1. 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
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
Height-5 column

6. Recursive Construction
Nonsquare AMMs
Square AMMs

7. Recursive Construction: Square AMMs

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

9. 8  8 Multiplier: Layout-Optimized
Apr. 2015
Computer Arithmetic, Multiplication
8  8 Multiplier: Layout-Optimized

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: