Slepian Sequences Research Articles

Slepian Transform Based Detectors for HF Skywave Massive MIMO-OFDM Systems

In this paper, we investigate signal detection for high frequency (HF) skywave massive multiple-input multiple-output (MIMO) systems with orthogonal frequency division multiplexing (OFDM) modulation. We first introduce the beam based channel model for HF skywave massive MIMO-OFDM channels and show the relationship of sparse supports between the beam domain channel and the Fourier spectrum of the space domain channel. Based on the sparse beam domain channel, we propose a separate Slepian transform (SST) based detector, where a set of modulated Slepian sequences are designed independently for user terminals (UTs). Before the minimum mean-squared error (MMSE) detection, the Slepian transform is performed for each UT to reduce the dimension of the observation vector and the channel matrix, thus avoiding high dimensional matrices multiplications and inversions. Due to the sparsity of the beam domain channel, the Slepian transform of channel vectors can be efficiently implemented by low-dimensional matrix-vector multiplications. To further reduce the computational complexity, we propose a joint Slepian transform (JST) based detector, where a fixed set of modulated Slepian sequences are designed. As a result, Slepian transforms of the observation vector can be efficiently implemented using low-dimensional fast Fourier transform (FFT). Simulation results demonstrate the advantages of high performance and low complexity of proposed detectors.

Two-Dimensional Transmission of Four-Dimensional LDPC-Coded Modulation with Slepian Sequences for DSP-Free 40 km Metro Network Applications.

The growing data demands are pushing researchers to pay more attention to spectrally efficient modulation formats. The four-dimensional (4D) signal constellation modulation format has been investigated for metro networks’ applications to achieve better power efficiency. To cope with such modulation formats, the requirement of better digital signal processing (DSP) is also increasing rapidly. More complicated DSPs bring us extra costs; thus, the DSP-free coherent receivers are also investigated because of the high-power consumption of conventional DSP-based receivers, but the transceivers upgrading also results in extra costs. In this invited paper we implement a 4-dimentional modulation format based on Slepian sequences. We applied LDPC coding and experimentally investigated the BER performance in a two-dimensional (2D) 40 km fiber link transmission and demonstrate that being error free is possible without employing the complicated DSP. We compared our proposed modulation scheme with regular 16QAM and found it outperforms 16QAM with DSP over back-to-back transmission by 3.8 dB improvement in OSNR when BER = 10−5, while over 40 km metro network communication link our proposed 4D modulation signals are still successfully transmitted, and the LDPC-coding still works properly with such a new transmission strategy. On the other hand, DSP-free transmission of LDPC-coded 16-QAM exhibits an early error floor phenomenon.

Prototype Filter for QAM-FBMC Systems Based on Discrete Prolate Spheroidal Sequences (DPSS)

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.

Multipath Suppression for Continuous Wave Radar via Slepian Sequences

In continuous wave (CW) radar systems, multiple signal copies impinge the receiver simultaneously. Often, undesired multipath and direct-path copies are many times stronger than potential targets. When applying matched filter signal processing techniques, the undesired signal components can mask weaker targets and decrease performance of post-processing techniques, such as target indication or estimation. In this manuscript, we propose a method of rejecting multipath-scattered returns over a continuous region in range and Doppler. We explore the computational cost of this method and additionally propose an approximate method of rejection which leverages the well-known discrete prolate spheroidal sequences (DPSS)–typically referred to as Slepian sequences–to gain a computational advantage. Results are shown to decrease the effective noise floor when applying matched filtering techniques as well as increase target signal-to-interference-plus-noise ratio (SINR) outside of an undesired multipath region. Comparisons are shown to traditional CW multipath removal in terms of rejection performance and run-time.

Proposal for Slepian-States-Based DV- and CV-QKD Schemes Suitable for Implementation in Integrated Photonics Platforms

Quantum key distribution (QKD) leverages underlying principles of quantum mechanics to realize distribution of keys with verifiable security. Despite appealing features of QKD, there are some fundamental and technical challenges that need to be solved prior to its widespread applications. First, QKD secret-key rate (SKR) is fundamentally limited by channel loss, as dictated by the rate-loss tradeoff. Quantum repeaters would be an ultimate solution to overcome this problem; however, they are well beyond the reach. The second challenge lies in the scalability and cost. Future's QKD systems must be suitable for mass production with low cost, reliable realignment-free operations, and small power consumption. To solve for these problems in a simultaneous manner, we propose to encode information in the orthogonal Slepian sequences’ bases. Such an approach is highly robust against turbulence effects in free-space optical links and dispersion effects/fiber nonlinearities in fiber-optics channels, thereby improving QKD distance. Moreover, exploiting multidimensional encoding space enables high spectral efficiency QKD so that the SKR can be significantly improved. Critically, generation, processing, and detection of Slepian states can be reliably implemented in an integrated quantum photonics platform, based on both reflective and transmissive waveguide Bragg gratings (WBGs). Proposed reflective/transmissive WBG-based Slepian states are applicable to both discrete variable and continuous variable QKD systems.

A multitaper spectral estimator for time-series with missing data

SUMMARY A multitaper estimator is proposed that accommodates time-series containing gaps without using any form of interpolation. In contrast with prior missing-data multitaper estimators that force standard Slepian sequences to be zero at gaps, the proposed missing-data Slepian sequences are defined only where data are present. The missing-data Slepian sequences are frequency independent, as are the eigenvalues that define the energy concentration within the resolution bandwidth, when the process bandwidth is $[ { - 1/2,\,\,\,1/2} )$ for unit sampling and the sampling scheme comprises integer multiples of unity. As a consequence, one need only compute the ensuing missing-data Slepian sequences for a given sampling scheme once, and then the spectrum at an arbitrary set of frequencies can be computed using them. It is also shown that the resulting missing-data multitaper estimator can incorporate all of the optimality features (i.e. adaptive-weighting, F-test and reshaping) of the standard multitaper estimator, and can be applied to bivariate or multivariate situations in similar ways. Performance of the missing-data multitaper estimator is illustrated using length of day, seafloor pressure and Nile River low stand time-series.

FBG-Based Weak Coherent State and Entanglement-Assisted Multidimensional QKD

In this paper, we propose several fiber Bragg gratings (FBGs) based on both weak coherent state (WCS) and entanglement-assisted (EA) multidimensional QKD (MQKD) protocols with mutually unbiased bases (MUBs) derived from the complex Hadamard matrices. We also propose how to implement them by using FBGs with orthogonal impulse responses, derived from Slepian sequences. The proposed FBG-based protocols are capable of simultaneously achieving unprecedented spectral efficiency and secret-key rates in long-distance quantum communication. Regarding the WCS protocols, random base selection and FGB-MUB-based protocols are introduced. Regarding EA protocols, we introduce a generic class of FBG-based N-dimensional protocols and describe the corresponding FBG-based encoders and decoders. The security analysis of the proposed MQKD protocols is further performed by taking into account FBG fabrication imperfections, modeled by the L-neighbor model. It has been shown that the proposed MQKD protocols significantly outperform the conventional two-dimensional QKD protocols, and impose less stringent bandwidth requirements compared to multidimensional time-frequency and time-bin protocols.

Quantum optimal control via gradient ascent in function space and the time-bandwidth quantum speed limit

A gradient ascent method for optimal quantum control synthesis is presented that employs a gradient derived with respect to the coefficients of a functional basis expansion of the control. Restricting the space of allowable controls to weighted sums of the Slepian sequences efficiently parameterizes the control in terms of bandwidth, resolution and pulse duration. A bound showing minimum time evolutions scaling with the inverse of the control bandwidth [S. Lloyd and S. Montangero, PRL, 113, 010502, (2014)] is recovered and the method is shown numerically to achieve the bound on entangling two-qubit quantum gates.

An evolutionary spectral representation for blind separation of biosignals

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.

An enhanced channel estimation technique for CDD OFDM system over time varying channels

Cyclic delay diversity (CDD), employing multiple antennas provides increased frequency-selectivity and thereby achieves frequency diversity using forward error correction in orthogonal frequency division multiplexing (OFDM). However, channel estimation of CDD OFDM becomes crucial over frequency-selective fading channels. In this paper, we propose a low-complexity channel estimation technique for CDD OFDM systems over time-varying channels using basis expansion model. We first analyze the frequency correlation of CDD OFDM channels and then design scattered pilot structure for estimating the channel coefficients. In the proposed channel estimator, Slepian sequences are employed estimate the channel impulse response by exploiting the frequency correlation of CDD OFDM over time-varying channel. Simulation results demonstrated that the proposed channel estimator achieves better mean square error and bit-error-rate performance compared with that of existing methods over time-varying channels.

Application of optimal band-limited control protocols to quantum noise sensing

Essential to the functionality of qubit-based sensors are control protocols, which shape their response in frequency space. However, in common control routines out-of-band spectral leakage complicates interpretation of the sensor’s signal. In this work, we leverage discrete prolate spheroidal sequences (a.k.a. Slepian sequences) to synthesize provably optimal narrowband controls ideally suited to spectral estimation of a qubit’s noisy environment. Experiments with trapped ions demonstrate how spectral leakage may be reduced by orders of magnitude over conventional controls when a near resonant driving field is modulated by Slepians, and how the desired narrowband sensitivity may be tuned using concepts from RF engineering. We demonstrate that classical multitaper techniques for spectral analysis can be ported to the quantum domain and combined with Bayesian estimation tools to experimentally reconstruct complex noise spectra. We then deploy these techniques to identify previously immeasurable frequency-resolved amplitude noise in our qubit’s microwave synthesis chain.

Multidimensional OAM-Based Secure High-Speed Wireless Communications

To address key challenges for beyond 5G wireless technologies in a simultaneous manner, we propose an orbital angular momentum (OAM)-based, secure, energy-efficient multidimensional coded modulation. The key idea is to employ all available degrees of freedom (DOFs) to convey the information over the wireless links, including amplitude, phase, polarization state, and spatial-domain DOFs. In particular, the OAM is associated with the azimuthal phase dependence of the wavefront, and represents an underutilized DOF. Given that OAM eigenstates are orthogonal, an arbitrary number of bits per symbol can be transmitted. Here, we propose utilizing OAM DOF not only to improve spectral and energy efficiencies, but also to significantly improve the physical-layer security of future wireless networks. To implement the OAM multiplexer and demultiplexer in the RF domain, we propose using properly designed antenna arrays. We also propose employing the Slepian sequences as either basis functions in baseband or impulse responses of antenna arrays in passband to further increase the dimensionality of the wireless system and enable beyond 1-Tb/s wireless transmission. Monte Carlo simulations demonstrate high tolerance to fading effects of LDPC-coded multidimensional signaling schemes compared with the conventional LDPC-coded QAM.

Design and performances evaluation of new Costas‐based radar waveforms with pulse coding diversity

Costas codes are a variant of pulse compression waveforms, largely studied for their attractive time-frequency properties. Their 'thumbtack-like' ambiguity function (AF) makes them highly suitable for delay and Doppler estimation, in radar and sonar applications. However, this behaviour depends heavily on the length of the code: the improvement in delay-Doppler resolutions and AF sidelobes level needs an increase in the size of the code. In this study, designs that allow good performance without increasing the size of the code are proposed. They are based on a modification of Costas codes by widening frequency separation between hops and replacing rectangular pulses by other waveforms. This will lead to a removal of autocorrelation function grating lobes that normally appear when frequency separation is increased. The originality of the work lies in the proposal of diversified pulse waveforms, such as phase codes, Slepian sequences, and other Costas codes, to encode main Costas pulses. A performance comparison of the proposed approaches is supplied. Such waveforms could also be of interest for applications where waveform diversity is desired.

Multidimensional Optical Transport Based on Optimized Vector-Quantization-Inspired Signal Constellation Design

An optimized vector-quantization-inspired signal constellation design (OVQ-SCD) suitable for multidimensional optical transport is proposed, in which signal constellation radii transformation function is optimized and near-uniform distribution of points is achieved. The proposed OVQ-SCD is used in a tandem with a hybrid multidimensional coded-modulation scheme employing Slepian sequences as electrical discrete-time basis functions, orthogonal prolate spheroidal wave functions as impulse responses of optical filters in orthogonal-division multiplexing, and spatial modes as optical continuous-time basis functions. It has been shown that the proposed multidimensional coded-modulation schemes based on OVQ-SCDs outperform corresponding counterparts and can be used to enable beyond 10 Pb/s serial optical transport over spatial division multiplexing (SDM) fibers as well as beyond 1 Pb/s transport over SMFs. The proposed OVQ-SCD-based hybrid multidimensional coded modulation scheme can simultaneously solve the problems related to the limited bandwidth of information-infrastructure, high energy consumption, and heterogeneity of network segments; while enabling elastic and dynamic bandwidth allocation.

Channel interpolation for LTE uplink systems with high mobility using Slepian sequences

Single Carrier Frequency Division Multiple Access (SC-FDMA) has been adopted for 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) uplink communication since it combines the low peak to average power ratio feature of single-carrier transmission systems with the multipath resistance and flexible frequency allocation of orthogonal frequency-division multiple access (OFDMA). Unfortunately, SC-FDMA transmission over time-varying multipath fading channels for very high speed environments the subcarrier orthogonality is destroyed giving rise to intercarrier interference (ICI) due to channel variation within an SC-FDMA symbol. This paper is, therefore, focused on the use of time-domain channel interpolator for LTE uplink systems in large Doppler spread scenarios to improve channel tracking performance. The resulting interpolation algorithm is based on the discrete prolate spheroidal wave functions (DPSWF) which is particularly well suited to represent the rapidly time-varying fading channel due to optimum finite time and frequency support feature. The variations of the channel taps are tracked first by a Kalman filter in time domain during training symbols. Then, a DPSWF based channel interpolator is applied to recover the time variation of channel taps between training symbols within two consecutive slots in a single subframe. The performance of the proposed interpolator is compared with a polynomial interpolator whose order is adjusted adaptively to achieve the best fit under particular user mobilities. Moreover, the analytical mean square error (MSE) performance of DPSWF is derived and compared with the simulations. The results show that the DPSWF represents the time-varying fading channel very effectively and the proposed algorithm has excellent channel estimation performance in high SNR even with a very small number of channel interpolation parameters employed in the algorithm.

Coherence of Ultrasound Radiofrequency Echoes from the Liver Estimated Using Multi-Taper Calculation.

Acoustic scattering from the microvasculature in the liver is frequently modeled by a quasi-periodic distribution of point scatterers. The received echoes from such an arrangement of scatterers has random but correlated phase at frequency components corresponding to the average spacing of the quasi-periodic scatterers. The extent of the correlation between different frequency components is measured by calculating signal coherence. To date in ultrasound, estimators for coherence have used windowed short-time FFTs of radiofrequency data where the windows selected have been chosen ad hoc. Using receiver operating characteristic analysis (ROC), we show that Slepian sequences provide an estimator which more reliably distinguishes between liver and ablated tissue in an ex vivo setting than estimators using the Hann, Hamming, or Blackman-Harris window. For a gate length of 7 mm or 21 wavelengths we achieve an area under the ROC curve of 0.93 with Slepian sequences for coherence calculations.

Orthogonal signals with jointly balanced spectra: Application to cdma transmissions

This paper presents a technique for generating orthogonal bases of signals with jointly optimized spectra, in the sense that they are made as close as possible. To this end, we propose a new criterion, the minimization of which leads to signals with close energy inside a set of prescribed subbands. Starting with the case of a single subband, we illustrate it by building orthogonal signals with maximum energy concentration in time and in frequency, with the same energy rate outside a fixed frequency interval or a fixed time interval, by resorting to Slepian sequences or Slepian functions, respectively. Then, we present spectrum balancing in a set of frequency intervals. We apply this method to Slepian sequences and Slepian functions, as well as to Walsh-Hadamard codes. On these examples, we point out a number of nice properties of the so-built orthogonal families that are of interest for signaling applications. PACS: signal processing techniques and tools; modulation techniques

Truncation-Error Reduction in Spherical Near-Field Scanning Using Slepian Sequences: Formulation for Scalar Waves

We discuss the error that results when the far field is reconstructed from spatially truncated near-field samples and present an effective mitigation technique based on the Slepian sequence for acoustic spherical near-field scanning. We show that the truncation error is inevitable whenever the far field is reconstructed using the classical near-field-to-far-field transformation. After discussing the Slepian sequence for a truncated spherical surface and its analytic and numerical properties, we apply it to expand truncated NF samples and derive the near-field-to-far-field transformation of the resulting expansion coefficients, from which the far field can be computed. We demonstrate the efficacy of this transformation by applying it to near-field scanning for bistatic scattering from a sphere and radiation from a current distribution.

Multiuser/MIMO Doubly Selective Fading Channel Estimation Using Superimposed Training and Slepian Sequences

We consider doubly selective multiuser/multiple-input-multiple-output (MIMO) channel estimation and data detection using superimposed training. The time- and frequency-selective fading channel is assumed to be well described by a discrete prolate spheroidal basis expansion model (DPS-BEM) using Slepian sequences as basis functions. A user-specific periodic (nonrandom) training sequence is arithmetically added (superimposed) at low power to each user's information sequence at the transmitter before modulation and transmission. A two-step approach is adopted, where, in the first step, we estimate the channel using only the first-order statistics of the observations. In this step, however, the unknown information sequence acts as interference, resulting in a poor signal-to-noise ratio (SNR). We then iteratively reduce the interference in the second step by employing an iterative channel-estimation and data-detection approach, where, by utilizing the detected symbols from the previous iteration, we sequentially improve the multiuser/MIMO channel estimation and symbol detection. Simulation examples demonstrate that, without incurring any transmission data rate loss, the proposed approach is superior to the conventional time-multiplexed (TM) training for uncoordinated users, where the multiuser interference in channel estimation cannot be eliminated and is competitive with the TM training for coordinated users, where the TM training design allows for multiuser-interference-free channel estimation.

Communicating over nonstationary nonflat wireless channels

We develop the concept of joint time-frequency estimation of wireless channels. The motivation is to optimize channel usage by increasing the signal-to-noise ratio (SNR) after demodulation while keeping training overhead at a moderate level. This issue is important for single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems but particularly so for the latter. Linear operators offer a general mathematical framework for symbol modulation in channels that vary both temporally and spectrally within the duration and bandwidth of one symbol. In particular, we present a channel model that assumes first-order temporal and spectral fluctuations within one symbol or symbol block. Discrete prolate spheroidal sequences (Slepian sequences) are used as pulse-shaping functions. The channel operator in the Slepian basis is almost tridiagonal, and the simple intersymbol interference pattern can be exploited for efficient and fast decoding using Viterbi's algorithm. To prove the concept, we use the acoustic channel as a meaningful physical analogy to the radio channel. In acoustic 2 /spl times/ 2 MIMO experiments, our method produced estimation results that are superior to first-order time-only, frequency-only, and zeroth-order models by 7.0, 9.4, and 11.6 db. In computer simulations of cellular wireless channels with realistic temporal and spectral fluctuations, time-frequency estimation gains us 12 to 18 db over constant-only estimation in terms of received SNR when signal-to-receiver-noise is 10 to 20 db. The bit error rate (BER) decreases by a factor of two for a binary constellation.