Low-cost constant time signed digit selection for most significant bit first multiplication

Serial binary multiplication is frequently used in many digital applications. In particular, left-to-right (aka online) manipulation of operands promotes the real-time generation of product digits for immediate utilization in subsequent online computations (e.g., successive layers of a neural network). In the left-to-right arithmetic operations, where a residual is maintained for digit selection, utilization of a redundant number system for the representation of outputs is mandatory, while the input operands and the residual may be redundant or non-redundant. However, when the input data paths are narrow (e.g., eight bits as in BFloat16), conventional non-redundant representations of inputs and residual provide some advantages. For example, the immediate and costless sign detection of the residual that is necessary for the next digit selection; a property not shared by redundant numbers. Nevertheless, digit selection, as practiced in the previous realizations, with both redundant and non-redundant inputs and/or residual, is slow and rather complex. Therefore, in this paper, we offer an imprecise, but faster digit selection scheme, with the required correction in the next cycle. Analytical evaluations and synthesis of the proposed circuits on FPGA platform, shows 30 % speedup and less cost with respect to both cases with redundant and non-redundant inputs and residual.