FPGA算法学习(1) -- Cordic（Verilog实现）

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
);/****************************************************/inputclk;
inputrst_n;
inputsigned[23:0]phi;outputsigned[23:0]phi_pre;//预处理后的角度值
output[1:0]quadrant_flag;//象限标记/****************************************************/parameterANGLE_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;/****************************************************/regsigned[23:0]phi_pre_r;
reg[1:0]quadrant_flag_r; /****************************************************/always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
phi_pre_r <= 24'sd0;
quadrant_flag_r <= 2'b00;
end
else if(phi >= ANGLE_0 && phi <= ANGLE_P90)//第一象限
begin
phi_pre_r <= phi;
quadrant_flag_r <= 2'b01;
end
else if(phi > ANGLE_P90 )//第二象限
begin
phi_pre_r <= phi - ANGLE_P90;
quadrant_flag_r <= 2'b10;
end
else if(phi < ANGLE_0 && phi >= ANGLE_N90)//第四象限
begin
phi_pre_r <= phi;
quadrant_flag_r <= 2'b00;
end
else
begin//第三象限
phi_pre_r <= phi - ANGLE_N90;
quadrant_flag_r <= 2'b11;
end
end/****************************************************/assignphi_pre= phi_pre_r;
assignquadrant_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：

/****************************************************/moduleCordic_Rotate(
clk,
rst_n,
phi_pre,
quadrant_flag,ret_x,
ret_y,
quadrant
);/****************************************************/inputclk;
inputrst_n;
inputsigned[23:0]phi_pre;
input[1:0]quadrant_flag;outputsigned[16:0]ret_x;
outputsigned[16:0]ret_y;
output[1:0]quadrant;
/****************************************************/parameterX_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
parameterANGLE_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;/****************************************************/regsigned[16:0]x_r[20:0];
regsigned[16:0]y_r[20:0];
regsigned[23:0]angle_remain[20:0];
regsigned[1:0]quadrant_r[20:0];/****************************************************/
//旋转的流水线过程
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[0] <= 17'sd0;
y_r[0] <= 17'sd0;
angle_remain[0] <= 24'sd0;
end
else
begin
x_r[0] <= X_ORIGN;
y_r[0] <= 17'sd0;
angle_remain[0] <= phi_pre;
end
end
//第1次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[1] <= 17'sd0;
y_r[1] <= 17'sd0;
angle_remain[1] <= 24'sd0;
endelse if(angle_remain[0] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第2次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[2] <= 17'sd0;
y_r[2] <= 17'sd0;
angle_remain[2] <= 24'sd0;
end
else if(angle_remain[1] > 24'sd0)//比较时，符号标记s必须带上，对结果有影响
begin
x_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;
end
else
begin
x_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;
end
end
//第3次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[3] <= 17'sd0;
y_r[3] <= 17'sd0;
angle_remain[3] <= 24'sd0;
end
else if(angle_remain[2] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第4次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[4] <= 17'sd0;
y_r[4] <= 17'sd0;
angle_remain[4] <= 24'sd0;
end
else if(angle_remain[3] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第5次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[5] <= 17'sd0;
y_r[5] <= 17'sd0;
angle_remain[5] <= 24'sd0;
end
else if(angle_remain[4] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第6次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[6] <= 17'sd0;
y_r[6] <= 17'sd0;
angle_remain[6] <= 24'sd0;
end
else if(angle_remain[5] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第7次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[7] <= 17'sd0;
y_r[7] <= 17'sd0;
angle_remain[7] <= 24'sd0;
end
else if(angle_remain[6] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第8次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[8] <= 17'sd0;
y_r[8] <= 17'sd0;
angle_remain[8] <= 24'sd0;
end
else if(angle_remain[7] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第9次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[9] <= 17'sd0;
y_r[9] <= 17'sd0;
angle_remain[9] <= 24'sd0;
end
else if(angle_remain[8] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第10次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[10] <= 17'sd0;
y_r[10] <= 17'sd0;
angle_remain[10] <= 24'sd0;
end
else if(angle_remain[9] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第11次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[11] <= 17'sd0;
y_r[11] <= 17'sd0;
angle_remain[11] <= 24'sd0;
end
else if(angle_remain[10] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第12次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[12] <= 17'sd0;
y_r[12] <= 17'sd0;
angle_remain[12] <= 24'sd0;
end
else if(angle_remain[11] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第13次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[13] <= 17'sd0;
y_r[13] <= 17'sd0;
angle_remain[13] <= 24'sd0;
end
else if(angle_remain[12] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第14次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[14] <= 17'sd0;
y_r[14] <= 17'sd0;
angle_remain[14] <= 24'sd0;
end
else if(angle_remain[13] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第15次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[15] <= 17'sd0;
y_r[15] <= 17'sd0;
angle_remain[15] <= 24'sd0;
end
else if(angle_remain[14] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第16次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[16] <= 17'sd0;
y_r[16] <= 17'sd0;
angle_remain[16] <= 24'sd0;
end
else if(angle_remain[15] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第17次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[17] <= 17'sd0;
y_r[17] <= 17'sd0;
angle_remain[17] <= 24'sd0;
end
else if(angle_remain[16] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第18次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[18] <= 17'sd0;
y_r[18] <= 17'sd0;
angle_remain[18] <= 24'sd0;
end
else if(angle_remain[17] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第19次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[19] <= 17'sd0;
y_r[19] <= 17'sd0;
angle_remain[19] <= 24'sd0;
end
else if(angle_remain[18] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
//第20次旋转
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
x_r[20] <= 17'sd0;
y_r[20] <= 17'sd0;
angle_remain[20] <= 24'sd0;
end
else if(angle_remain[19] > 24'sd0)
begin
x_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;
end
else
begin
x_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;
end
end
/****************************************************/
//每个phi值的所在现象的流水线延迟
always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
quadrant_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;
end
else
begin
quadrant_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：

/****************************************************/moduleCordic_Post(
clk,
rst_n,
ret_x,
ret_y,
quadrant,sin_phi,
cos_phi
);/****************************************************/
inputclk;
inputrst_n;
inputsigned[16:0]ret_x;
inputsigned[16:0]ret_y;
input[1:0]quadrant;outputsigned[16:0]sin_phi;
outputsigned[16:0]cos_phi;/****************************************************/regsigned[16:0]sin_phi_r;
regsigned[16:0]cos_phi_r;/****************************************************/always @(posedge clk or negedge rst_n)
begin
if(rst_n == 1'b0)
begin
sin_phi_r <= 17'sd0;
cos_phi_r <= 17'sd0;
end
else
case(quadrant)//根据原始角度所在象限，还原其三角函数值sin_phi和cos_phi
2'd01://若再乘上极径和模长补偿因子，则实现直角坐标系变换
begin
cos_phi_r <= ret_x;
sin_phi_r <= ret_y;
end
2'd10:
begin
cos_phi_r <= ~ret_y + 1'b1;
sin_phi_r <= ret_x;
end
2'd11:
begin
cos_phi_r <= ret_y;
sin_phi_r <= ~ret_x + 1'b1;
end
2'd00:
begin
cos_phi_r <= ret_x;
sin_phi_r <= ret_y;
end
default:
begin
sin_phi_r <= 17'sd0;
cos_phi_r <= 17'sd0;
end
endcaseend/****************************************************/
assignsin_phi= sin_phi_r;
assigncos_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)
begin
if( 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
begin
if((yn_r[21]))//yn最高位为1时，即坐标在第四象限，则逆时针旋转
begin
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;
end
else
begin//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;
end
end
else
begin
i <= i +1'b1;
end
end
else
begin
i <= 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。

参考

学习cordic算法所得（流水线结构、Verilog标准）