A high-speed VLSI multiplication algorithm internally using redundant binary representation is proposed. In n bit binary integer multiplication, n partial products are first generated and then added up pairwise by means of a binary tree of redundant binary adders. Since parallel addition of two n-digit redundant binary numbers can be performed in a constant time independent of n without carry propagation, n bit multiplication can be performed in a time proportional to log2 n. The computation time is almost the same as that by a multiplier with a Wallace tree, in which three partial products will be converted into two, in contrast to our two-to-one conversion, and is much shorter than that by an array multiplier for longer operands. The number of computation elements of an n bit multiplier based on the algorithm is proportional to n2. It is almost the same as those of conventional ones. Furthermore, since the multiplier has a regular cellular array structure similar to an array multiplier, it is suitable for VLSI implementation. Thus, the multiplier is excellent in both computation speed and regularity in layout. It can be implemented on a VLSI chip with an area proportional to n2 log2 n. The algorithm can be directly applied to both unsigned and 2's complement binary integer multiplication.

High-Speed VLSI Multiplication Algorithm with a Redundant Binary Addition Tree

This paper presents a dependence graph (DG) to visualize and describe a merged multiply-accumulate (MAC) hardware that is based on the modified Booth algorithm (MBA). The carry-save technique is used in the Booth encoder, the Booth multiplier, and the accumulator sections to ensure the fastest possible implementation. The DG applies to any MAC data word size and allows designing multiplier structures that are regular and have minimal delay, sign-bit extensions, and datapath width. Using the DG, a fast pipelined implementation is proposed, in which an accurate delay model for deep submicron CMOS technology is used. The delay model describes multi-level gate delays, taking into account input ramp and output loading. Based on the delay model, the proposed pipelined parallel MAC design is three times faster than other parallel MAC schemes that are based on the MBA. The speedup resulted from merging the accumulate and the multiply operations and the wide use of carry-save techniques.

A fast parallel multiplier-accumulator using the modified Booth algorithm

Array multipliers are well suited for VLSI implementation because of the regularity in their iterative structure. However, most VLSI circuits are difficult to test. This correspondence shows that, with appropriate cell design, array multipliers can be designed to be very easily testable. An array multiplier is called C-testable if all its adder cells can be exhaustively tested while requiring only a constant number of test patterns. The testability of two well-known array multiplier structures is studied in detail. The conventional design of the carry–save array multiplier is modified. The modified design is shown to be C-testable and requires only 16 test patterns. Similar results are obtained for the Baugh–Wooley two's complement array multiplier. A modified design of the Baugh–Wooley array multiplier is shown to be C-testable and requires 55 test patterns. The C-testability of two other array multipliers, namely the carry–propagate and the TRW designs, is also presented.

The Design of Easily Testable VLSI Array Multipliers

An XxY bit array multiplier/accumulator circuit (10), where X and Y are integers, for adding an input number having (X+Y) bits to an (X+Y) bit product of an X bit number and a Y bit number. Modified Booth's algorithm is implemented with an array structure which maintains a regular and systematic structure. The array structure uses adders (13) and multiplexers (12) in a predetermined column and row arrangement. Propagation delay is minimized while utilizing the modified Booth's algorithm by using a sum skipping technique and by using inverting logic properties of adders. Sign bit extension is provided by additional logic circuitry (14) and signed/unsigned modes of operation (15) are provided.

AN XxY BIT ARRAY MULTIPLIER/ACCUMULATOR CIRCUIT

A fast serial-parallel (FSP) multiplier design is derived from the carry-save add-shift (CSAS) multiplier structure. The CSAS technique accepts multiplier bits serially (lsb first) and produces outputs serially (lsb first). Multiplication of two n bit unsigned numbers requires 2n clock cycles to complete the process out of which n clocks are used for n-row carry-save additions, and the other n clocks are utilized only to propagate the remaining carries. This CSAS structure is modified so that it operates as a CSAS unit for the first n clocks and reconfigures itself as an n bit ripple-carry parallel adder at the (n + 1)st clock, thus allowing the carries to ripple through, eliminating the delay due to storage elements during the last n clocks. It is shown that this modification results in an about one-third increase in speed for an approximately one-third increase in hardware. The technique is extended to signed numbers represented in 2's complement form. Also, it is shown how these implementations can be modularized.

A Fast Serial-Parallel Binary Multiplier

Real-time digital signal processing requires very fast multiplication, which is now becoming possible using mathematical techniques to take advantage of single-chip multipliers.

S. Waser

Papers

High-Speed Monolithic Multipliers for Real-Time Digital Signal Processing