CORDIC算法——Verilog_verilog实现三角函数

CORIDC 技术并不是什么新鲜的东西。 事实上它可以追溯到 1957 年由 J. Volder 发表的一篇文章。 在上个世纪五十年代， 在大型实际的计算机中的实行移位相加受到了当时技术上的限制， 所以使用 CORDIC 变得非常必要。 到了七十年代， Hewlett Packard 和其他公司出产了手持计算器 , 许多计算器使用一个内部 CORDIC 单元来
计算所有的三角函数 ( 了解这件事的人们一定还记得，那时求一个角度的正切值需要延迟大约 1 秒中 )。
二十世纪八十年代，随着高速度乘法器与带有大存储量的通用处理器的出现， CORDIC 算法变得无关紧要了。
然而在二十一世纪的今天， 对于 FPGA 来说， CORDIC 一定是在 DSP 应用中 ( 诸如 多输入多输出 （ MIMO）,波束形成以及其他自适应系统 ) 计算三角函数的备选技术

ref
https://blog.csdn.net/qq_39210023/article/details/77456031

一、CORDIC的基本原理介绍

CORDIC算法是一个“化繁为简”的算法，将许多复杂的运算转化为一种“仅需要移位和加法”的迭代操作。CORDIC算法有旋转和向量两个模式，分别可以在圆坐标系、线性坐标系和双曲线坐标系使用，从而可以演算出8种运算，而结合这8种运算也可以衍生出其他许多运算。

问题情景

当已知一个点P1的坐标，并已知该点P1旋转的角度θ，则可以根据上述公式求得目标点P2的坐标。然后，麻烦来了，我们需要用FPGA去执行上述运算操作，而FPGA的Verilog语言根本不支持三角函数运算。因此，我们需要对上述式子进行简化操作，将复杂的运算操作转换为一种单一的“迭代位移”算法。那么，接下来我们介绍优化算法部分。

module Cordic_Test
(
	CLK_50M,RST_N,
	Phase,
	Sin,Cos,Error
);

input 						CLK_50M;
input 						RST_N;
input 		[31:0] 		Phase;
output		[31:0] 		Sin;
output		[31:0] 		Cos;
output		[31:0] 		Error;

`define rot0  32'd2949120    	//45бу*2^16
`define rot1  32'd1740992    	//26.5651бу*2^16
`define rot2  32'd919872    	//14.0362бу*2^16
`define rot3  32'd466944    	//7.1250бу*2^16
`define rot4  32'd234368    	//3.5763бу*2^16
`define rot5  32'd117312    	//1.7899бу*2^16
`define rot6  32'd58688    	//0.8952бу*2^16
`define rot7  32'd29312    	//0.4476бу*2^16
`define rot8  32'd14656    	//0.2238бу*2^16
`define rot9  32'd7360    		//0.1119бу*2^16
`define rot10 32'd3648    		//0.0560бу*2^16
`define rot11 32'd1856			//0.0280бу*2^16
`define rot12 32'd896    		//0.0140бу*2^16
`define rot13 32'd448    		//0.0070бу*2^16
`define rot14 32'd256    		//0.0035бу*2^16
`define rot15 32'd128    		//0.0018бу*2^16

parameter Pipeline = 16;
parameter K = 32'h09b74;	//K=0.607253*2^16,32'h09b74,

reg signed 	[31:0] 		Sin;
reg signed 	[31:0] 		Cos;
reg signed 	[31:0] 		Error;
reg signed 	[31:0] 		x0=0,y0=0,z0=0;
reg signed 	[31:0] 		x1=0,y1=0,z1=0;
reg signed 	[31:0] 		x2=0,y2=0,z2=0;
reg signed 	[31:0] 		x3=0,y3=0,z3=0;
reg signed 	[31:0] 		x4=0,y4=0,z4=0;
reg signed 	[31:0] 		x5=0,y5=0,z5=0;
reg signed 	[31:0] 		x6=0,y6=0,z6=0;
reg signed 	[31:0] 		x7=0,y7=0,z7=0;
reg signed 	[31:0] 		x8=0,y8=0,z8=0;
reg signed 	[31:0] 		x9=0,y9=0,z9=0;
reg signed 	[31:0] 		x10=0,y10=0,z10=0;
reg signed 	[31:0] 		x11=0,y11=0,z11=0;
reg signed 	[31:0] 		x12=0,y12=0,z12=0;
reg signed 	[31:0] 		x13=0,y13=0,z13=0;
reg signed 	[31:0] 		x14=0,y14=0,z14=0;
reg signed 	[31:0] 		x15=0,y15=0,z15=0;
reg signed 	[31:0] 		x16=0,y16=0,z16=0;
reg 			[ 1:0] 		Quadrant [Pipeline:0];

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x0 <= 1'b0; 						
		y0 <= 1'b0;
		z0 <= 1'b0;
	end
	else
	begin
		x0 <= K;
		y0 <= 32'd0;
		z0 <= Phase[15:0] << 16;
	end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x1 <= 1'b0; 						
		y1 <= 1'b0;
		z1 <= 1'b0;
	end
	else if(z0[31])
	begin
      x1 <= x0 + y0;
      y1 <= y0 - x0;
      z1 <= z0 + `rot0;
	end
	else
	begin
      x1 <= x0 - y0;
      y1 <= y0 + x0;
      z1 <= z0 - `rot0;
	end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x2 <= 1'b0; 						
		y2 <= 1'b0;
		z2 <= 1'b0;
	end
	else if(z1[31])
   begin
        x2 <= x1 + (y1 >>> 1);
        y2 <= y1 - (x1 >>> 1);
        z2 <= z1 + `rot1;
   end
   else
   begin
       x2 <= x1 - (y1 >>> 1);
       y2 <= y1 + (x1 >>> 1);
       z2 <= z1 - `rot1;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x3 <= 1'b0; 						
		y3 <= 1'b0;
		z3 <= 1'b0;
	end
	else if(z2[31])
   begin
       x3 <= x2 + (y2 >>> 2);
       y3 <= y2 - (x2 >>> 2);
       z3 <= z2 + `rot2;
   end
   else
   begin
       x3 <= x2 - (y2 >>> 2);
       y3 <= y2 + (x2 >>> 2);
       z3 <= z2 - `rot2;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x4 <= 1'b0; 						
		y4 <= 1'b0;
		z4 <= 1'b0;
	end
	else if(z3[31])
   begin
       x4 <= x3 + (y3 >>> 3);
       y4 <= y3 - (x3 >>> 3);
       z4 <= z3 + `rot3;
   end
   else
   begin
       x4 <= x3 - (y3 >>> 3);
       y4 <= y3 + (x3 >>> 3);
       z4 <= z3 - `rot3;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x5 <= 1'b0; 						
		y5 <= 1'b0;
		z5 <= 1'b0;
	end
	else if(z4[31])
   begin
       x5 <= x4 + (y4 >>> 4);
       y5 <= y4 - (x4 >>> 4);
       z5 <= z4 + `rot4;
   end
   else
   begin
       x5 <= x4 - (y4 >>> 4);
       y5 <= y4 + (x4 >>> 4);
       z5 <= z4 - `rot4;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x6 <= 1'b0; 						
		y6 <= 1'b0;
		z6 <= 1'b0;
	end
	else if(z5[31])
   begin
       x6 <= x5 + (y5 >>> 5);
       y6 <= y5 - (x5 >>> 5);
       z6 <= z5 + `rot5;
   end
   else
   begin
       x6 <= x5 - (y5 >>> 5);
       y6 <= y5 + (x5 >>> 5);
       z6 <= z5 - `rot5;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x7 <= 1'b0; 						
		y7 <= 1'b0;
		z7 <= 1'b0;
	end
	else if(z6[31])
   begin
       x7 <= x6 + (y6 >>> 6);
       y7 <= y6 - (x6 >>> 6);
       z7 <= z6 + `rot6;
   end
   else
   begin
       x7 <= x6 - (y6 >>> 6);
       y7 <= y6 + (x6 >>> 6);
       z7 <= z6 - `rot6;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x8 <= 1'b0; 						
		y8 <= 1'b0;
		z8 <= 1'b0;
	end
	else if(z7[31])
   begin
       x8 <= x7 + (y7 >>> 7);
       y8 <= y7 - (x7 >>> 7);
       z8 <= z7 + `rot7;
   end
   else
   begin
       x8 <= x7 - (y7 >>> 7);
       y8 <= y7 + (x7 >>> 7);
       z8 <= z7 - `rot7;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x9 <= 1'b0; 						
		y9 <= 1'b0;
		z9 <= 1'b0;
	end
	else if(z8[31])
   begin
       x9 <= x8 + (y8 >>> 8);
       y9 <= y8 - (x8 >>> 8);
       z9 <= z8 + `rot8;
   end
   else
   begin
       x9 <= x8 - (y8 >>> 8);
       y9 <= y8 + (x8 >>> 8);
       z9 <= z8 - `rot8;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x10 <= 1'b0; 						
		y10 <= 1'b0;
		z10 <= 1'b0;
	end
	else if(z9[31])
   begin
       x10 <= x9 + (y9 >>> 9);
       y10 <= y9 - (x9 >>> 9);
       z10 <= z9 + `rot9;
   end
   else
   begin
       x10 <= x9 - (y9 >>> 9);
       y10 <= y9 + (x9 >>> 9);
       z10 <= z9 - `rot9;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x11 <= 1'b0; 						
		y11 <= 1'b0;
		z11 <= 1'b0;
	end
	else if(z10[31])
   begin
       x11 <= x10 + (y10 >>> 10);
       y11 <= y10 - (x10 >>> 10);
       z11 <= z10 + `rot10;
   end
   else
   begin
       x11 <= x10 - (y10 >>> 10);
       y11 <= y10 + (x10 >>> 10);
       z11 <= z10 - `rot10;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x12 <= 1'b0; 						
		y12 <= 1'b0;
		z12 <= 1'b0;
	end
	else if(z11[31])
   begin
       x12 <= x11 + (y11 >>> 11);
       y12 <= y11 - (x11 >>> 11);
       z12 <= z11 + `rot11;
   end
   else
   begin
       x12 <= x11 - (y11 >>> 11);
       y12 <= y11 + (x11 >>> 11);
       z12 <= z11 - `rot11;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x13 <= 1'b0; 						
		y13 <= 1'b0;
		z13 <= 1'b0;
	end
	else if(z12[31])
   begin
       x13 <= x12 + (y12 >>> 12);
       y13 <= y12 - (x12 >>> 12);
       z13 <= z12 + `rot12;
   end
   else
   begin
       x13 <= x12 - (y12 >>> 12);
       y13 <= y12 + (x12 >>> 12);
       z13 <= z12 - `rot12;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x14 <= 1'b0; 						
		y14 <= 1'b0;
		z14 <= 1'b0;
	end
	else if(z13[31])
   begin
       x14 <= x13 + (y13 >>> 13);
       y14 <= y13 - (x13 >>> 13);
       z14 <= z13 + `rot13;
   end
   else
   begin
       x14 <= x13 - (y13 >>> 13);
       y14 <= y13 + (x13 >>> 13);
       z14 <= z13 - `rot13;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x15 <= 1'b0; 						
		y15 <= 1'b0;
		z15 <= 1'b0;
	end
	else if(z14[31])
   begin
       x15 <= x14 + (y14 >>> 14);
       y15 <= y14 - (x14 >>> 14);
       z15 <= z14 + `rot14;
   end
   else
   begin
       x15 <= x14 - (y14 >>> 14);
       y15 <= y14 + (x14 >>> 14);
       z15 <= z14 - `rot14;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		x16 <= 1'b0; 						
		y16 <= 1'b0;
		z16 <= 1'b0;
	end
	else if(z15[31])
   begin
       x16 <= x15 + (y15 >>> 15);
       y16 <= y15 - (x15 >>> 15);
       z16 <= z15 + `rot15;
   end
   else
   begin
       x16 <= x15 - (y15 >>> 15);
       y16 <= y15 + (x15 >>> 15);
       z16 <= z15 - `rot15;
   end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		Quadrant[0] <= 1'b0;
		Quadrant[1] <= 1'b0;
		Quadrant[2] <= 1'b0;
		Quadrant[3] <= 1'b0;
		Quadrant[4] <= 1'b0;
		Quadrant[5] <= 1'b0;
		Quadrant[6] <= 1'b0;
		Quadrant[7] <= 1'b0;
		Quadrant[8] <= 1'b0;
		Quadrant[9] <= 1'b0;
		Quadrant[10] <= 1'b0;
		Quadrant[11] <= 1'b0;
		Quadrant[12] <= 1'b0;
		Quadrant[13] <= 1'b0;
		Quadrant[14] <= 1'b0;
		Quadrant[15] <= 1'b0;
		Quadrant[16] <= 1'b0;
	end
	else
	begin
		Quadrant[0] <= Phase[17:16];
		Quadrant[1] <= Quadrant[0];
		Quadrant[2] <= Quadrant[1];
		Quadrant[3] <= Quadrant[2];
		Quadrant[4] <= Quadrant[3];
		Quadrant[5] <= Quadrant[4];
		Quadrant[6] <= Quadrant[5];
		Quadrant[7] <= Quadrant[6];
		Quadrant[8] <= Quadrant[7];
		Quadrant[9] <= Quadrant[8];
		Quadrant[10] <= Quadrant[9];
		Quadrant[11] <= Quadrant[10];
		Quadrant[12] <= Quadrant[11];
		Quadrant[13] <= Quadrant[12];
		Quadrant[14] <= Quadrant[13];
		Quadrant[15] <= Quadrant[14];
		Quadrant[16] <= Quadrant[15];
	end
end

always @ (posedge CLK_50M or negedge RST_N)
begin
	if(!RST_N)
	begin
		Cos <= 1'b0;
		Sin <= 1'b0;
		Error <= 1'b0;
	end
	else
	begin
		Error <= z16;
		case(Quadrant[16])
			2'b00: //if the Phase is in first Quadrant,the Sin(X)=Sin(A),Cos(X)=Cos(A)
				begin
					Cos <= x16;
					Sin <= y16;
				end
			2'b01: //if the Phase is in second Quadrant,the Sin(X)=Sin(A+90)=CosA,Cos(X)=Cos(A+90)=-SinA
				begin
					Cos <= ~(y16) + 1'b1;//-Sin
					Sin <= x16;//Cos
				end
			2'b10: //if the Phase is in third Quadrant,the Sin(X)=Sin(A+180)=-SinA,Cos(X)=Cos(A+180)=-CosA
				begin
					Cos <= ~(x16) + 1'b1;//-Cos
					Sin <= ~(y16) + 1'b1;//-Sin
				end
			2'b11: //if the Phase is in forth Quadrant,the Sin(X)=Sin(A+270)=-CosA,Cos(X)=Cos(A+270)=SinA
				begin
					Cos <= y16;//Sin
					Sin <= ~(x16) + 1'b1;//-Cos
				end
		endcase
	end
end

endmodule

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ref
https://wenku.baidu.com/view/6c623aa8910ef12d2bf9e732.html