Vijay K. Jain

High-Accuracy Analog Measurements via Interpolated FFT

By use of an interpolated fast-Fourier-transform (FFT) algorithms are developed for multiparameter measurements upon periodic signals. Eight pertinent measurements, such as fundamental frequency, phase, and amplitude, are made with enhanced accuracy compared to existing algorithms, including tapered-window-FFT algorithms. For the more general case of nonharmonic multitone signals also the method is shown to yield exact amplitudes and phases if the tone frequencies are known beforehand. These measurements are useful in a variety of applications ranging from analog testing of printed-circuit boards to measurement of Doppler signals in radar detection.

Quadrature mirror filter design in the time domain

A new technique for designing quadrature mirror filters is described. The formulation, carried out in the time domain, is shown to result in an optimization problem requiring minimization of a quartic multinomial. An iterative solution is suggested which involves (computation of) the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps. Our experiments show that convergence to the optimum tap weights is stable, and the accuracy of the final solution is limited only by the accuracy of the eigenvalue-eigenvector routine. As in an earlier technique, the user can specify the stop: band frequency the relative weights of the passband ripple energy and the stopband residual energy, and, of course, the number of filter taps.

Some mathematical considerations in dealing with the inverse problem

Many problems of mathematical physics can be formulated in terms of the operator equation Ax = y , where A is an integro-differential operator. Given A and x , the solution for y is usually straightforward. However, the inverse problem which consists of the solution for x when given A and y is much more difficult. The following questions relative to the inverse problem are explored. 1) Does specification of the operator A determine the set \{y\} for which a solution x is possible? 2) Does the inverse problem always have a unique solution? 3) Do small perturbations of the forcing function y always result in small perturbations of the solution? 4) What are some of the considerations that enter into the choice of a solution technique for a specific problem? The concept of an ill-posed problem versus that of a well-posed problem is discussed. Specifically, the manner by which an ill-posed problem may be regularized to a well-posed problem is presented. The concepts are illustrated by several examples.

A radix-8 wafer scale FFT processor

Wafer Scale Integration promises radical improvements in the performance of digital signal processing systems. This paper describes the design of a radix-8 systolic (pipeline) fast Fourier transform processor for implementation with wafer scale integration. By the use of the radix-8 FFT butterfly wafer that is currently under development, continuous data rates of 160 MSPS are anticipated for FFTs of up to 4096 points with 16-bit fixed point data.

Suboptimal approximation/identification of transient waveforms from electromagnetic systems by pencil-of-function method

A noniterative method for approximating signals by a linear combination of exponentials is presented. Although the technique results in a suboptimal approximation, the continuous dependence of the suboptimal exponents \sim{s}_{i} on the integral square error \epsilon is such that lim (\epsilon = 0) \sim{s}_{i} \rightarrow {s}_{i} , the best least squares exponents. The method is also useful for system identification, where the system is modeled by a black box and one has access only to the input and output terminals. A technique is demonstrated for finding the multiple poles of a system along with the residues at the poles when the system output to a known input is given. Advantages of the method are natural insensitivity to noise in the data and a capability for approximately determining signal order. Representative computations are made of the poles from the transient response of a conducting pipe tested at the ATHAMAS-I EMP simulator.

Speech parameter estimation by time-weighted-error Kalman filtering

In this paper, we discuss a method of improving the parameter-tracking performance of the Kalman filter for modeling time-varying signals. The Kalman filter is an effective means of recursively estimating the coefficients of an AR (or ARMA) model; however, its effectiveness is diminished by the weight which the filter gives to the history of the signal. With a view toward improved modeling of speech signals, we examine the use of a time-weighted error criterion to remedy this situation.

High-speed double precision computation of nonlinear functions

High-speed coprocessors for computing nonlinear functions are important for advanced scientific computing as well as real-time image processing. In this paper we develop an efficient interpolative approach to such coprocessors. Performed on suitable subintervals of the range of interest, our interpolation which uses third degree polynomial is adequate for many elementary functions of interest with double precision mantissas. Our method requires only one major multiplication, two minor multiplications and a few additions. The minor multiplications are for the second and third degree terms, and their significant bits are much fewer than those of the first degree term. This leads to a very fast and efficient VLSI architecture for such coprocessors. It appears that polynomial based interpolation can yield considerable benefits over previously used approaches, when execution time and silicon area are considered. Further, it is possible to combine the computation of multiple functions on a single chip, with most of the resources on the chip shared for several functions.<<ETX>>

Impulse response determination in the time domain-Theory

Two methods are presented for determining the impulse response of an object in the time domain when both the input and output time domain waveforms are specified. Pole extraction features can then be applied to the nonimpulsive portion of the impulse response to determine the singularity expansion method (SEM) parameters of the object. The first method involves synthetic division and is comparatively straightforward. The second technique is a least squares method which is computationally more stable. The effect of measurement errors in the input and output waveforms is evaluated for each method. An investigation is made as to what form of the input minimizes the noise variances in the computation. Finally a generalized least squares technique is presented, which yields a minimum variance unbiased estimate for the impulse response when the noise covariance matrix is known.

Neural network with maximum entropy constraint for nuclear medicine image restoration

A neural-network-based algorithm is proposed for the restoration of nuclear medicine images as required for antibody therapy. The method was designed to address the particular problem of restoration of planar and tomographic bremsstrahlung data acquired with a gamma camera. Restoration was achieved by minimizing the energy function of the Hopfield network using a maximum entropy constraint. The performance of the proposed algorithm was tested on simulated data and planar gamma camera images of pure p-emitting radionuclides used in radioimmunotherapy. The results were compared with those of previously reported restoration techniques based on neural networks or traditional filters. Qualitative and quantitative analysis of the data suggested that the neural network with the maximum entropy constraint has good overall restoration performance; it is stable and robust even in cases where the signal-to-noise ratio is poor and scattering effects are significant. This behavior is particularly important in imaging therapeutic doses of pure β emitters such as yttrium-90 in order to provide accurate in vivo estimates of the radiation dose to the target and/or the critical organs.

Square-root, reciprocal, sine/cosine, arctangent cell for signal and image processing

This paper discusses an efficient interpolation method for nonlinear function generation. Based on this, a 24 bit VLSI cell, capable of computing the (1) square root, (2) reciprocal, (3) sine/cosine, and (4) arctangent functions, is presented for single precision floating-point applications. A 53 bit version, suitable for double precision computations, is also presented. Finally, an extension of the method to the two dimensional nonlinear functions is briefly addressed.<<ETX>>