A Theoretical Analysis of Square versus Rectangular Component Multipliers in Recursive Multiplication Behrooz Parhami Department of Electrical and Computer Engineering University of California, Santa Barbara, USA parhami@ece.ucsb.edu 50th Asilomar Conference on Signals, Systems, and Computers Pacific Grove, CA, USA, November 6-9, 2016

Outline Introduction Additive Multiply Modules Design of AMMs Recursive Multiplication Basic Theory Delay-Optimized Layout-Optimized Theoretical Comparisons Special Case of Squaring Conclusion

Introduction: Multiplier Design Multiplication is a very important building block Different algorithms offer various trade-offs Naïve binary algorithm: O(k) cost; O(k log k) delay Basic binary with carry-save: O(k) cost; O(k) delay Radix-2h algorithm: O(hk) cost; O(k/h + log k) delay Partial-tree multiplier: Same as high-radix, with h = O(k) Tree multiplier: O(k2) cost; O(log k) delay Array multiplier: O(k2) cost; O(k) delay Recursive multiplier (our focus here)

Additive Multiply Modules AMM b-bit and c-bit multiplicative inputs b  c AMM b-bit and c-bit additive inputs (b + c)-bit output (2b – 1)  (2c – 1) + (2b – 1) + (2c – 1) = 2b+c – 1

Square vs. Nonsquare AMMs Height-5 column

Recursive Construction Nonsquare AMMs Square AMMs

Recursive Construction: Square AMMs

8  8 Multiplier Built 4  2 of AMMs Apr. 2015 Computer Arithmetic, Multiplication 8  8 Multiplier Built 4  2 of AMMs

8  8 Multiplier: Layout-Optimized Apr. 2015 Computer Arithmetic, Multiplication 8  8 Multiplier: Layout-Optimized

Other Examples of Non-Sqaure BBs

A Bound on Matrix Height Reduction xmax = max[0, min(h, g – 1 – hc/b)] g groups of b bits h groups of c bits

Example with No Height Reduction

Examples of Matrix Height Reduction Efstathiou, et al., 2004

Special Case of Squaring Non-square building blocks not beneficial because we won’t be able to use squarers A single multiplier Two squarers

Conclusion and Future Work Non-square components may be advantageous Closed-form formula for matrix height Formula valid in all cases of practical interest Fails in some corner cases that are uninteresting Choice of aspect ratio affects overall speed It also affects design complexity and regularity LUT schemes favor 4  2 and 3  3 modules Multi-level synthesis: 4  2  16  8  32  32

Questions? The PDF file of the final paper will be made available at: www.ece.ucsb.edu/~parhami/publications.htm