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(3) Hermite-based Communications A Time-Frequency Perspective. CornelisWillemKorevaar.

(4) Members of the graduation committee: prof. dr. ir. dr. ir. dr. ir. dr. ir. prof. dr. ir. prof. dr. ir. prof. dr. ir. prof. dr. ir.. G. J. M. Smit P. T. de Boer A. B. J. Kokkeler G. J. M. Janssen B. Nauta M. Siala R. N. J. Veldhuis P. M. G. Apers. University of Twente (promotor) University of Twente (co-promotor) University of Twente (co-promotor) Delft University of Technology University of Twente Ecole Supérieure des Communications de Tunis University of Twente University of Twente (chairman). Faculty of Electrical Engineering, Mathematics and Computer Science, Computer Architecture for Embedded Systems (CAES) chair.. %6+6. CTIT Ph.D. Thesis Series No. 16-397 Centre for Telematics and Information Technology PO Box 217, 7500 AE Enschede, The Netherlands. The presented work was supported by the Dutch TSP project. The project has joint funding from the European Regional Development Fund and the Dutch provinces of Gelderland and Overijssel.. Copyright © 2016 C. Willem Korevaar, Enschede, The Netherlands. All rights reserved. No part of this thesis may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the written permission of the author or, when appropriate, of the publishers of the publications. Printed by Gildeprint, The Netherlands, on FSC-certified paper. ISBN ISSN DOI. 978-90-365-4136-7 1381-3617 10.3990/1.9789036541367.

(5) Hermite-based Communications A Time-Frequency Perspective. Proefschrift. ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. H. Brinksma, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 3 juni 2016 om 14.45 uur. door Cornelis Willem Korevaar. geboren op 21 mei 1985 te Lelystad.

(6) Dit proefschrift is goedgekeurd door: prof. dr. ir. G. J. M. Smit dr. ir. P. T. de Boer dr. ir. A. B. J. Kokkeler. (promotor) (co-promotor) (co-promotor). Copyright © 2016 C. Willem Korevaar ISBN 978-90-365-4136-7.

(7) Contents. Summary. xi. Samenvatting. xvii. Acknowledgements. xxiii. 1. Introduction 1.1 1.2. A century of wireless communications . . . . . . . . Three fundamental problems with Fourier-based communications .. 3 5. . . . . . . . . .. 5 6. 1.2.1 1.2.2. 1.3.4. 1.4. 2. Spectral leakage limiting the (multi-user) spectrum efficiency Sensitivity to frequency offsets & dispersion . . . .. . Waveform design for multi-user communications . 1.3.1 Relevance – Spectrum and energy efficient communications . 1.3.2 Problem statement & research objective . . . . . 1.3.3 Design criteria, design methodology & limitations . . . 1.2.3. 1.3. 2. Peak powers limiting the energy efficiency. .. . .. . .. . .. Core concepts – Time-frequency analysis & Hermite functions. Outline of the thesis. .. .. .. .. .. .. .. .. .. .. The quest for minimizing time-frequency leakage 2.1 Introduction . . . . . . . . . . . . 2.2 Time-frequency analysis . . . . . . . . . 2.3 System model & spectral leakage . . . . . . . 2.4 2.5 2.6 2.7. 2.8 2.9. .. . . . Overview of conventional solutions to reduce spectral leakage . On the extremes of time-limited and band-limited . . . . A time-frequency localized signal basis . . . . . . . Hermite functions and their properties . . . . . . . 2.7.1 Orthonormality . . . . . . . . . . . 2.7.2 Eigenfunctions of the (fractional) Fourier transform . . 2.7.3 Time-frequency localization . . . . . . . . 2.7.4 Numerical evaluation . . . . . . . . . Hermite functions for wireless communications . . . . Conclusions . . . . . . . . . . . . . v. 6. 7 7 8 8 9. 10. 14 . . . . . . . . . . . . .. 15 16 18 20 21 22 24 24 25 25 26. 27 29.

(8) vi. 3. Contents. Closed-form expressions for mathematical operations involving Hermite functions 32 3.1 Introduction . . . . . . . . . . . . . . 33 34 3.2 Related work & outline . . . . . . . . . . . . 35 3.3 Definitions . . . . . . . . . . . . . . . . 35 3.4 The product, convolution and correlation of Hermite functions 37 3.5 Wigner distribution and ambiguity functions of Hermite functions . . . . . . . . . . . . . . 39 3.6 Generalizations. 3.7. 4. 6. One generalized expression for operations involving Hermite functions Generalizations to square-integrable functions . . . . .. 3.6.3. On observing Hermite functions in time-frequency. Conclusions. .. .. .. .. .. .. .. .. .. .. . .. . .. Equalization & synchronization in time-frequency: functions & spiral correlation 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Eigenstructures of wireless channels 4.2.1 The ideal channel . . . . . . . . . 4.2.2 Time dispersive and frequency dispersive channels . . 4.2.3 The time-frequency (doubly) dispersive channel . . 4.3 Synchronization in either time or frequency . . . . 4.4 Synchronization in time-frequency: Spiral correlation . . 4.5 4.6 4.7. 5. 3.6.1 3.6.2. . .. . .. Eigen. . . . . . .. Synchronization and matched filtering using the sunflower spiral Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . .. Peak power reduction by rotation of the time-frequency representation 5.1 Introduction . . . . . . . . . . . . . . 5.2 The PAPR of multi-carrier signals . . . . . . . . . . . . . . . . 5.3 OFDM, OTDM and Hermite signal sets . . . . . . 5.4 Time-frequency rotations to lower the PAPR 5.5 Simulation results . . . . . . . . . . . . . 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Conclusions Where Fourier meets Hermite: Fourier-Hermite functions 6.1 Introduction . . . . . . . . . . . . . . 6.2 The Fourier-Hermite signal set . . . . . . . . . . 6.3 Robustness against time-frequency offsets . . . . . . . . . . . . . 6.4 Variations on the Fourier-Hermite functions. 39 41 41. 42. 46 47 48 49 50 51. 51 52 55 57 58. 62 63 65 66 68 71 73 74. 78 79 80 82 83.

(9) Contents. 6.5 6.6 6.7 6.8. 7. Stacking ensembles of (Fourier-)Hermite functions . . . . Report on implementing & measurements with Hermite-based transceivers The correspondence between Hermite- and Fourier-based communica. . . . . . . . . . . . . . . . tions Conclusions . . . . . . . . . . . . . .. Hexagonal Hermite waveforms: Closing the gap with Balian-Low theorem 7.1 Introduction . . . . . . . . . . . . . 7.1.1 The Balian-Low theorem & outline . . . . . . 7.2 Definitions & system model . . . . . . . . . 7.3 7.4. 7.6. 8. 86 88 90 91. 94. . . .. 95. . . . . . . . . . .. 106 107. 95. 97 Design of (communication) waveforms with rotationally symmetric AFs 98 Results – Hexagonal Hermite waveforms . . . . . . . 103. . . Performance & implementation aspects . 7.5.1 Transmitter model . . . . . 7.5.2 Channel models . . . . . 7.5.3 Receiver model . . . . . . 7.5.4 Simulation results for doubly dispersive channels . 7.5.5 On multi-user communications with spectrum-scarcity . 7.5.6 Implementation aspects . . . . . . . . Conclusions . . . . . . . . . . . . 7.4.1 7.4.2. 7.5. the. vii. On the quasi-orthogonality . . On the time-frequency localization. Conclusions & recommendations 8.1 Conclusions . . . . . . 8.2 Contributions . . . . . . 8.3 Limitations & recommendations .. . . . . . .. . . . . . .. . . . . . .. . . . . . . .. . . . . . . . . . .. 108 108 109 110 111 112 114. 115. 118 . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 119 123 124. Acronyms. 129. List of symbols. 133. Bibliography. 137. List of publications. 149.

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(11) Summary in English Samenvatting in het Nederlands. Dankwoord.

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(13) Summary The radio spectrum is like a crowded house. Everyone is welcome. As more and more guests are joining and the space is limited, there may not be enough room for everyone. Entry fees may be charged and you need to speak loudly to make yourself heard in the overcrowded setting. The radio spectrum is such a crowded house. Wireless communications have become increasingly popular over the last century and wireless devices pop up literally everywhere. As there is limited bandwidth available for radio communications, a fierce competition for bandwidth takes place between the wireless devices. Bandwidth auctions have become billion dollar affairs. It requires more and more energy to transmit a message without errors in the overcrowded spectrum. To enable a future with even more wireless devices and datahungry applications, spectrum and energy efficient multi-user communications is of ever-increasing importance. The quest for time-frequency localized signals – Hermite functions Since the early days of communications we have relied upon the modulation of Fourier basis signals, better known as sinusoidal signals. These signals are usually limited in time which means that they are unlimited in bandwidth. That leads to spectral leakage and a degradation of the spectrum efficiency when there are multiple, unsynchronized spectrum users. Band-limited signals would eliminate spectral leakage, but are unlimited in time. Ideally, we would design signals which are both band-limited and time-limited. However, there is a fundamental bound, the uncertainty principle, which limits the extent to which one can design such signals. This thesis studies optimally time-frequency localized signals: Hermite functions. Hermite functions form an orthogonal signal basis, are eigenfunctions of the Fourier transform and are optimally localized in time-frequency. Their mathematical properties make Hermite functions – at least from a theoretical point of view – an interesting candidate for wireless communications, in particular for unsynchronized, multi-user communications. Criteria for multi-user communications – The need for time-frequency analysis Besides the aforementioned characteristics of Hermite functions, more criteria play a role in the design of an effective and efficient communication system. Effective means the extent to which the transmitter is able to send a message which is wellinterpreted by the receiver, whereas efficient means that the messages are transmitted with the least amount of time, bandwidth, energy, power and (computational) complexity. A list of criteria is formulated in this thesis for effective and efficient (multi-user) communications. Among the criteria are energy efficiency, power constraints, synchronization, equalization and the sensitivity to time-frequency offsets xi.

(14) xii. Summary. and dispersion. As most of the signal criteria impose similar constraints on the time representation as well as the frequency representation, an equal treatment of these domains seems justified. Hence, time-frequency analysis, the two-dimensional description and analysis of signals in time and frequency, plays a prominent role throughout this thesis. Mathematical operations involving Hermite functions – A generalized description Compared to the Fourier basis and Fourier-based communications, relatively little is known about Hermite functions and in particular Hermite-based communications. In this thesis, to assist signal analysis for Hermite functions, closed-form expressions are derived for popular mathematical operators involving Hermite functions, including the product, convolution, correlation, Wigner distribution function (WDF) and ambiguity function (AF). It was already known that these mathematical operations performed on Gaussian functions (Hermite functions of the zeroth-order) lead to a result which can again be expressed as a Gaussian function. We generalize this to Hermite functions of arbitrary order. The product, convolution, correlation, WDF and AF operations performed on two Hermite functions of order n and m lead to remarkably similar closed-form expressions, which can be expressed as a bounded sum of n + m Hermite functions. The connections and differences between all these mathematical operations are primarily determined by distinct phase changes of the weights of the Hermite functions in the result. The peak power problem – Time-frequency rotations to lower the PAPR In addition to spectral leakage, another fundamental problem in (multi-carrier) communications is the high peak-to-average power ratio (PAPR) of the transmit signals. Both Fourier-based and Hermite-based communications are characterized by transmit signals which can lead to high peak powers. These peak powers impose a demanding constraint on the dynamic range of analog building blocks, such as the power amplifier, and lead to a degradation of the overall energy efficiency of the communication system. In this thesis, a novel PAPR reduction technique is proposed, which is based on rotating the time-frequency presentation of the transmit signal. It effectively shifts the peak (powers) manifesting in the time domain to the frequency domain. Simulations and comparisons with existing techniques show that PAPR reduction by time-frequency rotations is an effective method to reduce peak powers. Synchronization in time-frequency – The spiral correlation method Hermite basis signals are neither stationary in time nor in frequency. As a consequence, synchronization needs to be established in both domains. A novel correlation procedure is proposed, called spiral correlation. Instead of correlating either in time or in frequency, spiral correlation is a one-dimensional formulation of correlation in time-frequency, starting at one point and following a trajectory according to the turns of the spiral. Spiral correlation, using the pattern of a sunflower, is simulated and evaluated. The proposed method leads to a significant reduction in the computational complexity necessary for synchronization. In addition, it is.

(15) Summary. xiii. shown that fractional delay filters – necessary for time synchronization – can be omitted by using fractional Fourier transform (FrFT) identities. Fourier-Hermite functions and actual (Fourier-)Hermite-based transceivers A Hermite function of order n is characterized by a time-bandwidth product and a time-frequency localization which is proportional to n. This causes a different sensitivity to time-frequency offsets and dispersion. To average the characteristics of individual Hermite functions, a new signal set is designed, called Fourier-Hermite functions, which is essentially a Fourier-weighted sum of Hermite functions. The robustness against time-frequency offsets is significantly better compared to Hermite functions. The inter-symbol interference (ISI), as a result of the non-orthogonality of symbols of (Fourier-)Hermite functions, is investigated in this thesis. The interference is insignificant in many scenarios, but can cause a degradation of the bit error rate (BER) in some use cases and low-noise regimes. The theoretical analysis of Hermite and Fourier-Hermite functions has been complemented by simulations and measurements on actual Hermite-based transceivers. The implementation of the theory in actual transceivers – using universal software radio peripherals (USRPs) – went smoothly and did not lead to problems. Introducing hexagonal Hermite waveforms – Closing the gap with the BLT To address the aforementioned concern of the non-orthogonality of symbols of (Fourier-)Hermite functions, we slightly move away from pure Hermite functions by constructing a new set of (quasi-)orthogonal waveforms. The aim is to maintain the attractive properties of Hermite functions and simultaneously arrive at an orthogonal, time-frequency localized and spectrum efficient set of waveforms. However, there is a fundamental theorem, called the Balian-Low theorem (BLT), which states the fundamental impossibility to design waveforms which 1) form an orthogonal set, 2) are time-frequency localized and 3) attain a critical waveform density. This thesis introduces hexagonal Hermite waveforms which close the gap between existing waveform designs and the BLT. The designed hexagonal Hermite waveforms are a weighted sum of (6n)th -order Hermite functions. The waveforms are quasi-orthogonal, time-frequency localized and achieve a density of up to 99% of the critical waveform density. Their performance is assessed by simulations and compared to conventional orthogonal frequency division multiplexing (OFDM), which show large benefits, in particular for time-frequency dispersive channels and multi-user scenarios. On Fourier- and Hermite-based communications This thesis discusses the resemblance and differences between Fourier-based and Hermite-based communications. Higher-order Hermite functions tend to behave increasingly similar to Fourier basis functions. The larger the signal sets, the smaller the differences between the two communication schemes. There are many use cases which justify the use of OFDM and orthogonal frequency division multiple access (OFDMA), avoiding many of the questions and topics addressed in this thesis. However, Hermite-based communications show significant benefits in situations where synchronization is difficult to achieve, in doubly dispersive channels and in multi-.

(16) xiv. Summary. user scenarios. In a spectrum-scarce world, with an ever increasing number of connected devices which are often not synchronized, this thesis provides insights to believe that Hermite functions will play a (more) dominant role in the future of wireless communications. The derived signal sets – including Hermite functions, Fourier-Hermite functions and hexagonal Hermite waveforms – as well as the newly developed methods proposed in this thesis – including spiral correlation, eliminating fractional delay filters with the FrFT, the closed-form expressions involving Hermite functions and peak-power reduction by time-frequency rotations – have been designed to pave the path towards that future..

(17) Summary in English Samenvatting in het Nederlands Dankwoord.

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(19) Samenvatting Het radiospectrum kan je zien als een open huis. Iedereen is welkom. Maar als steeds meer gasten arriveren en de ruimte beperkt is, kan het huis te klein zijn. Gevolg is dat er entree geheven moet worden en dat je hard moet praten om jezelf nog verstaanbaar te maken. In het radiospectrum speelt een vergelijkbaar probleem. Gedurende de laatste eeuw is draadloze communicatie steeds populairder geworden en overal zijn – hoewel je ze veelal niet ziet – draadloze apparaten. Omdat het bruikbare deel van het radiospectrum beperkt is, vindt er een felle competitie om bandbreedte plaats. In bandbreedteveilingen gaan tegenwoordig miljarden euro’s om. Tevens kost het steeds meer energie om een bericht zonder fouten verstuurd te krijgen in het overbevolkte spectrum. Om een toekomst met een groeiend aantal draadloze apparaten en data-intensieve applicaties mogelijk te maken, wordt spectrum- en energie-efficiënte communicatie steeds belangrijker. De zoektocht naar tijd-frequentie gelokaliseerde signalen – Hermite functies Sinds de eerste experimenten met communicatiesystemen hebben we vertrouwd op de modulatie van Fourier-basis signalen, beter bekend als sinusvormige signalen. Deze signalen worden veelal gelimiteerd in tijd, wat betekent dat ze ongelimiteerd zijn in bandbreedte. Dit leidt tot spectrale vervuiling en een degradatie van de spectrumefficiëntie als er meerdere, ongesynchroniseerde gebruikers van het spectrum zijn. Band-gelimiteerde signalen zouden spectrale vervuiling voorkomen, maar zijn ongelimiteerd in tijd. Idealiter zouden we signalen ontwerpen die zowel perfect band- als tijd-gelimiteerd zijn. Er is echter een fundamentele theorie, het onzekerheidsprincipe, die stelt dat dergelijke signalen niet kunnen bestaan. Dit proefschrift onderzoekt daarom de maximaal tijd-frequentie gelokaliseerde signalen: Hermite functies. Hermite functies vormen een orthogonale signaalbasis. Het zijn eigenfuncties van de Fourier transformatie en ze zijn optimaal gelokaliseerd in tijd-frequentie. Deze wiskundige eigenschappen maken Hermite functies – op zijn minst vanuit een theoretisch perspectief – een interessante kandidaat voor draadloze communicatie, in het bijzonder voor communicatie met veel ongesynchroniseerde, draadloze apparaten. Criteria voor multi-user communicatie – De noodzaak voor tijd-frequentie analyse Naast de bovengenoemde eigenschappen van Hermite functies, zijn er meer criteria van belang bij het ontwerpen van een effectief en efficiënt communicatiesysteem. Effectief betekent de mate waarin de zender in staat is om een bericht te versturen dat goed begrepen wordt door de ontvanger, terwijl efficiënt er op doelt dat het bericht verstuurd wordt in zo min mogelijk tijd en met zo weinig mogelijk bandbreedte, energie, vermogen en rekenkracht. In dit proefschrift is een lijst van xvii.

(20) xviii. Samenvatting. criteria opgesteld voor effectieve en efficiënte (multi-user) communicatie. Deze criteria hebben onder andere betrekking op energie-efficiëntie, eisen aan het piekvermogen, synchronisatie, equalisatie en de robuustheid tegen tijd-frequentie afwijkingen en dispersie. Omdat de meeste signaalcriteria zowel eisen opleggen aan de tijd- als aan de frequentie-representatie van signalen, is een gelijke behandeling van de beide domeinen gewenst. Daarom speelt tijd-frequentie analyse, de gelijktijdige beschrijving en analyse van signalen in tijd en frequentie, een belangrijke rol in dit proefschrift. Wiskundige bewerkingen met Hermite functies – Een generieke beschrijving In vergelijking met de Fourier basis en Fourier-gebaseerde communicatie is er relatief weinig bekend over Hermite functies en in het bijzonder Hermite-gebaseerde communicatie. Om de signaalanalyse met Hermite functies te vergemakkelijken, hebben we expressies afgeleid voor populaire wiskundige bewerkingen op Hermite functies waaronder het product, convolutie, correlatie, Wigner distribution function (WDF) en ambiguity function (AF) van twee Hermite functies. Het was reeds bekend dat deze wiskundige bewerkingen op Gaussische functies (Hermite functies van de nulde orde) leiden tot een resultaat dat wederom beschreven kan worden als een Gaussische functie. We generaliseren dit voor Hermite functies van willekeurige orde. Het berekenen van een product, convolutie, correlatie, WDF en AF op twee Hermite functies van orde n en m leidt tot verrassend gelijkende expressies, die beschreven kunnen worden door een som van n + m Hermite functies. De gelijkenissen en verschillen tussen al deze wiskundige bewerkingen worden voornamelijk bepaald door specifieke fase-veranderingen in de gewichten van de Hermite functies in het eindresultaat. Het probleem van piekvermogens – Tijd-frequentie rotaties om de PAPR te verlagen Naast spectrale vervuiling, is een ander fundamenteel probleem in (multi-carrier) communicatie de hoge peak-to-average power ratio (PAPR) van de zendsignalen. Zowel Fourier-gebaseerde als Hermite-gebaseerde communicatie worden gekarakteriseerd door zendsignalen die kunnen leiden tot hoge piekvermogens. Deze hoge piekvermogens leiden vervolgens tot hoge eisen aan het dynamisch bereik van de analoge componenten zoals de analoge versterker. Dit veroorzaakt een degradatie van de gehele energie-efficiëntie van het communicatiesysteem. Om dit te beperken, wordt in dit proefschrift een nieuwe PAPR reductietechniek geïntroduceerd die gebaseerd is op het roteren van de tijd-frequentie representatie van het zendsignaal. Simulaties en vergelijkingen met bestaande technieken tonen aan dat de voorgestelde PAPR reductie door tijd-frequentie rotaties een effectieve methode is om piekvermogens te reduceren. Synchronisatie in tijd-frequentie – De spiraalcorrelatie methode Hermite functies zijn niet stationair in tijd noch in frequentie. Een gevolg hiervan is dat de signalen in beide domeinen gesynchroniseerd dienen te worden. In dit proefschrift wordt een nieuwe correlatie methode geïntroduceerd, genaamd ‘spiraalcorrelatie’. In plaats van te correleren in óf tijd óf frequentie, is spiraalcorrelatie een herformulering van correlatie in tijd-frequentie, waarbij gestart wordt in één.

(21) Samenvatting. xix. punt en vervolgens de windingen van een spiraal gevolgd worden. Spiraalcorrelatie, waarbij gebruik is gemaakt van de zogeheten zonnebloemspiraal, is beschreven en gesimuleerd. De voorgestelde methode leidt tot een significante reductie in de benodigde rekenkracht voor het verkrijgen van synchronisatie. Bovendien laten we zien dat fractionele delay filters – die nodig zijn voor het verkrijgen van tijdsynchronisatie – uitgespaard kunnen worden door gebruik te maken van enkele fractional Fourier transform (FrFT) vergelijkingen. Fourier-Hermite functies en (Fourier-)Hermite gebaseerde zenders en ontvangers Een Hermite functie van orde n wordt gekarakteriseerd door een tijd-bandbreedte product en een tijd-frequentie lokalisatie die proportioneel is met n. Dit veroorzaakt een verschillende gevoeligheid voor tijd-frequentie afwijkingen en dispersie. Om de eigenschappen van individuele Hermite functies uit te middelen, wordt een nieuwe signaalset ontworpen, genaamd Fourier-Hermite functies, waarvan de signalen een Fourier-gewogen som van Hermite functies zijn. De proefschrift onderzoekt de inter-symbol interference (ISI) als gevolg van de niet-orthogonaliteit van symbolen van Fourier-Hermite functies. In veel gevallen is de verstoring niet significant, maar het kan leiden tot een degradatie van de bit error rate (BER) in sommige scenario’s en in omgevingen met weinig ruis. De theoretische analyses van Hermite en Fourier-Hermite functies worden ondersteund door simulaties en metingen aan daadwerkelijke Hermite-gebaseerde zenders en ontvangers. De implementatie van de theorie in zenders en ontvangers – gebruikmakende van universal software radio peripherals (USRPs) – verliep voorspoedig en heeft geen problemen aan het licht gebracht. Hexagonale Hermite signalen – Het benaderen van de Balian-Low theorem Om de eerdergenoemde aandachtspunten met betrekking tot de niet-orthogonaliteit van symbolen van (Fourier-)Hermite functies te adresseren, stappen we af van de pure Hermite functies door het ontwerp van een nieuwe set van (quasi-)orthogonale signalen. Het doel is om de aantrekkelijke eigenschappen van Hermite functies te behouden en tegelijkertijd te komen tot een set van orthogonale, tijd-frequentie gelokaliseerde en spectrum-efficiënte signalen. Echter, er is een fundamentele theorie, genaamd de Balian-Low theorem (BLT), die de beperking beschrijft om te komen tot signalen die 1) een orthogonale set vormen 2) tijd-frequentie gelokaliseerd zijn en 3) de kritieke signaaldichtheid behalen. Dit proefschrift introduceert hexagonale Hermite signalen die het gat dichten tussen huidige signaalconstructies en de BLT. De ontworpen signalen zijn quasi-orthogonaal, tijd-frequentie gelokaliseerd en halen een dichtheid tot 99% van de kritieke signaaldichtheid. De prestaties zijn bestudeerd met behulp van simulaties en vergeleken met conventionele orthogonal frequency division multiplexing (OFDM). Met de hexagonale Hermite signalen kunnen grote voordelen behaald worden, met name in kanalen met veel tijd-frequentie dispersie en in multi-user scenario’s. Over Fourier- en Hermite-gebaseerde communicatie Dit proefschrift richt zich tot slot op de overeenkomsten en verschillen tussen Fourier- en Hermite-gebaseerde communicatiesystemen. Hogere orde Hermite.

(22) xx. Samenvatting. functies worden meer en meer gelijkend aan Fourier basis functies. Des te groter de signaalsets, des te kleiner de verschillen tussen beide communicatieschema’s. Er zijn veel toepassingen en scenario’s die OFDM en de variant orthogonal frequency division multiple access (OFDMA) rechtvaardigen, waarmee veel van de vraagstukken en onderwerpen, die dit proefschrift behandelt, omzeild kunnen worden. Echter, Hermite-gebaseerde communicatie laat significante voordelen zien in toepassingen waar het moeilijk is om synchronisatie te verkrijgen, in tijd-frequentie dispersieve kanalen en in multi-user scenario’s. In een wereld waarin het spectrum schaars is, met een steeds verder toenemend aantal draadloze apparaten die veelal onderling niet gesynchroniseerd zijn, maakt dit proefschrift het aannemelijk dat Hermite functies een belangrijkere rol in de toekomst van draadloze communicatie zullen gaan spelen. De signaalsets – Hermite, Fourier-Hermite en hexagonale Hermite functies – alsmede de nieuw ontwikkelde methodieken – waaronder spiraalcorrelatie, het uitsparen van fractionele delay filters met de FrFT, de wiskundige expressies voor bewerkingen op Hermite functies en piek-vermogens reductie door tijd-frequentie rotaties – zijn ontworpen om die toekomst mogelijk te maken..

(23) Summary in English Samenvatting in het Nederlands Dankwoord.

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(25) Acknowledgements Just three words like ‘thinking of you!’ can add great value to someone’s day. And it has never been easier to send a beautiful message to any corner of the world, in a split second. Billions of messages, calls and documents are transmitted every day, wirelessly. Nowadays, more and more devices and data-hungry applications are joining the wireless party. As there is a limited availability of energy and radio spectrum, it has been my endeavor to design energy- and spectrum-efficient communications. All with the hope that one day – while you may never be aware of it – you are actually using one of the results as written down in this thesis. It has been an honor to work on the forefront of technology. I really enjoyed to revisit the assumptions underlying the status quo, envision the world of tomorrow and work towards that vision by developing new theories, technologies and businesses. We all know that tomorrow looks different than today. But what keeps us awake at night, what makes us wake up early, is to create a tomorrow which is not just different, but – and that’s at least what we strive for – in some way better. I am thankful for the support I received from my promotor, Gerard. He inspires and fosters new ideas in the earliest stages of development. I found an excellent complementary co-promotor in Pieter-Tjerk, who is a sparring partner to test and mature ideas into solid theorems. My other co-promotor André may not expect it, but I am regularly thinking about his advice to not just focus on the end-result, but to enjoy the ride. Those words are of great value for journeys, like a PhD, where there is a long road ahead, before arriving at the undiscovered destination. The university is a perfect playground. I love the environment where education, entrepreneurship and research come together. I got the chance to supervise and learn from students; thank you Mark, Vincent, Frank, Cristi, India and Preeti! I am thankful for the trust and support from funding partners, the secretaries and colleagues. I worked together and received valuable suggestions from Mark (Oude Alink), Mark (de Ruiter), Koen and Hermen. While staying overnight in Enschede, I enjoyed the warm hospitality and Thursday night beers with Bram & Dionne, Bram & Ellen, Bart and Berend. Last but not least, I am proud of my friends and colleagues at DOT.world: Rob, Martijn, Lars, Jonas, José, Joaquin, Olaf and Eric. Together, we created something unique in the world! Above all, I am thinking of my beloved family, including my sisters and paranymphs Walter and Chris, and my friends who were always interested and supportive, even after the years passed by. The things we achieve, we do together! Brains may be given, but perseverance is something valuable and without family and friends I would go nowhere. I am especially grateful for my parents, Hein & Reina, who have been a continuous, everlasting source of love and support on this journey. I am curious and excited how the journey continues... xxiii.

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(27) Chapter. 1. Introduction. Parts of this chapter have been published as: C.W. Korevaar, "A time-frequency localized signal basis for multi-carrier communication", Master’s thesis, University of Twente, The Netherlands, 2010..

(28) Abstract This chapter starts with an overview of the history and great achievements in (wireless) communications. Despite all the progress over the last century, some fundamental problems remain and limit the extent to which we are able to realize spectrum and energy efficient multi-user communications. These problems are often associated with Fourier-based communications and include spectral leakage, sensitivity to frequency offsets and dispersion, and a high peak-to-average power ratio (PAPR). A comprehensive list of design criteria is formulated which guides the process of designing waveforms, particularly suitable for unsynchronized, multi-user communications..

(29) 1.1 – A century of wireless communications. 1.1. 3. A century of wireless communications 1. It was the summer of 1858. Queen Victoria from the United Kingdom received an extraordinary message from James Buchanan, president of the United States. It was the first official communications over the newly laid transatlantic telegraph cable between the United Kingdom and the United States. Large celebrations with fireworks were organized to celebrate this technological breakthrough. That night, the fireworks set fire to the New York City Hall, which cast a shadow over the celebrations. Maybe even worse, the cable itself failed completely after six weeks. Actually, the cable never really worked well; the Queen’s message had taken 16-1/2 hours to transmit [1]. Despite the setbacks, the first transatlantic message marked the beginning of an era of wired communications which would drastically impact the lives of many generations to come, including ours today. During the early years that wired communications technology took off, in the year 1865, Maxwell published his work "A Dynamical Theory of the Electromagnetic Field" in which he set out four well-known equations based on the work of Gauss, Ampère and Faraday [3]. Studying the electromagnetic field theory of Maxwell, Hertz and Tesla showed the principle of radio communications in a laboratory environment. It was around 1900 that Marconi showed the world the use of radio waves by transmitting radio signals over the Atlantic Ocean [4]. Although it would take decades for wireless communications to become mainstream, the first experiments of these early scientists would pave the way for today’s wireless communications. In the 19th century the main application of communications was telegraphy. Transmission was achieved by making and breaking an electric contact resulting in audible short pulses. When multiple users used the same line, they were scheduled after. Figure 1.1 | Figures from U.S. patent no. 174.465, filed in 1876 by Bell, explaining the idea of FDM [2]. Waves A and B of different frequency are summed to A + B (left), sent over one single line, and excite a response in receiver A and receiver B which are tuned for waves of frequency A and B, respectively (right)..

(30) 4. Chapter 1 – Introduction. one another, which is nowadays known as time division multiple access (TDMA). One of the challenges of telegraph communications was to increase the capacity – in terms of users – of the lines. Bell studied the use of multiple frequencies to allow different telegraph users to communicate simultaneously. In 1876 he patented the idea of FDM [2]. In his patent, partly shown in Fig. 1.1, he describes a transmitter sending a sinusoidal wave giving a response by a telegraph machine tuned for that single frequency. By simultaneously sending several sinusoidal waves, each characterized by its own frequency, different telegraph connections are possible over a single line, at the same time. In traditional FDM transmission systems, the subchannels are separated in frequency by spectral guard spaces. Guard spaces are used to guarantee frequency isolation between the subchannels which are used by different users. Although these guard bands prevent inter-carrier interference (ICI), i.e., cross-talk between the different channels, the spectral efficiency is lowered as a result of the unused guard spaces. A solution has been found by means of orthogonal frequency division multiplexing (OFDM). The orthogonality among the signals allows for ICI-free communications without a guard space. Due to the reduced subcarrier spacing, communications using OFDM achieves a higher spectral efficiency than with traditional FDM. Important exploratory work on OFDM has been performed by Chang & Gibby [5] and Saltzberg [6] in the 1960s. They explored transmission systems using orthogonal waveforms. Full-cosine roll-off pulses, as shown in Fig. 1.2, were proposed by both authors and reduce the spacing between the carriers by 50%. Despite the conceptual beauty, OFDM had one important drawback: Its computational complexity. Cooley and Tukey presented their fast implementation of the discrete Fourier transform (DFT) in 1965 [7]. It marked a major turning point in discrete signal processing, though it turned out that the algorithm was actually a reinvention. It was already published in a slightly different form by Gauss 150 years before [8]. Hirosaki suggested the use of the inverse and forward fast Fourier transform (FFT) as modulator and demodulator for OFDM [9] such that the computational complexity for N sinusoidal carriers became proportional to N log2 N compared to N 2 for the earlier DFT realizations. The insight of using orthogonal signals in combination with the FFT implementa-. f f1. f2. Figure 1.2 | Illustration of overlapping orthogonal (full cosine roll-off ) pulses as proposed by Saltzberg in exploratory work on OFDM [6]..

(31) 1.2 – Three fundamental problems with Fourier-based communications. 5. tions enabled the widespread application of OFDM, which is nowadays the most popular multi-carrier scheme used in wireless local area networks (WLANs), digital video broadcasting (DVB), digital audio broadcasting (DAB) and in mobile communications (e.g., long-term evolution (LTE)).. 1.2. Three fundamental problems with Fourier-based communications. OFDM has become the dominant standard in wireless communications for a num-. ber of reasons [10–12]. First, the orthogonal subcarriers are easily generated and interpreted by using the (inverse) FFT. Second, due to the relatively long symbol times and by applying a cyclic prefix, the subcarriers become robust against missynchronizations and multipath effects. Third, the subcarriers can be individually modulated dependent on the (sub)channel conditions. Finally, interoperability between different wireless standards is eased as OFDM is nowadays so widely applied. In the upcoming chapters, OFDM is regularly referred to as Fourier-based communications, because of its close connection with the Fourier basis. Despite its popularity, conventional OFDM is also associated with three fundamental problems: 1) Spectral leakage, 2) High sensitivity of OFDM signals for frequency offsets and dispersion, and 3) Peak power problem (e.g., [11]). These topics are introduced in the next subsections and are important for the analysis and design of energy and spectrum efficient waveforms for multi-user communications.. 1.2.1. Spectral leakage limiting the (multi-user) spectrum efficiency. In an ideal scenario, a transmitter and a receiver communicate at one single wavelength. However, in general, signals spread out over a larger frequency band than just a single wavelength, which we refer to as spectral leakage. The main problem of spectral leakage is that it causes interference to other spectrum users, which leads to a drop in the overall spectrum efficiency in multi-user scenarios. Two major sources of spectral leakage can be identified. First, spectral leakage can be an inherent consequence of the signal basis. The strictly time-limited Fourier basis signals – as applied in OFDM – are characterized by a sinc-shaped power spectral density (PSD) where a significant portion of the energy is contained in the spectral sidelobes. These sidelobes slowly decrease over frequency and can cause significant interference to other spectrum users. Second, non-linear components such as filters and amplifiers cause intermodulation products. These can be inband, but can also fall out-of-band, which leads to undesirable interference to other spectrum users. While the importance of reducing intermodulation products is acknowledged, this thesis primarily focuses on the spectral leakage associated with the choice of the signal basis.. 1.

(32) 6. Chapter 1 – Introduction. Ideally, one aims to have strictly time-limited and band-limited signals which minimize energy spread both in time and in frequency. This is infeasible as strictly time-limited signals are band-unlimited and band-limited signals in turn have infinite duration. There is a fundamental bound – related to the quantum-mechanical uncertainty principle – which governs the extent to which we can design signals both localized in time and frequency. This bound is the primary topic of Chapter 2 and is the key to arrive at spectrum efficient, multi-user communications.. 1.2.2. Sensitivity to frequency offsets & dispersion. Once a communication signal has been transmitted, it propagates through a channel. It may become distorted due to absorptions and reflections by stationary and moving objects. In case the radio channel characteristics are perfectly known, waveforms can be designed such that they are not distorted during transmission. In practice, due to a lack of perfect channel knowledge, time-varying channels and to limit the computational complexity, assumptions are being made about the channel. One may therefore rely on standard channel models. The ‘eigenmessages’ (the messages which are not distorted) of some typical channel models are discussed in Chapter 4. Long symbol times – as typical for OFDM – result in narrowband subcarriers. As a consequence, a slight missynchronization in frequency (e.g., by a Doppler shift, phase noise or frequency offsets in the transceivers) leads to a loss of orthogonality among the subcarriers and can cause significant self-interference. To limit this interference, the waveforms – to be designed – need to be robust against frequency offsets and frequency dispersion. Instead of focusing just on the robustness in the frequency domain, the fundamental quest is to design signals which offer robustness both in time and in frequency. The robustness is essentially dependent on two factors: 1) The time-frequency distance between adjacent waveforms and 2) the time-frequency localization of the waveforms [13]. The aim is to have the largest distance between adjacent waveforms, achieve a high waveform-density in timefrequency (to enhance the spectrum efficiency) and to find the most time-frequency localized signals.. 1.2.3. Peak powers limiting the energy efficiency. There is a fundamental paradox in using multi-carrier signals and aiming for power efficient communications. The summation of many independent, modulated signals leads – as a consequence of the central limit theorem (CLT) – to a Gaussian or Rayleigh amplitude distribution. Such a distribution is characterized by a high peak-to-average power ratio (PAPR), which can theoretically become infinite. High peak powers require a large dynamic range of the transceiver components. This in turn leads to a high power consumption in, e.g., digital-to-analog converters.

(33) 1.3 – Waveform design for multi-user communications. 7. (DACs) and power amplifiers (PAs), which degrades the overall energy efficiency of the communication system. Shannon distinguished in his classical work [14] two cases: 1) Communications with an average power constraint (addressing the energy efficiency) and 2) communications with a peak-power constraint (addressing the power efficiency). The second case is less well-known, but the optimum amplitude distribution for a peak-power constrained signal is uniformly distributed. As multi-carrier communications leads to Gaussian amplitude distributions, a single-carrier scheme with uniform modulation seems favorable from a power efficiency viewpoint. A PAPR-reduction method can be used if, for other reasons, multi-carrier communications is still applied. A new method to reduce peak powers is introduced in Chapter 5.. 1.3. Waveform design for multi-user communications. 1.3.1. Relevance – Spectrum and energy efficient communications. In the pioneering work of Marconi et al., spark-gap transmitters were used for wireless communications. These transmitters are very energy inefficient and the signals transmitted occupy a large part of the electromagnetic spectrum. This may not have been a problem in the early days of Marconi, but given the limited amount of spectrum, the vast increase in wireless devices and more and more data-hungry applications, an efficient usage of the radio spectrum is essential nowadays. This has also been emphasized by the U.S. Federal Communications Commission (FCC): Our data shows there is a looming crisis. We may not run out of spectrum tomorrow or next month, but it is coming and we need to do something now. U.S. FCC Chairman [15]. Energy is another invisible, but scarce resource. Due to the limited capacity of batteries, the energy used by wireless devices is usually rather limited. In contrast, the whole infrastructure – including base stations – consumes a lot of energy as is underlined by the GreenTouch industry consortium: Mobile handsets themselves have a relatively small energy footprint - using a mobile phone for a year has the same emissions as driving an average European car for an hour. But that phone connects to a sprawling infrastructure that uses – and often wastes – massive amounts of electricity. Information technology and communications consume about 2% of the world’s energy, or roughly the same as the airline industry, and mobile networking represents between one half and one quarter of that total. GreenTouch industry consortium [16, 17]. 1.

(34) 8. Chapter 1 – Introduction. A large part of the energy consumption in base stations is a consequence of low efficiency (typical 20%..30%) of the PAs [18] and the associated cooling. It is both a societal and scientific challenge to reduce energy consumption and simultaneously enhance the spectrum efficiency, particularly in the context of multi-user communications. This forms the main motivation behind the presented research. 1.3.2. Problem statement & research objective. Section 1.2 outlined three problems associated with Fourier-based communications: 1) Spectral leakage, 2) High sensitivity of communication signals for frequency offsets and dispersion and 3) Peak power problem. These fundamental problems limit the spectrum and energy efficiency of today’s (unsynchronized) multi-user communications. It should be noted that many methods have been developed to deal with the problems. But instead of targeting the consequences resulting from the modulation of (time-limited) Fourier basis signals, one may question the signal basis itself. Hence, the main research objective of this thesis has been formulated as the design of spectrum and energy efficient waveforms for wireless communications with a focus on unsynchronized, multi-user scenarios where the available bandwidth is limited. 1.3.3. Design criteria, design methodology & limitations. As mentioned in preceding sections, some fundamental boundaries and problems limit the extent to which we are able to realize effective and efficient communications. Effective indicates the extent to which the transmitter is able to send a message to the receiver which is well-interpreted, whereas efficient means that the messages are transferred with the least amount of time, bandwidth, energy, power and (computational) complexity. In all cases, the environment may make it more challenging as it is likely to be a noisy environment, the signals may suffer from interference by other users, the signals may get attenuated due to the distance between transmitter and receiver, and the environment may distort the messages, e.g., due to reflections and/or moving objects. In general, an ideal communication system can be described as a transmitter and a receiver which: ⋆ Perfectly understand each other’s messages (adopt the same standard and share the same code dictionary). ⋆ Use the minimum amount of time and bandwidth necessary (attain a high spectrum efficiency). ⋆ Use the minimum amount of power and energy (a low peak-to-average-power ratio and the minimum energy / bit). ⋆ Listen at the right time instance and to the same wavelength(s) (are synchronized both in time and in frequency). ⋆ Use messages which are robust, even in the case of missynchronizations (waveforms are insensitive to time and frequency offsets)..

(35) 1.3.4 – Core concepts – Time-frequency analysis & Hermite functions. 9. ⋆ Use messages which are robust in noisy environments (waveforms are orthogonal and matched filtered∗ ). ⋆ Employ messages which are not distorted by the channel or are easily corrected (waveforms are ‘eigenmessages’ of the channel or are easily equalized∗∗ ). ⋆ Cause a minimum amount of cross-talk with other spectrum users (users need to be synchronized∗∗∗ and/or use waveforms localized in time-frequency). ⋆ Use messages which can be generated/interpreted by standard transceivers (numerical evaluation, discrete time, limited dynamic range, no DC component).. The requirements have been operationalized in mathematical terms as shown in italic. This list of 9 requirements for a communication system is based on the fundamental problems as discussed in Section 1.2 and the assessment of several signal properties, including continuity, linear dependency, orthogonality, correlation, crest factor, entropy, localization, energy, timing sensitivity and frequency sensitivity, which we have analyzed before and can be found in [10]. The criteria are discussed and analyzed in more detail in Chapter 2. There is no single, straightforward methodology to translate all these criteria to an optimum set of waveforms. Given the emphasis on spectrum efficient, multi-user communications, this thesis first addresses the problem of spectral leakage and afterwards addresses all other criteria to smaller or larger extent in the subsequent chapters. An evaluation takes place in the concluding Chapter 8. A communication system is designed to bring information from one point to another. In its most basic form, information is mapped onto a set of waveforms, which are summed together, mixed to radio frequency (RF), amplified and finally transmitted as an electromagnetic wave by the transmit antenna. The wave propagates, as described by the Maxwell equations [3], through a medium, which we assume to be air. After receiving the radio wave by a receive antenna, it is amplified, mixed to baseband, (matched) filtered and the original information is (hopefully correctly) recovered. The synthesis and analysis are assumed to take place in the digital domain. The advantage is that a proposed change of the signal basis doesn’t require any modification in the analog frontend. To avoid analysis at RF and to accommodate digital signal synthesis and analysis – unless otherwise stated – all signals described in this thesis are at baseband. 1.3.4. Core concepts – Time-frequency analysis & Hermite functions. Most signal criteria listed in Section 1.3.3 apply equally well to the time domain as to the frequency domain. Signals need to be robust, localized and synchronized in both domains. Hence, an equal treatment of both domains seems to be justified. * To achieve signal recovery in AWGN channels with minimum noise amplification, linear independent (or orthogonal) signals are required in combination with a matched filter [10]. ** Eigenmessages are those messages which are not distorted by the channel, see also Chapter 4. *** Which is the case for orthogonal frequency division multiple access (OFDMA), as explained in Chapter 2.. 1.

(36) 10. Chapter 1 – Introduction. Throughout this thesis, we therefore rely upon two-dimensional time-frequency descriptions and analysis. Hermite functions take a central role in this thesis. Hermite functions are a set of real functions named after the French mathematician Charles Hermite, who published on the polynomials in “Sur un nouveau développement en série de fonctions”, in the year 1864 [19]. Although the polynomials and functions are called after Hermite, Hermite was not the first to publish them. Before Hermite’s investigation, the polynomials were already described by Chebyshev [20] and had first appeared in the famous Mécanique Céleste by Laplace [21]. Hermite functions can be found in various fields of science: E.g., they constitute the stationary states of the quantum harmonic oscillator [22], are eigenmodes in multimode optical fibers [23], eigenfunctions of the (fractional) Fourier transform and possess maximum energy concentration in terms of their second-order moments in domains linked by the Fourier transform [24]. Applications can be found in image processing [25, 26], optics [27], electroencephalograph (EEG) processing [28], spectrum estimation [29] and communications. Communications with more spectrum efficient signal bases have been pursued before. Some popular fields of research have been concentrated on the prolate spheroidal wavefunctions (PSWFs) [24], Gabor frame theory [30], wavelets and filter bank multi-carrier systems [31]. Hermite functions play a role in all these different research fields: They are a ‘subset’ of the PSWFs [24], they have been analyzed as (super)frames for multi-carrier signals [30, 32] and as prototype filter for filter bank multi-carrier systems [33–35]. One of the first and only studies on the application of pure Hermite functions as a replacement for Fourier basis communications can be found in [36]. Despite previous work on the topic, quite some (theoretical) challenges and questions remain to be answered to arrive at Hermite-based communications. The aim of this thesis is to address these topics and to pay special attention to those matters which differ from Fourier-based communications.. 1.4. Outline of the thesis. In this chapter, a list of criteria has been formulated to enable spectrum and energy efficient multi-user communications. These criteria are addressed – to a smaller or larger extent – in the upcoming Chapters 2-7. Chapter 2 starts with the quest for finding signal sets which minimize spectral leakage, leading to Hermite functions as a candidate signal basis for communications. To facilitate signal analysis and synthesis using Hermite functions, closed-form expressions for common mathematical operators involving Hermite functions are derived in Chapter 3. Chapter 4 continues our quest for fulfilling all signal criteria by assessing the eigenfunctions, i.e., the signals which are not distorted by common radio channels. The chapter also contributes the spiral correlation method to achieve synchronization in time-frequency for non-stationary signals, including Hermite functions. Chapter 5.

(37) 1.4 – Outline of the thesis. 11. assesses the PAPR of Fourier- and Hermite-based transmit signals and introduces an effective PAPR reduction method based on the rotation of the time-frequency presentation. As Hermite functions have signal properties which are dependent on the order of the Hermite function, Chapter 6 introduces a new signal set called Fourier-Hermite functions which average out the order-dependent behavior. The chapter also discusses the orthogonality of ensembles of (Fourier-)Hermite functions. Moreover, the chapter includes a short report on simulations and real-world measurements of a (Fourier-)Hermite-based communication system. As ensembles of Hermite and Fourier-Hermite functions are not orthogonal which may – in some scenarios – lead to significant interference, Chapter 7 introduces a procedure to orthogonalize Hermite-based waveforms. The chapter introduces hexagonal Hermite (HH) waveforms, which are constructed by a sum of (6n)th -order Hermite functions. Finally, Chapter 8 lists and discusses the main results. Special attention is paid to the comparison between Fourier, Hermite, Fourier-Hermite and hexagonal Hermite waveforms for wireless communications, and in particular in the context of unsynchronized, multi-user communications.. 1.

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(39) Chapter. 2. The quest for minimizing time-frequency leakage. Parts of this chapter have been published as: C.W. Korevaar, "A time-frequency localized signal basis for multi-carrier communication", Master’s thesis, University of Twente, The Netherlands, 2010..

(40) Abstract The raison d’être of a communication system is to transmit information from one point to another. Ideally, a complete encyclopedia is sent in a split second using just one wavelength. However, the amount of time and bandwidth occupied by a signal cannot be arbitrarily small. There is a fundamental bound, called the uncertainty principle, which limits our ability to design so-called ‘time-frequency localized’ signals. This chapter treats the need and quest for these so-called time-frequency localized signals, especially in the context of unsynchronized, multi-user communications in spectrum-scarce environments. It leads to the proposition of using Hermite functions as a signal basis for (future) communications..

(41) 2.1 – Introduction. 2.1. 15. Introduction. A 4 year-old boy climbed behind the clavier of his dad. Joseph Haydn wrote about this boy that "posterity will not see such a talent again in 100 years." It was Wolfgang Amadeus Mozart. His father would soon start writing down the compositions of his son in the ‘Nannerl Nottenbuch’, as shown in Fig. 2.1. Mozart’s father wrote down the musical notes (frequencies) played by his son at specific moments in time. It can be regarded as a two-dimensional description of time and frequency. Musicians, including Mozart, heavily rely upon such two-dimensional time-frequency descriptions to describe their musical compositions. The question may arise why signal and radio engineers do not rely more on such time-frequency descriptions for their compositions for communications. Throughout this thesis, time-frequency analysis plays a prominent role and we will show that it proves beneficial for advanced signal constructions, achieving synchronization and lowering peak powers. Section 2.2 introduces the concept of time-frequency analysis. Section 2.3 presents the system model and provides an analysis of spectral leakage associated with OFDM. In Section 2.4, the conventional solutions to limit spectral leakage are discussed. The objective of finding time and frequency localized signals is formulated in Section 2.5 and Section 2.6 leads to Hermite functions as the optimally localized signal basis in time-frequency. The properties of Hermite functions are discussed in Section 2.7 and Section 2.8 assesses Hermite functions based on the list of signal criteria formulated in the previous chapter. Finally, Section 2.9 concludes this chapter.. Figure 2.1 | Composed at the age of 5, this is believed to be Mozart’s first work (Andante in C major, K.1a). The musical notes form a two-dimensional description of time-frequency with time on the x-axis and tones/frequencies on the y-axis.. 2.

(42) 16. Chapter 2 – The quest for minimizing time-frequency leakage. 2.2. Time-frequency analysis. A communication signal – as transmitted by an antenna – can be described by its temporal and spectral behavior. The temporal and spectral behavior of the signals are linked by the unitary, continuous-time Fourier transform (CTFT) and its inverse:. 1 Δ f (t) = √ 2π. ∫. ∞. −∞. 1 Δ F(ω) = √ 2π. F(ω) e jωt dω,. ∫. ∞. −∞. f (t) e − jωt dt, (2.1). where f (t) represents the time-varying signal, F(ω) the frequency representation Δ √ and j = −1. There are a couple of practical limitations with the equations listed before. First, the transform assumes the signals to be defined on the whole time and frequency domain, while in practice signals are defined only for a limited (symbol) time and frequency band. Second, as the concept of instantaneous frequency is not feasible, the spectrum F(ω) at time τ can only be found by localizing the function f (t) around τ, giving rise to the short-time Fourier transform (STFT): Δ. Fst (τ, ω) =. ∫. ∞. −∞. f (t)w(t − τ) e − jωt dt.. (2.2). While the integral still stretches from −∞ till ∞ in time, a windowing function w(t) has been introduced which is only nonzero for the region around t = τ. The STFT reduces to the CTFT when w(t) = 1. The STFT is strongly dependent upon the choice of the windowing function w(t) and each unique windowing function will lead to another time-frequency description. To avoid the problem of choosing a windowing function, one can resort to the Wigner distribution function (WDF). The WDF is a member of Cohen’s class of time-frequency distributions [37] and was originally developed in the field of quantum mechanics. For the WDF, w(t) in (2.2) equals the complex-conjugated, time reversed version of the function f (t). The WDF transforms a one-dimensional time-varying signal f (t) into a two-dimensional joint time-frequency description, being defined as [38]: Δ. W f , f (t, ω) =. ∫. ∞. −∞. f (t +. ′ t′ t′ ) ⋅ f (t − )e − jωt dt ′ , 2 2. (2.3). where the upper bar indicates the complex conjugate. The WDF may be regarded as a best-practice effort to make a time-frequency description out of a time-varying signal, similar to the musical notes as depicted in Fig. 2.1. It may be noted that the WDF is just one of a much larger class of bilinear time-frequency descriptions, each with its own characteristics, strengths and drawbacks [37]..

(43) 2.2 – Time-frequency analysis. ω W f , f (t, ω). 17. τ AF f , f (θ, τ) ψ. α t. θ. 2. Figure 2.2 | Coordinate system of the WDF (left) and AF (right).. Another useful tool in time-frequency analysis is the ambiguity function (AF), which gives the correlation between two functions which have a relative timefrequency offset. The AF finds application in time-frequency analysis, waveform design and radar signal processing [37]. The symmetric and narrowband AF of a function f (t) is defined as [38]: Δ. AF f , f (θ, τ) =. ∫. ∞. −∞. τ τ f (t + ) ⋅ f (t − )e jθ t dt, 2 2. (2.4). which is the Fourier dual of the WDF, with θ and τ representing the (radial) frequency and time lag, respectively. The coordinate systems for the WDF and AF are shown in Fig. 2.2. Time and frequency are two orthogonal axes of the time-frequency lattice. In addition, as we will show in upcoming chapters, it is advantageous to attain an intermediate representation between time and frequency. The intermediate representation between the axes of time and frequency can be obtained by means of the fractional Fourier transform (FrFT), which is a generalization of the Fourier transform. The transform was proposed by Namias in relation to quantum mechanics [39] and later found application in optics and signal processing. The unitary FrFT operator working on a function x(t) is defined as [39, 40]:. Δ. F α {x(t)} (u) =. ∫. ∞. −∞. Kα (t, u)x(t) dt,. (2.5). where Kα represents the kernel of the FrFT, i.e., √ 2 2 ⎧ 1− j cot(α) j u +t ⎪ 2 ⎪ ⋅ e ⎪ 2π Δ ⎪ Kα (t, u) = ⎨ δ(t − u) ⎪ ⎪ ⎪ ⎪ ⎩ δ(t + u). cot(α). ⋅ e − jut csc(α). if. α α α. ≠ p⋅π = p ⋅ 2π = p ⋅ 2π + π, p ∈ Z, (2.6).

(44) 18. Chapter 2 – The quest for minimizing time-frequency leakage. where cot (⋅), csc (⋅) and δ (⋅) refer to the cotangent, cosecant and the Dirac delta function, respectively. The FrFT leads to a rotation of the WDF by an angle α [38], as shown in Fig. 2.2. If α is equal to π/2 or −π/2 we obtain a counter-clockwise or clockwise rotation in time-frequency by 90 degrees and (2.5) reduces to the unitary forward and inverse Fourier transform, respectively. The FrFT, WDF and AF are extensively used in upcoming chapters for waveform design, interference analysis, synchronization and to analyze and lower peak powers.. 2.3. System model & spectral leakage. We describe a baseband multi-carrier transmit signal as an infinite sequence of symbols, each constructed by an ensemble of K different signals s k (t), i.e.,. Δ. ∞ K−1. x(t) = ∑ ∑ A k ,n s k (t − n ⋅ Ts ),. (2.7). n=0 k=0. where A k,n represents a real or complex value corresponding to the modulation scheme chosen and Ts is the symbol duration. During each symbol time Ts a symbol consisting of K waveforms is transmitted. In additive white Gaussian noise (AWGN) channels, orthogonality between the waveforms s k ′ and s k≠k ′ and among the symbols is a requirement to recover the information A k,n without interference and noise amplification [10]. In Chapter 1, OFDM has been introduced. OFDM is based on an orthogonal set of complex exponentials of increasing frequency, called subcarriers. A subcarrier of index k is defined as:. Δ. t. s k,OFDM (t) = e j2πk Ts. for −. Ts Ts <t≤ . 2 2. (2.8). Whereas complex exponentials are stationary signals which stretch over infinite time, the complex exponentials in OFDM are limited to the symbol time Ts . Due to different modulations per symbol, the transmit signal x(t) can change abruptly from symbol to symbol as illustrated for a single carrier in Fig. 2.3. It may be apparent that the sharp, unnatural signal transitions lead to many high-frequency components and cause problems. The band-limited transceiver components cannot perfectly accommodate these sharp transitions and the transmit signal is likely to become distorted, even before it is being transmitted. Moreover, the spectral leakage associated with these time-limited signals can cause significant interference to other spectrum users..

(45) 2.3 – System model & spectral leakage. 19. Amplitude. 1 0.5 0 −0.5. 2. −1 −1.5. −1. −0.5. 0 t/Ts. 0.5. 1. 1.5. Figure 2.3 | Illustration of an amplitude and phase modulated sinusoidal signal for three consecutive symbol times (one single carrier). The sharp transitions from symbol to symbol lead to a poor localization of the signal in the frequency domain.. A baseband transmit signal, based on one modulated complex exponential, can be described by: ∞. x k (t) = ∑ A k,n ⋅ e j(2π k t/Ts ) ⋅ rect((t − nTs )/Ts ),. (2.9). n=−∞. where rect (t ′ ) is the rectangular function which equals 1 for the interval ∣t ′ ∣ < 1/2 and 0 outside this interval. The frequency representation is obtained by means of the unitary CTFT and applying a few well-known properties of the Fourier transform: ∞ Ts ω πk X k (ω) = ∑ A k,n √ ⋅ sinc (( − ) Ts ) e − jωnTs ⋅ e j2πkn . 2 T 2π s n=−∞. (2.10). The frequency representation of an OFDM transmit signal of K carriers yields the summation of K frequency shifted sinc-shaped functions, mathematically given by: ∞. K−1. X(ω) = ∑ ∑ X k (ω).. (2.11). n=−∞ k=0. In conclusion, modulation of time-limited complex exponentials causes a typical squared sinc-shaped PSD, which decays only by 1/ω 2 . The normalized PSDs for five adjacent subcarriers are plotted in Fig. 2.4. Even a guard space of 100 subcarriers (based on a single subcarrier PSD) is not enough to limit interference to other spectrum users by 55dBc (a measure used by the FCC to allow unlicensed transmission in TV white spaces [41]). That means that for multi-carrier systems based on conventional OFDM, spectral guard spaces of hundreds of subcarriers should be used to reduce the interference to acceptable levels, limiting the overall spectrum efficiency significantly in multi-user scenarios. The next section discusses conventional solutions to limit this problem..

(46) Chapter 2 – The quest for minimizing time-frequency leakage. 0. Normalized PSD [dB/Hz]. Normalized PSD [dB/Hz]. 20. −10 −20 −30 −40 −6. −4. −2. 0. 2. 4. 6. Normalized frequency f Ts. 0 −10 −20 −30 −40 −50 −100. −80. −60. −40. −20. 0. 20. Normalized frequency f Ts. Figure 2.4 | Normalized PSDs of five adjacent OFDM subcarriers (left). Notice the slow decay of the sincshaped power spectra (right).. 2.4. Overview of conventional solutions to reduce spectral leakage. The five subcarriers in Fig. 2.4 can be transmitted by one user, but also by five different users. This is the concept of orthogonal frequency division multiple access (OFDMA). To prevent ICI between the users, it is essential that the users’ signals are synchronized when they arrive at a receiver. In the case of multiple transmitters, each with their own Doppler shift or frequency offset, this synchronization is difficult to achieve and requires additional signal processing steps at the receiver to minimize the ICI. In general, OFDMA only works well in the network downlink of a base station, where all the subcarriers are transmitted from the same point (the base station) and hence undergo the same Doppler frequency shift before reaching each receiver [12]. More on the synchronization of OFDMA can be found in [42]. As it is assumed that synchronization between different spectrum users, standards and devices is often not available or feasible, we will mainly focus on unsynchronized, multi-user communications in this thesis. For unsynchronized, multi-user communications, it is essential that spectral leakage is limited. We outline six general methods to deal with the problem of spectral sidelobes. First, the traditional solution to cope with the OFDM sidelobes is to use large spectral guard spaces, as discussed in previous sections. A guard space is some unused spectrum which allows for the OFDM sidelobes to decay to acceptable levels. Guard spaces are a simple method to ensure frequency isolation among spectral users. Second, a more advanced method is offered by active interference cancellation (AIC). Signals are added to the OFDM signals such that the inserted signals cancel the OFDM sidelobes [43]. A third method to suppress OFDM sidelobes is based on cancellation carriers (CCs). Some subcarriers are not used to carry information, but are modulated such that the sidelobes of these subcarriers nullify the sidelobes of the active subcarriers [44]. Fourth, sidelobes can also be minimized by weighting individual carriers [45]. The weights of the subcarriers are chosen such that the sidelobes of one subcarrier cancel another. Fifth, as sidelobes in OFDM are caused.

(47) 2.5 – On the extremes of time-limited and band-limited. 21. by abrupt constellation changes, smart mapping of data onto constellation points can provide smoother transitions than the ones shown in Fig. 2.3. Such constellation mappings are proposed in, e.g., [46] and [47]. Finally, most research has been dedicated to time domain pulse-shaping. The abrupt changes from symbol period to symbol period, as shown in Fig 2.3, are smoothened by a pulse-shaping filter. By adding an extra prefix and postfix to the symbol and using this to facilitate a smooth transition from symbol to symbol, the spectral sidelobes can be suppressed. This is also referred to as filtered OFDM and a common filter is the raised-cosine filter. An alternative is to filter each carrier individually by a filter, called filter bank multi-carrier (FBMC). A good overview of filter banks and implementations is given by Vaidyanathan [48] and an overview of the tradeoffs between OFDM, filtered OFDM and FBMC can be found in [31]. The choice for the best method is a trade-off between spectral leakage, noise performance, spectral efficiency, interference, robustness, power and/or computational complexity. Although effective to smaller or larger extent, all mentioned methods – except for FBMC which is discussed in Chapter 7 – have in common that they do not question the basis signals themselves. The methods modify/shape ‘the-not-sogood’ signals resulting from the inverse FFT modulator. It may be argued that the problems, i.e., the time-limited Fourier basis signals, should be revisited instead of dealing with the consequences. Hence, we aim for a signal basis which maintains the advantages of OFDM and inherently offers good spectral properties.. 2.5. On the extremes of time-limited and band-limited. It is not difficult to design signals without sidelobes in the frequency domain. Similarly, it is easy to design signals with no sidelobes in the time domain. However, it is fundamentally impossible to design real (communication) signals which are both band-limited and time-limited [49]. As a consequence, the signals should at least either be band-unlimited or unlimited in time. However, also this seems unreasonable. Slepian touched this debate in a lecture where he stated: "It is easy to argue that real signals must be band-limited. It is also easy to argue that they cannot be so. To assume that real signals must go on forever in time (a consequence of band-limitedness) seems just as unreasonable as to assume that real signals have energy at arbitrarily high frequencies (no band-limitation)" [24]. Consider the two extreme cases which are visualized in Fig. 2.5. On one hand, signals can be time-limited as is the case for conventional OFDM symbols. As discussed in Section 2.3, large parts of the spectrum are polluted by the corresponding sinc-shaped power spectra. On the other hand signals can also be strictly bandlimited, i.e., limited in frequency, which means that the signals occupy infinite time and are thereby non-causal. If no signal is both time-limited and band-limited, the question may arise what signal is maximally localized in the two-dimensional view of time-frequency? This. 2.

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