Wireless Transceiver Architecture

Transcription

1 Pierre Baudin Wireless Transceiver Architecture Bridging RF and Digital Communications

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3 WIRELESS TRANSCEIVER ARCHITECTURE

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5 WIRELESS TRANSCEIVER ARCHITECTURE BRIDGING RF AND DIGITAL COMMUNICATIONS Pierre Baudin

6 This edition first published John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Baudin, Pierre (Electrical engineer) Wireless transceiver architecture : bridging RF and digital communications / Pierre Baudin. pages cm Includes bibliographical references and index. ISBN (hardback) 1. Radio Transmitter-receivers. I. Title. TK B dc A catalogue record for this book is available from the British Library. ISBN Set in 10/12pt Times by Aptara Inc., New Delhi, India

7 To my children, Hugo and Chloé, and in memory of my parents

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9 Contents Preface List of Abbreviations Nomenclature xiii xvii xxi Part I BETWEEN MAXWELL AND SHANNON 1 The Digital Communications Point of View Bandpass Signal Representation RF Signal Complex Modulation Complex Envelope Concept Bandpass Signals vs. Complex Envelopes Bandpass Noise Representation Gaussian Components Phase Noise vs. Amplitude Noise Digital Modulation Examples Constant Envelope Complex Modulation Wideband Modulation First Transceiver Architecture Transmit Side Receive Side 69 2 The Electromagnetism Point of View Free Space Radiation Radiated Monochromatic Far-field Narrowband Modulated Fields Radiated Power Free Space Path Loss Guided Propagation Transmission Lines Amplitude Matching Power Matching 107

10 viii Contents 2.3 The Propagation Channel Static Behavior Dynamic Behavior Impact on Receivers The Wireless Standards Point of View Medium Access Strategies Multiplexing Users Multiplexing Uplink and Downlink Impact on Transceivers Metrics for Transmitters Respect for the Wireless Environment Transmitted Signal Modulation Quality Metrics for Receivers Resistance to the Wireless Environment Received Signal Modulation Quality 174 Part II IMPLEMENTATION LIMITATIONS 4 Noise Analog Electronic Noise Considerations on Analog Electronic Noise Thermal Noise Characterization of Noisy Devices Noise Temperatures Noise Factor Noise Voltage and Current Sources Cascade of Noisy Devices Illustration SNR Degradation LO Phase Noise RF Synthesizers Square LO Waveform for Chopper-like Mixers System Impact Linear Error Vector Magnitude Quantization Noise Quantization Error as a Noise Sampling Effect on Quantization Noise Illustration Conversion Between Analog and Digital Worlds Analog to Digital Conversion Digital to Analog Conversion 302

11 Contents ix 5 Nonlinearity Smooth AM-AM Conversion Smooth AM-AM Conversion Model Phase/Frequency Only Modulated RF Signals Complex Modulated RF Signals SNR Improvement Due to RF Compression Hard AM-AM Conversion Hard Limiter Model Hard Limiter Intercept Points SNR Improvement in the Hard Limiter AM-PM Conversion and the Memory Effect Device Model System Impacts Baseband Devices RF Impairments Frequency Conversion From Complex to Real Frequency Conversions Image Signal Reconsidering the Complex Frequency Conversion Complex Signal Processing Approach Gain and Phase Imbalance Image Rejection Limitation Signal Degradation Mixer Implementation Mixers as Choppers Impairments in the LO Generation Frequency Planning Impact of the LO Spectral Content Clock Spurs DC Offset and LO Leakage LO Leakage on the Transmit Side DC Offset on the Receive Side 492 Part III TRANSCEIVER DIMENSIONING 7 Transceiver Budgets Architecture of a Simple Transceiver Budgeting a Transmitter Review of the ZIF TX Problem Level Diagrams and Transmitter High Level Parameters Budgets Linked to Respect for the Wireless Environment Budgets Linked to the Modulation Quality Conclusion 531

12 x Contents 7.3 Budgeting a Receiver Review of the ZIF RX Problem Level Diagrams and Receiver High Level Parameters Budgets Linked to the Resistance to the Wireless Environment Budgets Linked to the Modulation Quality Conclusion Transceiver Architectures Transmitters Direct Conversion Transmitter Heterodyne Transmitter Variable-IF Transmitter Real-IF Transmitter PLL Modulator Polar Transmitter Transmitter Architectures for Power Efficiency Receivers Direct Conversion Receiver Heterodyne Receiver Low-IF Receiver PLL Demodulator Algorithms for Transceivers Transmit Side Power Control LO Leakage Cancellation P/Q Imbalance Compensation Predistortion Automatic Frequency Correction Cartesian to Polar Conversion Receive Side Automatic Gain Control DC Offset Cancellation P/Q Imbalance Compensation Linearization Techniques Automatic Frequency Correction 691 APPENDICES Appendix 1 Correlation 697 A1.1 Bandpass Signals Correlations 697 A1.2 Properties of Cross-Correlation Functions 703 A1.3 Properties of Autocorrelation Functions 704

13 Contents xi Appendix 2 Stationarity 707 A2.1 Stationary Bandpass Signals 707 A2.2 Stationary Complex Envelopes 710 A2.3 Gaussian Case 711 Appendix 3 Moments of Normal Random Vectors 713 A3.1 Real Normal Random Vectors 713 A3.2 Complex Normal Random Vectors 716 References 719 Index 723

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15 Preface The origins of this book lie in the frequent questions that I have been asked by colleagues in the different companies I have worked for about how to proceed in the dimensioning and optimization of a transceiver line-up. The recurrence of those questions, along with the problem of identifying suitable reference sources, made me think there could be a gap in the literature. There is indeed an abundant literature on the physical implementation of wireless transceivers (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 optimizing a transceiver line-up. Furthermore, the fact is that those questions were coming from two distinct categories of engineers. On the one hand, RF/analog designers are curious to understand how the specifications of their blocks are derived. On the other hand, the digital signal processing engineers in charge of the baseband algorithms need to understand the mechanisms involved in the degradation 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 attempt to overcome the communication problems between those two groups. Roughly speaking, you have on the one hand the baseband engineers that process complex envelopes while benchmarking their algorithms using AWGN, and on the other hand the RF/analog designers that optimize their designs based on the use of CW tones for evaluating the degradation of a signal expected to be modulated. Based on my experience, even if the difficulty in the discussion can be related somehow to the different nature of the technical issues addressed by the two communities, it is also closely related to the different formalisms traditionally used in those two domains of knowledge. I therefore wrote this book with two aims in mind. I first tried to detail the mindset required, at least based on my professional experience, to take care of the system design of a transceiver. Expressed like this, we understand that the purpose is not to be exhaustive about how to perform such dimensioning for all the architectures one can imagine. Rather the 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, there is a need to understand as far as possible the constraints we have to take into account in order to dimension a transceiver. Practically speaking, this means understanding the system design of transceiver line-up in its various aspects. We can, for instance, mention the need to understand the purpose of a transceiver from the signal processing perspective. Indeed, from the transmitted or received signal point of view, the transceiver implements nothing more than signal processing functions, mainly analog signal processing, but signal processing when all

16 xiv Preface is said and done. Alternatively, we can mention the need to understand the various limitations one 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 in the field of wireless transceivers. In practice, this means considering the digital communications formalism and the extensive use of the complex envelope concept for modeling modulated RF signals. As discussed above, the first goal was to make easier the link between RF and digital communications people who need to work together in order to optimize a line-up. This approach also happens to have many additional benefits. It allows us to correctly define RF concepts for modulated signals often introduced in a more intuitive way. It also allows us to perform straightforward analytical derivations in many situations of interest, as in nonlinearity for instance, while allowing explicit graphical representations in the complex plane. I am now fully convinced that this formalism is of much interest to RF problems and I would be pleased if 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 is expected from a transceiver. This part is composed of three chapters dedicated to the three areas that drive those requirements: (a) the digital communications theory itself, which allows us 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 electromagnetism theory, as theoretical results in the field of propagation allow us to explain some architectural constraints for transceivers; (c) the practical organization of wireless networks, as it drives most of the performance required from transceivers in practice. By the end of Part I we should thus have an understanding of the functionalities required in a transceiver as well as their associated performance. Part II is then dedicated to a review of the limitations we face in the physical implementation using electronic devices of the signal processing functions derived in Part I. Those limitations are sorted into three groups, leading to three chapters dedicated to: (a) the noise sources to be considered in a line-up; (b) the nonlinearity in RF/analog components; (c) what are classically labeled RF impairments. Part III then turns to the transceiver architecture and system design itself. We can now focus on how to dimension a transceiver that fulfills the requirements derived in Part I while taking into account the implementation limitations reviewed in Part II. Practically speaking, this is done through three chapters. The first of these is dedicated to the illustration of a transceiver budget 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 the various blocks of a given line-up in order to achieve the performance. The second chapter reviews different architectures of transceivers. In contrast to what is done in the previous chapter, we can see here how the fundamental limitations of a given line-up can be overcome by changing its architecture. The third chapter then examines some algorithms classically used for 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 reasoning 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 are involved in this area, are detailed in Parts I and II. As a result, I recommend that the reader should not embark on Part III without sufficient understanding of the phenomena discussed in Parts I and II.

17 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 way or another during this project, i.e. Alexis Bisiaux, Pascal Le Corre, Mikaël Guenais, Stéphane Paquelet, Arnaud Rigollé, Patrick Savelli, and in particular Larbi Azzoug and Anis Latiri. Then, I would like to warmly thank Marc Hélier, who taught me microwave engineering at Supélec some years ago, and who was kind enough to go through Chapter 2. Finally, I would like to thank Fabrice Belvèze, as it was all the good technical discussions we had during the old STMicroelectronics times that first convinced me that it was interesting to using the digital communications formalism for the system design of transceiver line-up. Things have changed since then, but the origins are there. Rennes, January 2014

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19 List of Abbreviations ABB ACPR ACLR ACS ADC ADPLL AFC AGC AM AWGN BBIC BER BTS CALLUM CCP CDMA CDE CDF CCDF CDP CF CMOS CORDIC CP CW DAC DC DNL DR DSB DtyCy EDGE EER EMF Analog Baseband Adjacent Channel Power Ratio Adjacent Channel Leakage Ratio Adjacent Channel Selectivity Analog to Digital Converter All Digital PLL Automatic Frequency Correction Automatic Gain Control Amplitude Modulation Additive White Gaussian Noise Baseband IC Bit Error Rate Base Transceiver Station Combined Analog Locked Loop Universal Modulator Cross-Compression Point Code Division Multiple Access Code Domain Error Cumulative Distribution Function Complementary Cumulative Distribution Function Code Domain Power Crest Factor Complementary Metal Oxide Semiconductor COordinate Rotation DIgital Computer Compression Point Continuous Wave Digital to Analog Converter Direct Current Differential NonLinearity Dynamic Range Double SideBand Duty Cycle Enhanced Data Rates for GSM Evolution Envelope Elimination and Restoration ElectroMotive Force

20 xviii List of Abbreviations EMI ERR ET EVM FDD FDMA FE FEM FIR FM FS GMSK GSM HPSK I/F IC ICP ICCP IF IIP IMD INL IP IPsat IRR ISI ISR LINC LNA LO LSB LTE LUT NCO NF OCP OFDM OIMD OIP OPsat OSR PA PAPR PGA PDF PFD ElectroMagnetic Interference Even Order Rejection Ratio Envelope Tracking Error Vector Magnitude Frequency Division Duplex Frequency Division Multiple Access Front-End Front-End Module Finite Impulse Response Frequency Modulation Full Scale Gaussian Minimum Shift Keying Global System for Mobile communications Hybrid Phase Shift Keying InterFace Integrated Circuit Input Compression Point Input Cross-Compression Point Intermediate Frequency Input Intercept Point Intermodulation Distortion Integral NonLinearity Intercept Point Input Saturated Power Image Rejection Ratio InterSymbol Interference Input Spurious Rejection LInear amplification using Nonlinear Component Low Noise Amplifier Local Oscillator Least Significant Bit Long-Term Evolution LookUp Table Numerically Controlled Oscillator Noise Figure Output Compression Point Orthogonal Frequency Division Multiplexing Output InterModulation Distortion Output Intercept Point Output Saturated Power OverSampling Ratio Power Amplifier Peak to Average Power Ratio Programmable Gain Amplifier Probability Density Function Phase Frequency Detector

21 List of Abbreviations xix PLL PM Psat PSD PSK QAM RF RFIC RL RMS RRC RX RXFE SEM SFDR SiNAD SNR SSB TDD TDMA TE TEM TM THD TX TRX UE VGA VSWR WCDMA WSS XM ZIF Phase Locked Loop Phase Modulation Saturated Power Power Spectral Density Phase Shift Keying Quadrature Amplitude Modulation Radio Frequency Radio Frequency Integrated Circuit Return Loss Root Mean Square Root Raised Cosine Receiver RX Front-End Spectrum Emission Mask Spurious Free Dynamic Range Signal to Noise and Distortion ratio Signal to Noise Power Ratio Single SideBand Time Division Duplex Time Division Multiple Access Transverse Electric Transverse Electromagnetic Transverse Magnetic Total Harmonic Distortion Transmitter Transceiver User Equipment Variable Gain Amplifier Voltage Standing Wave Ratio Wideband CDMA Wide Sense Stationarity Cross-Modulation Zero-IF

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23 Nomenclature t, τ time variables f frequency variable ω angular frequency variable (= 2πf ) x(t) continuous time signal x[n] discrete time signal X( f ), X(ω) frequency representations of x(t)orx[n] F {x(t)} ( f ), F {x(t)} (ω) Fourier transforms of x(t) v, i, j voltage, current, current density P,p (t) in-phase, in-phase component Q,q (t) quadrature, quadrature component j 1 Re{.}, Im{.} real part, imaginary part., arg{.} modulus, argument. complex conjugate convolution δ(.) Dirac delta distribution U(.) Heaviside unit step function x a (t) analytical signal associated with x(t) X a ( f ), X a (ω) frequency domain representations of x a (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 value E{.} stochastic expectation value γ x y (t 1, t 2 ) cross-correlation function ( = E { }) x t1 yt 2 γ x x (t 1, t 2 ) autocorrelation function γ x x (τ) autocorrelation function in stationary case ( = E { }) x t xt τ Γ x x ( f ), Γ x x (ω) power spectral densities of x(t) X, X vector, matrix. T transpose. Hermitian norm dot product cross product

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25 Part I Between Maxwell and Shannon

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27 1 The Digital Communications Point of View When detailing how to dimension a transceiver, it can seem natural to first clarify what is expected from such a system. This means understanding both the minimum set of functions that need to be implemented in a transceiver line-up as well as the minimum performance expected from them. In practice, these requirements come from different topics which can be sorted into three groups. We can indeed refer to the signal processing associated with the modulations encountered in digital communications, to the physics of the medium used for the propagation of the information, and to the organization of wireless networks when considering a transceiver that belongs to such system, or alternatively its coexistence with such systems. The last two topics are discussed in Chapter 2 and Chapter 3 respectively, while this chapter focuses on the consequences for transceiver architectures of the signal processing associated with the digital communications. In that perspective, a first set of functions to be embedded in such a system can be derived from the inspection of the relationship that holds between the modulating waveforms used in this area and the corresponding modulated RF signals to be propagated in the channel medium. As a side effect, this approach enables us to understand how information that needs a complex baseband modulating signal to be represented can be carried by a simple real valued RF signal, thus leading to the key concept of the complex envelope. It is interesting to see that this concept allows us to define correctly classical quantities used to characterize RF signals and noise, in addition to its usefulness for performing analytical derivations. It is therefore used extensively throughout this book. Finally, in this chapter we also review some particular modulation schemes that are representative 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 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.

28 4 The Digital Communications Point of View 1.1 Bandpass Signal Representation RF Signal Complex Modulation Digital modulating waveforms in their most general form are represented by a complex signal function of time in digital communications books [1]. But, even if we understand that this complex signal allows us to increase the number of bits per second that can be transmitted 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 at the transmitter (TX) output, is a real valued signal like any physical quantity that can be measured. Accordingly, we may wonder how the information that needs a complex signal to be represented can be carried by such an RF signal. Any RF engineer would respond by saying that an electromagnetic wave has an amplitude and a phase that can be modulated independently. Nevertheless, we can anticipate the discussion in Chapter 2, and in particular in Section 2.1.2, by saying that there is nothing in the electromagnetic theory that requires this particular structure for the time dependent part of the electromagnetic field. In fact, the right argument remains that this time dependent part, like any real valued signal, can be represented by two independent quantities that can be interpreted as its instantaneous amplitude and its instantaneous phase as long as it is a bandpass signal. Here, bandpass signal means that the spectral content of the signal has no low frequency component that spreads down to the zero frequency. In other words, the spectrum of the RF signal considered, whose positive and negative sidebands are assumed centered around ±ω c, must be non-vanishing only for angular frequencies in [ ω u ω c, ω c + ω l ] [+ω c ω l, +ω c + ω u ], with ω c, ω l and ω u defined as positive quantities, and with ω c >ω l. 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. We can assume that the spectrum of this signal spreads over [ ω l, +ω u ]. Such baseband signals with a non-vanishing DC component in their spectrum are called lowpass signals in contrast to the bandpass signals as given above. If we now wish to shift the spectrum of this signal around the central carrier angular frequency +ω c, we have to convolve its spectrum with the Dirac delta distribution δ(ω ω c ). In the time domain, this means multiplying the signal by the Fourier transform of this Dirac delta distribution, i.e. the complex exponential e +jω ct [2]. This results in the complex signal s a (t) defined by s a (t) = s(t)e +jω ct = (p (t) + jq (t))e +jω ct. (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{s a (t)} = p (t) cos(ω c t) q (t) sin(ω c t). (1.4)

29 Bandpass Signal Representation 5 But what is interesting to see is that even if we took only the real part of the upconverted initial complex lowpass signal transposed around +ω c, we have no loss of information compared to the initial complex baseband signal as long as ω c >ω l. Indeed, under that condition, the original complex modulating waveform s(t) can be reconstructed from the bandpass RF real signal s(t). To understand this, let us first consider the spectral content of the resulting bandpass signal s(t). What would be a good mathematical tool to choose for the spectral analysis? Dealing with digital modulations that are randomly modulated most of the time, the natural choice would be to use the stochastic approach to derive the signal power spectral density. The problem with this approach is that the power spectral density (PSD) of a signal is only linked to the modulus of the Fourier transform of the original signal. It thus leads to a loss of information compared to the time domain signal. As a result, in some cases of interest in this book, we need to keep the simple Fourier transform representation in order to be able to discuss the phase relationship between different sidebands present in the spectrum. Here, by sideband we mean a non-vanishing portion of spectrum of finite frequency support. This phase relationship is indeed required to understand the underlying phenomenon involved in concepts as reviewed in this chapter, but also in Chapter 6, for instance, when dealing with frequency conversion and image rejection. The existence of such Fourier transforms can be justified thanks to the practical finite temporal support of the signals of interest that ensures a finite energy. This is indeed the practical use case when dealing with the post-processing of a finite duration measurement or simulation result. The signals we deal with are therefore assumed to have a finite temporal support and a finite energy, i.e. they are assumed to belong to L 2 [0, T], the space of square-integrable functions over the bounded interval [0, T]. Nevertheless, when dealing with a randomly modulated signal, this approach means that we consider only the spectral properties of a single realization of the process of interest. Thus, even if this direct Fourier analysis is suitable for discussing some signal processing operations involved in transceivers, the power spectral analysis should be considered when possible for taking into account the statistical 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 between the spectral representation of s(t) and that of s(t), we can first expand the relationship between s(t) and the complex signal s a (t) given by equation (1.4). To do so, we use the general property that for any complex number s(t), we have Re{ s(t)} = 1 2 ( 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{s a (t)} = 1 2 (s a (t) + s a (t)). (1.6) Using the relationship between s a (t) and s(t) given by equation (1.2), we finally get that s(t) = 1 2 s(t)e+jω c t s (t)e jω c t. (1.7)

30 6 The Digital Communications Point of View It now remains to take the Fourier transform of this signal. For that, we can use two properties of the Fourier transform. The first states that for any signal, s(t), the Fourier transform of the complex conjugate, s (t), of such a signal can be related to that of s(t) through + F { s (t)}(ω) = s (t)e jωt dt [ + ] = s(t)e +jωt dt = [ F { s(t)} ( ω) ]. (1.8) We observe that this derivation remains valid when s(t) reduces to a real signal s(t). In that case, having s (t) = s(t) leads to having S ( ω) = S(ω). We then recover the classical property of real signals, i.e. the Hermitian symmetry of their spectrum. Then we can use the property that the Fourier transform of a product of signals is equal to the convolution of the Fourier transforms of each signal. Indeed, we get that i.e. that F { s1 (t) s 2 (t)} (ω) = + s 1 (t) s 2 (t)e jωt dt + = S 1 (ω + ) s 2 (t)e j(ω ω )t dtdω + = S 1 (ω ) S 2 (ω ω )dω, (1.9) F { s1 (t) s 2 (t)} (ω) = F { s 1 (t)} (ω) F { s 2 (t)}(ω). (1.10) Thus, using the two properties above, we get that the Fourier transform of equation (1.7) reduces to S(ω) = 1 2 S(ω) δ(ω ω c ) S ( ω) δ(ω + ω c ), (1.11) where 1 S(ω) stands for the Fourier transform of s(t) and where the Dirac delta distribution is the Fourier transform of the complex exponential. As this distribution is even, i.e. we have δ(ω + ω c ) = δ( ω ω c ), the spectrum of s(t) can be expressed as the sum of two components as 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 second component corresponds to the negative part of the spectrum, S (ω), and is therefore referred to 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 envelope s(t) and not for the complex envelope of the signal S(ω).