CORDIC overview

Description

http://opencores.org/project,cordic

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(X_new, Y_new) are:

X_new = X * cos(Z) - Y * sin(Z)
Y_new = Y * cos(Z) + X * sin(Z)

Or rewritten:

X_new / cos(Z) = X - Y * tan(Z)
Y_new / 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.

CORDIC implementations in hardware

• A sine computer in MyHDL (a Python-based HDL)
. CORDIC is used within the sine computer.
• A survey of CORDIC algorithms for FPGA based computers (PDF), by Ray Andraka, presented in FPGA 98.
• A very good tutorial overview of CORDIC PDF: CORDIC for Dummies
• Project report PDF of Verilog implementation by two students in George Mason University