Band-limited Signals Research Articles

Combined Power Ratio Calculation, Hadamard Transform and Least Mean Squares Algorithm for Channel Mismatch Calibration in Time-Interleaved ADCs

This paper presents a method for all-digital background calibration of multiple channel mismatches including offset, gain and timing mismatches in time-interleaved analog-to-digital converters (TIADCs). The average technique is used to remove offset mismatch at each channel. The gain mismatch is calibrated by calculating the power ratio of the sub-ADC over the reference ADC. The timing skew mismatch is calibrated by using Hadamard transform for error correction and LMS for timing mismatch estimation. The performance improvement of TIADCs employing these techniques is demonstrated through numerical simulations. Besides, achievement results on the field-programmable gate array (FPGA) hardware have demonstrated the effectiveness of the proposed techniques.
 KeywordsTime-interleaved analog-to-digital converter (TIADC), channel mismatches, all-digital background calibration
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Observing and tracking bandlimited graph processes from sampled measurements

A critical challenge in graph signal processing is the sampling of bandlimited graph signals; signals that are sparse in a well-defined graph Fourier domain. Current works focused on sampling time-invariant graph signals and ignored their temporal evolution. However, time can bring new insights on sampling since sensor, biological, and financial network signals are correlated in both domains. Hence, in this work, we develop a sampling theory for time varying graph signals, named graph processes, to observe and track a process described by a linear state-space model. We provide a mathematical analysis to highlight the role of the graph, process bandwidth, and sample locations. We also propose sampling strategies that exploit the coupling between the topology and the corresponding process. Numerical experiments corroborate our theory and show the proposed methods trade well the number of samples with accuracy.

The reweighted Landweber scheme for the extrapolation of band-limited signals-II

The extrapolation of a band-limited signal is in terms of its finite known segment in the time interval [−T,T] for T > 0 to determine the unknown part of the signal. Some of the singular values of the operator of the reconstruction equation are close to 1 while the rest are close to zero, and if T is small, more singular values are close to zero. In this paper, we theoretically improve the posedness of the extrapolation of a band-limited signal by using a reweighting method to improve the distribution properties of all eigenvalues in infinite dimensions and accordingly propose a reweighted Landweber scheme. In the implementation, a high accuracy numerical method of the operator of the equation for reconstruction is proposed to reduce the truncated errors of the reconstruction equation and thus the accumulative errors of the reweighting method. Numerical experiments validate the proposed method. Compared with our previous work and existing works, effective reconstructions are obtained with a much smaller T.

Compact CRB for delay, Doppler, and phase estimation – application to GNSS SPP and RTK performance characterisation

The derivation of tight estimation lower bounds is a key tool to design and assess the performance of new estimators. In this contribution, first, the authors derive a new compact Cramér–Rao bound (CRB) for the conditional signal model, where the deterministic parameter's vector includes a real positive amplitude and the signal phase. Then, the resulting CRB is particularised to the delay, Doppler, phase, and amplitude estimation for band-limited narrowband signals, which are found in a plethora of applications, making such CRB a key tool of broad interest. This new CRB expression is particularly easy to evaluate because it only depends on the signal samples, then being straightforward to evaluate independently of the particular baseband signal considered. They exploit this CRB to properly characterise the achievable performance of satellite-based navigation systems and the so-called real-time kinematics (RTK) solution. To the best of the authors’ knowledge, this is the first time these techniques are theoretically characterised from the baseband delay/phase estimation processing to position computation, in terms of the CRB and maximum-likelihood estimation.

A new class of superoscillatory functions based on a generalized polar coordinate system

Is it possible for a band-limited signal to possess oscillation that is arbitrarily higher than its highest Fourier component? Common knowledge assumed that the answer is ‘No.’ Counterintuitively, it turns out that there are band-limited functions that are able to oscillate arbitrarily faster than their fastest Fourier components. These are the superoscillatory functions. Since their discovery, superoscillations have been intriguing in the world of Fourier analysis, with a vast number of applications in quantum mechanics, optics, and radar theory, among other areas. A basic aim in the literature of superoscillations is to find new types of superoscillations that will be used for such technologies. In this paper, we introduce a geometrical-based method to construct a rich class of superoscillations using the concept of directional polar coordinates, developed in this research. We investigate their basic features and show how the proposed method allows generating superoscillations with an arbitrary number of superoscillatory regions, and with an arbitrary number of variables.

Time-Domain vs. Frequency-Domain Equalization for FTN Signaling

Faster-than-Nyquist (FTN) signaling has been recognized as a promising technique for next-generation high data rate communications. By intentionally reducing the symbol interval, FTN signaling is capable of transmitting more symbols than classic Nyquist signaling within the same time period and bandwidth. However, the intentional non-orthogonality of the bandlimited signaling pulses imposes severe inter-symbol interference (ISI), which requires powerful equalization at the receiver. Hence, we embark on the comparison of time- and frequency-domain equalization for FTN signaling both by theoretical analysis and numerical simulations. It is shown that frequency-domain equalization fails to reliably detect the FTN signal with a low FTN packing factor, while the time-domain equalization still performs well.

A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals

A modified modulated wideband converter (MWC)-based compressed sampling receiver and a parameter estimation algorithm in discrete time fractional Fourier domain (DTFrFD) to intercept and estimate the fractional bandlimited LFM signals are suggested in this work. Our proposed compressed sampling receiver has some advantages compared to the original MWC-based compressed sampling receiver. Firstly, the proposed receiver works in DTFrFD which is more effective to intercept the fractional bandlimited signals, while the original MWC-based digital receiver works in discrete time Fourier domain (DTFD). Secondly, the cross-channel signal can get better separation by the proposed digital receiver than by original MWC-based receiver because that the nonzero part of the signal’s available spectrum is narrower in DTFrFD than in DTFD. Then, with the data acquired from the digital receiver, we propose a novel parameter estimation algorithm for the intercepted fractional bandlimited LFM signal based on OMP and the differentiation spectrum in DTFrFD. The initial frequency and final frequency of the intercepted fractional bandlimited LFM signal can be extracted successfully without two-dimensional search and reconstruction, and only two iterations are needed according the algorithm. Both theoretical analysis and simulation results demonstrate that the proposed algorithm bears a relatively low complexity and its estimation precision is not only higher than both DFrFT-based search algorithm and DFrFT-based reconstruction algorithm but also close to Cramer–Rao lower bound.

Minimum Phase Conditions in Kramers-Kronig Optical Receivers

Theoretically, for continuous signals, Kramers-Kronig (KK) receivers should be able to precisely reconstruct a band-limited signal's phase from its envelope if the minimum phase (MP) condition is satisfied. However, simulations have shown that even if the MP condition is satisfied, for sampled signals the reconstructed signal can contain large errors. In this article, we explore the effectiveness of DSP-based KK receivers close to the MP condition using single-tone and multi-tones. For a single-tone signal, we use a precise mathematical analysis to demonstrate large distortions in the reconstructed signal even when the MP condition is clearly satisfied and illustrate this on the complex-plane. We then study multi-tone signals spectrally, and show that there are strong intermodulation distortion tones when MP is satisfied that either directly fall in-band, or could be aliased back in-band during down-sampling. Satisfying MP does, however, eliminate signal-signal beat interference, as expected with KK processing. High up-sampling ratios reduce the spectral density of these tones, which explains the improved performance seen in experimental systems. We then explore the error-vector magnitude close to the boundary of MP for various carrier-to-signal power ratios and up-sampling ratios, to illustrate that there is a soft transition around the MP condition. Overall, this work shows that the MP condition is not sufficient to guarantee error-free performance in noiseless conditions.

The ill-posedness of derivative interpolation and regularized derivative interpolation for band-limited functions by sampling

In this paper, the ill-posedness of derivative interpolation is discussed, and a regularized derivative interpolation for band-limited signals is presented. The ill-posedness is analyzed by the Shannon sampling theorem. The convergence of the regularized derivative interpolation is studied by the combination of a regularized Fourier transform and the Shannon sampling theorem. The error estimation is given, and high-order derivatives are also considered. The algorithm of the regularized derivative interpolation is compared with derivative interpolation using some other algorithms.

Jittered Sampling in Linear Canonical Domain

Linear canonical transform (LCT) based sampling has been attracted much attention in signal processing community. A variety of sampling theorems associated with the LCT are therefore derived. However, the random jittered sampling in linear canonical domain still remains unresolved. Starting from the Shannon's result for the LCT, a biased interpolated estimate for bandlimited signals in linear canonical domain from jittered samples is proposed. The inverse filter technique is then applied to correct the bias, giving rise to an unbiased estimator whose interpolation noise variance approaches zero as the jitter becomes less and less pronounced. To be specific, for bandlimited stationary stochastic signals in linear canonical domain the interpolation noise variance is increased, and this result is also verified through some examples and simulations.

Adaptive estimation and sparse sampling for graph signals in alpha-stable noise

In the graph signal processing literature, most methods were developed based on the assumption of Gaussian noise since it can lead to computationally efficient and mathematically tractable solutions. Unfortunately, the Gaussian distribution cannot capture the sharp spikes and tail heaviness of the signal noise in various natural phenomena. The α-stable distribution is a generalization of the Gaussian distribution and is a more appropriate model for such impulsive heavy-tailed noise. In this paper, we consider the problem of adaptive estimation and sparse sampling for signals defined over graphs in the presence of impulsive α-stable noise. To tackle the problems associated with α-stable noise, the graph signal estimation problem is formulated as a minimum dispersion (MD)-based optimization. A novel adaptive least mean pth power (LMP) algorithm is proposed for robust estimation of band-limited graph signals from partial observations in α-stable noise environments. The mean square performance of the proposed LMP algorithm is theoretically analyzed. To handle the case of band-limited graph signals with unknown and time-varying bandwidth and spectral contents, the sparse nature of the Graph Fourier transform of the signal is exploited to develop an adaptive graph sampling technique. Specifically, the proposed sparse sampling technique identifies the spectral signal support via sparse online estimation building on the iterative shrinkage-thresholding algorithm. Numerical simulation studies are presented to corroborate the performance advantages of the proposed adaptive estimation and sparse sampling algorithms.

An improved algorithm for timing error detection based on all-digital receiver

In view of the wide application of the Gardner algorithm in timing synchronization loop and its limitation of the band-limited signals, an enhanced Gardner timing error detection algorithm, which has a simple structure and can effectively reduce the deviation of the timing error and system self- noise, is proposed in this paper by comparison of the Gardner algorithm and its correction algorithm in both the pattern jitter and timing errors on the basis of the asynchronous sampling. The simulation results show that E-Gardner algorithm has a significant improvement in the performance of QPSK modulation signal’s clock capture and error detection in the case of a small roll-off factor under the Gaussian with Rayleigh fading channel.

Reconstruction of bandlimited graph signals from measurements

Graph signal reconstruction is a fundamental topic in graph signal processing. Recently, graph signal reconstruction in three typical sampling schemes has been studied in the literature. In this paper, we study the reconstruction of bandlimited graph signals in a general measurement scheme, which includes the three existing sampling schemes as its special cases. Theorems for perfect reconstruction are established, and reconstruction by random measurement matrices is studied. An iterative reconstruction algorithm based on convex projection is proposed, and numerical experiments are implemented to demonstrate the effectiveness of the algorithm.

Full-Spectrum Periodic Nonlinear Fourier Transform Optical Communication Through Solving the Riemann-Hilbert Problem

In this article, for the first time, a full-spectrum periodic nonlinear Fourier transform (NFT)-based communication system with the inverse transformation at the transmitter performed by using the solution of Riemann-Hilbert problem (RHP), is proposed and studied. The entire control over the nonlinear spectrum rendered by our technique, where we operate with two qualitatively different components of this spectrum represented, correspondingly, in terms of the main spectrum and the phases, allows us to design a time-domain signal tailored to the characteristics of the transmission channel. In the heart of our system is the RHP-based signal processing utilised to generate the time-domain signal from the modulated nonlinear spectrum. This type of NFT processing leads to a computational complexity that scales linearly with the number of time-domain samples, and we can process signal samples in parallel. In this article, we suggest the way of getting an exactly periodic signal through the correctly formulated RHP, and present evidence of the analogy between band-limited (in ordinary Fourier sense) signals and finite-band (in RHP sense) signals. Also, for the first time, we explain how to modulate the phases of individual periodic nonlinear modes. The performance of our transmission system is evaluated through numerical simulations in terms of bit error rate and Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -factor dependencies on the transmission distance and power, and the results demonstrate the good potential of the approach.

Low-Feedback Sampling Rate Digital Predistortion Using Deep Neural Network for Wideband Wireless Transmitters

In this paper, a low-feedback-sampling-rate digital predistortion (DPD) method is proposed for wideband wireless transmitters and radio-frequency power amplifiers (PAs). This DPD method inserts a non-ideal real band-pass filter into the feedback loop to limit the feedback bandwidth. Meanwhile, to recover the band-limited feedback signal, it introduces a signal-recovery module that is based on the deep neural network (DNN). A data-preprocessing technique is proposed to reduce the amount of input data for the DNN and thereby significantly reducing its structural complexity. Also, the use of DNN makes it feasible to implement off-line training. Both the simplicity of the proposed DNN and the availability of off-line training reduce the system complexity in DPD. Experimental validation was performed on a PA driven by wideband orthogonal-frequency-division-multiplexing (OFDM) signals. The results demonstrated the superiority of the proposed method in under-sampling applications. The proposed DPD method achieved the same linearization performance as the state-of-art DPD methods while requiring a sampling rate that was approximately 5% less. Meanwhile, the proposed DPD method has been validated to have stronger anti-noise and generalization capabilities.

Turing Computability of Fourier Transforms of Bandlimited and Discrete Signals

The Fourier transform is an important operation in signal processing. However, its exact computation on digital computers can be problematic. In this paper we consider the computability of the Fourier transform and the discrete-time Fourier transform (DTFT). We construct a computable bandlimited absolutely integrable signal that has a continuous Fourier transform, which is, however, not Turing computable. Further, we also construct a computable sequence such that the DTFT is not Turing computable. Turing computability models what is theoretically implementable on a digital computer. Hence, our result shows that the Fourier transform of certain signals cannot be computed on digital hardware of any kind, including CPUs, FPGAs, and DSPs. This also implies that there is no symmetry between the time and frequency domain with respect to computability. Therefore, numerical approaches which employ the frequency domain representation of a signal (like calculating the convolution by performing a multiplication in the frequency domain) can be problematic. Interestingly, an idealized analog machine can compute the Fourier transform. However, it is unclear whether and how this theoretical superiority of the analog machine can be translated into practice. Further, we show that it is not possible to find an algorithm that can always decide for a given signal whether the Fourier transform is computable or not.

A Novel Analytical OFDM Modulation Framework using Wavelet Transform with Window Function in the Hilbert space

Abstract Orthogonal frequency-division multiplexing (OFDM) technology has been applied in various wireless systems because OFDM approaches can transmit data with high capacity and high data rate. However, in the modern OFDM system, an efficient synchronization between transmitter and receiver is critical to avoid the inter symbol interference (ISI) for matching the huge demand of data in the Big Data era. The existing researches applied fast Fourier transform (FFT) for synchronization and symbol time recovery; however, these approaches resulted in a degraded performance in received signal quality, particularly in low-frequency carrier transmitting by losing the orthogonal characteristic of OFDM signal. To address this challenge, in this paper, we propose a novel analytical OFDM system that conforms to the IEEE 802.11 standard using the window function (rectangular waveform with roll-off characteristics) in the transmitter. The proposed OFDM framework eliminates the spectral sidelong of OFDM. Also, we design the timing recover system using Wavelet transform to extract the OFDM symbol frame signal in Hilbert space. The evaluation results demonstrate the bit error rate (BER) performance of the proposal in Additive white Gaussian noise (AWGN) for symbol data by using Wavelet transform to extract the band limited signal from the receiver side. In addition, we apply an IEEE 802.11 standard-conformed framing sequence and show that the BER performance is improved with the proposed OFDM transceiver synchronization model using Wavelet signal denoising in the same channel of AWGN.

Turing Meets Shannon: Computable Sampling Type Reconstruction With Error Control

The conversion of analog signals into digital signals and vice versa, performed by sampling and interpolation, respectively, is an essential operation in signal processing. When digital computers are used to compute the analog signals, it is important to effectively control the approximation error. In this paper we analyze the computability, i.e., the effective approximation of band limited signals in the Bernstein spaces B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> , 1 ≤ p <; ∞,andofthe corresponding discrete-time signals that are obtained by sampling. We show that for 1 <; p <; ∞, computability of the discrete-time signal implies computability of the continuous-time signal. For p = 1 this correspondence no longer holds. Further, we give a necessary and sufficient condition for computability and show that the Shannon sampling series provides a canonical approximation algorithm for p > 1. We discuss BIBO stable LTI systems and the time-domain concentration behavior of bandlimited signals as applications.

A New Class of Explicit Interpolatory Splines and Related Measurement Estimation

This paper proposes a new and rich class of cardinal splines called generalized MK-splines (GMK-splines), which manifest several beneficial properties (especially the interpolation property) and the ability to obtain a construction with a desired frequency response. The GMK-spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted B-splines of that degree in a unified framework, the GMK-spline bases form Riesz bases of their generating spaces, and the GMK-spline frequency responses meet Strang-Fix conditions. Furthermore, we provide search methods for minimal-support GMK-splines and local optimal GMK-spline frequency responses under different given passband frequencies. Several related estimates are theoretically assessed, including energy magnitude, maximum, and approximation error of GMK-spline interpolated signals. In signal processing applications, the GMK-splines behave as interpolating FIR filters with simple implementation, and some of them attain lower error than other traditional interpolation system responses, such as cubic cardinal B-spline and linear spline, when approximating to an ideal lowpass filter. Experimental results show that the GMK-spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction.

Intra-Pulse Modulation Recognition for Fractional Bandlimited Signals Based on a Modified MWC-Based Digital Receiver

In this paper, we present a modified modulated wideband converter (MWC)-based digital receiver for fractional bandlimited signals, and further propose an intra-pulse modulation recognition method in discrete time fractional Fourier domain (DTFrFD) for the intercepted signals. The proposed digital receiver can move the cross-channel signal to the baseband and since the nonzero part of its available spectrum is narrower in DTFrFD than in discrete time Fourier domain (DTFD), so a better separation can be achieved by the proposed digital receiver than by original MWC- based receiver. Then, with the data acquired from the digital receiver, we propose an intra-pulse modulation recognition method based on the optimal transformation order and the spectral kurtosis (SK) in DTFrFD for the six types of fractional bandlimited signals. In this algorithm, the optimal transformation order is tested to distinguish the encoded signals from the non-encoded signals, and then the SK is tested to determine the intra-pulse modulation type specially. The computational complexity of the proposed method is much lower than search-based methods. Meanwhile, since the SK of Gaussian signals is zero in DTFrFD, the proposed method shows better robustness than the original MWC discrete compressive sampling structure against SNR variation. Simulation results confirm the obtained analytical results.