Reduced Complexity Detection Methods for Continuous Phase Modulation

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Reduced Complexity Detection Methods for Continuous Phase Modulation Erik Samuel Perrins Brigham Young University - Provo Follow this and additional works at: Part of the Electrical and Computer Engineering Commons BYU ScholarsArchive Citation Perrins, Erik Samuel, "Reduced Complexity Detection Methods for Continuous Phase Modulation" (2005). All Theses and Dissertations This Dissertation 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 REDUCED COMPLEXITY DETECTION METHODS FOR CONTINUOUS PHASE MODULATION by Erik Samuel Perrins A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering Brigham Young University December 2005

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4 Copyright c 2005 Erik Samuel Perrins All Rights Reserved

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6 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a dissertation submitted by Erik Samuel Perrins This dissertation 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 Richard L. Frost Date Michael A. Jensen Date Wynn C. Stirling Date A. Lee Swindlehurst

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8 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the dissertation of Erik Samuel Perrins 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 REDUCED COMPLEXITY DETECTION METHODS FOR CONTINUOUS PHASE MODULATION Erik Samuel Perrins Electrical and Computer Engineering Doctor of Philosophy Continuous phase modulation (CPM) is often plagued by high receiver complexity. One successful method of dealing with this is the well-known pulse amplitude modulation (PAM) representation of CPM, which was first proposed by Laurent. It is shown that the PAM representation also applies to multi-h CPM and ternary CPM, two previously unconsidered cases. In both cases it is shown that many PAM components may be required to exactly represent the signal. This is especially true of partial-response systems where the memory of the signal is long. Therefore, approximations are proposed which require only a limited number of terms. These extensions of the PAM representation are used to construct reducedcomplexity detectors for CPM. These are generalizations of the detector first proposed by Kaleh. These detectors can be used in an optimal configuration, or in a suboptimal reduced-complexity configuration. The PAM complexity-reduction principle is shown explicitly. An exact expression is given for the pairwise error probability for the entire class of PAM-based CPM detectors, not just the extended cases proposed herein. The analysis is performed for the additive white Gaussian noise (AWGN) channel. The performance

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12 bound that results from this pairwise error probability is shown to be tighter than a previously published bound for PAM-based CPM detectors. The analysis shows that PAM-based detectors are a special case of the broad class of mismatched CPM detectors. However, it is shown that the metrics for PAM-based detectors accumulate distance in a different manner than metrics for other mismatched and suboptimal detectors. These distance properties are especially useful in applications with greatly reduced trellis sizes. The proposed detectors are included in two case studies. The first is for a multih CPM standard used in aeronautical telemetry. Many reduced-complexity detectors are studied in addition to PAM-based detectors. The second case study is for a ternary CPM known as shaped offset QPSK (SOQPSK). Here, the performance of serially concatenated coded SOQPSK is studied along with uncoded systems. It is shown that the coded systems achieve large gains over uncoded systems. However, the design proposed herein achieves these gains with less complexity than previously published designs.

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14 ACKNOWLEDGMENTS I would like to thank my wife, Kristi, and my children, Ethan and Bryn, for the support and encouragement I have received from them. I would also like to thank my advisor, Dr. Michael Rice, for excellent and accurate advice he has given me over the years.

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16 Contents Acknowledgments List of Tables List of Figures vii xiii xv 1 Introduction Previous Work on Reduced-Complexity Detectors for CPM Outline of the Chapters that Follow CPM Signal Model Optimal MLSD Detector for CPM PAM Representation of M-ary Multi-h CPM Key Points of the Chapter Introduction Derivation of the PAM Representation Binary Single-h Systems Binary Multi-h Systems M-ary Multi-h Systems Examples Binary Full-response 2-h systems Quaternary 3RC System with h={4/16,5/16} Autocorrelation of the Pseudo-symbols Mean-Square Approximation Examples using the Approximation ix

17 2.7.1 Binary Full-response 2-h systems Quaternary 3RC System with h={4/16,5/16} Conclusions PAM Representation of Ternary Single-h CPM Key Points of the Chapter Introduction Signal Model for Ternary CPM Equivalent PAM Representation Review of the Binary CPM Case PAM Representation of Ternary CPM Computing ν k,n and ρ k (t) Examples Full-response SOQPSK-MIL Partial-response SOQPSK-TG Conclusions A New Performance Bound for PAM-based CPM Detectors Key Points of the Chapter Introduction Multi-h CPM Signal Model Traditional Model PAM Model for Multi-h CPM Reduced State-Complexity of the Pseudo-symbols Reducing the Number of PAM Pulses PAM-based Detectors for Multi-h CPM Performance Analysis Pairwise Error Probability for PAM-based Detectors Probability of Bit Error Examples Quaternary 3RC with h={4/16,5/16} x

18 4.6.2 Quaternary 2RC with h=1/ Binary GMSK with L=4 and BT = 1/ Applications Conclusions A Survey of Detection Methods for ARTM CPM Key Points of the Chapter Introduction ARTM CPM Signal Model Maximum Likelihood Detection Complexity Reducing Techniques Frequency Pulse Truncation Reduced State Sequence Detection (RSSD) Pulse Amplitude Modulation (PAM) Technique Orthonormal Basis Functions Combined Techniques for Complexity Reduction Frequency Pulse Truncation and RSSDθ Pulse Amplitude Modulation and RSSD Orthogonal Basis Functions and RSSD Summary and Conclusions Applications to SOQPSK Key Points of the Chapter Introduction Signal Model Standard SOQPSK Precoder Recursive SOQPSK Precoder Detection Architectures CPM-based Detector for SOQPSK-MIL Near-Optimum Detectors for SOQPSK-TG xi

19 6.4.3 Complexity Comparison Performance Analysis of Uncoded Systems Suboptimal PAM-based detector for SOQPSK-TG Suboptimal PT-based detector for SOQPSK-TG Serially Concatenated Systems with Iterative Detection Conclusions Conclusions Contributions Areas of Future Study Bibliography 141 xii

20 List of Tables 2.1 Mapping from α n to γ l,n for the M = 4 case Pseudo-symbols, pulses, and pulse lengths for 4-ary L = 3 system Various PAM approximations for the 4-ary 3RC system Pseudo-symbols and pulses for the binary L = 2 case Intermediate pseudo-symbols and pulses for the ternary full-response case Final pseudo-symbols and pulses for the ternary full-response (L = 1) case Values of β k,i and D k for the binary L = 3 case Minimum-distance merger parameters for the examples Performance of reduced-complexity detectors for 4-ary 3RC Parameters defining ARTM CPM The performance/complexity trade-off for frequency pulse truncation The performance/complexity trade-off for RSSDα The performance/complexity trade-off for RSSDθ The performance/complexity trade-off for the PAM approximations The performance/complexity trade-off for the orthogonal basis function approximation The complexity-reduction properties of each proposed techniques The performance/complexity trade-off for the various combinations of reducedcomplexity detectors The relationship between the ternary branch symbol α l,n, the bits γ 1,l and γ 0,l, and the pseudo-symbols βk,n l for SOQPSK Coding gains for serially concatenated SOQPSK xiii

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22 List of Figures 1.1 Development of the PAM representation of CPM Frequency and phase pulses for L = 3 raised-cosine (3RC) example Optimal MLSD detector for CPM PAM-based CPM transmitter, including an expanded view of the k-th filter The signal pulses for binary 1REC, h = {3/8, 4/8} The 48 signal pulses for 4-ary 3RC system with h = {4/16, 5/16} The functions u i (t) for 3RC and h i = 4/16, 5/16, 8/16, and 10/ The matrices and vectors used in the minimum mean-squared error solution The signal pulses for binary 1REC, h = {6/8, 7/8} Residual mean-squared error over the 2-h plane for binary 1REC system Minimum distance loss (in db) over the 2-h plane for binary 1REC system The 3 pulses in the approximation for 4-ary 3RC system The two pulses for the exact PAM representation of SOQPSK-MIL The frequency and phase pulses for SOQPSK-TG The first two pulses for the PAM approximation of SOQPSK-TG Time-series comparison of the exact CPM signal with the PAM approximation for SOQPSK-TG The signal pulses for 4-ary 3RC with h = {4/16, 5/16} Grapical depiction of the binary pseudo-symbols in (4.11) PAM-based detector structure for multi-h CPM with expanded view of modulo-n h matched filter and delay Pulses for the PAM approximation of 4-ary 3RC with h = {4/16, 5/16} Performance of 4-ary 3RC with h = {4/16, 5/16} Performance of 4-ary 2RC with h = 1/ xv

23 4.7 Performance of binary GMSK with L = 4 and BT = 1/ The length-3t raised cosine (3RC) frequency pulse and corresponding phase pulse for ARTM CPM CPM detector showing matched filters and the use of sampled matched filter outputs for sequence detection Values of the squared distance for frequency pulse truncation The performance of detectors with frequency pulse truncation The performance of RSSDα-type detectors The performance of RSSDθ-type detectors Distance values for the PAM approximation The performance of PAM-type detectors with K = 48 (MLSD), K = 12, K = 3, and K = The performance of PAM-type detectors with averaged pulses Values of the squared distance for detectors with orthogonal basis functions The performance of detectors with orthogonal basis functions Values of the squared distance for the rogue merger Performance of detectors using a combination of frequency pulse truncation and RSSDθ Values of the squared distance for the 3-pulse PAM-RSSDθ approximation Values of the squared distance the 8-state PAM-RSSDθ-RSSDα detector Performance of detectors using a combination of PAM approximations, RSSDθ, and RSSDα Values of the squared distance for detectors with H = 3 orthogonal basis functions, RSSDθ, and RSSDα Performance of detectors using a combination of orthogonal basis functions, RSSDθ, and RSSDα Signal model for uncoded SOQPSK Four state time-varying trellis for SOQPSK The mapping between the trellis states and the phase state index Performance of optimal uncoded SOQPSK-MIL and -TG xvi

24 6.5 Squared distance values for the PAM-based detector for SOQPSK-TG Performance of the PAM-based detector for SOQPSK-TG Values of the squared distance for the PT-based detector for SOQPSK-TG Performance of the PT-based detector for SOQPSK-TG Block diagram of the serially concatenated system Performance of coded systems xvii

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26 Chapter 1 Introduction Continuous phase modulation (CPM) has two favorable characteristics that have motivated its widespread use. The first of these is that the transmitted signal has no variations in its amplitude (it is constant envelope). In applications where power supply constraints force the use of fully saturated, non-linear RF power amplifiers, the use of constant envelope modulations is a necessity. Another application where constant envelope modulations are useful is when simple and inexpensive transmitters are needed. Examples of this include digital FM land-mobile radio and Bluetooth systems. Overall, the constant envelope nature of the modulation makes CPM very transmitter-friendly. The second advantage CPM enjoys is bandwidth and power efficiency. A particular CPM format is specified by three parameters: the size of the data alphabet (binary, M-ary, etc.), the modulation index(es), and the duration and shape of the frequency pulse. By carefully selecting these parameters, one can control the power efficiency (minimum distance) and the spectrum of the signal. This makes CPM very versatile and spectrumfriendly. These advantages constant envelope, power efficiency, and spectral efficiency are favorable to the first two parts of the communications link: the transmitter and the transmission medium. However, the final part of the communications link, the receiver, is not treated favorably by CPM. The constant envelope nature of the signal means that it is nonlinear (i.e. the signal is not a linear function of the transmitted data). This makes the signal difficult to demodulate and difficult to synchronize. Furthermore, spectral efficiency is often achieved by increasing the size of the data alphabet and by using longer, smoother 1

27 frequency pulses. These also increase the complexity of the receiver. Therefore, the advantages of CPM are intertwined with the primary disadvantage of CPM, which is high complexity on the receiving end. In this work, we are interested in finding ways to reduce the complexity of the receiver without sacrificing in the area of power/detection efficiency. In other words, we are looking for simple near-optimum detectors for CPM. We point out that complexity can take on many forms. Here, when we speak of complexity we mean the number of trellis states and the number of matched filters (MFs) needed to detect the transmitted data. Other aspects of receiver complexity, such as synchronization, are outside the scope of this work Previous Work on Reduced-Complexity Detectors for CPM If we split the problem of receiver complexity into two parts, trellis states and MFs, there have been a number of important advances in each area over the years. Svensson, Sundberg, & Aulin [36] proposed a detector that was based on a simpler CPM scheme than the one used in the transmitter. The main idea was to truncate the frequency pulse to a shorter length, thereby reducing the number of trellis states and MFs simultaneously. This was called a mismatched detector. since the internal signal model of the detector is mismatched (different) with respect to the signal produced by the transmitter. There have been a number of advances, all variations on a common theme, that have dealt with reducing the number of MFs alone. These have been concerned with obtaining a set of orthonormal basis functions which can be used in place of the MFs. The detector uses only a limited number of these basis functions in order to reduce complexity. In [15] the Gram-Schmidt procedure was used to obtain the orthonormal basis functions. Other approaches have used sampling functions [6, 28], Walsh functions [37], and regularly spaced sinusoids [15]. Most recently, Moqvist & Aulin [24] used an eigenvalue/eigenvector based technique called the principal components method. Each of these various methods has its strengths and weaknesses. In broad terms, all of these detectors can also be viewed as mismatched detectors. In this instance the mismatch is due to the limited number of 1 However, the reduced-complexity techniques presented herein can be applied to synchronization and other problems. This is an area of future study. 2

28 basis functions used in the detector, rather than the truncation of the frequency pulse as in [36]. As for reducing the number of trellis states alone, there have also been variations on a common theme. In [29] a reduced search algorithm is used on the full trellis. In [35] decision feedback is used on a smaller trellis. Neither of these methods has any impact on the number of MFs. One very interesting viewpoint on CPM was introduced by Laurent in 1986 [19]. In his paper, Laurent showed that any binary CPM signal with a single modulation index (single-h CPM) can be exactly represented by a linear combination of pulse amplitude modulated (PAM) waveforms. The signal is not linearized completely, since the original binary data sequence α = {, α 0, α 1, } undergoes a nonlinear operation before being transmitted. However, the nonlinearity inherent in CPM is isolated by this nonlinear operation, which converts the original symbols into pseudo-symbols. From that point on, the signal can be viewed as a superposition of data-modulated pulses. The PAM representation of CPM, as Laurent s work is commonly known, has received a lot of attention in the literature. It has been applied in a number of synchronization problems [5, 8, 14, 17]. It has also been exploited by Kaleh in reduced-complexity detectors [16]. These detectors are based on a small subset of pseudo-symbols and pulses. The MFs in the PAM-based detector are simply time-reversed versions of the pulses. This means that the reduced number of pulses corresponds to a significant reduction in MFs. By coincidence, the small subset of pseudo-symbols which are retained in the detector can be described by a smaller trellis than the full set of pseudo-symbols. Therefore, PAM-based detectors achieve a simultaneous reduction in the number of MFs (pulses) and trellis states. Furthermore, these detectors have been shown to have only minor performance losses in spite of large complexity reductions, e.g. [16]. Laurent s original contribution was limited to binary single-h CPM, where h is not an integer. Due to the usefulness of the the PAM representation, it has since been extended to other cases. Mengali & Morelli [22] considered M-ary single-h CPM, where M is even. Huang & Li [14] considered single-h CPM with an integer modulation index. These extensions are summarized in Figure 1.1. The figure also shows two cases that have 3

29 Mengali & Morelli (1995) M-ary alphabet M-even single-h Chapter 2 M-ary alphabet M-even multi-h Laurent (1986) binary alphabet single-h h-noninteger Huang & Li (2003) single-h h-integer Developed here for the first time Chapter 3 M=3 (ternary) single-h h-noninteger Figure 1.1: Development of the PAM representation of CPM. not yet been considered, namely 1) M-ary CPM with multiple modulation indexes (multi-h CPM) where M is even, and 2) ternary (M = 3) single-h CPM. There are two communications standards with modulations that fall into these yet-to-be-considered categories. The UHF satellite communications standard MIL-STD [1] includes a specification for multi-h CPM and a specification for a ternary CPM known as shaped-offset quadrature phase-shift keying (SOQPSK). The aeronautical telemetry standard IRIG-106 [26] also includes recent specifications for multi-h CPM and SOQPSK formats that are a result of the Advanced Range Telemetry (ARTM) program. In the latter standard, the waveforms are highly bandwidth-efficient and highly complex. The existence of these standards and the need for complexity reduction motivates the development herein of simple, near-optimum, PAM-based detectors for these outlying cases. 4

30 1.2 Outline of the Chapters that Follow Before these reduced-complexity PAM-based detectors can be constructed, the PAM representation must be extended to handle these new cases. Chapter 2 gives the PAM representation of M-ary multi-h CPM and Chapter 3 gives the PAM representation of ternary single-h CPM. Although these extensions serve as building blocks toward constructing PAM-based detectors, they are significant undertakings in their own right. Chapter 4 applies these results to the design of reduced-complexity PAM-based detectors. The task of deriving the detectors themselves is a relatively straightforward extension of Kaleh s earlier work [16]. However, one important aspect of PAM-based CPM detectors that remains unstudied is their bit-error performance. To date, the performance of PAM-based CPM detectors has been determined only by computer simulations on a caseby-case basis. 2 There is a need for a general explanation of the performance characteristics of PAM-based detectors in order to facilitate the design of reduced-complexity detectors. This performance analysis taken up in Chapter 4 as well. Chapters 5 and 6 give in-depth application examples of the new PAM-based detectors. Specifically, in Chapter 5 we survey the performance of several different reducedcomplexity detectors that are available for the multi-h ARTM CPM standard in [26], among these are PAM-based detectors. In Chapter 6 we derive a simple PAM-based detector for SOQPSK that can be used in coded and uncoded systems. As an example, we use serially concatenated convolutional codes with iterative detection. Finally, we offer conclusions in Chapter 7. In the remaining sections of this chapter we introduce the standard notation for the CPM signal model. We also present the traditional form of the optimal maximum likelihood sequence detector (MLSD) for CPM, which serves as the benchmark detector for what follows. It serves as the performance benchmark since the reduced-complexity detectors cannot have better error performance than MLSD. It also serves as the complexity benchmark since the reduced-complexity detectors must have some simplification in order to justify their performance losses. 2 Kaleh did bound the performance loss for one simple binary CPM scheme in [16]; however, this bound is of no use for CPM in general. This is shown in greater detail in Chapter 4. 5

31 1.3 CPM Signal Model In this work we will use complex-baseband notation to represent the the various signals, with the understanding that the results are applicable in a carrier-modulated setting. The entire class of CPM signals can be expressed as [2] E s(t; α) = T exp{ jψ(t; α) } (1.1) where E is the symbol energy, T is the symbol duration, and ψ( ) is the phase of the signal. As (1.1) clearly shows, CPM signals have a constant envelope, which is one of the attractive features of this type of modulation. All the information is carried in the phase of the signal, which is given by ψ(t; α) 2π n i= α i h i q(t it ), nt t < (n + 1)T (1.2) where {h i } is a set of N h modulation indexes, α = {α i } are the data symbols drawn from an M-ary alphabet, and q(t) is the phase pulse. When M is even the symbol alphabet is { (M 1),, 3, 1, +1, +3,, (M 1)} and when M is odd it is { (M 1),, 2, 0, +2,, (M 1)}. The underlined subscript notation in (1.2) is defined as modulo-n h, i.e. i i mod N h. (1.3) We assume the modulation indexes are rational numbers of the form [27] h i K i /p. (1.4) We determine p by expressing all the modulation indexes as a fraction and taking p as the value of the smallest common denominator. The special and popular case where N h = 1 is called single-h CPM. The less common case where N h > 1 is called multi-h CPM. The phase pulse q(t) is usually thought of as the time-integral of a frequency pulse f(t) whose area is 1/2. The frequency pulse is zero outside the time interval (0, LT ). As an example, Figure 1.2 shows a L = 3 raised-cosine (3RC) frequency pulse f(t) and the corresponding phase pulse q(t). When L = 1 the signal is said to be full-response and when L > 1 it is said to be partial-response. 6

32 f(t), frequency pulse q(t), phase pulse Amplitude Normalized Time (t/t) Figure 1.2: Frequency and phase pulses for L = 3 raised-cosine (3RC) example. Considering that the modulation indexes are rational numbers and that q(t) has variations only in the finite interval (0, LT ), the phase signal (1.2) can be divided into three terms ψ(t; α) = θ(t; α n ) + θ n L + φ n. (1.5) The first of these terms is the correlative phase and is defined as n θ(t; α n ) 2π α i h i q(t it ) (1.6) i=n L+1 which is a function of the correlative state vector α n α n L+1,, α n 1, α n. (1.7) The correlative state vector contains the L most recent data symbols from (1.2) and is drawn from an alphabet of M L values. To obtain the second and third terms in (1.5) we manipulate the expression π n L i= α i h i (1.8) which contains the remainder of the data symbols from (1.2). We define the alternate data symbols [27] U i (α i + M 1)/2, U i {0, 1,, M 1} (1.9) and substitute α i = 2U i (M 1) into (1.8). This yields the phase state θ n L 2π 7 n L i= U i h i (1.10)

33 and a data-independent phase tilt which is given by the recursion φ n φ n 1 πh n (M 1). (1.11) Although (1.10) is a function of an infinite number of data symbols, it assumes only p unique values when taken modulo-2π due to the rational modulation index assumption in (1.4). In fact, (1.10) can be replaced by the modulo-p look-up table θ[υ] 2π p [ υ mod p ] (1.12) when the table is indexed by the phase state index ( n L ) I n L U i K i i= mod p. (1.13) The end result of these manipulations is that the CPM signal in (1.1) can be described by a trellis with a finite number of states (a finite state machine) where the input variable is α n and the state vector is the L-tuple (I n L, α n L+1,, α n 2, α n 1 ). (1.14) The state machine is also time-dependent and needs to know the value of (n mod N h ) = n. This could be handled by adding another state variable to (1.14), but instead it is addressed by using a time-varying trellis, with different sections for each value of n. Therefore, the final number of states needed to describe the CPM signal in (1.1) is [2, 27] N S = pm L 1 (1.15) which is simply the number of unique values that (1.14) can assume. 1.4 Optimal MLSD Detector for CPM The complex-baseband received signal model is r(t) = s(t; α) + n(t) (1.16) where n(t) is a complex valued additive white Gaussian noise (AWGN) process with onesided power spectral density N 0. Due to the AWGN assumption, the log-likelihood function 8

34 for (1.16), given a hypothetical data sequence α, is Λ( α) = r(t) s(t; α) 2 dt. (1.17) The objective of MLSD is to find the sequence α that maximizes (1.17). Since s(t; α) is constant envelope, maximizing (1.17) is equivalent to maximizing the correlation [2, Ch. 7] λ( α) = Re where ( ) is the complex conjugate. r(t)s (t; α) dt (1.18) An efficient method of computing (1.18) is to use the Viterbi Algorithm (VA) and the trellis discussed earlier. The trellis is organized as follows. There are N B = M N S = pm L (1.19) branches in the trellis, which are assigned a unique value of the index l {0,, N B 1}. We index the different time-varying sections of the trellis with n {0,, N h 1}. Therefore, a given (l, n) pair 3 indexes a specific and unique location (branch) in the trellis. There are a number of quantities associated with each branch. The branch hypothesis is the (L + 1)-tuple ) σ l,n (Ĩl n L, α n L+1, l, α n 1, l α n l ) = (Ĩl n L, α l n (1.20) (1.21) which can be thought of as an (L + 1)-tuple as in (1.20) or, equivalently, as a phase state index and a correlative state vector as in (1.21). Each branch also has a starting state S l,n {0, 1,, N S 1} (1.22) and an ending state E l,n {0, 1,, N S 1}. (1.23) Given these definitions, (1.18) can be computed recursively by λ n+1 (E l,n ) = λ n (S l,n ) + z(l, n) (1.24) 3 If we are given (l, n) then we of course know the value of n. For this reason we drop the pervasive use of modulo-n h notation and reserve it for places where it requires the most emphasis. 9

35 where λ n ( ) is the cumulative metric for a given state at index n. The branch metric increment is defined as [2, Ch. 7] z(l, n) Re (n+1)t nt r(t)e jψ(t; σ l,n) dt (1.25) which is simply one symbol interval s worth of the correlation in (1.18). Inserting (1.5) into (1.25) yields [2, 27] { } (n+1)t [ z(l, n) = Re e jθ[ĩl n L ] r(t)e jφ n ] e jθ(t; α l n) dt. (1.26) nt Figure 1.3 shows how (1.24) and (1.26) are implemented in a receiver. The received signal is first rotated by the data-independent phase tilt (1.11). The resulting signal is then fed to a bank of correlators (or MFs). The MFs are based on the correlative phase θ( ) in (1.6). The MF bank produces M L complex-valued outputs, one for each possible value of the correlative state vector (1.7). At first glance this would imply that 4M L real-valued MF operations are required to produce M L complex-valued outputs; however, half of these operations can be eliminated due to the identities sin( x) = sin(x) and cos( x) = cos(x). Therefore, the MF bank can be constructed with [2, Ch. 7] N MF = 2M L (1.27) real-valued MFs. Figure 1.3 shows that the set of M L MF outputs are then rotated by the p phase states given by (1.12) (only the real part of this phase rotation is computed). The resulting pm L real-valued outputs are input to the VA. The remainder of the VA operations are typical [2, Ch. 7]. Each ending state E l,n has M possible merging metrics. The one with the maximum value is declared as the survivor and is preserved for the next cycle. After some delay D the receiver traces back along the surviving path and outputs the branch symbol ˆα n D associated with this path. To illustrate the potential for high complexity in Figure 1.3 and (1.24), we briefly consider the ARTM CPM aeronautical telemetry standard [26] where M = 4, h = {4/16, 5/16}, and a 3RC frequency pulse is used. In this instance, the optimal detector requires N S = 256 trellis states and N MF = 128 MFs. This high complexity provides enough motivation for the work that follows. A PAM-based configuration that results in 10

36 M L L pm r(t) MF BANK... PHASE STATE ROTATION... SEQUENCE DETECTOR (VA) αˆn D j n e φ Figure 1.3: Optimal MLSD detector for CPM. Chapter 5 requires only 32 trellis states and 18 MFs 4 with a loss of only 0.08 db relative to MLSD. This is an appreciable reduction in complexity that results in a manageable performance loss. 4 Not all of the MFs considered herein have impulse responses of duration T, such as those in (1.26). Furthermore, some MFs are real-valued and others are complex-valued. In order to provide a meaningful basis for comparing computational complexity, MFs are always counted in terms of real-valued length-t filtering operations. 11

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38 Chapter 2 PAM Representation of M-ary Multi-h CPM 2.1 Key Points of the Chapter A summary is given of the two major building blocks for the PAM representation of M-ary multi-h CPM. The first is Laurent s original work, which applies only to binary single-h CPM. The second is Mengali & Morelli s work, which applies to multilevel (M-ary) single-h CPM. These existing concepts are generalized to the case of M-ary multi-h CPM. The PAM decomposition is presented in general terms as a function of the alphabet size, modulation indexes, and phase pulse of the CPM scheme. As with the previous work on this subject, it is shown that a potentially large number of PAM components are needed to construct the signal. With multi-h CPM, there is a further increase in the number of signal terms; this increase is in proportion to the number of modulation indexes. A minimum mean-squared error approximation is proposed which can significantly reduce the number of signal terms. This approximation can have two objectives: 1) to reduce the number of pulses in the same manner as has been proposed for single-h schemes, and/or 2) to reduce the number of multi-h pulses; the conditions are shown where this latter objective is most practical. This minimum mean-squared error approximation is compared to another method which was recently proposed for CPM. Numerical results on detection performance are also given which demonstrate the practicality of the proposed approximation. 13

39 2.2 Introduction The PAM representation of CPM has had a significant impact in the ongoing effort to reduce the complexity of CPM detectors. With this approach, the nonlinear nature of the signal is addressed head-on by decomposing the signal into a linear combination of PAM components. Recall that the CPM signal was defined in Section 1.3 as the constant-envelope waveform 1 s(t; α) = exp { jψ(t; α) }. (2.1) Laurent [19] showed that any binary single-h CPM signal can be exactly represented by a superposition of PAM waveforms Q 1 s(t; α) = b k,n c k (t nt ). (2.2) k=0 n Here the non-linearity inherent in CPM is moved to the pseudo-symbols b k,n (these are obtained from the binary information symbols), which are combined linearly to produce the CPM signal. The set of Q signal pulses c k (t) are obtained from the phase pulse of the CPM scheme. An important characteristic of the pulses c k (t) is that the signal energy is unevenly distributed between them. This is exploited in reduced complexity detectors [16] which are based on a small subset of these pulses and require a smaller filter bank and trellis size. In a subsequent paper, Mengali & Morelli [22] showed that M-ary single-h CPM waveforms can also be exactly represented by a PAM decomposition s(t; α) = N 1 k=0 a k,n g k (t nt ). (2.3) n The pseudo-symbols and pulses are obtained by viewing the M-ary waveform as a product of P binary waveforms, where P is on the order of log 2 M. Each of these binary waveforms has a PAM representation given by (2.2) and, by expanding their product, the PAM equivalent of M-ary CPM is given by (2.3). As with the binary case, the signal energy is unevenly distributed over the pulses which can be exploited in reduced-complexity detectors [5]. 1 In this chapter, the scale factor out in front of the CPM signal is unity in order to simplify the development. If desired, it is a simple matter to include the scale factor in Equation (1.1) once the PAM decomposition is accomplished. 14

40 Huang & Li [14] considered the special case of single-h CPM formats with integer modulation index (the Laurent decomposition [19] is not valid for this case, though Mengali & Morelli have shown another means of dealing with this limitation [22]). The PAM signal representation in [14] is slightly different than (2.2) and (2.3), though reducedcomplexity detectors are again obtained by discarding the signal pulses with the smallest amplitudes. In this chapter we show that any M-ary multi-h CPM waveform can be similarly viewed as a superposition of PAM waveforms s(t; α) = N 1 k=0 a k,n g k,n (t nt ) (2.4) n where the important distinction is that {g k,n (t)} is a set of N h N PAM pulses, N h being the number of modulation indexes. (While the number N h N is an apparent complexity increase over that required for single-h systems [22], in the detector this is not necessarily the case since only N matched filters are active in any given symbol interval regardless of the number of modulation indexes. This will be shown in Section 4.4). We derive a minimum mean-squared error approximation which can be used to reduce this (potentially) large number of signal pulses. The number of pulses can be reduced in two ways: 1) in the same manner as has been proposed for single-h schemes, which is to discard the less significant pulses using an optimal technique, and/or 2) to optimally average the multi-h pulses to produce the equivalent of single-h pulses. Depending on the circumstances, one objective or the other may be of interest, or both can be achieved simultaneously if so desired. While the second objective is optimal for an arbitrary set of modulation indexes, we show that it is most practical when max 0 n N h 1 {2P 1 h n } 1 2 (2.5) where {h n } are the N h modulation indexes. The approximation is also practical outside this range when the values of the modulation indexes are close to each other. We compare this minimum mean-squared error approximation with the method recently proposed in [24] and find that the cited technique yields a closer approximation than the proposed technique in terms of mean-squared error; however, we also give numerical results on the 15

41 detection performance of the proposed approximation which show that it preserves receiver performance to the same degree as the methods in [24]. This result alone establishes the usefulness of the proposed PAM approximation, which has the simultaneous benefit of reducing the required number of trellis states in the detector [16] (see also Section 4.3.3). The chapter proceeds as follows. In Section 2.3 we derive the multi-h PAM representation. We then provide some examples on constructing the signal in Section 2.4. In Section 2.5 we discuss the autocorrelation of the pseudo-symbols, which is used in Section 2.6 to derive the minimum mean-squared error approximation. We give examples on applying the approximation in Section 2.7 and present conclusions in Section 2.8. Before continuing, we briefly refresh the notation that was established in Chapter 1. The CPM signal is given by (2.1), where the phase is defined as ψ(t; α) 2π n α n h n q(t nt ) (2.6) and T is the symbol duration, {h n } is the set of N h modulation indexes, 2 and q(t) is the phase pulse. In this chapter it is explicitly assumed that the information symbols α = {, α 0, α 1, } are drawn from an M-ary alphabet {±1, ±3,, ±(M 1)} where M is even. The underlined subscript notation in (2.6) is defined as modulo-n h, i.e. n n mod N h. The phase pulse q(t) is the integral of the frequency pulse f(t). The frequency pulse is zero outside the time interval (0, LT ) and is scaled such that LT 2.3 Derivation of the PAM Representation 0 f(τ) dτ = q(lt ) = 1 2. (2.7) We observe that there are at least two approaches one could take to obtain the PAM decomposition of M-ary multi-h CPM. It is clear from (2.1) that the multi-h CPM waveform can be factored into a product of N h single-h waveforms, each with a signalingrate of 1/N h the original system. Mengali & Morelli s approach [22] could then be used to factor the signal yet again into a product of N h P binary single-h waveforms. The 2 The assumption in (1.4) that the modulation indexes are rational numbers is not a prerequisite for the PAM representation [19]. This assumption is necessary only in receivers or other settings where a finite state machine is needed to describe the signal in (2.1). Therefore, the results in this chapter also apply in cases where the modulation indexes are irrational numbers. 16

42 PAM representation in (2.2) could then be applied to each waveform, and the expansion of their product would yield the final result. However, Mengali found the product expansion time-consuming for the single-h case, even without the additional complexities introduced by the multi-h case. This approach was not taken here for those reasons. Instead, we first derive the desired multi-h result for the binary case. The binary result is then used as a building block for the M-ary case, where we show that Mengali & Morelli s final result can be applied with minor extension. We first discuss some of the characteristics of the binary single-h PAM representation Binary Single-h Systems Equation (2.2) is the superposition of Q = 2 L 1 pulses c k (t), which are scaled by the pseudo-symbols b k,n. These are derived from the actual symbols α n by the nonlinear mapping defined in [19]. The extension of this mapping to the more general multi-h case is trivial, and is given by { [ n L 1 ]} b k,n = exp jπ α m h m α n i h n i β k,i (2.8) m= i=0 where β k,i is the i-th bit in the radix-2 representation of k L 1 k = 2 i 1 β k,i, 0 k Q 1, (2.9) i=1 and β k,0 is always zero. The structure in (2.2) is shown in Figure 2.1 (ignoring the lower portion of the figure for the moment). The symbols α n are converted to pseudo-symbols and presented at the inputs to a bank of filters with impulse responses c k (t). The inner sum in (2.2) represents the pulse train at the output of each of the filters. This picture is helpful in examining the multi-h case Binary Multi-h Systems We start by extending the early steps of the derivation in [19] to account for the multi-h case. Laurent showed that, when α n {±1}, exp { j2πα n h n q(t nt ) } = sin ( πh n 2πh n q(t nt ) ) sin(πh n ) + exp{jπα n h n } sin( 2πh n q(t nt ) ) sin(πh n ). (2.10) 17

43 b 0,n b 0,n δ(t-nt) c 0 (t) b nc ( 0, 0 t nt ) n α n Nonlinear mapping b 1,n... b Q-1,n δ(t-nt) b 1,n δ(t-nt)... b Q-1,n δ(t-nt) c 1 (t)... c Q-1 (t) s(t,α) b Q nc ( 1, Q 1 t nt ) n c k,0 (t) b k,n δ(t-nt) c k,1 (t) c (, N 1 t h ) k c k (t) detail for filter c k (t) Figure 2.1: PAM-based CPM transmitter, including an expanded view of the k-th filter. 18

44 We note that (2.10) is not valid when h n is an integer. It was demonstrated in [14, 22] how this case is handled for single-h CPM schemes; however, it can be argued that this case is of less practical value since integer modulation indexes have weak minimum distance properties [2]. By inserting (2.10) into (2.1) and observing (2.7), the signal can be expressed as { s(t; α) = exp jπ η L m= } L 1 [ ] α m h m u i+l,η i (τ) + exp{jπα η i h η i }u i,η i (τ), (2.11) i=0 where τ = t mod T, ηt t < (η + 1)T, and the set of functions u j,i (τ) are nonzero in the interval 0 τ < T and are defined by ( u j,i (τ) sin 2πh i q(jt +τ) ) ( sin πh i 2πh i q((j L)T +τ) sin(πh i, 0 j L 1 ) ) sin(πh i ), L j 2L 1 0, otherwise. (2.12) Expanding the product in (2.11) yields a sum of 2 L terms. Laurent showed that a number of these terms are similar, and can be grouped into 2 L 1 pulses of varying lengths. At this point a specific example is in order. For convenience we write u j,i (τ) as u j,i. When L = 3, the product in (2.11) expands to 8 terms s(t, α) =b 0,η u 0,η u 1,η 1 u 2,η 2 + b 0,η 1 u 1,η 1 u 2,η 2 u 3,η + b 0,η 2 u 2,η 2 u 3,η u 4,η 1 + b 0,η 3 u 3,η u 4,η 1 u 5,η 2 + b 1,η u 0,η u 4,η 1 u 2,η 2 + b 1,η 1 u 1,η 1 u 5,η 2 u 3,η + (2.13) b 2,η u 0,η u 1,η 1 u 5,η 2 + b 3,η u 0,η u 4,η 1 u 5,η 2 where (2.8) has been applied. We will focus on the first four terms, which involve the four pseudo-symbols b 0,n for n = η 3, η 2, η 1, η. For the single-h case in (2.2), these four terms would compose the pulse c 0 (t). However, there are two important differences for the multi-h case we are considering. The first is that the modulo-n h index means these terms vary from one symbol time to the next. Thus there is not one pulse, but rather a set of N h distinct pulses. The second observation is that one of these pseudo-symbols, say b 0,η, modulates a pulse for 4 symbol times. This implies that the terms in (2.13) at index η are from different pulses and are interleaved together. Thus for this L = 3 example, the 19

45 set of pulses described by these four terms are referred to as c 0,n (t), where the modulo-n h subscript indexes the distinct pulses in the set. These pulses are given by u 0,n (τ)u 1,n 1 (τ)u 2, n 2 (τ), 0 t < T u 1,n+1 1 (τ)u 2,n+1 2 (τ)u 3,n+1 (τ), c 0,n (t) = u 2,n+2 2 (τ)u 3,n+2 (τ)u 4,n+2 1 (τ), T t < 2T 2T t < 3T (2.14) u 3,n+3 (τ)u 4,n+3 1 (τ)u 5,n+3 2 (τ), 3T t < 4T 0, elsewhere where τ = t mod T. The subscript indexes in (2.14) are left in an unsimplified form to expose the underlying pattern. Since (2.14) describes one pulse, we note that its length-t segments are not the same as those in (2.13), which come from multiple pulses interleaved together. We can obtain the first four terms in (2.13) by expanding the summation b 0,n c 0,n (t nt ). (2.15) n Additional analysis for this L = 3 example produces these same observations for all the pulses in the set {c k,n (t)}. The transmitter in Figure 2.1 applies to the multi-h case with the exception that the pulse filters must be generalized, as shown in the expanded view in the lower portion of the figure. The pseudo-symbols are fed into a commutator, which cycles through the set of N h pulses and interleaves the pseudo-symbols accordingly. The outputs of all N h filters are summed to form the final output. signal pulses We generalize (2.14) for all values of L and k to arrive at a expression for the c k,n (t) = L 1 j=0 u v(k,j,t),w(n,j,t) (τ), 0 n N h 1 (2.16) v(k, j, t) j + m + Lβ k,j (2.17) w(n, j, t) (n + m (j + m) mod L ) mod N h (2.18) t τ = t mod T, m =. (2.19) T The index v(k, j, t) was originally reported by Laurent [19]. For the special case of single-h (N h = 1), the index w(n, j, t) is always zero and may be disregarded, which reduces (2.16) 20

46 Table 2.1: Mapping from α n to γ l,n for the M = 4 case α n γ 1,n γ 0,n to the expression reported by Laurent [19]. The pulses have a duration of D k = min i { L(2 βk,i ) i }, 0 i L 1 (2.20) which is unchanged from that in [19]. The binary multi-h signal can be expressed as M-ary Multi-h Systems Q 1 s(t; α) = b k,n c k,n (t nt ). (2.21) k=0 n Mengali & Morelli [22] showed that a M-ary CPM signal can be represented as the product of P binary CPM waveforms, where the integer P satisfies the conditions 2 P 1 < M 2 P. (2.22) This is accomplished by representing α n {±1, ±3,, ±(M 1)} as a set of binary coefficients P 1 α n = γ l,n 2 l, γ l,n {±1}. (2.23) l=0 Table 2.1 gives an example of (2.23) for the case where M = 4. Inserting (2.23) into (2.1) produces where h (l) i s(t; α) = = P 1 l=0 P 1 Q 1 { exp j2π i l=0 k=0 n } γ l,i h (l) i q(t it ) (2.24) b (l) k,n c(l) k,n (t nt ) (2.25) = 2 l h i, and (2.25) is obtained by applying (2.21) to each of the binary waveforms in (2.24). We get b (l) k,n by redefining (2.8) in terms of the l-indexed quantities {γ l,i} 21

47 and h (l) i and likewise for c (l) k,n (t) with [ n b (l) k,n {jπ exp m= L 1 ]} γ l,m h (l) m γ l,n i h (l) n i β k,i i=0 (2.26) L 1 k,n (t) u (l) v(k,j,t),w(n,j,t) (τ), 0 n N h 1 (2.27) c (l) j=0 where u (l) j,i (τ) ( sin 2πh (l) i q(jt +τ) sin(πh (l) i ) ( sin πh (l) i ) 2πh (l) i q((j L)T +τ) sin(πh (l) i ), 0 j L 1 ), L j 2L 1 0, otherwise. (2.28) The final step in the derivation is to evaluate the products in (2.25). This is a procedure which Mengali & Morelli indicated was time-consuming and the details were not given in [22]. However, the final result in [22] for the single-h case contains terms of the form b (l) k,n i c(l) k (t + it ), where i is some integer. A close inspection of (2.25) and its counterpart in [22] shows that the only difference is the modulo-n h index n on c (l) k,n (t). Taking this additional index into account, it can be shown that Mengali & Morelli s final result can be extended to contain the terms b (l) k,n i c(l) k,n i (t + it ), This results in s(t; α) = N 1 k=0 where the pseudo-symbols and pulses are obtained by a k,n g k,n (t nt ), N = Q P (2 P 1) (2.29) n a k,n = g k,n (t) = P 1 l=0 b (l) d j,l,n e (m) j,l P 1 c (l) d j,l,n e (m) j,l l=0 (2.30) (t + e (m) j,l T ). (2.31) The quantities d j,l and e (m) j,l are used exactly as in [22, Sec. III-C] and their lengthy definitions are not reproduced here. Figure 2.1 also applies to the signal in (2.29) with b k,n, c k,n (t), and Q interchanged with a k,n, g k,n (t), and N respectively. 22