在數位訊號處理過程中，常常會需要用到可以運算特定數學函數的硬體單元，例如：求倒數、開根號、取指數或對數、三角函數…等運算。而這種函數計算單元，通常採用以查表為主的函數近似方法來實作，如此可避免因為運算繁雜而導致效能不佳，但是隨著精準度提高，表格所占的面積卻將大幅增加。本論文針對兩種不同的函數近似演算法（Piecewise與Multipartite）加以改良，可有效減少在高精準度要求下的表格面積，並實際應用在三維繪圖的點著色引擎（Vertex Shaders）。

In many digital signal processing applications, we often need some special function units that can compute complicated arithmetic functions such as reciprocal, square-root, base-2 logarithm, power of 2, trigonometric functions, etc. The most popular design approaches to compute these single-value functions are based on look-up tables (LUT) with interpolation. In general, there are two different types of LUT-based method: piecewise and multipartite. As the required bit accuracy increases, the size of LUT increases exponentially. In this thesis, we will develop a generator that can automatically synthesize suitable hardware to compute these special arithmetic functions given the required bit accuracy. In particular, higher-order piecewise method will be supported to reduce the table size for high-accuracy applications. The synthesized arithmetic units are used in the design of a vertex shader for 3D graphics application.

Chapter 1 導論 5
1.1 研究動機 5
1.2 論文架構 6
Chapter 2 研究背景與相關研究 7
2.1 查表法(Table-based Methods)簡介 7
2.2 函數近似方法之簡介與分類 10
2.3 Piecewise Table Look-up Method 12
2.4 Bipartite and Multipartite Table Methods 15
Chapter 3 函數運算之相關算術單元 17
3.1 捨棄式乘法單元(Truncated Multiplier) 17
3.2 平方器(Squarer) 22
Chapter 4 High-Order Piecewise Table Method 25
4.1 High-Order Piecewise Table Method的函數近似方法 28
4.2 High-Order Piecewise Table Method資料路徑(datapath)的設計 35
Chapter 5 相關合成結果以及在三維繪圖之應用 39
5.1 函數產生器的合成數據 40
5.2 函數產生器在三維繪圖之應用實例 52
Chapter 6 結論與未來展望 55
6.1 結論 55
6.2 未來展望 56
參考文獻 57

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