Center for New Music and Audio Technologies

The CORDIC-Algorithm for Computing a Sine

In 1959 Jack E. Volder [7] described the COordinate Rotation DIgital Computer or CORDIC for the calculation of trigonometric functions, multiplication, division and conversion between binary and mixed radix number systems. The CORDIC-algorithm provides an iterative method of performing vector rotations by arbitrary angles using only shifts and adds. Volder's algorithm is derived from the general equations for vector rotation. If a vector with components is to be rotated through an angle a new vector with components is formed by:

Figure 1.2 illustrates the rotation of a vector by the angle .

Figure 1.2: Rotation of a vector V by the angle

The individual equations for and can be rewritten as [8]:

and rearranged so that:

The multiplication by the tangent term can be avoided if the rotation angles and therefore are restricted so that . In digital hardware this denotes a simple shift operation. Furthermore, if those rotations are performed iteratively and in both directions every value of is representable. With the cosine term could also be simplified and since it is a constant for a fixed number of iterations. This iterative rotation can now be expressed as:

where and . The product of the 's represents the so-called K factor [9]:

This factor can be calculated in advance and applied elsewhere in the system. A good way to implement the factor is to initialize the iterative rotation with a vector of length which compensates the gain inherent in the CORDIC algorithm. The resulting vector is the unit vector as shown in Figure 1.3.

Figure 1.3: Iterative vector rotation, initialized with V0

Equations 1.7 and 1.8 can now be simplified to the basic CORDIC-equations:

The direction of each rotation is defined by and the sequence of all 's determines the final vector. This yields to a third equation which acts like an angle accumulator and keeps track of the angle already rotated. Each vector can be described by either the vector length and angle or by its coordinates and . Following this incident, the CORDIC algorithm knows two ways of determining the direction of rotation: the rotation mode and the vectoring mode. Both methods initialize the angle accumulator with the desired angle . The rotation mode, determines the right sequence as the angle accumulator approaches 0 while the vectoring mode minimizes the y component of the input vector^1.2.

The angle accumulator is defined by:

where the sum of an infinit number of iterative rotation angles equals the input angle [10]:

Those values of can be stored in a small lookup table or hardwired depending on the way of implementation. Since the decision is which direction to rotate instead of whether to rotate or not, is sensitive to the sign of . Therefore can be described as:

   . (1.14)

With equation 1.14 the CORDIC algorithm in rotation mode is described completely. Note, that the CORDIC method as described performs rotations only within and . This limitation comes from the use of for the tangent in the first iteration. However, since a sine wave is symmetric from quadrant to quadrant, every sine value from 0 to can be represented by reflecting and/or inverting the first quadrant appropriately.

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