SPTM-12.1

Fast and Recursive Algorithms for Magnitude Retrieval from DTFT Phase at Irregular Frequencies
Andrew E Yagle (University of Michigan)

We derive two new algorithms for reconstructing a discrete time 1-D signal from the phase of its discrete-time Fourier transform (DTFT) at irregular frequencies. Previous algorithms for this problem have either required the computation of a matrix nullspace, requiring O(N^3) computations, or have been iterative in nature; for the latter, the irregularity of the frequency samples precludes use of the fast Fourier transform. Our first algorithm requires only O(N^2) computations (O(Nlog^3 N) asymptotically). In the special case of equally-spaced frequency samples, it is related to a previous algorithm. The second algorithm is recursive--at each recursion a meaningful magnitude-retrieval problem is solved. This is useful for updating a solution; it also allows checking of the result at each recursion, avoiding any errors due to computational roundoff error and ill-conditioning of the problem.

SPTM-12.2

On frequency estimation from oversampled quantized observations
Peter Handel (Department of Signals, Sensors and Systems), Anders Host-Madsen (TRLabs/University of Calgary)

The effect of sampling and quantization on frequency estimation for a single sinusoid is investigated. Asymptotic Cramer-Rao bounds (CRB) for 1-bit quantization and for non-ideal filters are derived, which are simpler to calculate than the exact CRB while still relatively accurate. It is further investigated how many bits should be used in quantization to avoid the problems of 1-bit quantization, and it turns out that 3-4 bits are enough. Finally, oversampled 1-bit quantization is investigated. It is determined how much the signal should be oversampled, and in addition sigma-delta modulators are investigated.

SPTM-12.3

Estimating the offset parameters of a mixture in the Fourier domain.
Sandrine Vaton (ENST 46 rue Barrault 75013 Paris), Thierry Chonavel (ENST Bretagne BP 832 29285 Brest Cedex France)

In this contribution we present an algorithm for estimating some parameters of offset in the case of incomplete data. This estimation cannot be performed directly with an EM or SEM method because the density of local extrema in the likelihood map grows exponentially with the number of observations and because the SEM method provides a monotonic sequence of estimates so that bad initialization cannot be recovered. We perform the estimation in the Fourier domain. The offsets in time domain are transformed into pulsations in the Fourier domain. We minimize a quadratic distance between the parametric and empirical sampled Fourier transform with an EM method. Contrary to the problems encountered in the time domain the asymptotic loglikelihood of the sampled empirical Fourier transform is continuous w.r.t. the parameters of offset. We discuss the influence of the frequencies at which the Fourier transform is sampled and we present a numerical study of convergence of the proposed algorithms.

SPTM-12.4

Optimal Phase Parameter Estimation of Random Amplitude Linear FM Signals Using Cyclic Moments
Mark R. Morelande (Cooperative Research Centre for Satellite Systems)

This paper considers the problem of estimating the phase parameters of a linear FM signal which is modulated by a random process and is embedded in additive noise. In particular, we consider the use of cyclic moments and derive variance expressions for the phase parameter estimates for all values of the lag parameter of the second order cyclic moment, tau. It is seen that the accuracy of the phase parameter estimates depends greatly on tau. This allows the definition of an optimal value of tau, in the sense that it minimises the phase parameter estimation variance.

SPTM-12.5

On estimating random amplitude chirp signals
Olivier BESSON (ENSICA), Mounir GHOGHO (University of Strathclyde), Ananthram SWAMI (Army Research Lab)

This paper considers the problem of estimating the parameters of chirp signals with randomly time-varying amplitude. Two methods for solving this problem are presented. First, a nonlinear least-squares approach (NLS) is proposed. It is shown that by minimizing the NLS criterion with respect to all samples of the time-varying amplitude, the problem reduces to a two-dimensional maximization problem. A theoretical analysis of the NLS estimator is presented and an expression for its asymptotic variance is derived. It is shown that the NLS estimator has a variance very close to the Cramer-Rao Bound. The second approach combines the principles behind the High-Order Ambiguity Function (HAF) and the NLS approach. It provides a computationally simpler but suboptimum estimator. A statistical analysis of this estimator is also carried out. Numerical examples attest to the validity of the theoretical analysis and establish a comparison between the two proposed methods.

SPTM-12.6

On Subspace Based Sinusoidal Frequency Estimation
Martin Kristensson, Magnus Jansson, Bj�rn Ottersten (Royal Institute of Technology, Sweden)

Subspace based methods for frequency estimation rely on a low-rank system model that is obtained by collecting the observed scalar valued data samples into vectors. Estimators such as MUSIC and ESPRIT have for some time been applied to this vector model. Also, a statistically attractive Markov-like procedure [1] for this class of methods has been proposed in the literature. Herein, the Markov estimator is re-investigated. Several results regarding rank, performance, and structure are given in a compact manner. The results are used to establish the large sample equivalence of the Markov estimator and the Approximate Maximum Likelihood (AML) algorithm proposed by Stoica et. al..

SPTM-12.7

Least squares estimation of polynomial phase based on finite step stochastic tree search
Dawei Huang, Simon Sando, Lian Wen (CiSSaIM, Queensland University of Technology, Australia)

Estimating the parameters for a constant amplitude, polynomial-phase signal with additive Gaussian noise is considered. The difficulty of this problem is that there are many unobserved integers when a linear regression model is used for wrapped phases [1]. Analyzing the least squares target function based on the regression model, we use the differencing approach [3] to simplify it. Thus, a tree-search algorithm can be used to find the solution of the least square problem. To reduce the computational complexity, statistical inference methods are applied. Then an attractable recursive algorithm is derived. Simulation results show that this algorithm works at a lower SNR than that for existing methods.

SPTM-12.8

Multiple Frequency Estimation in Additive and Multiplicative Colored Noises
Martial Coulon, Jean-Yves Tourneret (INPT / ENSEEIHT)

This paper addresses the problem of estimating sinusoidal frequencies in additive and multiplicative colored noises. Specific Yule-Walker equations yield second-order statistic-based estimates. The frequency estimates are shown to be asymptotically normally distributed. Their asymptotic covariance is derived.

SPTM-12.9

Cramer-Rao Bounds and Parameter Estimation for Random Amplitude Phase Modulated Signals
Mounir Ghogho, Asoke K Nandi (Strathclyde University, Dept of EEE, UK), Ananthram Swami (US Army Research Lab, USA)

The problem of estimating the phase parameters of a phase modulated signal in the presence of coloured multiplicative noise (random amplitude modulation) and additive white noise, both Gaussian, is addressed. Closed-form expressions for the exact and large-sample Cramer-Rao Bounds (CRB) are derived. It is shown that the CRB is not significantly affected by the colour of the modulating process, especially when the signal-to-noise ratio is high. Hence, maximum likelihood type estimators which ignore the noise colour and optimize a criterion with respect to only the phase parameters are proposed. These estimators are shown to be equivalent to the nonlinear least squares estimators which consist of matching the squared observations with a constant amplitude phase modulated signal when the mean of the multiplicative noise is forced to zero. Closed-form expressions are derived for the efficiency of these estimators, and are verified via simulations.

SPTM-12.10

Cramer-Rao Lower Bounds for Atomic Decomposition
Jeffrey C O'Neill (Boston University), Patrick Flandrin (Ecole Normale Superieure de Lyon)

In a previous paper we presented a method for atomic decomposition with chirped, Gabor functions based on maximum likelihood estimation. In this paper we present the Cramer-Rao lower bounds for estimating the seven chirp parameters, and the results of a simulation showing that our sub-optimal, but computaitionally tractable, estimators perform well in comparison to the bound at low signal-to-noise ratios. We also show that methods based on signal dictionaries will require much higher computations to perform well in low signal-to-noise ratios.

SPTM-12.11

Harmonic Retrieval in Nonstationary Noise
G. Tong Zhou, Yongsub Kim (Georgia Institute of Technology)

Harmonic retrieval is a classical signal processing problem but it has been almost invariably assumed that the additive noise is stationary. In this paper, we abandon this requirement and allow the additive noise to be nonstationary (but also non-cyclostationary in order to distinguish it from the information bearing signal). We show that various FFT based approaches can still be used on a single record of data to yield frequency estimates that have the $O(T^{-3})$ variance, where $T$ is the data length. Stationary multiplicative noise is also permitted in the model. Numerical examples illustrate the key concepts of the paper.

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Last Update: February 4, 1999 Ingo Höntsch