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.     

The predictability of continuous-time, bandlimited processes

In communications and signal processing, we can find examples of applications that could benefit from the prediction of a bandlimited random process. We consider a continuous-time linear predictor applied to a bandlimited process. We show that if the past values of the process are known over an interval of arbitrary positive length, the mean squared prediction error may be made arbitrarily small, regardless of how far in the future we wish to make the prediction. We also show that this is no longer true when a certain energy constraint is applied to the predictor. Furthermore, we discuss what this means for the case in which the prediction is based on past values that are corrupted by estimation errors     

Superoscillations: Faster Than the Nyquist Rate

It is commonly assumed that a signal bandlimited to$mu /$2 Hz cannot oscillate at frequencies higher than$mu $Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called superoscillations, can only occur in signals that possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared$L^2$norm) as a function of the superoscillation's frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and time-frequency analysis, for example. It also shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations' period.     

Bounds for truncation error in sampling expansions of finite energy band-limited signals (Corresp.)

An upper bound is established for the magnitude of the truncation error incurred when a real-valued finite energy signal that is bandlimited to-pi r leq omega leq pi r, 0 < r < 1, is approximated by2N + 1terms from its Shannon sampling series expansion with the samples taken at the integer points. The derived bound is an improvement on known results.     

自适应小波的构造及其在信号处理中的应用

基于带限正交小波基的特性,本文采用匹配方法构造了一种新的带限尺度函数,它与信号的主要能量所在频带具有相似的特征.匹配尺度滤波器是优化滤波器,它锁定能量聚集的频带,使得所需频率分量的输出增大.原始信号能够用优化的小波实时处理.通过对脑电图信号的处理,证明本文算法是有效的.     

频带有限信号非一致采样重建的谱方法

本文研究带限信号的非一致采样重建问题。我们利用信号频带分割方法,导出带限信号周期非一致采样情 况下的具体重建公式,并由此给出带限信号有限采样在区间上的插值公式并对它进行了数值实验。最后利用频谱分割方法, 讨论了多带限信号一些采样策略。     

Parseval's relationship for nonuniform samples of signals withseveral variables

The author shows that the Parseval's relationship as described by F.A. Marvasti and L. Chuande (see IEEE Trans. Acoust., Speech, Sign. Processing, vol.38, no.6, p.1061-3, 1990) only holds true for bandlimited signals of finite energy if the irregular sampling sequence is relatively separated, e.g. if there is a positive minimal distance between two sampling points. The author presents a correct proof of the result for bandlimited functions of any number of variables, if this condition is satisfied     

Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions

Shannon's sampling theory is based on the reconstruction of bandlimited signals which requires infinite number of uniform time samples. Indeed, one can only have finite number of samples for numerical implementation. In this paper, as a dual of the bandlimited reconstruction, a solution for time-limited signal reconstruction from nonuniform samples is proposed. The system model we present is based on the idea that time-limited signals which are also nearly bandlimited can be well approximated by a low-dimensional subspace. This can be done by using prolate spheroidal wave functions as the basis. The order of the projection on this basis is obtained by means of the time–frequency dimension of the signal, especially in the case of non-stationary signals. The reconstruction requires the estimation of the nonuniform sampling times by means of an annihilating filter. We obtain the reconstruction parameters by solving a linear system of equations and show that our finite-dimensional model is not ill-conditioned. The practical aspects of our method including the dimensionality reduction are demonstrated by processing synthetic as well as real signals.     

Sampling and Reconstruction of Sparse Signals in Fractional Fourier Domain

Sampling theory for continuous time signals which have a bandlimited representation in fractional Fourier transform (FrFT) domain-a transformation which generalizes the conventional Fourier transform-has blossomed in the recent past. The mechanistic principles behind Shannon's sampling theorem for fractional bandlimited (or fractional Fourier bandlimited) signals are the same as for the Fourier domain case i.e. sampling (and reconstruction) in FrFT domain can be seen as an orthogonal projection of a signal onto a subspace of fractional bandlimited signals. As neat as this extension of Shannon's framework is, it inherits the same fundamental limitation that is prevalent in the Fourier regime-what happens if the signals have singularities in the time domain (or the signal has a nonbandlimited spectrum)? In this paper, we propose a uniform sampling and reconstruction scheme for a class of signals which are nonbandlimited in FrFT sense. Specifically, we assume that samples of a smoothed version of a periodic stream of Diracs (which is sparse in time-domain) are accessible. In its parametric form, this signal has a finite number of degrees of freedom per unit time. Based on the representation of this signal in FrFT domain, we derive conditions under which exact recovery of parameters of the signal is possible. Knowledge of these parameters leads to exact reconstruction of the original signal.     

Efficient Pulse Shaping Using MSK or PSK Modulation

The increasing requirement for high data rate, bandwidth efficient digital radio systems has led to the development of MSK-type modulation methods designed to achieve a compact signal spectrum. These modulation methods include sinusoidal frequency shift keying (SFSK), special MSK-type pulse shapes, and multiamplitude minimum shift keying (MAMSK). When more compact signal spectra are required, bandlimited filtering must be introduced. This note considers the use of conventional MSK or PSK modulators followed by newly developed bandlimited pulse shaping filters. With this approach, it is shown that MSK and offset QPSK modulators yield identical signals on the channel when filtered by properly designed bandlimited pulse shaping networks.     

Adaptive trellis-coded modulation for bandlimited meteor burstchannels

The throughput performances of three adaptive information rate techniques on the bandlimited meteor burst channel are investigated. Closed-form expressions for throughput are derived based on the channel model commonly used in the literature. The throughput performance is compared to the conventional fixed information rate modem and upper bounds on throughput improvement over the fixed rate modem are derived. It is shown that an adaptive technique that uses trellis-coded modulation (TCM) with three phase-shift keyed (PSK) signal sets can increase throughput over the conventional fixed rate modem by more than a factor of 3. Data from the US Air Force High Latitude Meteor-Scatter Test Bed confirm the superiority of the adaptive TCM technique. A practical implementation is suggested that uses a single rate 1/2 convolutional code for all three PSK signal sets. The use of this single code, versus the three best Ungerboeck codes, results in a throughput loss of less than 2%. An expression for the theoretical information capacity of the bandlimited meteor burst channel is derived     

Transmission of an analog signal over a fixed bit-rate channel

The transmission of a nonbandlimited analog signal over a digital channel with a fixed bit-rate is considered. The trade-off between the mean-square error due to quantizing and the mean-square error due to the process of sampling and reconstructing the signal is investigated. Simple approximations to these errors, which are valid in most practical situations, are derived, and simple expressions are obtained from which the optimum sampling interval and number of bits per sample can be calculated. Results for first-, second-, and third-order Butterworth and fiat bandlimited spectra, together with the zero-order hold and the linear point connector, are included. The resulting mean-square error goes to zero with large channel bit-rates in a slower manner than the Shannon limit, which assumes a strictly bandlimited signal and perfect reconstruction.     

Sampling signals with finite rate of innovation

The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems     

Geometric properties of a large set of random signals

For a bandwidth-limited, power-limited communication channel corrupted by additive white Gaussian noise, Shannon (1949) showed that, at any information rate below the channel capacity, an arbitrarily low error probability can be obtained when the transmitted signal is selected from a set a of M independent, white-Gaussian-noise-like waveforms having the prescribed power and bandwidth. He utilized a space whose points represent transmitted or received signals. Since for each possible received signal there is a most likely transmitted signal, this space is divided into decision regions {R_t}, one associated with each member s_t (i=1, 2,..., M) of set s_t This article investigates some of the geometrical properties of these decision regions: their shape, size and number of faces, distances from s_t to R_t's nearest face or edges of various sorts, etc., thus providing insight into how random coding succeeds in achieving arbitrarily good performance     

Capacity penalty due to ideal zero-forcing decision-feedbackequalization

We consider the capacity C of a continuous-time channel with frequency response H(f) and additive white Gaussian noise. If H(f)|^-2 behaves like a polynomial of order ρ at high frequencies, we show that the per-symbol capacity approaches ρ/2 nats per channel use at high signal powers. If the receiver uses an ideal zero forcing decision-feedback equalizer (DFE) consisting of a sampled whitened-matched filter followed by a zero-forcing tail canceler that is free of error propagation, the overall system is free of intersymbol interference and has a well-defined capacity C_ZF. By comparing this capacity with the capacity C of the underlying channel, we quantify the loss of information inherent in the tail-canceling operation that typifies zero-forcing DFE and zero-forcing precoding systems. For strictly bandlimited channels, we find that the capacity penalty approaches zero in the limit of large signal power. On the other hand, for nonstrictly bandlimited channels, the asymptotic penalty is nonzero; however, with bandwidth optimization, the asymptotic penalty is at most 0.59 dB, and the asymptotic ratio C_ZF/C is at least 93.6%, depending on the asymptotic order ρ of the channel response     

An eight-dimensional 64-state trellis code for transmitting 4 bitsper 2-D symbol

An eight-dimensional, 64-state, 90° rotationally invariant trellis code for transmitting 4 bits/baud over a bandlimited channel is described. The 2-D constellation contains 20 points. The code achieves a 5.23-dB coding gain over the uncoded 4×4 QAM (quadrature amplitude modulation) constellation and a 1.23-dB gain over the standard CCITT V32 trellis code. Simulation results are presented that verify these coding gains. Simulation results showing symbol error probability versus signal/noise ratio and trellis depth are also presented     

Blind compensation of nonlinear distortion for bandlimited signals

Nonlinear distortion of bandlimited signals results in spectral spreading. This paper develops a blind nonlinear compensation method for bandlimited signals by suppressing the spectral content of the distorted signal above the original signal bandwidth by means of adaptive nonlinear filtering. The nonlinear compensator is constructed using a power series filter with adaptive coefficients. The adaptive coefficients are identified blindly by applying a least-squares criterion to the out-of-band spectral content of the nonlinear compensator output. The extraction of the out-of-band signal is efficiently performed by the discrete cosine transform. The effectiveness of the blind nonlinear compensation method is demonstrated by way of simulation examples involving periodic, colored noise, and bandlimited speech signals.     

Microring resonator channel dropping filters

Microring resonators side coupled to signal waveguides provide compact, narrow band, and large free spectral range optical channel dropping filters. Higher order filters with improved passband characteristics and larger out-of-band signal rejection are realized through the coupling of multiple rings. The analysis of these devices is approached by the novel method of coupling of modes in time. The response of filters comprised of an arbitrarily large dumber of resonators may be written down by inspection, as a continued fraction. This approach simplifies both the analysis and filter synthesis aspects of these devices     

A general sampling theory for nonideal acquisition devices

The authors first describe the general class of approximation spaces generated by translation of a function ψ(x), and provide a full characterization of their basis functions. They then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle. The theory puts no restrictions on the system input which can be an arbitrary finite energy signal; bandlimitedness is not required. In contrast to previous approaches, this formulation allows for an independent specification of the sampling (analysis) and approximation (synthesis) spaces. In particular, when both spaces are identical, the theorem provides a simple procedure for obtaining the least squares approximation of a signal. They discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited, and polynomial spline signal representations. They also define a spectral coherence function that measures the “similarity” between the sampling and approximation spaces, and derive a relative performance bound for the comparison with the least squares solution