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## Details

Name: cordic
Created: Sep 25, 2001
Updated: Jan 8, 2013
SVN Updated: Mar 10, 2009

## Other project properties

Category: Arithmetic core
Language: VHDL
Development status: Stable
Additional info: Design done , FPGA proven
WishBone Compliant: No
License: GPL

## Description

The CORDIC algorithm is an iterative algorithm to evaluate many mathematical functions, such as trigonometrically functions, hyperbolic functions and planar rotations.

## Core Description

As the name suggests the CORDIC algorithm was developed for rotating coordinates, a piece of hardware for doing real-time navigational computations in the 1950's. The CORDIC uses a sequence like successive approximation to reach its results. The nice part is it does this by adding/subtracting and shifting only. Suppose we want to rotate a point(X,Y) by an angle(Z). The coordinates for the new point(Xnew, Ynew) are:

Xnew = X * cos(Z) - Y * sin(Z) Ynew = Y * cos(Z) + X * sin(Z)
Or rewritten:
Xnew / cos(Z) = X - Y * tan(Z) Ynew / cos(Z) = Y + X * tan(Z)
It is possible to break the angle into small pieces, such that the tangents of these pieces are always a power of 2. This results in the following equations:
X(n+1) = P(n) * ( X(n) - Y(n) / 2^n) Y(n+1) = P(n) * ( Y(n) + X(n) / 2^n) Z(n) = atan(1/2^n)
The atan(1/2^n) has to be pre-computed, because the algorithm uses it to approximate the angle. The P(n) factor can be eliminated from the equations by pre-computing its final result. If we multiply all P(n)'s together we get the aggregate constant.
P = cos(atan(1/2^0)) * cos(atan(1/2^1)) * cos(atan(1/2^2))....cos(atan(1/2^n))
This is a constant which reaches 0.607... Depending on the number of iterations and the number of bits used. The final equations look like this:
Xnew = 0.607... * sum( X(n) - Y(n) / 2^n) Ynew = 0.607... * sum( Y(n) + X(n) / 2^n)
Now it is clear how we can simply implement this algorithm, it only uses shifts and adds/subs. Or in a program-like style:
For i=0 to n-1
If (Z(n) >= 0) then
X(n + 1) := X(n) – (Yn/2^n); Y(n + 1) := Y(n) + (Xn/2^n); Z(n + 1) := Z(n) – atan(1/2^i);
Else
X(n + 1) := X(n) + (Yn/2^n); Y(n + 1) := Y(n) – (Xn/2^n); Z(n + 1) := Z(n) + atan(1/2^i);
End if;
End for;
Where 'n' represents the number of iterations.

## Implementation

See the on-line documentation for the theory behind and information about the available CORDIC cores.

## Status

- Design is available in VHDL from OpenCores CVS via cvsweb or via cvsget
- ToDo: finish documentation