Convolution-Based Trigonometric Interpolation of Band-Limited Signals

This paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band-limited signal from a finite sequence of samples. The polynomial delivers accurate approximations in the range covered by the sequence, except for a short frame close to the range limits. Besides, its accuracy increases exponentially with the frame width. The method is based on using a band-limited window in order to reduce the truncation error of a convolution series. It is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete Fourier transform (DFT). Specifically, two basic procedures are presented, one based on the fast Fourier transform (FFT), and another based on a recursive update algorithm for the short-time FFT. The paper contains three applications. The first is a variable fractional delay (VFD) filter, which consists of a short-time FFT combined with the evaluation of a trigonometric polynomial. This filter has low complexity and can be implemented using CORDIC rotations. The second is the interpolation of nonuniform Fourier summations, where the proposed method eliminates the need to interpolate any kernel sample. Finally, the third can be viewed as a generalization of the FFT convolution algorithm and makes it possible to interpolate the output of an finite-impulse-response (FIR) filter efficiently.      

High-Rate Interpolation of Random Signals From Nonideal Samples

We address the problem of reconstructing a random signal from samples of its filtered version using a given interpolation kernel. In order to reduce the mean squared error (MSE) when using a nonoptimal kernel, we propose a high rate interpolation scheme in which the interpolation grid is finer than the sampling grid. A digital correction system that processes the samples prior to their multiplication with the shifts of the interpolation kernel is developed. This system is constructed such that the reconstructed signal is the linear minimum MSE (LMMSE) estimate of the original signal given its samples. An analytic expression for the MSE as a function of the interpolation rate is provided, which leads to an explicit condition such that the optimal MSE is achieved with the given nonoptimal kernel. Simulations confirm the reduction in MSE with respect to a system with equal sampling and reconstruction rates.      

Block-Based Methods for the Reconstruction of Finite-Length Signals From Nonuniform Samples

Two novel block-based algorithms are presented for the reconstruction of uniform samples given the nonuniform samples. The first algorithm uses a sinc interpolator whereas the second one uses a DFT-based interpolator. It is shown that the proposed algorithms are stable and the error due to noise and sampling jitter is bounded by the corresponding error norms of noise and jitter, respectively. We show that both of the block-based algorithms provide nearly perfect reconstruction for a class of practically time and bandlimited signals. Boundary effects are considered and single and multiblock processing is discussed. A modified block-based algorithm is developed by using the windowing technique in order to improve the mean-squared error (MSE) performance for nonbandlimited signals. It is shown that this algorithm performs better than a group of alternative algorithms, including Yen's third algorithm, for a variety of signal, noise, and sampling grids     

Nonuniform Interpolation of Noisy Signals Using Support Vector Machines

The problem of signal interpolation has been intensively studied in the information theory literature, in conditions such as unlimited band, nonuniform sampling, and presence of noise. During the last decade, support vector machines (SVM) have been widely used for approximation problems, including function and signal interpolation. However, the signal structure has not always been taken into account in SVM interpolation. We propose the statement of two novel SVM algorithms for signal interpolation, specifically, the primal and the dual signal model based algorithms. Shift-invariant Mercer's kernels are used as building blocks, according to the requirement of bandlimited signal. The sine kernel, which has received little attention in the SVM literature, is used for bandlimited reconstruction. Well-known properties of general SVM algorithms (sparseness of the solution, robustness, and regularization) are explored with simulation examples, yielding improved results with respect to standard algorithms, and revealing good characteristics in nonuniform interpolation of noisy signals.     

周期非均匀采样信号的重建中的优化子频带划分

对于周期非均匀采样,由于每个均匀采样流的采样率通常都是小于Nyquist率的,因此,采样信号频谱中会发生频率混叠。这表明在采样信号的重建频带内将会有多于1的谱分量。因为不是所有的与重建频带相交的谱分量都会覆盖整个重建频带,所以有必要将重建频带分成若干个子频带进行分析,构造内插函数,这样有利于减小重建所必需的最小采样率。本文提出一种优化的子频带划分方法,通过该方法在重建频带上定义子频带,能在保证重建所需的采样率最低的情况下使子频带的个数最少,这对于简化内插滤波器的结构有重要的意义。     

Reconstruction of Multidimensional Signals From Irregular Noisy Samples

We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the MarČenko–Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.      

Resampling of Band-Limited Gaussian Random Signals With Flat Power Spectrum, Available Through 1-Bit Quantized Samples

A theoretical analysis for the evaluation of the probability of error occurring in resampling a noise-like band-limited Gaussian signal with flat power spectrum available through its digitized samples, is presented. The analysis assumes the use of an ideal sine-based interpolation algorithm for the digitized signal reconstruction, which is proved to be optimum for the considered class of signals and quantization functions. The particular case of lowest order, i.e., 1 bit, quantization function, is fully treated in analytical terms and a theoretical prediction for the error probability is derived. Validation of the presented analysis is made through a comparison with numerical simulations.     

A Versatile Iterative Framework for the Reconstruction of Bandlimited Signals from Their Nonuniform Samples

In this paper, we study a versatile iterative framework for the reconstruction of uniform samples from nonuniform samples of bandlimited signals. Assuming the input signal is slightly oversampled, we first show that its uniform and nonuniform samples in the frequency band of interest can be expressed as a system of linear equations using fractional delay digital filters. Then we develop an iterative framework, which enables the development and convergence analysis of efficient iterative reconstruction algorithms. In particular, we study the Richardson iteration in detail to illustrate how the reconstruction problem can be solved iteratively, and show that the iterative method can be efficiently implemented using Farrow-based variable digital filters with few general-purpose multipliers. Under the proposed framework, we also present a completed and systematic convergence analysis to determine the convergence conditions. Simulation results show that the iterative method converges more rapidly and closer to the true solution (i.e. the uniform samples) than conventional iterative methods using truncation of sinc series.     

Energy Optimization of Band-Limited Nyquist Signals in the Time Domain

This paper presents a theoretical consideration of the optimal design of band-limited Nyquist-type signal shapes for data transmission, which maximizes its energy in a given time interval and which generates no intersymbol interference at the periodic sampling instants. A method based on a completely analytical approach is given for design of such signals. The optimal signal is a solution of an inhomogeneous linear integral equation of Fredholm type. A technique for solving this equation is given. The computation is straightforward and involves the determination of eigenvalues and eigenfunctions of a positive definite and symmetric kernel in terms of prolate spheroidal wave functions. The constraint for intersymbol interference is shown to be easily included into the problem. Finally, some numerical examples are given and the performance of the optimal signal shapes is compared to that resulting from the use of the "raised-cosine" type of signals. It is also concluded that especially for small values of rolloff factor, the optimal signals, thus obtained, are almost maximally immune to small timing offsets at the sampling instants.     

On Optimal Detection of Band-Limited PAM Signals with Excess Bandwidth

PAM data transmission receivers accomplishing maximum likelihood sequence detection (MLSD) usually require a matched filter prefilter, a sampler at the symbol rate, and a Viterbi algorithm detector. When the channel is unknown or slowly changing, one must use an adaptive matched filter prefilter. We examine an alternative optimum receiver whose optimality is independent of the matched filter prefilter and which is applicable when the channel is effectively band-limited. The sampler in the proposed receiver operates at a rate faster than the data symbol rate, enabling one to replace the matched filter by a fixed low-pass filter and still ensure that the maximum likelihood detector is supplied with a set of sufficient statistics. It is shown that the matched filter is incorporated within a modified Viterbi detector without increasing the number of states in the algorithm, although the Viterbi detector must perform computations at approximately twice the usual rate. Simulations support the optimality of the new receiver and quantitatively indicate the degradation in performance experienced by some adaptive receivers previously proposed.     

On Sampling of Band-Limited Signals Associated With the Linear Canonical Transform

Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.      

Multichannel Sampling of Signals Band-Limited in Offset Linear Canonical Transform Domains

The aim of the multichannel sampling is the reconstruction of a band-limited signal f(t), from the samples of the responses of M linear time invariant systems, each sampled by the 1/Mth Nyquist rate. As the offset linear canonical transform (OLCT) has been found wide applications in signal processing and optics fields, it is necessary to consider the multichannel sampling based on offset linear canonical transform. In this paper, we develop a multichannel sampling theorem for signals band-limited in offset linear canonical transform domains. Moreover, by designing different OLCT filters, reconstruction formulas for uniform sampling from the signal, from the signal and its first derivative or its generalized Hilbert transform are obtained based on the derived multichannel sampling theorem. Since recurrent nonuniform sampling for the signal has valuable applications, reconstruction expression for recurrent nonuniform samples of the signal band-limited in the offset linear canonical transform domain is also obtained by using the derived multichannel sampling theorem and the properties of the offset linear canonical transform.     

一维带限信号正则外推的快速算法

本文给出用离散正则化方法进行一维带限信号外推的快速算法,其基本思想是将正则化方法与离散Fourier变换(DFT)结合起来,而正则参数的选取则基于偏差原理和作者提出的三阶收敛算法来实现.这样,可将计算量由原米的O(n3/3)量级减少到O(12n2)量级(当采样点n为偶数时)乃至O(12/2nlog2n/2)量级(当n=2*,p为正整数时),分析和数值试验表明,新算法具有快速、稳定和抑制高频噪音干扰等优点.     

Generic Weighted Filtering of Stochastic Signals

In this paper, a new theory of optimal weighted nonlinear filtering is presented. Two filter models are considered. The first model is based on a representation of the filter in the polynomial-like form with $q$ terms where each term consists of weighted matrices and the matrix determined from the error minimization problem. The second model extends the first one to the case of the filter concatenation. The filter models are given in terms of pseudo-inverse matrices, i.e., the requirement of invertibility for covariance matrices is omitted. Thus, our filters always exist. We develop methods which allow us to exploit advantages associated with the proposed nonlinear filter models. The methods consist of the orthogonalization procedure and the reduction of the original problem to $q$ individual minimization problems for smaller matrices. This leads to a considerable reduction in the required computational work. The error associated with the first filter model decreases when the number $q$ of terms of filter increases. Its compression ratio can be adjusted by varying a particular value of ranks in each of its $q$ terms. This means that the proposed filer structure provides the two degrees of freedom. The second filter model provides another degree of freedom, a number $k$ of filters in the concatenation. Variations of the degrees of freedom improve the performance of the proposed filters. In particular, the error associated with the filter concatenation decreases as the filter number $k$ in the concatenation increases.      

Interpolation of Bounded Bandlimited Signals and Applications

A method to interpolate a bounded bandlimited signal from its own samples with minimum complexity is presented that guarantees an a priori specified accuracy. The method is the result of combining a fast-converging sampling expansion with the Farrow interpolator technique. It provides important complexity reductions in various signal processing applications, from which three cases are studied. For the Farrow interpolator, it assures that the interpolation error will be smaller than a known bound, both in the time and frequency domains. For delay estimation, the method allows one to decouple a given estimation/detection algorithm in two steps. In the first one, a few finite convolutions are carried out in order to compute a set of interpolation coefficients. In the second, the algorithm is executed but with a complexity independent of the length of the temporal observation interval. Besides, it is shown how to introduce this decoupling in the matched filter, MUSIC, and conditional maximum-likelihood delay estimators. Finally, the method is employed to give a simple solution to an important efficiency problem in the simulation of communications systems: the generation of Rayleigh processes.     

二维带限信号正则重构和外推的快速算法

本文给出用离散正则化方法进行二维带限信号重构和外推的快速算法.其基本思想是将正则化方法与快速Fourier变换(FFT)结合起来,而正则参数的选取则基于偏差原理和作者提出的三阶收敛算法来实现,并进行数值了模拟.计算结果表明本文给出的新算法具有快速、稳定和抑制高频噪音干扰等优点.     

用FrFT插值实现LFM信号的参数估计

分数阶傅立叶域的分辨率由信号时宽带宽积决定,有限的分辨率影响了利用FrFT(分数阶傅立叶变换)估计LFM(线性调频)信号参数的精度.针对此问题,提出了一种利用FrFT插值的LFM信号参数估计方法.在分析引起峰值点偏差的因素后,得到FrFT在峰值点的分数阶傅立叶域的函数表达式,基于此,在分数阶傅立叶域进行插值计算,突破了原有分辨率的限制,提高了参数估计的精度.仿真实验结果验证了算法的有效性.     

基于小波变换的非均匀采样信号频谱的研究

该文提出基于小波变换的非均匀采样信号频谱的检测方法,给出变换函数关系使得非均匀采样信号满足小波变换的两个基本条件。文中说明了小波的非均匀化过程,从均匀小波得到非均匀小波,以非均匀小波分析非均匀采样信号,得到非均匀采样信号的频谱。文中还说明了非均匀小波变换的抗混叠的原理以及对信号频谱的检测方法,最后给出实验结果。理论和实验表明,非均匀采样信号的小波变换方法是一种行之有效的非均匀采样信号的频率检测方法,使用该方法处理信号可以得到准确的频率估计效果。     

Reconstruction of Two-Periodic Nonuniformly Sampled Band-Limited Signals Using a Discrete-Time Differentiator and a Time-Varying Multiplier

This brief considers the problem of reconstructing a band-limited signal from its two-periodic nonuniformly spaced samples. We propose a novel reconstruction system where a finite-impulse response filter designed as differentiator followed by a time-varying multiplier recovers the uniformly spaced from the nonuniformly spaced samples. The system roughly doubles the signal-to-noise ratio with relatively few filter coefficients. The main advantage is that once the differentiator has been designed, it can be implemented with fixed multipliers, and only the coefficients of the time-varying multiplier have to be adapted when the sampling pattern changes; this reduces implementation costs substantially. In addition, the system allows an efficient polyphase implementation.