本文是用於記錄在了解和學習CORDIC算法期間的收獲，以供日后自己及他人參考；並且附上了使用Verilog實現CORDIC算法求解角度的正弦和余弦的代碼、簡單的testbench測試代碼、以及在Modelsim下的仿真結果。
本文主要參考了：
【1】https://www.cnblogs.com/aikimi7/p/3929592.html （cordic算法的verilog實現及modelsim仿真）
【2】https://www.cnblogs.com/qiweiwang/archive/2010/07/28/1787021.html（CORDIC算法--流水線結構）
【3】https://www.cnblogs.com/yuphone/archive/2010/09/21/1832217.html（[文檔].艾米電子 - 算術運算電路，Verilog）
【4】https://wenku.baidu.com/view/54b251aaa98271fe900ef97e（如何理解CORDIC算法）
【5】https://wenku.baidu.com/view/60a0a07831b765ce050814b7（針對正弦余弦計算的CORDIC算法優化及其FPGA實現）
感謝！
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1、算法簡介
CORDIC（Coordinate Rotation Digital Computer）算法即坐標旋轉數字計算方法，是J.D.Volder1於1959年首次提出，主要用於三角函數、雙曲線、指數、對數的計算。該算法通過基本的加和移位運算代替乘法運算，使得矢量的旋轉和定向的計算不再需要三角函數、乘法、開方、反三角、指數等函數，計算向量長度並能把直角坐標系轉換為極坐標系。因為Cordic 算法只用了移位和加法，很容易用純硬件來實現，非常適合FPGA實現。
CORDIC算法是天平稱重思想在數值運算領域的傑出范例。核心的思想是把非線性的問題變成了線性的迭代問題【4】。
CORDIC算法完成坐標或向量的平面旋轉（下圖以逆時針旋轉為例）。
旋轉后，可得如下向量：
旋轉的角度θ經過多次旋轉得到的（步步逼近，接近二分查找法），每次旋轉一小角度。單步旋轉定義如下公式：
公式（2）提取cosθ，可修改為：
修改后的公式把乘法次數從4次改為3次，剩下的乘法運算可以通過選擇每次旋轉的角度去除，將每一步的正切值選為2的指數（二分查找法），除以2的指數可以通過右移操作完成（verilog）。
每次旋轉的角度可以表示為：
所有迭代角度累加值等於最終需要的旋轉角度θ：
這里Sn為1或者-1，根據旋轉方向確定（后面有確定方法，公式（15）），順時針為-1，逆時針為1。
可以得到如下公式：
結合公式（3）和（7），得到公式（8）：
到這里，除了余弦值這個系數，算法只要通過簡單的移位和加法操作完成。而這個系數可以通過預先計算最終值消掉。首先重新重寫這個系數如下：
第二步計算所有的余弦值並相乘，這個值K稱為增益系數。
由於K值是常量，我們可以先忽略它。
到這里我們發現，算法只剩下移位和加減法，這就非常適合硬件實現了，為硬件快速計算三角函數提供了一種新的算法。在進行迭代運算時，需要引入一個新的變量Z，表示需要旋轉的角度θ中還沒有旋轉的角度。
這里，我們可以把前面提到確定旋轉方向的方法介紹了，就是通過這個變量Z的符號確定。
通過公式（5）和（15），將未旋轉的角度變為0。
一個類編程風格的結構如下，反正切值是預先計算好的。
1.1 旋轉模式
旋轉模式下，CORDIC算法驅動Z變為0，結合公式（13）和（16），算法的核心計算如下：
一種特殊情況是，另初始值如下：
因此，旋轉模式下CORDIC算法可以計算一個輸入角度的正弦值和余弦值。
1.2 向量模式
向量模式下，有兩種特例：
因此，向量模式下CORDIC算法可以用來計算輸入向量的模和反正切，也能開方計算，並可以將直角坐標轉換為極坐標。
算法介紹：http://en.wikipedia.org/wiki/Cordic，http://blog.csdn.net/liyuanbhu/article/details/8458769
2、硬件算法實現
根據【5】，可以看到CORDIC迭代算法的一種直接實現方式是反饋結構，此結構只設計一級CORDIC運算迭代單元、然后在系統時鍾的驅動下，將本級的輸出作為本級的輸入，通過同一級迭代完成運算。這種方法硬件開銷小、但控制相對復雜。
所以根據【1】、【2】，使用流水線結構實現了CORDIC迭代算法、求取了角度的正弦和余弦值。
下面分段介紹下各部分代碼：
首先是角度的表示，進行了宏定義，360讀用16位二進制表示2^16，每一度為2^16/360。
//360°--2^16,phase_in = 16bits (input [15:0] phase_in) //1°--2^16/360 `define rot0 16'h2000 //45 `define rot1 16'h12e4 //26.5651 `define rot2 16'h09fb //14.0362 `define rot3 16'h0511 //7.1250 `define rot4 16'h028b //3.5763 `define rot5 16'h0145 //1.7899 `define rot6 16'h00a3 //0.8952 `define rot7 16'h0051 //0.4476 `define rot8 16'h0028 //0.2238 `define rot9 16'h0014 //0.1119 `define rot10 16'h000a //0.0560 `define rot11 16'h0005 //0.0280 `define rot12 16'h0003 //0.0140 `define rot13 16'h0002 //0.0070 `define rot14 16'h0001 //0.0035 `define rot15 16'h0000 //0.0018
然后是流水線級數定義、增益放大倍數以及中間結果位寬定義。流水線級數16，為了滿足精度要求，有文獻指出流水線級數必須大於等於角度位寬16（針對正弦余弦計算的CORDIC算法優化及其FPGA實現）。增益放大2^16，為了避免溢出狀況中間結果（x，y，z）定義為17為，最高位作為符號位判斷，1為負數，0為正數。
module cordic ( input clk, input [15:0] phase_in, output reg signed [16:0] eps, output reg signed [16:0] sin, output reg signed [16:0] cos ); parameter PIPELINE = 16; parameter K = 16'h9b74; //gian k=0.607253*2^16,9b74,n means the number pipeline //pipeline 16-level //maybe overflow,matlab result not overflow //MSB is signed bit,transform the sin and cos according to phase_in[15:14] reg signed [16:0] x0=0,y0=0,z0=0; reg signed [16:0] x1=0,y1=0,z1=0; reg signed [16:0] x2=0,y2=0,z2=0; reg signed [16:0] x3=0,y3=0,z3=0; reg signed [16:0] x4=0,y4=0,z4=0; reg signed [16:0] x5=0,y5=0,z5=0; reg signed [16:0] x6=0,y6=0,z6=0; reg signed [16:0] x7=0,y7=0,z7=0; reg signed [16:0] x8=0,y8=0,z8=0; reg signed [16:0] x9=0,y9=0,z9=0; reg signed [16:0] x10=0,y10=0,z10=0; reg signed [16:0] x11=0,y11=0,z11=0; reg signed [16:0] x12=0,y12=0,z12=0; reg signed [16:0] x13=0,y13=0,z13=0; reg signed [16:0] x14=0,y14=0,z14=0; reg signed [16:0] x15=0,y15=0,z15=0; reg signed [16:0] x16=0,y16=0,z16=0;
還需要定義memory型寄存器數組並初始化為0，用於寄存輸入角度高2位的值。
reg [1:0] quadrant [PIPELINE:0]; integer i; initial begin for(i=0;i<=PIPELINE;i=i+1) quadrant[i] = 2'b0; end
接着，是對輸入角度象限處理，將角度都轉換到第一象限，方便處理。輸入角度值最高兩位賦值0，即轉移到第一象限[0°,90°]。此外，完成x0，y0和z0的初始化，並增加一位符號位。
always @ (posedge clk)//stage 0,not pipeline begin x0 <= {1'b0,K}; //add one signed bit,0 means positive y0 <= 17'b0; z0 <= {3'b0,phase_in[13:0]};//control the phase_in to the range[0-Pi/2] end
接下來根據剩余待旋轉角度z的符號位進行16次迭代處理，即完成16級流水線處理。
always @ (posedge clk)//stage 1 begin if(z0[16])//the diff is negative so clockwise begin x1 <= x0 + y0; y1 <= x0 - y0; z1 <= z0 + `rot0; end else begin x1 <= x0 - y0;//x1 <= x0; y1 <= x0 + y0;//y1 <= x0; z1 <= z0 - `rot0;//reversal 45 end end always @ (posedge clk)//stage 2 begin if(z1[16])//the diff is negative so clockwise begin x2 <= x1 + (y1>>>4'd1); y2 <= y1 - (x1>>>4'd1); z2 <= z1 + `rot1;//clockwise 26 end else begin x2 <= x1 - (y1>>>4'd1); y2 <= y1 + (x1>>>4'd1); z2 <= z1 - `rot1;//anti-clockwise 26 end end always @ (posedge clk)//stage 3 begin if(z2[16])//the diff is negative so clockwise begin x3 <= x2 + (y2>>>4'd2); //right shift n bits,divide 2^n y3 <= y2 - (x2>>>4'd2); //left adds n bits of MSB,in first quadrant x or y are positive,MSB =0 ？？ z3 <= z2 + `rot2;//clockwise 14 //difference of positive and negtive number and no round(4,5) end else begin x3 <= x2 - (y2>>>4'd2); y3 <= y2 + (x2>>>4'd2); z3 <= z2 - `rot2;//anti-clockwise 14 end end always @ (posedge clk)//stage 4 begin if(z3[16]) begin x4 <= x3 + (y3>>>4'd3); y4 <= y3 - (x3>>>4'd3); z4 <= z3 + `rot3;//clockwise 7 end else begin x4 <= x3 - (y3>>>4'd3); y4 <= y3 + (x3>>>4'd3); z4 <= z3 - `rot3;//anti-clockwise 7 end end always @ (posedge clk)//stage 5 begin if(z4[16]) begin x5 <= x4 + (y4>>>4'd4); y5 <= y4 - (x4>>>4'd4); z5 <= z4 + `rot4;//clockwise 3 end else begin x5 <= x4 - (y4>>>4'd4); y5 <= y4 + (x4>>>4'd4); z5 <= z4 - `rot4;//anti-clockwise 3 end end always @ (posedge clk)//STAGE 6 begin if(z5[16]) begin x6 <= x5 + (y5>>>4'd5); y6 <= y5 - (x5>>>4'd5); z6 <= z5 + `rot5;//clockwise 1 end else begin x6 <= x5 - (y5>>>4'd5); y6 <= y5 + (x5>>>4'd5); z6 <= z5 - `rot5;//anti-clockwise 1 end end always @ (posedge clk)//stage 7 begin if(z6[16]) begin x7 <= x6 + (y6>>>4'd6); y7 <= y6 - (x6>>>4'd6); z7 <= z6 + `rot6; end else begin x7 <= x6 - (y6>>>4'd6); y7 <= y6 + (x6>>>4'd6); z7 <= z6 - `rot6; end end always @ (posedge clk)//stage 8 begin if(z7[16]) begin x8 <= x7 + (y7>>>4'd7); y8 <= y7 - (x7>>>4'd7); z8 <= z7 + `rot7; end else begin x8 <= x7 - (y7>>>4'd7); y8 <= y7 + (x7>>>4'd7); z8 <= z7 - `rot7; end end always @ (posedge clk)//stage 9 begin if(z8[16]) begin x9 <= x8 + (y8>>>4'd8); y9 <= y8 - (x8>>>4'd8); z9 <= z8 + `rot8; end else begin x9 <= x8 - (y8>>>4'd8); y9 <= y8 + (x8>>>4'd8); z9 <= z8 - `rot8; end end always @ (posedge clk)//stage 10 begin if(z9[16]) begin x10 <= x9 + (y9>>>4'd9); y10 <= y9 - (x9>>>4'd9); z10 <= z9 + `rot9; end else begin x10 <= x9 - (y9>>>4'd9); y10 <= y9 + (x9>>>4'd9); z10 <= z9 - `rot9; end end always @ (posedge clk)//stage 11 begin if(z10[16]) begin x11 <= x10 + (y10>>>4'd10); y11 <= y10 - (x10>>>4'd10); z11 <= z10 + `rot10; end else begin x11 <= x10 - (y10>>>4'd10); y11 <= y10 + (x10>>>4'd10); z11 <= z10 - `rot10; end end always @ (posedge clk)//stage 12 begin if(z11[16]) begin x12 <= x11 + (y11>>>4'd11); y12 <= y11 - (x11>>>4'd11); z12 <= z11 + `rot11; end else begin x12 <= x11 - (y11>>>4'd11); y12 <= y11 + (x11>>>4'd11); z12 <= z11 - `rot11; end end always @ (posedge clk)//stage 13 begin if(z12[16]) begin x13 <= x12 + (y12>>>4'd12); y13 <= y12 - (x12>>>4'd12); z13 <= z12 + `rot12; end else begin x13 <= x12 - (y12>>>4'd12); y13 <= y12 + (x12>>>4'd12); z13 <= z12 - `rot12; end end always @ (posedge clk)//stage 14 begin if(z13[16]) begin x14 <= x13 + (y13>>>4'd13); y14 <= y13 - (x13>>>4'd13); z14 <= z13 + `rot13; end else begin x14 <= x13 - (y13>>>4'd13); y14 <= y13 + (x13>>>4'd13); z14 <= z13 - `rot13; end end always @ (posedge clk)//stage 15 begin if(z14[16]) begin x15 <= x14 + (y14>>>4'd14); y15 <= y14 - (x14>>>4'd14); z15 <= z14 + `rot14; end else begin x15 <= x14 - (y14>>>4'd14); y15 <= y14 + (x14>>>4'd14); z15 <= z14 - `rot14; end end always @ (posedge clk)//stage 16 begin if(z15[16]) begin x16 <= x15 + (y15>>>4'd15); y16 <= y15 - (x15>>>4'd15); z16 <= z15 + `rot15; end else begin x16 <= x15 - (y15>>>4'd15); y16 <= y15 + (x15>>>4'd15); z16 <= z15 - `rot15; end end
其中使用到了算數右移（>>>）運算、所以在之前聲明時將相應的reg/wire均聲明為signed類型。這一點在【1】的最后也有說明。
需要注意的是這里的算數移位運算（這一運算的詳細過程在【3】中進行了說明），與之區分的是邏輯移位運算。
二者規則為：
邏輯左移和右移，空出的位均補零。
算數左移與邏輯左移相同，都在低位補零；算數右移、移出的高位比特使用符號位填充（0正1負）
舉例說明，對8'b_1011_0111進行邏輯、算數移位的結果如下圖所示：
比如這里的原數值是8'b10110111、為負數（補碼形式）、換算成十進制為-73.
算數右移一位之后的結果是8'b11011011、由補碼換算成原碼再換算為十進制為-37.
由於進行了象限的轉換，最終流水結果需要根據象限進行轉換為正確的值。這里寄存17次高2位角度輸入值，配合流水線結果用於象限判斷，並完成轉換。
//according to the pipeline,register phase_in[15:14] always @ (posedge clk) begin quadrant[0] <= phase_in[15:14]; 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
最后，根據寄存的高2位角度輸入值，利用三角函數關系，得出最后的結果，其中負數進行了補碼操作。
//alter register, according to quadrant[16] to transform the result to the right result always @ (posedge clk) eps <= z16; always @ (posedge clk) begin case(quadrant[16]) //or 15 2'b00:begin //if the phase is in first quadrant,the sin(X)=sin(A),cos(X)=cos(A) cos <= x16; sin <= y16; end 2'b01:begin //if the phase is in second quadrant,the sin(X)=sin(A+90)=cosA,cos(X)=cos(A+90)=-sinA cos <= ~(y16) + 1'b1;//-sin sin <= x16;//cos end 2'b10:begin //if the phase is in third quadrant,the sin(X)=sin(A+180)=-sinA,cos(X)=cos(A+180)=-cosA cos <= ~(x16) + 1'b1;//-cos sin <= ~(y16) + 1'b1;//-sin end 2'b11:begin //if the phase is in forth quadrant,the sin(X)=sin(A+270)=-cosA,cos(X)=cos(A+270)=sinA cos <= y16;//sin sin <= ~(x16) + 1'b1;//-cos end endcase end
完整代碼：
//360°--2^16,phase_in = 16bits (input [15:0] phase_in) //1°--2^16/360 `define rot0 16'h2000 //45 `define rot1 16'h12e4 //26.5651 `define rot2 16'h09fb //14.0362 `define rot3 16'h0511 //7.1250 `define rot4 16'h028b //3.5763 `define rot5 16'h0145 //1.7899 `define rot6 16'h00a3 //0.8952 `define rot7 16'h0051 //0.4476 `define rot8 16'h0028 //0.2238 `define rot9 16'h0014 //0.1119 `define rot10 16'h000a //0.0560 `define rot11 16'h0005 //0.0280 `define rot12 16'h0003 //0.0140 `define rot13 16'h0002 //0.0070 `define rot14 16'h0001 //0.0035 `define rot15 16'h0000 //0.0018 module cordic ( input clk, //input rst_n, //input ena, input [15:0] phase_in, output reg signed [16:0] eps, output reg signed [16:0] sin, output reg signed [16:0] cos ); parameter PIPELINE = 16; //parameter K = 16'h4dba;//k=0.607253*2^15 parameter K = 16'h9b74;//gian k=0.607253*2^16,9b74,n means the number pipeline //pipeline 16-level //maybe overflow,matlab result not overflow //MSB is signed bit,transform the sin and cos according to phase_in[15:14] reg signed [16:0] x0=0,y0=0,z0=0; reg signed [16:0] x1=0,y1=0,z1=0; reg signed [16:0] x2=0,y2=0,z2=0; reg signed [16:0] x3=0,y3=0,z3=0; reg signed [16:0] x4=0,y4=0,z4=0; reg signed [16:0] x5=0,y5=0,z5=0; reg signed [16:0] x6=0,y6=0,z6=0; reg signed [16:0] x7=0,y7=0,z7=0; reg signed [16:0] x8=0,y8=0,z8=0; reg signed [16:0] x9=0,y9=0,z9=0; reg signed [16:0] x10=0,y10=0,z10=0; reg signed [16:0] x11=0,y11=0,z11=0; reg signed [16:0] x12=0,y12=0,z12=0; reg signed [16:0] x13=0,y13=0,z13=0; reg signed [16:0] x14=0,y14=0,z14=0; reg signed [16:0] x15=0,y15=0,z15=0; reg signed [16:0] x16=0,y16=0,z16=0; reg [1:0] quadrant [PIPELINE:0]; integer i; initial begin for(i=0;i<=PIPELINE;i=i+1) quadrant[i] = 2'b0; end always @ (posedge clk)//stage 0,not pipeline begin x0 <= {1'b0,K}; //add one signed bit,0 means positive y0 <= 17'b0; z0 <= {3'b0,phase_in[13:0]};//control the phase_in to the range[0-Pi/2] end always @ (posedge clk)//stage 1 begin if(z0[16])//the diff is negative so clockwise begin x1 <= x0 + y0; y1 <= x0 - y0; z1 <= z0 + `rot0; end else begin x1 <= x0 - y0;//x1 <= x0; y1 <= x0 + y0;//y1 <= x0; z1 <= z0 - `rot0;//reversal 45 end end always @ (posedge clk)//stage 2 begin if(z1[16])//the diff is negative so clockwise begin x2 <= x1 + (y1>>>4'd1); y2 <= y1 - (x1>>>4'd1); z2 <= z1 + `rot1;//clockwise 26 end else begin x2 <= x1 - (y1>>>4'd1); y2 <= y1 + (x1>>>4'd1); z2 <= z1 - `rot1;//anti-clockwise 26 end end always @ (posedge clk)//stage 3 begin if(z2[16])//the diff is negative so clockwise begin x3 <= x2 + (y2>>>4'd2); //right shift n bits,divide 2^n y3 <= y2 - (x2>>>4'd2); //left adds n bits of MSB,in first quadrant x or y are positive,MSB =0 ？？ z3 <= z2 + `rot2;//clockwise 14 //difference of positive and negtive number and no round(4,5) end else begin x3 <= x2 - (y2>>>4'd2); y3 <= y2 + (x2>>>4'd2); z3 <= z2 - `rot2;//anti-clockwise 14 end end always @ (posedge clk)//stage 4 begin if(z3[16]) begin x4 <= x3 + (y3>>>4'd3); y4 <= y3 - (x3>>>4'd3); z4 <= z3 + `rot3;//clockwise 7 end else begin x4 <= x3 - (y3>>>4'd3); y4 <= y3 + (x3>>>4'd3); z4 <= z3 - `rot3;//anti-clockwise 7 end end always @ (posedge clk)//stage 5 begin if(z4[16]) begin x5 <= x4 + (y4>>>4'd4); y5 <= y4 - (x4>>>4'd4); z5 <= z4 + `rot4;//clockwise 3 end else begin x5 <= x4 - (y4>>>4'd4); y5 <= y4 + (x4>>>4'd4); z5 <= z4 - `rot4;//anti-clockwise 3 end end always @ (posedge clk)//STAGE 6 begin if(z5[16]) begin x6 <= x5 + (y5>>>4'd5); y6 <= y5 - (x5>>>4'd5); z6 <= z5 + `rot5;//clockwise 1 end else begin x6 <= x5 - (y5>>>4'd5); y6 <= y5 + (x5>>>4'd5); z6 <= z5 - `rot5;//anti-clockwise 1 end end always @ (posedge clk)//stage 7 begin if(z6[16]) begin x7 <= x6 + (y6>>>4'd6); y7 <= y6 - (x6>>>4'd6); z7 <= z6 + `rot6; end else begin x7 <= x6 - (y6>>>4'd6); y7 <= y6 + (x6>>>4'd6); z7 <= z6 - `rot6; end end always @ (posedge clk)//stage 8 begin if(z7[16]) begin x8 <= x7 + (y7>>>4'd7); y8 <= y7 - (x7>>>4'd7); z8 <= z7 + `rot7; end else begin x8 <= x7 - (y7>>>4'd7); y8 <= y7 + (x7>>>4'd7); z8 <= z7 - `rot7; end end always @ (posedge clk)//stage 9 begin if(z8[16]) begin x9 <= x8 + (y8>>>4'd8); y9 <= y8 - (x8>>>4'd8); z9 <= z8 + `rot8; end else begin x9 <= x8 - (y8>>>4'd8); y9 <= y8 + (x8>>>4'd8); z9 <= z8 - `rot8; end end always @ (posedge clk)//stage 10 begin if(z9[16]) begin x10 <= x9 + (y9>>>4'd9); y10 <= y9 - (x9>>>4'd9); z10 <= z9 + `rot9; end else begin x10 <= x9 - (y9>>>4'd9); y10 <= y9 + (x9>>>4'd9); z10 <= z9 - `rot9; end end always @ (posedge clk)//stage 11 begin if(z10[16]) begin x11 <= x10 + (y10>>>4'd10); y11 <= y10 - (x10>>>4'd10); z11 <= z10 + `rot10; end else begin x11 <= x10 - (y10>>>4'd10); y11 <= y10 + (x10>>>4'd10); z11 <= z10 - `rot10; end end always @ (posedge clk)//stage 12 begin if(z11[16]) begin x12 <= x11 + (y11>>>4'd11); y12 <= y11 - (x11>>>4'd11); z12 <= z11 + `rot11; end else begin x12 <= x11 - (y11>>>4'd11); y12 <= y11 + (x11>>>4'd11); z12 <= z11 - `rot11; end end always @ (posedge clk)//stage 13 begin if(z12[16]) begin x13 <= x12 + (y12>>>4'd12); y13 <= y12 - (x12>>>4'd12); z13 <= z12 + `rot12; end else begin x13 <= x12 - (y12>>>4'd12); y13 <= y12 + (x12>>>4'd12); z13 <= z12 - `rot12; end end always @ (posedge clk)//stage 14 begin if(z13[16]) begin x14 <= x13 + (y13>>>4'd13); y14 <= y13 - (x13>>>4'd13); z14 <= z13 + `rot13; end else begin x14 <= x13 - (y13>>>4'd13); y14 <= y13 + (x13>>>4'd13); z14 <= z13 - `rot13; end end always @ (posedge clk)//stage 15 begin if(z14[16]) begin x15 <= x14 + (y14>>>4'd14); y15 <= y14 - (x14>>>4'd14); z15 <= z14 + `rot14; end else begin x15 <= x14 - (y14>>>4'd14); y15 <= y14 + (x14>>>4'd14); z15 <= z14 - `rot14; end end always @ (posedge clk)//stage 16 begin if(z15[16]) begin x16 <= x15 + (y15>>>4'd15); y16 <= y15 - (x15>>>4'd15); z16 <= z15 + `rot15; end else begin x16 <= x15 - (y15>>>4'd15); y16 <= y15 + (x15>>>4'd15); z16 <= z15 - `rot15; end end //according to the pipeline,register phase_in[15:14] always @ (posedge clk) begin quadrant[0] <= phase_in[15:14]; 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 //alter register, according to quadrant[16] to transform the result to the right result always @ (posedge clk) eps <= z16; always @ (posedge clk) begin case(quadrant[16]) //or 15 2'b00:begin //if the phase is in first quadrant,the sin(X)=sin(A),cos(X)=cos(A) cos <= x16; sin <= y16; end 2'b01:begin //if the phase is in second quadrant,the sin(X)=sin(A+90)=cosA,cos(X)=cos(A+90)=-sinA cos <= ~(y16) + 1'b1;//-sin sin <= x16;//cos end 2'b10:begin //if the phase is in third quadrant,the sin(X)=sin(A+180)=-sinA,cos(X)=cos(A+180)=-cosA cos <= ~(x16) + 1'b1;//-cos sin <= ~(y16) + 1'b1;//-sin end 2'b11:begin //if the phase is in forth quadrant,the sin(X)=sin(A+270)=-cosA,cos(X)=cos(A+270)=sinA cos <= y16;//sin sin <= ~(x16) + 1'b1;//-cos end endcase end endmodule
testbench測試代碼：
`timescale 1 ps/ 1 ps module cordic_tb; // test vector input registers reg arst; reg clk; reg [15:0] phase = 16'h2000; // wires wire signed [16:0] cosine_out; wire signed [7:0] eps_out; wire signed [16:0] sine_out; // localparam coef=1000; // assign statements (if any) cordic i1 ( // port map - connection between master ports and signals/registers .clk(clk), .cos(cosine_out), .eps(eps_out), .phase_in(phase), .sin(sine_out) ); initial begin clk=0; #(10000*coef) $stop; end always #(5*coef) clk=~clk; // always @(negedge clk) begin phase=phase+16'h0100; end endmodule
3、Modelsim仿真結果
仿真結果的補充說明：
（1）程序全程未使用復位信號，testbench中第一個計算的角度為16'h2000也就是45度，如果以圖示時刻為0時刻、仿真結果對應的波形即分別為sin(x+π/4)和cos(x+π/4)的波形。作為參考，0.5*√2*65535≈46340.
（2）關於運算過程中的位數溢出
根據仿真結果，本測試例下，x4出現過16位位數溢出。
（3）關於流水線設計的理解
前文提到過實現CORDIC迭代算法時可以使用反饋結構（只使用一級）、也可以使用流水線結構（多級），如果任務是只單獨計算一個角度的正弦或者余弦值，那么所需要的迭代次數或者說消耗的時鍾周期數量其實是相同的，本設計中為16個時鍾。
流水線結構的威力是在連續、源源不斷地計算一組多個角度的正余弦值的時候才展現出來，當初始流水線被填滿之后，每經過一個時鍾周期、都會在輸出上獲得一個更新的角度的正余弦結果值，上圖仿真結果圖中黃色cursor左側的時間段內、流水線即被逐步填滿。
換句話說，如果現在的任務是要計算n個角度的正余弦值、計算一個角度需要的迭代次數為x，反饋結構需要的時長為（n*x）個時鍾周期，流水線結構只需要（n+x-1）個時鍾周期。