Roland Best Costas Loops Theory, Design, and Simulation Costas Loops Roland Best Costas Loops Theory, Design, and Simulation 123 RolandBest Oberwil Switzerland Additional material tothis bookcanbedownloaded from http://www.springerlink.com/978-3-319-72008-1 ISBN978-3-319-72007-4 ISBN978-3-319-72008-1 (eBook) https://doi.org/10.1007/978-3-319-72008-1 LibraryofCongressControlNumber:2017959923 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The Costas loop has been invented in 1956 by an American engineer J. P. Costas. Hisoriginalcircuitwasusedforthesynchronousdemodulationofdouble-sideband amplitude-modulated signals with suppressed carrier. What Costas designed that time was a variant of the phase-locked loop (PLL), a circuit that had been known long before. We will see later why the conventional PLL failed in such an appli- cation.Today,Costasloopsaremainlyusedforthedetectionofsignalsbymaking use of digital modulation techniques, such as binary phase shift keying (BPSK). It showed up that a BPSK signal has very similar properties like the formerly men- tioned amplitude-modulated signal. Later, the Costas loop was extended for the applicationinquadraturephaseshiftkeying(QPSK)andalsoinm-aryPSK.Costas loopsarealsofoundtodayinthedemodulationofquadratureamplitudemodulation (QAM) signals. LikethePLL,theCostasloopisasynchronizingdevice.Theincomingsignalin both systems is usually a carrier having frequency f that is modulated with the C transmitted signal. Both Costas loop and PLL incorporate a local oscillator, oper- ating at frequency f , and this frequency is controlled in such a way that it locks Loc ontothecarrierinbothfrequencyandphase,hencethename“phase-lockedloop.” When a data transmission starts or when a PLL or Costas loop is switched on, the initial frequency f is not yet synchronized to the carrier frequency, but it must Loc firstgetlockedtothatfrequency.Thisprocessisreferredtoasacquisitionprocess. With the PLL, two different acquisition processes have been defined: (1) the rel- atively fast lock-in process and (2) the slower pull-in process. For the PLL, a quantity called lock range f has been defined. When the initial frequency of the L local oscillator is within that lock range, the system will lock within at most one beat note between carrier frequency f and initial local oscillator frequency f . C Loc ThetimetogetlockediscalledlocktimeT .Whentheinitialfrequencyofthelocal L oscillatorisoutsidethelockrangebutwithinanotherrangecalledpull-inrangef , P acquisitionwillstilltakeplacebutismuchslower.Thetimerequiredforthepull-in process is called pull-in time T . The dynamic performance has been extensively P investigated in case of the PLL; here, the designer can make use of equation enabling to compute all these parameters (lock range, lock time, pull-in range, v vi Preface pull-in time) explicitly as a function of loop parameters such as natural frequency f , damping factor f and gain factors of building blocks such as phase detector or n voltage-controlled oscillator (VCO). Such equations enable the designer to tailor his/her device in order to fulfill a number of given requirements, e.g., locking within, say, 20 ls. It is surprising that this dynamic analysis has never been performed for the Costas loop, although it has been described in many textbooks and papers. A possible reason for that could be the higher complexity of mathematics. When I triedfirsttodevelopsuchdesignequationsfortheCostasloop,Igotawarethatthe Costas loop presents more nonlinearities than the PLL, which complicates the mathematical treatment considerably. Only after introducing a number of simpli- fications and linearizations, I was finally able to get explicit mathematical expres- sions for lock range, lock time, pull-in range, and pull-in time for the Costas loop. The mathematical treatment is even more aggravated because different analyses must be performed for the different types of Costas loop. The corre- sponding design equations will be presented in this textbook. AnotheraspectoftheCostasloopoverlookedbyalmostallauthorsisthedesign of “modified” Costas loops, i.e., of Costas loops that operate with so-called pre-envelope signal, also referred to as “analytical” signal. Operatingwiththepre-envelopehasadramaticimpactontheperformanceofthe Costas loop. First, it is easily shown that the lowpass filters used in conventional Costas loops are no longer required. It can be demonstrated that this greatly improves the dynamic performance of the loop, i.e., the pull-in range of such modified loops becomes much larger. When the loop filter is implemented by a PI filter (proportional + integral filter), the pull-in range becomes even infinite. Of course, this is only of “academic” interest; however in a real circuit, the loop can lock onto every frequency that can be generated by the local oscillator. Anotherpromisingtechnologythathasbeenwidelydiscardedbymostauthorsis theuseof“phasorrotators”inCostasloops.Insuchsystems,thelocaloscillatoris notrealizedasanoscillatorwhosefrequencycanbecontrolledbyacontrolsignal, butasasimpleoscillatorthatgeneratesaconstantfrequency.Inordertogetlocked, thetwooutputsignalsoftheCostasloop—itwillbeshownthattherearetwosuch signals in each Costas loop—are considered to form a “phasor,” a complex quantity. Acquisition is obtained by a rotation of that phasor. Such a design offers someadvantages:thecomplexityofthehigh-frequencyportionisreduced,andthe rotating circuits are easily implemented from some logic circuits. The book is organized as follows: Chapter 2 gives a short introduction to the Costas loop. It concentrates on the differences between phase-locked loop and Costas and shows by some simple exampleswherethePLLcanbeusedandwherethePLLfailstodotherequiredjob and should be replaced by the Costas loop. Chapters 3 and 4 discuss the conventional Costas loops for BPSK and QPSK, where “conventional” means a loop that operates with real input signals and not with the pre-envelope signal. After theoretical investigation, design procedures are presented, including case studies for the design of analog and digital circuits. Preface vii Finally, Simulink models are shown (all MATLAB files on attached CD), which enable the designer to verify the design. In Chaps. 5 and 6 modified Costas loops for BPSK and QPSK are discussed, including design procedure and simulation. These systems operate with the pre-envelope signal. Chapter 7 presents theory and design of a Costas loop for m-ary PSK demod- ulation, with design procedure and simulation. Chapter 8 presents Costas loop for BPSK using phasor rotation circuit with design procedure, case study for designing a digital Costas loop, and simulation. Chapter 9 presents Costas loop for QPSK using phasor rotation circuit with design procedure, case study for designing a digital Costas loop, and simulation. Chapter 10 presents Costas loop for demodulation of quadrature amplitude modulation (QAM) signals with theory, design procedure, and simulation. Oberwil, Switzerland Roland Best Contents 1 Simulink Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Model Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Note on MATLAB/Simulink File Types. . . . . . . . . . . . . . . . 2 1.3 Downloading and Installing the Simulink Models. . . . . . . . . . . 2 2 Introduction: From Phase-Locked Loop to Costas Loop . . . . . . . . 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Conventional Costas Loop for BPSK, Dynamic Analysis, Design Procedure, and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 The Linear Model of the Costas Loop . . . . . . . . . . . . . . . . . . . 14 3.2 Lock Range DxL and Lock Time TL . . . . . . . . . . . . . . . . . . . . 18 3.3 Nonlinear Model of the Costas Loop . . . . . . . . . . . . . . . . . . . . 19 3.4 Pull-in Range DxP and Pull-in Time TP. . . . . . . . . . . . . . . . . . 26 3.5 Design Procedures for Conventional Costas Loop for BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5.1 Case Study 1: Designing an Analog Costas Loop for BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.5.2 Case Study 2: Designing a Digital Costas Loop for BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.6 Simulating the Costas Loop for BPSK. . . . . . . . . . . . . . . . . . . 32 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Conventional Costas Loop for QPSK . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Linear Model and Frequency Response . . . . . . . . . . . . . . . . . . 35 4.2 Lock Range DxL and Lock Time TL . . . . . . . . . . . . . . . . . . . . 37 4.3 Nonlinear Model in the Unlocked State . . . . . . . . . . . . . . . . . . 39 4.4 Pull-in Range DxP and Pull-in Time TP. . . . . . . . . . . . . . . . . . 40 4.5 Design Procedure for Costas Loop for QPSK. . . . . . . . . . . . . . 43 4.6 Simulating Costas Loops for QPSK. . . . . . . . . . . . . . . . . . . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ix x Contents 5 Modified Costas Loop for BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Linear Model and Frequency Response . . . . . . . . . . . . . . . . . . 50 5.2 Lock Range DxL and Lock Time TL . . . . . . . . . . . . . . . . . . . . 54 5.3 Nonlinear Model for the Unlocked State . . . . . . . . . . . . . . . . . 55 5.4 Pull-in Range and Pull-in Time of the Modified Costas Loop for BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.5 Design Procedure for Modified Costas Loop for BPSK. . . . . . . 57 5.6 Simulating Modified Costas Loops for BPSK. . . . . . . . . . . . . . 59 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6 Modified Costas Loop for QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1 Operating Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2 The Transfer Function of the Modified Costas Loop for QPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.3 Lock Range DxL and Lock Time TL . . . . . . . . . . . . . . . . . . . . 69 6.4 NonLinear Model for the Unlocked State. . . . . . . . . . . . . . . . . 70 6.5 Pull-in Range and Pull-in Time of the Modified Costas Loop for QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.6 Design Procedure for Modified Costas Loop for QPSK. . . . . . . 73 6.7 Simulating the Digital Costas Loop for BPSK . . . . . . . . . . . . . 75 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7 Costas Loop for m-ary Phase Shift Keying (mPSK) . . . . . . . . . . . . 77 7.1 Operating Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.2 Transfer Function of the modified Costas Loop for mPSK . . . . 78 7.3 Lock Range DxL and Lock Time TL . . . . . . . . . . . . . . . . . . . . 80 7.4 Nonlinear Model for the Unlocked State . . . . . . . . . . . . . . . . . 82 7.5 Pull-in Range and Pull-in Time of the Modified Costas Loop for QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.6 Design Procedure for Costas Loop for mPSK. . . . . . . . . . . . . . 84 7.7 Simulink Model for Costas Loop for mPSK. . . . . . . . . . . . . . . 86 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8 Costas Loop for BPSK Using Phasor Rotator Circuit. . . . . . . . . . . 93 8.1 Operating Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 Design Procedure for Costas Loop Using Phasor Rotator . . . . . 99 8.3 Simulating the Costas Loop for BPSK Using Phasor Rotator. . . 100 8.4 Modified Costas Loop for BPSK Using Phasor Rotator. . . . . . . 103 8.5 Simulating the Modified Costas Loop for BPSK Using Phasor Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9 Costas Loop for QPSK Using Phasor Rotator Circuit . . . . . . . . . . 107 9.1 Operating Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9.2 Design Procedure for Costas Loop for QPSK Using Phasor Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Contents xi 9.3 Simulating the Costas Loop for QPSK Using Phasor Rotator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.4 Modified Costas Loop for QPSK Using Phasor Rotator . . . . . . 113 9.5 Simulating the Modified Costas Loop for QPSK Using Phasor Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 10 Costas Loop for Quadrature Amplitude Modulation . . . . . . . . . . . 117 10.1 QAM Signal Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10.2 Structure of a Costas Loop for QAM. . . . . . . . . . . . . . . . . . . . 124 10.2.1 Nyquist Filtering of Input Signal S(T) (RRCF1 and RRCF2). . . . . . . . . . . . . . . . . . . . . . . . . 124 10.2.2 Automatic Gain Control (AGC) . . . . . . . . . . . . . . . . . 125 10.2.3 Phase Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.2.4 Estimator (Estim) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.2.5 Clock Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10.2.6 LF (Loop Filter). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.2.7 Voltage-Controlled Oscillator (VCO). . . . . . . . . . . . . . 132 10.3 Design Procedure for Costas Loop for QAM . . . . . . . . . . . . . . 133 10.3.1 Blocks RRCF1 and RRCF2 (Root Raised Cosine Filters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.3.2 Clock Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 10.3.3 Frequency Control Loop (Blocks RRCF, Phase Detector, LF, VCO) . . . . . . . . . . . . . . . . . . . . . 138 10.3.4 Phase Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.3.5 Preamble and Acquisition Process. . . . . . . . . . . . . . . . 143 10.3.6 Automatic Gain Control (AGC) . . . . . . . . . . . . . . . . . 144 10.4 Simulating the Costas Loop for QAM . . . . . . . . . . . . . . . . . . . 145 10.4.1 Simulations with Model QAM16_Nyq_mod1C . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 153