Timing estimation for a filtered Poisson process in Gaussian noise

The problem of estimation of time shift of an inhomogeneous casually filtered Poisson process in the presence of additive Gaussian noise is discussed. Approximate expressions for the likelihood function, the MAP estimator, and the MMSE estimator that becomes increasingly accurate as the per-unit-time density of superimposed filter responses becomes small are obtained. The optimal MAP estimator takes the form of a cascade of linear and memoryless nonlinear components. For smooth point process intensities, the performance of the MAP estimator is studied via local bias and local variance. A rate distortion type lower bound on the MSE of any estimator of time delay is then derived by identification of a communications channel that accounts for the mapping from time delay to observation process. Results of numerical studies of estimator performance are presented. Based on the examples considered it is concluded: (1) the small-error MSE of the nonlinear MAP estimator can be significantly better than the small-error MSE of the optimal linear estimator: (2) the rate distortion lower bound can be significantly tighter than the Poisson limited bounds determined in previous studies     

Bounds on the accuracy attainable in the estimation of continuous random processes

The problem of estimating a message,a(t), which is a sample function from a continuous Gaussian random process is considered. The message to be estimated may be contained in the transmitted signal in a nonlinear manner. The signal is corrupted by additive noise before observation. The received waveform is available over some observation interval[T_{i}, T_{f}]. We want to estimatea(t)over the same interval. Instead of considering explicit estimation procedures, we find bounds on how well any procedure The principle results are as follows: 1) a lower bound on the mean-square estimation error. This bound is a generalization of bounds derived previously by Cramer, Rao, and Slepian for estimating finite sets of parameters. 2) The bound is evaluated for several practical examples. Possible extension and applications are discussed briefly.     

RKHS approach to detection and estimation problems--V: Parameter estimation

Using reproducing-kernel Hilbert space (RKHS) techniques, we obtain new results for three different parameter estimation problems. The new results are 1) an explicit formula for the minimum-variance unbiased estimate of the arrival time of a step function in white Gaussian noise, 2) a new interpretation of the Bhattacharyya bounds on the variance of an unbiased estimate of a function of regression coefficients, and 3) a concise formula for the Cramér-Rao bound on the variance of an unbiased estimate of a parameter determining the covariance of a zero-mean Gaussian process.     

Lower bounds on the mean square estimation error

A general class of lower bounds on the mean square error (mse) in random parameter estimation is formulated. These bounds are generated using functions of the parameter and the data that are orthogonal to the data. A particular choice in the class yields a new lower bound which is superior to both the Cramer-Rao and Bobrovsky-Zakai lower bounds.     

Capacity and error exponent for the direct detection photonchannel. I

The capacity and the lower bound on the error exponent of a direct detection optical channel are derived. The channel input in a T-second interval is a waveform satisfying certain peak and average power constraints for the optical signal. The channel output is a Poisson process with a given intensity parameter; the intensity parameter takes account of the dark-current signal component. An explicit construction for an exponentially optimum family of codes is also presented     

On the computation of an asymptotic bound for estimating autoregressive signals in white noise

The Cramer-Rao lower bound (CRLB) provides a useful reference for evaluating the performance of parameter estimation techniques. This paper considers the problem of estimating the parameters of an autoregressive signal corrupted by white noise. An explicit formula is derived for computing the asymptotic CRLB for the signal and noise parameters. Formulas for the asymptotic CRLB for functions of the signal and noise parameters are also presented. In particular, the center frequency, bandwidth and power of a second order process are considered. Some numerical examples are presented to illustrate the usefulness of these bounds in studying estimation accuracy.     

Time-delay estimation for filtered Poisson processes using anEM-type algorithm

We develop a modified EM algorithm to estimate a nonrandom time shift parameter of an intensity associated with an inhomogeneous Poisson process N_t, whose points are only partially observed as a noise-contaminated output X of a linear time-invariant filter excited by a train of delta functions, a filtered Poisson process. The exact EM algorithm for computing the maximum likelihood time shift estimate generates a sequence of estimates each of which attempt to maximize a measure of similarity between the assumed shifted intensity and the conditional mean estimate of the Poisson increment dN_t. We modify the EM algorithm by using a linear approximation to this conditional mean estimate. The asymptotic performance of the modified EM algorithm is investigated by an asymptotic estimator consistency analysis. We present simulation results that show that the linearized EM algorithm converges rapidly and achieves an improvement over conventional time-delay estimation methods, such as linear matched filtering and leading edge thresholding. In these simulations our algorithm gives estimates of time delay whose mean square error virtually achieves the CR lower bound for high count rates     

On the use of Cramer-Rao-like bounds in the presence of randomnuisance parameters

We focus on the use of Cramer-Rao-like bounds for the estimation of a deterministic parameter in the presence of random nuisance parameters. As a working example, we consider the problem of estimating the carrier frequency offset affecting a linearly modulated waveform received through a Rician fading channel. We calculate the relevant Cramer-Rao bound (CRB) and also a set of Cramer-Rao-like bounds proposed in the literature, among which we mention the hybrid CRB, the modified CRB and the Miller-Chang (1978) bound. An extension of the latter bound is also provided. The relative merits of these bounds are discussed, both in terms of their tightness and ease of calculation. A few useful inequalities involving the above bounds are also established     

Information of partitions with applications to random access communications

The minimum amount of information and the asymptotic minimum amount of entropy of a random partition which separates the points of a Poisson point process are found. Related information theoretic bounds are applied to yield an upper bound to the throughput of a random access broadcast channel. It is shown that more information is needed to separate points by partitions consisting of intervals than by general partitions. This suggests that single-interval conflict resolution algorithms may not achieve maximum efficiency.     

Asymptotic global confidence regions for 3-D parametric shape estimation in inverse problems.

This paper derives fundamental performance bounds for statistical estimation of parametric surfaces embedded in R3. Unlike conventional pixel-based image reconstruction approaches, our problem is reconstruction of the shape of binary or homogeneous objects. The fundamental uncertainty of such estimation problems can be represented by global confidenceregions, which facilitate geometric inference and optimization ofthe imaging system. Compared to our previous work on global confidence region analysis for curves [two-dimensional (2-D) shapes], computation of the probability that the entire surface estimate lies within the confidence region is more challenging because a surface estimate is an inhomogeneous random field continuously indexed by a 2-D variable. We derive an asymptotic lower bound to this probability by relating it to the exceedence probability of a higher dimensional Gaussian random field, which can, in turn, be evaluated using the tube formula due to Sun. Simulation results demonstrate the tightness of the resulting bound and the usefulness of the three-dimensional global confidence region approach.     

A Comment on the Weiss–Weinstein Bound for Constrained Parameter Sets

The Weiss–Weinstein bound (WWB) provides a lower limit on the mean-squared error (MSE) achievable by an estimator of an unknown random parameter. In this correspondence, it is shown that some previously proposed simplified versions of the bound do not always hold for constrained parameters, i.e., parameters whose distribution has finite support. These simplifications can produce results which are no longer lower bounds on the MSE. Sufficient conditions are provided for the reductions to be valid.      

Cramer-Rao bounds for parametric shape estimation in inverse problems

We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown gray level, or color, and boundary parameterization, on an arbitrary known background. While such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the Cramer-Rao lower bounds, very few results have been reported due to the difficulty of computing the derivatives of a functional with respect to shape deformation. We provide a general formula for computing Cramer-Rao lower bounds in inverse problems where the observations are related to the object by a general linear transform, followed by a possibly nonlinear and noisy measurement system. As an illustration, we derive explicit formulas for computed tomography, Fourier imaging, and deconvolution problems. The bounds reveal that highly accurate parametric reconstructions are possible in these examples, using severely limited and noisy data.     

Relations between Belini-Tartara, Chazan-Zakai-Ziv, and Wax-Zivlower bounds

A class of lower bounds on the mean-square error in parameter estimation is presented that are based on the Belini-Tartara lower bound (1974). The Wax-Ziv lower bound (1977) is shown to be a special case in the class. These bounds often are significantly tighter than the Chazan-Zakai-Ziv lower bound (1975) when the parameter to be estimated is subject to ambiguity and threshold effects     

Phase Estimators with Predetection Feedback for Optical Communication

Causal, minimum mean-square error (MMSE) estimators of a Gauss-Markov process observed through a conditional Poisson process whose rate parameter is a linear function of the estimation error are presented. Although the conditional estimation performance is data dependent, precomputable upper bounds on the average estimation performance are obtained. Approximate expressions are also presented for the Cramer-Rao lower bound, and it is shown that the estimator performance achieves that bound with equality when the estimator is operating considerably "above threshold." The estimator structure is applied to the problem of phase-tracking receivers for optical communication. Two receiver structures that use predetector phase feedback are considered: one uses local reference fields to allow the detector to observe phase error (homodyne/heterodyne), while the other is a novel direct detection receiver that depends explicitly on the closed-loop nature of the phase estimator. It is concluded that a large local oscillator amplitude is desirable to improve the phase-tracking performance in the homodyne/heterodyne case, and that as few as 4/8 detected signal photons per phase coherence time are required to keep the estimator above threshold. The direct detection scheme achieves the same performance as the homodyne system only in the limit of no dark current-background noise counts, and in general may require considerably more signal photons to keep the estimator "locked."     

Finite-memory universal prediction of individual sequences

The problem of predicting the next outcome of an individual binary sequence under the constraint that the universal predictor has a finite memory, is explored. In this analysis, the finite-memory universal predictors are either deterministic or random time-invariant finite-state (FS) machines with K states (K-state machines). The paper provides bounds on the asymptotic achievable regret of these constrained universal predictors as a function of K, the number of their states, for long enough sequences. The specific results are as follows. When the universal predictors are deterministic machines, the comparison class consists of constant predictors, and prediction is with respect to the 0-1 loss function (Hamming distance), we get tight bounds indicating that the optimal asymptotic regret is 1/(2K). In that case of K-state deterministic universal predictors, the constant predictors comparison class, but prediction is with respect to the self-information (code length) and the square-error loss functions, we show an upper bound on the regret (coding redundancy) of O(K/sup -2/3/) and a lower bound of /spl Theta/(K/sup -4/5/). For these loss functions, if the predictor is allowed to be a random K-state machine, i.e., a machine with random state transitions, we get a lower bound of /spl Theta/(1/K) on the regret, with a matching upper bound of O(1/K) for the square-error loss, and an upper bound of O(logK/K) Throughout the paper for the self-information loss. In addition, we provide results for all these loss functions in the case where the comparison class consists of all predictors that are order-L Markov machines.     

Cramer-Rao bounds for wavelet transform-based instantaneous frequency estimates

We use an asymptotic integral approximation of a wavelet transform as a model for the estimation of instantaneous frequency (IF). Our approach allows the calculation of the Cramer-Rao bound for the IF variance at each time directly, without the need for explicit phase parameterization. This is in contrast to other approaches where the Cramer-Rao bounds rely on a preliminary decomposition of the IF with respect to a (usually polynomial) basis. Attention is confined to the Morlet wavelet transform of single-component signals corrupted with additive Gaussian noise. Potential computationally and statistically efficient IF extraction algorithms suggested by the analysis are also discussed.     

The true Crame/spl acute/r-Rao lower bound for data-aided carrier-phase-independent time-delay estimation from linearly Modulated waveforms

In this paper, we present the derivation and analysis of the true Crame/spl acute/r-Rao lower bound (CRB) for the variance of unbiased, data-aided (DA) symbol-timing estimates, obtained from a block of K samples of a linearly modulated information signal, transmitted through an additive white Gaussian noise channel with random carrier phase. We consider a carrier-phase-independent time-delay estimation scenario wherein the carrier phase is viewed as an unwanted or nuisance parameter. The new bounds require only a moderate computational effort and are tighter than the CRB for the variance of unbiased time-delay estimates obtained under the assumption that the carrier phase is known. These bounds are particularly useful to assess the ultimate accuracy that can be achieved by pilot-assisted symbol synchronizers. Conversely, they may be used to evaluate data sequence suitability for the purpose of time-delay estimation. Comparison of the actual variance of a DA feedforward timing estimator with the new bounds show that these are attainable by practical synchronizers.     

Bounds on the capacity of a spectrally constrained Poisson channel

An upper bound is derived on the capacity of a Poisson channel that has a stationary input process of a given spectrum and is subjected to peak and average power constraints. The bound is shown to be asymptotically tight with the relaxation of the spectral constraints. Its maximization over a given set of admissible spectra is closely related to an analogous problem in the AWGN regime. The results are used for bounding the capacity of a Poisson channel under a second-moment-bandwidth constraint, as well as the capacity under a strict bandwidth constraint. Asymptotically tight lower bounds on the channel capacity for the above two cases are also presented. The approach for lower bounding the capacity for the latter case yields, as a by-product, improved bounds on the bit-error probability in uncoded amplitude shift keying (and on-off modulation as a special case) operating over a Poisson channel impaired by intersymbol interference     

Bounds on Error in Signal Parameter Estimation

In this correspondence, the problem of lower bounds on mean-square error in parameter estimation is considered. Lower bounds on mean-square error can be used, for instance, to bound the performances, namely the attainable output signal-to-noise ratio, of pulse modulation transmission systems, such as pulse-position modulation (PPM) or pulse-frequency modulation (PFM). The tightest lower bounds to mean-square error previously known are the Ziv-Zakai bounds; the analysis carried out in this paper, which is based on an inequality first obtained by Kotel'nikov, leads to lower bounds tighter than previously known bounds.     

A Constructive Capacity Lower Bound of the Inhomogeneous Wireless Networks

Many research results in the direction of wireless network capacity are based on the homogeneous Poisson node process and random homogeneous traffic. However, most of the realistic wireless networks are inhomogeneous. And for this kind of networks, this paper gives a constructive capacity lower bound, which may be effective on network designing. To ensure significant inhomogeneities, we select both inhomogeneous node process and traffic. We divide the transmission into two parts: intra-cluster transmission and inter-cluster transmission. Within each distinct cluster, a circular percolation model is proposed and the highway system is established. Different with regular rectangle percolation model, the highway in our model is in the radial direction or around the circle. Based on this model, we propose a routing strategy and get the intra-cluster per-node rate. In the following, among these clusters, we set many “information pipes” connecting them. By getting the results of per-node transmission rate of each part, we can find that the bottleneck of the throughput capacity is caused by the difference of the node density all over the network region. Specially, the lower bound interval of the capacity can be easily obtained when the traffic is inhomogeneous.