High-radix number systems enable higher information storage density, less complexity, fewer system components, and fewer cascaded gates and operations. A simple one-step fully parallel high-radix signed-digit arithmetic is proposed for paralleloptical computing based on new joint spatial encodings. This reduces hardware requirements and improves throughput by reducing the space-bandwidth product needed. The high-radix signed-digit arithmetic operations are based on classifying the neighboringinput digit pairs into various groups to reduce the computation rules. A new joint spatial encoding technique is developed to present both the operands and the computation rules. This technique increases the spatial bandwidth product (SBWP) of the spatial light modulators (SLMs) of the system. An optical implementation of the proposed high-radix signed-digit arithmetic operations is also presented. It is shown that our one-step trinary signed-digit (TSD) and quarternary signed-digit (QSD) arithmetic unitsare much simpler and better than all previously reported high-radix signed-digit techniques.
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