Optical linear algebra processors: noise and error-source modeling

Matrix-vector multiplication using digital partitioning for more accurate optical computing

Digital partitioning offers a flexible means of increasing the accuracy of an optical matrix-vector processor. This algorithm can be implemented with the same architecture required for a purely analog processor, which gives optical matrix-vector processors the ability to perform high-accuracy calculations at speeds comparable with or greater than electronic computers as well as the ability to perform analog operations at a much greater speed. Digital partitioning is compared with digital multiplication by analog convolution, residue number systems, and redundant number representation in terms of the size and the speed required for an equivalent throughput as well as in terms of the hardware requirements. Digital partitioning and digital multiplication by analog convolution are found to be the most efficient algorithms if coding time and hardware are considered, and the architecture for digital partitioning permits the use of analog computations to provide the greatest throughput for a single processor. To our knowledge this is the first study to propose the use of digital partitioning for optical matrix processing.

Design and performance analysis of fiber-optic infinite-impulse response filters

Design of direct forms of infinite-impulse response digital filters using radiatively tapped fiber-optic delay lines is discussed. Infinite-impulse response filters are attractive because they can provide better magnitude characteristics and sharp transition bands in their frequency response for a smaller number of taps than finite-impulse response filters can. The designs described here use spatial-light modulators and thus can be incorporated into optical adaptive filters for processing light-wave signals. Models of error sources in the fiber-optical realization and their effects on both time- and frequency-response characteristics are explained. Cascading of lower-order subsystems to obtain higher-order filters is discussed. A possible way of reducing the insertion loss using erbium-doped fiber-optic amplifiers is derived.

Design and performance optimization of fiber optic adaptive filters

There is a great need for easy-to-fabricate and versatile fiber optic signal processing systems in which optical fibers are used for the delay and storage of wideband guided lightwave signals. We describe the design of the least-mean-square algorithm-based fiber optic adaptive filters for processing guided lightwave signals in real time. Fiber optic adaptive filters can learn to change their parameters or to process a set of characteristics of the input signal. In our realization we employ as few electronic devices as possible and use optical computation to utilize the advantages of optics in the processing speed, parallelism, and interconnection. Many schemes for optical adaptive filtering of electronic signals are available in the literature. The new optical adaptive filters described in this paper are for optical processing of guided lightwave signals, not electronic signals. We analyzed the convergence or learning characteristics of the adaptive filtering process as a function of the filter parameters and the fiber optic hardware errors. From this analysis we found that the effects of the optical round-off errors and noise can be reduced, and the learning speed can be comparatively increased in our design through an optimal selection of the filter parameters. A general knowledge of the fiber optic hardware, the statistics of the lightwave signal, and the desired goal of the adaptive processing are enough for this optimum selection of the parameters. Detailed computer simulations validate the theoretical results of performance optimization.

Acoustooptic linear heterodyned complex-valued matrix-vector processor

A new analog (linear) optical vector-inner-product and matrix-vector processor architecture is described. It employs linear input laser-diode arrays and a multichannel acoustooptic cell. Attention is given to the selection of the operating modes for all devices, error correction, the ability of the system to operate on complex-valued data, and its stability with respect to temperature, data history, and component variations. Laboratory data are provided verifying the potential of the system architecture to achieve 9-10-bit accuracy and complex-valued multiplication.

Effects of error sources on the parallelism of an optical matrix-vector processor

The error sources in a high accuracy optical matrix-vector processor are analyzed by numerical simulation in terms of their effects on the parallelism and speed of the processor. These effects are detailed for radices -2, -4 and -8. Radix -4 is shown to provide maximum parallel processing capabilities under the effects of the system's error sources. Processing speed is shown to be a function of matrix partitioning and the number of parallel processing channels. Consequently, radix -4 operation provides a higher processing speed than radix -2 and -8 for most matrix-vector multiplications when error source effects are considered.

Noise limitations in optical linear algebra processors

A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.

Accuracy requirements of optical linear algebra processors in adaptive optics imaging systems

A ground-based adaptive optics imaging telescope system attempts to improve image quality by measuring and correcting for atmospherically induced wavefront aberrations. The necessary control computations during each cycle will take a finite amount of time, which adds to the residual error variance since the atmosphere continues to change during that time. Thus an optical processor may be well-suited for this task. This paper investigates this possibility by studying the accuracy requirements in a general optical processor that will make it competitive with, or superior to, a conventional digital computer for adaptive optics use.

Realization of preconditioned Lanczos and conjugate gradient algorithms on optical linear algebra processors

Lanczos and conjugate gradient algorithms are important in computational linear algebra. In this paper, a parallel pipelined realization of these algorithms on a ring of optical linear algebra processors is described. The flow of data is designed to minimize the idle times of the optical multiprocessor and the redundancy of computations. The effects of optical round-off errors on the solutions obtained by the optical Lanczos and conjugate gradient algorithms are analyzed, and it is shown that optical preconditioning can improve the accuracy of these algorithms substantially. Algorithms for optical preconditioning and results of numerical experiments on solving linear systems of equations arising from partial differential equations are discussed. Since the Lanczos algorithm is used mostly with sparse matrices, a folded storage scheme to represent sparse matrices on spatial light modulators is also described.

Reduced sensitivity algorithm for optical processors using constraints and ridge regression

Optical linear algebra processors that involve solutions of linear algebraic equations have significant potential in adaptive and inference machines. We present an algorithm that includes constraints on the accuracy of the processor and improves the accuracy of the results obtained from such analog processors. The constraint algorithm matches the problem to the accuracy of the processor. Calculation of the adaptive weights in a phased array radar is used as a case study. Simulation results prove the benefits advertised. The desensitization of the calculated weights to computational errors in the processor is quantified. Ridge regression isused to determine the parameter needed in the algorithm.

Matrix preconditioning: a robust operation for optical linear algebra processors

Analog electrooptical processors are best suited for applications demanding high computational throughput with tolerance for inaccuracies. Matrix preconditioning is one such application. Matrix preconditioning is a preprocessing step for reducing the condition number of a matrix and is used extensively with gradient algorithms for increasing the rate of convergence and improving the accuracy of the solution. In this paper, we describe a simple parallel algorithm for matrix preconditioning, which can be implemented efficiently on a pipelined optical linear algebra processor. From the results of our numerical experiments we show that the efficacy of the preconditioning algorithm is affected very little by the errors of the optical system.