A generalized multibit recoding of two's complement binary numbers and its proof with application in multiplier implementations

High-Speed and Low-Power PID Structures for Embedded Applications

“…Based on the multibit recoding algorithm presented in [6], the equation (2.1.2) of [6] is rewritten in a simpler hardware-friendly form as follows: In this general case, the multiplier Y is divided into n/r slices, each of r+1 bits. Each pair of two contiguous slices has one overlapping bit.…”

“…There is no need to prove equation (4) since it is a combination of equations (3) and modified Booth algorithm (MBA) which were both already proven in [6] and [13], respectively.…”

“…Its speed and power requirements are two critical factors limiting the whole system performances (PID in our case). Since the publication of Booth s algorithm in 1951, a huge number of improvement attempts were proposed, especially after the publication of a generalized version of modified Booth algorithm accompanied with its proof [6]. Most of the proposals aimed to reduce the number of partial products either by employing digital optimization techniques [7][8] [9] or by using larger slices (higher radices) [10].…”

Minimal weight digit set conversions

“…The authors of [15], [16] and others mention that the modified Booth technique (s-bit groups overlapped by 1 bit) can be extended to groups of any size. General treatments appear in [17] and [18] in which the criteria to be met by all correct uniform overlapped multiple-bit scanning techniques or generalized multibit recoding techniques are derived.…”

A New Recursive Multibit Recoding Algorithm for High-Speed and Low-Power Multiplier

“…Theoretically, only the signed multibit recoding multiplication algorithm [7] is capable of a drastic reduction (n/r) of the partial product number, given that r+1 is the number of bits of the multiplier that are simultaneously treated (1≤r≤n). Unfortunately, this algorithm requires the pre-computation of a number of odd multiples of the multiplicand (until (2 r-1 -1).X) that scales linearly with r. The large number of odd multiples not only requires a considerable amount of multiplexers to perform the necessary complex recoding into PPG, but dramatically increases the routing density as well.…”