# cordicrotate

Rotate input using CORDIC-based approximation

## Syntax

```v = cordicrotate(theta,u) v = cordicrotate(theta,u,niters) v = cordicrotate(theta,u,Name,Value) v = cordicrotate(theta,u,niters,Name,Value) ```

## Description

`v = cordicrotate(theta,u)` rotates the input `u` by `theta` using a CORDIC algorithm approximation. The function returns the result of `u` .* `e`^(`j`*`theta`).

`v = cordicrotate(theta,u,niters)` performs `niters` iterations of the algorithm.

`v = cordicrotate(theta,u,Name,Value)` scales the output depending on the Boolean value, `b`.

`v = cordicrotate(theta,u,niters,Name,Value)` specifies both the number of iterations and the `Name,Value` pair for whether to scale the output.

## Input Arguments

 `theta` `theta` can be a signed or unsigned scalar, vector, matrix, or N-dimensional array containing the angle values in radians. All values of `theta` must be in the range [–2π 2π). `u` `u` can be a signed or unsigned scalar value or have the same dimensions as `theta`. `u` can be real or complex valued. `niters` `niters` is the number of iterations the CORDIC algorithm performs. This argument is optional. When specified, `niters` must be a positive, integer-valued scalar. If you do not specify `niters`, or if you specify a value that is too large, the algorithm uses a maximum value. For fixed-point operation, the maximum number of iterations is the word length of `u` or one less than the word length of `theta`, whichever is smaller. For floating-point operation, the maximum value is 52 for double or 23 for single. Increasing the number of iterations can produce more accurate results, but it also increases the expense of the computation and adds latency.

### Name-Value Pair Arguments

Optional comma-separated pairs of `Name,Value` arguments, where `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside single quotes (`''`).

 `'ScaleOutput'` `ScaleOutput` is a Boolean value that specifies whether to scale the output by the inverse CORDIC gain factor. This argument is optional. If you set `ScaleOutput` to `true` or `1`, the output values are multiplied by a constant, which incurs extra computations. If you set `ScaleOutput` to `false` or `0`, the output is not scaled. Default: true

## Output Arguments

 `v` `v` contains the approximated result of the CORDIC rotation algorithm. When the input `u` is floating point, the output `v` has the same data type as the input. When the input `u` is a signed integer or fixed point data type, the output `v` is a signed `fi` object. This `fi` object has a word length that is two bits larger than that of `u`. Its fraction length is the same as the fraction length of `u`. When the input u is an unsigned integer or fixed point, the output `v` is a signed `fi` object. This `fi` object has a word length that is three bits larger than that of `u`. Its fraction length is the same as the fraction length of `u`.

## Examples

Run the following code, and evaluate the accuracy of the CORDIC-based complex rotation.

```wrdLn = 16; theta = fi(-pi/3, 1, wrdLn); u = fi(0.25 - 7.1i, 1, wrdLn); uTeTh = double(u) .* exp(1i * double(theta)); fprintf('\n\nNITERS\tReal\t ERROR\t LSBs\t\tImag\tERROR\tLSBs\n'); fprintf('------\t-------\t ------\t ----\t\t-------\t------\t----\n'); for niters = 1:(wrdLn - 1) v_fi = cordicrotate(theta, u, niters); v_dbl = double(v_fi); x_err = abs(real(v_dbl) - real(uTeTh)); y_err = abs(imag(v_dbl) - imag(uTeTh)); fprintf('%d\t%1.4f\t %1.4f\t %1.1f\t\t%1.4f\t %1.4f\t %1.1f\n',... niters, real(v_dbl),x_err,(x_err * pow2(v_fi.FractionLength)), ... imag(v_dbl),y_err, (y_err * pow2(v_fi.FractionLength))); end fprintf('\n'); ```

The output table appears as follows:

```NITERS Real ERROR LSBs Imag ERROR LSBs ------ ------- ------ ---- ------- ------ ------ 1 -4.8438 1.1800 4833.5 -5.1973 1.4306 5859.8 2 -6.6567 0.6329 2592.5 -2.4824 1.2842 5260.2 3 -5.8560 0.1678 687.5 -4.0227 0.2560 1048.8 4 -6.3098 0.2860 1171.5 -3.2649 0.5018 2055.2 5 -6.0935 0.0697 285.5 -3.6528 0.1138 466.2 6 -5.9766 0.0472 193.5 -3.8413 0.0746 305.8 7 -6.0359 0.0121 49.5 -3.7476 0.0191 78.2 8 -6.0061 0.0177 72.5 -3.7947 0.0280 114.8 9 -6.0210 0.0028 11.5 -3.7710 0.0043 17.8 10 -6.0286 0.0048 19.5 -3.7590 0.0076 31.2 11 -6.0247 0.0009 3.5 -3.7651 0.0015 6.2 12 -6.0227 0.0011 4.5 -3.7683 0.0017 6.8 13 -6.0237 0.0001 0.5 -3.7666 0.0001 0.2 14 -6.0242 0.0004 1.5 -3.7656 0.0010 4.2 15 -6.0239 0.0001 0.5 -3.7661 0.0005 2.2```

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### CORDIC

CORDIC is an acronym for COordinate Rotation DIgital Computer. The Givens rotation-based CORDIC algorithm is one of the most hardware-efficient algorithms available because it requires only iterative shift-add operations (see References). The CORDIC algorithm eliminates the need for explicit multipliers. Using CORDIC, you can calculate various functions, such as sine, cosine, arc sine, arc cosine, arc tangent, and vector magnitude. You can also use this algorithm for divide, square root, hyperbolic, and logarithmic functions.

Increasing the number of CORDIC iterations can produce more accurate results, but doing so also increases the expense of the computation and adds latency.

## Algorithms

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### Signal Flow Diagrams

#### CORDIC Rotation Kernel

X represents the real part, Y represents the imaginary part, and Z represents theta. This algorithm takes its initial values for X, Y, and Z from the inputs, `u` and `theta`.

### fimath Propagation Rules

CORDIC functions discard any local `fimath` attached to the input.

The CORDIC functions use their own internal `fimath` when performing calculations:

• `OverflowAction``Wrap`

• `RoundingMethod``Floor`

The output has no attached `fimath`.

## References

[1] Volder, JE. “The CORDIC Trigonometric Computing Technique.” IRE Transactions on Electronic Computers. Vol. EC-8, September 1959, pp. 330–334.

[2] Andraka, R. “A survey of CORDIC algorithm for FPGA based computers.” Proceedings of the 1998 ACM/SIGDA sixth international symposium on Field programmable gate arrays. Feb. 22–24, 1998, pp. 191–200.

[3] Walther, J.S. “A Unified Algorithm for Elementary Functions.” Hewlett-Packard Company, Palo Alto. Spring Joint Computer Conference, 1971, pp. 379–386. (from the collection of the Computer History Museum). www.computer.org/csdl/proceedings/afips/1971/5077/00/50770379.pdf

[4] Schelin, Charles W. “Calculator Function Approximation.” The American Mathematical Monthly. Vol. 90, No. 5, May 1983, pp. 317–325.