Optimized FPGA-Based 16-bits Squarers Using LUTs and a Divide-and-Conquer Approach
Bibliographic record
Abstract
Small squarers are needed in real-life applications. Field Programmable Gate Arrays (FPGAs), like the modern 7-series Xilinx ones, come with DSP blocks that contain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$25\mathrm{x}18$</tex> bits hardwired multipliers. For practical reasons, Look-Up Tables (LUTs) in FPGAs can be used to realize small squarers instead of using these embedded hardwired multipliers. In this paper, we show how divide-and-conquer techniques can be used to perform 8-bits and 9-bits squarers and use these squarers to build 16-bits squarers using LUTs in FPGAs. The proposed 16-bits squarers require small squarers and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$8\mathrm{x}8$</tex> bits multipliers. In our first proposed approach, these <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$8\mathrm{x}8$</tex> bits multipliers are synthesized as LUTs, and are realized using squarers in our second proposed approach. The proposed approaches address both unsigned and signed squarers, and are experimentally tested using Xilinx Artix-7 FPGAs and the Vivado 2020.2 synthesis tool. Experimental results show that the first proposed approach is able to reduce the clock period and the number of LUTs compared to the traditional approach. For the second proposed approach, the clock period has been increased but the area has been reduced. Notice that the speed and the area are two conflicting objectives; increasing the speed leads to increasing the area, and vice versa.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".