https://www.youtube.com/watch?v=4HxdV5xEe9k

Coordinate Rotation Digital Computer

• Technique to evaluate trigonometric, hyperbolic and other mathematical functions
  • Digit-by-digit algorithm that produces one additional digit of precision per iteration
  • We can tune the accuracy by modifying the number of iterations (time), or by increasing the number of resources
• Uses only addition, subtraction, bit-shifting and table lookups - hardware efficient

Rotation a vector by an angle can be expressed as a matrix multiplication.
By selecting specific values of 2, we can rely on basic binary operations to rotate the vector

CORDIC revolves around the idea of “rotating” the phase of a complex number, by multiplying it by a succession of constant values. However, the multiples can all be powers of 2, so in binary arithmetic they can be done using just shifts and adds; no actual multiplier is needed.

--> Cordic Phase - angle which we rotated :: arctan(2^-1) ???

Unoptimised CORDIC algorithm

A scaling factor is inadvertently induced, so we need to compensate for it

float values

Floats consume a large number of resources. When targetting FPGAs it is better to use fixed point arithmetic types to represent fractional numbers. The parameters are customised to suit the specific use case.

• Start with float type
• Then optimise after

ap_fixed<size, int_bits> and ap_ufixed<size, int_bits>

• Also have a rounding mode - floor, ceil, trunc, round
• https://docs.xilinx.com/r/en-US/ug1399-vitis-hls/Arbitrary-Precision-Fixed-Point-Data-Types

Fixed-point and Optimised CORDIC

Optimisations

• Fixed point types
• Multiplications -> Bit Shifts
• Loop iterations
• Unrolling, pipelining, etc