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C4I Publication Abstracts
Signal Processing

C3I-5001

Improved Ziv-Zakai Lower Bound for
Vector Parameter Estimation

Authors: K. L. Bell, Y. Steinberg, Y. Ephraim, and H. L. Van Trees

Ziv-Zakai bounds on the mean square error (MSE) in parameter estimation are some of the tightest available bounds. These bounds relate the MSE in the estimation problem to the probability of error in a binary hypothesis testing problem. The original Bayesian version derived by Ziv and Zakai, and improvements by Chazan-Zakai-Ziv and Bellini-Tartara, are applicable to scalar random variables with uniform prior distributions. This bound was recently extended by Bell-Ephraim-Steinberg-Van Trees to vectors of random variables with arbitrary prior distributions. The goal of this paper is to present an improvement to the new vector bound, explore some properties of the bound, and present further generalizations.

Proceedings of 1994 IEEE-IMS Workshop on Information Theory and Statistics, Alexandria, VA, October 1994.

C3I-5002

Improved Bellini-Tartara Lower Bound for
Parameter Estimation

Authors: K. L. Bell, Y. Ephraim, Y. Steinberg, and H. L. Van Trees

The Chazan-Zakai-Ziv or Bellini-Tartara lower bound on the mean square error in parameter estimation is one of the tightest available bounds. It is a Bayesian bound applicable to estimation of scalar random variables with uniform prior distributions.The goal of this paper is to extend the Bellini-Tartara bound to vectors of random variables with arbitrary prior distributions and to establish a simple proof for the bound.

Proceedings of 1994 International Symposium on Information Theory, Trondheim, Norway, June 1994.

C3I-5003

Ziv-Zakai Lower Bounds for Vector Parameter Estimation

Authors: K. L. Bell, Y. Steinberg, Y. Ephraim, and H. L. Van Trees

Lower bounds on the minimum mean square error (MSE) in estimating a set of random parameters from noisy observations are studied. In particular, the Bayesian Ziv-Zakai bound and its improvements by Chazan-Zakai-Ziv and Bellini-Tartara are considered. These bounds were originally derived for a scalar random variable with uniform prior distribution. In this paper, the bounds are extended for vectors of parameters with arbitrary prior distributions. Similar to the original bound, the vector bound relates the MSE in the estimation problem to the probability of error in a binary hypothesis testing problem. Further extensions include a tighter bound which uses the probability of error in an M-ary hypothesis testing problem, a hybrid bound for vector parameters in which some of the components are random and some are deterministic, and bounds for a large class of distortion measures other than mean square error. The new bounds are compared with existing bounds in some examples, and are shown to be the tightest available bounds in the threshold and asymptotic regions.

Submitted to IEEE Transactions on Information Theory.

C3I-5004

Ziv-Zakai Lower Bounds in Bearing Estimation

Authors: K. L. Bell, Y. Ephraim, and H. L. Van Trees

Bounds on the mean square error in estimating the bearing of a planewave signal is of considerable interest in many fields. Of particular importance is the ability of a bound to closely characterize performance in the small error or asymptotic region, and the large error or ambiguity region, and to accurately predict the location of the threshold between the regions. In this paper, the vector Ziv-Zakai bound is applied to the problem of estimating two-dimensional bearing with planar arrays of arbitrary geometry. The bound is calculated for square and circular arrays, and compared with the Weiss- Weinstein bound. The Ziv-Zakai bound is shown to be tighter than the Weiss-Weinstein bound in the threshold and asymptotic regions.

Proceedings of 1995 IEEE International Conference on Acoustics, Speech, and Signal Processing, Detroit, MI, May 1995.

C3I-5005

Enhancement of Noisy Speech for
the Hearing Impaired Using the Signal Subspace Approach

Authors: Yariv Ephraim, Harry L. Van Trees, and Sigfrid D. Soli (House Ear Institute)

One of the greatest challenges in developing modern hearing aids is the suppression of undesired signals and noise to improve quality and intelligibility of desired signals. Our research applies the newly developed signal subspace approach for noisy speech enhancement to hearing aid signal processing. This single-microphone approach capitalizes on the fact that speech signals occupy only a subspace of the Euclidean space of the noisy signals. Hence, the space of the noisy signal is first decomposed into a noise subspace and a signal plus noise subspace. Then, the noise subspace is removed and the speech signal is estimated from the remaining signal subspace. This decomposition can be performed by applying the Karhunen-Loeve transform (KLT) to the noisy signal, and is approximated with the discrete Fourier transform (DFT). The signal is estimated using a perceptually meaningful criterion which aims at masking the residual noise by the speech signal.

Preliminary listening tests with normal-hearing listeners have been performed at the HEI. These tests have shown that the subspace approach produces improvements in estimated sound quality and intelligibility for normal hearing listeners. For example, an improvement of 15-20% in estimated intelligibility was obtained for both flat and speech spectrum-shaped noise at +5 dB initial S/N ratio. An improvement of only 10% was obtained when the spectral subtraction approach was used. In terms of quality improvement, listeners reported that processing noise was audible and distracting with spectral subtraction, but was almost inaudible with the signal subspace approach. We are currently testing the subspace approach with hearing impaired subjects. We will report the results of these intelligibility and sound quality tests for speech contaminated with spectrally flat noise and with speech spectrum-shaped noise at different S/N ratios.

C3I-5006

A Spectrally-Based Signal Subspace Approach for
Speech Enhancement

Authors: Yariv Ephraim and Harry L. Van Trees

The signal subspace approach for enhancing speech signals degraded by uncorrelated additive noise is studied. The underlying principle is to decompose the vector space of the noisy signal into a signal plus noise subspace and a noise subspace. Enhancement is performed by removing the noise subspace and estimating the clean signal from the remaining signal subspace. The decomposition can theoretically be performed by applying the Karhunen-Loeve transform (KLT) to the noisy signal. Linear estimation of the clean signal is performed using a perceptually meaningful estimation criterion. The estimator is designed by minimizing signal distortion for a fixed desired spectrum of the residual noise. This criterion enables masking of the residual noise by the speech signal. The filter is implemented as a gain function which modifies the KLT components corresponding to the signal subspace. The gain function is solely dependent on the desired spectrum of the residual noise. Listening tests indicate that 14 out of 16 listeners strongly preferred the proposed approach over the spectral subtraction approach.

Presented at the IEEE Int. Conf. on Acoust., Speech, and Signal Processing, Detroit, May 1995.

C3I-5007

Recurrent Neural Network Training
with Feedforward Complexity

Author: Oluseyi Olurotimi

This paper presents a training method that is of no more than feedforward complexity for fully recurrent networks. The method is not approximate, but rather depends on an exact transformation that reveals an embedded feedforward structure in every recurrent network. It turns out that given any unambiguous training data set, such as samples of the state variables and their derivatives, we need only to train this embedded feedforward structure. The necessary recurrent network parameters are then obtained by an inverse transformation that consists only of linear operators. As an example of modeling a representative nonlinear dynamical system, the method is applied to learn Bessel's differential equation, thereby generating Bessel functions within, as well as outside the training set.

IEEE Transactions on Neural Networks, Vol. 5, No. 2, March 1994.