CORDIC Algorithm Basics

This page covers the basics of the CORDIC algorithm, including its function and common uses. CORDIC stands for Coordinate Rotation Digital Computer.

It’s a clever algorithm used to calculate trigonometric functions (like sine and cosine) and hyperbolic functions (like the inverse tangent). It relies on the concept of a lookup table.

Let’s say you want to calculate sin(φ) for some angle φ. A simple way to do this is to store all the values of sin(φ) for every possible value of φ in memory (RAM). Then, based on the input φ, you simply output the corresponding stored value.

CORDIC becomes especially useful when a hardware multiplier isn’t available. It’s frequently used in FPGAs (Field-Programmable Gate Arrays) and DSPs (Digital Signal Processors) where only subtraction, addition, table lookup, and bit-shift operations are readily available.

CORDIC is employed for operations such as sine, cosine, polar-to-rectangular coordinate conversion, general rotation, arctangent, vector magnitude, rectangular-to-polar conversion, arcsine, arccosine, and more.

It can also be adapted for linear functions, hyperbolic functions, square rooting, logarithms, and exponentials.

Operation	Mode	Initialize	Direction
Sine, Cosine	Rotation	x = 1/Aₙ , y = 0, z = α	Reduce z to Zero
Polar to Rect.	Rotation	x = (1/Aₙ)X mag , y = 0, z = X phase	Reduce z to Zero
General Rotation	Rotation	x = (1/Aₙ)X₀ , y = (1/Aₙ)y₀ , z = α	Reduce z to Zero
Arctangent	Vector	x = (1/Aₙ)X₀ , y = (1/Aₙ)y₀ , z = 0	Reduce y to zero
Vector Magnitude	Vector	x = (1/Aₙ)X₀ , y = (1/Aₙ)y₀ , z = 0	Reduce y to zero
Rect. to Polar	Vector	x = (1/Aₙ)X₀ , y = (1/Aₙ)y₀ , z = 0	Reduce y to zero
Arcsine, Arccosine	Vector	x = (1/Aₙ), y = 0, arg = sin(α) or cos(α)	Reduce y to value in arg register