1 / 16

Behrooz Parhami Department of Electrical and Computer Engineering

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.

robertjjones
Télécharger la présentation

Behrooz Parhami Department of Electrical and Computer Engineering

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related