Xilinx LogiCORE IP CORDIC：高级旋转数字计算机算法核心
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Codic IP是Xilinx LogiCORE系列中的一个重要组件，专为FPGA（Field-Programmable Gate Array）设计，特别是针对Virtex-7、Kintex-7、Virtex-6、Virtex-5、Virtex-4、Spartan-6、Spartan-3/XA、Spartan-3A/XA/AN/3ADSP以及Spartan-3E/XA等系列的FPGA平台。CORDIC（Coordinate Rotation Digital Computer）算法是一种用于数字信号处理中进行角度计算和向量操作的高效算法，其核心思想是通过迭代和简单的算术运算来实现旋转和转换。
该Codic IP模块提供了多种功能配置，包括但不限于：
1. 矢量旋转：允许将极坐标数据转换为直角坐标，或反之，这对于在需要角度映射或测量的应用中非常有用，如图像处理、导航系统和无线通信。
2. 矢量平移：能够执行从直角坐标到极坐标或反之的变换，这在需要根据位置或方向进行调整的场景中必不可少。
3. 正弦和余弦计算：用于获取角度的正弦和余弦值，对于信号分析、滤波器设计以及频率响应计算等数学运算至关重要。
4. 反正弦和反正切：提供Atan和Atanh功能，用于计算角度的反正切和双曲反正切，这对信号处理中的相位检测和信号分析很有帮助。
5. 平方根：实现数值的平方根运算，常用于信号处理中的噪声去除、滤波和数据归一化。
6. 粗粒度旋转模块：扩展了Codic IP的旋转范围，使得算法可以处理从正四分之一圆周到负四分之一圆周，甚至整个圆周的角度计算。
7. 幅度补偿缩放模块：考虑到Codic算法输出的幅度缩放问题，此模块提供了额外的补偿功能，确保输出结果的准确性。
8. 低级接口和高级接口：提供不同的接口选项，以便灵活地集成到各种设计中，满足不同性能和功耗的需求。
这些功能使得Codic IP成为FPGA项目中处理角度计算和向量操作的理想选择，尤其是在需要高性能、低功耗或者定制化功能的嵌入式系统和数字信号处理应用中。通过利用Codic IP，设计师可以简化复杂计算，提高系统的效率和可靠性。
DS249 March 1, 2011 www.xilinx.com 6
Product Specification
LogiCORE IP CORDIC v4.0
The CORDIC Scale Factor
The outputs of the CORDIC algorithm, equations 4 and 5, are equivalent to a vector rotation or vector translation
scaled by a constant Z
n
. The constant Z
n
is known as the CORDIC scale factor.
The CORDIC Scale Factor Equation 6
6a)
The Taylor series expansion of acos (atan (2
-i
) ) is (1 + 2
-2i
)
-1/2
. Hence, the constant Z
n
can be expressed as
6b)
The CORDIC scale factor, Z
n
, is only dependent on the number of iterations, n. Only functional configurations
Rotate, Translate, Rectangular to Polar, and Polar to Rectangular are affected by the CORDIC scale factor. When
these functional configurations are selected, the CORDIC core provides the option of multiplying by 1 / Z
n
to
cancel out the scaling factor. See Advanced Configuration Parameters for detailed information.
Output Quantization Error
The Output Quantization Error can be split into two components; the Output Quantization Error due to the Input
Quantization (OQEIQ) and the Output Quantization Error due to Internal Precision (OQEIP).
OQEIQ is due to the 1/2 lsb of quantization noise on the X,Y and Phase inputs. In a vector rotation this input
quantization noise results in OQEIQ of 1/2 an lsb on both the X and Y outputs. In a vector translation this input
quantization noise results in OQEIQ of 1/2 an lsb on the X output however OQEIQ on the phase output is
dependant on the ratio (Y/ X). Thus for small X inputs the effect of input quantization noise on OQEIQ is greatly
magnified.
OQEIP is due to the limited precision of internal calculations. In the CORDIC core the default internal precision is
set such that the accumulated OQEIP is less than 1/2 the OQEIQ. The internal precision can be manually set to
(input width + output width + log
2
(output_width)). This reduces OQEIP to 1/ 2 an lsb (the phase is calculated to full
precision regardless of the magnitude input vector).
The Output Quantization Error, for a CORDIC core with default internal precision, is dominated by OQEIQ.
OQEIQ can only be reduced by increasing the number of significant magnitude bits in the input vector (X,Y).
Increasing the internal precision or zero padding X and Y inputs only affects OQEIP and has minimal effect on the
total output quantization error.
The effect of input quantization and internal quantization on the CORDIC phase output quantization error is
illustrated in the following examples.
Example 1a: The quantization error in phase output for a small input vector, (Xin_small, Yin_small).
Xin_small : “0000000001” => 1/256.
Yin_small : “0000000001” => 1/256.
Vector translation with no input quantization:
Xin_ideal : “0000000001” => 1/256.
Yin_ideal : “0000000001” => 1/256.
Z
n
=
1
2
i–
()atan()acos
n
∏
-----------------------------------------------------
Z
n
=
12
2i–
+()
12/
i 1=
n
∏
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