Hermite-based communications: a time-frequency perspective

Cornelis Willem Korevaar

    Research output: ThesisPhD Thesis - Research UT, graduation UT

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    The radio spectrum is like a crowded house. Everyone is welcome, but when more and more guests are entering and the space is limited, there might not be enough room for everyone. Entry fees may be charged and you need to speak loud to make yourself heard in the overcrowded setting.The radio spectrum is such a crowded house. Wireless communications has become increasingly popular over the last century and wireless devices pop-up literally verywhere. 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. Furthermore, it is costing more and more energy to transmit a message without errors in the overcrowded spectrum. To enable a future with even more wireless devices and data-hungry applications, spectrum- and energy-efficient communications is of ever-increasing importance. Since the early days of communications we have relied upon the modulation of (time-limited) Fourier basis signals, better known as sinusoidal signals. Timelimited signals are frequency-unlimited which 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 like to 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 and are optimally localized in time-frequency. Their mathematical properties make Hermite functions – from a theoretical point of view – an interesting candidate for wireless communications; in particular for unsynchronized, multi-user communications. Besides mentioned characteristics of Hermite functions, more criteria play a role in the design of an effective and efficient communication system. 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 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 the energy efficiency, power constraints, synchronization, equalization and the sensitivity for time-frequency dispersion/offsets. As most of the signal criteria impose similar constraints on the time representation as well as the frequency representation, a more equal treatment of these domains seems justified. Therefore, time-frequency analysis, the two-dimensional description and analysis of signals in time-frequency, plays a prominent role throughout this thesis. Compared to the Fourier basis and Fourier-based communications, relatively little is known about Hermite functions and in particular Hermite-based communications. To assist signal analysis for Hermite functions, in this thesis 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 be expressed as a Gaussian function again. We generalize this reciprocity 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-formexpressions, which can be expressed as a bounded sum of n + m Hermite functions.The connection and difference between all these mathematical operations is primarily determined by distinct phase changes of the weights of the Hermite functions in the result. 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 like 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 intuitive and effective method to reduce peak powers. Hermite-basis signals are neither stationary in time nor in frequency. Therefore, 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 can lead to a significant reduction in the computational complexity necessary for synchronization. In addition, it is shown that fractional delay filters – necessary for synchronization – can be omitted by using fractional Fourier transform (FrFT) identities. When designing a Hermite-based communication system, more criteria are relevant and require attention, including numerical evaluation, sampling, sensitivity for time-frequency offsets, eliminating the DC component and stacking multiple symbols ofHermite functions.These criteria have been assessed by theoretical analysis, simulations and actual measurements on the first Hermite-based transceivers (for as far as known by the author).The conclusion is that Hermite functions are an ideal candidate as a signal basis for tomorrow’s transceivers. However, two potential barriers are identified that block the widespread adoption today. A first concern are the signal characteristics of Hermite functions, which are dependent on the order of the Hermite function. E.g., both the time-bandwidth product and the time-frequency localization of Hermite functions are linearly dependent on the order of the Hermite function, causing a different robustness against time-frequency offsets and dispersion. It has been partly overcome by the introduction of a new set of signals; Fourier-Hermite functions. Fourier-Hermite functions average out the properties of individual Hermite functions, but not to the full extent. A second concern which remains is the non-orthogonality of time-shifted symbols of Hermite functions.The interference caused by overlapping Hermite-based symbols is insignificant in many scenarios, but can cause a degradation of the bit error rate (BER) in some use cases and low-noise regimes. To address aforementioned concerns, we slightly move away from ‘pure’ Hermite functions by constructing a new set of waveforms.The aim is tomaintain 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 the fundamental limit 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 waveforms designs and the BLT.The designed Hexagonal Hermite waveforms are quasi-orthogonal, time-frequency localized and achieve a density up to 99% of the critical waveform density. The robustness in doubly dispersive channels and the efficiency for multi-user scenarios are discussed and compared to conventional orthogonal frequency division multiplexing (OFDM) which show large benefits, in particular for time-frequency dispersive channels and multi-user scenarios. This thesis also investigates the resemblance and differences between Fourier-based and Hermite-based communications. Related to the quantum-mechanical correspondence principle, higher-order Hermite functions tend to behave more and more 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 usage of OFDM and orthogonal frequency division multiple access (OFDMA), avoiding many of the questions and topics addressed in this thesis. However, Hermite-based communications shows significant gains in situations where synchronization is difficult to achieve, in doubly dispersive channels and in multi-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 by 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.
    Original languageEnglish
    QualificationDoctor of Philosophy
    Awarding Institution
    • University of Twente
    • Smit, G.J.M., Supervisor
    • de Boer, Pieter-Tjerk, Co-Supervisor
    • Kokkeler, Andre B.J., Co-Supervisor
    Award date3 Jun 2016
    Place of PublicationEnschede
    Print ISBNs978-90-365-4136-7
    Publication statusPublished - 3 Jun 2016


    • Pulse shaping
    • Orthogonal frequency division multiplexing (OFDM)
    • Hexagonal lattice communications
    • Wireless communications
    • Hermite functions
    • Time frequency localisation


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