On the performance of cyclic moments-based parameter estimators ofamplitude modulated polynomial phase signals

Parameter estimation of amplitude-modulated polynomial phase signals embedded in additive white Gaussian noise is considered. The amplitude modulation is modeled as the sum of a real-valued deterministic function and a zero mean correlated stationary random process. It is shown that cyclic moments-based estimators, previously proposed for parameter estimation of polynomial phase signals modulated by stationary random processes, can be adapted to the more general signal model considered here. The covariance matrices of the cyclic moments-based amplitude and phase parameter estimators are derived for large sample lengths. Using this result, it is shown how the lags can be chosen to minimize the large-sample variances of the cyclic moments-based phase parameter estimators. Comparisons with the Cramer-Rao bounds are performed under the assumption of a Gaussian modulating process. The theoretical derivations are confirmed by simulation results     

Shape and duration of clicks in modulated FM transmission

A Slepian model is derived for the shape of dicks in modulated FM transmission. The modulated signal may be a deterministic periodic function or a stationary stochastic process, not necessarily Gaussian. To describe the clicks one first derives the stochastic properties of noise and signal at the clicks. (For a periodic signal this amounts to giving the relative number of clicks in any given part of the period.) Then the conditional distributions of noise and signal near clicks are derived in explicit form, using the noise and signal values at the click as (random) parameters, and finally these conditional distributions are mixed into a total model for the click shapes. For increasing carrier-to-noise power ratioA^{2}/2, the Slepian model converges after normalization to a rational function with random coefficients. The amplitude of a click is shown to be of the orderA^{2}, in contrast to the unmodulated ease, where click amplitudes are of the orderA.     

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.     

On the existence of the solution for one-dimensional discrete phase retrieval problem

We consider the discrete form of the one-dimensional phase retrieval (1-D DPhR) problem from the point of view of input magnitude data. The direct method can provide a solution to the 1-D DPhR problem if certain conditions are satisfied by the input magnitude data, namely the corresponding trigonometric polynomial must be nonnegative. To test positivity of a trigonometric polynomial a novel DFT-based criterion is proposed. We use this DFT criterion for different sets of input magnitude data to evaluate whether the direct method applied to the 1-D DPhR problem leads to a solution in all explored cases.     

Solution of the two-dimensional spectral factorization problem

An approximation theorem is proven which solves a classic problem in two-dimensional (2-D) filter theory. The theorem shows that any continuous two-dimensional spectrum can be uniformly approximated by the squared modulus of a recursively stable finite trigonometric polynomial supported on a nonsymmetric half-plane.     

Robust least-squares estimation with a relative entropy constraint

Given a nominal statistical model, we consider the minimax estimation problem consisting of finding the best least-squares estimator for the least favorable statistical model within a neighborhood of the nominal model. The neighborhood is formed by placing a bound on the Kullback-Leibler (KL) divergence between the actual and nominal models. For a Gaussian nominal model and a finite observations interval, or for a stationary Gaussian process over an infinite interval, the usual noncausal Wiener filter remains optimal. However, the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty. On the other hand, standard causal least-squares estimators are not optimal, and a characterization is provided for the causal estimator and the corresponding least favorable model. The causal estimator takes the form of a risk-sensitive estimator with an appropriately selected risk sensitivity coefficient.     

Group codes for phase- and amplitude-modulated signals on a Gaussian channel

A group code is a set of vectors, generated by a group of orthogonal matrices transforming a starting vector. Ingemarsson [5] has shown that, if the group is commutative, the code is equivalent to a set of phase-modulated signals. Codes generated by noncommutative groups have been described by Slepian [9]. In this paper another class of codes generated by noncommutative groups is described. This class is called group codes for PHAMP signals and can be described as a combination of phase- and amplitude-modulated signals. The performance of these codes, when communicating over the Gaussian channel, is analyzed, and it is shown that some of them have a performance close to the bounds given by Slepian [8].     

The effects of a finite time-bandwidth product on the variance of the sample mean of a low-pass stationary random process (Corresp.)

Expressions are derived for the variance of the sample mean and the variance of the differences of sample means for an ideal Iow-pass Gaussian stationary random process. The effects of a finite data time-bandwidth product are discussed.     

Information transmission with time-continuous random processes

Shannon's definition for the information content of a Gaussian, time-continuous process in Gaussian noise is extended to the case where the observation interval is finite, and where the processes may be nonstationary, in a straightforward way. The extension is based on a generalization of the Karhunen-Loeve Expansion, which allows both the signal and noise processes to be expanded in the same set of functions, with uncorrelated coefficients. The resultant definition is consistent with that of Gel'fand and Yaglom, and avoids the difficulties posed by Good and Doog to Shannon's original definition. This definition is shown to be useful by applying it to the calculation of the information content of some cases of stationary signals in stationary noise, with different spectra, and to one case where both are nonstationary. Limiting relations are derived, to show that this reduces to previously established results in some cases, and to enable one to obtain rule-of-thumb estimates in others. In addition, both the matched filter and the Wiener filter are related to the information; the matched filter in a very direct way, in that it converts a time-continuous process to a set of random variables while conserving the information.     

On Divergence-Power Inequalities

Expressions for (entropy-power inequality (EPI) Shannon type) divergence-power inequalities (DPIs) in two cases (time-discrete and time-continuous) of stationary random processes are given. The new expressions connect the divergence rate of the sum of independent processes, the individual divergence rate of each process, and their power spectral densities. All divergences are between a process and a Gaussian process with same second-order statistics, and are assumed to be finite. A new proof of the Shannon EPI, based on the relationship between divergence and causal minimum mean-square error (CMMSE) in Gaussian channels with large signal-to-noise ratio, is also shown     

An information-theoretic proof of Burg's maximum entropy spectrum

It is known that the maximum entropy stationary Gaussian stochastic process, subject to a finite number of autocorrelation constraints, is the Gauss-Markov process of appropriate order. The associated spectrum is Burg's maximum entropy spectral density. We pose a somewhat broader entropy maximization problem, in which stationarity and normality are not assumed, and shift the burden of proof from the previous focus on the calculus of variations and time series techniques to a string of information-theoretic inequalities. This results in an elementary proof of greater generality.     

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.      

Commutative group codes for the Gaussian channel

A class of codes for the Gaussian channel is analyzed. The code class is a subclass of the group codes for the Gaussian channel described by Slepian. Using the vector model for the Gaussian channel, the code vectors are obtained by transformations of an initial vector. The class of codes in which the transformations form a commutative group is called the class of commutative group codes. The performance of the codes is evaluated using the union bound on the error probability as a performance measure. The union bound is shown to be closely related to the moments of the scalar product between the code vectors. Commutative group codes are described. It is shown that linear algebraic codes may be represented as commutative group codes. The code class is also shown to include simplex and biorthogonal codes in all dimensions.     

On the rate-distortion function for the nonstationary Gaussian autoregressive process (Corresp.)

A general form for the rate-distortion function is presented for the nonstationary Gaussian autoregressive (AR) process and is shown to differ from the well-known form for the asymptotically stationary process in a term corresponding to the log rate of the variance growth if the process has exponentially growing variance.     

On the existence of group codes for the Gaussian channel

A number of theorems are given, which give insight into the algebraic structure of group codes for the Gaussian channel, as defined by Slepian. The problem of the existence of group codes for a given pair of defining parameters (i.e., number of codewordsM, and dimension of the signal spacen) is partially solved.     

Power spectrum parameter estimation

The power spectrum Of a zero-mean stationary Gaussian random process is assumed to be known except for one or more parameters which are to be estimated from an observation of the process during a finite time interval. The approximation is introduced that the coefficients of the Fourier series expansion of a realization of long-time duration are uncorrelated. Based on this approximation maximum likelihood estimates are derived and lundamental limits on the variances attainable are found by evaluation of the Cramér-Rao lower bound. Parameters specifically considered are amplitude, center frequency, and frequency scale factor. Also considered is ripple frequency which refers to the cosine factor in the spectrum produced by the addition of a delayed replica of the random process. The dual problem of estimating parameters of the time-varying power level of a nonstationary baud-limited white noise process is examined.     

Instantaneous frequency estimation of nonlinear frequency-modulatedsignals in the presence of multiplicative and additive noise

This paper addresses the problem of estimating the instantaneous frequency (IF) of monocomponent nonlinear, not necessarily polynomial, frequency modulated (FM) signals affected by stationary multiplicative and additive noise. Both noise processes are assumed to be complex circular Gaussian and independent. The peak of the polynomial Wigner-Ville distribution (PWVD) is proposed here as an IF estimator. We derive analytical expressions for the bias and asymptotic variance of the estimator and propose an algorithm to select the optimal window length to resolve the bias-variance tradeoff in the IF estimation. Simulation results are presented to confirm the theoretical results     

PKCS: A Polynomial Kernel Family With Compact Support for Scale- Space Image Processing

In a scale-space framework, the Gaussian kernel has some properties that make it unique. However, because of its infinite support, exact implementation of this kernel is not possible. To avoid this drawback, there exist two different approaches: approximating the Gaussian kernel by a finite support kernel, or defining new kernels with properties closed to the Gaussian. In this paper, we propose a polynomial kernel family with compact support which overcomes the Gaussian practical drawbacks while preserving a large number of the useful Gaussian properties. The new kernels are not obtained by approximating the Gaussian, though they are derived from it. We show that, for a suitable choice of kernel parameters, this family provides an approximated solution of the diffusion equation and satisfies some other basic constraints of the linear scale-space theory. The construction and properties of the proposed kernel are described, and an application in which handwritten data are extracted from noisy document images is presented.     

Estimating random amplitude polynomial phase signals: acyclostationary approach

Modeling of a class of nonstationary signals with randomly time-varying amplitude and parametric polynomial phase is addressed. A novel approach is proposed for the estimation of the time-varying phase by exploiting the higher order cyclostationarity of these signals. The method does not require nonlinear search, is easy to implement, and yields consistent estimates for the parameters. The resulting algorithms are theoretically tolerant to a large class of noises including additive stationary non-Gaussian noise and any Gaussian noise. Simulation examples supporting the theory are provided     

A trellis code construction and coding theorem for stationary Gaussian sources (Corresp.)

For stationary discrete-time Gaussian sources and the squared-error distortion measure, a trellis source code is constructed. The encoder consists of a Karhunen-Loeve transform on the source output followed by a search on a trellis structured code, where the decoder is a time-variant nonlinear filter. The corresponding code theorem is proved using the random coding argument. The proof technique follows that of Viterbi and Omura, who proved the trellis coding theorem for memoryless sources. The resultant coding scheme is implementable and applicable at any nonzero rate to a stationary Gaussian source with a bounded and continuous power spectrum. Therefore. for stationary sources, it is more general than Berger's tree coding scheme, which is restricted to autoregressive Gaussian sources in a region of high rate (low distortion).