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# Prolate spheroidal wave functions, an introduction to the Slepian series and its properties

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## Abstract

For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.

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... The spectrum {γ n | n ∈ N 0 } of an operator belonging to this class is compact, countable, real positive and satisfies ∞ n=0 γ n < ∞ [102,103]. The spectrum of the sine kernel (3.2) is non degenerate, it satisfies the following bounds [74,104] 0 < γ n < 1 (3.6) and its trace reads ∞ n=0 γ n = 2 π η = a 2π . ...
... The eigenvalues and eigenfuctions in (3.1) can be expressed in terms of the prolate spheroidal wave functions (PSWFs), which have been introduced as solutions of the Helmoltz wave equation in spheroidal coordinates [109][110][111]. In particular, the eigenvalues can be written through the radial PSWFs of zero order R 0n as follows [69,74,104] γ n (η) = 2η π R 0n (η, 1) 2 n ∈ N 0 (3.8) and the eigenfunctions in terms of the angular PSWFs of zero order S 0n as ...
... In (3.8) and (3.9), the normalisation chosen by Wolfram Mathematica for R 0n (η, x) and S 0n (η, x) is compatible with the normalisation induced by the standard inner product of L 2 [−1, 1] for the eigenfunctions f n (η; x), which has been imposed above. Notice that different normalisations for the PSWFs have been introduced in the literature [69,104]. ...
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We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, which is a non relativistic model with Lifshitz exponent $z=2$. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The numerical results show that the entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic $C$ function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent $z$, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.
... It is known that the Slepian solution for the discrete-time problem, called discrete prolate spheroidal sequences (DPSS), is the equivalent version of PSWFs in discrete-time domain [26], consequently they have the same properties. Both DPSS and PSWFs have been widely studied [27]- [31]. ...
... , can be defined as the normalized eigenfunctions of the following integral equation [26], [27]: ...
... PSWFs should not be confused with the solution of the discrete-time energy concentration problem, the DPSS. DPSS are much simpler to calculate, and they are known only in a finite interval [27]. ...
Article
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In conventional wireless communications, cyclic-prefix orthogonal frequency division multiplexing (CP-OFDM) has been adopted as the baseline multicarrier scheme. Despite their corroborated merits, they are explored only in the asynchronous and strictly orthogonal scenario. To overcome the limitations observed in CP-OFDM, to support the constraints imposed by different 5G scenarios, and also to improve robustness against channel impairments, several waveforms have been investigated. Quadrature amplitude modulation associated with filter-bank multicarrier (QAM-FBMC) has been an auspicious technology for 5G communication systems and beyond. The main feature of QAM-FBMC is its capacity for high spectral confinement, which is possible thanks to the per-subcarrier filtering. Aiming to improve the QAM-FBMC performance, in this paper, we propose a prototype filter design based on the discrete prolate spheroidal sequences (DPSS), also known as Slepian sequences. The presented filters were obtained by optimizing the weights that are used to compose the desired prototype filter to such an extent that they minimize the intrinsic interference of the system. At the same time, these weights keep spectral confinement of the filter for a previously determined bandwidth limitation. Simulation results show that the optimized filters achieve higher intrinsic interference attenuation than the competitors. Indeed, we can confirm the improvement in the system performance brought by the usage of the optimized filters through the bit error rate (BER) evaluation. Furthermore, the proposed method is flexible thanks to the suitability of its parameters.
... commutes with the finite Fourier transform F c [14,21]: ...
... where U (ξ; T x ) is the operator defined in (14) and f is the solution of ...
... The properties discussed in this subsection can be found in [14,15,21]. First, we note that the prolate spheroidal wave functions ψ c n , n ∈ N, satisfy the following eigenvalue equation ...
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Time and band limiting operators are expressed as functions of the confluent Heun operator arising in the spheroidal wave equation. Explicit formulas are obtained when the bandwidth parameter is either small or large and results on the complete Fourier transform are recovered.
... It is known that the Slepian solution for the discrete-time problem, called discrete prolate spheroidal sequences (DPSS), is the equivalent version of PSWFs in discrete-time domain [78], consequently they have the same properties. Both of them, the DPSS and PSWFs, have been widely studied together with their applications [79][80][81][82]. ...
... The PSWFs, ψ p (t, b), concentrated in the interval [−T 0 , T 0 ], can be defined as the normalized eigenfunctions of the following integral equation [78,79]: ...
... In our application we are interested in the sequences φ PSWFs should not be confused with the solution of the discrete-time energy concentration problem, the DPSS. DPSS are much simpler to calculate, and they are known only in a finite interval [79]. ...
Thesis
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Le système Filter-Bank Multi-Carrier (FBMC) est une technologie prometteuse pour permettre de futures communications sans fil, par exemple 5G/6G. Par rapport au multiplexage par répartition orthogonale de la fréquence (OFDM), les systèmes FBMC ont une meilleure localisation de fréquence et, par conséquent, causent moins d'interférences aux systèmes voisins. Les systèmes FBMC peuvent être associés à différentes modulations telles que la modulation d'amplitude en quadrature, qui nous conduit au système QAM-FBMC, ou offset QAM appelé (OQAM-FBMC). Dans les deux cas, la conception du filtre prototype est une clé pour garantir de bonnes performances, car elle sera utilisée pour construire tous les filtres du banc de filtres. Dans cette thèse, nous abordons le problème de la conception de filtres prototypes pour les systèmes OQAM-FBMC et QAM-FBMC, à la lumière des procédures d'optimisation. Nous procédons à un choix judicieux de la fonction objectif à minimiser pour réduire les interférences intrinsèques tout en conservant le confinement spectral. Nous étudions également les performances des systèmes à porteuses multiples grâce à leur probabilité d'erreur sur les bits (BEP). Pour évaluer les performances des systèmes QAM-FBMC, nous proposons une expression mathématique de son BEP. Nous étudions comment le choix du filtre prototype affecte les interférences du système et par conséquent le BEP. Bien que plusieurs filtres prototypes aient été conçus pour les systèmes QAMFBMC, une interférence résiduelle est toujours observée et dégrade les performances du système. Par conséquent, un récepteur plus élaboré est nécessaire. Ainsi, nous menons également une recherche sur les récepteurs afin de réduire les interférences et nous proposons un récepteur spécifiquement conçu pour les systèmes QAM-FBMC.
... where Π(·) is the gating function and f n = n Ts . In the proposed scheme, the signal-bearing "subcarriers" are LPFs [3]: ...
... Similarly as OFDM, the proposed scheme extends the pulses in the time domain and offers resilience against chromatic dispersion with the possibility of additional compensation in the digital domain. At the receiver, in the baseband domain, using the analysis equation (over the finite interval) for signal extrapolation with LPFs [3] the symbols are recovered using a bank of correlators: ...
Conference Paper
This work develops the multicarrier modulation (MCM) scheme tailored to optical communications where the transceiver is composed of: (i) radio frequency (RF) MCM sending end, (ii) electro-optical (RF-To-Optical, RTO) up converter, (iii) optical transmission link, (iv) Optical-To-RF (OTR) down converter, and (v) RF MCM receiver [1]. The approach follows the system model of Optical Orthogonal Frequency Division Multiplexing (OFDM) using sinusoidal subcarriers with this paper contribution in the deployment of the linear prolate functions (LPFs) as “subcarriers”. This is to increase bandwidth efficiency and provide resilience against imperfections in optical medium. Specifically, the proposed generalization of OFDM takes advantage of unique features of orthogonal and bandlimited bases of LPFs which are known to be maximally concentrated in time as well as in frequency and invariant to Fourier Transform.
... • Setting the bandwidth per symbol to be B = B LP in (30). (30) and (31). Hence the achievable rate of FSEE-OFDM is written as follows: ...
... Thus, instead of rectangular pulses, we choose to use prolate spheroidal wave functions (PSWFs) [31]. Those are considered optimal in terms of their time-frequency concentration properties. ...
Article
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In this paper, we comprehensively investigate the achievable rates of selected band-limited intensity modulation schemes, which are important for optical wireless communication (OWC) applications, while accounting for the specific nature of their signal construction (non-negative, real and baseband), and imposing identical bandwidth and average optical power constraints. Furthermore, we identify/devise methods to effectively trade between these parameters.Three variants of orthogonal frequency division multiplexing (OFDM), namely Asymmetrically Clipped Optical OFDM (ACO-OFDM), spectrally and energy efficient OFDM (SEE-OFDM), and DC biased optical OFDM (DCO-OFDM), and single-carrier pulse amplitude modulation (SC-PAM) are studied. The clipping noise in ACO-OFDM and SEE-OFDM is found to consume a large excess bandwidth. The detrimental effects of this excess bandwidth on the achievable rate are evaluated. For SEE-OFDM, the problem of optimal power allocation among its components is formulated and solved using the Karush-Kuhn-Tucker (KKT) method. For DCO-OFDM, the clipping noise is modeled and incorporated in the analysis. Among the existing schemes, DCO-OFDM yields the best overall performance, due to its compact spectrum. In order to improve the achievable rate, we propose and analyze two improved distortionless variants, filtered ACO-OFDM (FACO-OFDM) and filtered SEE-OFDM (FSEE-OFDM), which yield better spectral efficiency than ACO-OFDM and SEE-OFDM, respectively. FSEE-OFDM, being the most spectrally efficient, outperforms all schemes.
... These sequences are also known as Slepian sequences. PSWF have been used in many applications such as analysis of nonstationary and nonlinear time series [38], communication theory [39] and their mathematical properties and computation are presented in [40]. The PSWF are real-valued, finite support functions with maximum energy concentration in a given bandwidth. ...
... For a given integer K ≤ N , we can get N × K matrix formed by taking the first K columns of φ N,Ω . When K ≈ 2N Ω, it is a highly efficient basis that captures most of the signal energy [39,40]. ...
Article
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Time–frequency (TF) representations are very important tools to understand and explain circumstances, where the frequency content of nonstationary signals varies in time. A variety of biosignals such as speech, electrocardiogram (ECG), electroencephalogram (EEG), and electromyogram (EMG) show some form of non-stationarity. Considering Priestley’s evolutionary (time-dependent) spectral theory for analysis of non-stationary signals, the authors defined a TF representation called evolutionary Slepian transform (EST). The evolutionary spectral theory generalises the definition of spectra while avoiding some of the shortcomings of bilinear TF methods. The performance of the EST in the representation of biosignals for the blind source separation (BSS) problem to extract information from a mixture of sources is studied. For example, in the case of EEG recordings, as electrodes are placed along the scalp, what is actually observed from EEG data at each electrode is a mixture of all the active sources. Separation of these sources from a mixture of observations is crucial for the analysis of recordings. In this study, they show that the EST can be used efficiently in the TF-based BSS problem of biosignals. © 2018 Institution of Engineering and Technology.All right reserved.
... The PSWF have been used in time series analysis Moghtaderi et al. (2009). Indeed, PSWF have been used in many applications, one example is in communication theory Moore and Cada (2004), Wavelet-like properties is in Simons et al. (2018) and their mathematical properties and computation are presented in Walter and Shen (2003). Discrete form of the PSWF i.e., DPSS resulted from the work of Slepian about the problem of concentrating a signal jointly in temporal and spectral domains Slepian (1962). ...
... Left: First four Slepian sequences for chosen N=512 and N Ω =3; right: energy concentrations i.e., eigenvalues the signals are represented using the DPSS. Details on how many DPSS are needed for optimum representation can be found inOh et al. (2010);Moore and Cada (2004);Walter and Shen (2003).In general, a signal x[n] can be represented in terms of an orthogonal basis { k [n]} as, We showed inOh et al. (2010) that rewriting x[n] as follows: ...
Article
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Blind source separation (BSS) methods are used to separate sources from a mixed observations with very little prior knowledge of the mixing coefficients or sources. In this paper we propose an evolutionary spectral representation to implement BSS. Introduced by Priestley, evolutionary spectral theory generalizes the definition of spectrum for nonstationary processes. Under certain conditions, the evolutionary spectrum at each instant of time can be estimated from a single realization of a process such that it is possible to study processes with changing spectral patterns. In particular we are interested in the problem of separation of individual biosignals from electrophysiological recordings mixed by volume conduction. As biosignals such as electrocardiogram and electroencephalogram recordings are prime examples of nonstationary signals, evolutionary spectral representations can be used for the analysis of them. Our proposed evolutionary spectral representation is based on the discrete prolate spheroidal sequences (DPSS). Also known as Slepian sequences, the DPSS are defined to be the sequences with maximum spectral concentration for a given duration and bandwidth. Using the relation between discrete evolutionary transform and evolutionary periodogram, we derive the Slepian evolutionary spectrum. After the evolutionary spectrum is computed, we implement it for the BSS problem and compare with the well known time-frequency methods (Wigner-Ville distribution and S-transform) for performance evaluation.
... due to the orthogonality of the PSWF [21], then ...
... The discretized values of PSWF are derived with equidistant values of dt/T=0.001 as per the procedure described in Appendix A. The Fourier invariance of PSWF [21] indicates the following relationships, ...
Article
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Predicting the wave-induced response in the near-future is of importance to ensure safety of ships. To achieve this target, a possible method for deterministic and conditional prediction of future responses utilizing measured data from the most recent past has been developed. Herein, accurate derivation of the autocorrelation function (ACF) is required. In this study, a new approach for deriving ACFs from measurements is proposed by introducing the Prolate Spheroidal Wave Functions (PSWF). PSWF can be used in two ways: fitting the measured response itself or fitting the sample ACF from the measurements. The paper contains various numerical demonstrations, using a stationary heave motion time series of a containership, and the effectiveness of the present approach is demonstrated by comparing with both a non-parametric and a parametric spectrum estimation method; in this case, Fast Fourier Transformation (FFT) and an Auto-Regressive (AR) model, respectively. The present PSWF-based approach leads to two important properties: (1) a smoothed ACF from the measurements, including an expression of the memory time, (2) a high frequency resolution in power spectrum densities (PSDs). Finally, the paper demonstrates that a fitting of the ACF using PSWF can be applied for deterministic motion predictions ahead of current time.
... Therefore, the spectrogram obtained with this optimal window can improve the diagnostic decision process, compared with the use of non-optimal windows. The type of windows that are optimally and maximally concentrated, for a finite duration and bandwidth, are the Slepians [58,65]. Accordingly, in this paper, the Slepian window is proposed for the fault diagnosis of IMs. ...
... The Slepians have the remarkable property of orthogonality, both over an infinite and a finite range of the independent variable [65]. Due to the fact that the functions ϕ n (t) form a complete set of orthonormal functions, band-limited functions y(t) can be expanded in terms of the Slepians with the same bandwidth as ...
Article
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The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in the transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration, the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as the analysis window. In this paper, the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15-MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources.
... (cf. [60]). This open question was answered by David Slepian (1923Slepian ( -2007 et al. at Bell Laboratories in a series of seminal papers dated back to 1960s (see e.g., [53,83,86]). ...
... (i) Study of their analytic and asymptotic properties, numerical evaluations, prolate quadrature, interpolation and related issues (see, e.g., [4,6,8,25,37,40,43,60,66,70,73,91,[99][100][101]); ...
... From Eq. (19), it is clear that we can define a local frequency operator κ(x), which is of the form (20) and takes the function g(x) as its argument and provides a local value of the spatial frequency at the point x. This operator can be used to find the local superoscillatory regions of a waveform, and it has become a standard tool in such investigations. ...
... In hindsight, superoscillations have been hiding in plain sight for decades, in the form of prolate spheroidal wave functions (PSWFs), as pointed out by Ferreira [14]. The PSWFs were studied extensively in the 1960s by Landau, Pollack and Slepian [18,19]; a modern discussion of their behavior was written by Moore and Cada [20]. Let us consider functions in x, k space. ...
Article
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It is now well-appreciated that a bandlimited wave can possess oscillations much more rapidly than those predicted by the bandlimit itself, in a phenomenon known as superoscillation. Such superoscillations are required to be of dramatically smaller amplitude than the signal they are embedded in, and this has initially led researchers to consider them of limited use in applications. However, this view has changed in recent years and superoscillations have been employed in a number of systems to beat the limits of conventional diffraction theory. In this review, we discuss the current state of research on superoscillations in terms of superresolved imaging and subwavelength focusing, including the use of special non-diffracting and Airy beams to carry transverse superoscillating patterns. In addition, we discuss recent analogous works on using superoscillations to break the temporal resolution limit, and also consider the recently introduced inverse of superoscillations, known as suboscillations.
... Their discovery as being solutions of an integral operator came later as a result of the efforts by researchers from Bell Laboratories while trying to answer the question originally posed by Claude Shannon: "To what extent are functions, which are confined to a finite bandwidth, also concentrated in the time domain?" [3]. Later works followed, outlining properties of PSWFs PSWFs and finding expressions describing the eigenvalue distribution, which has the distinct characteristic of clustering around the two values 0 and 1. Asymptotic expressions were derived in [1] and non-asymptotic expressions in [6]. ...
... Section 3 outlines the major novel results of this work which consist of Lemma 1 and Theorems 1-4. The proof of each theorem is given 40 in a subsection with the corresponding theorem number under Section 4. Section 5 consists of assisting theorems (5,6) and lemmas (2)(3)(4) for DPSWFs that can be quickly derived from their PSWF counterparts and thus are considered minor yet novel contributions. Section 6 provides numerical results to validate the bound given by Theorem 4. For the sake of completeness, Appendices I and II provide simplifications of certain expressions and derivations of standard formulae used in this work. ...
... It is known that every bandlimited signal can be represented by a linear superposition of PSWF, see e.g. Xiao et al. [22], while the phase components of the signal can be explicitly obtained, thanks to the Fourier invariance of PSWF [23]. Finer frequency resolution than the FFT can be attained even if the response time length is short, by estimating numerical values in PSWF together with the eigenvalues of PSWF in advance, e.g. by using the Legendre polynomials-based approach [22]. ...
... Here ψj denote the PSWF, c the Slepian frequency [20], and Ne the number of PSWF to be considered. The orthogonality of PSWF [23] offers rapid derivation of the coefficients aj,k as follows: ...
Article
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This paper presents a new approach to attain estimates of the sea state based on short-time sequences of wave-induced ship responses. The present sea state estimation method aims at reconstructing the incident wave profiles in time domain. In order to identify phase components of the incident waves, the Prolate Spheroidal Wave Functions (PSWF) are employed. The use of PSWF offers an explicit expression of phase components in the measured responses and incident waves, indicating that estimations can be efficiently attained. A method to estimate the relative wave heading angle based on the response measurements and pre-computed transfer functions of the responses is also proposed. The method is tested with numerical simulations and experimental measurements of ship motions, i.e. heave, pitch, and roll, together with vertical bending moment and local pressure in a post-panamax size containership. Validation is made by comparing the reconstructed wave profiles with the incident waves. The accuracy and efficiency of the present approach are promising. At the same time, it is shown that the use of responses, which are more broad-banded in their frequency characteristics, is an effective means to cope with high frequency noise in reconstructed waves.
... Although there are other functions which are their own infinite Fourier transform, only the prolate functions enjoy the property for the finite transform: this property uniquely defines the prolate functions [1]. Associated with each function, there is an eigenvalue ( ) n c λ and a free parameter c which is a useful descriptor of system performance [2]. Some of the mentioned mathematical properties make the prolate functions easily applicable to optics [3]. ...
Article
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We propose a low complexity iterative algorithm for band limited signal extrapolation. The extrapolation method is based on the decomposition of finite segments of the signal via truncated series of real-valued linear prolate functions. Our theoretical derivation shows that given a truncated series (up to a selectable value) of prolate functions, it is possible to extrapolate the band limited function elsewhere if each extrapolated portion of the function is subject only to moderate truncation errors that we quantify in this paper. The effects of different sources of errors have been analyzed via extensive simulations. We have investigated a property of the signal decomposition formula based on linear prolate functions whereby the integration interval does not need to be symmetric with respect to the origin while time-shifted prolate functions are used in the series.
... In this form, the STFT can be thought of as an inner product between the signal and a series of windowed sinusoids given by h(t Ϫ )e j t , which are the basis functions onto which the signal is projected. For example, Figure 2A shows a zero-order Slepian (also known as a discrete prolate spheroidal sequence, or DPSS) window (Slepian, 1978;Moore and Cada, 2004) of 100 ms duration (this particular window is used in MTM, as explained in more detail below), while Figure 2B shows the corresponding basis functions with three different center frequencies at 20 (black), 50 (red), and 100 Hz (green). The spectrum of such windowed sinusoids is obtained by convolution of the spectra of the sinusoids (which is a delta function) and the window function, which, by the sifting theorem, is the spectrum of the window centered at . ...
Article
Signals recorded from the brain often show rhythmic patterns at different frequencies, which are tightly coupled to the external stimuli as well as the internal state of the subject. In addition, these signals have very transient structures related to spiking or sudden onset of a stimulus, which have durations not exceeding tens of milliseconds. Further, brain signals are highly nonstationary because both behavioral state and external stimuli can change on a short time scale. It is therefore essential to study brain signals using techniques that can represent both rhythmic and transient components of the signal, something not always possible using standard signal processing techniques such as short time fourier transform, multitaper method, wavelet transform, or Hilbert transform. In this review, we describe a multiscale decomposition technique based on an over-complete dictionary called matching pursuit (MP), and show that it is able to capture both a sharp stimulus-onset transient and a sustained gamma rhythm in local field potential recorded from the primary visual cortex. We compare the performance of MP with other techniques and discuss its advantages and limitations. Data and codes for generating all time-frequency power spectra are provided.
... An application of linear prolate functions with high precision to numerical analysis and synthesis of signals was studied for the first time in [9]. ...
Article
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Digital signals transmitted over a communication channel are mostly affected by noise. To reduce the detrimental effects of noise, a band-limited filter is used at the receiver, which results a phenomenon known as Inter-Symbol Interference. To avoid Inter-Symbol Interference, filters with greater bandwidth can be used. However, this causes high frequency noise to interfere with the transmitted information signal. This paper illustrates an innovative way to reduce Inter-Symbol Interference in the received baseband signal. This is achieved by making use of the processing bandwidth of a special filter designed by using Linear Prolate Functions. The result of the signal reconstruction capabilities of a prolate filter are compared with those of an ideal low pass filter in this paper.
... Detailed procedure for the KLE representation of a sea state in which the PSWF are used can be found in Sclavounos (2012). The properties of the PSWFs are discussed in Osipov and Rokhlin (2014), their application in signal processing have been discussed in Moore and Cada (2004) while detailed procedure for their evaluation is outlined in Xiao, Rokhlin, and Yarvin (2001). ...
Article
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This paper proposes an approach for evaluation of time dependent reliability of Jackup structures. An approach for signal processing using prolate spheroidal wave functions is combined with stochastic field representation method to represent ocean waves with least number of independent sources of uncertainty. First passage probability for dynamical systems subject to stochastic loading was then used in the formulation of the reliability approach. A simplified Jackup was modelled and used to demonstrate the time dependent reliability approach by propagating the uncertain wave load on the unit. In-house computer codes were developed for the analysis of the stochastic response in time-domain to obtain time dependent failure probabilities. The results were compared with those of a similar model in which the statistical method is used. © 2017 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.
... The PSWFs possess many interesting properties, see e.g. [34,33,23]. ...
Article
We consider the problem of imaging extended reflectors in waveguides using partial-aperture array, i.e. an array that does not span the whole depth of the waveguide. For this imaging, we employ a method that back-propagates a weighted modal projection of the usual array response matrix. The challenge in this setup is to correctly define this projection matrix in order to maintain good energy concentration properties for the imaging method, which were obtained previously by Tsogka et al (2013 SIAM J. Imaging Sci. 6 2714–39) for the full-aperture case. In this paper we propose a way of achieving this and study the properties of the resulting imaging method.
... Slepian et al. [8] naturally extended them to higher dimension and discussed their approximation in some special case in the following years. After that, the works on this functions are slowly developed until 1980s a large number of engineering applied this functions to signal processing, such as bandlimited signals extrapolation, filter designing, reconstruction and so on [16,17,18]. ...
Article
Quaternionic Linear Canonical Transforms (QLCTs) are a family of integral transforms, which generalized the quaternionic Fourier transform and quaternionic fractional Fourier transform. In this paper, we extend the energy concentration problem for 2D hypercomplex signals (especially quaternionic signals). The most energy concentrated signals both in 2D spatial and quaternionic linear canonical frequency domains simultaneously are recently recognized to be the quaternionic prolate spheroidal wave functions (QPSWFs). The improved definitions of QPSWFs are studied which gave reasonable properties. The purpose of this paper is to understand the measurements of energy concentration in the 2D spatial and quaternionic linear canonical frequency domains. Examples of energy concentrated ratios between the truncated Gaussian function and QPSWFs intuitively illustrate that QPSWFs are more energy concentrated signals.
... Les premières résolutions de l'équation avec ces coordonnées sont faites par Lamé en 1837, puis s'ensuivent de nombreuses études qui aboutissent aux deux ouvrages de référence par Flammer et Meixner [42,84], auxquelles on ajoutera les développements asymptotiques de Slepian [125]. Notons que ces fonctions peuvent être utilisées comme base d'une série entière [91] ou d'une quadrature [160] et possèdent une propriété de double orthogonalité [110]. La liste des propriétés utiles pour l'acoustique est, entre autres, donnée par [60]. ...
Thesis
Le rayonnement d'un haut-parleur monté sur une enceinte est généralement caractérisé par des mesures ou une simulation par la méthode des éléments finis. Cependant, ces méthodes de référence restent très coûteuses et ne permettent pas une interprétation physique des résultats. Dans ce manuscrit, nous proposons deux modèles analytiques pour prédire ce rayonnement, dans le cadre d'une application à une barre de son. Le premier modèle consiste à assimiler la géométrie de la barre de son à un sphéroïde. De cette manière, les variables sont séparées et on peut trouver une solution analytique sous la forme d'une somme d’harmoniques sphéroïdales. On décrira chaque étape de la méthode, ainsi que la comparaison des résultats à ceux des méthodes de référence. On développera particulièrement le calcul pour un haut-parleur circulaire sur le sphéroïde et la mise en place d'un critère de troncature des harmoniques. Ce modèle fonctionne bien en basse-fréquence, mais ne peut pas rendre compte de tous les phénomènes de diffraction par l'enceinte en haute fréquence. Pour celles-ci, la diffraction du champ sonore par les arêtes de l'enceinte devient non négligeable. Nous avons donc développé un second modèle analytique, basé sur une formulation intégrale de cette diffraction. Celle-ci est vue comme un ensemble de sources secondaires localisées sur les arêtes. On montrera comment établir ce modèle et on donnera des détails sur son implémentation. L'application de ce modèle permet d'interpréter physiquement le rayonnement d'une barre de son, et donc de conclure sur la validité du modèle sphéroïdal. On montre également comment les arêtes du baffle d'une enceinte entraînent des effets d'interférences constructives, qui peuvent induire un gain allant jusqu’à 3 dB. Ces effets entraînent également un phénomène contre-intuitif : si le haut-parleur n'est pas centré sur le baffle, la direction du maximum d'intensité du rayonnement tend à se décaler vers le côté opposé au décalage du haut-parleur sur le baffle.
... So as to make the most of the presented SVD filtering technique our further aim is to find the connection with conventional deconvolution methods which obtain exactly the same effect but the filtering factors can be recalculated in each iteration tailored to the given back projection system matrix. The special form (PSWF which is strongly connected to Fourier-transform 22,23 ) of the singular vectors makes this direction promising. ...
Article
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Background In emission tomography maximum likelihood expectation maximization reconstruction technique has replaced the analytical approaches in several applications. The most important drawback of this iterative method is its linear rate of convergence and the corresponding computational burden. Therefore, simplifications are usually required in the Monte Carlo simulation of the back projection step. In order to overcome these problems, a reconstruction code has been developed with graphical processing unit based Monte Carlo engine which enabled full physical modelling in the back projection. Materials and methods Code performance was evaluated with simulations on two geometries. One is a sophisticated scanner geometry which consists of a dodecagon with inscribed circle radius of 8.7 cm, packed on each side with an array of 39 × 81 LYSO detector pixels of 1.17 mm sided squares, similar to a Mediso nanoScan PET/CT scanner. The other, simplified geometry contains a 38,4mm long interval as a voxel space, detector pixels are assigned in two parallel sections each containing 81 crystals of a size 1.17×1.17 mm. Results We have demonstrated that full Monte Carlo modelling in the back projection step leads to material dependent inhomogeneities in the reconstructed image. The reasons behind this apparently anomalous behaviour was analysed in the simplified system by means of singular value decomposition and explained by different speed of convergence. Conclusions To still take advantage of the higher noise stability of the full physical modelling, a new filtering technique is proposed for convergence acceleration. Some theoretical considerations for the practical implementation and for further development are also presented.
... Here f denotes the Fourier transform of f. Due to their excellent properties, the PSWFs have found many applications in many areas such as communications, see [5] and in signal processing, see for example [6][7][8] . This explains the great interest of many authors for the computation of PSWFs and their corresponding eigenvalues, see for example [1,[9][10][11][12]. ...
Article
In this paper, we show that the generalized prolate spheroidal wave functions (GPSWFs), called sometimes Slepian’s functions, can be defined as the most concentrated functions among functions in some reproducing kernel Hilbert spaces (RKHS for short). As a consequence, we use them the GPSWFs to approximate the sets of K-bandlimited functions that are essentially time limited to an interval (a, b). © 2018, African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature.
... Recently, Pei and Ding Pei and Ding [12] used prolate spheroidal wave functions for analyzing the properties of the finite extension Fourier transform. Moore and Cada [13] exploited the orthogonal properties of prolate spheroidal wave functions in the form of a new orthogonal expansion which we have named the Slepian series. Wang [14] reviewed the prolate spheroidal wave functions and their variants from the viewpoint of spectral/spectral-element approximations using such functions as basic functions. ...
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The principal aim of this paper is to investigate the thermoelastic responses in a generalised prolate spheroid subjected to the radiation type boundary conditions on the inner and outer curved surfaces. The recent research activity motivates the work carried out on a different spheroid (oblate or prolate) to avoid structural failure due to thermal stress response. The method of integral transformation approach is used to obtain an exact solution of heat conduction equation subjected to the generation of heat within the body. The relations obtained in this paper can be applied to any arbitrary boundary and initial conditions. Some results which are derived using computational tools are accurate enough for the practical purpose were depicted graphically. MSC: 35B07 • 35G30 • 35K05 • 44A10
... The PSWFs, also known as Slepian The one-dimensional PSWFs have so far mainly been developed in two different directions. One is that fast and highly accurate methods have been deeply developed for the approximation of the PSWFs and their eigenvalues [7,8,9,10,11]. From a computational point of view, the PSWFs provide a natural and efficient tool for computing with bandlimited functions defined on an interval [12,13]. ...
Article
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One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions (QPSWFs). The purpose of this paper is to present the definition and fundamental properties of the QPSWFs and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem.
... As a result, each time slot may include six or seven CP-OFDM symbols, depending on the length of the cyclic prefix, which is determined by the transmission settings. Eventually, the LTE transmission bandwidth capacity will be determined in relation to the total number of subcarriers used for transmission, with each pair of twelve contiguous subcarriers forming a so-called bodily resource block (PRB) or, for convenience, a useful resource block (RB) [42][43][44][45][46][47][48]. The final filtered multi-service (UFMC), which also belongs to the sub-band based entirely filtering group, was another inspiration as a recommended option for 5G. ...
Article
The need for internet of things (IoT) and machine-to-machine communication (MTC) has been growing rapidly all across the world. To meet the client's needs, many literature reviews were undertaken in several countries. Orthogonal frequency division multiplexing (OFDM), Universal Filtered Multi-Carrier (UFMC), filter-bank multicarrier offset construction amplitude modulation (FBMC-OQAM), generalized frequency division multiplexing (GFDM), and others are candidates for LTE, LTE advance, and 5G, according to the majority of the researchers. However, because it is sensitive to propagation and noise, such as amplitude, with a huge dynamic range, it requires RF power amplifiers with a high peak to average power quantitative relationship; consequently, it is not recommended for LTE, LTE advance, or 5G. As a result, the same concerns were addressed by introducing innovative type filtered orthogonal frequency division multiplexing (F- OFDM), which was the subject of this study. In addition, F-OFDM mathematical models were constructed and simulated in the MATLAB software environment. To validate the proposed innovative F-OFDM, OFDM was compared. For innovative F-OFDM, the simulated result was 0.00083333 bit error rate (BER). Furthermore, the bit error rate (BER) of F-OFDM over OFDM was 89.4 percent, and the peak to average power ratio was 17 percent. The simulation results unmistakably show that the suggested innovative F-OFDM is the greatest fit for LTE, LTE advanced, and 5G contenders.
... x ∈ −1, 1 (3.5) whose eigenvalues and eigenfuctions can be expressed in terms of the prolate spheroidal wave functions (PSWF) [77][78][79]. The eigenvalues in (3.5) can be written in terms of the radial PSWF of zero order R 0n [48,53,80] γ n = 2η π R 0n (η, 1) 2 n ∈ N 0 (3. 6) while the corresponding eigenfunctions are expressed through the angular PSWF of zero order S 0n as follows ...
Preprint
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We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schr\"odinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored.
... Prolate spheroidal wave function: ψ τ0 (τ ) (see Fig. 2 Table 1: Values of the minimal separation γ for commonly encountered point spread functions. Herein, J 0 (·) denotes the Bessel function of the first kind, and ψ τ0 (·) refers to the prolate spheroidal wave function (PSWF) of order 0 for the temporal concentration band [−τ 0 , τ 0 ] [18,19], i.e., the function g(·) with a frequency band − 1 2 , 1 2 and with g L2 = 1 which maximizes the integral ...
Preprint
The stability of spike deconvolution, which aims at recovering point sources from their convolution with a point spread function (PSF), is known to be related to the separation between those sources. When the observations are noisy, it is critical to ensure support stability, where the deconvolution does not lead to spurious, or oppositely, missing estimates of the point sources. In this paper, we study the resolution limit of stably recovering the support of two closely located point sources using the Beurling-LASSO estimator, which is a convex optimization approach based on total variation regularization. We establish a sufficient separation criteria between the sources, depending only on the PSF, above which the Beurling-LASSO estimator is guaranteed to return a stable estimate of the point sources, with the same number of estimated elements as of the ground truth. Our result highlights the impact of PSF on the resolution limit in the noisy setting, which was not evident in previous studies of the noiseless setting.
... (cf. [16,20]). Any square integrable function f (ξ) is bandlimited, if its Fourier transform ψ(t) has a finite support [−c, c] such that f (ξ) = (1.1) ...
Article
In this paper, we introduce one family of vectorial prolate spheroidal wave functions of real order α > −1 on the unit ball in R 3 , which satisfy the divergence free constraint, thus are termed as divergence free vectorial ball PSWFs. They are vectorial eigenfunctions of an integral operator related to the finite Fourier transform, and solve the divergence free constrained maximum concentration problem in three dimensions, i.e., to what extent can the total energy of a band-limited divergence free vectorial function be concentrated on the unit ball? Interestingly, any optimally concentrated divergence free vectorial functions, when represented in series in vector spherical harmonics, shall be also concentrated in one of the three vectorial spherical harmonics modes. Moreover, divergence free ball PSWFs are exactly the vectorial eigenfunctions of the second order Sturm-Liouville differential operator which defines the scalar ball PSWFs. Indeed, the divergence free vectorial ball PSWFs possess a simple and close relation with the scalar ball PSWFs such that they share the same merits. Simultaneously, it turns out that the divergence free ball PSWFs solve another second order Sturm-Liouville eigen equation defined through the curl operator ∇× instead of the gradient operator ∇.
... This observation allowed for the efficient computation of PSWFs and the eventual incorporation of their digital counterparts (the discrete PSWFs) in computer hardware. Analyticity and asymptotic properties of PSWFs, including numerical evaluations and applications to quadrature and interpolation are investigated in [4,5,13,18,[23][24][25]31]. Other applications of PSWFs can be found in [7,12,14,16,19,20,22,26,30]. ...
Preprint
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In the present paper, we introduce the multidimensional Clifford prolate spheroidal wave functions (CPSWFs) defined on the unit ball as eigenfunctions of a Clifford differential operator and provide a Galerkin method for their computation as linear combinations of Clifford-Legendre polynomials. We show that these functions are eigenfunctions of the truncated Fourier transformation. Then we investigate the role of the CPSWFs in the spectral concentration problem associated with balls in the space and frequency domains, the behaviour of the eigenvalues of the time-frequency limiting operator and their spectral accumulation property.
... signal processing [21], [22], geophysics [23]- [25], medical imaging [26]). In particular, these functions have been used in solving partial differential equations [27], [28], inverse problems [29], [30], interpolation [31], [32] and extrapolation [33]. In fields with spatially limited observations, the functions have become the dominant spatial or spectral windows for regularisation of quadratic inverse problems of power spectral estimation [23]. ...
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This work presents the construction of a novel spherical wavelet basis designed for incomplete spherical datasets, i.e. datasets which are missing in a particular region of the sphere. The eigenfunctions of the Slepian spatial-spectral concentration problem (the Slepian functions) are a set of orthogonal basis functions which exist within a defined region. Slepian functions allow one to compute a convolution on the incomplete sphere by leveraging the recently proposed sifting convolution and extending it to any set of basis functions. Through a tiling of the Slepian harmonic line one may construct scale-discretised wavelets. An illustration is presented based on an example region on the sphere defined by the topographic map of the Earth. The Slepian wavelets and corresponding wavelet coefficients are constructed from this region, and are used in a straightforward denoising example.
... To cope with, prolate spheroidal wave functions (PSWFs) were proposed. They are time-limited filters with small side-lobes [11], [12]. The Hermite−Gaussian functions are special cases of PSWFs which provide optimum TF concentration. ...
Preprint
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The perspective for the next generation of wireless communications prognosticates that the communication should take place with high heterogeneity in terms of services and specifications. The emergence of new communication networks, especially between devices like internet of things (IoT) networks, machine type communications (MTC) and unmanned aerial vehicles (UAV), makes the redesign of the radio air-interfaces even higher significant. High data rate, efficient use of the spectrum, low latency, flexibility, scalable ability, energy efficiency and reduced complexity are in the vanguard of the challenging needs. This highlights the necessity to overrides classical waveforms and propose new air interface that best fits these requirements. Several attempts were conducted to propose the most suitable prototype filters and their associated waveforms that support the future applications with these crucial prerequisites. In the present paper, we analyze the most important marked prototype filters reported in literature, and their corresponding waveforms proposed as prominent candidates for wireless air interface. We will show that the filter properties impact the waveform performances and that a close filter-waveform interrelationship is there. The principal contribution is to give a deep insight on the structure of these filters and waveforms and to come up with a fair comparison regarding different criteria and distinct figures of merit.
... Indeed, the largest 2c/π eigenvalues are dominant. The significant values of |λj| become |λj| 2 /c  (Moore and Cada, 2004). ...
Article
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This paper studies real-time deterministic prediction of wave-induced ship motions using the autocorrelation functions (ACFs) from short-time measurements, namely the instantaneous ACFs. The Prolate Spheroidal Wave Functions (PSWF) are introduced to correct the large lag time errors in the instantaneous sample ACF, together with a modification of the autocorrelation (AC) matrix for ensuring its positive definiteness. The validity of the PSWF-based ACFs is first examined by using the ship motion measurements from model experiment under stationary wave excitations. It is shown that the use of PSWF-based ACFs leads to better prediction accuracy than direct use of sample ACFs. The validation is then extended to ship motion prediction using in-service data from a container ship, and an improvement of the prediction accuracy by PSWF-based ACFs is again found. Finally, the effectiveness of use of the instantaneous ACFs for non-stationary wave-induced responses is highlighted by comparing with the prediction results based on the ACFs from long-time measurements.
... Prolate Spheroidal Wave Functions (PSWFs) have excellent basic characteristics such as high energy aggregation in the time-frequency domain, biorthogonality in the time domain, approximate time band limitation, and controllable frequency spectrum [1][2][3][4], since its introduction, it has received extensive attention from academia, and it has been widely used in signal analysis and communication systems [5][6][7]. Non-sinusoidal orthogonal modulation system based on prolate spheroidal wave function [8] is a typical non-sinusoidal multi-carrier communication system. ...
Article
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Aiming at the problems of non-sinusoidal time-domain multiple quadrature modulation signals based on Prolate Spheroidal Wave Functions (PSWFs), as the number of pulses increases, the bandwidth efficiency of the modulation signal frequency band decreases, the Peak-to-Average Power Ratio (PAPR) and the implementation complexity are high, a method for optimal design of PSWFs pulse groups based on DFT precoding is proposed. The PSWFs pulse signal frequency domain generation method is adopted. By introducing the precoding matrix, the multi-channel PSWFs pulse group signals are optimized and designed, and then the input data is redistributed to obtain new PSWFs pulse group signals. Experimental results show that this method can further improve the power efficiency of the modulated signal and effectively reduce the peak-to-average power ratio of the modulated signal without changing the orthogonality of the original PSWFs subcarriers. As the number of carriers increases, the suppression effect on the peak-to-average power ratio becomes more obvious.
... Slepian and Pollak [24] have proposed optimal functions to simultaneously maximise α 2 and β 2 . A set of band-limited functions limited in the band ( − Ω, Ω), namely prolate spheroidal wave functions (PSWFs) are introduced [24], out of which the zero-order PSWF [25][26][27] denoted by ψ 0 (t), possesses interesting optimality property of achieving maximum energy concentration in the time interval ( − T /2, T /2). The maximum achievable concentration is denoted by λ 0 (TΩ) and is the maximum eigenvalue of the following associated eigenfunction equation ...
Article
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For numerous applications in the field of signal processing, it is desired to design a compact window that can simultaneously concentrate maximum energy in finite time interval and frequency band. Although zero-order discrete prolate spheroidal sequence (DPSS) meets this requirement, it is of infinite support. Limiting this sequence to finite support no longer guarantees the optimality property. This study aims at designing a discrete finite length window that can maximise the energy simultaneously in narrow time interval and frequency band. A multi-objective optimisation approach is adopted to obtain the upper bound of maximum achievable time and frequency domain energy concentrations for finite length sequences. The optimal sequence thus obtained is termed as the optimal window with finite support (OWFS), and its various associated properties are discussed. It is shown analytically that as the support of OWFS approaches infinity, it converges to zero-order DPSS. In order to illustrate its optimality, the compactness of the proposed OWFS is compared with those of various window functions. Extending the proposed OWFS, this study also discusses the formulation and associated properties of the zero-order periodic DPSS. Further, the closed form expression for the upper bound of achievable time and frequency domain energy concentrations is also derived.
... (cf. [16,20]). Any square integrable function f (ξ) is bandlimited, if its Fourier transform ψ(t) has a finite support [−c, c] such that f (ξ) = (1.1) ...
Preprint
In this paper, we introduce one family of vectorial prolate spheroidal wave functions of real order $\alpha>-1$ on the unit ball in $R^3$, which satisfy the divergence free constraint, thus are termed as divergence free vectorial ball PSWFs. They are vectorial eigenfunctions of an integral operator related to the finite Fourier transform, and solve the divergence free constrained maximum concentration problem in three dimensions, i.e., to what extent can the total energy of a band-limited divergence free vectorial function be concentrated on the unit ball? Interestingly, any optimally concentrated divergence free vectorial functions, when represented in series in vector spherical harmonics, shall be also concentrated in one of the three vectorial spherical harmonics modes. Moreover, divergence free ball PSWFs are exactly the vectorial eigenfunctions of the second order Sturm-Liouville differential operator which defines the scalar ball PSWFs. Indeed, the divergence free vectorial ball PSWFs possess a simple and close relation with the scalar ball PSWFs such that they share the same merits. Simultaneously, it turns out that the divergence free ball PSWFs solve another second order Sturm-Liouville eigen equation defined through the curl operator $\nabla\times$ instead of the gradient operator $\nabla$.
Article
In this paper, we propose an orthogonal time-domain multiplexing scheme based on parallel transmission of data symbols modulating linear prolate functions (LPFs). As in orthogonal frequency-division multiplexing (OFDM) systems, equalization is performed in the frequency domain and the cyclic prefix is introduced. As shown in the paper, the overlapping of LPFs in time (and frequency) provides higher degree of sidelobes suppression compared with OFDM. Moreover, it is possible sending more parallel streams in the same bandwidth and time interval (with respect to an OFDM system), by accepting a slight increase of the spectrum "leakage". The paper also proposes the use of these additional streams either to reduce the Peak-to-Average Power Ratio (PAPR) or to estimate the channel. Performance comparisons between the proposed system and OFDM are shown for both an additive white Gaussian noise (AWGN) channel and a highly frequency selective channel.
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Superoscillations are making a growing impact on an ever-increasing number of real-world applications, as early theoretical analysis has evolved into wide experimental realisation. This is particularly true in optics: the first application area to have extensively embraced superoscillations, with much recent growth. This review provides a tool for anyone planning to expand the boundaries in an application where superoscillations have already been used, or to apply superoscillations to a new application. By reviewing the mathematical methods for constructing superoscillations, including their considerations and capabilities, we lay out the options for anyone wanting to construct a device that uses superoscillations. Superoscillations have inherent trade-offs: as the size of spot reduces, its relative intensity decreases as high-energy sidebands appear. Different methods provide solutions for optimising different aspects of these trade-offs, to suit different purposes. Despite numerous technological ways of realising superoscillations, the mathematical methods can be categorised into three approaches: direct design of superoscillatory functions, design of pupil filters and design of superoscillatory lenses. This categorisation, based on mathematical methods, is used to highlight the transferability of methods between applications. It also highlights areas for future theoretical development to enable the scientific and technological boundaries to be pushed even further in real-world applications.
Preprint
The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as analysis window. In this paper the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15 MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources.
Article
In this study, a method for predicting the extreme value distribution of the Vertical Bending Moment (VBM) in a flexible ship under a given short-term sea state is presented. The First Order Reliability Method (FORM) is introduced to evaluate the Probability of Exceedances (PoEs) of extreme VBM levels. The Karhunen-Loeve (KL) representation of stochastic ocean wave is adopted in lieu of the normal wave representation using the trigonometric components, by introducing the Prolate Spheroidal Wave Functions (PSWFs) to formulate the wave elevations. By this means, reduction of the number of stochastic variables to reproduce ocean wave is expected, which in turn the number of computations required during FORM based prediction phases is significantly reduced. In this study, the Reduced Order Model (ROM), which was developed in our previous studies, is used to yield the time-domain VBMs along with the hydroelastic (whipping) component in a ship. Two different short-term sea states, moderate and severe ones, are assumed. The FORM based predictions using PSWF for normal wave-induced VBM are then validated by comparing with those using the normal trigonometric wave representation and Monte Carlo Simulations (MCSs). Through a series of numerical demonstrations, the computational efficiency of the FORM based prediction using PSWF is presented. Then, the validation is extended to the severe sea state where the whipping vibration contributes to the extreme VBM level to a large degree, and finally the conclusions are given.
Article
We consider the problem of representing a finite-energy signal with a finite number of samples. When the signal is interpolated via sinc function from the samples, there will be a certain reconstruction error since only a finite number of samples are used. Without making any additional assumptions, we derive a lower bound for this error. This error bound depends on the number of samples but nothing else, and is thus represented as a universal curve of error versus number of samples. Furthermore, the existence of a function that achieves the bound shows that this is the the tightest such bound possible.
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MathSciNet Review of the related paper, https://www.researchgate.net/publication/257312852_Effective_band-limited_extrapolation_relying_on_Slepian_series_and_regularization , avaiable on RG.
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The stability of spike deconvolution, which aims at recovering point sources from their convolution with a point spread function (PSF), is known to be related to the separation between those sources. When the observations are noisy, it is critical to ensure support stability, where the deconvolution does not lead to spurious, or oppositely, missing estimates of the point sources. In this paper, we study the resolution limit of stably recovering the support of two closely located point sources using the Beurling-LASSO estimator, which is a convex optimization approach based on total variation regularization. We establish a sufficient separation criterion between the sources, depending only on the PSF, above which the Beurling-LASSO estimator is guaranteed to return a stable estimate of the point sources, with the same number of estimated elements as that of the ground truth. Our result highlights the impact of PSF on the resolution limit in the noisy setting, which was not evident in previous studies of the noiseless setting. Towards the end, we show that the same resolution limit applies to resolving two close-located sources in conjunction of other well-separated sources.
Thesis
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In a baseband system, the digital information signal required to be transmitted over a channel, has a wide frequency spectrum. This wide frequency spectrum gets attenuated due to the band limited response of the filters utilized for communication. This band limiting of the signal pulses, causes them to expand outward thereby interfering with each other and causing a phenomenon known as Inter-Symbol Interference. The known techniques used for avoiding this limitation have used bandwidth greater than the minimum required bandwidth for no interference given by Nyquist. The physical increase in bandwidth of the filters can cause high frequency noise to interfere with the transmitted information signal. This thesis presents an innovative way to reduce Inter-Symbol Interference in the received signal by utilising a special filter designed by using Linear Prolate Functions. The results compare the signal reconstruction capabilities of a prolate filter with those of an ideal low pass filter.
Conference Paper
Information-theoretic and signal processing aspects of some modulation schemes designed for intensity modulation/direct detection (IM/DD) optical wireless communication (OWC) systems are studied. Due to the constraints of IM/DD signals (non-negative real and baseband signals), the construction of these signals along with their time and frequency characteristics differ from their RF counterparts. This necessitates a careful study of such schemes under practical constraints. Three schemes are studied in this paper, namely, single carrier pulse amplitude modulation (SC-PAM), asymmetrically clipped optical OFDM (ACO-OFDM), and DC biased optical OFDM (DCO-OFDM). Our aim is to carry out a comparative study of these schemes in the presence of identical constraints on bandwidth and average optical power. The study reveals that the clipping operation required in ACO-OFDM significantly reduces its information rate, and as a result, it is outperformed by SC-PAM. Such a limitation does not apply to DCO-OFDM which has a higher information rate, even though part of the available optical power is expended in the non-information-bearing DC bias.
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In this work, we present a generalized formulation of the Slepian concentration problem on the sphere for finding band-limited functions with an optimal concentration in the spatial domain. By introducing weighting functions in the formulation of classical Slepian concentration problem and assigning different values to these weighting functions, we present two variants of the concentration problem namely the differential and the weighted Slepian concentration problem. In the differential Slepian concentration problem, we consider two regions on the sphere and find band-limited functions such that the energy is maximized in one region at the expense of the energy in the other region. We propose non-negative weighting using a spatial window function to formulate and solve the weighted Slepian concentration problem. Each problem can be solved by formulating it in the harmonic domain as an eigenvalue problem, the solution of which yields eigenfunctions that serve as alternative basis functions for the representation of band-limited signals and are referred to as Slepian functions. We also present and analyse the properties of the Slepian functions. To support the applications in acoustics and cosmology, we also provide a demonstration for the use of the proposed Slepian functions for the robust signal modeling and the estimation of the energy spectrum of red and white stochastic processes on the sphere.
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In this work, we propose a prototype filter design for Filter Bank MultiCarrier (FBMC) systems based on convex opti- mization, aiming superior spectrum features while maintaining a high symbol reconstruction quality. Initially, the proposed design is written as a non-convex Quadratically Constrained Quadratic Programming (QCQP), which is relaxed into a convex QCQP guided by a line search. Through the resulting problem, we design three prototype filters: Type-I, II and III. In particular, the Type-II filter shows a slightly better performance than the classical Mirabasi-Martin design, while Type-I and III filters offer a much more contained spectrum than most of the prototype filters suitable for FBMC applications. Furthermore, numerical results corroborate the effectiveness of the designed filters as the proposed filters offer fast decay and contained spectrum while not jeopardizing symbol reconstruction in practice.
Preprint
The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as analysis window. In this paper the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15 MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources.
Article
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The prolate spheroidal wave functions (PSWFs) are used in sampling of bandlimited signals. Several formulae based on integer values of these PSWFs are derived and used to replace the sinc function in sampling theorems. They are also used to construct analysis and synthesis filter banks for sampled values. A type of multiresolution analysis based on PSWFs similar to that of wavelet theory is shown to exist. Several numerical examples illustrating the theory are presented.
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Prolate spheroidal wave functions (PSWFs) arise as solutions of an integral equation. This makes them bandlimited functions in a Paley-Wiener space, but because they are also solutions to a Sturm-Liouville problem, they behave very much like polynomials locally. Chromatic series are series expansions in which the coefficients are linear combinations of derivatives of a function. They were introduced by Ignjatovic as a replacement for Taylor's series and are based on orthogonal polynomials. Since the PSWFs are close to orthogonal polynomials they can be used to replace them in the Ignjatovic theory. The theory can be extended further to prolate spheroidal wavelet series that then combine chromatic series with sampling series. This leads to an overdetermined system which can use either local or global data to approximate the original function.
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It has been proven that the prolate-spheroidal wave function (PSWF) window is the optimal high-resolution window in the classical spectral estimation. However, the implementation of the PSWF window is difficult in the past due to the lack of a closed form solution. The numerical computation of the PSWF window using the Legendre function is considered and the performance of this window compared with other windows is discussed. The theoretical analysis and simulation results showed that under the same conditions, the PSWF window which is approximated by using the proposed method is superior to other commonly used windows
Conference Paper
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The optimal window, the time limited sequence whose energy is most concentrated in a finite frequency interval, is related to a particular discrete prolate spheroidal sequence. The optimal window is actually a family of windows with many degrees of freedom. The Kaiser (1974) window is an approximation to this optimal window. Kaiser used this approximation because the standard method employed to compute the optimal window is numerically ill-conditioned. We show the actual optimal window can be efficiently computed by using an alternative formulation of the discrete prolate spheroidal sequences. We then give a set of design formulas to generate the optimal window for the desired window length, mainlobe width, and relative peak sidelobe height
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This chapter discusses applications, specifically dealing with the performance characteristics of optical systems. The linear prolate functions are a set of band limited functions, which like the trigonometric functions, are orthogonal and complete over a finite interval. However, unlike the trig functions, they are also complete and orthogonal over the infinite interval. Fourier transform of a linear prolate function is proportional to the same prolate function. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in the chapter. The optical applications arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate ability of lens and lensless systems to form high quality images. There is one physical phenomenon that unifies these applications—diffraction at a finite aperture—and it can be said that this phenomenon is optimally analyzed by use of the prolate functions.
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The purpose of this paper is to examine the mathematical truth in the engineering intuition that there are approximately 2WT independent signals ϕi of bandwidth W concentrated in an interval of length T. Roughly speaking, the result is true for the best choice of the ϕi (prolate spheroidal wave functions), but not for sampling functions (of the form sin t/t). Some typical conclusions are: Let f(t), of total energy 1, be band‐limited to bandwidth W, and let (Formula Presented.) Then (Formula Presented.) is true for all such f with N = 0, C = 12, if the ϕn are the prolate. spheroidal wave functions; false for some such f for any finite constants N and C if the ϕn are sampling functions.
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This paper not only reviews the various methodologies for evaluating the angular and radial prolate and oblate spheroidal functions and their eigenvalues, but also presents an efficient algorithm which is developed with the software package Mathematica. Two algorithms are developed for computation of the eigenvalues λmn and coefficients dmnr. Important steps in programming are provided for estimating eigenvalues of the spheroidal harmonics with a complex argument c. Furthermore, the starting and ending points for searching for the eigenvalues by Newton's method are successfully obtained. As compared with the published data by Caldwell [J. Phys. A 21, 3685 (1988)] or Press et al. [Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992)] (for a real argument) and Oguchi [Radio Sci. 5, 1207 (1970)] (for a complex argument), the spheroidal harmonics and their eigenvalues estimated using this algorithm are of a much higher accuracy. In particular, a lot of tabulated data for the intermediate coefficients dmnρ\|r, the prolate and oblate radial spheroidal functions of the second kind, and their first-order derivatives, as obtained by Flammer [Spheroidal Wave Functions (Stanford University Press, Stanford, CA, 1987)], are found to be inaccurate, although these tabulated data have been considered as exact referenced results for about half a century. The algorithm developed here for evaluating the spheroidal harmonics with the Mathematica program is also found to be simple, fast, and numerically efficient, and of a much better accuracy than the other results tabulated by Flammer and others, being able to produce results of 100 significant digits or more.
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Glossary of symbols 1. Introduction to spectral analysis 2. Stationary stochastic processes 3. Deterministic spectral analysis 4. Foundations for stochastic spectral theory 5. Linear time-invariant filters 6. Non-parametric spectral estimation 7. Multiple taper spectral estimation 8. Calculation of discrete prolate spheroidal sequences 9. Parametric spectral estimation 10. Harmonic analysis References Appendix: data and code via e-mail Index.
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The volume is one of a series of six volumes published by the Naval Research Laboratory containing tabulation of explicit values of radial spheroidal wave functions of both oblate and prolate kinds over extended ranges of parameters. It is designed to provide the mathematical physicist and research engineer with accurate values of important but not easily calculated functions needed to solve boundary value problems of radiation, scattering, and propagation of scalar or vector waves in spheroidal coordinates. This series vastly extends the scope and accuracy of existing tabulations of radial spheroidal wave functions. The presence of many of the entries was made possible only through adoption of calculation techniques involving extreme precision. This was particularly true in the calculation of the characteristic values for the radial equation resulting from separation of the Helmholtz wave equation in spheroidal coordinates, a knowledge of which is essential in the calculation of spheroidal angle functions. The present document consists of Volume 1-prolate m=0. (Author)
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A set of tables of spheroidal wave functions designed to simplify the computation of acoustic and electromagnetic scattering from spheroids. The tables were computed to five-place accuracy on the Whirlwind digital computer, and automatically tabulated. An introduction discusses the mathematical properties of the functions and describes some of their applications.
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Polynomials are one of the principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc, it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well developed apparatus of prolate spheroidal wavefunctions to construct quadratures, interpolation and differentiation formulae, etc, for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc, the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.
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In two earlier papers* in this series, the extent to which a square-integrable function and its Fourier transform can be simultaneously concentrated in their respective domains was considered in detail. The present paper generalizes much of that work to functions of many variables. In treating the case of functions of two variables whose Fourier transforms vanish outside a circle in the two-dimensional frequency plane, we are led to consider the integral equation It is shown that the solutions are also the bounded eigenfunctions of the differential equation a generalization of the equation for the prolate spheroidal wave functions. The functions ϕ (called “generalized prolate spheroidal functions”) and the eigenvalues of both (i) and (ii) are studied in detail here, and both analytic and numerical results are presented. Other results include a general perturbation scheme for differential equations and the reduction to two dimensions of the case of functions of D > 2 variables restricted in frequency to the D sphere.
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A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum concentrated on a subinterval of the fundamental period of the spectrum. Key to the analysis are certain sequences, called discrete prolate spheroidal sequences, and certain functions of frequency called discrete prolate spheroidal functions. Their mathematical properties are investigated in great detail, and many applications to signal analysis are pointed out.
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The theory developed in the preceding paper is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.
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The problem of extrapolating discrete-index bandlimited signals from a finite number of samples is addressed in this paper. The algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time-limited is approximated well by a low-dimensional linear subspace. This fact, which is well known for one-dimensional (1-D) signals with contiguous passbands and time-concentration intervals, is established for a more general class of multidimensional (m-D) signals with discontiguous passbands and discontiguous time-concentration regions. A criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband, time-concentration regions, energy concentration factor, and bounds on the tolerable extrapolation error. The extrapolation is constrained to lie in this subspace, and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations. For certain sampling patterns, the system is ill conditioned, and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error. A novel criterion for rank selection based on known bounds on noise power and modeling error is presented. The effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations
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Linear prolate functions (LPFs) are a set of bandlimited functions constructed to be invariant to the Fourier transform and orthonormal on the real line for the given bandwidth. Their unique properties make LPFs useful in signal processing. A method is described to evaluate the LPs by solving the eigensystem of the corresponding differential equation. The eigenvectors of this system provide the coefficients of the representation of the required functions into a series of spherical Bessel functions. The method omits several cumbersome steps inherent to previous algorithms without loss of accuracy
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A fast band-limited signal extrapolation technique is presented where the total extrapolation process is achieved by a single matrix operation. The proposed technique and its implementation has many advantages over known extrapolation techniques in terms of computational savings and accuracy of the results, and it can he operated on a realtime basis.
• S Zhang
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Prolate spheroidal wave functions, Fourier analysis and uncertainty-II
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