Recently published results demonstrated that the Lévy Flight Firefly Algorithm (LFFA) can be effectively applied to IIR adaptive filter structures designed with parallel or cascade second order sections. Recently published results also demonstrated that the LFFA can be effectively used for IIR adaptive filters designed with second order coupled form sections. This paper now proposes that bio-inspired optimization algorithms can be applied to FIR and IIR adaptive lattice-ladder structures, which have been well known throughout the literature for many past decades. Experimental results are presented to demonstrate that the LFFA performs effectively on several different types of adaptive lattice-ladder structures.

In this paper the performance of the Genetic Algorithm (GA), the Particle Swarm Optimization (PSO) algorithm, the Modified Particle Swarm Optimization (MPSO) algorithm, and the Lévy Flight Firefly Algorithm (LFFA) are compared for system identification with various types of nonlinear systems. When performing system identification with Volterra nonlinear adaptive structures, matched-order fixed-nonlinearity LNL filter structures, reduced-order adaptable-nonlinearity LNL filter structures and neural networks the LFFA generally demonstrates faster convergence rates and lower minimum mean square errors (MMSE) compared to the GA, PSO, and MPSO algorithms. This work includes performance comparisons of these algorithms when applied to nonlinear system identification.

The mathematical theory of Residue Number System (RNS) arithmetic existed in the mathematical literature for thousands of years, having first been published by Sun Tzu in 100 A.D. In the mid 1900's RNS number theory began to evolve into many engineering applications as the evolution of digital computers began. In recent times RNS arithmetic has again emerged extensively in cryptography, cyber security, machine learning, fault tolerant signal processing, biomedical signal processing, etc. This paper reviews the history of how over many decades RNS arithmetic has emerged into rapidly developing digital signal processing applications to provide efficient DSP solutions.

Due to the fact that the two authors of this paper were both graduate students during the time when Dr. Sanjit Mitra was approximately ten years into his early academic career, both of the authors' technical careers evolved along trajectories similar to Dr. Mitra's, but with a time delay of approximately ten years. This scenario has enabled the authors to acquire a clear picture of Dr. Mitra's life-long career and they are pleased to have this opportunity to explain their interactive experiences with Dr. Mitra and present a broad historical overview of his professional activities.

Dictionary learning algorithms have been successfully used for both reconstructive and discriminative tasks, where an input signal is represented with a sparse linear combination of dictionary atoms. While these methods are mostly developed for single-modality scenarios, recent studies have demonstrated the advantages of feature-level fusion based on the joint sparse representation of the multimodal inputs. In this paper, we propose a multimodal task-driven dictionary learning algorithm under the joint sparsity constraint (prior) to enforce collaborations among multiple homogeneous/heterogeneous sources of information. In this task-driven formulation, the multimodal dictionaries are learned simultaneously with their corresponding classifiers. The resulting multimodal dictionaries can generate discriminative latent features (sparse codes) from the data that are optimized for a given task such as binary or multiclass classification. Moreover, we present an extension of the proposed formulation using a mixed joint and independent sparsity prior, which facilitates more flexible fusion of the modalities at feature level. The efficacy of the proposed algorithms for multimodal classification is illustrated on four different applications--multimodal face recognition, multi-view face recognition, multi-view action recognition, and multimodal biometric recognition. It is also shown that, compared with the counterpart reconstructive-based dictionary learning algorithms, the task-driven formulations are more computationally efficient in the sense that they can be equipped with more compact dictionaries and still achieve superior performance.

Dictionary learning algorithms have been successfully used in both reconstructive and discriminative tasks, where the input signal is represented by a linear combination of a few dictionary atoms. While these methods are usually developed under ℓ1sparsity constrain (prior) in the input domain, recent studies have demonstrated the advantages of sparse representation using structured sparsity priors in the kernel domain. In this paper, we propose a supervised dictionary learning algorithm in the kernel domain for hyperspectral image classification. In the proposed formulation, the dictionary and classifier are obtained jointly for optimal classification performance. The supervised formulation is task-driven and provides learned features from the hyperspectral data that are well suited for the classification task. Moreover, the proposed algorithm uses a joint (ℓ12) sparsity prior to enforce collaboration among the neighboring pixels. The simulation results illustrate the efficiency of the proposed dictionary learning algorithm.

Fetal electrocardiograms (ECGs) are useful for monitoring fetal health during pregnancy. Previously published results demonstrated that a sequential combination of adaptive linear prediction (LPC), adaptive noise cancellation (ANC), and IIR comb filtering (CF) is an effective strategy to remove maternal interference from fetal ECGs. This paper presents a theory of how LPC-ANC sequential processing is capable to extract fetal signals from noninvasive abdominal measurements. Results are presented to demonstrate that multi-channel adaptive processing is more effective than signal channel processing. Finally, results illustrate that sequential processing followed by single stage of comb filtering is more effective than direct comb filtering.

Recent studies have demonstrated advantages of information fusion based on sparsity models for multimodal classification. Among several sparsity models, tree-structured sparsity provides a flexible framework for extraction of cross-correlated information from different sources and for enforcing group sparsity at multiple granularities. However, the existing algorithm only solves an approximated version of the cost functional and the resulting solution is not necessarily sparse at group levels. This paper reformulates the tree-structured sparse model for multimodal classification task. An accelerated proximal algorithm is proposed to solve the optimization problem, which is an efficient tool for feature-level fusion among either homogeneous or heterogeneous sources of information. In addition, a (fuzzy-set-theoretic) possibilistic scheme is proposed to weight the available modalities, based on their respective reliability, in a joint optimization problem for finding the sparsity codes. This approach provides a general framework for quality-based fusion that offers added robustness to several sparsity-based multimodal classification algorithms. To demonstrate their efficacy, the proposed methods are evaluated on three different applications - multiview face recognition, multimodal face recognition, and target classification.

Based on the understanding of liberal arts educations of the past and a vivid awareness of advancing technology of the future it is now becoming clear that engineering education will become increasingly popular as it provides an alternate form of liberal arts education in the highly technical society of the future. This evolutionary trend does not mean that the old fashioned liberal arts will disappear, nor does it mean that engineering education will be the dominant liberal arts education in the future. But it does mean that engineering education will be superimposed on society and engineers will no longer be geeks living on islands within our society, but rather an engineering education will stimulate interactions among groups of frequently networking citizens who have opportunities to proceed along many professional paths in a technically sophisticated world.

Previously published results showed that when an FFT-based transform domain Fault Tolerant Adaptive Filter (FTAF) operates on real-valued signals, parameter redundancy inherent in the complex arithmetic provides fault tolerant capabilities. It was shown experimentally that the Modified Discrete Fourier Transform (MFFT) generates frequency domain samples that are not strictly real-valued or constrained to have conjugate symmetry in the transform domain. This paper provides a more detailed analysis of the fault tolerant capabilities of MFFT-FTAF architectures and further demonstrates how the MFFT-FTAF architecture is able to overcome certain fault conditions that cannot be properly handled in a conventional FFT-based FTAF architecture.

Previously it was shown that when an FFT-based transform domain Fault Tolerant Adaptive Filter (FTAF) operates on real-valued signals parameter redundancy inherent in the complex arithmetic provides fault tolerant capabilities for adaptive filters implemented in highly scaled nano-technologies that are prone to both soft errors and hardware faults. This work replaces the conventional DFT with the Modified Discrete Fourier Transform (MDFT) to provide frequency domain samples that are not constrained to be strictly real-valued. It is shown that the MDFT-FTAF architecture resulting from the MDFT provides considerable fault tolerant capabilities by means of the parameter redundancy in the complex arithmetic.

Recently the Quadratic Modified Fermat Number Transform (QMFNT) based on Left-angle and Right-angle Circular Convolution (LCC and RCC) was extended to define a new Modified Discrete Fourier Transform (MDFT) that relies on a similar combination of RCC and LCC. It was shown that the MDFT enables overlap-add FFT block processing to be implemented without zero padding, resulting in reduced computational complexity and potentially reduced power requirements for nanoscale VLSI implementations. This paper extends the MDFT into two dimensions and analyzes how the 2-D extension manages two-dimensional wrap-around effects while implementing 2-D overlap-add block processing without zero padding.

Previously it was shown that the Modified Fermat Number Transform (MFNT) reduces computational requirements for an adaptive filter architecture based on convolution achieved by FNT block processing (FNTBP). The work of this paper extends the MFNT to a Quadratic MFNT (QMFNT) by introducing Left-angle circular convolution and interpreting the result as a quadratic representation of the convolution output obtained by combining the Left-angle and Right-angle circular convolutions. The resulting computational complexity is similar to the MFNT but provides a significant reduction of the dynamic range constraints on the input sequences. This results in an adaptive filter architecture that has increased computational efficiency and potentially lower power requirements when used to realize nano-scale adaptive filters.

Recently the previously reported Modified Fermat Number Transform (MFNT) based on Right Circular Convolution (RCC) was extended to form a Quadratic MFNT (QMFNT) by introducing Left-angle Circular Convolution (LCC) and interpreting the combined result as a quadratic representation of the resulting convolution output. This paper introduces a new Modified Discrete Fourier Transform (MDFT) that relies on a similar combination of RCC and LCC. The MDFT enables overlap-add FFT block processing to be implemented without zero padding, resulting in reduced computational complexity and potentially reduced power requirements in nanoscale VLSI implementations.

Previous work explored the Walsh-Hadamard transform (WHT) as a computationally efficient way of achieving adaptive fault tolerance in transform domain adaptive filters (TDAF). Since the WHT requires only addition and subtraction it is compatible with residue number system (RNS) arithmetic that has powerful error detection and correction capabilities. This paper investigates the hybrid combination of RNS error detection/correction with WHT-based adaptive fault tolerance to achieving a high level of fault tolerant performance that requires a modest amount of hardware overhead.

Transform domain fault tolerant adaptive filters (FTAFs) rely on inherent learning capabilities of the adaptive process to compensate for transient (soft) or permanent (hard) errors in the hardware implementation. In this paper it is shown that an FFT-based transform domain FTAF operating on real valued signals can provide a considerable degree of fault tolerance without introducing redundant hardware. When the filter operates on real valued signals the added complexity of complex arithmetic provides additional fault tolerant capabilities that may be useful for adaptive filters implemented in highly scaled nano-technology circuits where soft and hard errors are of increasing concern.

This paper proposes fault tolerant signal processing strategies for achieving reliable performance in VLSI signal processors that are prone to transient errors due to increasingly smaller feature dimensions and supply voltages. The proposed methods are based on residue number system (RNS) coding, involving either hardware redundancy or multiple execution redundancy (MER) strategies designed to identify and overcome transient errors. RNS techniques provide powerful low-redundancy fault tolerance properties that must be introduced at VLSI design levels, whereas MER strategies generally require higher degrees of redundancy that can be introduced at software programming levels.

Recently particle swarm optimization (PSO) has been studied for use in adaptive filtering problems where the mean squared error (MSE) surface is ill-conditioned. Although the swarm generally converges to a limit point, when the population of the swarm is small the entire swarm often stagnates before reaching the global minimum on the MSE surface. This paper examines enhancements designed to improve the performance of the conventional PSO algorithm. It is shown that an enhanced PSO algorithm, called the Modified PSO (MPSO) algorithm, is quite effective in achieving global convergence for IIR and nonlinear adaptive filters

The paper investigates the application of a modified particle swarm optimization technique to nonparametric density estimation based independent component analysis (ICA). Nonparametric ICA has the advantage over traditional ICA techniques in that its performance is not dependent upon prior assumptions about the source distributions. Particle swarm optimization (PSO) is similar to the genetic algorithm in that it utilizes a population based search suitable for optimizing multimodal error surfaces where gradient-based algorithms tend to fail, such as those generated by nonlinear entropy maximization schemes used in ICA algorithms.

This paper introduces the application of particle swarm optimization techniques to infinite impulse response (IIR) adaptive filter structures. Particle swarm optimization (PSO) is similar to the genetic algorithm (GA) in that it performs a structured randomized search of an unknown parameter space by manipulating a population of parameter estimates to converge on a suitable solution. Unlike the genetic algorithm, particle swarm optimization has not emerged in adaptive filtering literature. Both techniques are independent of the adaptive filter structure and are capable of converging on the global solution for multimodal optimization problems, which makes them especially useful for optimizing IIR and nonlinear adaptive filters. This paper outlines PSO and provides a comparison to the GA for IIR filter structures.

This paper investigates the application of particle swarm optimization techniques to infinite impulse response (IIR) adaptive phase equalizers. Particle swarm optimization (PSO) is similar to the genetic algorithm (GA) in that it utilizes a population based search suitable for optimizing multimodal error surfaces where gradient-based algorithms tend to fail, such as those generated by IIR adaptive filters. This paper investigates PSO for the phase equalization of minimum phase surface acoustic wave (SAW) filters used in CDMA receivers.

It has been shown previously that adaptive filters can be designed with hardware redundancy so they can continue operating effectively after the occurrence of certain types of hardware failures, thereby achieving reliability through a mechanism known as adaptive fault tolerance. This paper examines the capabilities of adaptive fault tolerance in cases of single and multiple bit errors in the adaptive filter coefficients using floating-point number representation.

Reliable performance is very important for high speed channel equalizers and echo cancellers used in high speed communications channels. A common type of hardware fault occurs when the coefficients get "stuck" at an uncontrollable value. Such faults lead to larger overall mean square errors, and generally poor performance. Redundancy can provide the ability to compensate for these types of faults if the proper design is introduced into the adaptive filter structure. Unfortunately this form of redundancy can lead to poor convergence performance for the adaptive filter after the fault occurrence. This paper examines the use of affine projection and row projection techniques to improve the convergence performance of the fault tolerant adaptive filtering structure. Algorithms are developed for two cases: fault knowledge and no fault knowledge incorporated in the adaptive filtering update. These algorithms are introduced in this paper and simulations are presented to illustrate the effectiveness of these approaches.

This paper presents the design details of an experimental ASIC for an all-digital adaptive equalizer. In this design, the LMS algorithm is chosen because of its simplicity. The adaptive equalizer design, which is based on an RNS architecture, consists of an RNS multiplier, an RNS adder, an RNS filter, a binary-to-residue converter, a residue-to-binary converter, and an update algorithm. The design is verified by a high level hardware simulation tool. The designs of all these units are discussed in this paper.

The input autocorrelation matrix for a third order (cubic) Volterra adaptive filter for general colored Gaussian input processes is analyzed to determine how to best formulate a computationally efficient fast adaptive algorithm. When the input signal samples are ordered properly within the input data vector, the autocorrelation matrix of the cubic filter inherits a block diagonal structure, with some of the sub-blocks also having diagonal structure. A computationally efficient adaptive algorithm is presented that takes advantage of the sparsity and unique structure of the correlation matrix that results from this formulation.

Adaptive fault tolerance (AFT) has been developed for achieving reliable performance of FIR adaptive filters in the presence of certain types of hardware failures. Previous studies limited the use of AFT to one-dimensional FIR filter structures with simple "stuck-at" fault models that account for errors introduced into the adaptive process by faulty coefficients that cease to adjust properly. This paper considers a broader class of hardware errors which, in addition to "stuck-at" errors in the multiplier inputs, can be modeled by errors in the outputs of both multipliers and adders. By including a simple circuit that removes erroneous induced constants in the output of the adaptive filter, adaptive fault tolerance can be extended to a broader class of hardware failures.

The method of preconditioned conjugate gradients (PCG) is introduced as an accelerator for simple IIR algorithms to significantly increase their rate of convergence without dramatically adding to their complexity. This paper develops a block Newton-type adaptive IIR digital filter with complexity O(log(N)). The proposed algorithm exploits a structured approximation to the Hessian which permits the application of a fast preconditioned conjugate gradient optimization method. In this novel formulation, the identification problem of the IIR coefficients separates into two subproblems, each of which may be solved by application of fast adaptive FIR techniques. Present IIR algorithms require greater computational cost, or converge more slowly. It is the adoption of fast PCG which permits the development of an O(log(N)) adaptive algorithm. The PCG method manipulates an approximation of the Hessian matrix to form an orthogonalizing update term for the IIR LMS algorithm. Rapid convergence follows, and the method is robust with respect to fixed-point instability. The use of preconditioned conjugate gradients in the Gauss-Newton update leads naturally to the application of the planar least squares inverse to bound the poles of the adaptive system by projecting an unstable denominator onto a stable polynomial. This technique is invoked whenever the output of the adaptive filter exceeds a certain threshold. This approach provides a computationally efficient means to ensure robust IIR adaptive behavior.

Three constant modulus algorithms (CMA), the fast quasi-Newton CMA, the transform domain CMA, and the genetic search based CMA are proposed in this paper. The performances of these three algorithms are compared with each other via computer simulation. It is shown that the fast quasi-Newton CMA and the transform domain CMA achieve much faster convergence rate than the constant modulus algorithm based on the LMS algorithm. This fact shows that the whitening technique is not only useful but also necessary for the CMA.

This paper presents two new design techniques for adaptive equalizers based on a constant modulus error criterion: (1) the fast quasi-Newton CMA and (2) the conjugate gradient CMA. The algorithms for these new equalization techniques are summarized and their performance is demonstrated through computer simulation.

Adaptive fault tolerance has been applied to adaptive filters to provide fault tolerance at low hardware cost. However, these filter structures have suffered from poor post-fault convergence characteristics. Also, the computational burden required by these filters has not been properly addressed. In this paper, a new fault tolerant FIR adaptive filter based on the FFT is introduced. This filter structure has superior convergence behavior over previously introduced structures, and since it is based on the FFT, it is efficient to compute, as well as conducive to an additional degree of fault tolerance capability. Computer simulations are given to demonstrate the robust performance of the new filter.>

Several different two-dimensional adaptive filter structures and algorithms have been developed, the most recent of which is a 2D IIR adaptive filter. Two-dimensional IIR adaptive filters can be used in a variety of image and video processing applications. Of primary concern with IIR adaptive filters, whether 1D or 2D, is the possible existence of multiple stationary points on the mean-squared error surface. Gradient based adaptive algorithms can get stuck at suboptimal local minima, rendering themselves less effective. Results show that the 2D IIR mean-squared-error performance surface characteristics are similar to those of the 1D IIR adaptive filter with respect to unimodality of the error surface. The authors explore the mathematical characteristics of the 2D IIR MSE minimization problem. Some simple special cases are examined, and experimental evidence is discussed.>

The FFT-based TDFTAF has been shown to have desirable structural and convergence properties when being trained with a white noise input signal. However, for colored noise training signals, which frequently arise in adaptive filtering scenarios, some of the good convergence properties do not hold. In this paper we demonstrate the convergence behavior of the FFT-based TDFTAF with colored noise input signals and provide a computationally efficient method to address the poor convergence rate problem.>

Two-dimensional adaptive filters are useful in a wide variety of image and video processing applications. Several different two-dimensional structures and algorithms have been developed, the most recent of which is a fast 2-D FIR quasi-Newton adaptive filter. Quasi-Newton search algorithms can compensate for the eigenvalue disparity of the autocorrelation matrix in colored signal environments. Learning curves show dramatic improvement in the rate of convergence of 2D adaptive filters with colored noise.>

This paper proposes combining system-level modular redundancy with the arithmetic modularity of residue number system (RNS) arithmetic to achieve fault tolerance in high speed digital signal processing (DSP) systems. Double, triple, and quadruple modular redundancy are combined with RNS modularity for realizing important DSP computational kernels. The discussion includes the development of the serial-by-modulus (SBM) RNS architecture in which residue digits are processed sequentially in circuits that handle only one modular operation at a given time, thereby sacrificing speed for circuit simplicity. As a potential application of the SBM concept, a variable-word-length sum-of-products signal processing kernel is developed based on a serial-by-modulus RNS architecture. Because the RNS is not a weighted number representation, if the instantaneous dynamic range requirement can be estimated it may be possible to perform the computation with only enough residue digits to provide the necessary dynamic range.>

Convolution backprojection (CBP) image reconstruction has been proposed as a means of producing high-resolution synthetic-aperture radar (SAR) images by processing data directly in the polar recording format which is the conventional recording format for spotlight mode SAR. The CBP algorithm filters each projection as it is recorded and then backprojects the ensemble of filtered projections to create the final image in a pixel-by-pixel format. CBP reconstruction produces high-quality images by handling the recorded data directly in polar format. The CBP algorithm requires only 1-D interpolation along the filtered projections to determine the precise values that must be contributed to the backprojection summation from each projection. The algorithm is thus able to produce higher quality images by eliminating the inaccuracies of 2-D interpolation, as well as using all the data recorded in the spectral domain annular sector more effectively. The computational complexity of the CBP algorithm is O(N (3)).

The authors introduce a family of rapidly converging IIR (infinite impulse response) adaptive algorithms with O(N) computational complexity, where N is the filter order. By observing the similarity between the numerical solution of partial differential equations and the IIR adaptive filtering problem, results from the solution of systems of sparse linear equations may be employed. In this formulation the identification problem of the IIR coefficients separates into two subproblems, each of which may be solved by application of fast adaptive FIR (finite impulse response) techniques. Present IIR algorithms require greater computational cost or converge more slowly.>

The use of the chirp Z transform (CZT) is proposed to improve the rate of convergence of transform domain adaptive output error algorithms. The use of the CZT is novel and is shown to possess advantages over the DFT. Other attempts at incorporating prior knowledge of input spectra use frequency sampling structures and their generalizations. Such structures require pole-zero cancellation, which is inexact with finite word length and may lead to instability if a zero occurs on the unit circle. Use of the CZT eliminates these difficulties and provides a stable, rapidly converging solution.>

A 2-D adaptive filter structure based on the McClellan transformation design technique for 2-D FIR (finite impulse response) filters is proposed as a means of achieving improved performance. Performance is compared with that of a direct-form 2-D LMS (least mean squares) structure in terms of learning characteristics and computational efficiency. It is shown that if the transformation structure is constrained by a priori knowledge of contour shapes in the frequency domain, the 2-D adaptive algorithm greatly reduces computational requirements and improves the learning characteristics, as compared with the direct form. The transformation filter is then generalized by including the contour parameters in the adaptive parameter set to eliminate the constraints on the frequency domain contours. It is shown how an orthogonal transformation can be included to improve the convergence rate of the constrained transformation structure.>

An adaptive filtering algorithm is introduced which is largely immune to the deleterious effects of colored inputs, yet requires only O(N) computation. Simulation results are presented which demonstrate that the proposed algorithm's performance is comparable to RLS (recursive least squares), and that it is quite robust with respect to finite-wordlength implementation.>

Adaptive digital filters are currently used in many communication systems for echo cancellation, channel equalization, and adaptive noise cancellation. Most practical applications presently use adaptive finite impulse response (FIR) digital filters because they are well behaved in terms of convergence and stability properties. This paper presents an adaptive algorithm that operates with a parallel form infinite impulse response (IIR) filter structure for applications in which an FIR filter requires too much computation. The new algorithm is derived mathematically and the results of computer experiments are presented to demonstrate its performance.

Several well known imaging techniques operate by recording samples of the Fourier transform of the object function and then reconstructing the object function by means of the 2D inverse FFT. A central problem arises in interpolating from the inherent polar raster to a rectangular raster, so the inverse FFT can be properly applied. This paper considers the use of nearest neighbor and inverse distance interpolation when the angular recording interval is relatively small. The objective is to obtain a computationally simple reconstruction algorithm that achieves good resolution in the final image plane.

We compare the performance of three algorithms for polar-to-cartesian interpolation in spotlight mode synthetic aperture radar. The original polar data in this case lies far from the origin in a region of very small angular extent.

A design is presented for an experimental device which converts data from residue representation to mixed radix representation while simultaneously checking for single digit errors. The experimental system has a high speed pipelined architecture and operates with five 5-bit moduli, two of which are redundant. Experience gained through experimentation with this system led to the design of a universal building block that is proposed for monolithic fabrication.

More sensors are usually required in order to enhance the resolution of a beamformer and to improve its beam pattern. Due to physical and/or economical constraints this requirement cannot be met. This paper presents the results of an analytical and experimental study of the use of a signal extrapolation method in order to "synthetically" (without adding more physical sensors) enhance the beamformer resolution. A simple model for the extrapolation error has been obtained experimentally. The beam pattern has been extensively improved. Further improvements can be achieved by combining other techniques such as windowing, temporal interpolation, etc. to the spatial extrapolation.

During the last few years residue number (RNS) arithmetic has gained increasing importance for providing high speed fault tolerant performance in dedicated digital signal processors. One factor that has limited the use of redundant RNS theory in practice is the hardware complexity of the error checker. This paper presents a mathematical analysis of the error correction algorithm which suggests a new design with considerably reduced hardware complexity. A hardward architecture for a high speed pipelined error checker is proposed.

Multidimensional imaging of three dimensional (3-D) objects is often complicated by physical constraints imposed by the observation technique. An example of this problem occurs in 3-D imaging of specimens with electron-microscopes. In this case the range of observation angle (perspective) is typically less than π/2 radians due to the limited angle through which the specimen can be rotated relative to the electron beam. This paper presents three novel algorithms that significantly enhance the resolution of techniques limited by lack of perspective. The methods are based on a combination of Gerchberg's spectral error energy reduction algorithm (1) and tomographic image reconstruction techniques.

Residue number system product codes are proposed for the design of self-checking digital filters. It is shown that product codes require redundancy that is similar to, but slightly less than that of systematic codes for simultaneous overflow detection and single error correction. The design of a product code error checker is presented, and compared to a systematic code error checker. Also, a new concept called expanded mixed radix conversion is presented to simplify the error checker hardware design.

This paper presents two approaches to the design of residue-to-analog converters to be used with residue number digital filters. The first approach, based on the associated mixed radix number system, is effective when it is desired to translate the residue samples directly into analog form. The second approach, based on the Chinese Remainder Theorem, is a more direct method for translating from residue to weighted binary representation. A hardware experiment is described that was constructed to demonstrate the conversion principles.