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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
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Outline Introduction Additive Multiply Modules Design of AMMs
Recursive Multiplication Basic Theory Delay-Optimized Layout-Optimized Theoretical Comparisons Special Case of Squaring Conclusion
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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)
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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
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Square vs. Nonsquare AMMs
Height-5 column
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Recursive Construction
Nonsquare AMMs Square AMMs
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Recursive Construction: Square AMMs
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8 8 Multiplier Built 4 2 of AMMs
Apr. 2015 Computer Arithmetic, Multiplication 8 8 Multiplier Built 4 2 of AMMs
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8 8 Multiplier: Layout-Optimized
Apr. 2015 Computer Arithmetic, Multiplication 8 8 Multiplier: Layout-Optimized
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Other Examples of Non-Sqaure BBs
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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
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Example with No Height Reduction
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Examples of Matrix Height Reduction
Efstathiou, et al., 2004
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Special Case of Squaring
Non-square building blocks not beneficial because we won’t be able to use squarers A single multiplier Two squarers
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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
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Questions? The PDF file of the final paper will be made available at:
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