Space-Time Coding with Offset Modulations

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1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Space-Time Coding with Offset Modulations N. Thomas Nelson Brigham Young University - Provo Follow this and additional works at: Part of the Electrical and Computer Engineering Commons BYU ScholarsArchive Citation Nelson, N. Thomas, "Space-Time Coding with Offset Modulations" (2007). 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.

2 SPACE-TIME CODING WITH OFFSET MODULATIONS by Tom Nelson 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 2007

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4 Copyright c 2007 Tom Nelson All Rights Reserved

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6 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a dissertation submitted by Tom Nelson 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 Michael A. Jensen Date David G. Long Date Brian D. Jeffs Date Wynn C. Stirling

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8 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the dissertation of Tom Nelson 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 J. Wirthlin Graduate Coordinator Accepted for the College Alan R. Parkinson Dean, Ira A. Fulton College of Engineering and Technology

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10 ABSTRACT SPACE-TIME CODING WITH OFFSET MODULATIONS Tom Nelson Department of Electrical and Computer Engineering Doctor of Philosophy In this dissertation it is shown that the telemetry versions of Feher-patented QPSK (FQPSK-JR) and shaped offset QPSK (SOQPSK-TG) can be interpreted as both crosscorrelated, trellis-coded quadrature modulation (XTCQM) and continuous phase modulation (CPM). Based on these representations, both modulations can be detected with near optimal bit error rate performance using a common detector that is formulated as either an XTCQM detector, a traditional CPM detector, or a pulse amplitude modulation (PAM) detector (due to the PAM decomposition of the CPM representations of these modulations). In addition it is shown that the complexity of the XTCQM detector for SOQPSK-TG can be reduced by a factor of 128 with only a 0.2 db loss in detection efficiency relative to the optimum detector. Three decoders for STC encoded OQPSK are presented. One decoder has a bit error rate performance that matches the SISO case but with much higher complexity than that of the QPSK decoder. A second decoder matches the simplicity of the decoder for STC encoded non-offset QPSK but with a loss of 3 db relative to the single-input, single-output (SISO) case. A third decoder matches SISO performance with lower complexity than the first one.

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12 These results for STC encoded OQPSK are extended to STC SOQPSK. It is shown that the maximum likelihood decoder is not computationally feasible. Two suboptimal decoders based on the STC OQPSK decoders are presented. These decoders have much higher complexity than their OQPSK counterparts, and they provide inferior bit error rate performance. In addition, a least squares decoder for STC encoded SOQPSK is presented which is less complex and has better performance (within 1 db of the SISO bound) than the previous two decoders. This decoder also handles the differential delays that can occur on aeronautical telemetry channels.

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14 ACKNOWLEDGMENTS I would like to thank my advisor, Professor Michael Rice for his valuable advice and encouragement throughout my doctoral program. I would also like to thank my wife, Lisa, and my children: Karianne, Susan, Nathan, and Megan, for their encouragement and patience while I have pursued my PhD. I could not have accomplished this without their support.

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16 Table of Contents Title Page i Acknowledgements xiii List of Tables xix List of Figures xxii 1 Introduction Background ARTM Tier 1 Modulations The Two Antenna Problem on Aeronautical Telemetry Channels Contributions Organization Near Optimal Common Detection Techniques for Shaped Offset QPSK and Feher s QPSK Introduction Interoperable Modulations SOQPSK-TG FQPSK-JR Symbol-by-Symbol Detection Common XTCQM Detector xv

17 2.4 Common CPM Detector Common Detector Based on Complex Exponential Representation Common Detector Based on PAM Representation Conclusions Reduced Complexity Detection of Tier 1 Waveforms Introduction An XTCQM Representation of SOQPSK-TG Optimum XTCQM Detection Reduced Complexity XTCQM Detector Reduced Waveform XTCQM Representation of SOQPSK-TG Probability of Bit Error of Reduced Complexity Detectors Improved 8-Waveform XTCQM Detector Conclusions Detection of Offset QPSK with Unitary Space-Time Block Codes Introduction Alamouti Space-Time Coding for Quadrature Modulations Non-offset QPSK Offset QPSK Maximum-Likelihood Detection of STBC-OQPSK Alternate decoders for STBC-OQPSK Half-Symbol Decoder for Alamouti STBC Encoded OQPSK Offset-Decode Detection of Alamouti STBC Encoded OQPSK Rayleigh Fading Channel Extension to Other Unitary STBCs Conclusion xvi

18 5 Detection of Shaped Offset QPSK with Unitary Space-Time Block Codes Introduction Alamouti Space-Time Coding for Quadrature Modulations QPSK and Offset QPSK Shaped Offset QPSK Maximum-Likelihood Detection of STBC-SOQPSK-MIL Alternate Decoders for STBC-SOQPSK-MIL Decoder Based on Approximate Representation of SOQPSK-MIL Offset-Decode Detector SOQPSK-TG Conclusion A Space-Time Coded Receiver for Aeronautical Telemetry Introduction Least Squares Detection of Tier 1 Modulations SISO Least Squares Detection Least Squares Detection of Alamouti Encoded Tier 1 Modulations Application of Space-Time Coding to Aeronautical Telemetry Conclusions Conclusion Contributions Areas of Future Research Bibliography 137 xvii

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20 List of Tables 2.1 Comparison of Common Detectors Alamouti Encoding Rules for Non-Offset and Offset QPSK Number of operations required for the HS, Cavers, and OD decoders to detect a block of four bits for STC-OQPSK Number of operations required for the ML-ASM and OD decoders to detect each block of four bits for STC-SOQPSK xix

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22 List of Figures 1.1 A typical 2 1 MIMO system in aeronautical telemetry Overview of the topics investigated in this dissertation Frequency and phase pulses for SOQPSK-TG and FQPSK-JR BER for the I&D and common symbol-by-symbol detectors Block diagram of the maximum likelihood XTCQM detector Distance spectra for three common XTCQM detectors BER for the common XTCQM detector ML CPM detector Modified distances d 2 for the three candidate CPM detectors (a) when FQPSK-JR is transmitted and (b) when SOQPSK-TG is transmitted BER for the common CPM detector Principal PAM pulses for SOQPSK-TG and FQPSK-JR ML Pam detector Modified distances d 2 for the three candidate PAM detectors (a) when FQPSK-JR is transmitted and (b) when SOQPSK-TG is transmitted BER for the common PAM detector Block diagram of the maximum likelihood XTCQM detector Reduced complexity XTCQM detector BER performance Modified distances for reduced complexity detectors Modified distances for the 8-waveform detector xxi

23 3.5 BER for the 8-waveform detector Block diagram of the improved 8-waveform detector P b for the improved 8-waveform detector Block diagrams of STC-OQPSK decoders BER for the ML and HS decoders BER for OD-ML and OD-LS decoders BER for ML, HS, OD-ML, and OD-LS decoders with Rayleigh fading Waveforms for SOQPSK-MIL Quadrature detector block diagram and detection filter Detection filter output for SOQPSK-MIL Detection filter output for SOQPSK-MIL, second plot Simulation results for the ML-ASM detector Detection filter output for the SOQPSK-MIL OD detector Detection filter output for the SOQPSK-MIL OD detector, second plot Simulation results for the Offset Decode (OD) detector Simulation results for the ML-ASM SOQSK-TG detector Simulation results for the OD SOQSK-TG detector BER for least squares detector for various values of θ Block diagram of the MISO detector for SOQPSK-TG Detection filter output samples when τ > Detection filter output samples when τ < Detector trellis when τ > Detector trellis when τ < BER for the LS detector for various values of τ xxii

24 Chapter 1 Introduction 1.1 Background The work presented in this dissertation is motivated by efforts to apply space-time coding to aeronautical telemetry. Space-time codes (STCs) are well known in the literature and have proven to be useful in situations where multi-path fading impairs wireless communication channels. Systems employing STCs use multiple transmit and/or receive antennas to combat the effects of that fading. Much work has been done to find codes that provide full transmit and receive diversity (see, for example, [1 5]) and to find ways of increasing the capacity of wireless communications links. The application of STC to aeronautical telemetry channels presents unique challenges that have not been addressed in the STC literature. These challenges arise due to the modulations used in aeronautical telemetry, as explained below. In addition, some of the specific details of the telemetry system such as antenna placement and high bit rate also complicate the application of STC to this situation. These challenges are explained in the following sections ARTM Tier 1 Modulations Power and bandwidth constraints present challenges to modulation design. The constant envelope constraint is also imposed when operation through a fully saturated nonlinear RF power amplifier is required. Examples include commercial and military satellite communication links, digital mobile telephony (i.e., Gaussian minimum shift keying (GMSK) for the Global System for Mobile communications (GSM) [6]), and aeronautical telemetry [7]. 1

25 s 1 ( t ) h s ( t ) 1 1 τ 1 h 1 s 0 ( t ) h ( t ) 0 s 0 τ 0 h 0 Figure 1.1: A typical 2 1 MIMO system in aeronautical telemetry. Aeronautical telemetry is an interesting case study since the solution to this problem resulted in the adoption of two interoperable waveforms known as Feher-patented quadrature phase shift keying (FQPSK) and shaped offset quadrature phase shift keying (SOQPSK). From the 1970s, pulse code modulation/frequency modulation (PCM/FM) has been the dominant modulation used for test and evaluation on government test ranges in the USA, Europe, and Asia. (PCM/FM is binary continuous phase modulation (CPM) with a digital modulation index h = 0.7 and a frequency pulse which is a rectangular pulse with a duration of one bit time that has been low-pass filtered.) In the USA, the main spectral allocations for aeronautical telemetry are L-band ( MHz), lower S-band ( MHz), and upper S-band ( MHz). Increasing data rate requirements along with an ever increasing number of test flights put tremendous pressure on these spectral allocations in the 1980s and 1990s. The situation was further exacerbated in 1997 when the lower portion of upper S-band from 2310 to 2360 MHz was reallocated in two separate auctions 1. In response to this situation, the Advanced Range Telemetry (ARTM) Group of the Range Commander s Council adopted a more bandwidth efficient modulation as part of its MHz was reallocated for digital audio radio in one auction while MHz and MHz were allocated to wireless communications services in the other auction. 2

26 Interrange Instrumentation Group (IRIG) standard, IRIG-106 [8], in 2000 [9]. This modulation, known as FQPSK-B, was a proprietary version of FQPSK described in [10 12]. Efforts to reduce some aspects of the implementation complexity resulted in a non-proprietary version, known as FQPSK-JR [13] which was adopted as part of IRIG-106 in Also in 2004, a version of SOQPSK, known as SOQPSK-TG [14], was adopted as a licensefree, fully interoperable alternative in the IRIG-106 standard. Together these modulations are designated as ARTM Tier 1 modulations because they have twice the bandwidth efficiency of PCM/FM (which is the Tier 0 modulation) while maintaining the same detection efficiency as PCM/FM. These modulations are described in more detail in Section 2.2. Briefly, FQPSK (and its variants) is a linear modulation whose inphase and quadrature components are drawn from a set of waveforms in a constrained way. The set of waveforms, called wavelets in the original patents [15], are defined to produce a quasi-constant envelope modulated carrier. (The quadrature waveforms can be defined as delayed versions of the inphase waveforms thereby giving the modulation the look and feel of an offset modulation.) Simon s pioneering analysis of this waveform revealed that the waveform selection constraints can be formulated as a trellis code and termed this, and the general class of modulations, crosscorrelated trellis-coded quadrature modulation or XTCQM. SOQPSK-TG is defined as a constrained ternary partial response CPM with modulation index h = 1/2 and was derived from the full response version of SOQPSK defined in the military UHF satellite communication standard MIL-STD [16]. The fact that these two different modulations can be used interchangeably creates some difficulty in applying STC to the telemetry channel. One could design an STC detector for FQPSK-JR and another STC detector for SOQPSK- TG, but it would be better to have a single detector that works well for both modulations. Besides the fact that two separate modulations are used interchangeably in aeronautical telemetry, the fact that both of these modulations have an offset nature also complicates the use of STC with them. Traditionally STCs have been used with non-offset modulations. They can be used with offset modulations, but decoders for STC encoded offset modulations are more complex than the decoders for the same STCs when non-offset modulations are used. 3

27 1.1.2 The Two Antenna Problem on Aeronautical Telemetry Channels In addition to the use of constant envelope offset modulations, antenna placement on test vehicles also creates difficulties for the use of STC on telemetry channels. Historically, aircraft links were composed of a transmit antenna mounted on the underside of the fuselage and a fixed ground station equipped with a tracking antenna. However, certain aircraft maneuvers can block the required line-of-sight propagation path. The traditional solution has been the use of two antennas radiating the same signal, with the typical placement of one antenna on the bottom and another on the top of the fuselage, to maintain the link for these maneuvers. While this configuration is effective for overcoming data outages due to signal obstruction, difficulties arise when both antennas have a clear view to the ground station. In this case, the two signals arrive at the receiver with different angle-dependent phase-shifts resulting in destructive interference for certain vehicle attitudes. The two-antenna system therefore behaves as a single array with an undesirable gain pattern [17 19]. When the antennas are separated both longitudinally and vertically on the fuselage, the antenna gain pattern variations are dramatic in azimuth and elevation angles, leading to significant signal fluctuations for simple turns as well as more aggressive maneuvers such as rolls and loops. The resulting signal outages during flight often require test flights to be performed multiple times, inserting costly delays in the system evaluation process. It has been shown that the 2 1 Alamouti space-time code provides sufficient transmit diversity to overcome this problem [20 22]. The basic system configuration is illustrated in Figure 1.1. Alamouti-encoded SOQPSK-TG signals s 0 (t) and s 1 (t) are transmitted from the lower and upper transmit antennas, respectively. Each signal experiences a different line-of-site gain, denoted h 0 and h 1. The unusual feature of this application is the different propagation delays from the two transmit antennas to the receive antenna. Traditional MIMO and space-time coding research assumes that the combination of antenna geometry (i.e., spatial configuration) and data rates are such that any differences in path delay are negligibly small. In this case, the data rate is high (10 20 Mbits/s) and the antenna separation, defined by the dimensions of the air frame, is large enough to produce differential delays τ 1 τ 0 that can be a significant fraction of the bit interval T b. As a 4

28 Aeronautical Telemetry Channel Constant Envelope, Bandwidth Efficient Modulation SOQPSK-TG and FQPSK-JR Dual Antenna Problem STC Solution Common Detectors STC for OQPSK Reduced Complexity Detector STC for SOQPSK STC Telemetry System Figure 1.2: Overview of the topics investigated in this dissertation. Shaded blocks indicate contributions of this work. result, it is necessary to take this differential delay into account when designing algorithms to detect STC encoded telemetry signals. 1.2 Contributions In order to apply the STC solution to the two antenna problem when the modulation is one of the ARTM Tier 1 modulations, several issues need to be resolved. The contribution of this dissertation is the resolution of these issues and the formulation of an Alamouti STC Tier 1 detector. Specifically, this dissertation addresses the following: A common representation of the ARTM Tier 1 modulations. The fact that two different modulations are used interchangeably on aeronautical telemetry channels complicates the detection of Tier 1 signals. Before considering space-time coding 5

29 with these modulations, we first develop three common detector architectures that can detect either modulation well without knowing which modulation is used. A reduced complexity detector for SOQPSK-TG. The optimum detector for signals modulated as SOQPSK-TG has high complexity due to the large amount of memory in this signal. The addition of space-time coding will further increase the complexity of an STC decoder. In anticipation of that increased complexity we present a detection algorithm for SOQPSK-TG that has significantly lower complexity than the optimum detector with only a small reduction in bit error rate performance. Space-time coding and unshaped OQPSK. Because the Tier 1 modulations can be interpreted as offset modulations, it is helpful to understand how the combination of space-time coding and the offset affect the detection of such signals. We present detection algorithms designed for STC encoded unshaped OQPSK. Space-time coding and shaped OQPSK. We also present detection algorithms for space-time encoded SOQPSK. These are based on the results with unshaped OQPSK and provide good bit error rate performance, but the increased memory inherent in the Tier 1 modulations increases the complexity of these detectors when compared to the corresponding detectors for OQPSK. An STC aeronautical telemetry receiver. The results of the preceding contributions are extended to deal with the complications that arise due to the presence of differential delays that can occur on aeronautical telemetry channels. A differential delay is the result of the antenna placement on the aircraft in test. The antenna spacing is such that the signals from the two antennas can experience different delays as they travel to the receive antenna. Additional complexity is necessary in the receiver to deal with the differential delays. 6

30 1.3 Organization This dissertation is organized as follows. In Chapter 2 the difficulty presented by the fact that two different (though inter-operable) modulations have been adopted into the standard for aeronautical telemetry is addressed. While an obvious solution to this problem is to develop a separate detector for each modulation, in this chapter it is shown that a single common detector can be used to detect both SOQPSK-TG and FQPSK-JR with bit error rates that are very close to optimal for each modulation. This detector is used to develop a common Tier 1 STC detector in Chapter 6. In Chapter 3 reduced complexity detection of SOQPSK-TG is explored. The complexity reduction technique described in Chapter 2 to allow a common XTCQM detector is extended to further reduce the XTCQM detector for SOQPSK-TG. This complexity reduction applies to FQPSK-JR as well and is also used in the Tier 1 STC detector. Before considering space-time coding with the Tier 1 modulations, STC with unshaped OQPSK is investigated in Chapter 4. The results of this chapter are beneficial to the effort of developing a Tier 1 STC detector because both SOQPSK-TG and FQPSK-JR can be interpreted as variations of OQPSK, so results for the simpler modulation can be extended to the Tier 1 modulations. In Chapter 5 the results on Tier 1 detection in Chapters 2 and 3 along with the results in Chapter 4 are combined to explore detection algorithms for STC encoded Tier 1 modulations. Then in Chapter 6 preceding results are brought together to design a Tier 1 STC detector that can deal with the non-ideal aspects of the aeronautical telemetry channel, including the difficulties that arise in consequence of the differential delays that can occur in this situation. Conclusions and suggestions for further work are presented in Chapter 7. 7

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32 Chapter 2 Near Optimal Common Detection Techniques for Shaped Offset QPSK and Feher s QPSK 2.1 Introduction In this chapter we explore solutions to the fact that two separate modulations are used interchangeably on aeronautical telemetry channels, as explained in Chapter 1. An effective method of detecting either modulation without prior knowledge of which is being transmitted is helpful in the formulation of a Tier 1 space-time coding processor, which is presented in Chapter 6. In most situations, a trellis-coded linear modulation and a CPM are not adopted as equivalent transmission techniques in a standard. In this case, the difficulties of adopting a standard with proprietary components and the challenges of licensing patented technology proved the dominant factors in arriving at this odd situation. FQPSK-JR and SOQPSK-TG can coexist in a standard because both have essentially the same bandwidth and the same bit error rate performance when detected using a simple integrate-and-dump offset quadrature phase shift keying (OQPSK) detector. In the absence of errors, the simple symbolby-symbol detector produces exactly the same sequence when the transmitter uses either FQPSK-JR or SOQPSK-TG. It is in this sense that the modulations are considered fully interoperable. The simple symbol-by-symbol detector has two attractive features: 1) low complexity, and 2) it does not have to know which modulation is used by the transmitter. These features are achieved at the expense of detection efficiency: the bit error rate performance of this simple detector is about 2 db worse than what could be achieved with maximum likelihood detection for either modulation. Since SOQPSK-TG is a CPM and FQPSK-JR is an XTCQM, it is natural to assume that the optimal detector must be equipped with two 9

33 different detection algorithms and endowed with the knowledge of which modulation is used by the transmitter. In this chapter we show how a single detection algorithm can be used for both modulations and that this algorithm does not have to know which modulation is used. We refer to this detector as a common detector and show that its bit error rate performance for both SOQPSK-TG and FQPSK-JR is within 0.1 db of the maximum likelihood performance for each. While considering candidates for the common detector we compare each detector s bit error rate performance for the two modulations in question to the optimum performance for each of those modulations. SOQPSK-TG and FQPSK-JR have similar distance properties and hence their optimum detectors have similar probabilities of bit error. To facilitate these comparisons for SOQPSK-TG, we define the performance metric S = SNR C,S db (2.1) where SNR C,S is the signal to noise ratio (SNR, which is E b /N 0 in db where E b is the energy per bit and N 0 is the variance of the complex noise) that the common detector requires to achieve probability of bit error P b = 10 5 when SOQPSK-TG is transmitted and db is the SNR required for the optimum detector to achieve the same probability of bit error 1. For FQPSK-JR we define F = SNR C,F db (2.2) where SNR C,F is the SNR required for the common detector to achieve P b = 10 5 when FQPSK-JR is transmitted and db is the SNR required for the maximum likelihood FQPSK-JR detector to achieve P b = The values S and F quantify the detection efficiency loss for the candidate common detectors and are used as figures of merit in this chapter. 1 The number db comes from (2.10) which is an approximation to the union bound. For both SOQPSK-TG and FQPSK-JR (for which the number is db), the inclusion of two terms in (2.10) and (2.13), respectively, is sufficient to obtain a very good approximation even at SNR values as low as 3 db, as confirmed by simulations reported later in this chapter. 10

34 The major contribution of this chapter is to show that a given maximum likelihood detector can be tuned to give near optimal performance for two different modulation types. In particular, We derive the equivalent XTCQM representation for SOQPSK-TG and use this representation to produce a common approximate XTCQM representation that applies to both SOQPSK-TG and FQPSK-JR. The detection performance of SOQPSK-TG and FQPSK-JR using the maximum likelihood XTCQM detector based on this approximation is analyzed and simulated. This development is detailed in Section 2.3 where it is shown that the common XTCQM detector has S = 0.14 db and F = 0.08 db. We derive an approximate CPM representation for FQPSK-JR and use this representation to produce a common approximate CPM representation for both SOQPSK-TG and FQPSK-JR. The detection performance of SOQPSK-TG and FQPSK-JR using the maximum likelihood CPM detector based on this approximation is analyzed and simulated. The CPM approximation for FQPSK-JR is derived and the performance analysis is presented in Section where it is shown that S = 0.21 db and F = 0.01 db. The common CPM representation suggests an equivalent PAM representation which can be used as the basis for a common detector. The PAM representation is presented in Section The corresponding analysis and simulation of the common PAM detector shows that S = 0.11 db and F = 0.09 db. We use the preceding results to propose a common detector for SOQPSK-TG and FQPSK- JR in Section Interoperable Modulations SOQPSK-TG SOQPSK-TG is defined as a CPM of the form s S (t, α) = 2Eb T b exp [j (φ(t, α) + φ 0 )] (2.3) 11

35 where T b is the bit time (or reciprocal of the bit rate) and E b is the energy per bit in the signal. The phase is t φ(t, α) = 2πh = 2πh n= n= α n g S (τ nt b )dτ α n q S (t nt b ) (2.4) where g S (t) is the frequency pulse; q S (t) = t g S(τ)dτ is the phase pulse; φ 0 is an arbitrary phase which, without loss of generality, can be set to 0; h = 1/2 is the modulation index; and α n { 1, 0, 1} are the ternary symbols which are related to the binary input symbols a n { 1, 1} by [23] α n = ( 1) n+1 a n 1 (a n a n 2 ). (2.5) 2 The frequency pulse for SOQPSK-TG is a spectral raised cosine windowed by a temporal raised cosine [14]: g S (t) = C 1 4 ( πρbt cos 2T b ) ( ρbt 2T b ) 2 ( ) πbt 2T b ( ) w(t) (2.6) πbt 2T b sin for 1 0 t 2T b < T1 ( ( )) w(t) = cos π t 2 2 T 2 2T b T 1 T 1 t 2T b T1 + T 2. (2.7) 0 T 1 + T 2 < t 2T b For SOQPSK-TG, the parameters are 2 ρ = 0.7, B = 1.25, T 1 = 1.5, and T 2 = 0.5. The constant C is chosen to make q S (t) = 1/2 for t 2(T 1 + T 2 )T b. The frequency pulse and corresponding phase pulse for this case are shown in Figure 2.1. Observe that these 2 In the original publication [14], two versions of SOQPSK were described: SOQPSK-A defined by ρ = 1, B = 1.35, T 1 = 1.4, and T 2 = 0.6 and SOQPSK-B defined by ρ = 0.5, B = 1.45, T 1 = 2.8, and T 2 = 1.2. SOQPSK-A has a slightly narrower bandwidth (measured at the -60 db level) and slightly worse detection efficiency than SOQPSK-B. The Telemetry Group of the Range Commanders Council adopted the compromise waveform, designated SOQPSK-TG in

36 q S (t) 0.5 q F (t) Amplitude g S (t) g F (t) t/t b Figure 2.1: The frequency pulse and phase pulse for both SOQPSK-TG (g S (t) and q S (t)) and the CPM approximation of FQPSK-JR (g F (t) and q F (t)). values of ρ, B, T 1 and T 2 make SOQPSK-TG a partial response CPM spanning L = 8 bit intervals. An analysis of maximum likelihood detection of SOQPSK was performed by Geoghegan [24 26] following the standard union bound technique based on pairwise error probabilities [27]. The binary-to-ternary mapping (2.5) contributes an extra step to the analysis. Let a =... a k 3, a k 2, a k 1, a k, a k+1, a k+2, a k+3,... (2.8) represent a generic binary symbol sequence with a k { 1, +1}. The minimum distance error event occurs between the waveforms corresponding to two binary symbol sequences whose difference satisfies a 1 a 2 = ± [..., 0, 0, 0, 2, 0, 0, 0,...] (2.9) where the difference (or erroneous symbol) occurs at index k. As it turns out, there are two ways a pair of binary sequences can produce (2.9). Half of the sequence pairs have a k 1 = a k+1 and are characterized by waveforms separated by a normalized squared 13

37 Euclidean distance of The other half of the sequence pairs have a k 1 = a k+1 and are characterized by waveforms separated by a normalized squared Euclidean distance of Since these error events produce one bit error, the probability of error is well approximated by P b Q ( 1.60 E b N 0 ) Q ( 2.58 E b N 0 ). (2.10) Using this expression, P b = 10 5 is achieved at SNR = db FQPSK-JR FQPSK-JR is defined as an OQPSK modulation of the form s F (t) = n s I,m (t nt s ) + js Q,m (t nt s T s /2) (2.11) with data dependent pulses s I,m (t) and s Q,m (t) each drawn in a constrained way from a set of 16 waveforms [13]. The waveform index m is determined by the modulating data bits as explained in [28]. The 16 pulses have a duration of 2T b = T s and are listed in [13] and [28]. Simon showed that the original version of FQPSK has an XTCQM interpretation from which the optimum maximum likelihood detector followed [28]. This representation consists of 16 waveforms for the inphase component and 16 waveforms for the quadrature component for a total of 32 possible complex-valued waveforms when the constraints on possible combinations are taken into account. The XTCQM representation of FQPSK-JR is the same as that for the original FQPSK except that three of the waveforms are modified. Consequently the optimum detector for FQPSK-JR has the same form as that described by Simon [28] for FQPSK. For the purposes of comparison with the XTCQM representation of SOQPSK-TG, it is advantageous to re-express FQPSK-JR in the form s F (t) = k I F (t kt s ; a 2k,..., a 2k 4 ) + jq F (t kt s ; a 2k,..., a 2k 4 ). (2.12) 14

38 Five information bits are used to select an in-phase waveform I F (t; ) and a quadrature waveform Q F (t; ) which are transmitted during an interval of 2T b seconds. The next waveform is determined by clocking in two new bits (and discarding the two oldest bits) to form a new group of 5 bits that select the waveform. This slightly different, but equivalent, point of view represents the memory in the modulated carrier using a sliding window that is five bits wide and strides through the input data 2 bits at a time. Note that I F (t; ) is drawn from a set of 16 waveforms (the s I,m (t) of (2.11) which are uniquely specified using 4 bits) but that the addressing uses 5 bits. This is explained as follows: when FQPSK is expressed in the form (2.12), the list of 32 waveforms for I F (t; ) consists of the 16 waveforms s I,m (t) of (2.11) each repeated twice. The same applies to Q F (t; ) together with s Q,m (t) of (2.11). This double listing is required to accommodate the waveform indexing scheme to produce the proper 32 waveforms. The representation is largely notional and is used to conceptualize the relationships between FQPSK and SOQPSK. The asymptotic performance of maximum likelihood detection of FQPSK has been analyzed by Simon [28]. In concept, maximum likelihood detection of FQPSK organizes the outputs of 32 matched filters (one filter matched to each of the 32 possible transmitted waveforms) in a trellis and performs maximum likelihood sequence estimation. The standard union bound composed of pairwise error probabilities is used to quantify the bit error rate performance of this modulation. The minimum distance error events span three trellis states. Over this span, every trellis state is reachable by every trellis state via two paths. Thus there are = 2048 such paths over three steps. Each of these paths has one competing path associated with it that contributes a single bit error of these path pairs are separated by a normalized Euclidean distance of 1.56 and 1024 of these path pairs are separated by a normalized Euclidean distance of As such, the probability of bit error is well approximated by ( ) ( ) P b Q 1.56 E b N Q 2.59 E b. (2.13) N 0 (Why the coefficients are expressed this way will become evident in Section 2.3.) Using this expression, P b = 10 5 is achieved at SNR = db. 15

39 2.2.3 Symbol-by-Symbol Detection SOQPSK-TG and FQPSK-JR are considered to be interoperable because of their essentially identical bandwidth and similar bit error rate performance with symbol-by-symbol detection using an integrate-and-dump detection filter. Using an unshaped OQPSK detector with FQPSK (and its variants) is natural since FQPSK is defined as an offset modulation with data dependent pulse shapes. The use of this detection technique with SOQPSK- TG is motivated by the well established connection between CPM with modulation index h = 1/2 and OQPSK [29 32]. Symbol-by-symbol detection has been thoroughly investigated for SOQPSK-TG by Geoghegan [24] and for FQPSK by Simon [28]. Our own simulation results are shown in Figure 2.2 where we see that S 2.0 db and F 2.2 db. Symbol-by-symbol detection with better detection filters has also been investigated for SOQPSK-TG in [24] and for FQPSK in [28]. The XTCQM representations for both modulations can be used to define detection filters for use with a symbol-by-symbol detector (the method for determining the detection filter is defined in Section 2.3). The bit error rate performance of SOQPSK-TG and FQPSK-JR using the improved detection filter is also plotted in Figure 2.2. Observe that the improved performance reduces S to about 1.5 db and F to about 1.6 db. The performance of this approach falls well short of that of maximum likelihood detection since symbol-by-symbol detection ignores the memory inherent in the waveforms. The fact that these losses are still significant motivates the search for common detectors that perform better than the symbol-by-symbol detector. 2.3 Common XTCQM Detector A generic maximum-likelihood XTCQM detector is illustrated in Figure 2.3. The in-phase component of the noisy received waveform is filtered by a bank of filters matched to the N X possible in-phase waveforms. Likewise, the quadrature component of the received waveform is filtered by a bank of filters matched to the possible quadrature waveforms. These matched filter outputs are sampled, once per symbol, and used by a maximum likelihood sequence estimator operating on a trellis with 2N X states. For FQPSK and FQPSK-JR, N X =

40 bit error rate Theory (Tx=SOQPSK TG) Theory (Tx=FQPSK JR) I&D (Tx=SOQPSK TG) Common det. filter (Tx=SOQPSK TG) I&D (Tx=FQPSK JR) Common det. filter (Tx=FQPSK JR) E /N (db) b 0 Figure 2.2: Bit error rates for SOQPSK-TG and FQPSK-JR for the integrate-and-dump (I&D) detector and the common symbol-by-symbol detector along with the theoretical curves for each modulation. For the I&D detector S = 2.0 db and F = 2.2 db while the average matched filter detector has S = 1.5 db and F = 1.6 db. I( t;0) t = 2kT b in-phase component I M ( t; NX 1) Q( t;0) t = 2kT b Trellis 2 N X states quadrature component Q M ( t; NX 1) Figure 2.3: Block diagram of the maximum likelihood XTCQM detector. Each filter is a real-valued filter of length 2T b. The indexes in the filter impulse responses are the decimal equivalents of the binary symbol patterns that define the waveforms. 17

41 In order to formulate such a detector for SOQPSK-TG, an XTCQM representation of SOQPSK-TG is needed. Such a representation is developed here. In order to obtain this representation of SOQPSK-TG we need to determine the data dependent pulses I S (t; a n,...) and Q S (t; a n,...) for this modulation which are analogous to those for FQPSK-JR in (2.12). These waveforms have a duration of 2T b, which causes the resulting XTCQM signal trellis to be time invariant 3. In order to obtain length 2T b quadrature waveforms for SOQPSK-TG, we begin by examining the phase (2.4) during the interval nt b t (n + 1)T b. φ(t, α) during this interval can be written as φ(t, α) = 2πh = π 2 n L k= n L k= α k q s (t kt b ) + 2πh α k + π n k=n L+1 n k=n L+1 α k q S (t kt b ) α k q s (t kt b ), nt b t (n + 1)T b. (2.14) The duration of interest can be extended from T b to 2T b by extending the upper index on the second summation by one to produce φ(t; α) = θ n + π n+1 l=n L+1 α l q S (t lt b ), nt b t (n + 2)T b (2.15) where θ n = π 2 n L l= α l, (2.16) which is known as the phase state. The value n is now constrained to be even. We make this constraint explicit by setting n = 2k. Inserting (2.5) into (2.15) results in φ(t; a 2k ) = θ 2k + π 2k+1 i=2k L+1 ( 1) i+1 a i 1 (a i a i 2 ) q S (t it b ) (2.17) 2 3 Aulin s quadrature representation of CPM has waveforms with a duration of T b and the resulting signal trellis is time varying [33]. Rimoldi incorporated a tilted phase into Aulin s length T b representation [34] to obtain a time invariant trellis. Simon took a different approach to obtain a quadrature representation with a time invariant trellis: he extended the duration of the waveforms to 2T b [23]. Our quadrature representation of SOQPSK-TG is a combination of Aulin s approach and Simon s approach. It has a time invariant trellis due to the fact that the waveforms have a duration of 2T b similar to Simon but it does not have an encoder separate from a waveform mapper as Simon s representation does. 18

42 ] where a 2k = [a 2k L 1 a 2k L a 2k+1. Note that the term π 2k+1 i=2k L+1 ( 1) i+1 a i 1 (a i a i 2 ) q S (t it b ) (2.18) 2 in (2.17) is known as the correlative state vector in the CPM literature. With L = 8T b there are 9 terms in the sum in (2.17) and 11 bits that contribute to φ(t, a 2k ) during the interval 2kT b t (2k + 2)T b. The phase state θ 2k does not introduce a dependency on any additional bits. Therefore the number of waveforms is determined solely by the number of bits that contribute to the sum in (2.17). As a result, 2048 complex waveforms are needed to exactly represent SOQPSK-TG as an XTCQM. The I and Q waveforms are given by I S (t; a 2k ) = cos (φ(t; a 2k )), Q S (t; a 2k ) = sin (φ(t; a 2k )). (2.19) Then the XTCQM representation of SOQPSK-TG can be expressed as s S (t) = k I S (t kt s ; a 2k ) + jq S (t kt s ; a 2k ). (2.20) This representation consists of 1024 in-phase waveforms I S (t) and 1024 quadrature waveforms Q S (t) indexed by a sliding window that is 11 bits wide and strides through the bits 2 at a time. The XTCQM representation of SOQPSK-TG requires 2048 complex waveforms and is an exact representation of this modulation. The XTCQM view of SOQPSK-TG suggests an alternative form for the optimal detector: an XTCQM detector. The maximum-likelihood XTCQM detector is that of Figure 2.3 with N X = Since this detector performs maximum likelihood detection, the bit error rate performance of this detector is given by (2.10). In the case of the optimum XTCQM detector, the constant that scales each term in (2.10), 1/2, is obtained as follows. The minimum distance error event spans 6 trellis stages and produces a single bit error. Over this interval, each of the 2048 states is reachable by every state via two paths, so there are a total of = 2 23 possible pairs of paths to consider of these path pairs are separated by a normalized Euclidean distance of 1.60 and the other 2 22 are 19

43 separated by a normalized Euclidean distance of Thus (2.10) can be rewritten as ( ) ( ) P b Q 1.60 E b Q 2.58 E b. (2.21) 2 23 N N 0 This form of the expression for the probability of bit error will be helpful in the discussion below about suboptimal detectors. Since the number of matched filters and the number of trellis states is different for FQPSK-JR and SOQPSK-TG, the forgoing formulation does not permit the detector in Figure 2.3 to operate as a common detector for these modulations. This issue is resolved by identifying a set of 32 waveforms suitable for use with both FQPSK-JR and SOQPSK- TG in Figure 2.3. The first step is to reduce the number of waveforms required to represent SOQPSK-TG to 32. As explained in Section 2.3, the common detector requires a representation for SOQPSK that uses 32 waveforms instead of the 2048 required by (2.20). The number of waveforms required by the XTCQM representation of SOQPSK-TG can be reduced by averaging the waveforms that differ in the first and last bits. These are the waveforms that are most similar. (This technique was used by Simon [35] to reduce the number of waveforms required to represent FQPSK.) This averaging technique is illustrated for the inphase waveforms I S (t; ) below. Application to the quadrature waveforms Q S (t; ) is straight forward. The number of inphase waveforms is reduced from 2048 to 512 by averaging together the four waveforms that differ in the first and last bits to get waveforms that are a function of 9 bits. This is accomplished by using the expression I S,512 (t; a 2k 8,..., a 2k ) = 1 4 I S(t; 1, a 2k 8,..., a 2k, 1) I S(t; 1, a 2k 8,..., a 2k, +1) I S(t; +1, a 2k 8,..., a 2k, 1) s S(t; +1, a 2k 8,..., a 2k, +1). (2.22) 20

44 Performing the same averaging process on the 512 waveforms results in the 128 waveforms given by I S,128 (t; a 2k 7,..., a 2k 1 ) = 1 4 I S,512(t; 1, a 2k 7,..., a 2k 1, 1) I S,512(t; 1, a 2k 7,..., a 2k 1, +1) I S,512(t; +1, a 2k 7,..., a 2k 1, 1) I S,512(t; +1, a 2k 7,..., a 2k 1, +1) (2.23) and averaging those waveforms results in the 32 waveforms given by I S,32 (t; a 2k 6,..., a 2k 2 ) = 1 4 I S,128(t; 1, a 2k 6,..., a 2k 2, 1) I S,128(t; 1, a 2k 6,..., a 2k 2, +1) I S,128(t; +1, a 2k 6,..., a 2k 2, 1) I S,128(t; +1, a 2k 6,..., a 2k 2, +1). (2.24) The same procedure can be used to reduce the number of quadrature waveforms as well, giving Q S,512 (t; a 2k 9,..., a 2k 1 ), Q S,128 (t; a 2k 8,..., a 2k 3 ), Q S,32 (t; a 2k 7,..., a 2k 3 ). By setting ĨS(t; ) = I S,32 (t; ) and Q S (t; ) = Q S,32 (t; ), the 32 waveform approximation of SOQPSK-TG can be written as s S (t) k Ĩ S (t kt s ; a 2k 6,..., a 2k 2 ) + j Q S (t kt s ; a 2k 6,..., a 2k 2 ). (2.25) The in-phase waveform ĨS(t; ) is drawn from a set of 16 waveforms while the quadrature waveform Q S (t; ) is drawn from a different set of 16 waveforms. The total number of complex-valued waveforms is 32, and 5 bits are used to select the waveform. Note that with a simple redefinition of the bit indexes, the representation (2.25) is identical in form to the XTCQM representation of FQPSK-JR given by (2.12). Before proceeding with the development of that common detector, we first note that the complexity reduction process can be continued. The reason for doing this is to obtain 21

45 approximations for SOQSK-TG that can be used to formulate low complexity detectors. Repeating the averaging process for the inphase waveforms gives the 8 waveforms I S,8 (t; a 2k 6, a 2k 5, a 2k 4 ) = 1 4 I S,32(t; 1, a 2k 6, a 2k 5, a 2k 4, 1) I S,32(t; 1, a 2k 6, a 2k 5, a 2k 4, +1) I S,32(t; +1, a 2k 6, a 2k 5, a 2k 4, 1) I S,32(t; +1, a 2k 6, a 2k 5, a 2k 4, +1), (2.26) and repeating the process one last time gives the 2 waveforms I S,2 (t; a 2k 5 ) = 1 4 I S,8(t; 1, a 2k 5, 1) I S,8(t; 1, a 2k 5, +1) I S,8(t; +1, a 2k 5, 1) I S,8(t; +1, a 2k 5, +1). (2.27) The corresponding quadrature waveforms are Q S,8 (t; a 2k 6, a 2k 5, a 2k 4 ) and Q S,2 (t; a 2k 5 ). Note that I S,2 (t; a 2k 5 ) consists of a single length-2t b waveform and its negative. This waveform was averaged with the corresponding average of the FQPSK-JR waveforms to produce the common waveform used by the common symbol-by-symbol detector discussed in Section The XTCQM detector based on the 8-waveform approximation I S,8 (t; a 2k 6, a 2k 5, a 2k 4 ) and Q S,8 (t; a 2k 6, a 2k 5, a 2k 4 ) will be examined in detail in Chapter 3. Since the sets of waveforms from which the waveforms I F (t; ) and ĨS(t; ) are drawn are different from the waveforms in (2.25), a common set of waveforms must be identified in order to produce a common detector of the form shown in Figure 2.3. The common waveforms define the matched filters and trellis connections. Three possibilities were explored: 1) the FQPSK-JR waveforms I F (t; ) and Q F (t; ); 2) the SOQPSK-TG waveforms ĨS(t; ) and Q S (t; ); and 3) average waveforms I avg (t; ) = I F(t; ) + ĨS(t; ) 2 and Q avg (t; ) = Q F(t; ) + Q S (t; ). (2.28) 2 22

46 A number of other possibilities could be envisioned (e.g., waveforms that minimize the average squared error). However, the performance results, summarized below, show that a detector based on the average waveforms is on the order of 1/10 of a db from optimum. This suggests there is very little to be gained by using waveforms based on more elaborate criteria. Since the set of waveforms used by the detector is different from the set of waveforms used by the modulator, the mismatched receiver analysis technique, described in [27, 36, 37], can be used to evaluate the performance of each of these options. We first examine the case where the transmitted signal is FQPSK-JR and examine the performance of the three detectors for this modulations. Then we will examine their performance when SOQPSK-TG is transmitted. When FQPSK-JR is produced by the modulator and the detector is based on I avg (t; ) and Q avg (t; ), the mismatch is a result of the fact that the detector s model for the transmitted signal is based on the 32 waveforms defined by (2.28) rather than on the actual 32 FQPSK-JR waveforms. Each error event included in (2.13) involves a pair of waveforms defined by bit sequences a 1 and a 2. Let s(t; a 1 ) and s(t; a 2 ) represent the corresponding signals produced by the transmitter and let s(t; a 1 ) and s(t; a 2 ) represent the corresponding signal used by the detector based on its set of waveforms. The probability of the error event is where and Q ( ) d E 2 b N 0 (2.29) 1 d 1 d d = 2, (2.30) 2E b d3 d 1 = s(t; a 1 ) s(t; a 2 ) 2 dt, (2.31) R d 2 = s(t; a 1 ) s(t; a 1 ) 2 dt, (2.32) d 3 = R R s(t; a 1 ) s(t; a 2 ) 2 dt; (2.33) 23

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