math - Why we use CORDIC gain? -

i'm studying cordic. , found cordic gain. k=0.607xxx.

from cordic, k_i = cos(tan^-1(2^i)).

as know k approched 0.607xxx.when going infinity

this value come k multiplying.

i understand reason of exist each k. curioused used ? why use value k=0.607xx?

the scale factor rotation mode of circular variant of cordic can established first principles. idea behind cordic take point on unit circle , rotate it, in steps, through angle u sine , cosine want determine.

to end define set of incremental angles a₀, ..., a_n-1, such a_k = atan(0.5^k). sum these incremental angles appropriately partial sum of angles s_k, such s_n ~= u. let y_k = cos(s_k) , x_k = sin(s_k). if in given step k rotate a_k, have

y_k+1 = cos (s_k+1) = cos (s_k + a_k)
x_k+1 = sin (s_k+1) = sin (s_k + a_k)

we can compute x_k+1 , y_k+1 x_k , y_k follows:

y_k+1 = y_k * cos (a_k) - x_k * sin (a_k)
x_k+1 = x_k * cos (a_k) + y_k * sin (a_k)

considering may both add , subtract a_k, , tan(a_k) = sin(a_k)/cos(a_k), get:

y_k+1 = cos (a_k) * (y_k ∓ x_k * tan(a_k)) = cos (s_k+1)
x_k+1 = cos (a_k) * (x_k ± y_k * tan(a_k)) = sin (s_k+1)

to simplify computation, can leave out multiplication cos(a_k) in every step, gives our cordic iteration scheme:

y_k+1 = y ∓ x_k * tan(a_k)
x_k+1 = x ± y_k * tan(a_k)

because of our choice of a_k, multiplications tan(a_k) turn simple right shifts if compute in fixed-point arithmetic. because left off factors cos(a_k), wind with

y_n ~= cos(u) * (1 / (cos (a₀) * cos (a₁) * ... * cos (a_n))
x_n ~= sin(u) * (1 / (cos (a₀) * cos (a₁) * ... * cos (a_n))

the factor f = cos (a₀) * cos (a₁) * ... * cos (a_n) 0.607..., noted. incorporate computation setting starting values

y₀ = f * cos(0) = f
x₀ = f * sin(0) = 0

here c code shows entire computation in action, using 16-bit fixed-point arithmetic. input angles scaled such 360 degrees correspond 2¹⁶, while sine , cosine outputs scaled such 1 corresponds 2¹⁵.

#include <stdio.h> #include <stdlib.h> #include <math.h>  /* round (atand (0.5**i) * 65536/360) */ static const short a[15] =  {     0x2000, 0x12e4, 0x09fb, 0x0511,      0x028b, 0x0146, 0x00a3, 0x0051,      0x0029, 0x0014, 0x000a, 0x0005,      0x0003, 0x0001, 0x0001 };  #define swap(a,b){a=a^b; b=b^a; a=a^b;}  void cordic (unsigned short u, short *s, short *c) {     short x, y, oldx, oldy, q;     int i;      x = 0;     y = 0x4dba;  /* 0.60725 */     oldx = x;     oldy = y;      q = u >> 14;    /* quadrant */     u = u & 0x3fff; /* reduced angle */     u = -(short)u;      = 0;     {         if ((short)u < 0) {             x = x + oldy;             y = y - oldx;             u = u + a[i];         } else {             x = x - oldy;             y = y + oldx;             u = u - a[i];         }         oldx = x;         oldy = y;         i++;         /* right shift of signed negative number implementation defined in c */         oldx = (oldx < 0) ? (-((-oldx) >> i)) : (oldx >> i);         oldy = (oldy < 0) ? (-((-oldy) >> i)) : (oldy >> i);     } while (i < 15);      (i = 0; < q; i++) {         swap (x, y);         y = -y;     }      *s = x;     *c = y; }  int main (void) {     float angle;     unsigned short u;     short s, c;      printf ("angle in degrees [0,360): ");     scanf ("%f", &angle);     u = (unsigned short)(angle * 65536.0f / 360.0f + 0.5f);     cordic (u, &s, &c);     printf ("sin = % f  (ref: % f)  cos = % f (ref: % f)\n",             s/32768.0f, sinf(angle/360*2*3.14159265f),              c/32768.0f, cosf(angle/360*2*3.14159265f));     return exit_success; }