# cordiccexp

CORDIC-based approximation of complex exponential

## Syntax

``y = cordiccexp(theta)``
``y = cordiccexp(theta,niters)``

## Description

````y = cordiccexp(theta)` computes `cos`(`theta`) + j*`sin`(`theta`) using a CORDIC algorithm approximation and returns the approximated complex result in `y`.```

example

````y = cordiccexp(theta,niters)` performs `niters` iterations of the algorithm.```

## Examples

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This example illustrates the effect of the number of iterations on the result of the `cordiccexp` approximation.

```wl = 8; theta = fi(pi/2,1,wl); output_type = fi([], 1,wl,wl - 2); results_array = zeros(wl - 1,1,'like',output_type)'; for niters = 1:(wl - 1) cis = cordiccexp(theta,niters); fl = cis.FractionLength; x = real(cis); y = imag(cis); x_dbl = double(x); y_dbl = double(y); x_err = abs(x_dbl - cos(double(theta))); y_err = abs(y_dbl - sin(double(theta))); result = [niters,y_dbl,y_err,(y_err*pow2(fl)),... x_dbl,x_err,(x_err*pow2(fl))]; results_array = [results_array; result]; end results_table = array2table(results_array,'VariableNames',{'NITERS','Y (SIN)','Y ERROR','Y LSBs','X (COS)','X ERROR','X LSBs'})```
```results_table = 8×7 table NITERS Y (SIN) Y ERROR Y LSBs X (COS) X ERROR X LSBs ______ _______ ________ ______ ________ ________ _______ 0 0 0 0 0 0 0 1 0.70312 0.29688 1.9844 -0.70312 0.70312 1.9844 1.9844 0.9375 0.0625 1.9844 -0.3125 0.3125 1.9844 1.9844 0.96875 0.03125 1.9844 -0.0625 0.0625 1.9844 1.9844 0.96875 0.03125 1.9844 0.0625 0.0625 1.9844 1.9844 0.98438 0.015625 1 0 0 0.46875 1.9844 0.98438 0.015625 1 0.03125 0.03125 1.9844 1.9844 1 0 0 0.015625 0.015625 1.4688```

## Input Arguments

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Input array, specified as a signed or unsigned scalar, vector, matrix, or multidimensional array. All values of `theta` must be real and in the range `[-2π 2π)`.

If the input is a `fi` object, it must use binary-point scaling.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fi`

Number of iterations the CORDIC algorithm performs, specified as a positive scalar integer. Increasing the number of iterations can produce more accurate results but also increases the expense of the computation and adds latency.

If you do not specify `niters`, or if you specify a value that is too large, the algorithm uses a maximum value based on the data type of the inputs:

• Fixed-point inputs — The maximum number of iterations is one less than the word length of `theta`.

• Floating-point inputs — The maximum value is `52` for `double` or `23` for `single`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fi`

## Output Arguments

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Approximated complex exponential e^(j*`theta`), returned as a scalar, vector, matrix, or multidimensional array. The data type of the output depends on the input:

• When the input `theta` is floating point, the output data type is the same as the input type.

• When the input `theta` is fixed point, the output has the same word length as the input and a fraction length equal to the word length minus 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 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. The accuracy of the CORDIC rotation kernel depends on the choice of initial values for X, Y, and Z. This algorithm uses the following initial values:

• z0 is initialized to the θ input argument value

• x0 is initialized to 1/AN

• y0 is initialized to 0

### `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`.

## Version History

Introduced in R2010a