在本論文中，將傳統的定點數(fixed-point)座標軸數位旋轉計算器(Coordinate Rotation Digital Computer, CORDIC)演算法之原理，推廣到浮點數(floating-point) CORDIC，以執行高精確度和高數值範圍之超越函數計算，如三角函數、指數、對數等。我們也根據不同演算法，實現了兩種高速 (high-throughput)管線式浮點數CORDIC架構硬體，其中一種架構中每個pipeline stage的移位運算是採用硬體移位器 (Barrel Shifter)設計，另一種架構中的移位運算是利用直接的拉線(hardwired)來實現。另外，本論文也根據不同的浮點數CORDIC演算法和架構，使用數學式分析其誤差，並與實際軟體程式之執行結果做比較。最後，本論文探討如何應用浮點數CORDIC硬體，來實現 3D 電腦圖形之旋轉相關運算。

In this thesis, the traditional fixed-point CORDIC algorithm is extended to floating-point version in order to calculate transcendental functions (such as sine/cosine, logarithm, powering function, etc.) with high accuracy and large range. Based on different algorithm derivations, two different floating-point high-throughput pipelined CORDIC architectures are proposed. The first architecture adopts barrel shifters to implement the shift operations in each pipelined stage. The second architecture uses pure hardwired method for the shifting operations. Another key contribution of this thesis is to analyze the execution errors in the floating-point CORDIC architectures and make comparison with the execution resulting from pure software programs. Finally, the thesis applies the floating-point CORDIC to realizing the rotation-related operations required in 3D graphics applications.

第1章 導論 9
1.1 研究動機 9
1.2 本文架構 9
第2章 CORDIC原理與相關研究 11
2.1 CORDIC原理 11
2.1.1 演算法 11
2.1.2 架構 13
2.2 CORDIC相關研究 15
2.2.1 CORDIC演算法及架構之改進 15
2.2.2 CORDIC演算法之誤差分析 17
2.2.3 Floating-Point CORDIC 18
2.2.4 CORDIC應用 20
第3章 浮點數CORDIC誤差分析 22
3.1 簡易浮點數CORDIC 22
3.2 向量旋轉Vectoring 24
3.3 角度旋轉Rotation 26
3.4 模擬結果 29
3.5 結論 40
第4章 浮點數Redundant CORDIC及實作 42
4.1 浮點數CORDIC演算法 42
4.2 Pipeline浮點數Redundant CORDIC實作 45
4.2.1 前(後)置處理與前(後)置旋轉 45
4.2.2 內部Redundant CORDIC實作 47
4.2.2.1 移位運算使用硬體移位器(Barrel Shifter) 47
4.2.2.2 移位運算使用直接拉線(Hardwired Shifter) 50
4.3 驗證與實驗數據 55
4.3.1 浮點數Redundant CORDIC驗證 55
4.3.2 各項實驗數據 56
第5章 新架構浮點數CORDIC誤差分析 61
5.1 浮點數CORDIC使用Barrel Shifter 61
5.1.1 向量旋轉Vectoring 62
5.1.2 角度旋轉Rotation 65
5.1.2.1 小角度旋轉時Post Normalization的影響 66
5.1.2.2 大角度旋轉時Post Normalization的影響 69
5.2 浮點數CORDIC使用Hardwired Shifter 75
5.2.1 向量旋轉Vectoring 76
5.2.2 角度旋轉Rotation 79
5.3 模擬結果 82
5.3.1 FP. CORDIC使用Barrel Shifter模擬結果 82
5.3.2 FP. CORDIC使用Hardwired Shifter模擬結果 84
5.4 結論 88
第6章 Redundant CORDIC FPU之應用 90
6.1 Redundant CORDIC FPU應用在旋轉運算 90
6.1.1 旋轉運算簡介 90
6.1.2 設計與架構 95
6.2 效能比較 97
6.3 其他應用 98
第7章 結論 102
參考文獻 103

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