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Behrooz Parhami Department of Electrical and Computer Engineering

Explore the efficiency trade-offs between square and rectangular multipliers in recursive multiplication, with detailed theoretical comparisons and implications for multiplier design optimization.

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Behrooz Parhami Department of Electrical and Computer Engineering

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  1. A Theoretical Analysis of Square versus Rectangular Component Multipliers in Recursive Multiplication • BehroozParhami • 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

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

  3. 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)

  4. 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

  5. Square vs. Nonsquare AMMs AMM AMM Height-5 column

  6. Recursive Construction Nonsquare AMMs Square AMMs

  7. Recursive Construction: Square AMMs

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

  9. 8  8 Multiplier: Layout-Optimized Computer Arithmetic, Multiplication

  10. Other Examples of Non-Sqaure BBs

  11. 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

  12. Example with No Height Reduction

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

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

  15. 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

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

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