Aloysius A. Beex

Sensitivity analysis of digital filter structures

A reasonable coefficient sensitivity measure for state space, recursive, finite wordlength, digital filters is the sum of the

A time-varying Prony method for instantaneous frequency estimation at low SNR

L_2

An informational approach to the convergence of output error adaptive IIR filter structures

norm of all first-order partial derivatives of the system function with respect to the system parameters. This measure is actually a linear lower bound approximation to the output quantization noise power. An important feature of this measure is that it can be broken down into evaluations of ARMA auto- and cross-covariance sequences, both of which can be computed efficiently and in closed form. This efficient closed form computation is a big improvement over the computational methods used by previous researchers. Their limited methods produced only approximations to the sensitivity measure and wasted computer time (i.e., these methods are open form solutions). The direct form II sensitivity, which is shown to be approx-imately inversely proportional to the sum of products of system pole and zero distances, can, as a result, usually be reduced by the judicious placement of ad...

The effect of identifier structure on parameter convergence

Most available instantaneous frequency (IF) estimation methods for frequency modulated signals in white noise deteriorate dramatically when SNR falls below some threshold. We present a time-varying Prony method for IF estimation from low SNR data. It is an extension of a frequency estimation method for stationary processes which was shown to be less complicated than, yet close in performance to, the best available approaches based on singular value decomposition or the principle of maximum likelihood. First, the time-varying autoregressive order is set higher than that needed for pure signal components so that the extra poles capture part of the noise. Then we choose signal poles based on a subset selection procedure. The performance improvement at low SNR over the TVAR method without subset selection is evidenced through simulation experiments.

Convergence analysis results for the class of affine projection algorithms

The convergence of previously described output error identification procedures is examined. The convergence analysis uses the eigeninformation of the parameter correlation matrix (really its inverse, the Fisher information matrix) for the identified structure. Convergence rates are important in digital adaptive equalizer design, for example. The eigeninformation of the parameter information matrix related the system sensitivity and numerical conditioning in a manner which provides insight into the identification process. An interesting sideline of this work is that balanced-coordinate state-space structures appear to consistently have good information properties. The relevant eigeninformation is combined in a proposed scalar convergence time constant. An important result is that identification of the usually identified direct form II parameters (the standard ARMA parameters) does not necessarily yield the fastest parameter convergence for the system being identified.<<ETX>>

Restricted geometry acoustic arrays for highly directional patterns

The effects of a chosen system structure on the identification of its parameters are considered. In particular, convergence rates are compared for the parameters of three state-space structures: direct form II, parallel, and dual generalized Hessenberg representation structures. It is shown that the chosen structure does influence identification algorithms, and that this influence is measured by examining the information contained in the structural parameters. A conjecture about the relative convergence of the parameters is offered, and evidence in its support is provided. An important result is that identification of the usually identified direct from II parameters (the standard ARMA parameters) does not necessarily yield the fastest parameter convergence for the system being identified. >

Model structure incorporated into recursive partial realization strategies

Over the last decade, a class of equivalent algorithms called the affine projection class of algorithms, which accelerate the convergence of the normalized LMS (NLMS) algorithm, has been discovered independently. The APA algorithms update weight estimates on the basis of multiple input signal vectors. In this paper, we present the results of the convergence analysis of the APA class of algorithms using a simple model for the input signal vectors. Conditions for convergence of the algorithms are presented. The convergence rate of APA is exponential, and it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms exhibit less misadjustment (steady state error) than NLMS. Simulation results are provided to corroborate the analytical results.

Restricted geometry, adaptive acoustic arrays for directional interference nulling

Abstract We consider the design and performance of a restricted geometry, 2- or 4- element (termed ‘small’) acoustic array as a beamformer. These arrays are designed to be mobile, for individuals who need to communicate in noisy environments. An acoustic array is made as directional as possible (thus ‘forming the beam’), so as to reduce (or eliminate) any unwanted narrowband acoustic signals which impinge on the array from undesirable directions. The directionality of multi-element arrays, and thus the capability for interference rejection, is greatly superior to that possible with a single-element array. Enhanced directionality comes from the extra knowledge gained when acoustic signals are spatially sampled. From intuition, we expect such a result since humans have 2 ears to hear with, and we do not have the ‘extra’ one merely for redundancy. That we know more about the incoming signal comes at the cost of a higher complexity in handling the signal itself. This added expense includes not only the cost of the added sensors, but also the cost of the interconnections required by the multi-element array. We introduce some measures of directionality, which we use to show that 4-element acoustic arrays are much more directional than 2-element arrays. A sensitivity analysis shows that for human speech, the narrowband beamformer can be implemented for speech-band use with only a slight decrease in the array directionality.

Model order, convergence rates and information content in noisy partial realizations

Abstract We present strategies for choosing from various canonical identifier structures in recursive partial realization algorithms for identification of state-space models. The different model structures considered in this paper have different estimation convergence rates. The relative convergence rates of the proposed deterministic algorithms are conjectured to be consistent with our (proposed) measure of the chosen model structure parameter information content. This measure, which we call the relative structural-algorithm time constant, is determined from the Fisher information matrix. This description of relative convergence rates apparently is independent of model order, so that either over- or under-determined model parameter convergence rates can be characterized by our measure of parameter information. In the proposed stochastic identifiers, the model parameter sensitivity itself is shown to have a significant effect on the initial convergence of the model parameter estimates, while final convergence rates are again strongly connected to the measure of parameter information. Some results on the minimization of the identification criterion are examined, especially those dealing with model structure influence on location and occurrence of the criterion minima. We present a strategy for increasing model convergence rates and estimation accuracy using our relative structural-algorithm time constant. Finally, we compare our method with a recently described iterative method.

Fast generalized affine projection algorithm

Abstract We examine the design and performance of a restricted geometry, adaptive 2-element (termed “small”) acoustic array used for directional hearing enhancement. These arrays are designed to be portable and lightweight. They may be used in noisy environments to block out unwanted sounds. The directionality of multi-element arrays, and thus the capability for interference rejection, is greatly superior to that possible with a 1-element device. Enhanced directionality comes from information gained when acoustic signals are spatially sampled. Human ears are designed in a similar arrangement, and we do not have the “extra” one merely for redundancy. This information comes at the expense of increased requirements for hardware, real-time computing and communications. We explore the balance between directional improvement and hardware requirements. Making the array adaptive is shown to enhance the array output SNR above that achievable by a fixed array.