Cordic算法——verilog实现

转载自：https://www.cnblogs.com/rouwawa/p/7102173.html

上两篇博文Cordic算法——圆周系统之旋转模式、Cordic算法——圆周系统之向量模式做了理论分析和实现，但是所用到的变量依然是浮点型，而cordic真正的用处是基于FPGA等只能处理定点的平台。只需将满足精度的浮点数，放大2^n倍，取整，再进行处理。

1. 旋转模式

假设要通过FPGA计算极坐标（55.6767°,1）的直角坐标。首先，角度值为浮点数，需要进行放大处理，放大10000倍。则预设的旋转角度同样要放大10000倍。

实现伪旋转（忽略模长补偿因子）的代码如下所示，注意，因为是整型运算，起始旋转时x放大了2^15，放大倍数决定计算精度，满足需求即可。最后得到的x,y在缩小2^15，即得到伪旋转后的x,y。最后进行模长波长运算（因为是浮点，同样需要放大）。

#include <stdio.h>
#include <stdlib.h>int cordic_c(int a,int r);
int x = 32768, y = 0;       //以X轴为旋转起始点,放大倍数2^15int main(viod)
{int remain = cordic_c(556767,1);        //极坐标值（极角，极径）printf("旋转角度误差：%d, 直角坐标：x = %d, y = %d\n",remain,x,y);return 0;
}int cordic_c(int a,int r)
{const int theta[] = {450000,265651,140362,71250,35763,17899,8952,4476,2238,1119,560,280,140,70,35,17,9,4,2,1}; //旋转角度int i = 0;int x_temp = 0, y_temp = 0;int angle_new = 0;      //旋转后终止角度int angle_remain = a;   //旋转后，剩余角度char detection;             //旋转方向for( i=0; i<20;i++){if(angle_remain > 0){angle_new = angle_new + theta[i];angle_remain = a - angle_new;x_temp = (x - (y >>i));y_temp = (y + (x >> i));x = x_temp;y = y_temp;detection = '+';}else{angle_new = angle_new - theta[i];angle_remain = a - angle_new;x_temp = (x + (y>>i));y_temp = (y - (x>>i));x = x_temp;y = y_temp;detection = '-'; }printf(" x = %-8d, y = %-8d, 旋转次数 = %-8d 旋转角度 = %-12d  旋转方向：%-8c  终点角度 = %-8d\n", x,y,i+1, theta[i],detection,angle_new);}x = r*x;y = r*y;return angle_remain;
}

完整的FPGA实现过程，包含预处理和后处理，支持{-π，π}的角度，采用流水线方式实现，Verilog完整代码如下，注意在移位过程中要用算术移位（>>>），才能保证带符号的数正确移位：


/****************************************************/
//预处理
module Cordic_Pre(clk,rst_n,phi,    phi_pre,quadrant_flag);/****************************************************/input                   clk;input                   rst_n;input   signed  [23:0]  phi;    output  signed  [23:0]  phi_pre;                        //预处理后的角度值output          [1:0]   quadrant_flag;                  //象限标记/****************************************************/parameter               ANGLE_P90   = 24'sd90_0000,     //输入角度范围{-pi,pi}，角度值放大了10000倍ANGLE_N90   = -24'sd90_0000,ANGLE_0     = 24'sd00_0000;/****************************************************/reg     signed  [23:0]  phi_pre_r;reg             [1:0]   quadrant_flag_r;     /****************************************************/always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginphi_pre_r <= 24'sd0;quadrant_flag_r <= 2'b00;endelse if(phi >= ANGLE_0 && phi <= ANGLE_P90)     //第一象限beginphi_pre_r <= phi;quadrant_flag_r <= 2'b01;           endelse if(phi > ANGLE_P90 )                       //第二象限          beginphi_pre_r <= phi - ANGLE_P90;quadrant_flag_r <= 2'b10;endelse if(phi < ANGLE_0 && phi >= ANGLE_N90)      //第四象限          beginphi_pre_r <= phi;quadrant_flag_r <= 2'b00;endelsebegin                                       //第三象限phi_pre_r <= phi - ANGLE_N90;quadrant_flag_r <= 2'b11;endend/****************************************************/assign  phi_pre         = phi_pre_r;assign  quadrant_flag   = quadrant_flag_r;/****************************************************/endmodule

我的设计要求精度较高，所以采用20次旋转，旋转过程的代码如下：

/****************************************************/module  Cordic_Rotate(clk,rst_n,phi_pre,quadrant_flag,ret_x,ret_y,quadrant);/****************************************************/input                   clk;input                   rst_n;input   signed  [23:0]  phi_pre;input           [1:0]   quadrant_flag;output  signed  [16:0]  ret_x;output  signed  [16:0]  ret_y;output          [1:0]   quadrant;
/****************************************************/parameter               X_ORIGN     = 17'sd32768;   //旋转时x的起始大小，根据精度要求而定。//每次旋转的固定角度值parameter               ANGLE_1     = 24'sd450000,  ANGLE_2     = 24'sd265651,ANGLE_3     = 24'sd140362,  ANGLE_4     = 24'sd71250,ANGLE_5     = 24'sd35763,   ANGLE_6     = 24'sd17899,ANGLE_7     = 24'sd8952,    ANGLE_8     = 24'sd4476,ANGLE_9     = 24'sd2238,    ANGLE_10    = 24'sd1119,ANGLE_11    = 24'sd560,     ANGLE_12    = 24'sd280,ANGLE_13    = 24'sd140,     ANGLE_14    = 24'sd70,ANGLE_15    = 24'sd35,      ANGLE_16    = 24'sd17,ANGLE_17    = 24'sd9,       ANGLE_18    = 24'sd4,ANGLE_19    = 24'sd2,       ANGLE_20    = 24'sd1;/****************************************************/reg     signed  [16:0]  x_r             [20:0];reg     signed  [16:0]  y_r             [20:0];reg     signed  [23:0]  angle_remain    [20:0];reg     signed  [1:0]   quadrant_r      [20:0];/****************************************************/
//旋转的流水线过程always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[0] <= 17'sd0;y_r[0] <= 17'sd0;angle_remain[0] <= 24'sd0;endelsebeginx_r[0] <= X_ORIGN;               y_r[0] <= 17'sd0;angle_remain[0] <= phi_pre;endend//第1次旋转 always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[1] <= 17'sd0;y_r[1] <= 17'sd0;angle_remain[1] <= 24'sd0;endelse if(angle_remain[0] > 24'sd0)beginx_r[1] <= x_r[0] - y_r[0];y_r[1] <= y_r[0] + x_r[0];angle_remain[1] <= angle_remain[0] - ANGLE_1;endelsebeginx_r[1] <= x_r[0] + y_r[0];y_r[1] <= y_r[0] - x_r[0];angle_remain[1] <= angle_remain[0] + ANGLE_1;endend//第2次旋转     always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[2] <= 17'sd0;y_r[2] <= 17'sd0;angle_remain[2] <= 24'sd0;endelse if(angle_remain[1] > 24'sd0)                           //比较时，符号标记s必须带上，对结果有影响beginx_r[2] <= x_r[1] - (y_r[1] >>> 1);              y_r[2] <= y_r[1] + (x_r[1] >>> 1);                  //二元加的优先级高于算术移位angle_remain[2] <= angle_remain[1] - ANGLE_2;endelsebeginx_r[2] <= x_r[1] + (y_r[1] >>> 1);y_r[2] <= y_r[1] - (x_r[1] >>> 1);angle_remain[2] <= angle_remain[1] + ANGLE_2;endend//第3次旋转     always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[3] <= 17'sd0;y_r[3] <= 17'sd0;angle_remain[3] <= 24'sd0;endelse if(angle_remain[2] > 24'sd0)beginx_r[3] <= x_r[2] - (y_r[2] >>> 2);y_r[3] <= y_r[2] + (x_r[2] >>> 2);angle_remain[3] <= angle_remain[2] - ANGLE_3;endelsebeginx_r[3] <= x_r[2] + (y_r[2] >>> 2);y_r[3] <= y_r[2] - (x_r[2] >>> 2);angle_remain[3] <= angle_remain[2] + ANGLE_3;endend//第4次旋转     always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[4] <= 17'sd0;y_r[4] <= 17'sd0;angle_remain[4] <= 24'sd0;endelse if(angle_remain[3] > 24'sd0)beginx_r[4] <= x_r[3] - (y_r[3] >>> 3);y_r[4] <= y_r[3] + (x_r[3] >>> 3);angle_remain[4] <= angle_remain[3] - ANGLE_4;endelsebeginx_r[4] <= x_r[3] + (y_r[3] >>> 3);y_r[4] <= y_r[3] - (x_r[3] >>> 3);angle_remain[4] <= angle_remain[3] + ANGLE_4;endend//第5次旋转     always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[5] <= 17'sd0;y_r[5] <= 17'sd0;angle_remain[5] <= 24'sd0;endelse if(angle_remain[4] > 24'sd0)beginx_r[5] <= x_r[4] - (y_r[4] >>> 4);y_r[5] <= y_r[4] + (x_r[4] >>> 4);angle_remain[5] <= angle_remain[4] - ANGLE_5;endelsebeginx_r[5] <= x_r[4] + (y_r[4] >>> 4);y_r[5] <= y_r[4] - (x_r[4] >>> 4);angle_remain[5] <= angle_remain[4] + ANGLE_5;endend//第6次旋转     always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[6] <= 17'sd0;y_r[6] <= 17'sd0;angle_remain[6] <= 24'sd0;endelse if(angle_remain[5] > 24'sd0)beginx_r[6] <= x_r[5] - (y_r[5] >>> 5);y_r[6] <= y_r[5] + (x_r[5] >>> 5);angle_remain[6] <= angle_remain[5] - ANGLE_6;endelsebeginx_r[6] <= x_r[5] + (y_r[5] >>> 5);y_r[6] <= y_r[5] - (x_r[5] >>> 5);angle_remain[6] <= angle_remain[5] + ANGLE_6;endend//第7次旋转 always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[7] <= 17'sd0;y_r[7] <= 17'sd0;angle_remain[7] <= 24'sd0;endelse if(angle_remain[6] > 24'sd0)beginx_r[7] <= x_r[6] - (y_r[6] >>> 6);y_r[7] <= y_r[6] + (x_r[6] >>> 6);angle_remain[7] <= angle_remain[6] - ANGLE_7;endelsebeginx_r[7] <= x_r[6] + (y_r[6] >>> 6);y_r[7] <= y_r[6] - (x_r[6] >>> 6);angle_remain[7] <= angle_remain[6] + ANGLE_7;endend//第8次旋转 always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[8] <= 17'sd0;y_r[8] <= 17'sd0;angle_remain[8] <= 24'sd0;endelse if(angle_remain[7] > 24'sd0)beginx_r[8] <= x_r[7] - (y_r[7] >>> 7);y_r[8] <= y_r[7] + (x_r[7] >>> 7);angle_remain[8] <= angle_remain[7] - ANGLE_8;endelsebeginx_r[8] <= x_r[7] + (y_r[7] >>> 7);y_r[8] <= y_r[7] - (x_r[7] >>> 7);angle_remain[8] <= angle_remain[7] + ANGLE_8;endend//第9次旋转 always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[9] <= 17'sd0;y_r[9] <= 17'sd0;angle_remain[9] <= 24'sd0;endelse if(angle_remain[8] > 24'sd0)beginx_r[9] <= x_r[8] - (y_r[8] >>> 8);y_r[9] <= y_r[8] + (x_r[8] >>> 8);angle_remain[9] <= angle_remain[8] - ANGLE_9;endelsebeginx_r[9] <= x_r[8] + (y_r[8] >>> 8);y_r[9] <= y_r[8] - (x_r[8] >>> 8);angle_remain[9] <= angle_remain[8] + ANGLE_9;endend//第10次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[10] <= 17'sd0;y_r[10] <= 17'sd0;angle_remain[10] <= 24'sd0;endelse if(angle_remain[9] > 24'sd0)beginx_r[10] <= x_r[9] - (y_r[9] >>> 9);y_r[10] <= y_r[9] + (x_r[9] >>> 9);angle_remain[10] <= angle_remain[9] - ANGLE_10;endelsebeginx_r[10] <= x_r[9] + (y_r[9] >>> 9);y_r[10] <= y_r[9] - (x_r[9] >>> 9);angle_remain[10] <= angle_remain[9] + ANGLE_10;endend//第11次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[11] <= 17'sd0;y_r[11] <= 17'sd0;angle_remain[11] <= 24'sd0;endelse if(angle_remain[10] > 24'sd0)beginx_r[11] <= x_r[10] - (y_r[10] >>> 10);y_r[11] <= y_r[10] + (x_r[10] >>> 10);angle_remain[11] <= angle_remain[10] - ANGLE_11;endelsebeginx_r[11] <= x_r[10] + (y_r[10] >>> 10);y_r[11] <= y_r[10] - (x_r[10] >>> 10);angle_remain[11] <= angle_remain[10] + ANGLE_11;endend//第12次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[12] <= 17'sd0;y_r[12] <= 17'sd0;angle_remain[12] <= 24'sd0;endelse if(angle_remain[11] > 24'sd0)beginx_r[12] <= x_r[11] - (y_r[11] >>> 11);y_r[12] <= y_r[11] + (x_r[11] >>> 11);angle_remain[12] <= angle_remain[11] - ANGLE_12;endelsebeginx_r[12] <= x_r[11] + (y_r[11] >>> 11);y_r[12] <= y_r[11] - (x_r[11] >>> 11);angle_remain[12] <= angle_remain[11] + ANGLE_12;endend//第13次旋转always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[13] <= 17'sd0;y_r[13] <= 17'sd0;angle_remain[13] <= 24'sd0;endelse if(angle_remain[12] > 24'sd0)beginx_r[13] <= x_r[12] - (y_r[12] >>> 12);y_r[13] <= y_r[12] + (x_r[12] >>> 12);angle_remain[13] <= angle_remain[10] - ANGLE_13;endelsebeginx_r[13] <= x_r[12] + (y_r[12] >>> 12);y_r[13] <= y_r[12] - (x_r[12] >>> 12);angle_remain[13] <= angle_remain[12] + ANGLE_13;endend//第14次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[14] <= 17'sd0;y_r[14] <= 17'sd0;angle_remain[14] <= 24'sd0;endelse if(angle_remain[13] > 24'sd0)beginx_r[14] <= x_r[13] - (y_r[13] >>> 13);y_r[14] <= y_r[13] + (x_r[13] >>> 13);angle_remain[14] <= angle_remain[13] - ANGLE_14;endelsebeginx_r[14] <= x_r[13] + (y_r[13] >>> 13);y_r[14] <= y_r[13] - (x_r[13] >>> 13);angle_remain[14] <= angle_remain[13] + ANGLE_14;endend//第15次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[15] <= 17'sd0;y_r[15] <= 17'sd0;angle_remain[15] <= 24'sd0;endelse if(angle_remain[14] > 24'sd0)beginx_r[15] <= x_r[14] - (y_r[14] >>> 14);y_r[15] <= y_r[14] + (x_r[14] >>> 14);angle_remain[15] <= angle_remain[14] - ANGLE_15;endelsebeginx_r[15] <= x_r[14] + (y_r[14] >>> 14);y_r[15] <= y_r[14] - (x_r[14] >>> 14);angle_remain[15] <= angle_remain[14] + ANGLE_15;endend //第16次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[16] <= 17'sd0;y_r[16] <= 17'sd0;angle_remain[16] <= 24'sd0;endelse if(angle_remain[15] > 24'sd0)beginx_r[16] <= x_r[15] - (y_r[15] >>> 15);y_r[16] <= y_r[15] + (x_r[15] >>> 15);angle_remain[16] <= angle_remain[15] - ANGLE_16;endelsebeginx_r[16] <= x_r[15] + (y_r[15] >>> 15);y_r[16] <= y_r[15] - (x_r[15] >>> 15);angle_remain[16] <= angle_remain[15] + ANGLE_16;endend //第17次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[17] <= 17'sd0;y_r[17] <= 17'sd0;angle_remain[17] <= 24'sd0;endelse if(angle_remain[16] > 24'sd0)beginx_r[17] <= x_r[16] - (y_r[16] >>> 16);y_r[17] <= y_r[16] + (x_r[16] >>> 16);angle_remain[17] <= angle_remain[16] - ANGLE_17;endelsebeginx_r[17] <= x_r[16] + (y_r[16] >>> 16);y_r[17] <= y_r[16] - (x_r[16] >>> 16);angle_remain[17] <= angle_remain[16] + ANGLE_17;endend //第18次旋转always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[18] <= 17'sd0;y_r[18] <= 17'sd0;angle_remain[18] <= 24'sd0;endelse if(angle_remain[17] > 24'sd0)beginx_r[18] <= x_r[17] - (y_r[17] >>> 17);y_r[18] <= y_r[17] + (x_r[17] >>> 17);angle_remain[18] <= angle_remain[17] - ANGLE_18;endelsebeginx_r[18] <= x_r[17] + (y_r[17] >>> 17);y_r[18] <= y_r[17] - (x_r[17] >>> 17);angle_remain[18] <= angle_remain[17] + ANGLE_18;endend //第19次旋转always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[19] <= 17'sd0;y_r[19] <= 17'sd0;angle_remain[19] <= 24'sd0;endelse if(angle_remain[18] > 24'sd0)beginx_r[19] <= x_r[18] - (y_r[15] >>> 18);y_r[19] <= y_r[18] + (x_r[15] >>> 18);angle_remain[19] <= angle_remain[18] - ANGLE_19;endelsebeginx_r[19] <= x_r[18] + (y_r[18] >>> 18);y_r[19] <= y_r[18] - (x_r[18] >>> 18);angle_remain[19] <= angle_remain[18] + ANGLE_19;endend //第20次旋转    always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginx_r[20] <= 17'sd0;y_r[20] <= 17'sd0;angle_remain[20] <= 24'sd0;endelse if(angle_remain[19] > 24'sd0)beginx_r[20] <= x_r[19] - (y_r[19] >>> 19);y_r[20] <= y_r[19] + (x_r[19] >>> 19);angle_remain[20] <= angle_remain[19] - ANGLE_20;endelsebeginx_r[20] <= x_r[19] + (y_r[19] >>> 19);y_r[20] <= y_r[19] - (x_r[19] >>> 19);angle_remain[20] <= angle_remain[19] + ANGLE_20;endend
/****************************************************/
//每个phi值的所在现象的流水线延迟always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginquadrant_r[0] <= 2'b00;         //不能合并着写quadrant_r[1] <= 2'b00;quadrant_r[2] <= 2'b00;quadrant_r[3] <= 2'b00;quadrant_r[4] <= 2'b00;quadrant_r[5] <= 2'b00;quadrant_r[6] <= 2'b00;quadrant_r[7] <= 2'b00;quadrant_r[8] <= 2'b00;quadrant_r[9] <= 2'b00;quadrant_r[10] <= 2'b00;quadrant_r[11] <= 2'b00;quadrant_r[12] <= 2'b00;quadrant_r[13] <= 2'b00;quadrant_r[14] <= 2'b00;quadrant_r[15] <= 2'b00;quadrant_r[16] <= 2'b00;quadrant_r[17] <= 2'b00;quadrant_r[18] <= 2'b00;quadrant_r[19] <= 2'b00;quadrant_r[20] <= 2'b00;endelsebeginquadrant_r[0]   <= quadrant_flag;quadrant_r[1]   <= quadrant_r[0];quadrant_r[2]   <= quadrant_r[1];quadrant_r[3]   <= quadrant_r[2];quadrant_r[4]   <= quadrant_r[3];quadrant_r[5]   <= quadrant_r[4];quadrant_r[6]   <= quadrant_r[5];quadrant_r[7]   <= quadrant_r[6];quadrant_r[8]   <= quadrant_r[7];quadrant_r[9]   <= quadrant_r[8];quadrant_r[10]  <= quadrant_r[9];quadrant_r[11]  <= quadrant_r[10];quadrant_r[12]  <= quadrant_r[11];quadrant_r[13]  <= quadrant_r[12];quadrant_r[14]  <= quadrant_r[13];quadrant_r[15]  <= quadrant_r[14];quadrant_r[16]  <= quadrant_r[15];quadrant_r[17]  <= quadrant_r[16];quadrant_r[18]  <= quadrant_r[17];quadrant_r[19]  <= quadrant_r[18];quadrant_r[20]  <= quadrant_r[19];endend/****************************************************/  assign ret_x    = x_r[20];assign ret_y    = y_r[20];assign quadrant = quadrant_r[20];
/****************************************************/
endmodule

后处理将象限变换过的坐标还原：

/****************************************************/module  Cordic_Post(clk,rst_n,ret_x,ret_y,quadrant,sin_phi,cos_phi);/****************************************************/input                   clk;input                   rst_n;input   signed  [16:0]  ret_x;input   signed  [16:0]  ret_y;input           [1:0]   quadrant;output  signed  [16:0]  sin_phi;output  signed  [16:0]  cos_phi;/****************************************************/  reg     signed  [16:0]  sin_phi_r;reg     signed  [16:0]  cos_phi_r;/****************************************************/always @(posedge clk or negedge rst_n)beginif(rst_n == 1'b0)beginsin_phi_r <= 17'sd0;cos_phi_r <= 17'sd0;endelsecase(quadrant)                  //根据原始角度所在象限，还原其三角函数值sin_phi和cos_phi2'd01:                      //若再乘上极径和模长补偿因子，则实现直角坐标系变换begincos_phi_r <= ret_x;sin_phi_r <= ret_y;                         end2'd10:begincos_phi_r <= ~ret_y + 1'b1;sin_phi_r <= ret_x;end2'd11:begincos_phi_r <= ret_y;sin_phi_r <= ~ret_x + 1'b1;end 2'd00:begincos_phi_r <= ret_x;sin_phi_r <= ret_y;enddefault:beginsin_phi_r <= 17'sd0;cos_phi_r <= 17'sd0;endendcaseend/****************************************************/assign  sin_phi = sin_phi_r;assign  cos_phi = cos_phi_r;
/****************************************************/
endmodule

在四个象限分别选取一个角度进行仿真，仿真结果如下图所示：

角度从输入到转换完毕，一共延时21个时钟周期。正好是预处理（1个周期）+旋转（20个周期）的结果。

2. 向量模式

至于向量模式，只要理解的算法思想，在编程上大同小异，如果要处理的坐标比较少，可以不采用流水线的方式。FPGA上用非流水线方式实现，Verilog主要代码片段如下，这次是用ROM存着旋转的固定角度值，利用addr地址线来读取相应的旋转角度：

3'd2:if(times < 5'd16)beginif( yn_r !==22'd0)                          //当旋转到y=0时，提前结束，否则继续旋转反而影响精度beginif((yn_r[21]))                      //yn最高位为1时，即坐标在第四象限，则逆时针旋转beginxn_r <= xn_r - (yn_r >>> times);yn_r <= yn_r + (xn_r >>> times);addr_r <= addr_r + 1'd1;times <= addr_r;zn_r <= zn_r-angle;i <= 3'd2;endelsebegin                                   //反之，坐标在第一象限，则顺时针旋转xn_r <= xn_r + (yn_r >>> times);yn_r <= yn_r - (xn_r >>> times);addr_r <= addr_r + 1'd1;times <= addr_r;zn_r <= zn_r+angle;i <= 3'd2;endendelsebegini <= i +1'b1;   endendelsebegini <= i +1'b1;end

至此，cordic基于圆周系统的算法总结完毕，至于还有基于线性系统、双曲系统来实现其它运算，等有机会了再学习。