David C. Farden

Thinning the impulse response of FIR digital filters

The design and performance of finite duration impulse response (FIR) digital filters with nonuniform tap spacings is examined. It is shown that a reduction in filter order can be achieved with a relatively small sacrifice in performance by thinning a high order uniformly spaced impulse response. The process of optimally selecting the tap spacings to minimize the statistical mean-squared estimation error is discussed and alternative procedures for implementation of the selection mechanism are mentioned. The results apply to more general estimation problems where control can be exercised over the selection of data to be incorporated into the estimator structure.

A direct approach to time-varying modelling

We consider the on-line identification of continuously adaptive autoregressive (AR) models for observed data records which are realizations of non-stationary stochastic processes. Emphasis is placed on the treatment of arbitrary non-stationarities and the use of realistic assumptions in this operation. Because of these two objectives, usual adaptation procedures or description techniques are not well suited to the problem, which motivates the investigation of a new (direct) approach to time-varying parameter estimation. The cost criterion considered is a constrained least squares cost functional which incorporates with equal weight all instantaneous errors up to the current time of observation. The constraint is specified from limited a priori knowledge about the nature of the non-stationarity, namely the expected maximum rate of change (MRC) of the model parameters.

Statistical design of cascade finite wordlength FIR digital filters

This paper addresses the problem of efficiently designing cascade-form FIR digital filters, particularly relevant to applications that require very short wordlength coefficients. A technique based on the statistical mean-squared error criterion is presented to sequentially construct cascade-form filters one stage at a time, accounting for the effects of previous stages in the design specifications for each new stage. The statistical method has the unique ability to incorporate filter structure and parameter quantization into the approximation phase of the design process without requiring vast amounts of computer time characteristic of discrete optimization problems. Updating of design specifications is accomplished with FFTs and efficient Toeplitz matrix inversions, while coefficients are quantized by either simple coefficient rounding or, optionally, by very low-order local search methods in the latter design stages.

Time-varying modelling of arbitrary non-stationary signals

A novel approach to time-varying signal modeling is presented to reliably identify rational models in arbitrary nonstationary environments. It is motivated by the shortcomings of both adaptation methods, which cannot handle arbitrary nonstationarities, and description techniques, which tend to use unjustifiable assumptions on the observed data. Only limited a priori knowledge about the nonstationarity, namely, the expected maximum rate of change of the model parameters, is necessary to estimate these parameters online. The criterion considered is a constrained-least-squares cost functional which incorporates with equal weight all instantaneous errors up to the time of observation. An appropriate algorithm is developed and its performance is discussed to illustrate the increased confidence in time-varying model estimation which results from the approach.<<ETX>>

Continuously adaptive linear predictive coding of speech

A continuously adaptive approach to speech encoding is presented. In contrast with other adaptation methods, it provides reliable modeling of the transition between two phonemes, and, unlike the usual block-stationary techniques, it eliminates the need to detect these transitions. Continuously adaptive linear predictive coding takes into account the inherent nonstationarity of the speech signal by using the expected minimum rate of change of the model parameters as a constraint in the recursive estimation of these parameters. The criterion considered is a constrained-least-squares cost functional which incorporates with equal weight all instantaneous errors up to the time of observation. An appropriate algorithm is given, and simulations are presented to illustrate the basic cost-performance tradeoffs involved in the approach.<<ETX>>

A general linear prediction approach to the complex split inversion algorithm

A general linear prediction framework is developed for the derivation of the complex split Levinson algorithm, which leads naturally to a structural interpretation of the redundancy present in the classical Levinson-Durbin algorithm. A novel order-recursive predictor structure is presented. It is conceptually simpler than a lattice realization since it propagates only one (generalized) prediction error. When the usual prediction error is desired, it can be recovered through a simple terminal stage. >

A fast algorithm for the recursive design of linear phase filters

One appropriate technique for the recursive design of linear phase filters is via a minimum mean square error estimation procedure. This paper presents a different parameterization of the problem, which enforces the linear phase structure at all stages of the procedure. The computational complexity of the resulting filter design algorithm is approximately one-half that of existing algorithms, when measured as the overall number of multiplications required. This algorithm in turn leads to a new linear phase realization, simpler than a linear phase lattice/ladder structure and exhibiting better numerical properties than a direct form implementation. The improvement in finite wordlength effects is illustrated by the example design of a lowpass filter.

A microprogrammed signal processor

The implementation of a microprogrammable signal processor which utilizes a bit-sliced microprocessor and a nanoprogrammed convolution computer is described. The processor was designed primarily for the implementation of adaptive filtering algorithms, and is controlled by a microprogram which resides in a writable control store. This primary microcode controls the operations performed by a bit-sliced ALU, the access to the data bus, I/O, memory access, and the initiation of the next lower level of microcode called nanocode. The convolution computer is implemented using two VLSI multiplier-accumulators, has two parallel data paths to memory which are independent of the main data bus, and has its own addressing logic. This allows convolutions to be done at an effective multiply-accumulation time of 100 nanoseconds per weight while the bit-slice ALU is doing I/O or computing parameters for the filter. The adaption algorithm is also implemented in nanocode in the convolution computer.

Constrained Time-Varying System Modelling

Tools are presented to reliably identify a time-varying autoregressive (AR) model for a realization of a stochastic process with an arbitrary non-stationarity. Only limited a priori knowledge about the nature of the non-stationarity, namely the expected maximum rate of change of the model parameters, is necessary to estimate these parameters on-line. The criterion considered is a constrained least squares cost functional which incorporates with equal weight all instantaneous errors up to the time of observation. The constraint is specified from the maximum rate of change using a (non-unique) backward state-space description for the parameter variation. A doubly recursive algorithm based on smoothing theory is derived to find a quasi-optimal solution to the recursive parameter estimation problem. Associated trade-offs are discussed for various non-stationary environments.

Bounds on tracking errors for adaptive signal processing algorithms

New upper bounds on the norm of the error between a parameter vector obtained by an adaptive signal processing algorithm and the desired parameter vector are presented. The family of algorithms treated includes the Widrow-Hoff LMS algorithm. The results are applicable to heavily correlated training data satisfying very mild covariance decay-rate conditions. The main result is a new proof for the almost sure exponential convergence of matrix products which arise in the analysis of adaptive signal processing algorithms. This result includes an estimate for the convergence rate of such products.