 3 4 5 6−40−30−20−10010The parameter TAmplitude(a) 0 1 2 3 4 5 6−80−60−40−20020The parameter TAmplitude(b) Real part of symbol 0Real part of symbol 1Imag part of symbol 0Imag part of symbol 1Figure 3.9: Waveforms of binary spiral modulation with time reversal when T = 6 (a) Realpart waveform (b) Imaginary part waveformThe BER plots for modulation order of 8 and 16 are shown in Figures 3.10 - 3.11.353.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −5 0 5 10 15 2010−510−410−310−210−1100SNR, dBBit Error Rate simulation−8Spiral−T=1simulation−8Spiral−T=2simulation−8Spiral−T=3simulation−8PSKtheoretical−8PSKFigure 3.10: BER for 8PSK and 8-ary spiral modulation with time reversal−10 −5 0 5 10 15 2010−510−410−310−210−1100SNR, dBBit Error Rate simulation−16Spiral−T=1simulation−16Spiral−T=2simulation−16Spiral−T=3simulation−16QAMtheoretical−16QAMFigure 3.11: BER for 16QAM and 16-ary spiral modulation with time reversalIt can be seen from Figures 3.10 - 3.11 that spiral modulation has a better BER per-formance than conventional modulations when T is increased. In Figure 3.10, when T = 1,363.2. BER Comparison Between Conventional Modulations and Spiral Modulation8-ary spiral modulation with time reversal is about 3.5 dB worse than 8PSK at BER of10−5. When T = 2, 8-ary spiral modulation with time reversal is already better than8PSK. 8-ary spiral modulation with T = 3 almost has the same performance when T = 2and it is about 2 dB better than 8PSK at BER of 10−5. It can be shown numerically thatfurther increasing T does not contribute to the performance improvement. The same resultcan be observed in Figure 3.11. But in Figure 3.11, 16-ary spiral modulation with timereversal has a better BER performance than 16QAM even when T = 1. When T increasesto 2 or 3, 16-ary modulation with time reversal is about 4.5 dB better than 16QAM atBER of 10−5. Based on these results, we can conclude that by using the general spiralmodulation formula, 8-ary (or 16-ary) spiral modulation with time reversal has better BERperformance than 8PSK (or 16QAM). In the other words, 8-ary (16-ary) spiral modulationwith time reversal signals can be distinguished easier from each other than conventionalsignals in the presence of AWGN.The Parameter mIn Astrapi’s patent, the company only uses m = 3 as an example to modulate signals.In (2.3), when m > 2, it can produce different spirals. Therefore, in this section, we explorethe BER performance of different m values. We choose m = 2.1, 3, 4 and 5 for binaryspiral modulation with time reversal. The last section shows that binary spiral modulationwith time reversal has the same performance as that of BPSK when T = 6. Therefore, weonly choose the scenario when T = 6.373.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−2Spiral−m=2.1simulation−2Spiral−m=3simulation−2Spiral−m=4simulation−2Spiral−m=5Figure 3.12: BER for binary spiral modulation with time reversal for different m valuesFrom Figure 3.12, it can be seen that when m = 2.1 and m = 3 binary spiral modulationwith time reversal has the same BER performance. But when m > 3, the BER performanceof binary spiral modulation with time reversal gets worse. The BER performance can alsobe shown numerically for 4-ary, 8-ary, 16-ary and 64-ary spiral modulations with timereversal that spiral modulation has the best BER performance when m = 3. The reasonmight be that as shown in Figure 2.1, when m increases, the frequency of a complex spiralis decreasing. This decreasing frequency causes the complex spiral becomes more and moreslowly varying during one symbol duration. Therefore, the receiver is difficult to distinguishthe symbols when m is large. The same simulation can also be done for spiral modulationwith rotation reversal and spiral modulation with both reversals. For spiral modulationwith rotation reversal, different m values have the same BER performance. For spiralmodulation with both reversals, when m = 3 it has the best BER performance. Therefore,we only choose m = 3 to modulate signals throughout this thesis.383.2. BER Comparison Between Conventional Modulations and Spiral Modulation3.2.3 Spiral Modulation With Rotation ReversalIn this section, we present the BER performance simulation of spiral modulation withrotation reversal. According to the general spiral modulation formula in (2.3), when onlyrotation reversal is applied and we also choose m = 3 to form a spiral, eq. (2.3) becomesf3(t) = K0eiω0etiK2eiω2 (22−3). (3.29)Eq. (3.29) can be used to generate symbol waveform equations. Table 3.3 shows theparameter values of spiral modulation with rotation reversal for different modulation orders.Table 3.3: Parameter values of spiral modulation with rotation reversalModulationOrderRotationReversalPhases AmplitudeCorrespondingconventionalmodulations2K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1n/a n/a BPSK4K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1ω0 = 0, pi n/a 4PSK8K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1ω0 = 0,pi2 , pi,3pi2n/a 8PSK16K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1ω0 = 0,pi4 ,pi2 ,3pi4 , pi,5pi4 ,3pi2 ,7pi4K0 = 1,√2 16QAM393.2. BER Comparison Between Conventional Modulations and Spiral ModulationAccording to Table 3.3, we can generate waveforms for different modulation orders.The symbol waveform equations are shown in Table 3.4.Table 3.4: Symbol waveform equations of spiral modulation with rotation reversalModulationOrderSymbol Waveform Equations2 S0(t) = eti(2−1), S1(t) = eti−(2−1)4S00(t) = eti(2−1), S01(t) = eti−(2−1), S11(t) = eipieti(2−1),S10(t) = eipieti−(2−1)8S000(t) = eti(2−1), S001(t) = eipi2 eti−(2−1),S011(t) = eipi2 eti(2−1), S010(t) = eipieti−(2−1),S110(t) = eipieti(2−1), S111(t) = ei 3pi2 eti−(2−1),S101(t) = ei 3pi2 eti(2−1), S100(t) = eti−(2−1)16S0000(t) = eti−(2−1), S0001(t) = eti(2−1),S0011(t) =√2eipi4 eti−(2−1), S0010(t) =√2eipi4 eti(2−1),S0110(t) = eipi2 eti−(2−1), S0111(t) = eipi2 eti(2−1),S0101(t) =√2ei3pi4 eti−(2−1), S0100(t) =√2ei3pi4 eti(2−1),S1100(t) = eipieti−(2−1), S1101(t) = eipieti(2−1),S1111(t) =√2ei5pi4 eti−(2−1), S1110(t) =√2ei5pi4 eti(2−1),S1010(t) = ei 3pi2 eti−(2−1), S1011(t) = ei 3pi2 eti(2−1),S1001(t) =√2ei7pi4 eti−(2−1), S1000(t) =√2ei7pi4 eti(2−1)The resulting complex plane plot diagram of spiral modulation with rotation reversalfor 0 < t < 1 is shown in Figure 3.13. In Figure 3.13, the solid line represents non rotationreversed spiral and the dashed line represents rotation reversed spiral. It can be seen thatthe amplitude of a signal is time-varying and it reaches to its maximum value when treaches T . The BER comparison with conventional modulations is shown in Figures 3.14- 3.17.403.2. BER Comparison Between Conventional Modulations and Spiral Modulation0 0.5 1 1.5 2 2.5−1.5−1−0.500.511.5Real axisImag axis 01(a) Binary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5−1−0.500.511.5Real axisImag axis 1011 0100(b) 4-ary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52real axisimag axis 000001011010110111100101(c) 8-ary−3 −2 −1 0 1 2 3−3−2−10123Real axisImag axis 00010011001001100101010011001101111111101010 10111001100001110000(d) 16-aryFigure 3.13: Complex plane plot of spiral modulation with rotation reversal413.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−2Spiral−T=1simulation−2Spiral−T=2simulation−2Spiral−T=3simulation−BPSKtheoretical−BPSKFigure 3.14: BER for BPSK and binary spiral modulation with rotation reversal−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−4Spiral−T=1simulation−4Spiral−T=2simulation−4Spiral−T=3simulation−4PSKtheoretical−4PSKFigure 3.15: BER for 4PSK and 4-ary spiral modulation with rotation reversal423.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −5 0 5 10 15 2010−510−410−310−210−1100SNR, dBBit Error Rate simulation−8Spiral−T=1simulation−8Spiral−T=2simulation−8Spiral−T=3simulation−8PSKtheoretical−8PSKFigure 3.16: BER for 8PSK and 8-ary spiral modulation with rotation reversal−10 −5 0 5 10 1510−510−410−310−210−1100SNR, dBBit Error Rate simulation−16Spiral−T=1simulation−16Spiral−T=2simulation−16Spiral−T=3simulation−16QAMtheoretical−16QAMFigure 3.17: BER for 16QAM and 16-ary spiral modulation with rotation reversalFrom Figures 3.14 - 3.17, it can be seen that binary and 4-ary spiral modulation withrotation reversal have the same BER performance as that of BPSK and 4PSK under differ-433.2. BER Comparison Between Conventional Modulations and Spiral Modulationent T values. For binary spiral modulation with rotation reversal, the real parts of symbol0 and 1 are the same, which means only the imaginary parts carry information. In BPSK,only the in-phase component carries information. Therefore, binary spiral modulation withrotation reversal is actually identical to BPSK. For a 4-ary spiral modulation, the real partbetween symbol 00 and symbol 01, symbol 10 and symbol 11 are the same, and the imag-inary part between symbol 00 and symbol 10, symbol 11 and symbol 01 are the samewhich is similar to the 4PSK modulation that two neighbouring symbols have the samein-phase or quadrature component. Therefore, the 4-ary spiral modulation with rotationreversal has the same BER performance as that of 4PSK. But in Figure 3.16, it can beseen that when T = 1, the performance of 8-ary spiral modulation with rotation reversalhas the same performance with 8PSK and is better than 8PSK when T is increased. WhenT = 2 and T = 3, the 8-ary spiral modulation with rotation reversal is about 1.5 dB and5 dB better than 8PSK at BER of 10−5. In Figure 3.17, the similar observation can beobtained. When T = 1, the 16-ary spiral modulation with rotation reversal is about 1 dBbetter than 16QAM at BER of 10−5. The BER performances of 16-ary spiral modulationwith rotation reversal are the same when T = 2 and T = 3. It can also be seen that 16-aryspiral modulation with rotation reversal is about 3 dB better than 16QAM when T = 2 andT = 3. Therefore, we conclude that different from spiral modulation with time reversal,both binary and 4-ary spiral modulation with rotation reversal have the same BER perfor-mance as that of BPSK and 4PSK even when T = 1. For a modulation order of 8 or 16,spiral modulation with rotation reversal has better BER performance than conventionalmodulations.3.2.4 Spiral Modulation With Both ReversalsIn this section, we present the BER performance simulation of spiral modulation withboth time reversal and rotation reversal. According to the general spiral modulation for-mula in (2.3), when both time reversal and rotation reversal are applied, and we also choosem = 3, eq. (2.3) becomesf3(t) = K0eiω0e[K1eiω1 ](t+t0)i[K2eiω2 ](2(2−3)). (3.30)443.2. BER Comparison Between Conventional Modulations and Spiral ModulationEq. (3.30) inherits all the parameters from (2.3). Table 3.5 shows the parameter values ofspiral modulation with both reversals for different modulation orders.Table 3.5: Parameter values of spiral modulation with both reversalsModulationOrderTime ReversalRotationReversalPhases AmplitudeCorre-spondingconven-tionalmodula-tions4K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1n/a n/a 4PSK8K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1ω0 = 0, pi, n/a 8PSK16K1 = ±1 withω1 = 0, orω1 = 0, pi withK1 = 1K2 = ±1 withω2 = 0, orω2 = 0, pi withK2 = 1ω0 = 0,pi2 ,pi, 3pi2 ,n/a 16QAMIn Table 3.5, there is no binary case. Because when both rotation reversal and timereversal are applied, eq. (3.30) generates at least 4 complex spiral signals. The symbolwaveform equations are shown in Table 3.6.453.2. BER Comparison Between Conventional Modulations and Spiral ModulationTable 3.6: Symbol waveform equations of spiral modulation with both reversalsModulationOrderSymbol Waveform Equations4S00(t) = eti(2−1), S01(t) = e−(t+t0)i(2−1) , S11(t) = eti−(2−1),S10(t) = e−(t+t0)i−(2−1)8S000(t) = eti(2−1), S001(t) = e−(t+t0)i(2−1) ,S011(t) = eipie−(t+t0)i(2−1), S010(t) = eipieti(2−1),S110(t) = eipieti−(2−1), S111(t) = eipie−(t+t0)i−(2−1),S101(t) = e−(t+t0)i−(2−1) , S100(t) = eti−(2−1)16S0000(t) = eti−(2−1), S0001(t) = e−(t+t0)i(2−1) ,S0011(t) = eipi2 e−(t+t0)i−(2−1), S0010(t) = eipi2 eti−(2−1),S0110(t) = eipi2 eti(2−1), S0111(t) = eipi2 e−(t+t0)i(2−1),S0101(t) = eipie−(t+t0)i−(2−1), S0100(t) = eipieti−(2−1),S1100(t) = eipieti(2−1), S1101(t) = eipie−(t+t0)i(2−1),S1111(t) = ei 3pi2 e−(t+t0)i−(2−1), S1110(t) = ei 3pi2 e−ti(2−1),S1010(t) = ei 3pi2 eti(2−1), S1011(t) = ei 3pi2 e−(t+t0)i(2−1),S1001(t) = e−(t+t0)i−(2−1) , S1000(t) = eti−(2−1)The resulting complex plane plot diagram of spiral modulation with rotation reversalfor 0 < t < 1 is shown in Figure 3.18. In Figure 3.18, the solid line represents non rotationreversed spiral, the dashed line represents rotation reversed spiral and the dots representstime reversed spiral. It can be seen that 16-ary spiral modulation with both reversals doesnot use amplitude modulation which is differnt from 16-ary spiral modulation with timereversal (or rotation reversal). The BER comparison with conventional modulations isshown in Figures 3.19 - 3.22.463.2. BER Comparison Between Conventional Modulations and Spiral Modulation0 0.5 1 1.5 2 2.5−1.5−1−0.500.511.5Real axisImag axis line−00Circle−01line−11 Circle−10(a) 4-ary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5−1−0.500.511.5Real axisImag axis Circle−011Line−010Line−110Circle111Line−000Line−100Circle−001Circle−101(b) 8-ary−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52Real axisImag axis Line−0000Line−0010Line−0110Line−0100Circle−1101Circle−0011Circle−0101Circle−0111Circle−0001Circle−1111Line−1110 Line−1010Circle−1001Circle−1011Line−1100Line−1000(c) 16-aryFigure 3.18: Complex plane plot of spiral modulation with both reversals473.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−4Spiral−T=1simulation−4Spiral−T=2simulation−4Spiral−T=3simulation−4PSKtheoretical−QPSKFigure 3.19: BER for BPSK and binary spiral modulation with both reversals−10 −8 −6 −4 −2 0 2 4 6 8 1010−510−410−310−210−1100SNR, dBBit Error Rate simulation−4Spiral−T=6simulation−4PSKtheoretical−4PSKFigure 3.20: BER for 4PSK and 4-ary spiral modulation with both reversals when T = 6483.2. BER Comparison Between Conventional Modulations and Spiral Modulation−10 −5 0 5 10 15 2010−510−410−310−210−1100SNR, dBBit Error Rate simulation−8Spiral−T=1simulation−8Spiral−T=2simulation−8Spiral−T=3simulation−8PSKtheoretical−8PSKFigure 3.21: BER for 8PSK and 8-ary spiral modulation with both reversals−10 −5 0 5 10 15 2010−510−410−310−210−1100SNR, dBBER simulation−16Spiral−T=1simulation−16Spiral−T=2simulation−16Spiral−T=3simulation−16QAMtheoretical−16QAMFigure 3.22: BER for 16QAM and 16-ary spiral modulation with both reversalsIn Figure 3.19, it can be seen that the BER performance of 4-ary spiral modulationwith both reversals is worth than 4PSK. But when T is increased, the BER performance493.3. Summaryof 4-ary spiral modulation with both reversals improves. In Figure 3.20, when T = 6, 4-aryspiral modulation with both reversals has the same BER performance as 4PSK. Similarto 4-ary spiral modulation with time reversal, further increase of T does not improve theBER performance which means 4-ary spiral modulation with both reversals can not exceedthe BER performance of 4PSK. In Figure 3.21, when T = 1, 2 and 3, the 8-ary spiralmodulation with both reversals is about 0.5 dB, 2.5 dB and 3.5 dB better than 8PSK atBER of 10−5 respectively. In Figure 3.22, when T = 1, the BER performance of 16-aryspiral modulation with both reversals is the same with 16QAM. But when T = 2 and3, the 16-ary spiral modulation is about 1 dB and 2 dB better than 16QAM at BER of10−5 recpectively. Also, further increasing T does not contribute to the BER performance.Therefore, we conclude that when the modulation order is equal to or larger than 8, spiralmodulation with both reversals has better BER performance than the conventional mod-ulations. A possible explanation is that when the modulation order becomes larger, thereceiver can distinguish spiral modulation signals easier than the conventional modulations.3.3 SummaryIn this chapter, we presented the transmitter and receiver structure of spiral mod-ulation. Then, a BER analysis of spiral modulation was provided by using MATLABsimulation. The symbol waveform equations and complex plot of spiral modulation werealso presented. From the comparison between spiral modulation and conventional modu-lations, it can be seen that spiral modulation has better BER performance than PSK andQAM when the modulation order is equal to or larger than eight.50Chapter 4Spectral Efficiency AnalysisIn this chapter, we study the spectral efficiency of the spiral modulation. We first studythe spectrum and the bandwidth of spiral modulation signals. Then a spectral efficiencyplane is plotted to analyze the spectral efficiency of spiral modulation.4.1 Spectrum AnalysisIn Chapter 2, we have showed that the BER performance of spiral modulation is betterthan that of PSK and QAM when the modulation order is equal to or larger than 8. Butin the simulation, we do not consider that signal is bandlimited. Therefore, we need toanalyze its spectral efficiency in order to consider signal bandwidth of the spiral modulatedsignals.In (2.3), it is difficult to obtain a closed-form for its Fourier transform. Therefore, weneed to use some numerical methods to compute the spectrum of spiral modulation. Amethod called the periodogram can be used to compute the power spectral density and itis given by [14]Sxx(f) =1N∣∣∣∣∣N−1∑n=0x[n]e−j2pifn∣∣∣∣∣2. (4.1)wher x[n] is a finite sequence and n = 0, 1, . . . , N − 1. This is the simplest technique toestimate the spectrum of a data stream. However by using this method, the spectrumof a finite-length signal exhibits side-lobes due to discontinuities at the endpoints [15].Therefore, in order to obtain a better view of the spectrum of spiral modulation signals,windows are used to smooth the periodogram method [14]. For a finite duration window514.1. Spectrum Analysissequence of length N − 1 then, the smoothed periodogram isSxx(f) =1N∣∣∣∣∣N−1∑n=0w[n]x[n]e−j2pifn∣∣∣∣∣2(4.2)where w[n] is a selected window function. There are many different window functions. Thesimplest window is the rectangular window as shown below [16].w[n] = 1 for n = 0, 1, . . . , N − 10 elsewhere (4.3)where N − 1 is the total window width. Among all the windows, Blackman window is theone given by a sum of cosine terms and can minimize the maximum side lobe and have aprecise amplitude display by varying the number and coefficients of the terms [17]. TheBlackman window is shown below [18].w[n] = a0 − a1 cos(2pinN − 1)+ a2 cos(4pinN − 1)n = 0, 1, . . . , N − 1 (4.4)where N − 1 is the total window width; a0 = (1 − α)/2; a1 = 0.5 and a2 = α/2. Whenα = 0.16, it is called ”exact Blackman” [17] which gives low values at the third and fourthsidelobes. Therefore, in order to have a better view of the main lobe, Blackman windowis chosen for the periodogram method to compute the spectrum of spiral modulation. Wefirst apply this method to PSK signals which use rectangular pulse. In [13], for PSKmodulation, the rectangular pulses is used as pulse-shaping which is defined asg(t) =√ET 0 ≤ t ≤ T0 otherwise(4.5)where E is the symbol energy and T is the symbol duration. Then the baseband powerspectral density is defined as [13]S(f) =2E sin2(piTf)piTf. (4.6)Then the theoretical PSD of PSK signal using rectangular pulse is plotted and comparedwith simulated PSD.524.1. Spectrum Analysis−5 −4 −3 −2 −1 0 1 2 3 4 5−150−100−500Normalized frequency (Hz), fT Normalized power spectral density (dB) Estimated PSD with blackman windowTheoretical PSK signal PSDFigure 4.1: Comparison between theoretical PSD and simulated PSD of rectangular pulsesIn Figure 4.1, the theoretical curve can fit the simulated PSD very well. Therefore,we can use the periodogram function with Blackman window in MATLAB to compute thespectrum of spiral modulation signals. Eq. (2.3) gives the signal waveform equation ofspiral modulation. Without any modulation parameters and m = 3, eq. (2.3) becomesf3(t) = eti(2(2−3))0 < t < T. (4.7)Then we can produce a sequence of this waveform in MATLAB and compute the spectrum.The spectrum of spiral modulation signals is shown in Figures 4.2 - 4.4.534.1. Spectrum Analysis−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100 Normalized frequency, fTbNormalized power spectral densityFigure 4.2: PSD of spiral modulation signals when T = 1−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100 Normalized frequency, fTbNormalized power spectral densityFigure 4.3: PSD of spiral modulation signals when T = 2544.1. Spectrum Analysis−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100 Normalized frequency, fTbNormalized power spectral densityFigure 4.4: PSD of spiral modulation signals when T = 3In Figure 4.2, it can be seen that the spectrum of spiral modulation signals when T = 1is similar to the spectrum of a rectangular pulse. This is because when T = 1, both the realpart and the imaginary part waveforms of spiral modulation have small amplitude changeand suddenly drop off at the end of the waveform which are similar to a rectangular pulseas shown in Figures 4.5-4.6.554.1. Spectrum Analysis0 1 2 3 4 5 6 7 8 9 1011.11.21.31.41.51.61.7 Number of SymbolsAmplitudeFigure 4.5: Waveform of real part of spiral modulation when T = 10 1 2 3 4 5 6 7 8 9 1000.20.40.60.811.21.4 Number of SymbolsAmplitudeFigure 4.6: Waveform of imaginary part of spiral modulation when T = 1From Figure 4.3 to Figure 4.4, the main lobe of the spectrum becomes blurring, scat-tering to the side. It is not easy to define the bandwidth of spiral modulation from thesespectrums. Because they are not the same as ideal sinc pulse or raised-cosine pulse wherebandwidth is limited in a certain range. When T increases, it is harder to read the main564.1. Spectrum Analysislobe from the power spectrum. In order to compare it with conventional techniques anddue to the similarity with rectangular pulse, we define the bandwidth as the same as thebandwidth definition of a rectangular pulse. For a rectangular pulse, the most popularmeasure is called null-to-null [1], which is the width of the main spectral lobe. But forspiral modulation, it is difficult to read the main lobe when T increases. Therefore, wecalculate the percentage that the main lobe occupies in a spectrum of rectangular pulse.After calculation, the main lobe is 91.98% of the whole spectrum. Therefore, we start fromthe middle of the spectrum and calculate 91.98% of the spectrum of spiral modulation.Then the 91.98% bandwidth is shown in Figures 4.7 - 4.9.−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100Normalized frequency, fTNormalized power spectral density PSD of spiral modulation91.98% PSDFigure 4.7: 91.98% bandwidth of spiral modulation signals when T = 1574.1. Spectrum Analysis−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100Normalized frequency, fTNormalized power spectral density PSD of spiral modulation91.98% PSDFigure 4.8: 91.98% bandwidth of spiral modulation signals when T = 2−5 −4 −3 −2 −1 0 1 2 3 4 5−80−70−60−50−40−30−20−100Normalized frequency, fTNormalized power spectral density PSD of spiral modulation91.98% PSDFigure 4.9: 91.98% bandwidth of spiral modulation signals when T = 3From Figures 4.7 - 4.9, the 91.98% bandwidth is increasing with increased T . For T = 1,2, and 3, 91.98% bandwidth is 2.65 Hz, 3.42 Hz and 4.4 Hz, respectively. In [1], there aremany different bandwidth criteria. For example, 3 dB bandwidth is the interval betweenfrequencies at which PSD has dropped to half-pwer, or 3 dB below the peak value. Another584.1. Spectrum Analysisuseful criterion is called fractional power containment bandwidth. This criterion has beenadopted by the Federal Communications Commission (FCC Rules and Regulations Section2.202). This criterion states that the occupied bandwidth is the band that leaves exactly0.5% of the signal power above the upper and the lower band limit [1]. Therefore it is alsocalled 99% bandwidth. With these different criteria, we plot the effect of the parameter Ton the signal bandwidth of the spiral modulation in Figure 4.10.0 0.5 1 1.5 2 2.5 3 3.5 401234567891011Parameter TNormalized bandwidth in Hz 3 dB bandwidth90.60% bandwidth99% bandwidthFigure 4.10: The effect of parameter T on spiral modulation signal under different band-width criteriaIn Figure 4.10, the 3 dB bandwidth gives the smallest bandwidth, but this bandwidthcriterion lacks precision since most of the powers are not included. the 99% bandwidth hasthe largest bandwidth and precision because most of the powers are included. But for thiskind of spectrum which is similar to the spectrum of rectangular pulse, as stated above,we choose 91.98% bandwidth to define the bandwidth of spiral modulation signals. Incomparison with conventional pulses, we also plotted the spectrum of ideal sinc pulse andrectangular pulse in Figure 4.11 [19]. It can be seen that ideal sinc pulse has a bandwidthof 1 Hz and rectangular pulse has a bandwidth of roughly 2 Hz as shown in Figure 4.11.594.2. Spectrum-Efficiency PlaneTherefore, it also can be seen that the signal of spiral modulation needs more bandwidththan conventional pulses.−5 0 5−160−140−120−100−80−60−40−200Normalized frequencyNormalized power spectral density recatangularsincFigure 4.11: PSD of sinc pulse and rectangular pulse4.2 Spectrum-Efficiency PlaneIn [1], a bandwidth-efficiency plane is constructed for a BER of 10−5. After definingthe bandwidth of spiral modulation, the bandwidth-efficiency plane of spiral modulationcan be constructed.604.2. Spectrum-Efficiency Plane−2024681012141618200.250.5124816Energy to noise ratio, Eb/No dBSpectral efficiency, R/W bit/sec/Hz Capacity boundary16SM−T: T=2BPSK8SM−T: T=316SM−T: T=18PSK16QAM8SM−T: T=116PSK16SM−T:T=32SM−T: T=24PSK2SM−T: T=32SM−T: T=64SM−T: T=24SM−T: T=18SM−T: T=22SM−T: T=14SM−T: T=34SM−T: T=6SM−T represents SpiralModulation with Time reversalFigure 4.12: Spiral modulation with time reversal:Spectral efficiency vs energy to noise ratio for bit error probability at 10−5614.2. Spectrum-Efficiency Plane−2024681012141618200.250.5124816Energy to noise ratio, Eb/No dBSpectral efficiency, R/W bit/sec/Hz Capacity boundaryBPSK4PSK16SM−R: T=28SM−R: T=38SM−R: T=116SM−R: T=18PSK16QAM16PSKSM−R represents SpiralModulation with Rotation reversal2SM−R: T=22SM−R: T=316SM−R:T=38SM−R: T=24SM−R: T=12SM−R: T=14SM−R: T=24SM−R: T=3Figure 4.13: Spiral modulation with rotation reversal:Spectral efficiency vs energy to noise ratio for bit error probability at 10−5624.2. Spectrum-Efficiency Plane−2024681012141618200.250.5124816Energy to noise ratio, Eb/No dBSpectral efficiency, R/W bit/sec/Hz Capacity boundary4PSKBPSK8SM−B: T=38PSK16QAM16PSK16SM−B: T=216SM−B:T=3SM−B represents SpiralModulation with Bothreversals4SM−B: T=14SM−B: T=24SM−B: T=34SM−B: T=38SM−B: T=28SM−B: T=116SM−B: T=1Figure 4.14: Spiral modulation with both reversals:Spectral efficiency vs energy to noise ratio for bit error probability at 10−5634.2. Spectrum-Efficiency PlaneFigures 4.12 - 4.14 show the spectral efficiency of spiral modulation versus the conven-tional modulations. The derivation of the capacity boundary can be shown as follows. Thenormalized version of SNR, Eb/No can be written as [1]EbNo=STbN/W(4.8)=S/RN/W(4.9)=SNoC(4.10)where S is the signal power, Tb is the bit time, R is the bit rate, N is the noise power andW is the bandwidth. For the case where transmission bit rate is equal to channel capacity,R = C, we can use the identity presented in (4.8) to write [1]SNoC=EbNo. (4.11)Hence, we can rewrite (2.6) asCW= log2[1 +EbN0(CW)](4.12)EbNo=WC(2C/W − 1). (4.13)By using (4.13), we can plot the capacity boundary. For Nyquist (ideal rectangular) fil-tering at baseband, the bandwidth efficiency of PSK and QAM is R/W = log2M , whereM is the modulation order [1]. Because the bandwidth of spiral modulation signals isvarying, the bandwidth efficiency of spiral modulation is R/W = log2M/Wspiral, whereWspiral is the bandwidth of spiral modulation signals. Then we plotted the spectral ef-ficacy of each modulation at a BER of 10−5. It can be seen from the figures that spiralmodulation requires more bandwidth than conventional modulations. In Figures 4.12-4.14,the spectral efficiency of spiral modulation is worse than PSK and QAM under differentmodulation orders. In these spectral efficiency plots, the capacity boundary shows thatspiral modulation can not exceed the Shannon limit, which is contrasting to the statementthat Astrapi claimed about the spiral modulation can exceed the Shannon limit. But inthe term of power efficiency which is Eb/No, spiral modulation requires less power to meeta BER target of 10−5 for a modulation order of 8 or 16. This observation is consistent with644.3. Summarythe BER analysis in Chapter 3. By comparing these three figures, it can be seen that spiralmodulation with time reversal has the best power efficiency. In other words, spiral mod-ulation has a better power efficiency than conventional modulations but a worse spectralefficiency. In today’s communications, spectrum management is a growing problem due tothe growing number of spectrum use. The techniques with fewer occupying bandwidth areoften more desirable. But for satellite communications, power efficiency is also importantdue to the long link distance between the ground station and the satellite [20]. Therefore,spiral modulation may be a viable technology for satellite communications.4.3 SummaryIn this chapter, we studied the spectral efficiency of the spiral modulation. We firstpresented the periodogram method to compute the spectrum of spiral modulation and de-fined the bandwidth of spiral modulation signals. The PSD of the spiral modulation signalshowed that the bandwidth is not bandlimited. Based on the spectrum efficiency plane, itcan be concluded that the spectrum efficiency of spiral modulation is worse than the con-ventional modulation techniques, and the spiral modulation cannot exceed the Shannoncapacity boundary.65Chapter 5ConclusionsIn this chapter, we conclude the thesis by summarizing the accomplished work andsuggesting some potential research topics related to spiral modulation.5.1 Summary of Accomplished WorkIn this thesis, we investigated performance evaluation for spiral modulation. The perfor-mance evaluation was performed in both BER performance analysis and spectral efficiencyanalysis. In Chapter 1, we provided the motivation of this thesis and a literature reviewof Astrapi’s patents. In Chapter 2, we gave a detailed background knowledge on the spi-ral modulation, including generalization of Euler’s formula, the general spiral modulationformula and Gray coding for spiral modulation. A detailed modulation method of spiralmodulation is presented. In Chapter 3, we presented the transmitter and receiver struc-ture which was not presented by Astrapi company in its white paper and patents. Then,for the first time, we compared the BER performance between the spiral modulation andconventional modulations from low modulation order to high modulation order. The resultshows that the BER performance of spiral modulation is better than FSK and QAM whenthe modulation order is equal to or larger than 8. In Chapter 4, we studied the the spectralefficiency of spiral modulation. First we described the periodogram which is later used tocompute the PSD of spiral modulation and defined the bandwidth of spiral modulationaccording to its PSD plot. Then, for the first time we plotted spectral efficiency plane ofspiral modulation and conventional modulations. The result shows that the spectral effi-ciency of spiral modulation is worse than conventional modulations and it requires muchmore bandwidth to achieve a better BER. It also shows that it can not exceed the Shannonlimit. But the power efficiency of spiral modulation is better than conventional modula-665.2. Suggested Future Worktion when the modulation order is equal to or larger than 8. The accomplished work issummarized as follows:− A receiver structure of spiral modulation is proposed in Chapter 3.− A BER performance analysis is presented in Chapter 3 and shows that the BERperformance of spiral modulation is better than PSK and QAM when the modulationorder is equal to or larger than 8 if the bandwidth is not taken into consideration.− A spectral efficiency analysis is presented in Chapter 4 and reveals that the spectralefficiency of spiral modulation is worse than the conventional digital modulations.− From the spectrum efficiency plane plot, it can be concluded that spiral modulationcan not exceed the Shannon limit.5.2 Suggested Future WorkIn this thesis, we showed that spiral modulation has a poor spectral efficiency whichmakes this modulation scheme impractical. In the future work, it will be interesting toinvestigate how to strictly limit the bandwidth of spiral modulation signal in order toachieve a good spectral efficiency. Also, if we can have a bandlimited signal of spiralmodulation, its waveform must be changed. In this thesis, investigation is based on thewhite paper and patents that Astrapi published online. Astrapi may also have some ownresearch progress. Therefore, it is of importance to work along with Astrapi to find out thepotential applications of spiral modulation. Astrapi’s white paper and patent have veryfew mathematic analysis. Thus, in the future work, it will be important and valuable ifsome mathematical analysis can be performed for spiral modulation.67Bibliography[1] B. Sklar, Digital Communications Fundamentals and Apllications. New Jersey:Prentice-Hall, 1988. → pages 1, 57, 58, 59, 60, 64[2] A. Goldsmith, Wireless Communications. Cambridge: Cambridge University press,2005. → pages 19, 26[3] F. Xiong, Digital Modulation Techniques. Norwood: Artech House, INC., 2000. →pages 1[4] C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IEEE,vol. 37, pp. 10–21, Jan. 1949. → pages 2, 20[5] “Astrapi corporation website,” Astrapi Corporation, 2016. [Online]. Available:http://www.astrapi-corp.com/ → pages 2[6] “The shannon law for non-periodic channels,” J. D. Prothero, 2012.[Online]. Available: http://www.astrapi-corp.com/wp-content/uploads/2014/02/R-2012-1-The-Shannon-Law-for-Non-Periodic-Channels-v2.pdf → pages 2, 6, 20[7] “Telecommunication signaling using nonlinear functions,” J. D. Prothero, 2013.[Online]. Available: http://ip.com/pat/US20130251018 → pages 2, 3, 7, 13, 21, 22,27[8] “Methods and systems for communicating,” J. D. Prothero, 2014. [Online]. Available:http://ip.com/pat/US20120263031 → pages 3, 7[9] “Method for gray coding a symbol alphabet,” J. D. Prothero, 2014. [Online].Available: http://www.patentsencyclopedia.com/app/20140254728 → pages 3, 1768Bibliography[10] “Software defined radio,” J. D. Prothero, 2014. [Online]. Available: http://ip.com/pat/US8995546 → pages 3, 14[11] J. Stillwell, Mathematics and Its History, 3rd ed. New York: Springer, 2010. →pages 6[12] H. T. Nguyen, J. B. Andersen, G. F. Pedersen, P. Kyritsi, and P. C. F. Eggers,“Time reversal in wireless communications: A measurement-based investigation,”IEEE Transactions on Wireless Communications, vol. 5, 2006. → pages 8[13] S. Haykin, Communication Systems. New Jersey: John Wiley and Sons, Inc, 2014.→ pages 19, 23, 52[14] “Power spectra estimation,” National Semiconductor Apllication Note 255, 1980.[Online]. Available: http://www.ti.com/lit/an/snoa719/snoa719.pdf → pages 51[15] “Power spectrum and correlation,” S. V. Vaseghi, 2000. [Online]. Available:http://dsp-book.narod.ru/302.pdf → pages 51[16] “The fundamentals of signal analysis,” Hewlett Packard, 1994. [Online]. Available:http://www.hpmemoryproject.org/an/pdf/an 243.pdf → pages 52[17] “Spectrum and spectral density estimation by the discrete fourier transform(dft), including a comprehensive list of window functions and some new at-topwindows,” G. Heinzel, A. Rudiger, R. Schilling, 2002. [Online]. Available:http://edoc.mpg.de/395068 → pages 52[18] P. Podder, T. Z. Khan, M. H. Khan, and M. M. Rahman, “Comparative performanceanalysis of hamming, hanning and blackman window,” International Journal of Com-puter Applications, vol. 96, Jun. 2014. → pages 52[19] M. Owen, Practical Signal Processing. Cambridge: Cambridge University Press, 2007.→ pages 59[20] B. R. Elbert, Introduction to Satellite Communication. Norwood: Artech House,INC., 2008. → pages 6569
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Is spiral modulation really useful?
Text