在許多信號處理應用，如繪圖處理器的特殊函數單元，往往需要計算複雜函
數值此種重要的算術計算，以硬體實現函數近似值的計算通常包含查表和一些
簡單的乘法或是加法器單元，其中表格面積有時佔了整體面積的很大比例，尤
其是當高精確度或是多種函數共用算術單元但是仍有個別自己的表格時。本論
文主要是針對表格和加法之函數值計算方法，提出表格最佳化之分解。此種方
法的表格可分成兩大類：存初始值的表格(Table of Initial values) 和存位移值的表
格(Tables of Offset)。在過去的文獻中，Multipartite table method (MP) 相較於更早
提出的symmetric bipartite table methods (SBTM) 和symmetric table addition method
(STAM) 方法，在中低精確度的應用上，有更小的表格面積。本論文將提出一個
廣義的MP 法，稱為階層式(Hierarchical) MP（HMP），經由多層的MP 表格分解，
能找出最省整體表格面積的表格分解方式，並且搭配提出的誤差綜合考量方法，
找出最佳化的位元寬度，達到最省面積的硬體設計。此外，本論文也改善最近發
表的無失真表格壓縮的方法，套用在TI 表格，在不增加額外的硬體面積和時間
延遲的情況下，更進一步的降低整體表格面積。ASIC 和FPGA 實驗證明，本論文
提出的表格和加法的函數求值計算單元設計，可有效率的降低整體硬體面積。

Function evaluation is an important arithmetic computation in many signal processing
applications, such as special function units in modern graphics processing units (GPUs).
Hardware implementations of function evaluation usually consists of lookup tables (LUT)
and some simple arithmetic units of multipliers and/or adders. LUT usually takes a significant
portion of total area cost, especially when function evaluators are allowed to compute
several different arithmetic functions with shared arithmetic units where evaluation of each
function needs separate LUT. In this thesis, we focus on the category of table-lookup-andaddition
(TA) function evaluators that are composed of two types of LUT: table of initial
values (TI) and table of offset values (TO), followed by a multi-operand adder. It has
been shown that multipartite table method (MP) has significant improvement over prior
similar designs such as symmetric bipartite table methods (SBTM) and symmetric table
addition methods (STAM) for applications with low-to-medium precision requirements.
This thesis presents an extension of MP, called hierarchical multipartite (HMP), which
further reduces total table size by applying several levels of table decompositions. Furthermore,
we perform the bit-width optimization by jointly considering the impacts of all
error sources during the search of best table decompositions, leading to more efficient
hardware design. Besides, a new lossless decomposition of TI is presented, resulting
in additional saving of table size without incurring any extra errors. Experimental results
show that the proposed design can efficiently reduce the total area cost in ASIC and FPGA
implementations.

目錄
論文口試委員審定書. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
圖目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
表目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
第一章緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 研究動機. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 論文架構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
第二章研究背景與相關研究. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 函數近似方法分類. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Table-Lookup-and-Addition (TA) . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Bipartite Table Methods (BP) . . . . . . . . . . . . . . . . . . . 4
2.2.2 Symmetric Bipartite Table Methods (SBTM) . . . . . . . . . . . 7
2.2.3 Symmetric Table Addition Methods (STAM) . . . . . . . . . . . 9
2.2.4 Multipartite Table Methods (MP) . . . . . . . . . . . . . . . . . 11
2.3 Piecewise Polynomial Approximation (PPA) . . . . . . . . . . . . . . . . 15
2.4 整合誤差方法(Combined Error Methods) . . . . . . . . . . . . . . . . . 17
2.5 無損表格壓縮(Lossless ROM Compression) . . . . . . . . . . . . . . . 19
2.5.1 Two-Table Decomposition Scheme . . . . . . . . . . . . . . . . 19
iv
2.5.2 Three-Table Decomposition Scheme . . . . . . . . . . . . . . . . 21
第三章Hierarchical Multipartite (HMP). . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 函數的定義域(domain) 與值域(range) . . . . . . . . . . . . . . . . . . 25
3.2 MP 取樣方法及誤差分配(Error Budget) . . . . . . . . . . . . . . . . . 27
3.2.1 Approximation Error . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Quantization Error and Final Round Error . . . . . . . . . . . . . 30
3.3 HMP 方法概述. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 整合誤差(Combined Error) 與窮舉搜尋(Exhaustive Search) . . . . . . 41
3.5 低代價(low cost) 的無損(lossless) 壓縮方法. . . . . . . . . . . . . . 45
第四章實驗結果與比較. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
第五章結論與未來展望. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
v

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