Performance bounds for polynomial phase parameter estimation withnonuniform and random sampling schemes

Estimating the parameters of a cisoid with an unknown amplitude and polynomial phase using uniformly spaced samples can result in ambiguous estimates due to Nyquist sampling limitations. It has been shown previously that nonuniform sampling has the advantage of unambiguous estimates beyond the Nyquist frequency; however, the effect of sampling on the Cramer-Rao bounds is not well known. This paper first derives the maximum likelihood estimators and Cramer-Rao bounds for the parameters with known, arbitrary sampling times. It then outlines two methods for incorporating random sampling times into the lower variance bounds, describing one in detail. It is then shown that for a signal with additive white Gaussian noise the bounds for the estimation with nonuniform sampling tend toward those of uniform sampling. Thus, nonuniform sampling overcomes the ambiguity problems of uniform sampling without incurring the penalty of an increased variance in parameter estimation     

Sub‐Nyquist rate ADC sampling‐based compressive channel estimation

To realize high‐speed communication, broadband transmission has become an indispensable technique in the next‐generation wireless communication systems. Broadband channel is often characterized by the sparse multipath channel model, and significant taps are widely separated in time, and thereby, a large delay spread exists. Accurate channel state information is required for coherent detection. Traditionally, accurate channel estimation can be achieved by sampling the received signal with large delay spread by analog‐to‐digital converter (ADC) at Nyquist rate and then estimate all of channel taps. However, as the transmission bandwidth increases, the demands of the Nyquist sampling rate already exceed the capabilities of current ADC. In addition, the high‐speed ADC is very expensive for ordinary wireless communication. In this paper, we present a novel receiver, which utilizes a sub‐Nyquist ADC that samples at much lower rate than the Nyquist one. On the basis of the sampling scheme, we propose a compressive channel estimation method using Dantzig selector algorithm. By comparing with the traditional least square channel estimation, our proposed method not only achieves robust channel estimation but also reduces the cost because low‐speed ADC is much cheaper than high‐speed one. Computer simulations confirm the effectiveness of our proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

基于新限速抽样的超宽带通信系统信道时延估计

提出了利用新限速抽样率理论解决信道估计问题,建立了既可用于宽带也可用于窄带信道估计的框架,此框架的抽样率低于奈奎斯特速率。通过实例展示了利用奈奎斯特速率抽样而工作在低于奈奎斯特抽样率的算法性能,减少了功耗,也降低了算法的复杂性。     

Sampling, data transmission, and the Nyquist rate

The sampling theorem for bandlimited signals of finite energy can be interpreted in two ways, associated with the names of Nyquist and Shannon. 1) Every signal of finite energy and bandwidth W Hz may be completely recovered, in a simple way, from a knowledge of its samples taken at the rate of 2W per second (Nyquist rate). Moreover, the recovery is stable, in the sense that a small error in reading sample values produces only a correspondingly small error in the recovered signal. 2) Every square-summable sequence of numbers may be transmitted at the rate of 2W per second over an ideal channel of bandwidth W Hz, by being represented as the samples of an easily constructed band-limited signal of finite energy. The practical importance of these results, together with the restrictions implicit in the sampling theorem, make it natural to ask whether the above rates cannot be improved, by passing to differently chosen sampling instants, or to bandpass or multiband (rather than bandlimited) signals, or to more elaborate computations. In this paper we draw a distinction between reconstructing a signal from its samples, and doing so in a stable way, and we argue that only stable sampling is meaningful in practice. We then prove that: 1) stable sampling cannot be performed at a rate lower than the Nyquist, 2) data cannot be transmitted as samples at a rate higher than the Nyquist, regardless of the location of sampling instants, the nature of the set of frequencies which the signals occupy, or the method of construction. These conclusions apply not merely to finite-energy, but also to bounded, signals.     

Importance sampling for tanner trees

This correspondence presents an importance sampling (IS) simulation scheme for the soft iterative decoding on loop-free multiple-layer trees. It is shown that this scheme is asymptotically efficient in that, for an arbitrary tree and a given estimation precision, the required number of samples is inversely proportional to the noise standard deviation. This work has its application in the simulation of low-density parity-check (LDPC) codes.     

An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition]

Conventional approaches to sampling signals or images follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (Nyquist rate). In the field of data conversion, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation - the signal is uniformly sampled at or above the Nyquist rate. This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use.     

Reconstruction of band-limited periodic nonuniformly sampled signals through multirate filter banks

A band-limited signal can be recovered from its periodic nonuniformly spaced samples provided the average sampling rate is at least the Nyquist rate. A multirate filter bank structure is used to both model this nonuniform sampling (through the analysis bank) and reconstruct a uniformly sampled sequence (through the synthesis bank). Several techniques for modeling the nonuniform sampling are presented for various cases of sampling. Conditions on the filter bank structure are used to accurately reconstruct uniform samples of the input signal at the Nyquist rate. Several examples and simulation results are presented, with emphasis on forms of nonuniform sampling that may be useful in mixed-signal integrated circuits.     

基于矩阵分解的压缩感知算法研究

奈奎斯特采样定律是长久以来具有指导意义的经典信号处理技术,它提出信号在采样过程中,当且仅当采样率大于信号带宽的2倍时,才能精确重构信号。压缩感知理论突破了奈奎斯特采样定理对信号采样率的限制,以更低采样率采样信号,并通过适当的重构算法恢复信号。文中以压缩感知理论为基础,结合目前广泛采用的正交匹配追踪算法,基于矩阵分解思想,提出2种改进算法,在运算复杂度方面取得优化,并且满足信号处理时对重构精度的要求。     

Time-domain analysis of cross correlation for time delay estimationwith an autocorrelated signal

The estimation accuracy of the time difference of arrival (TDOA) of an exponentially autocorrelated signal at two sensors in white noise is analyzed. The estimate is obtained by cross correlating samples taken as short-term integrals of the noisy signals from the two sensors. This technique avoids ambiguities in the cross correlation, and it is shown that the best sampling rate is double the Nyquist rate, for which the Cramer-Rao lower bound (CRLB) is met in practice     

Interpolation and extrapolation in nonuniform sampling sequences with average sampling rates below the Nyquist rate

Coefficients for interpolation and extrapolation, including average sampling rates below the Nyquist rate, have been derived. The method is applicable for nonuniform patterns obtained through omission of samples in a uniform pattern with a sampling rate slightly higher than the Nyquist rate. The interpolations and extrapolations are valid for band-limited signals, except those theoretical signals that are zero valued at all sampling points.     

Vector sampling expansion

The vector sampling expansion (VSE) is an extension of Papoulis' (1977) generalized sampling expansion (GSE) to the vector case. In VSE, N bandlimited signals, all with the same bandwidth B, are passed through a multi-input-multi-output (MIMO) linear time invariant system that generates M (M⩾N) output signals. The goal is to reconstruct the input signals from the samples of the output signals at a total sampling rate of N times the Nyquist rate, where the Nyquist rate is B/π samples per second. We find necessary and sufficient conditions for this reconstruction. A surprising necessary condition for the case where all output signals are uniformly sampled at the same rate (N/M times the Nyquist rate) is that the expansion factor M/N must be an integer. This condition is no longer necessary when each output signal is sampled at a different rate or sampled nonuniformly. This work also includes a noise sensitivity analysis of VSE systems. We define the noise amplification factor, which allows a quantitative comparison between VSE systems, and determine the optimal VSE systems     

Bandpass sampling criteria for nonlinear systems

Sampling criteria for nonlinear systems with a band-pass input are developed in this paper. It is well known that nonlinear systems may produce an output signal with a larger bandwidth than that of their input signal. According to the Nyquist sampling theorem, the sampling rate needs to be at least twice the maximum frequency of the output signal; otherwise, the sampled output would be aliased. However, if the input is a bandpass signal, the spectrum of the output signal often occupies multiple frequency bands. In this case, it is possible, by using the bandpass sampling concept, to sample the output signal at a rate much lower than the Nyquist sampling frequency. In this paper, all conditions in which bandpass sampling can be achieved are derived for nonlinear systems up to the third order. Furthermore, for nonlinear systems higher than the third order, some conditions in which bandpass sampling can be guaranteed are derived. The result can be used to choose an appropriate sampling frequency for nonlinear systems of an arbitrary order     

Reconstruction of two-dimensional signals from level crossings

Recent results indicate that reconstruction of two-dimensional signals from crossings of one level requires, in theory and practice, extreme accuracy in positions of the samples. The representation of signals with one-level crossings can be viewed as a tradeoff between bandwidth and dynamic range, in the sense that if the available bandwidth is sufficient to preserve the level crossings accurately, then the dynamic range requirements are significantly reduced. On the other hand, representation of signals by their samples at the Nyquist rate can be considered as requiring relatively small bandwidth and large dynamic range, because, at least in theory, amplitude information at prespecified points is needed, to infinite precision. An overview of existing results in zero crossing representation is presented, and a number of new results on sampling schemes for reconstruction from multiple-level threshold crossing are developed. The quantization characteristics of these sampling schemes appear to lie between those of Nyquist sampling and one-level crossing representations, thus bridging the gap between explicit Nyquist sampling and implicit one-level crossing sampling strategies     

An error bound for Lagrange interpolation of low-pass functions (Corresp.)

The well-known error formula for Lagrange interpolation is used to derive an expression for a truncation error bound in terms of the sampling rate and Nyquist frequency for regular samples and central interpolation. The proof is restricted to pulse-type functions possessing a Fourier transform. The formula finds application to the estimation of convergence rate in iterative interpolation, thus providing a criterion for the choice of sampling rate to achieve a specified truncation error level in a given number of steps. The formula can also be used as a guide when the samples are not regular but fairly evenly distributed.     

基于有限新息率的THz脉冲信号采样和恢复

为了降低硬件设计的难度,采用有限新息率(FRI)理论,通过选择合适的采样核函数,对太赫兹脉冲信号以高于信号的新息率的速率进行采样,进而利用子空间算法对它的自由参量进行估计,重建出原始信号。一般信号的新息率远远低于信号的带宽,这样就大大降低了采样速率。通过延时估计误差,验证FRI采样理论对太赫兹脉冲信号采样的正确性以及子空间算法对信号重建的有效性。     

A dual-rate self-tuning pole-placement controller

A dual-rate self-tuning pole-placement controller is proposed. It allows the use of a larger sampling interval for on-line parameter estimation and a smaller sampling interval for control. The conversion of model parameters from one sampling interval to another can be readily performed using a pole-zero mapping technique. The major problem encountered is that the process zeros at the slow sampling rate cannot be estimated accurately by using input-output data obtained at the fast sampling rate. The proposed solution of fixing the zeros at z=-1 while keeping the DC gain the same is supported by analysis of the Nyquist plot. The main merits of the dual-rate self-tuning control system are in the simultaneous achievement of robust estimation and reduced computation load and in improved performance in the regulation of deterministic and stochastic load disturbances. Its performance is substantiated by digital simulation and experiments on a pilot plant     

Power spectral density of unevenly sampled data by least-squareanalysis: performance and application to heart rate signals

This work studies the frequency behavior of a least-square method to estimate the power spectral density of unevenly sampled signals. When the uneven sampling can be modeled as uniform sampling plus a stationary random deviation, this spectrum results in a periodic repetition of the original continuous time spectrum at the mean Nyquist frequency, with a low-pass effect affecting upper frequency bands that depends on the sampling dispersion. If the dispersion is small compared with the mean sampling period, the estimation at the base band is unbiased with practically no dispersion. When uneven sampling is modeled by a deterministic sinusoidal variation respect to the uniform sampling the obtained results are in agreement with those obtained for small random deviation. This approximation is usually well satisfied in signals like heart rate (HR) series. The theoretically predicted performance has been tested and corroborated with simulated and real HR signals. The Lomb method has been compared with the classical power spectral density (PSD) estimators that include resampling to get uniform sampling. The authors have found that the Lomb method avoids the major problem of classical methods: the low-pass effect of the resampling. Also only frequencies up to the mean Nyquist frequency should be considered (lower than 0.5 Hz if the HR is lower than 60 bpm). It is concluded that for PSD estimation of unevenly sampled signals the Lomb method is more suitable than fast Fourier transform or autoregressive estimate with linear or cubic interpolation. In extreme situations (low-HR or high-frequency components) the Lomb estimate still introduces high-frequency contamination that suggest further studies of superior performance interpolators. In the case of HR signals the authors have also marked the convenience of selecting a stationary heart rate period to carry out a heart rate variability analysis     

High‐resolution compressive channel estimation for broadband wireless communication systems

Broadband channel is often characterized by a sparse multipath channel where dominant multipath taps are widely separated in time, thereby resulting in a large delay spread. Accurate channel estimation can be done by sampling received signal with analog‐to‐digital converter (ADC) at Nyquist rate and then estimating all channel taps with high resolution. However, these Nyquist sampling‐based methods have two main disadvantages: (i) demand of the high‐speed ADC, which already exceeds the capability of current ADC, and (ii) low spectral efficiency. To solve these challenges, compressive channel estimation methods have been proposed. Unfortunately, those channel estimators are vulnerable to low resolution in low‐speed ADC sampling systems. In this paper, we propose a high‐resolution compressive channel estimation method, which is based on sampling by using multiple low‐speed ADCs. Unlike the traditional methods on compressive channel estimation, our proposed method can approximately achieve the performance of lower bound. At the same time, the proposed method can reduce communication cost and improve spectral efficiency. Numerical simulations confirm our proposed method by using low‐speed ADC sampling. Copyright © 2012 John Wiley & Sons, Ltd.     

分布式压缩感知视频编码中CS帧的二次修正准则研究

分布式视频编码是新的视频编码体系,与传统的视频编码体系相比,具有编码端相对简单、解码端相对复杂的特点。此外,压缩感知突破了奈奎斯特采样定理,降低了信号的采样率。将压缩感知理论应与分布式视频编码相结合,使编码端复杂度降低。在一些分布式压缩视频编码研究中,CS帧是由边信息和发送端传送的信息联合重建的,由于不同CS帧的边信息的预测准确度不同,导致不同CS帧恢复质量相差较大。为了解决这个问题,本文对CS帧的二次修正准则的进行研究,首先从理论上推导出方差作为修正准则的可行性,并在实验中加以验证。实验可知,本文提出的方法在一定程度上改善了这些帧的重建质量。      

Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence{r(n)}, wherer(n) = E[x(l_{m+n}) overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if{t_n}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort leq t_oort geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known{ r(n) }.