IS SPIRAL MODULATION REALLY USEFUL?

Transcription

1 IS SPIRAL MODULATION REALLY USEFUL? by Haobing Chu B.Eng., Birmingham City University, UK, 212 M.Sc., The University of Warwick, UK, 213 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) December 216 c Haobing Chu, 216

2 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/216 (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

3 Abstract 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. ii

4 Table of Contents Abstract ii Table of Contents iii List of Tables v List of Figures List of Acronyms Acknowledgements Dedication vi ix x xi Chapter 1: Introduction Background and Motivation Literature Review Thesis Organization and Contributions Chapter 2: Background Generalization of the Euler s Formula The General Spiral Modulation Formula Gray Coding for Spiral Modulation Shannon-Hartley Theorem and Astrapi s Claims Summary Chapter 3: BER Performance Analysis iii

5 TABLE OF CONTENTS 3.1 System Structure Transmitter Structure Receiver Structure BER Comparison Between Conventional Modulations and Spiral Modulation Phase Shift Keying and Quadrature Amplitude Modulation Spiral Modulation With Time Reversal Spiral Modulation With Rotation Reversal Spiral Modulation With Both Reversals Summary Chapter 4: Spectral Efficiency Analysis Spectrum Analysis Spectrum-Efficiency Plane Summary Chapter 5: Conclusions Summary of Accomplished Work Suggested Future Work Bibliography iv

6 List of Tables Table 2.1 Conventional modulation sets for an alphabet of eight [1] Table 2.2 Spiral modulation sets for an alphabet of eight [1] Table 3.1 Parameter values of spiral modulation with time reversal Table 3.2 Symbol waveform equations of spiral modulation with time reversal. 29 Table 3.3 Parameter values of spiral modulation with rotation reversal Table 3.4 Symbol waveform equations of spiral modulation with rotation reversal 4 Table 3.5 Parameter values of spiral modulation with both reversals Table 3.6 Symbol waveform equations of spiral modulation with both reversals 46 v

7 List of Figures Figure 2.1 Complex spirals of the generalized Euler s formula Figure 2.2 Time reversal representation Figure 2.3 (a) Real part waveform when K 1 = 1 or ω 1 = (b) Imaginary part waveform when K 1 = 1 or ω 1 = Figure 2.4 (a) Real part waveform when K 1 = 1 or ω 1 = π (b) Imaginary part waveform when K 1 = 1 or ω 1 = π Figure 2.5 Rotation reversal representation Figure 2.6 (a) Real part waveform when K 2 = 1 or ω 2 = (b) Imaginary part waveform when K 2 = 1 or ω 2 = Figure 2.7 (a) Real part waveform when K 2 = 1 or ω 2 = π (b) Imaginary part waveform when K 2 = 1 or ω 2 = π Figure 2.8 (a) Real part waveform for different m values (b) Imaginary part waveform for different m values Figure 2.9 Generalized Euler s formula when m = 3 and T = 1, 2 and Figure 2.1 Generalized Euler s formula when m = 3 and T = 1, 2 and 3 (a) Real part waveform (b) Imaginary part waveform Figure 2.11 Complex plane plot of 8-ary Spiral modulation with K 1 = ±1 for < t < Figure 3.1 The transmitter structure of the spiral modulation [2] Figure 3.2 Receiver of spiral modulation Figure 3.3 Complex plane plot of spiral modulation with time reversal Figure 3.4 BER for BPSK and binary spiral modulation with time reversal.. 32 Figure 3.5 BER for 4PSK and 4-ary spiral modulation with time reversal vi

8 LIST OF FIGURES Figure 3.6 BER for BPSK and binary spiral modulation with time reversal when T = Figure 3.7 BER for 4PSK and 4-ary spiral modulation with time reversal when T = Figure 3.8 Waveforms of binary spiral modulation with time reversal when T = 1 (a) Real part waveform (b) Imaginary part waveform Figure 3.9 Waveforms of binary spiral modulation with time reversal when T = 6 (a) Real part waveform (b) Imaginary part waveform Figure 3.1 BER for 8PSK and 8-ary spiral modulation with time reversal Figure 3.11 BER for 16QAM and 16-ary spiral modulation with time reversal. 36 Figure 3.12 BER for binary spiral modulation with time reversal for different m values Figure 3.13 Complex plane plot of spiral modulation with rotation reversal.. 41 Figure 3.14 BER for BPSK and binary spiral modulation with rotation reversal 42 Figure 3.15 BER for 4PSK and 4-ary spiral modulation with rotation reversal. 42 Figure 3.16 BER for 8PSK and 8-ary spiral modulation with rotation reversal. 43 Figure 3.17 BER for 16QAM and 16-ary spiral modulation with rotation reversal 43 Figure 3.18 Complex plane plot of spiral modulation with both reversals Figure 3.19 BER for BPSK and binary spiral modulation with both reversals. 48 Figure 3.2 BER for 4PSK and 4-ary spiral modulation with both reversals when T = Figure 3.21 BER for 8PSK and 8-ary spiral modulation with both reversals.. 49 Figure 3.22 BER for 16QAM and 16-ary spiral modulation with both reversals. 49 Figure 4.1 Comparison between theoretical PSD and simulated PSD of rectangular pulses Figure 4.2 PSD of spiral modulation signals when T = Figure 4.3 PSD of spiral modulation signals when T = Figure 4.4 PSD of spiral modulation signals when T = Figure 4.5 Waveform of real part of spiral modulation when T = vii

9 LIST OF FIGURES Figure 4.6 Waveform of imaginary part of spiral modulation when T = Figure % bandwidth of spiral modulation signals when T = Figure % bandwidth of spiral modulation signals when T = Figure % bandwidth of spiral modulation signals when T = Figure 4.1 The effect of parameter T on spiral modulation signal under different bandwidth criteria Figure 4.11 PSD of sinc pulse and rectangular pulse Figure 4.12 Spiral modulation with time reversal: Figure 4.13 Spiral modulation with rotation reversal: Figure 4.14 Spiral modulation with both reversals: viii

10 List of Acronyms Acronyms Definitions 4PSK 8PSK 16PSK 64PSK AWGN BER BPSK PSD QAM QPSK SDR SNR 4-ary Phase-Shift Keying 8-ary Phase-Shift Keying 16-ary Phase-Shift Keying 64-ary Phase-Shift Keying Additive White Gaussian Noise Bit-Error Rate Binary Phase-Shift Keying Power Spectral Density Quadrature Amplitude Modulation Quadrature Phase-Shift Keying Software Defined Radio Signal-to-Noise Ratio ix

11 Acknowledgements I would like to appreciate many people for their help. First and foremost of all, I would like 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 effort and significant amount of time to serve on my M.A.Sc. committee. I would also like to express my thanks to Dr. Yang Cao for serving as my university examiner. I really appreciate their valuable time and constructive comments on my thesis. My thanks to all my friends at UBC Okanagan for their help and friendship. Special thanks to my dear colleagues Changle Zhu, Chunpu Wang, Fang Fang, Guanshan Ye, Hao Liu, Hui Ma, and Yanjie Dong for their help on study and life at UBC. I would also like to thank all my friends in Kelowna, Vancouver and Hefei. Finally, I would like to special thank to my dear parents for their support during these years at UBC. But more importantly, thanks for their greatest love and support. x

12 To My Parents xi

13 Chapter 1 Introduction 1.1 Background and Motivation Digital modulation techniques are essential in many digital communication systems, whether it is a satellite communication system or a mobile cellular communication system. In the past decades, modulation techniques have been well studied and researched thoroughly, and many promising results have been developed. In digital modulation, information bits are transformed into waveforms that are compatible with the characteristics of the channel. The information bits are binary digits taking on values of either 1 or. In digital modulation techniques, the modulation process consists of mapping the information bits into an analog signal for transmission over the channel, and the demodulation process 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 of digital symbols are used to create a square pulse waveform with certain features such as pulse amplitude, pulse width, and pulse position. The square pulse waveform then can be 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 the parameters of a high-frequency sinusoidal signal. Sinusoidal waveform is used as carrier for transmission propose. Therefore, according to the three features of sinusoid, bandpass modulation includes amplitude modulation, frequency modulation, phase modulation or a combination of them. A variety of modulation schemes can be derived from these combinations. Quadrature amplitude modulation (QAM) and phase-shift keying (PSK) are standard bandpass modulation techniques, and they are widely used [1 3]. In nowadays communications, people are demanding rapid data transmission. The use 1

14 1.2. Literature Review of digital modulation provides more capacity, higher data security, better quality communications, and quicker system availability. But due to the fact of maximum channel capacity that was introduced by Shannon [4], none of these traditional modulation methods can exceed this limitation. Recently, a new modulation technique called spiral modulation has been developed by a company called Astrapi Corporation [5]. On its website, the company claims itself the pioneer of a revolutionary method of spiral-based modulation, a game-changing technology in communications technology [5]. Astrapi Corporation holds several key patents on spiral modulation. The company claims that spiral modulation provides better performance than the conventional modulation techniques. More suspiciously, it claims that spiral modulation can exceed the Shannon limit [6]. Moreover, the company doesn t provide any performance study of the spiral modulation. In order to verify these claims on spiral modulation, we study in this thesis the performance of the spiral modulation in additive white Gaussian noise (AWGN) channels. 1.2 Literature Review The telecommunication industry faces a fundalmental problem with the exponential growth of data transmission demand. Shannon limit provides a sharp limit on the maximum rate 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 213 by Jerrold Prothero. Dr. Jerrold Prothero received his Ph.D. in mechanical engineering from University of Washington in 1998 and is currently the CEO of Astrapi Corporation. He developed a generalization 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 spiral modulation. Astrapi claims that this spiral modulation can increase spectral efficiency dramatically and can even exceed the Shannon limit. Astrapi owns several patents on spiral 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 s 2

15 1.2. Literature Review formula that corresponds to a rotating circle in the complex plane, the generalization of Euler 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 structure generates a complex waveform for transmission purpose. The details of this structure is illustrated in Chapter 3. In Astrapi s patents and white papers, no receiver structure is given. Therefore, we will apply the traditional correlator receiver to analyze the performance of spiral modulation in Chapter 3. Reference [8] provides the general spiral modulation formula. This formula can be used to map information bits into a complex waveform. In the traditional modulation techniques, only amplitude, frequency and phase are used for signal modulation. But in the general spiral modulation formula, in addition to amplitude, frequency and phase, spiral modulation also uses time reversal, rotation reversal and a parameter m to represent different 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 can allow more modulating waves than the classical modulations. In [9], a Gray coding method for spiral modulation is presented. In the spiral modulation, the concept of signal constellation doesn t apply. Therefore we can not code symbols according to a constellation diagram using a Gray mapping rule. In [9], the author provides 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 be illustrated in Chapter 2. In [1], a software defined radio (SDR) implementation for spiral modulation is presented. 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 modulation combinations to accommodate channel conditions and receives analog signals from an analog-to-digital convertor, A digital-to-analog convertor that transforms digital signals to analog signals, 3

16 1.3. 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 and feeds the signal back to the microprocessor. 1.3 Thesis Organization and Contributions This thesis consists of five chapters. A summary of each chapter and its contributions are presented as follows. Chapter 1 reviews some background on modulation techniques. Traditional modulations have been studied thoroughly and the well-known Shannon-Hartley theorem has been established for over sixty years. Recently, however, a new modulation called spiral modulation has been developed by Astrapi and the company claims spiral modulation can exceed the Shannon capacity boundary. The motivation of this thesis is to verify the claims made by Astrapi and explore the potentials of spiral modulation. A literature review of the spiral modulation is provided. Finally, a summary of contributions of this thesis is presented. Chapter 2 presents the background on spiral modulation, including generalization of Euler s formula, the general spiral modulation formula, and Gray coding for spiral modulation. In addition to amplitude modulation, frequency modulation and phase modulation, spiral modulation also uses time reversal, rotation reversal and the parameter m to modulate signals. At last, we review the Shannon-Hartley theorem. Several key claims of Astrapi are stated and they will be verified in the following chapters. Chapter 3 first describes the transmitter and receiver structures of the spiral modulation. Then a bit error rate (BER) comparison between the conventional modulations and the spiral modulation is presented. In this comparison, we compare spiral modulation with two conventional modulations: PSK and QAM. Chapter 3 shows how a data stream can be modulated 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 modulation orders. Finally, a summary of the BER performance of spiral modulation is presented. The 4

17 1.3. Thesis Organization and Contributions results show that the BER performance of spiral modulation is better than the traditional modulations when the modulation order is equal to or larger than eight, assuming that bandwidth is not taken into consideration. In Chapter 4, we study the spectral efficiency of spiral modulation. A comprehensive spectrum analysis is presented. In this analysis, we compute the spectrum of spiral modulation 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 spectral efficiency of the spiral modulation is worse than the traditional modulations. We conclude that, unlike the claim made by Astrapi, the spiral modulation cannot exceed Shannon s capacity boundary. Finally, we summarize the contents and the thesis contributions in Chapter 5, and suggest some further research topics related to the spiral modulation. 5

18 Chapter 2 Background In 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 summarise some Astrapi s key claims. 2.1 Generalization of the Euler s Formula The Euler s formula [11] e it = cos(t) + i sin(t) (2.1) where i 2 = 1, is the most remarkable equation in mathematics and it has wide applications in 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 s formula corresponds to a unit circle in the complex plane as t ranges through the real numbers. In the unit circle, t is the angle between a line connecting the origin with a point on 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] e ti(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 complex plane. 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. 6

19 2.2. The General Spiral Modulation Formula 1 m = 3 5 t m = 2.5 t = Imag axis 5 m = Real axis Figure 2.1: Complex spirals of the generalized Euler s formula In Figure 2.1, there are three spirals and their trajectories for different values of m become longer when the time variable t takes lager values. When m becomes larger, the spiral becomes more and more outwards. In [8], the author uses this generalization of the Euler s formula to design signal modulation. 2.2 The General Spiral Modulation Formula The general spiral modulation formula is given as [7] f m (t) = K e iω e [K 1e iω 1 ](t+t )i [K 2 eiω 2 ](2 (2 m) ) < t < T. (2.3) In (2.3), T is the symbol duration and the first level is specified by K and ω such that K can be used for amplitude modulation and ω can be used for phase modulation. The second level is specified by K 1, ω 1 and t such that K 1 can be used for frequency modulation, and K 1 and ω 1 can be used for time reversal. The parameter t can be used 7

20 2.2. The General Spiral Modulation Formula for time shift when time reversal is applied. The term time reversal refers to that the spiral direction can be either outwards or inwards. For example, time reversal can be achieved by assuming K 1 = ±1 with ω 1 =, or equivalently ω 1 = and ω 1 = π with K 1 = 1. An example of a complex spiral signal with time reversal is illustrated in Figure Outward spiral when K 1 = 1 or ω 1 = Inward spiral when K 1 = 1 or ω 1 = π 5 Imag axis 5 Inward Outward Starting location for outward spiral (solid line) 1 Starting location for inward spiral (circle) Real axis Figure 2.2: Time reversal representation In Figure 2.2, the solid line represents an outward spiral and the circle represents an inward spiral. In the time domain, it indicates whether the waveform is starting from high amplitude to low amplitude or vise versa. The real part and the imaginary part are plotted versus t in Figures It can be seen that the real part waveform in Figure 2.3 are time reversed version of the real part waveform in Figure 2.4. The imaginary part waveform in Figure 2.3 are time reversed version of the imaginary part waveform in Figure 2.4. Time reversal in the spiral modulation is different from the concept of time reversal in wireless communications. In wireless communications, the main purpose of time reversal in such applications is to focus energy in space and time on the point of interest by filtering the signal through the complex conjugate and time inverted channel impulse response [12]. 8

21 2.2. The General Spiral Modulation Formula 2 (a) Amplitude Time variable t 1 (b) Amplitude Time variable t Figure 2.3: (a) Real part waveform when K 1 = 1 or ω 1 = (b) Imaginary part waveform when K 1 = 1 or ω 1 = 2 (a) Amplitude Time variable t 1 (b) Amplitude Time variable t Figure 2.4: (a) Real part waveform when K 1 = 1 or ω 1 = π (b) Imaginary part waveform when K 1 = 1 or ω 1 = π 9

22 2.2. The General Spiral Modulation Formula The third level is specified by K 2, ω 2 and m such that K 2 and ω 2 can be used for rotation reversal. The term of rotation reversal indicates whether the spiral form is clockwise or anticlockwise. For example, rotation reversal can be achieved by using K 2 = ±1 with ω 2 =, or equivalently ω 2 = and ω 2 = π with K 2 = 1. An examples is shown in Figure Anti clockwise spiral when K 2 = 1 or ω 2 = Clockwise spiral when K 2 = 1 or ω 2 = π Anti clockwise 5 Imag axis 5 1 Clockwise Real axis Figure 2.5: Rotation reversal representation From Figure 2.5 one observes that the two curves are both outwards, but the solid one is anti-clockwise and the dotted one is clockwise. The real part and the imaginary part are plotted versus t in Figures

23 2.2. The General Spiral Modulation Formula 2 (a) Amplitude Time variable t 1 (b) Amplitude Time variable t Figure 2.6: (a) Real part waveform when K 2 = 1 or ω 2 = (b) Imaginary part waveform when K 2 = 1 or ω 2 = 2 (a) Amplitude Time variable t 2 (b) Amplitude Time variable t Figure 2.7: (a) Real part waveform when K 2 = 1 or ω 2 = π (b) Imaginary part waveform when K 2 = 1 or ω 2 = π From Figures , it can be seen that the two real parts are the same and the two imaginary parts are reflections to each other with respect to the horizontal axis. But not 11

24 2.2. The General Spiral Modulation Formula every rotation reversal paired signals have the same real parts and different imaginary parts. Some rotation reversal paired signals have the same imaginary parts but different imaginary parts. Some rotation reversal paired signals have different real parts and different imaginary 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, the three spirals in Figure 2.1 are complex spirals. Then the real part and the imaginary part can be plotted versus t separately to have a better understanding of parameter m. 1 (a) Amplitude m = 2.1 m = 2.5 m = Parameter t 1 (b) Amplitude m = 2.1 m = 2.5 m = Parameter t Figure 2.8: (a) Real part waveform for different m values (b) Imaginary part waveform for different m values Figures 2.8 shows that both real part and imaginary part are not periodic waveforms and the amplitude increases when t is increased. When m becomes larger, the amplitude changes rapidly versus t. Therefore, this kind of waveforms have been named as non- 12

25 2.2. The General Spiral Modulation Formula periodic amplitude varying waveforms by Astrapi Corporation [7]. At last, t represents the time variable which lasts from < t < T where T is the symbol duration which also indicates the length of a spiral trajectory in the complex plane [7]. In spiral modulation, when m > 2, eq. (2.2) corresponds to a spiral in the complex plane and Astrapi uses this spiral to modulate signal, resulting the amplitude of spiral modulated signal changes with time. For example, setting m = 3 and T = 1, 2, and 3 in (2.2), the corresponding spiral signal can be plotted in Figure T = 1 T = 2 T = Imag axis Real axis Figure 2.9: Generalized Euler s formula when m = 3 and T = 1, 2 and 3 The real part and the imaginary part are plotted versus t in Figure

26 2.2. The General Spiral Modulation Formula 2 (a) Amplitude 2 4 T = 1 T = 2 T = Parameter t (b) 8 Amplitude T = 1 T = 2 T = Parameter t Figure 2.1: Generalized Euler s formula when m = 3 and T = 1, 2 and 3 (a) Real part waveform (b) Imaginary part waveform Figure 2.1 shows when t increases, the maximum amplitude also increases. After understanding these parameters, we know that (2.3) can be used for signal modulation. Different from conventional modulation formats that use amplitude, frequency and phase to modulate the signal, spiral modulation has three new parameters that the conventional modulation 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 can be modulated to a complex waveform with K 2 = 1 (anti-clockwise spiral) and bit 1 can be modulated to a complex waveform with K 2 = 1 (clockwise spiral). Because of time reversal and rotation reversal, spiral modulation has more options to modulate signals. In one of Astrapi s patents [1], all possible modulation sets of an alphabet size of 8 between conventional modulation and spiral modulation are listed. 14

27 2.2. The General Spiral Modulation Formula Table 2.1: Conventional modulation sets for an alphabet of eight [1] Phases Amplitudes Frequencies Symbols

28 2.3. Gray Coding for Spiral Modulation Table 2.2: Spiral modulation sets for an alphabet of eight [1] Phases Amplitudes Frequencies Rotation Time Symbols Table 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 time reversal and rotation reversal to modulate the signal. In Table 2.2, it can be seen that for an alphabet of 8, we can use time reversal, rotation reversal or both reversals to modulate the signal to form 8 different symbols. 2.3 Gray Coding for Spiral Modulation Similar to conventional modulation methods, spiral modulation also uses Gray coding. A method for performing Gray coding of spiral modulation has been introduced in 16

29 2.3. 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 the spiral modulation. Reference [9] provides an algorithm to determine which waveforms are most likely to be confused with each other, and applies a Gray encoding technique to the symbols. In this section, this algorithm is described. Algorithm 1: Compute the information on the likelihood of inter-symbol confusion of spiral modulation input : Totally M waveforms for a modulation order of M output: A matrix with the information on the likelihood of inter-symbol confusion Step 1: For each waveform s i (t), n points are sampled such that each waveform is discrete in time and denoted as s i (n) ; Step 2: Create normalized version of s i (n) such that ŝ i (n) = s i (n)/ s i (n), where denotes the norm operation; Step 3: Form a matrix A i,j such that A i,j = ŝ i (n) s j (n), where i denotes the row in a matrix, j denotes the column in a matrix and denotes the inner product operation; Step 4: Take the diagonal elements of A i,j such that d = diag(a i,j ), where diag( ) denotes diagonal elements of a diagonal matrix; Step 5: Form a matrix D i,j such that each row is the vector d and there are totally M rows; Step 6: Form a matrix B i,j such that B i,j = A i,j D i,j ; In Algorithm 1, step 2 provides correlation for each waveform. In step 3, each element of A i,j represents the strength of the match for the ith waveform with the jth waveform, and therefore provides information on the likelihood of inter-symbol confusion. In step 6, the matrix B i,j has zero along the diagonal and all positive values elsewhere. Smaller non-zero values in B i j,j imply a greater risk that the receiver will incorrectly identify a transmitted symbol i as the symbol j. Therefore, we can apply Gray coding to the waveforms according to this matrix B i,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 phase 17

30 2.3. Gray Coding for Spiral Modulation Algorithm 2: Apply Gray code to spiral modulation input : The matrix B i,j output: Gray coded waveforms Step 1: Create an empty ordered list L ; Step 2: Find the column j with the smallest non-zero value in the matrix B i,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 is most likely to be confused with the jth waveform; Step 5: Remove column j from the matrix B i,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 bit sequences that differ by one bit; shift is shown in Figure imag axis real axis Figure 2.11: Complex plane plot of 8-ary Spiral modulation with K 1 = ±1 for < t < 1 In Figure 2.11, the time variable t is set as < t < 1 and the parameter m is equal 18

31 2.4. Shannon-Hartley Theorem and Astrapi s Claims to 3. The solid line represents rotation reversed spiral and the dashed line represents non rotation reversed spiral. Because there is no constellation for spiral modulation, we can only show Gray coding in a complex plane with all 8 complex waveforms plotted. After applied Gray code method as described above, each spiral is assigned to a symbol. Two successive values differ in only one bit are mapped to two more likely confused spirals. 2.4 Shannon-Hartley Theorem and Astrapi s Claims In radio communication systems, communication channels play an important role in studying and designing modulation schemes. Different modulation techniques are chosen according to different channel characteristics in order to optimize their performance. In digital 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 fading does not exist. Therefore, the received signal can be expressed as r(t) = s(t) + n(t) (2.4) where s(t) is the transmitted signal and n(t) is the additive white Gaussian noise [2]. The power spectral density (PSD) of AWGN is [13] S N (f) = N /2, < f < (2.5) where N is the noise power per unit of bandwidth and the term 1/2 indicates that power is symmetric across both positive and negative frequencies. Eq. (2.5) implies that AWGN is a random process with a flat PSD over all frequencies. In this thesis, AWGN channel model is adopted for simulation purpose. In information theory, for such an AWGN channel with a finite bandwidth W, the capacity C of this AWGN channel can be expressed in terms of available bandwidth and the signal-to-noise ratio. The capacity relationship can be stated as [13] C = W log 2 ( 1 + S = W log 2 ( 1 + S N N W ) ) (2.6) (2.7) 19

32 2.5. Summary where S is the average received signal power and N = N W is the average noise power. Eq. (2.7) is the well-known Shannon-Hartley theorem [4]. This theorem shows that the maximum rate at which information can be reliably transmitted over an AWGN channel with a specified bandwidth. It is the most fundamental theorem in information theory and has been standing for over sixty years. However Astrapi claims that spiral modulation can exceed the Shannon limit [6]. Astrapi also has several other claims about spiral modulation in their patents. In this thesis, some key claims are singled out to be verified. These claims are listed below. Claim 1: The performance of spiral modulation is better than conventional modulations [6]. Claim 2: Spiral modulation has a better spectral efficiency than conventional modulations [6]. Claim 3: Spiral modulation can exceed the Shannon limit [6]. 2.5 Summary In this chapter, we presented the background knowledge of spiral modulation, including generalization of the Euler s formula, the general spiral modulation formula, Gray coding for 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 were stated and they are to be verified in the next two chapters. 2

33 Chapter 3 BER Performance Analysis In this chapter, we first describe the transmitter and receiver structure of spiral modulation. Then we use the general spiral modulation formula to analyze the BER of spiral modulation by using MATLAB simulation. We also present the symbol waveform equations 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 Structure The transceiver of spiral modulation is more complicated than the conventional modulation schemes. In this section, the transmitter and receiver structures of spiral modulation are described. In Astrapi s patents, no receiver structure is provided. Therefore, we apply the traditional receiver structure to analyze spiral modulation Transmitter Structure According to (2.3), every spiral modulated signal is a complex signal, i.e. every modulated signal has a real part and an imaginary part. In [7], Astrapi claimes that a twocomponent transmission is used for spiral modulation. Here, the two-component transmission means that the transmitter generates the real part signal and the imaginary part signal separately. From (2.3), by setting K = K 1 = K 2 = 1 and ω = ω 1 = ω 2 =, eq (2.3) becomes f m (t) = e ti(2(2 m) ) = e t(eiπ/2 ) (2(2 m) ) = e tei(2(1 m) π) (3.1) (3.2) (3.3) 21

34 3.1. System Structure = e t[cos(2(1 m) π)+i sin(2 (1 m) π)] (3.4) = e t cos(2(1 m) π) [cos(t sin(2 (1 m) π)) + i sin(t sin(2 (1 m) π)]. (3.5) From (3.5), it can be seen that both the real part and the imaginary part of a spiral modulation can be distinguished from each other. In [7], a block diagram of the transmitter structure is given to describe how the spiral waveforms are generated and are multiplied with carrier signals with a carrier frequency at least twice times the baseband signal frequency. The structure is shown in Figure

35 3.1. System Structure Data m m-factor π*2 (1-m) sin & cos sin cos X Clock t-value X sin & cos of t*sin(π2 (1-m) ) exp of t*cos(π2 (1-m) ) cos exp sin exp X X Imaginary part Real part Im{f(t)} Re{f(t)} sin(2πfct) X X cos(2πfct) + s(t) Figure 3.1: The transmitter structure of the spiral modulation [2] Receiver Structure In Astrapi s patents and white papers, no receiver structure is given. Therefore, we will apply the traditional correlator receiver to analyze the performance of spiral modulation. In MATLAB simulation, correlation detection is applied [13]. The receiver structure diagram is shown in Figure

36 3.1. System Structure cos(2πf ct) Correlator Detection X LPF Re{f(t)} Find the correlation between f(t) and r(t) = Re{f(t)} cos(2πf c t) + Im{f(t)} sin(2πf c t) each reference signals. Then make decision X LPF Im{f(t)} sin(2πf ct) Figure 3.2: Receiver of spiral modulation In the transmitter structure, the information signal f(t) can be treated as an envelope of a carrier signal. Therefore, we first need to perform the envelope detection. The envelope detection process is given in the following description. A received signal r(t), which has a real part and an imaginary part, goes through each multiplication operator by multiplying each carrier waveform. The mathematical process is shown below. For the upper branch, we have Upperbranch = r(t) cos(2πf c t) (3.6) = Re {f(t)} cos 2 (2πf c t) + Im {f(t)} sin(2πf c t) cos(2πf c t) (3.7) = 1 2 Re {f(t)} Re {f(t)} cos(4πf ct) Im {g(t)} sin(4πf ct). (3.8) For the lower branch, we have Lowerbranch = r(t) sin(2πf c t) (3.9) = Im {f(t)} sin 2 (2πf c t) + Re {f(t)} cos(2πf c t) sin(2πf c t) (3.1) = 1 2 Im {f(t)} 1 2 Im {f(t)} cos(4πf ct) Re {g(t)} sin(4πf ct). (3.11) The output is passed through a lowpass filter, and the real part and imaginary parts can 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, f i (t) (i = 1,..., M). The correlation detection includes two steps. In the first step, the detected envelope f(t) is correlated with each reference signals to form a set of random variable, 24

37 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation z i (T ) (i = 1,..., M) at the output of the correlators at time t = T, where T is the symbol duration. The mathematical process is shown below z i (T ) = T f(t)f i (t)dt i = 1,..., M. (3.12) In the second step, a symbol decision is made. A reasonable decision rule is to choose the waveform, f i (t) that matches best or has the largest correlation with f(t). In other words, the decision rule is Choose the f i (t) whose index corresponds to the max z i (T ) (3.13) 3.2 BER Comparison Between Conventional Modulations and Spiral Modulation In this section, the performance of spiral modulation is investigated by using MATLAB simulation and it is compared with several conventional modulations. Table 2.2 shows that the spiral modulation has more modulation combinations than the conventional modulations. According to Table 2.2, we can divide the spiral modulation into three different schemes 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 8 symbol waveforms. But for spiral modulation, spiral modulation with rotation reversal (or time reversal) uses 4 different phases to form 4 symbol waveforms and then apply rotation reversal (time reversal) to these waveforms to form another 4 rotation-reversed (timereversed) symbol waveforms. Also, spiral modulation with both reversals uses 2 different phases to form 2 symbol waveforms and then apply rotation reversal to form another 2 waveforms and apply time reversal to these 4 waveforms to form another 4 waveforms. In this section, we will study the BER performance of each spiral modulation scheme and compare it with that of the conventional modulations. 25

38 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation Phase Shift Keying and Quadrature Amplitude Modulation In this section, we describe two of the most commonly used conventional modulations: phase shift keying and quadrature amplitude modulation. In the following sections, these two 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] Si P SK (t) = Ag(t) cos [ 2π(i 1) M ] cos 2πf c t Ag(t) sin [ 2π(i 1) M ] sin 2πf c t t T (3.14) where A is typically a function of the signal energy; g(t) is the pulse shaping function; f c is the carrier frequency; T is the symbol duration; M is the modulation order and i = 1;..., M. In MATLAB simulation, for the convenience purpose, we only simulate the baseband transmission. All PSK transmitted signals Si P SK (t) have equal energy [2] E S P SK i = T (S P SK i (t)) 2 dt (3.15) = A 2. (3.16) For QAM, the information is encoded in both the amplitude and phase of the transmitted signal. Thus, the QAM transmitted signal is given by [2] S QAM i (t) = A i g(t) cos(θ i ) cos 2πf c t A i g(t) sin(θ i ) sin 2πf c t t T (3.17) where A i is typically a function of the signal energy; g(t) is the pulse shaping function and must maintain the orthonormal properties of basis functions; f c is the carrier frequency; T is the symbol duration; i = 1,..., M and M is the modulation order. The signal energy between symbols with different amplitude are not the same. The energy of each symbol is [2] E S QAM i = T (S QAM i (t)) 2 dt (3.18) = A 2 i. (3.19) 26

39 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation Then the average signal energy is M Ē QAM i=1 EP SK S S = i. (3.2) M Once we have the signal energy, we can define the SNR in the simulation. In order to make conventional modulations have the same SNR definition with spiral modulation, the SNR is defined as SNR[db] = E b N o [db] (3.21) = E s N o [db] + 1 log(log 2 (M)) (3.22) where E b is the energy per bit; E s is the energy per symbol; N o is the noise power and E s N o = E SP SK i N o for PSK ( Es N o = Ē S QAM i N o for QAM) Spiral Modulation With Time Reversal The reason why spiral modulation can have more modulation combination than conventional modulations is that it introduces two additional parameters to modulate the signal. One is time reversal and the other one is rotation reversal. In spiral modulation with time reversal, only time reversal is applied. Therefore, according to the general spiral modulation formula in (2.3), when only time reversal is applied, eq. (2.3) becomes f m (t) = K e iω e K 1e iω 1 (t+t )i (22 m ). (3.23) In Astrapi s patent [7], they use m = 3 to illustrate spiral modulation. choose m = 3 to form a spiral. Then, eq. (3.23) becomes Therefore, we f 3 (t) = K e iω e K 1e iω 1 (t+t )i (22 3 ). (3.24) Then, we can use eq. (3.24) to modulate signals. Table 3.1 shows the parameter values of spiral modulation with rotation reversal for different modulation orders. 27

40 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation Table 3.1: Parameter values of spiral modulation with time reversal Modulation Order Time Reversal Phases Amplitude K 1 = ±1 with Corresponding conventional modulations 2 4 ω 1 =, or ω 1 =, π with K 1 = 1 K 1 = ±1 with ω 1 =, or ω 1 =, π with K 1 = 1 K 1 = ±1 with n/a n/a BPSK ω =, π n/a 4P SK 8 ω 1 =, or ω 1 =, π with ω =, π 2, π, 3π 2 n/a 8P SK K 1 = 1 16 K 1 = ±1 with ω 1 =, or ω 1 =, π with K 1 = 1 ω =, π 4, π 2, 3π 4, π, 5π 4, 3π 2, 7π 4 K = 1, 2 16QAM According 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 each modulation order. After computing the information on the likelihood of inter-symbol confusion and applying Gray coding, the symbol waveform equations are shown in Table

41 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation Table 3.2: Symbol waveform equations of spiral modulation with time reversal Modulation Order Symbol Waveform Equations 2 S (t) = e ti(2 1 ), S 1 (t) = e (t+t )i (2 1 ) S (t) = e ti(2 1 ), S 1 (t) = e (t+t )i (2 1 ), S 11 (t) = e iπ e ti(2 1 ), S 1 (t) = e iπ e (t+t )i (2 1 ) S (t) = e ti(2 1 ), S 1 (t) = e (t+t )i (2 1 ), S 11 (t) = e i π 2 e ti(2 1 ), S 1 (t) = e i π 2 e (t+t )i (2 1 ), S 11 (t) = e iπ e ti(2 1 ), S 111 (t) = e iπ e (t+t )i (2 1 ), S 11 (t) = e i 3π 2 e ti(2 1 ), S 1 (t) = e i 3π 2 e (t+t )i (2 1 ) S (t) = e ti (2 1 ), S 1 (t) = e (t+t )i (2 1 ), S 11 (t) = 2e i π 4 e (t+t )i (2 1 ), S 1 (t) = 2e i π 4 e ti(2 1 ), S 11 (t) = e i π 2 e (t+t )i (2 1 ), S 111 (t) = e i π 2 e ti(2 1 ), S 11 (t) = 2e i 3π 4 e (t+t )i (2 1 ), S 1 (t) = 2e i 3π 4 e ti(2 1 ), S 11 (t) = e iπ e (t+t )i (2 1 ), S 111 (t) = e iπ e ti(2 1 ), S 1111 (t) = 2e i 5π 4 e (t+t )i (2 1 ), S 111 (t) = 2e i 5π 4 e ti(2 1 ), S 11 (t) = e i 3π 2 e (t+t )i (2 1 ), S 111 (t) = e i 3π 2 e ti(2 1 ), S 11 (t) = 2e i 7π 4 e (t+t )i (2 1 ), S 1 (t) = 2e i 7π 4 e ti(2 1 ) The resulting complex plane plot diagram of spiral modulation with time reversal for < t < 1 is shown in Figure 3.3. Note that the complex plane plot is not the same as a conventional constellation diagram in which symbols are represented by points. In the complex plane plot of spiral modulation, symbols are represented by complex spirals. In Figure 3.3, the solid line represents non time reversed spiral and the circle represents time reversed spiral. It can be seen that the signals which are represented by circles are time reversed version of the signals which are represented by solid line. 29

42 3.2. BER Comparison Between Conventional Modulations and Spiral Modulation Imag axis Circle 1 Outward Inward Line Imag axis line 11 Circle 1 Circle 1 line Real axis Real axis (a) Binary (b) 4-ary Line 11 Circle 111 Circle Line 1 Circle 11 Circle 11 Line 1 Circle 111 Circle 1 Line 11 Imag axis Circle 1 Line Imag axis Line 11 Line Line 11 Line 11 Circle Circle 111 Circle 1111 Line 111 Line 11 Line 1 Circle 11 Circle Real axis Real axis (c) 8-ary (d) 16-ary Figure 3.3: Complex plane plot of spiral modulation with time reversal 3