臺灣博碩士論文加值系統

CORDIC(Coordinate Rotation Digital Computer)演算法，利用座標旋轉原理，執行一連串的疊代運算，可以產生一些強大的數學函數功能，而本身的硬體結構只是簡單的移位器及加法器而已，因此非常適合使用現今VLSI技術製作高性能的晶片應用在大量及快速的函數運算需求之數位訊號處理及影像處理領域。在過去30年間CORDIC實作及理論應用，被廣泛地探討，但實用後必須探討之運算誤差文獻非常少，而該項卻非常重要。因為充分的瞭解誤差的所在，就可以做誤差上的預知及硬體上的設計，以達到最經濟的效能和預期的精確結果。
本篇論文在於探討CORDIC的運算誤差，將CORDIC誤差分門別類作有系統的分析推導。就擴充輸入範圍的有無分為擴充前及擴充後兩大類，又根據標度因子的補償分為有無補償及疊代前或疊代後補償三大類，共六大類。每一大類依照CORDIC的特性分旋轉和向量兩模式，再以誤差種類分為近似誤差和截位誤差兩種，截位誤差又分固點式及浮點式，然後分別就三種不同座標系統分析誤差，共產生108個誤差分析式，以整體誤差的表現分為72式，並以表列的方式表現函數功能參考誤差表及容許誤差範圍的設計建議表。
本篇論文完成一個涵蓋領域相當廣而且完整的CORDIC誤差分析，不僅考量及改良一些以往忽略的誤差和表示方法，更將誤差分析結果表示成只與疊代次數和位元寬度有關的表示式。在誤差分析的討論上也得到一些重要的發現。此外，特別舉例說明CORDIC應用之誤差計算，將誤差分析成果，直接應用在CORDIC的設計發展上。

CORDIC(Coordinate Rotation Digital Computer), is an algorithm for performing a sequence of iteration computation using the coordinate rotation. It can generate some powerful elementary function only realized by a simple set of adder and shifter. As such, it is suitable to implement high performance chips which are applied in digital signal processing and image processing fields with high speed and massive function computation using VLSI technology. In the duration of the past thirty years, the CORDIC applications of implementation and algorithm had been discussed in different fields, but there were few papers proposed which were very important about computation error using CORDIC. As knowing where the errors are, we can design the hardware with the error consideration in order to get the best cost-performance and the desired outcomes.
In this thesis, we had discussed the computation error of CORDIC. We split the error of CORDIC into different kinds in order to analyze and derive from it systematically. There were split into before and after expansion according to the expansion of input range, and then split into before and after iteration according to the compensation of scale factor which was applied or not. Every kind split into rotation and vector mode according to the characteristic of CORDIC, then split into approximation error and truncation error for kind of error. The truncation error split into fixed point and floating point. We had analyzed all errors to generate 108 formulas of error analysis or 72 formulas from overall view in three different coordinate system. We also revealed the reference error with functional base and some suggestions on design with the error tolerance in tables.
In this thesis, we had got a far and wide and perfect error analysis of CORDIC. We had not only considered and promoted some neglected errors and formulas, but also revealed the outcomes of error analysis in the form which involved the iterative times and word-length only. We also had got some important discovery in the discussion of error analysis. Beside, we took examples to explain the CORDIC application for error computation. It was applied directly in the design and implementation of CORDIC from the outcomes of error analysis.