Symbols and Abbreviations

Symbols

α Distortion coefficientA0 Low-frequency gainB Number of bitsBW Band-WidthC CapacitanceCMRR Common-Mode Rejection RatioCR Cross-over RateENOB Effective Number Of BitsFoM Figure of Meritfs Sampling frequencyG Conductancegout Output conductance of transistorGBW Gain–Band-Widthgm Transconductance of transistorHDm Harmonic Distortion of mth orderINL Integral Non-LinearityIP3 Intercept Point of 3rd orderIRR Image Rejection RatioNF Noise FigureMR Mutation RateOSR Over-Sampling RatioR ResistanceSNDR Signal-to-Noise-and-Distortion RatioSNR Signal-to-Noise Ratioτ Time constantZ Impedance

262 Symbols and Abbreviations

Mathematical notations

Numbers

B Set of boolean numbersC Set of complex numbersR Set of real numbersZ Set of integersj Imaginary unit (j2 = −1)c Complex conjugate of complex number cc Real part of complex number cc Imaginary part of complex number c|x| Absolute value of number x‖x‖ Norm of quantity xx Floor function of xx Ceiling function of x∏n

i=1 ai Product of a1, . . . , an∑ni=1 ai Sum of a1, . . . , an(

nk

)Binomial coefficient

φ[a1, . . . , am] Occurrence function (5.72)R1‖R2 Resistance of parallel connection of resistors R1 and R2

Sets

∅ Empty set∈ Element symbol⊂ Subset symbolA ∪ B Union of sets A and BA × B Cartesian product of sets A and B2A Power set of set A|A| Cardinality of set AdimS Dimension of space S

Logic

∀ Universal quantifier∃ Existential quantifier∨ Logical or∧ Logical and

Functions

C0[R] Set of continuous-time signalsFC0[R] Set of frequency-domain signalsf(x) ∗ g(x) Convolution of the functions f(x) and g(x)

Symbols and Abbreviations 263

F s(t) Fourier transform of signal s(t)H s(t) Hilbert transform of signal s(t)df(t)

dt Derivative of function f(t)∂f(x,...)

∂x Partial derivative of function f(x, . . .) to variable x∫ b

af(t) dt Integral of function f(t) for t = a to t = b

O( · ) Landau symbolδ(t) Dirac pulseu(f) Unit step function in frequency domainsgn(x) Sign functionTi(x) Chebyshev polynomial of order i

E t0;sλ,m(t) E -function of signal s(t) with eigenvalue λ, order m and

start point t0C t0;s

α,β,m(t) C -function of signal s(t) with eigenvalue α + jβ, order mand start point t0

S t0;sα,β,m(t) S -function of signal s(t) with eigenvalue α + jβ, order m

and start point t0W (f) Window functionp(t) Pulse functionψk(t) Base function of order k[a, b] Interval from a to b, with a and b inclusive

Matrices

v Vector symbolM Matrix symbolT Tensor symbolRn Set of n-dimensional vectors with elements of RRm×n Set of matrices with m rows and n columns with elements

of RIN Identity matrix of order Ndiag(a1, . . . , an) n × n diagonal matrix with elements a1, . . . , an

diagn(a) n × n diagonal matrix with n times element aAT Transpose of matrix AA† Conjugate transpose of complex matrix AeA Exponential of matrix AA⊗B Kronecker product of matrices A and B⟨T , v

⟩n

Inner product between tensor T and vector v over index nusing normal product⟨

T , v⟩∗n

Inner product between tensor T and vector v over index nusing convolution

v (m) mth Hadamard power of vector vv(x) w(x) Hadamard convolution between function vectors v(x) and

w(x)v w Hadamard product of vectors v and w

264 Symbols and Abbreviations

nullA Null space of matrix Aλ Real eigenvalueα + jβ Complex eigenvalue

Statistics

E · Expectation operatorσ2 VarianceRxy(τ) Cross-correlation function between processes x(t) and y(t)Rn(τ) Correlation matrixK Covariance matrix

Design flows

D Description levelA Abstraction levelE Alphabet of elementary elementsI Alphabet of interconnection typesRD Refinement operationSD Simplification operationTD,D′ Translation operationTD Transformation operationp Parameterp(D) Set of parameters at description level DP Performance valueP(A) Set of performance values at abstraction level AΠD,A(p) Performance functionΦD,A(P ) Feasibility function

Generic behavioral models

M Formal machineΣ Input alphabetΥ Noise alphabetΨ Output alphabetQout Output state alphabetΩ Feedback alphabetQfb Feedback state alphabetQc Continuous-time state alphabetQ Discrete-time state alphabetΛQ Alphabet of sets with elements of alphabet QQp Phase alphabetΘN,A Internal signals alphabetΞN,A Internal noise signals alphabet

Symbols and Abbreviations 265

u Input signaln Vector of noise signalsy Vector of output signalsqout Vector of output statesv Vector of feedback signalsq fb Vector of feedback statessc Vector of continuous-time statess Vector of discrete-time statesLs Set of vectors si

σ Clock signalP(σ) Set of pairs of adjacent time points of σqp Phasev0 Initial values for vector vλ Output/Sampling functionφ Feedback functionδ Next-state functionδL Linear next-state functionδNL Nonlinear next-state functionε Switching functionκin Input conversion functionκout Output conversion functionζ High-frequency filter functionξ Low-frequency filter functionρ Re-arrangement function

Time-domain characterization

qn(t) State vector of nth building blockxn(t) Output of nth building blockAn System matrix of nth building blockEn,i Input matrix from output of building block i to building

block nFn Input matrix from feedback signal to building block nCn Output matrix of nth building blockgn,i Transmission matrix from output of building block i to

building block nH n Transmission matrix from feedback signal to building

block nθn(t) Vector of control variables of building block nTn Transformation matrix of building block n

R[k]n Input matrix of kth nonlinearities for building block n

S [k]n Transmission matrix of kth nonlinearities for building

block n

266 Symbols and Abbreviations

Phase-Frequency Transfer model

xL(t) Low-pass equivalent signal in the time domainXL(f) Low-pass equivalent signal in the frequency domaineL Base vector for polyphase signalE Polyphase baseN Number of phasesBN Base transformation matrix of order Nx (t) Polyphase signal in the time domainX (f) Polyphase signal in the frequency domainx L(t) Low-pass equivalent polyphase signal in the time domainX L(f) Low-pass equivalent polyphase signal in the frequency

domainH(f) Polyphase transfer matrixHL(f) Low-pass equivalent polyphase transfer matrixF Set of fundamental frequenciesA Number of fundamental frequenciesH (f) Harmonic Transfer MatrixPA,N Permutation matrix corresponding to A fundamental

frequencies and N phasesx L(t) Polyphase Harmonic Signal in the time domainX L(f) Polyphase Harmonic Signal in the frequency domainCc Operator for mirroring a signal componentCc1,...,cm

Operator for mirroring multiple signal componentsHL(f) Polyphase Harmonic Transfer MatrixDL

(f) Polyphase Distortion TensorSn(f) Power Spectral Density MatrixRL

n(t + τ, t) Polyphase Harmonic Correlation MatrixSL

n(f, t) Polyphase Harmonic Power Spectral Density Matrix

Heterogeneous synthesis

D Design structureD PopulationC Performance calculatorC Set of calculatorsPD Set of performances for design DE EvaluatorE Set of evaluatorswE Weight factor of evaluator EpE

i Probability of evaluator E for transformation Ti

SD Set of satisfaction levels for design DSE

D Satisfaction level of evaluator E for design DT Transformation

Symbols and Abbreviations 267

τ Transformation contextR Context requirement functionO Expected outcome functionI Set of invariant conditionsTD Set of transformations for design DT E

D Transformations of evaluator E for design D

Abbreviations

ABS Anti-lock Braking SystemAC Alternating CurrentADC Analog-to-Digital ConverterA/D Analog-to-DigitalAMS Analog and Mixed-SignalAM Amplitude ModulationASCII American Standard Code for Information InterchangeB.C. Before ChristCAD Computer-Aided DesignCMOS Complementary Metal-Oxide-SiliconCOL collocationCPU Central Processing UnitCT continuous-timeDAC Digital-to-Analog ConverterD/A Digital-to-AnalogDC Direct CurrentDDD Determinant Decision DiagramDECT Digital Enhanced Cordless TelecommunicationsDSP Digital Signal ProcessingDT discrete-timeEDA Electronic Design Automatione.g. exempli gratiaESC Electronic Stability Controletc et ceteraFD Frequency-DomainFFT Fast Fourier TransformFM Frequency ModulationFSM Finite State MachineGA Genetic AlgorithmGMSK Gaussian Minimum Shift KeyingGP Geometric ProgramGSM Global System for Mobile communicationGaAs Gallium ArsenideHB Harmonic BalanceHDL Hardware Description Language

268 Symbols and Abbreviations

HTM Harmonic Transfer MatrixIC Integrated CircuitIEEE Institute of Electrical and Electronics EngineersIFFT Inverse Fast Fourier TransformIF Intermediate FrequencyIP Intellectual PropertyIQ In- and Quadrature-phaseISSCC International Solid-State Circuits ConferenceITRS International Technology Roadmap for SemiconductorsInP Indium PhosphideI In-phasei.e. id estLNA Low Noise AmplifierLPF Low-Pass FilterLTI Linear Time-InvariantMC-CDMA Multi-Carrier Code Division Multiple AccessMCM Multi-Chip-ModuleMEMS Micro-Electro-Mechanical SystemMINLP Mixed Integer NonLinear ProgramMNA Modified Nodal AnalysisMOSFET MOS Field-Effect TransistorMOS Metal-Oxide-SiliconMP3 MPEG-1 Audio Layer 3MPEG Moving Picture Experts GroupNRTZ non-return-to-zeroNWR Newton-Waveform-Relaxationop amp operational amplifierOTA Operational Transconductance Amplifierp. pagePCB Printed Circuit BoardPC Personal ComputerPDT Polyphase Distortion TensorPFT Phase-Frequency TransferPHCM Polyphase Harmonic Correlation MatrixPHPSDM Polyphase Harmonic Power Spectral Density MatrixPHS Polyphase Harmonic SignalPHTM Polyphase Harmonic Transfer MatrixPLL Phase-Locked LoopPM Phase ModulationPSDM Power Spectral Density MatrixPWJ pulse-width jitterQAG Quadrature, Adaptive, General-purposeQ Quadrature-phaseRF Radio FrequencyRMS Root Mean Square

Symbols and Abbreviations 269

RTZ return-to-zeroSD sampled-dataSIA Semiconductor Industry AssociationSPICE Simulation Program with Integrated Circuit EmphasisSQP Sequential Quadratic ProgrammingSiGe Silicon-GermaniumSiP System-in-PackageSi SiliconSoC System-on-ChipTD Time-DomainTM time-marchingVASE VHDL-AMS Synthesis EnvironmentVCCS Voltage-Controlled Current-SourceVCO Voltage Controlled OscillatorVHDL-AMS VHSIC Hardware Description Language – Analog and

Mixed-SignalVHSIC Very High-Speed Integrated CircuitsVLSI Very Large Scale IntegrationVerilog-AMS Verification and logic – Analog and Mixed-SignalWLAN Wireless Local Area NetworkWRN waveform-relaxation-NewtonWR waveform relaxationw.r.t. with respect toWSTS World Semiconductor Trade StatisticsWWII World War II

Index

abstraction level, 22abstraction–description plane, 22,

23–25, 39alphabet, 69

∆Σ modulator, 96–98RF receiver, 188–189sampled-data system, 130–131

analog-to-digital converterfigure of merit, 44frequency domain, see sampling,

frequency domainfunctional description, 227, 245functionality description, 245synthesis, 243–252transformations, 222, 235, 237,

247–250analysis, 12, 23

symbolic, see symbolic analysisanalytical equations

automatically, see symbolic analysisuser-defined, 27

analytical integration, 108annealing, 50architecture, 24

base transformationPHPSDM, 173PHS, 160PHTM, 167polyphase filter, 161polyphase signal, 156

baseband equivalent, see equivalentlow-pass transformation

behavioral modeldescription level, 21, 64, 67generic, see generic behavioral modelspecific, 68, 124, 226, 248

C++, 243C -function, 106calculator, 219, 228, 246charge injection, 130, 134Chebyshev polynomial, 115, 116, 126circuit level, 21, 40, 64class, 71

∆Σ modulator, 95RF receiver, 187sampled-data system, 129

clock feedthrough, 130, 134–138cluster, 223, 236, 250collocation, 91, 114–117common mode, 155, 157, 161complex filter, see symmetrical

polyphase filtercomputer-aided design, 4, 6, 11–12concurrent task, see parallelismconstraints, 25, 48–50, 233

transformation, 31, 56convergence, 46, 221cost function, 48, 234crossover, see recombination

∆Σ modulator, 92clock, 97loop filter, 96, 102–118, 120–122quantizer, 95, 118–119

272 Index

description level, 19, 19–21, 64design flow

bottom-up, 30, 59–60, 63flat, 30, 52, 63generic, 28–29top-down, 31, 55–56, 63, 67, 214–215,

219–221design knowledge, see rulesdesign structure, 222differential mode, 155, 157, 161discrete-time, see sampled-data systemdistortion tensor, see polyphase

distortion tensor

E -function, 106embryonic

circuit, 62design, 220, 225

equivalent low-pass transformation,148, 159, 164

error function, see cost functionevaluator, 219, 228, 232, 234, 235, 247event-driven simulation, 94, 100, 122,

132evolution, 51, 67

evolutionary algorithm, 54, 55, 59,218–219

genetic algorithm, 51, 55genetic programming, 51, 62, 223

exploration, 24, 30, 40architectural, 57–62, 219–243parametric, 44–51, 239, 250

feasibility function, 28, 43–44figure of merit, 44fitness, 51flat

design methodology, 30template, 54–55

frequency set, 157conversion, see re-arrangement of

frequenciesfunctional description, 21, 55, 64,

225–228, 245fuzzy, 43, 216, 233

generic behavioral model, 68∆Σ modulator, 93–102RF receiver, 188–192sampled-data system, 130–132

generic function, 69∆Σ modulator, 98–99RF receiver, 189–190sampled-data system, 131

genetic algorithm, see evolution, geneticalgorithm

genetic programming, see evolution,genetic programming

geometric program, 49global optimum, 45, 242greedy, 50, 230

hardware description language, 21, 71,225

SystemC, 122–123, 136VHDL-AMS, 55, 124–129

harmonic balance, 147harmonic distortion

seeweak nonlinearities, 138harmonic transfer matrix, 167, 194heuristics, 43, 45, 48hierarchical

decomposition, 65, 91template, 53–54

hysteresis, 95, 118

image rejection ratio, 199–202, 237–238inner product, 182integrator, 102–105, 111–112interaction model, 69

∆Σ modulator, 100–102RF receiver, 190–192sampled-data system, 132

intercept point, 177, 202inverse Laplace transformation, 89IQ architecture, 152, 153, 161

jitter, 90, 94, 119–120, 128–129

layout, 11description level, 19, 48, 64generation tools, 48, 49, 51, 63

linear distortion∆Σ modulator, 94, 102–109, 125frequency domain, 152PFT model, 160–171RF receiver, 199–202sampled-data system, 134

linear response, 106

Index 273

machine, 70∆Σ modulator, 102RF receiver, 192sampled-data system, 132

macro-model, 21, 27, 64integrator, 104, 111

MATLAB, 198matrix box plot, 169meet-in-the-middle, 32, 63, 67mixed integer nonlinear program, 55model of computation, 70modeling, 11, 32

behavioral, see behavioral modelformal, 70frequency domain, 145–151functional, see functional descriptiongeneric, 67macro-model, see macro-modelmodel generation, 23performance, see performance,

performance modelrestrictions, 45time domain, 86–92

Moore’s law, 5multi-objective, 41, 45, 49, 51multi-rate multi-carrier model, 149mutation, 51, 238–240, 250

negative frequencies, 156, 159, 165, 178noise, see thermal noisenoise figure, 153, 238

objective function, see cost functionobjective-driven selection, 240offset, 95, 118optimization, 40

architecture, 58–59heterogeneous, 52–55, 59, 62, 67,

213–254parameters, 44–51

orthogonal polynomial, 115

parallelism, 217, 228, 230parallel virtual machine, 244

parameter, 25performance

calculator, see calculatorestimation, 56–57heterogeneity, 66, 216

performance function, 26performance model, 27, 103performance value, 25, 228, 233

perturbation, 113–117phase converter, 192phase-converter, 162phase-frequency transfer model,

151–186polyphase base, 155

transformation, see base transforma-tion

polyphase distortion tensor, 177polynomial nonlinearity, 178second-order nonlinearity, 180–181third-order nonlinearity, 181–184Volterra kernel, 184

polyphase filter, 161polyphase harmonic correlation matrix,

172polyphase harmonic polyphase density

matrixfiltering, 174–177

polyphase harmonic power spectraldensity matrix, 172

polyphase harmonic signal, 157polyphase harmonic transfer matrix,

165graphical, 169–171LTI filter, 167, 192–193mixer, 167operations, 168–169polyphase mixer, 175, 193–195sampled-data system, 197sampling, 195–196

polyphase signal, 154population, 219, 222–224posynomial, 49power spectral density matrix, 172productivity gap, 9pseudo-spectral, see collocationpulse

digital-to-analog converter, 94, 97,119, 128–129

Dirac, 109, 135sampling, 187, 195

re-arrangement of frequenciesPHS, 160PHTM, 167

274 Index

recombination, 51, 62, 238–240, 250refinement

∆Σ modulator, 125formal, 23, 29frequency-domain signals, 159, 189

relaxationfrequency domain, 150, 169, 192time domain, 90–91, 114

RF systemtransformations, 237–238

rules, 43, 46–237heuristic, see heuristicssizing rules method, 45, 48

S -function, 106sampled-data system, 129

frequency domain, 157, 163–164, 197time-domain, 130–138

samplingfrequency domain, 195–196, 202–203time domain, 97, 120

satisfaction function, 229, 233satisfaction level, 229–231, 232,

240–242, 252saturation

frequency domain, 186time domain, 117–118, 127

selectionbefore or after dimensioning, 41–51during dimensioning, 51–55evolutionary operation, 51, 240–242

simplificationformal, 23, 28frequency-domain signals, 190population, 221, 223, 232

simulated annealing, see annealingsimulation, 11, 26, 67, 71

calculator, 246∆Σ modulator, 93–94, 100–102,

123–129discrete-time ∆Σ converter, 89frequency domain, 145–151phase-locked loop, 89RF receiver, 190–192, 199–203RF system, 149, 150sampled-data system, 89–90, 132,

136–138switched-capacitor filter, 89, 137time domain, 86–92

single-phases, 155

SPICE, 11, 88

state-space model

linear, 102, 103, 135–136

weakly nonlinear, 110, 112

strong nonlinearities

∆Σ modulator, 94, 117–118, 127

frequency domain, 148, 149

PFT model, 186

sampled-data system, 134

switched-capacitor filter, see sampled-data system

switches, 134

symbolic analysis, 12, 27, 169, 200

symmetrical components, 155, 156,161, 163, 170, 199

symmetrical polyphase filter, 163

synthesis, 12, 32

high-level, 13, 213–254

synthesis tools

amplifier, 49, 51, 62

analog cells, 48–50, 62

analog filter, 49, 58

analog-to-digital converter, 49, 51, 59

∆Σ modulator, 42, 44, 51, 54, 55

inductor, 49

layout-aware, 39, 48

low-noise amplifier, 42, 49

mixer, 42, 51

op amp, 43, 44, 48–52, 54, 55

phase-locked loop, 42, 49

power amplifier, 51

RF system, 42, 51

statistical variation, 49, 50

systems, 59

voltage reference, 51

voltage-controlled oscillator, 42, 51

system, 17

SystemC, see hardware descriptionlanguage, SystemC

table-based functions, 26

template

architectural, 52, 52–55

generic function, 102–122, 132–136,192–197

performance model, 27

Index 275

thermal noise

∆Σ modulator, 95, 97, 120–122

PFT model, 171–177

time tyranny, 86

time-marching, 87–88

tolerance, see synthesis tools, statisticalvariation

topology, 24

transfer matrix, 183

transformation

architectural, 59–60

collection, 219, 235

context, 235

∆Σ modulator, 119, 127

during synthesis, 222, 228, 234, 240

equivalent low-pass, see equivalentlow-pass transformation

formal, 24, 28

heterogeneity, 216–217, 236

of control variables, see transforma-tion matrix

polyphase base, see base transforma-tion

selection, 240–242

stochastic, 238–240

transformation matrix, 110, 111

transition function, 89, 99

translationarchitectural mapping, 57–59embryonic design, 225–228formal, 23, 29

treedecision tree, 43design tree, 223, 225, 249, 251–252genetic programming, 51, 62

user-defined equations, see analyticalequations, user-defined

verification, 12, 70VHDL-AMS, see hardware description

language, VHDL-AMSVolterra series, 90, 150, 184

waveform relaxation, see relaxation,time domain

weak nonlinearities∆Σ modulator, 94, 109–117, 126frequency domain, 148–150, 152PFT model, 177–186RF receiver, 202sampled-data system, 134, 137

yield, see synthesis tools, statisticalvariation