CORDIC Architectures: A Survey

Table 1

Realization of some functions using CORDIC Algorithm.

$m$	Mode	Initialization	Output
1 (Circular)	Rotation	$x_0=x_{\text{in}}$	$x_n=K_m·(x_{\text{in}}\cos \hspace{0.17em}\theta -y_{\text{in}}\sin \theta )$
		$y_0=y_{\text{in}}$	$y_n=K_m·(y_{\text{in}}\cos \hspace{0.17em}\theta +x_{\text{in}}\sin \theta )$
		$z_0=\theta$	$z_n=0$
		$x_0=1/K_m$	$x_n=\cos \hspace{0.17em}\theta$
		$y_0=0$	$y_n=\sin \theta$
		$z_0=\theta$	$z_n=0$
		$x_0=1$	$x_n=\sqrt{1+a^2}$
		$y_0=a$	$y_n=\sin \theta$
		$z_0=\pi /2$	$z_n=0$
1 (Circular)	Vectoring	$x_0=x_{\text{in}}$	$x_n=K_m·\mathrm{sign}\hspace{0.17em}(x_0)·(x_{\text{in}}^2+y_{\text{in}}^2{)}^{1/2}$
		$y_0=y_{\text{in}}$	$y_n=0$
		$z_0=0$	$z_n={\tan }^{-1}(y_{\text{in}}/x_{\text{in}})$
0 (Linear)	Rotation	$x_0=x_{\text{in}}$	$x_n=x_{\text{in}}$
		$y_0=y_{\text{in}}$	$y_n=y_{\text{in}}+x_{\text{in}}·z$
		$z_0=z$	$z_n=0$
0 (Linear)	Vectoring	$x_0=x_{\text{in}}$	$x_n=x_{\text{in}}$
		$y_0=y_{\text{in}}$	$y_n=0$
		$z_0=z$	$z_n=z+y_{\text{in}}/x_{\text{in}}$
$-1\mathrm{\hspace{0.17em}\hspace{0.17em}}$(Hyperbolic)	Rotation	$x_0=x_{\text{in}}$	$x_n=K_m·(x_{\text{in}}\cosh \hspace{0.17em}\theta +y_{\text{in}}\sinh \hspace{0.17em}\theta )$
		$y_0=y_{\text{in}}$	$y_n=K_m·(y_{\text{in}}\cosh \hspace{0.17em}\theta +x_{\text{in}}\sinh \hspace{0.17em}\theta )$
		$z_0=\theta$	$z_n=0$
		$x_0=1/K_m$	$x_n=\cosh \hspace{0.17em}\theta$
		$y_0=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_0=\theta$	$z_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{y}_n=\sinh \hspace{0.17em}\theta$
		$x_0=a$	$x_n=a{e}^{\theta }$
		$y_0=a,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_0=\theta$	$z_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{y}_n=a{e}^{\theta }$
$-1\mathrm{\hspace{0.17em}\hspace{0.17em}}$(Hyperbolic)	Vectoring	$x_0=x_{\text{in}}$	$x_n=K_m·\mathrm{sign}\hspace{0.17em}(x_0)·(x_{\text{in}}^2-y_{\text{in}}^2{)}^{1/2}$
		$y_0=y_{\text{in}}$	$y_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_n=\theta +{\text{tanh}}^{-1}(y_{\text{in}}/x_{\text{in}})$
		$x_0=a$	$x_n=\sqrt{a^2-1}$
		$y_0=0$	$y_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_n={\coth \hspace{0.17em}}^{-1}a$
		$x_0=a+1$	$x_n=2\sqrt{a}$
		$y_0=a-1$	$y_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_n=0.5\ln \hspace{0.17em}(a)$
		$x_0=a+b$	$x_n=2\sqrt{ab}$
		$y_0=a-b$	$y_n=0,\mathrm{\hspace{0.17em}\hspace{0.17em}}{z}_n=0.5\ln \hspace{0.17em}(a/b)$