Page 516 - Introduction to Information Optics

9.4. Parallel Signed-Digit Arithmetic 50 i

The proposed signed-digit algorithms can be operated on all operand digits
in parallel. This parallelism allows the aforementioned algorithms to be
efficiently mapped on the parallel cellular optical computing architecture
[17]. Consequently, these algorithms have been integrated in the parallel
architectures via symbolic substitution, logic array processing, and matrix
matrix multiplication, all of which are powerful techniques for two-dimensional
array data processing. For optoelectronic implementation, truth-table-based
content-addressable memory (CAM), shared CAM, coherent and incoherent
optical correlators, and nonlinear logic gates have been employed. In these
systems, the signed digits can be encoded by spatial position, intensity, and
polarization states.
Clever utilization of parallel optical cellular array architectures may lead to
an even more powerful optoelectronic computing system. Spatial-light modu-
lators (SLMs) with high processing speed, high resolution, and high contrast
are being developed for processing the input, intermediate operands, and
output operands. In the near future, the ultrahigh processing speed, parallelism,
massive data rates from optical memory, and superior processing capabilities
of optoelectronic technology can be fused to produce the next generation of
optoelectronic computers.

9.4. PARALLEL SIGNED-DIGIT ARITHMETIC

9.4.1. GENERALIZED SIGNED-DIGIT NUMBER SYSTEMS

The aforementioned MSD, TSD, and QSD representations are all subsets
of the previously defined signed-digit number system where the radix r can be
greater than 2. With the introduction of the NSD number system, it is
necessary to define a generalized signed-digit (GSD) number representation in
which the radix r can be either positive or negative. In the GSD system, a
decimal number X can be defined as

X= x ; r',^e{-a,..., -1,0, l,...,a), (9.23)

where a is a positive integer. In this system, the number of elements of the digit
set is usually greater than |r| resulting in redundant representation; that is, there
is more than one representation for each number. The redundancy also
depends on the selection of a and the minimum and maximum values of a are