CORDIC Algorithm COordinate Rotation DIgital Computer Method for Elementary Function Evaluation (e.g., sin(z), cos(z), tan -1 (y)) Originally Used for Real-time Navigation (Volder 1956) Idea is to Rotate a Vector in Cartesion Plane by Some Angle Complexity Comparable to Division

Rotations in the Plane (1,0) (0.707,0.707)  (cos45°, sin45°) x y 45° 0.707

Rotations in the Plane-Simplifications Factoring the term cos  from the above Equations: Restrict the Rotation Angles Such that: Above Multiplication is Reduced to a Simple Shift!!

Iterations Over Angles Arbitrary Angles Obtained After Series of Smaller Rotations (increasing i values) At Each Iteration Rotate Up or Rotate Down Possible Rotation Angles, e (i) = tan –1 2 -i (degrees):

Rotations in the Plane-Simplifications Restrict the Rotation Angles Such that: Substituting into Previous Equations:

Rotations in the Plane-Simplifications Using the Identity: Previous Equations: Are Further Simplified as:

Scaling Constant – CORDIC Gain Factor Gain Factor Becomes: This Value is Pre-computed for a Given i Value: Can Remove the K i Value and Apply Elsewhere in System Can Pre-scale (x, y) by A n

Angle Accumulator, z i Overall Rotation Angle Defined by Sequence of Iterations: Use Angle Accumulator to Keep Track of Rotations:

Binary Angular Measurement - BAM Angle Accumulator can Represent Angles as BAM Encode d i ={-1,+1} as Bit Values {0,1} Example: -1 Represented by 0 and +1 Represented by 1 LSb Represents d 0 Content z=01011 z=+45    +7.1  -3.6  =61.1  Can Simplify CORDIC Circuitry for Some Modes May Need BAM encode/decode – Can Use Lookup Table

CORDIC - Rotation Mode Input is Angle,  – Initialized in Angle Accumulator Vector Initialized to Lie on x-axis Each Iteration d i Chosen by Sign of Angle Attempt to Bring Angle to Zero Result is x Register Contains ~cos  Result is y Register Contains ~sin  Also Polar to Rectangular if x Register Initialized to Magnitude

CORDIC - Vector Mode Input is (Pre-scaled) Vector in (x,y) Registers Angle,  – Initialized to Zero Each Iteration d i Chosen to Move Vector to Lie Along Positive x-axis (Want to Reduce y Register to Zero) Result is Original Vector Angle in Angle Accumulator Can be Used for sin -1  and cos -1  Also Rectangular to Polar Conversion Magnitude in x Register

CORDIC – Rotation/Vector Modes Rotation Mode: Vector Mode:

Rotation Angle Limits Rotation/Vector Algorithms Limited to  90  Due to Use of  =tan(2 0 ) for First Iteration Several Ways to Extend Range One Way is to Use Additional Rotation for Angles Outside Range This Rotation is Initial  90  Rotation

CORDIC Uses Can Use CORDIC For Others Also: – Linear Functions – Hyperbolic Functions – Square Rooting – Logarithms, Exponentials

CORDIC Hardware

Iterative CORDIC Structure* *Taken from “A Survey of CORDIC Algorithms for FPGA Based Computers”, R. Andraka, FPGA’98

Bit-serial CORDIC Structure* *Taken from “A Survey of CORDIC Algorithms for FPGA Based Computers”, R. Andraka, FPGA’98