The use of the twos complement binary representation in performing matrix-matrix multiplication using an optical engagement array architecture is presented. Twos complement arithmetic offers a convenient means for handling bipolar numbers, avoids the need for matrix partitioning when the matrices are real, and offers a means to improve accuracy over conventional optical analog techniques.
Until recently, all optical processing architectures described for performing either matrix-vector or matrix-matrix multiplication have been limited to accuracies typically in the 8-10-bit range. This is because matrix and vector element information has been encoded using conventional analog representations.
However, attempts are presently being made to increase the accuracy of optical processors by using either residue or binary arithmetic representations. A noteworthy example is an approach' 2 based on the mergence of performing matrix-matrix multiplication using outer products 3 and a technique for multiplying two numbers in binary form via convolution. The outer product between two vectors can be performed using optical techniques by crossing two linear-array light modulators as described in detail in Ref. 4. The binary multiplication via convolution technique was first suggested by Whitehouse and Speiser 5 and later described by Psaltis et al. from an optical implementation point of view. 6 This convolutional technique is novel in that binary numbers may be added without carries if the output is allowed to be represented in a mixed binary format. In the mixed binary format, like binary arithmetic, each digit is weighted by a power of 2, but unlike binary arithmetic the value of each digit can be >2. It is the elimination of the need for carries that makes this technique particularly attractive in terms of optical implementation. Discussions concerning this technique have been limited primarily to matrices with real-positive elements only. One exception, described