Wireless Transceiver Architecture

Bridging RF and Digital Communications

Pierre Baudin

WIRELESS TRANSCEIVERARCHITECTURE

WIRELESS TRANSCEIVERARCHITECTUREBRIDGING RF AND DIGITALCOMMUNICATIONS

Pierre Baudin

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Library of Congress Cataloging-in-Publication Data

Baudin, Pierre (Electrical engineer)Wireless transceiver architecture : bridging RF and digital communications / Pierre Baudin.

pages cmIncludes bibliographical references and index.ISBN 978-1-118-87482-0 (hardback)

1. Radio–Transmitter-receivers. I. Title.TK6564.3.B38 2014621.384–dc23

2014011413

A catalogue record for this book is available from the British Library.

ISBN 9781118874820

Set in 10/12pt Times by Aptara Inc., New Delhi, India

1 2015

To my children, Hugo and Chloe, and in memory of my parents

Contents

Preface xiii

List of Abbreviations xvii

Nomenclature xxi

Part I BETWEEN MAXWELL AND SHANNON

1 The Digital Communications Point of View 31.1 Bandpass Signal Representation 4

1.1.1 RF Signal Complex Modulation 41.1.2 Complex Envelope Concept 81.1.3 Bandpass Signals vs. Complex Envelopes 13

1.2 Bandpass Noise Representation 321.2.1 Gaussian Components 341.2.2 Phase Noise vs. Amplitude Noise 38

1.3 Digital Modulation Examples 441.3.1 Constant Envelope 441.3.2 Complex Modulation 501.3.3 Wideband Modulation 56

1.4 First Transceiver Architecture 661.4.1 Transmit Side 671.4.2 Receive Side 69

2 The Electromagnetism Point of View 732.1 Free Space Radiation 73

2.1.1 Radiated Monochromatic Far-field 742.1.2 Narrowband Modulated Fields 812.1.3 Radiated Power 892.1.4 Free Space Path Loss 94

2.2 Guided Propagation 982.2.1 Transmission Lines 982.2.2 Amplitude Matching 1052.2.3 Power Matching 107

viii Contents

2.3 The Propagation Channel 1152.3.1 Static Behavior 1162.3.2 Dynamic Behavior 1262.3.3 Impact on Receivers 134

3 The Wireless Standards Point of View 1453.1 Medium Access Strategies 145

3.1.1 Multiplexing Users 1453.1.2 Multiplexing Uplink and Downlink 1463.1.3 Impact on Transceivers 149

3.2 Metrics for Transmitters 1513.2.1 Respect for the Wireless Environment 1523.2.2 Transmitted Signal Modulation Quality 161

3.3 Metrics for Receivers 1673.3.1 Resistance to the Wireless Environment 1673.3.2 Received Signal Modulation Quality 174

Part II IMPLEMENTATION LIMITATIONS

4 Noise 1834.1 Analog Electronic Noise 184

4.1.1 Considerations on Analog Electronic Noise 1844.1.2 Thermal Noise 184

4.2 Characterization of Noisy Devices 1864.2.1 Noise Temperatures 1864.2.2 Noise Factor 1914.2.3 Noise Voltage and Current Sources 1994.2.4 Cascade of Noisy Devices 2104.2.5 Illustration 2144.2.6 SNR Degradation 229

4.3 LO Phase Noise 2314.3.1 RF Synthesizers 2324.3.2 Square LO Waveform for Chopper-like Mixers 2434.3.3 System Impact 252

4.4 Linear Error Vector Magnitude 2634.5 Quantization Noise 266

4.5.1 Quantization Error as a Noise 2674.5.2 Sampling Effect on Quantization Noise 2784.5.3 Illustration 282

4.6 Conversion Between Analog and Digital Worlds 2874.6.1 Analog to Digital Conversion 2874.6.2 Digital to Analog Conversion 302

Contents ix

5 Nonlinearity 3075.1 Smooth AM-AM Conversion 308

5.1.1 Smooth AM-AM Conversion Model 3085.1.2 Phase/Frequency Only Modulated RF Signals 3135.1.3 Complex Modulated RF Signals 3395.1.4 SNR Improvement Due to RF Compression 377

5.2 Hard AM-AM Conversion 3925.2.1 Hard Limiter Model 3935.2.2 Hard Limiter Intercept Points 3945.2.3 SNR Improvement in the Hard Limiter 398

5.3 AM-PM Conversion and the Memory Effect 4025.3.1 Device Model 4025.3.2 System Impacts 407

5.4 Baseband Devices 413

6 RF Impairments 4176.1 Frequency Conversion 417

6.1.1 From Complex to Real Frequency Conversions 4176.1.2 Image Signal 4216.1.3 Reconsidering the Complex Frequency Conversion 4236.1.4 Complex Signal Processing Approach 426

6.2 Gain and Phase Imbalance 4376.2.1 Image Rejection Limitation 4376.2.2 Signal Degradation 442

6.3 Mixer Implementation 4536.3.1 Mixers as Choppers 4536.3.2 Impairments in the LO Generation 455

6.4 Frequency Planning 4826.4.1 Impact of the LO Spectral Content 4836.4.2 Clock Spurs 487

6.5 DC Offset and LO Leakage 4896.5.1 LO Leakage on the Transmit Side 4906.5.2 DC Offset on the Receive Side 492

Part III TRANSCEIVER DIMENSIONING

7 Transceiver Budgets 4977.1 Architecture of a Simple Transceiver 4977.2 Budgeting a Transmitter 499

7.2.1 Review of the ZIF TX Problem 4997.2.2 Level Diagrams and Transmitter High Level Parameters 5057.2.3 Budgets Linked to Respect for the Wireless Environment 5117.2.4 Budgets Linked to the Modulation Quality 5247.2.5 Conclusion 531

x Contents

7.3 Budgeting a Receiver 5327.3.1 Review of the ZIF RX Problem 5327.3.2 Level Diagrams and Receiver High Level Parameters 5397.3.3 Budgets Linked to the Resistance to the Wireless Environment 5547.3.4 Budgets Linked to the Modulation Quality 5667.3.5 Conclusion 580

8 Transceiver Architectures 5838.1 Transmitters 583

8.1.1 Direct Conversion Transmitter 5848.1.2 Heterodyne Transmitter 5888.1.3 Variable-IF Transmitter 5928.1.4 Real-IF Transmitter 5948.1.5 PLL Modulator 5968.1.6 Polar Transmitter 6028.1.7 Transmitter Architectures for Power Efficiency 612

8.2 Receivers 6298.2.1 Direct Conversion Receiver 6298.2.2 Heterodyne Receiver 6328.2.3 Low-IF Receiver 6358.2.4 PLL Demodulator 639

9 Algorithms for Transceivers 6439.1 Transmit Side 643

9.1.1 Power Control 6449.1.2 LO Leakage Cancellation 6509.1.3 P/Q Imbalance Compensation 6549.1.4 Predistortion 6619.1.5 Automatic Frequency Correction 6699.1.6 Cartesian to Polar Conversion 672

9.2 Receive Side 6759.2.1 Automatic Gain Control 6759.2.2 DC Offset Cancellation 6809.2.3 P/Q Imbalance Compensation 6839.2.4 Linearization Techniques 6899.2.5 Automatic Frequency Correction 691

APPENDICES

Appendix 1 Correlation 697A1.1 Bandpass Signals Correlations 697A1.2 Properties of Cross-Correlation Functions 703A1.3 Properties of Autocorrelation Functions 704

Contents xi

Appendix 2 Stationarity 707A2.1 Stationary Bandpass Signals 707A2.2 Stationary Complex Envelopes 710A2.3 Gaussian Case 711

Appendix 3 Moments of Normal Random Vectors 713A3.1 Real Normal Random Vectors 713A3.2 Complex Normal Random Vectors 716

References 719

Index 723

Preface

The origins of this book lie in the frequent questions that I have been asked by colleaguesin the different companies I have worked for about how to proceed in the dimensioningand optimization of a transceiver line-up. The recurrence of those questions, along with theproblem of identifying suitable reference sources, made me think there could be a gap in theliterature. There is indeed an abundant literature on the physical implementation of wirelesstransceivers (e.g. the RF/analog CMOS design), or on digital communications theory itself (e.g.the signal processing required), but little on how to proceed for dimensioning and optimizinga transceiver line-up.

Furthermore, the fact is that those questions were coming from two distinct categories ofengineers. On the one hand, RF/analog designers are curious to understand how the specifi-cations of their blocks are derived. On the other hand, the digital signal processing engineersin charge of the baseband algorithms need to understand the mechanisms involved in thedegradation of the wanted signal along the line-up for optimizing their processing. Obviously,it is the job of an RFIC architect to make the link between the two communities and to attemptto overcome the communication problems between those two groups. Roughly speaking, youhave on the one hand the baseband engineers that process complex envelopes while bench-marking their algorithms using AWGN, and on the other hand the RF/analog designers thatoptimize their designs based on the use of CW tones for evaluating the degradation of a signalexpected to be modulated. Based on my experience, even if the difficulty in the discussioncan be related somehow to the different nature of the technical issues addressed by the twocommunities, it is also closely related to the different formalisms traditionally used in thosetwo domains of knowledge.

I therefore wrote this book with two aims in mind. I first tried to detail the mindsetrequired, at least based on my professional experience, to take care of the system designof a transceiver. Expressed like this, we understand that the purpose is not to be exhaustiveabout how to perform such dimensioning for all the architectures one can imagine. Ratherthe goal is to explain the spirit of it, and how to initiate such work in practice. Conversely,in order to be able to react correctly whatever the architecture under consideration, thereis a need to understand as far as possible the constraints we have to take into account inorder to dimension a transceiver. Practically speaking, this means understanding the systemdesign of transceiver line-up in its various aspects. We can, for instance, mention the need tounderstand the purpose of a transceiver from the signal processing perspective. Indeed, fromthe transmitted or received signal point of view, the transceiver implements nothing more thansignal processing functions, mainly analog signal processing, but signal processing when all

xiv Preface

is said and done. Alternatively, we can mention the need to understand the various limitationsone can face in the implementation of this analog signal processing using electronic devices.Those limitations can indeed be encountered whatever the architecture implemented.

I then also tried to unify the formalisms used in the various domains of knowledge involved inthe field of wireless transceivers. In practice, this means considering the digital communicationsformalism and the extensive use of the complex envelope concept for modeling modulatedRF signals. As discussed above, the first goal was to make easier the link between RF anddigital communications people who need to work together in order to optimize a line-up. Thisapproach also happens to have many additional benefits. It allows us to correctly define RFconcepts for modulated signals often introduced in a more intuitive way. It also allows us toperform straightforward analytical derivations in many situations of interest, as in nonlinearityfor instance, while allowing explicit graphical representations in the complex plane. I am nowfully convinced that this formalism is of much interest to RF problems and I would be pleasedif this book can help to propagate its use.

As a result, this book consists of three parts. Part I focuses on the explanation of what isexpected from a transceiver. This part is composed of three chapters dedicated to the threeareas that drive those requirements: (a) the digital communications theory itself, which allowsus to define the minimum set of signal processing functions to be embedded in a transceiver,as well as introducing key concepts such as complex envelopes; (b) the electromagnetismtheory, as theoretical results in the field of propagation allow us to explain some architecturalconstraints for transceivers; (c) the practical organization of wireless networks, as it drivesmost of the performance required from transceivers in practice. By the end of Part I we shouldthus have an understanding of the functionalities required in a transceiver as well as theirassociated performance.

Part II is then dedicated to a review of the limitations we face in the physical implementationusing electronic devices of the signal processing functions derived in Part I. Those limitationsare sorted into three groups, leading to three chapters dedicated to: (a) the noise sources to beconsidered in a line-up; (b) the nonlinearity in RF/analog components; (c) what are classicallylabeled RF impairments.

Part III then turns to the transceiver architecture and system design itself. We can now focuson how to dimension a transceiver that fulfills the requirements derived in Part I while takinginto account the implementation limitations reviewed in Part II. Practically speaking, this isdone through three chapters. The first of these is dedicated to the illustration of a transceiverbudget for a given architecture. This shows how a practical line-up budget can be done, i.e.how the constraints linked to the implementation limitations can be balanced between thevarious blocks of a given line-up in order to achieve the performance. The second chapterreviews different architectures of transceivers. In contrast to what is done in the previouschapter, we can see here how the fundamental limitations of a given line-up can be overcomeby changing its architecture. The third chapter then examines some algorithms classically usedfor improving or optimizing the performance of transceiver line-ups.

At this stage, I need to highlight that, due to the organization of the book, only the rea-soning used for the architecture and system design of transceivers is discussed in Part III.All the theoretical results, as well as the description of the elementary phenomena that areinvolved in this area, are detailed in Parts I and II. As a result, I recommend that the readershould not embark on Part III without sufficient understanding of the phenomena discussed inParts I and II.

Preface xv

To conclude, I would like to thank all those people who helped in completing this project.First of all, I would like to thank my former colleagues at Renesas who participated in one wayor another during this project, i.e. Alexis Bisiaux, Pascal Le Corre, Mikael Guenais, StephanePaquelet, Arnaud Rigolle, Patrick Savelli, and in particular Larbi Azzoug and Anis Latiri.Then, I would like to warmly thank Marc Helier, who taught me microwave engineering atSupelec some years ago, and who was kind enough to go through Chapter 2. Finally, I wouldlike to thank Fabrice Belveze, as it was all the good technical discussions we had during theold STMicroelectronics times that first convinced me that it was interesting to using the digitalcommunications formalism for the system design of transceiver line-up. Things have changedsince then, but the origins are there.

Rennes, January 2014

List of Abbreviations

ABB Analog BasebandACPR Adjacent Channel Power RatioACLR Adjacent Channel Leakage RatioACS Adjacent Channel SelectivityADC Analog to Digital ConverterADPLL All Digital PLLAFC Automatic Frequency CorrectionAGC Automatic Gain ControlAM Amplitude ModulationAWGN Additive White Gaussian NoiseBBIC Baseband ICBER Bit Error RateBTS Base Transceiver StationCALLUM Combined Analog Locked Loop Universal ModulatorCCP Cross-Compression PointCDMA Code Division Multiple AccessCDE Code Domain ErrorCDF Cumulative Distribution FunctionCCDF Complementary Cumulative Distribution FunctionCDP Code Domain PowerCF Crest FactorCMOS Complementary Metal Oxide SemiconductorCORDIC COordinate Rotation DIgital ComputerCP Compression PointCW Continuous WaveDAC Digital to Analog ConverterDC Direct CurrentDNL Differential NonLinearityDR Dynamic RangeDSB Double SideBandDtyCy Duty CycleEDGE Enhanced Data Rates for GSM EvolutionEER Envelope Elimination and RestorationEMF ElectroMotive Force

xviii List of Abbreviations

EMI ElectroMagnetic InterferenceERR Even Order Rejection RatioET Envelope TrackingEVM Error Vector MagnitudeFDD Frequency Division DuplexFDMA Frequency Division Multiple AccessFE Front-EndFEM Front-End ModuleFIR Finite Impulse ResponseFM Frequency ModulationFS Full ScaleGMSK Gaussian Minimum Shift KeyingGSM Global System for Mobile communicationsHPSK Hybrid Phase Shift KeyingI/F InterFaceIC Integrated CircuitICP Input Compression PointICCP Input Cross-Compression PointIF Intermediate FrequencyIIP Input Intercept PointIMD Intermodulation DistortionINL Integral NonLinearityIP Intercept PointIPsat Input Saturated PowerIRR Image Rejection RatioISI InterSymbol InterferenceISR Input Spurious RejectionLINC LInear amplification using Nonlinear ComponentLNA Low Noise AmplifierLO Local OscillatorLSB Least Significant BitLTE Long-Term EvolutionLUT LookUp TableNCO Numerically Controlled OscillatorNF Noise FigureOCP Output Compression PointOFDM Orthogonal Frequency Division MultiplexingOIMD Output InterModulation DistortionOIP Output Intercept PointOPsat Output Saturated PowerOSR OverSampling RatioPA Power AmplifierPAPR Peak to Average Power RatioPGA Programmable Gain AmplifierPDF Probability Density FunctionPFD Phase Frequency Detector

List of Abbreviations xix

PLL Phase Locked LoopPM Phase ModulationPsat Saturated PowerPSD Power Spectral DensityPSK Phase Shift KeyingQAM Quadrature Amplitude ModulationRF Radio FrequencyRFIC Radio Frequency Integrated CircuitRL Return LossRMS Root Mean SquareRRC Root Raised CosineRX ReceiverRXFE RX Front-EndSEM Spectrum Emission MaskSFDR Spurious Free Dynamic RangeSiNAD Signal to Noise and Distortion ratioSNR Signal to Noise Power RatioSSB Single SideBandTDD Time Division DuplexTDMA Time Division Multiple AccessTE Transverse ElectricTEM Transverse ElectromagneticTM Transverse MagneticTHD Total Harmonic DistortionTX TransmitterTRX TransceiverUE User EquipmentVGA Variable Gain AmplifierVSWR Voltage Standing Wave RatioWCDMA Wideband CDMAWSS Wide Sense StationarityXM Cross-ModulationZIF Zero-IF

Nomenclature

t, 𝜏 time variablesf frequency variable𝜔 angular frequency variable (= 2𝜋f )x(t) continuous time signalx[n] discrete time signalX( f ), X(𝜔) frequency representations of x(t) or x[n]ℱ{x(t)}( f ), ℱ{x(t)}(𝜔) Fourier transforms of x(t)v, i, j voltage, current, current densityP , p (t) in-phase, in-phase componentQ , q (t) quadrature, quadrature component

j√−1

Re{.}, Im{.} real part, imaginary part|.|, arg{.} modulus, argument.∗ complex conjugate⋆ convolution𝛿(.) Dirac delta distributionU(.) Heaviside unit step functionxa(t) analytical signal associated with x(t)Xa( f ), Xa(𝜔) frequency domain representations of xa(t)x(t) Hilbert transform of x(t)X( f ), X(𝜔) frequency domain representations of x(t)x(t) complex envelope associated with x(t)X( f ), X(𝜔) frequency domain representations of x(t)(.) time average valueE{.} stochastic expectation value𝛾x×y(t1, t2) cross-correlation function

(=E

{xt1

y ∗t2

})𝛾x×x(t1, t2) autocorrelation function𝛾x×x(𝜏) autocorrelation function in stationary case

(=E

{xtx

∗t−𝜏

})Γx×x( f ), Γx×x(𝜔) power spectral densities of x(t)X, X vector, matrix.T transpose‖.‖ Hermitian norm⋅ dot product× cross product

Part IBetween Maxwelland Shannon

1The Digital CommunicationsPoint of View

When detailing how to dimension a transceiver, it can seem natural to first clarify what isexpected from such a system. This means understanding both the minimum set of functionsthat need to be implemented in a transceiver line-up as well as the minimum performanceexpected from them. In practice, these requirements come from different topics which canbe sorted into three groups. We can indeed refer to the signal processing associated withthe modulations encountered in digital communications, to the physics of the medium usedfor the propagation of the information, and to the organization of wireless networks whenconsidering a transceiver that belongs to such system, or alternatively its coexistence withsuch systems.

The last two topics are discussed in Chapter 2 and Chapter 3 respectively, while this chapterfocuses on the consequences for transceiver architectures of the signal processing associatedwith the digital communications. In that perspective, a first set of functions to be embeddedin such a system can be derived from the inspection of the relationship that holds between themodulating waveforms used in this area and the corresponding modulated RF signals to bepropagated in the channel medium.

As a side effect, this approach enables us to understand how information that needs acomplex baseband modulating signal to be represented can be carried by a simple real valuedRF signal, thus leading to the key concept of the complex envelope. It is interesting to see thatthis concept allows us to define correctly classical quantities used to characterize RF signalsand noise, in addition to its usefulness for performing analytical derivations. It is thereforeused extensively throughout this book.

Finally, in this chapter we also review some particular modulation schemes that are rep-resentative of the different statistics that can be encountered in classical wireless standards.These schemes are then used as examples to illustrate subsequent derivations in this book.

Wireless Transceiver Architecture: Bridging RF and Digital Communications, First Edition. Pierre Baudin.© 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.

4 The Digital Communications Point of View

1.1 Bandpass Signal Representation

1.1.1 RF Signal Complex Modulation

Digital modulating waveforms in their most general form are represented by a complexsignal function of time in digital communications books [1]. But, even if we understandthat this complex signal allows us to increase the number of bits per second that can betransmitted by working on symbols using this two-dimensional space, a question remains.The final RF signal that carries the information, like the RF current or voltage generated atthe transmitter (TX) output, is a real valued signal like any physical quantity that can bemeasured. Accordingly, we may wonder how the information that needs a complex signalto be represented can be carried by such an RF signal. Any RF engineer would respond bysaying that an electromagnetic wave has an amplitude and a phase that can be modulatedindependently. Nevertheless, we can anticipate the discussion in Chapter 2, and in particularin Section 2.1.2, by saying that there is nothing in the electromagnetic theory that requires thisparticular structure for the time dependent part of the electromagnetic field. In fact, the rightargument remains that this time dependent part, like any real valued signal, can be representedby two independent quantities that can be interpreted as its instantaneous amplitude and itsinstantaneous phase as long as it is a bandpass signal. Here, “bandpass signal” means thatthe spectral content of the signal has no low frequency component that spreads down to thezero frequency. In other words, the spectrum of the RF signal considered, whose positive andnegative sidebands are assumed centered around ±𝜔c, must be non-vanishing only for angularfrequencies in [−𝜔u − 𝜔c,−𝜔c + 𝜔l] ∪ [+𝜔c − 𝜔l,+𝜔c + 𝜔u], with 𝜔c, 𝜔l and 𝜔u defined aspositive quantities, and with 𝜔c > 𝜔l.

To understand this behavior, let us consider the complex baseband signal s(t) expressed as

s(t) = p (t) + jq (t), (1.1)

where p (t) and q (t) are respectively the real and imaginary parts of this complex signal. Wecan assume that the spectrum of this signal spreads over [−𝜔l,+𝜔u]. Such baseband signalswith a non-vanishing DC component in their spectrum are called lowpass signals in contrastto the bandpass signals as given above. If we now wish to shift the spectrum of this signalaround the central carrier angular frequency +𝜔c, we have to convolve its spectrum with theDirac delta distribution 𝛿(𝜔 − 𝜔c). In the time domain, this means multiplying the signal bythe Fourier transform of this Dirac delta distribution, i.e. the complex exponential e+j𝜔ct [2].This results in the complex signal sa(t) defined by

sa(t) = s(t)e+j𝜔ct = (p (t) + jq (t))e+j𝜔ct. (1.2)

Suppose now that we take the real part of this signal. Using

e+j𝜔t = cos(𝜔t) + j sin(𝜔t), (1.3)

we get the classical form of the resulting RF signal s(t) we are looking for,

s(t) = Re{sa(t)} = p (t) cos(𝜔ct) − q (t) sin(𝜔ct). (1.4)

Bandpass Signal Representation 5

But what is interesting to see is that even if we took only the real part of the upconverted initialcomplex lowpass signal transposed around +𝜔c, we have no loss of information comparedto the initial complex baseband signal as long as 𝜔c > 𝜔l. Indeed, under that condition, theoriginal complex modulating waveform s(t) can be reconstructed from the bandpass RF realsignal s(t).

To understand this, let us first consider the spectral content of the resulting bandpass signals(t). What would be a good mathematical tool to choose for the spectral analysis? Dealing withdigital modulations that are randomly modulated most of the time, the natural choice wouldbe to use the stochastic approach to derive the signal power spectral density. The problemwith this approach is that the power spectral density (PSD) of a signal is only linked to themodulus of the Fourier transform of the original signal. It thus leads to a loss of informationcompared to the time domain signal. As a result, in some cases of interest in this book, we needto keep the simple Fourier transform representation in order to be able to discuss the phaserelationship between different sidebands present in the spectrum. Here, by “sideband” wemean a non-vanishing portion of spectrum of finite frequency support. This phase relationshipis indeed required to understand the underlying phenomenon involved in concepts as reviewedin this chapter, but also in Chapter 6, for instance, when dealing with frequency conversionand image rejection. The existence of such Fourier transforms can be justified thanks to thepractical finite temporal support of the signals of interest that ensures a finite energy. Thisis indeed the practical use case when dealing with the post-processing of a finite durationmeasurement or simulation result. The signals we deal with are therefore assumed to have afinite temporal support and a finite energy, i.e. they are assumed to belong to L2[0, T], the spaceof square-integrable functions over the bounded interval [0, T]. Nevertheless, when dealingwith a randomly modulated signal, this approach means that we consider only the spectralproperties of a single realization of the process of interest. Thus, even if this direct Fourieranalysis is suitable for discussing some signal processing operations involved in transceivers,the power spectral analysis should be considered when possible for taking into account thestatistical properties of the modulating process of interest, as done in “Power spectral density”(Section 1.1.3).

Let us therefore derive the Fourier transform of s(t). As the aim is to make the link betweenthe spectral representation of s(t) and that of s(t), we can first expand the relationship betweens(t) and the complex signal sa(t) given by equation (1.4). To do so, we use the general propertythat for any complex number s(t), we have

Re{s(t)} = 12

(s(t) + s∗(t)), (1.5)

where s∗(t) stands for the complex conjugate of s(t). This means that we can write

s(t) = Re{sa(t)} = 12

(sa(t) + s∗a (t)). (1.6)

Using the relationship between sa(t) and s(t) given by equation (1.2), we finally get that

s(t) = 12

s(t)e+j𝜔ct + 12

s∗(t)e−j𝜔ct. (1.7)

6 The Digital Communications Point of View

It now remains to take the Fourier transform of this signal. For that, we can use two propertiesof the Fourier transform. The first states that for any signal, s(t), the Fourier transform of thecomplex conjugate, s∗(t), of such a signal can be related to that of s(t) through

ℱ{s∗(t)}(𝜔) = ∫+∞

−∞s∗(t)e−j𝜔tdt

=[∫

+∞

−∞s(t)e+j𝜔tdt

]∗=

[ℱ{s(t)}(−𝜔)

]∗. (1.8)

We observe that this derivation remains valid when s(t) reduces to a real signal s(t). In thatcase, having s∗(t) = s(t) leads to having S∗(−𝜔) = S(𝜔). We then recover the classical propertyof real signals, i.e. the Hermitian symmetry of their spectrum. Then we can use the propertythat the Fourier transform of a product of signals is equal to the convolution of the Fouriertransforms of each signal. Indeed, we get that

ℱ{s1(t)s2(t)}(𝜔) = ∫+∞

−∞s1(t)s2(t)e−j𝜔tdt

= ∫+∞

−∞S1(𝜔′)∫

+∞

−∞s2(t)e−j(𝜔−𝜔′)tdtd𝜔′

= ∫+∞

−∞S1(𝜔′)S2(𝜔 − 𝜔′)d𝜔′, (1.9)

i.e. that

ℱ{s1(t)s2(t)}(𝜔) = ℱ{s1(t)}(𝜔) ⋆ℱ{s2(t)}(𝜔). (1.10)

Thus, using the two properties above, we get that the Fourier transform of equation (1.7)reduces to

S(𝜔) = 12

S(𝜔) ⋆ 𝛿(𝜔 − 𝜔c) + 12

S∗(−𝜔) ⋆ 𝛿(𝜔 + 𝜔c), (1.11)

where1 S(𝜔) stands for the Fourier transform of s(t) and where the Dirac delta distribution isthe Fourier transform of the complex exponential. As this distribution is even, i.e. we have𝛿(𝜔 + 𝜔c) = 𝛿(−𝜔 − 𝜔c), the spectrum of s(t) can be expressed as the sum of two componentsas illustrated in Figure 1.1. The first component corresponds to the positive frequencies part,denoted S+(𝜔), and is referred to as the positive sideband of the spectrum of s(t). The secondcomponent corresponds to the negative part of the spectrum, S−(𝜔), and is therefore referredto as the negative sideband of the spectrum of s(t). As s(t) is assumed to be bandpass, there is

1 Recall our convention in this book that S(𝜔) stands for the spectral domain representation of the complex envelopes(t) and not for the complex envelope of the signal S(𝜔).