A PAM Decomposition of Weak CPM

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations A PAM Decomposition of Weak CPM Mason B. Wardle Brigham Young University - Provo Follow this and additional works at: Part of the Electrical and Computer Engineering Commons BYU ScholarsArchive Citation Wardle, Mason B., "A PAM Decomposition of Weak CPM" (2005). All Theses and Dissertations This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 A PAM DECOMPOSITION OF WEAK CPM by Mason B. Wardle A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical and Computer Engineering Brigham Young University August 2005

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4 Copyright c 2005 Mason B. Wardle All Rights Reserved

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6 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Mason B. Wardle This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Michael D. Rice, Chair Date Michael A. Jensen Date A. Lee Swindlehurst

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8 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the thesis of Mason B. Wardle in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Michael D. Rice Chair, Graduate Committee Accepted for the Department Michael A. Jensen Graduate Coordinator Accepted for the College Alan R. Parkinson Dean, Ira A. Fulton College of Engineering and Technology

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10 ABSTRACT A PAM DECOMPOSITION OF WEAK CPM Mason B. Wardle Department of Electrical and Computer Engineering Master of Science The Enhanced Flight Termination System uses weak CPM as its modulation scheme and a limiter-discriminator as its demodulation scheme. A PAM representation of weak CPM was developed which representation provided the necessary componenents to build a simplified PAM-based receiver that outperformed the EFTS limiter-discriminator, even in the presence of phase noise. The PAM representation also provided a new perspective into the negative characteristics of weak CPM.

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12 ACKNOWLEDGMENTS I would like to acknowledge first and foremost my Heavenly Father Who constantly provided me with motivation and energy to complete this work. He also gave me a small yet crucial burst of inspiration that sent me in the right direction to develop a PAM representation for weak CPM. Next, I would like to thank my wife for her love and support, who gave me motivation to work hard. She helped me continue when I was depressed and felt I could not continue. She has simply made my life better ever since she became a part of it. I would also like to thank my family for their help in urging me to be a good student and to work hard to be succesful. My father deserves special thanks for my success as he helped me learn to work. My mother has also been there for me as I struggled my way through school. Last but not least, I would like to mention the tremendous support I received while at BYU from the Electrical Engineering (EE) Department. First in this list is my professor and advisor, Professor Michael Rice, who so generously supported me, both financially and academically, as I pursued my master s degree. He also gave freely of his time to help me understand and make progress on my research as well as gave me the space and time I needed to ultimately make the findings I have. I would also like to recognize the support of so many of the professors in the EE department as well as the terrific staff, including Ann Tanner. Their friendship and help was greatly appreciated. Also not to be neglected are the many friends who are studying in the Electrical Engineering department including the folks in the Telemetry Lab.

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14 Contents Acknowledgments List of Tables List of Figures vi ix xi 1 Introduction Definition of CPM PAM Decomposition of CPM Exact Representation Approximate Representation PAM Representation of Weak CPM Weak CPM PAM Signal Representation The Need For a New Approach General PAM Decomposition of Weak CPM Derivation of PAM Representation Simplifying the Exact Representation PAM Representation of a Special Important Case Method for Forming a PAM Representation of Weak CPM Application to EFTS Power Spectrum of Weak CPM Using the PAM Representation Weak CPM Receivers Maximum Likelihood Detection ML Detection Using the Complex Exponential Representation vii

15 3.1.2 ML Detection Using the PAM Representation Sub-Optimum Detection Reduced-Complexity Detection Using the PAM Representation Reduced-Complexity Detection Using an FM Limiter-Discriminator Detector Application to EFTS CPM Modulator MLSE and Sub-Optimum Receivers FM Limiter-Discriminator EFTS Receiver Performance and Complexity Conclusion 47 A Derivation of a PAM Decomposition of CPM and Weak CPM 51 A.1 CPM A.2 Weak CPM A.3 Pulses That are Zero: the Full Derivation B Phase Noise and Phase-Lock Loop Models 63 Bibliography 70 viii

16 List of Tables 3.1 Receiver Complexity ix

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18 List of Figures 1.1 Weak Frequency and Phase Pulse EFTS PAM Pulses EFTS PAM Pulse Spectra EFTS Trellis EFTS Receivers Phase Noise Simulation Setup Distance Analysis BER Plots of Receiver Performance B.1 Phase Noise Generator xi

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20 Chapter 1 Introduction Most range safety programs use a Flight Termination System (FTS) to bring stray airborne test vehicles to crash at a preselected location, by bringing the vehicle into a state of zero lift and zero thrust. Methods used to meet this goal include parachute deployment and detonation of explosive charges that destroy the test vehicle [1]. The test range initiates flight termination by sending a radio signal on a dedicated channel to the stray test vehicle. The first FTS, brought into use in the 1950s, modulates a frequency modulation (FM) carrier with different frequency audio tones [2]-[4]. The aircraft completes the arm and terminate commands after the arrival of a predefined sequence of these tones from the ground-based transmitter [2]. The modified high-alphabet system [5] adds a security feature by using a predefined sequence of tone pairs to encode the commands. With the increase in flight altitudes came an incident where the terminate signal sent by one test range inadvertently terminated the flight of a vehicle at a nearby test range [1, 6]. In response to this incident, the Range Safety Group of the Range Commanders Council created a committee in April 2000 to enumerate the requirements of the next generation FTS that would search for techniques to deal with the aforementioned situation. In January 2002, this group chose bi-phase pulse-coded modulation/frequency modulation (PCM/FM) as the modulation for the next generation FTS, named Enhanced Flight Termination System (EFTS) [7]. The justification behind this choice of modulation included the important reason that the ground-based FTS transmitter (i.e., the current hardware infrastructure) required an AC-coupled input to the FM modulator [8]. The EFTS standard chose a more digital route for the EFTS waveform which opened the door for better security (3DES encryption) and a higher level of reliability (Reed-Solomon error control coding). 1

21 The selection of bi-phase (also known as Manchester) PCM/FM (also known as continuous-phase modulation or CPM) included the assumption that the airborne vehicle would use a FM limiter-discriminator for a demodulator because the FM limiter-discriminator has a history of reliability in aeronautical applications. Unfortunately, this FM limiterdiscriminator detector, (i.e., a simple FM-demodulator followed by a comparator) has a much poorer bit-error rate (BER) than an optimal receiver. An optimum receiver, on the other hand, requires a maximum-likelihood sequence detector and full phase coherency which is difficult to attain in the presence of excessive amounts of phase noise resulting from the high levels of shock and vibration typical in airborne vehicles. In spite of this difficulty, consider such an optimum receiver. One method for designing an optimal receiver utilizes the Viterbi algorithm to effect maximum likelihood sequence estimation (MLSE). This approach yields a much lower BER at the expense of increased complexity, which comprises part of the difficulty of taking an optimal approach in EFTS. Anderson, Aulin, and Sundberg suggested a handful of techniques that facilitate the construction of simplified receivers for CPM which receivers have sub-optimal performance [9, chapter 8]; however, they can perform better than an FM limiter-discriminator. Pierre Laurent also presented a method for simplified receiver design. He showed that CPM can be reformulated as a sum of several PAM pulses [10]. (His results were later extended to M-ary CPM by Mengali [11] 1 and to multi-h CPM by Perrins and Rice [13].) Using all of the pulses leads to MLSE detection; using fewer pulses results in a simplified receiver, but still with better expected performance than an FM demodulator. The design of a simplified PAM receiver for EFTS-based CPM that outperforms, and maintains roughly the same level of complexity of, the current EFTS receiver would be possible; 2 this design goal motivated my research. Before diving into an explanation of my research, a review of CPM and its PAM representation will be presented. 1 Mengali also developed a method to form a PAM representation of CPM when the modulation index is an integer. Another approach to forming a PAM representation for an integer modulation index can be found in [12] 2 This assumes that full phase coherency is achievable in the presence of such severe phase noise as that found in EFTS applications 2

22 1.1 Definition of CPM The complex-envelope of the transmitted CPM signal, s(t), is defined as ( ) t N s(t) = exp j2πh α(n)f(τ nt) dτ τ= n= (1.1) where f(t) is termed the frequency pulse and has a region of support (spans an interval of time) typically restricted to 0 < t < LT where T is the symbol time and L is the number of symbol times spanned by the frequency pulse. Also, α(n) represents the n-th binary data symbol and the modulation index, h, which dictates how much the phase changes for each symbol. (Note that the frequency pulse dictates how the frequency of the complex exponential changes as a function of time.) Reversing the order of the integral and the summation in (1.1) and performing integration yields s(t) = exp(j 2πh φ(t)) (1.2) where N φ(t) = α(n)g(t nt) t (N + 1)T (1.3) g(t) = n= t f(τ) dτ. (1.4) Three design parameters available in CPM are the choice of the frequency pulse and the choice of the symbol alphabet. The modulation index, h, is set depending on the application. Henceforth, the alphabet will be assumed to be binary. The choice of frequency pulse (or phase pulse, depending on the perspective), can dramatically affect the behavior of the CPM signal. The phase pulseis most often defined as: 0 t 0 g(t) = (1.5) 1 t LT 2 which implies that the phase pulse has an infinite length, although it is constant for t > LT. Weak CPM In rare cases, the frequency pulse is defined such that the phase pulse has a finite length (EFTS being one such case). Looking at (1.4) and (1.5), one can see that the 3

23 1 0.8 f(t) g(t) t/t Figure 1.1: The frequency and phase pulses for the EFTS weak CPM. (Note: the pulses have been normalized to unit peak amplitude for display.) frequency pulse must integrate to zero in such a case. The phase pulse is thus defined for this case as g(t) = 0 t 0 and t LT. (1.6) Figure 1.1 presents the plot of one example of such a frequency-pulse/phase-pulse pair. A frequency pulse of this type has been termed weak [9, page 64]. Thus, from this point forward, CPM based on such a frequency pulse will be termed weak CPM, whereas CPM based on the more common frequency pulse, whose phase pulse is defined in (1.5), will simply be termed CPM. The reason behind naming the frequency pulse weak comes from the effect a weak frequency pulse has on the frequency spectrum and on the minimum Euclidean distance of the CPM signal; these effects will be investigated in Sections 2.4 and 3.1, respectively. References [14]-[22] present some of the limited research completed for weak CPM. 4

24 1.2 PAM Decomposition of CPM The PAM representation presented by Laurent facilitates the design of simplified receivers since there has been extensive research performed on linear 3 modulations (modulations which include PAM); examples of this research include synchronization methods and optimal receivers [23] Exact Representation where Laurent showed that a binary CPM signal, s(t), can be reformulated as s(t) = C K (t) = N= L 1 i=0 2 L 1 1 K=0 a K (N)C K (t NT) (1.7) u(t + [i + β K (i)l]t) (1.8) u(t) = sin[ψ(t)] sin(πh) 2πhg(t) Ψ(t) = πh 2πhg(t LT) a K (N) = J A K(N) A K (N) = N n= t < LT t LT (1.9) (1.10) (1.11) L 1 α(n) α(n i) β K (i) (1.12) i=1 L 1 K = 2 i 1 β K (i) 0 K 2 L 1 1 (1.13) i=1 J = e jπh (1.14) and β K (i) is the i-th coefficient for the radix-2 decomposition of an integer K. (Appendix A presents a complete derivation of this PAM decomposition.) Equation (1.7) contains a sum of PAM pulses for each N. In Laurent s derivation, the sums of PAM pulses are divided into 3 PAM representations of CPM transform the non-linear CPM into a linear PAM modulation by pushing the non-linearity out of the modulation into the data symbols. The most important reason for making this representation is that it allows the design of a demodulator to perform sub-optimal detection as explained later. 5

25 groups. Each group consists of all PAM pulses that start in the interval NT < t < (N+1)T for a given value of N. This grouping becomes useful in understanding the derivation of the PAM representation in Chapter Approximate Representation Generally speaking, in a PAM decomposition, one pulse or perhaps a few pulses comprise most of the energy in the CPM signal. The use of only the most energetic PAM pulses provides a receiver with acceptable performance. This ease of approximation helps the PAM representation finds its niche: the PAM representation opens up a path to build a simplified receiver. (See reference [24].) The approximation gives a simple way to reduce complexity, while limiting the inevitable increase in BER relative to that of an optimal receiver. Simualtion results presented in Chapter 3 showed that a simplified PAM-based receiver greatly outperforms an FM demodulator PAM Representation of Weak CPM In order to build a simplified receiver, a PAM representation of weak CPM, which was not provided by Laurent, must be developed; my research has focused on such a development. Weak CPM is shown to have a PAM representation, just as CPM does. Such a representation provided a new perspective to the failings of weak CPM. Most importantly, this PAM representation gave a way to build a simplified EFTS receiver by following the work of Ghassan Kaleh [24]; this simplified receiver performed significantly better than an FM limiter-discriminator, even in the presence of phase noise that arises in EFTS applications. Chapter 2 derives a PAM representation for weak CPM. Chapter 3 shows how to build weak CPM receivers and presents simulation results for EFTS PAM-based receivers. 6

26 Chapter 2 Weak CPM PAM Signal Representation Laurent s 1986 paper did not consider the rarely used weak CPM. The whole idea of Laurent s work aimed to find a linear PAM decomposition, but Laurent s method applied directly to weak CPM yields a non-linear PAM decomposition. However, my work has derived a linear PAM representation for weak CPM using an approach to Laurent s work similar to that taken by Mengali and Morelli to create a PAM representation for M-ary CPM [11]. Section 2.1 demonstrates the need for an approach different than that of Laurent s. Section 2.2 derives the PAM representation of weak CPM; the end of this section presents a summary of an approach to calculate a general PAM representation for weak CPM Finally, Section 2.3 finds the PAM pulses for EFTS-based CPM, and Section 2.4 shows how discrete lines surface in the spectrum of weak CPM. 2.1 The Need For a New Approach A comparison of Laurent PAM decompositions of CPM and weak CPM demonstrates the need for a new approach in formulating a PAM representation of weak CPM. An extended example (used throughout this chapter) illustrates this need in a straightforward way as well as logically develops a new approach. Example for L=2 Part 1 Appendix A shows that exp [ jπhα(n)g(t nt) ] = u ( t + [L n]t ) + J α(n) u(t nt), (2.1) 7

27 with u(t) defined in (1.9). Equation (2.1) allows s(t), the CPM signal, to be expressed by the following equation: s(t) = J L 1 (N L) [ n= α(n) u ( t + [i + L N]T ) + J α(n i) u ( t + [i N]T )]. (2.2) i=0 (For a full derivation, consult Appendix A). Using the time interval NT < t < (N + 1)T for N = 0, 1, 2 and L = 2, equation (2.2) becomes: 0 < t < T : s(t) = J 2 n= α(n) [C 0 (t + 2T) + J α( 1) C 0 (t + T) + J α(0)+α( 1) C 0 (t) + J α(0) C 1 (t) ] T < t < 2T : s(t) = J 2 n= α(n) [J α( 1) C 0 (t + T) + J α(0)+α( 1) C 0 (t) + J α(1)+α(0)+α( 1) C 0 (t T) + J α(1)+α ( 1) C 1 (t T) ] 2T < t < 3T : s(t) = J 2 n= α(n) [J α(0)+α( 1) C 0 (t) + J α(1)+α(0)+α( 1) C 0 (t T) + J α(2)+α(1)+α(0)+α( 1) C 0 (t 2T) + J α(2)+α(0)+α( 1) C 1 (t 2T) ]. (2.3) The coefficents of shifted versions of C K terms for a given value of K are all the same. For example, the coefficient for C 0 (t) is J α(0)+α( 1) for 0 < t < 3T and the coefficient for C 0 (t + T) is J α( 1) for 0 < t < 2T. ( C 0 (t + T) is zero for t > 2T. ) This characteristic ultimately allows the reformation of CPM into linear PAM. This reformation, found in (1.7) and repeated here for convenience, expands (2.3) to include all N: s(t) = where C K (t) = N= L 1 i=0 2 L 1 1 K=0 a K (N)C K (t NT) (2.4) u ( t + [i + β K (i)l]t ) (2.5) u(t) = sin[ Ψ(t) ] sin(πh) 2πhg(t) Ψ(t) = πh 2πhg(t LT) 8 t < LT t LT (2.6) (2.7)

28 a K (N) = J A K(N) (2.8) N L 1 A K (N) = α(n) α(n i) β K (i) (2.9) n= i=1 L 1 K = 2 i 1 β K (i) 0 K 2 L 1 1 (2.10) i=1 J = e jπh. (2.11) On the other hand, applying weak CPM to (2.2) for N = 0, 1, 2 yields: 0 < t < T : s(t) = [ C 0 (t + 2T) + J α( 1) C 0 (t + T) + J α(0)+α( 1) C 0 (t) + J α(0) C 1 (t) ] T < t < 2T : s(t) = [ C 0 (t + T) + J α(0) C 0 (t) + J α(1)+α(0) C 0 (t T) + J α(1) C 1 (t T) ] (2.12) 2T < t < 3T : s(t) = [ C 0 (t) + J α(1) C 0 (t T) + J α(2)+α(1) C 0 (t 2T) + J α(2) C 1 (t 2T) ]. Now in this case, unlike generic CPM shown in (2.3), the coefficients of the C K terms change with each N. For example, consider again C 0 (t). For 0 < t < T, C 0 (t) has a coefficient of J α(0)+α( 1) ; for T < t < 2T, this coefficient becomes J α(0). This makes it impossible, using this form, to create a linear PAM expression for s(t) for all t. Thus, Laurent s PAM approach applied to weak CPM does not yield a linear result, and so another approach must be taken to find a linear PAM representation of weak CPM. 2.2 General PAM Decomposition of Weak CPM In order to derive a PAM decomposition for weak CPM, the signal will be split into the product of two different CPM signals. This method of rewriting CPM as the product of two CPM signals can also be found in [11] Derivation of PAM Representation Consider a frequency pulse, f(t), of length LT, that integrates to zero. In calculating the PAM decomposition for CPM based on this frequency pulse, the first step is to 9

29 divide the frequency pulse into two pulses: f(t) = f + (t) + f (t) (2.13) where f(t) f(t) > 0 f + (t) = 0 otherwise f(t) f(t) < 0 f (t) =. 0 otherwise (2.14) The next step is to calculate the phase pulses g + (t) and g (t), which are equal to the integral of f + (t) and the integral of f (t), respectively. Thus g(t), which is the integral of f(t), as defined in (1.4), can be expressed as the sum of two shorter-length phase pulses: g(t) = g + (t) + g (t). (2.15) Clearly, if g + (LT) = 1/2, then g (LT) = 1/2, since for weak CPM, g(lt) = 0. The PAM representation of g + (t) can be formulated using (2.4) and assuming g + (t) is the same length as g(t). With a slight modification to Laurent s original development, we can in a similar manner find the PAM representaion for g (t). To find the PAM representaion for g (t), note that CPM based on a frequency pulse that integrates to 1/2 is just like CPM with a frequency pulse that integrates to A but has each α i negated. Making this modification to (2.9), (2.4) becomes s (t) = N= 2 L 1 1 K=0 a K (N)D K(t nt) (2.16) where denotes complex-conjugation and the D K s are the PAM pulses for g (t). (For a derivation of this modification see Appendix A.) Recalling (1.2) and using f(t) as defined in (2.14), s(t) can now be rewritten as s(t) = exp j[ n= α(n)g + (t nt)] exp j[ 10 n= α(n)g (t nt)]. (2.17)

30 Applying Laurent s PAM decomposition to both terms in this product yields s(t) = N= 2 L 1 1 K=0 Example for L = 2 Part 2 a K (N)C K (t NT) N = 2 L 1 1 K =0 a K (N )D K (t N T). (2.18) Consider an expansion of the product of sums in (2.18), where the group of PAM pulses that are nonzero in the interval 0 < t < T will be considered. Thus (2.18) for 0 < t < T is: s(t) = a 0 (0) a 0(0)C 0 (t)d 0 (t) + a 0 (0) a 1 (0)C 0(t)D 1 (t) + a 0 (0) a 0( 1)C 0 (t)d 0 (t + T) + a 0 (0) a 0 ( 2)C 0(t)D 0 (t + 2T) + a 1 (0) a 0(0)C 1 (t)d 0 (t) + a 1 (0) a 1 (0)C 1(t)D 1 (t) + a 1 (0) a 0 ( 1)C 1(t)D 0 (t + T) + a 1 (0) a 0 ( 2)C 1(t)D 0 (t + 2T) + a 0 ( 1) a 0 (0)C 0(t + T)D 0 (t) (2.19) + a 0 ( 1) a 1 (0)C 0(t + T)D 1 (t) +a 0 ( 1) a 0 ( 1)C 0(t + T)D 0 (t + T) +a 0 ( 1) a 0 ( 2)C 0(t + T)D 0 (t + 2T) + a 0 ( 2) a 0 (0)C 0(t + 2T)D 0 (t) + a 0 ( 2) a 1(0)C 0 (t + T)D 1 (t) +a 0 ( 2) a 0 ( 1)C 0(t + 2T)D 0 (t + T) +a 0 ( 2) a 0 ( 2)C 0(t + 2T)D 0 (t + 2T). The terms of (2.19) in bold face are shifted versions of other pulses already present in (2.19). For example if q 0 (t) = C 0 (t)d 0 (t) and d 0 (0) = a 0 (0)a 0 (0), then a 0( 2) a 0 ( 2)C 0(t + 2T)D 0 (t + 2T) = d 0 ( 2)q 0 (t + 2T). This term came into the picture in the time interval 2T < t < T. Any pulse q i (t + NT), where its region of support began in the interval 11

31 NT < t < (N + 1)T N 0, will be removed from this group and placed in the group of pulses that also started in the interval NT < t < (N + 1)T N 0. Thus the terms in bold face belong in different groups; the first three belong in the group of pulses that start in the interval T < t < 0 and the fourth pulse belongs with pulses that start in the interval 2T < t < T. (Laurent also regrouped the PAM pulses along the same lines; see Section for further explanation.) Thus neglecting these terms, s(t) for the group of PAM pulses for the interval 0 < t < T can be written as: s(t) = a 0 (0) a 0 (0)C 0(t)D 0 (t) + a 0 (0) a 1 (0)C 0(t)D 1 (t) + a 0 (0) a 0 ( 1)C 0(t)D 0 (t + T) + a 0 (0) a 0 ( 2)C 0(t)D 0 (t + 2T) + a 1 (0) a 0 (0)C 1(t)D 0 (t) + a 1 (0) a 1 (0)C 1(t)D 1 (t) + a 1 (0) a 0 ( 1)C 1(t)D 0 (t + T) (2.20) + a 1 (0) a 0 ( 2)C 1(t)D 0 (t + 2T) + a 0 ( 1) a 0(0)C 0 (t + T)D 0 (t) + a 0 ( 1) a 1 (0)C 0(t + T)D 1 (t) + a 0 ( 2) a 0(0)C 0 (t + 2T)D 0 (t) + a 0 ( 2) a 1 (0)C 0(t + T)D 1 (t). Thus, there are twelve unique PAM pulses in weak CPM with L = Simplifying the Exact Representation and Calculating the Number of PAM Pulses At no point in the preceding derivation were approximations made. Therefore, (2.20) is an exact representation of weak CPM for L = 2, which equation can be simplified without deferring to an approximate representation. 12

32 Example for L = 2 Part 3 An examination of (2.20) shows that some of the pseudo-symbol products are equal. Since these products are coefficients of the PAM pulses, any products that are equal allow the combining of the PAM pulses corresponding to these products, thus reducing the number of PAM pulses without compromising the exact PAM representation in (2.20). This reduction will be performed using (2.8) which states that a K (N) = J A K(N) ; this means the simplification process will simply involve taking the difference between sums of various combinations of α(n). The pseudo-symbol products are: a 0 (0) a 0 (0) = 1 (2.21) a 0 (0) a 1(0) = J α( 1) (2.22) a 0 (0) a 0 ( 1) = Jα(0) (2.23) a 0 (0) a 0( 2) = J α(0)+α( 1) (2.24) a 1 (0) a 0 (0) = J α( 1) (2.25) a 1 (0) a 1(0) = 1 (2.26) a 1 (0) a 0 ( 1) = Jα(0) α( 1) (2.27) a 1 (0) a 0( 2) = J α(0) (2.28) a 0 ( 1) a 0 (0) = J α(0) (2.29) a 0 ( 1) a 1 (0) = J α(0)+α( 1) (2.30) a 0 ( 2) a 0 (0) = J α(0) α( 1) (2.31) a 0 ( 2) a 1 (0) = J α(0). (2.32) Clearly a 0 (0) a 0(0) = a 1 (0) a 1(0) (2.33) a 0 (0) a 0 ( 1) = a 1(0) a 0 ( 2) (2.34) a 0 ( 1) a 0(0) = a 0 ( 2) a 1(0). (2.35) 13

33 This results in the simplification of s(t) for 0 < t < T, where the newly combined pulses are displayed in bold face: s(t) = [C 0 (t)d 0 (t) + C 1 (t)d 1 (t)] + J α( 1) C 0 (t)d 1 (t) + J α(0) [C 0 (t)d 0 (t + T) + C 1 (t)d 0 (t + 2T)] + J α(0)+α( 1) C 0 (t)d 0 (t + 2T) + J α( 1) C 1 (t)d 0 (t) (2.36) + J α(0) α( 1) C 1 (t)d 0 (t + T) + J α(0) [C 0 (t + T)D 0 (t) + C 0 (t + T)D 1 (t)] + J α(0)+α( 1) C 0 (t + T)D 1 (t) + J α(0) α( 1) C 0 (t + 2T)D 0 (t). Maximum Number of PAM Pulses Just as done in transforming (2.3) into (2.4), (2.36) can be generalized to include all time as follows: s(t) = P 1 N= K=0 d K (N)q K (t NT) (2.37) where P = 3 2, d K (N) represents the new pseudo-symbols, and q K (t) is defined as the products of C K (t) and D K (t), each shifted by various amounts. As an example of q K (t), one might define the following: q 0 (t) = C 0 (t)d 0 (t) + C 1 (t)d 1 (t) q 1 (t) = C 0 (t)d 1 (t). (2.38) (In actuality, q 0 (t) and q 1 (t) will be assigned different pulse-product combinations than those in (2.38); criteria for such assignment will be give later on.) The pseudo-symbols in (2.36) are functions only of the current and previous data symbols and their coefficients: d K (N) = J ν(n)α(n)+ν(n 1)α(n 1) (2.39) where α(i) represents the binary data symbols and the coefficients ν(i) {±1, 0} are taken from a ternary alphabet. This gives 3 2 possible data-symbol combinations (as opposed to 14

34 the number of different values the pseudo-symbols can assume which is much smaller). If an example were completed for L = 3, similar to the example just worked, the pseudosymbols would end up being d K (N) = J ν(n)α(n)+ν(n 1)α(n 1)+ν(n 2)α(n 2) (2.40) yielding 3 3 possible pseudo-symbols. For general L, each of the pseudo-symbols are of the general form d K (N) = L 1 i=0 J ν(n i)α(n i). (2.41) It can thus be seen that there are 3 L possible pseudo-symbols since there are L different binary data symbols in the product in (2.41) and since ν(i) are taken from a ternary alphabet. Therefore, weak CPM requires no more than 3 L PAM pulses for an exact PAM representation PAM Representation of a Special Important Case The foregoing development uses a generic weak frequency pulse. Greater reduction in the number of pulses arises when the weak frequency pulse for an important case is used. Pulses That are Zero Many of the PAM pulses become zero if the frequency pulse, f(t), is defined in a way that represents an important case of weak CPM. In the following derivation, only the intervals to which each pulse is time-limited will be considered as opposed to the actual behavior of the pulse during that interval. Consider a definition of a frequency pulse, f(t), of length LT, that integrates to zero: f(t) > 0 f(t) < 0 0 < t < κlt κlt < t < LT, (2.42) where 0 < κ < 1. Referring to (2.14) it is seen that f + (t), which integrates to 1/2, is zero for κlt < t < LT, and f (t), which integrates to -1/2, is zero for 0 < t < κlt. g + (t) 15

35 and g (t) thus become: 0 t < 0 g + (t) = 1 t > κlt 2 0 t < κlt g (t) = t > LT 1 2. (2.43) With g + (t) and g (t) in hand, Ψ + (t) and Ψ (t) can both be determined which in turn determines u + (t) and u (t). u + (t) 0 u (t) 0 0 < t < (κl + L)T κlt < t < 2LT. (2.44) The latter equation will ultimately show that some PAM pulses are zero. PAM pulse lengths Using (2.44) and (2.5) (the latter repeated in (2.45) for convenience), the length of each C K (t) and D K (t) can be calculated. C K (t) = L 1 i=0 u + (t + [i + β K (i)l]t) (2.45) L 1 D K (t) = u (t + [i + β K (i)l]t) (2.46) i=0 L 1 K = 2 i 1 β K (i) 0 K 2 L 1 1 (2.47) i=1 For general L, consider the intervals to which C K (t) and D K (t) are time-limited: C < t < (κl + 1)T (2.48) D 0 0 κlt < t < (L + 1)T (2.49) C < t < (κl 1)T (2.50) D 1 0 κlt < t < (L 1)T (2.51) C 2, C < t < (κl 2)T (2.52) D 2, D 3 0 κlt < t < (L 2)T (2.53) C 4, C 5, C 6, C < t < (κl 3)T (2.54) 16

36 D 4, D 5, D 6, D 7 0 κlt < t < (L 3)T (2.55). (2.56) C 2 /2,...,C L 1 2 L < t < (κl (L 1))T (2.57) D 2 /2,...,D L 1 2 L κlt < t < (L (L 1))T. (2.58) Example for L = 2 Part 4 Consider once again the example for L = 2 by recalling (2.36): s(t) =[C 0 (t)d 0 (t) + C 1 (t)d 1 (t)] + J α( 1) C 0 (t)d 1 (t) + J α(0) [C 0 (t)d 0 (t + T) + C 1 (t)d 0 (t + 2T)] + J α(0)+α( 1) C 0 (t)d 0 (t + 2T) + J α( 1) C 1 (t)d 0 (t) (2.59) + J α(0) α( 1) C 1 (t)d 0 (t + T) + J α(0) [C 0 (t + T)D 0 (t) + C 0 (t + T)D 1 (t)] + J α(0)+α( 1) C 0 (t + T)D 1 (t) + J α(0) α( 1) C 0 (t + 2T)D 0 (t). At this point (2.48) is used to find PAM pulses that are zero. Defining the weak phase pulse as found in (2.43), it can be shown that C < t < (2κ + 1)T D 0 0 2κT < t < 3T C < t < (2κ 1)T D 1 0 2κ < t < T. (2.60) (2.61) 17

37 The results in (2.60) and (2.61) cause many of the pulses in (2.59) go to zero thus reducing the number of pulses in (2.36) to 4: C 0 (t)d 0 (t + T) + C 1 (t)d 0 (t + 2T) 0 C 0 (t)d 0 (t + 2T) 0 max[0, (2κ 1)T] < t < min[(2κ + 1)T, 2T 0 < t < T C 0 (t)d 0 (t) + C 1 (t)d 1 (t) 0 2κT < t < (2κ + 1)T C 0 (t)d 1 (t) 0 2κT < t < T. (2.62) Thus the final PAM pulses for this special case of weak CPM are as follows: q 0 (t) = C 0 (t)d 0 (t + T) + C 1 (t)d 0 (t + 2T) q 1 (t) = C 0 (t)d 0 (t + 2T) q 2 (t) = C 0 (t)d 1 (t) (2.63) q 3 (t) = C 0 (t)d 0 (t) + C 1 (t)d 1 (t). It can be seen from (2.36) that the pseudo-symbols corresponding to these PAM pulses in (A.73) are a 0 (n) = J α(n) a 1 (n) = J α(n)+α(n 1) a 2 (n) = J α(n 1) (2.64) a 3 (n) = J 0 = 1. A similar process can be applied when L is larger which will yield a similarly extensive reduction in the number of pulses. It turns out that for this special case, the number of pulses reduces to 2L if κ = 1 (a common occurrence), which is comparable L to the number of PAM pulses in the PAM decomposition of binary CPM which is equal to 2 L 1. Thus the number of PAM pulses increases linearly with L for weak CPM but increases exponentially with L for CPM. Indeed, 2L = 2 L 1 for L = 4 but 2L < 2 L 1 for L > 4! 18

38 2.2.4 Method for Forming a PAM Representation of Weak CPM Considerable ground has been covered in order to arrive at a PAM representation for weak CPM. The final results are summarized in this section. The general PAM representation is virtually identical to Laurent s orginal PAM expression: s(t) = P 1 N= K=0 d K (N)g K (t NT). (2.65) A substantial amount of work has been put into finding a closed form for the pseudosymbols, d K (N), and the PAM pulses, g K (t). For example, Mengali presented an extensive algorithm used to calculate the pseudo-symbols and PAM pulses. In the case of weak CPM this algorithm, which was used to derive the PAM representation, leaves open the question of the length of each pulse and of the form of the pseudo-symbols. This extra level of complexity beyond that required to form Laurent s PAM decomposition, is no greater than the complexity required to calculate the pulses by hand. An approach to computing the PAM pulses and pseudo-symbols of a complexity comparable to Mengali s algorithm is therefore a brute-force approach. Once the pulses are calculated, simplification of the exact representation can then be completed. As discussed in the process of the derviation, at most, there will be 3 L PAM pulses for a given representation. A further reduction in the number of pulses comes when weak CPM uses an important common form of weak frequency pulses, as discussed in Section Indeed, all indications suggest that the number of PAM pulses for weak CPM can be reduced to as few as 2L, which is comparable to (and less than for larger values of L) the number of pulses in Laurent s PAM representations of CPM. The algorithm chosen is as follows. First, calculate the PAM pulses C K (t) (based on g + (t)) and D K (t) (based on g (t)). This requires that g + (t) and g (t) be calculated first, 19

39 and therefore also requires formation of f + (t) and f (t), using the following equations: f(t) f(t) > 0 f + (t) = 0 otherwise (2.66) f(t) f(t) < 0 f (t) = 0 otherwise g ± (t) = t f ± (τ) dτ. (2.67) This allows the calculation of Ψ + (t) and Ψ (t) and subsequently u + (t) and u (t), the latter two functions being the basis of the PAM representation: g + (t) 0 < t < LT Ψ + (t) = (2.68) 2πh πhg + (t) LT < t < 2LT and g (t) Ψ (t) = 2πh + πhg (t) 0 < t < LT LT < t < 2LT (2.69) u + (t) = sin[ψ+ (t)] sin(πh) (2.70) u (t) = sin[ψ (t)] sin(πh). (2.71) Finally, the PAM pulses based on u + (t) and u (t) can be formed: C K (t) = L 1 i=0 u + (t + [i + β K (i)l]t) (2.72) L 1 D K (t) = u (t + [i + β K (i)l]t). (2.73) i=0 β K (i) is the i-th coefficient for the radix-2 decomposition of an integer K, as seen in the following: L 1 K = 2 i 1 β K (i) 0 K 2 L 1. (2.74) i=1 20

40 expanded: s(t) = Next, the product of the PAM representations of two CPM signals is formed and N= 2 L 1 1 K=0 a K (N)C K (t NT) N = 2 L 1 1 K =0 a K (N )D K (t N T). (2.75) New pulses are thus formed as products of C K (t+nt) and D K (t+n T) for all possible permutations of K, K, N, and N, as demonstrated in (2.19). A particular bit time is chosen and any pulses whose region of support began in previous bit times will be discarded, as explained in Section in conjuction with (2.19). Pulses with identical pseudo-symbol coefficients are grouped to effect simplifications possible regardless of the frequency pulse used in defining the weak CPM signal. At this point, there will be no more than 3 L unique PAM pulses. If f(t) is as defined in (2.42), further simplification is possible. The time-limited nature of C K (t) and D K (t), as shown in (2.48), can be examined along with (2.42) to discover and eliminate pulses that are zero; these two equations are repeated here for convenience: C < t < (κl + 1)T D 0 0 κlt < t < (L + 1)T C < t < (κl 1)T D 1 0 κlt < t < (L 1)T C 2, C < t < (κl 2)T D 2, D 3 0 κlt < t < (L 2)T (2.76) C 4, C 5, C 6, C < t < (κl 3)T D 4, D 5, D 6, D 7 0 κlt < t < (L 3)T. C 2 L 1 /2,...,C 2 L D 2 /2,...,D L 1 2 L f(t) > 0 f(t) < 0 0 < t < (κl (L 1))T κlt < t < (L (L 1))T 0 < t < κlt κlt < t < LT. (2.77) 21

41 This elimination results in limiting the number of PAM pulses to as few as 2L where 2L is reached when κ = 1/L. 2.3 Application to EFTS The EFTS signal has the following design requirements. The peak frequency deviation, f d, is 60 khz. The bit rate, R b, is 7200 bits per second (bps). The modulation type is continuous-phase modulation, using a Manchester frequency pulse. Before the frequency pulse train is input into an FM modulator to create the CPM signal, EFTS dictates the frequency-pulse train is to be filtered by a 4-pole low-pass Bessel filter with cut-off frequency F c = 15 khz. These parameters need to be converted into the model used for simulating the CPM modulator modeled in Chapter 1. The model requires the modulation index, h, defined in [23, section 4.3.3] as h = 2f d T (2.78) where T is the symbol time, which for a binary modulation is equal to the bit time, T b. The bit time can be calculated as T b = 1 bit symbol ( R bits ) = 1 ; (2.79) b R second b therefore, h = (2.80) The model chosen to represent the low-pass Bessel filter was a Gaussian low-pass filter with F c = 15 khz. This choice was made for two reasons. First, the Gaussian lowpass filter is used as the pre-modulation filter for GMSK [25] where GMSK is CPM based on an NRZ frequency pulse, filtered by a low-pass Gaussian filter with a time-bandwidth BT, and with h = 1. This is interesting because GMSK provides approximately the same 2 spectral shaping as the recommended EFTS pre-modulation filter. Second, the Gaussian low-pass filter produces a somewhat more tractable mathematical expression for the frequency, and phase pulses, compared to when using a Bessel filter. 22

42 The frequency and phase pulses may be expressed as f(t) = χ [ ( ) t LT/2 2πσh 2Q σ ( ) ( )] t (L 1)T/2 t (L + 1)T/2 Q Q σ σ [ ( g(t) = χπh σ e 1 2( t (L+1)T/2 σ ) 2 e ((L+1)T) 2σ ) 2) + ( ) ( ) (L + 1)T t (L + 1)T/2 2π t Q 2σ σ 2 + ( ) (L + 1)T (L + 1)T 2π Q 2 2σ ( + σ e 1 2( t (L 1)T/2 σ ) 2 e ((L 1)T) 2σ ) 2) + ( ) ( ) (L 1)T t (L 1)T/2 2π t Q 2σ σ 2 + ( ) (L 1)T (L 1)T 2π Q 2 2σ ( ) 2σ e 1 2( t LT/2 σ ) 2 LT e ( 2σ ) 2 2 ( 2π t LT ) ( ) t LT/2 Q 2 2π LT2 ( 2σ σ Q LT )] 2 2σ (2.81) (2.82) where Q(x) = 1 e u2 /2 du, (2.83) 2π x ln 2 σ = 2πB, (2.84) and χ is a constant required to produce the desired frequency deviation. As stated, the modulation for this model is similar to Gaussian minimum-shift keying (GMSK). Recall that L is the number of symbol periods spanned by the frequency pulse. Tsai and Lui [26] give the following approximation for L in GMSK as a function of BT : L 1 BT (2.85) 23

43 where denotes the integer closest to and larger than ( ). For the model used by EFTS, BT = B T b 15 khz = R (2.86) b 15 khz = 7.2 kbps = It turns out that a slightly different approximation should be used for EFTS, as can be seen in Figure 1.1, which is a Manchester frequency pulse filtered by a Gaussian low-pass filter with BT = This figure shows that the pulse spans somewhere between one and two symbol times. Thus for this case, L can be calculated as, 1 L 2 = 2. (2.87) BT Since L = 2 and κ = 1 2 (as used in (2.77)), (2.62) shows that q 0 (t + nt) = C 0 (t + nt)d 0 (t + (n + 1)T) 0 0 < t < 2T, q 1 (t + nt) = C 0 (t + nt)d 0 (t + (n + 2)T) 0 0 < t < T, (2.88) and q 2 (t + nt) = C 0 (t + nt)d 0 (t + nt) 0 T < t < 2T are the appropriate PAM pulses for an exact representation of EFTS-based CPM, shown in Figure 2.1 with their spectra shown in Figure 2.2. It can be seen from (2.64) that the respective pseudo-symbols for these PAM pulses are a 0 (n) = J α(n) a 1 (n) = J α(n)+α(n 1) (2.89) a 2 (n) = J 0 = 1. Thus, the PAM representation of EFTS PAM is s(t) = (e jπhα(n) q 0 (t NT) + e jπh[α(n)+α(n 1)] q 1 (t NT) + q 2 (t NT). (2.90) N= 2.4 Power Spectrum of Weak CPM Using the PAM Representation The PAM representation of weak CPM aids in understanding the undesirable frequency spectrum characteristics in weak CPM. Anderson, Aulin and Sundberg state that 24

44 2 q 0 (t) t/t 1 q 1 (t) t/t 2 q 2 (t) t/t Figure 2.1: The three PAM pulses q 0 (t), q 1 (t), and q 2 (t) in the PAM representation of the EFTS waveform. q 0 (t) q 1 (t) q 2 (t) magnitude (db) ft (cycles/bit) magnitude (db) ft (cycles/bit) magnitude (db) ft (cycles/bit) Figure 2.2: The spectra of the three PAM pulses q 0 (t), q 1 (t), and q 2 (t) in the PAM representation of the EFTS waveform. 25

45 weak CPM has discrete lines in its power spectral density (PSD). They further state that these lines are undesirable, because they represent wasted transmit power since they contain no information [9, page 154]. A periodic function has discrete lines, or impulses, in its Fourier Transform; the magnitude of each of these impulses is proportional to the Fourier-series coefficients of the periodic function [27, Section 4.2]. Thus, if an autocorrelation function has a periodic component, the PSD, which is the Fourier transform of the autocorrelation function, will have discrete lines. This understanding combined with the PAM representation provides a way to discover the presence of discrete lines in weak CPM, different from that found in [9]. As seen in (2.36), when L = 2 one of the PAM components is symbol-independent, which is deterministic; such a pulse surfaces in the PAM representation of weak CPM regardless of the value of L. A function made up of such a deterministic pulse repeated at regular intervals is periodic. Since one of the PAM pulses of weak CPM is always symbol-independent, (2.65) shows that there is a time-shifted version of this symbol-independent pulse in every symbol interval, thus giving s(t) a periodic component. The presence of this periodic component also implies the presence of a periodic component in the autocorrelation function, ultimately resulting in discrete lines in the PSD. Example for L = 2 Part 5 Consider the symbol-independent pulse associated with the L = 2 example: q i (t) = C 0 (t)d 0 (t) + C 1 (t)d 1 (t) (2.91) where i will be chosen based on the number of pulses that are present in the PAM representation. In forming the autocorrelation of the CPM signal, φ ss (t, t + θ) = E {s(t)s (t + θ)}, (2.92) q i (t) will remain symbol-independent only when placed in a product with shifted versions of itself. 26

46 Assuming the frequency pulse is defined as f(t) > 0 0 < t < LT 2 (2.93) LT f(t) < 0 2 < t < LT which is of the form of (2.42), q i (t) is time-limited to T < t < 2T. The periodic pulse present in the autocorrelation in this special case of L = 2 is therefore φ ss,periodic (t, t + θ) = N q i (t + NT){q i (t + θ + NT) + q i (t + θ + [N + 1]T)} (2.94) where 0 θ < T. The region of support of this periodic pulse is NT < t < (N + 1)T, (2.95) and so has a period of T. The autocorrelation function, φ ss (t, t+θ), represents the periodic part of a cyclo-stationary random process, and thus φ ss,periodic (t, t + θ) makes up a part of a cyclo-stationary random process. In order to calculate the PSD by taking the Fourier transform of the autocorrelation function, the autocorrelation function must represent a stationary random process (i.e. it must be a function of only one variable). The common practice used to convert the autocorrelation function of a cyclo-stationary random process into the autocorrelation function of a stationary random process is to average over one period of the autocorrelation function [23]. Thus φ ss,periodic (θ) = 1 T (N+1)T NT φ ss,periodic (t, t + θ) dt (2.96) represents a stationary random process. The discrete lines due to (2.94) can thus be calculated using the following equation [27], X(f) = b k δ(f 1/T) k= where b k = 1 (2.97) φ ss,periodic (θ)e j k T θ dθ. T T The latter part of (2.97) represents the Fourier series coefficients of φ ss,periodic (θ). A PAM representation for weak CPM has now been developed; the next step is to see how this PAM representation can be used to build a PAM-based receiver and how to predict the performance of such a receiver, and finally, predict and simulate the receiver s performance. 27

47 28

48 Chapter 3 Weak CPM Receivers The main interest in a PAM representation of CPM arises in the design of the receiver, which seeks to detect the transmitted data sequence from the received signal r(t): r(t) = s(t) + w(t). (3.1) This received signal consists of the transmitted signal, s(t), plus noise w(t), modeled as additive white Gaussian noise (AWGN). An optimal way, in terms of bit-error rate (BER), to effect this detection is through the use of a maximum likelihood sequence detector (MLSD), which is a detector that performs maximum likelihood sequence estimation (MLSE). This detector requires a sometimes prohibitively high complexity; it is therefore desirable to use a sub-optimum detector which requires a lower level of complexity. The use of a PAMbased representation of CPM is one method that allows reduction in the complexity of the receiver. (For other methods, see [9, chapter 8].) The act of reformulating weak CPM as a PAM signal does not alone provide for a reduced complexity receiver. Indeed, a receiver built using all of the PAM pulses to detect the signal is just another formulation of MLSE [24], which is of roughly the same complexity as the MLSE detector for CPM described in [9, page 249]. On the other hand, using fewer PAM pulses than the number required for an exact representation (i.e. using an approximate representation) does decrease complexity without a significant drop in BER; even a receiver using only the most energetic PAM pulse greatly outperforms an FM limiter-discriminator detector. Section 3.1 reviews maximum-likelihood (ML) detectors and Section 3.2 outlines the design of sub-optimum detectors. Section 3.3 applies the results of the first two sections 29

49 to the EFTS waveform. Finally, Section 3.4 presents simulation results and complexity comparisons of the various EFTS detectors. 3.1 Maximum Likelihood Detection In order to understand how a MLSD works, it will be helpful to describe CPM using a phase-state trellis, which is defined based on the phase states, correlative states, and state transition of CPM. States In weak CPM, the trellis arises when the frequency pulse spans more than one symbol time. Within the trellis structure, there are phase states, which represent accumulated phase due to previous data symbols which are no longer transitioning; in weak CPM, there is only one phase state, since the accumulated phase is zero. There are also correlative states based on previous symbols that are still transitioning; there are thus 2 L 1 (M L 1 for M-ary CPM) possible correlative states. The correlative states and the phase states are combined into a state vector; each trellis state is described by a unique state vector. There are two paths leaving each state (M paths for M-ary CPM). The state transition, which is based on the current symbol, describes these paths; the current data symbol determines to which of the two possible states these paths lead. Using the state vector and the state transition, a trellis can be constructed to show all possible states and state transitions. This will be done for EFTS later on in this chapter ML Detection Using the Complex Exponential Representation A MLSD searches for the path through the phase state trellis which most closely represents the path taken by the transmitted signal. Most closely resembles in this case (signal+awgn) means that the signal representing the path chosen as the most likely path is less different to the received signal in terms of squared Euclidean distance than a signal representing any other path through the trellis. A metric that performs this search is { [ arg max ln pr(t) α (r(t) α) ]} { } = arg min r(t) s(t, α) 2 dt α α where α represents a possible transmitted sequence. (3.2) (Equation 3.2 is simply the loglikelihood function for some signal in the AWGN channel.) The squared term in (3.2) 30

50 is expanded and only the pieces that vary with α are considered. This yields an equation equivalent to (3.2): arg max α { } Λ( α) = arg max Re r(t)s (t, α) dt. (3.3) α This metric assumes the whole received signal is in hand before a decision is made. The Viterbi Algorithm, on the other hand, calculates this metric piece-meal so as to enable the making of decisions as the signal arrives. The Viterbi Algorithm The Viterbi Algorithm splits up the decision by incorporating newly received information one symbol time at a time as follows: arg max α Re{Λ j (n; α)} = arg max λ m,i (n; α n ) = α (n+1)t nt { } Λ i (n 1; α) + Re[λ m,i (n; α n )] (3.4) r(t)s m (t, α n) dt (3.5) where α n = {α(n), α(n 1),...,α(n [L 1])} is a possible data sequence based on the symbols from time n (L 1) up to and including time n. Two comments about (3.4) will be instructive. First, the integral in (3.4) simply represents a matched filtering of r(t) by filters matched to the possible CPM waveforms in the interval nt < t < (n + 1)T. The second comment regards the metrics uses in (3.4). λ m,i is the recursive metric and Λ is the cumulative metric [9, page 249]. Λ represents the metric for a given state, and since there are 2 L 1 trellis states, there are 2 L 1 different Λ metrics, thus i and j in the subscripts of (3.4) each take on a different value in the range 0 i, j 2 L 1 1. Also, λ m,i represents the metrics for each of the two paths leaving each of the states. Equation (3.4) shows that λ m,i is a function of the previous L 1 symbols and the current symbol, so there are 2 L different recursive metrics to be calculated each symbol time. In order to allow for 2 L combinations, m ranges as 0 m 1, since i is in the range 0 i 2 L 1 1; i.e., since there are 2 L 1 trellis states and there are 2 paths leaving each state, there must be 2 L different paths and so there must be an equal number of metrics to describe these paths. 31