cordicpol2cart

CORDIC-based approximation of polar-to-Cartesian conversion

Syntax

```[x,y] = cordicpol2cart(theta,r) [x,y] = cordicpol2cart(theta,r,niters) [x,y] = cordicpol2cart(theta,r,Name,Value) [x,y] = cordicpol2cart(theta,r,niters,Name,Value) ```

Description

`[x,y] = cordicpol2cart(theta,r)` returns the Cartesian xy coordinates of `r`* e^(j*`theta`) using a CORDIC algorithm approximation.

`[x,y] = cordicpol2cart(theta,r,niters)` performs `niters` iterations of the algorithm.

`[x,y] = cordicpol2cart(theta,r,Name,Value)` scales the output depending on the Boolean value of `b`.

`[x,y] = cordicpol2cart(theta,r,niters,Name,Value)` specifies both the number of iterations and `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). `r` `r` contains the input magnitude values and can be a scalar or have the same dimensions as `theta`. `r` must be real 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 `r` 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 also increases the expense of the computation and adds latency.

Name-Value 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

 `[x,y]` `[x,y]` contains the approximated Cartesian coordinates. When the input `r` is floating point, the output `[x,y]` has the same data type as the input. When the input `r` is a signed integer or fixed point data type, the outputs `[x,y]` are signed `fi` objects. These `fi` objects have word lengths that are two bits larger than that of `r`. Their fraction lengths are the same as the fraction length of `r`. When the input `r` is an unsigned integer or fixed point, the outputs `[x,y]` are signed `fi` objects. These `fi` objects have word lengths are three bits larger than that of `r`. Their fraction lengths are the same as the fraction length of `r`.

Examples

Run the following code, and evaluate the accuracy of the CORDIC-based Polar-to-Cartesian conversion.

 ```wrdLn = 16; theta = fi(pi/3, 1, wrdLn); u = fi( 2.0, 1, wrdLn); fprintf('\n\nNITERS\tX\t\t ERROR\t LSBs\t\tY\t\t ERROR\t LSBs\n'); fprintf('------\t-------\t ------\t ----\t\t-------\t ------\t ----\n'); for niters = 1:(wrdLn - 1) [x_ref, y_ref] = pol2cart(double(theta),double(u)); [x_fi, y_fi] = cordicpol2cart(theta, u, niters); x_dbl = double(x_fi); y_dbl = double(y_fi); x_err = abs(x_dbl - x_ref); y_err = abs(y_dbl - y_ref); fprintf('%d\t%1.4f\t %1.4f\t %1.1f\t\t%1.4f\t %1.4f\t %1.1f\n',... niters,x_dbl,x_err,(x_err * pow2(x_fi.FractionLength)),... y_dbl,y_err,(y_err * pow2(y_fi.FractionLength))); end fprintf('\n'); NITERS X ERROR LSBs Y ERROR LSBs ------ ------- ------ ---- ------- ------ ---- 1 1.4142 0.4142 3392.8 1.4142 0.3178 2603.8 2 0.6324 0.3676 3011.2 1.8973 0.1653 1354.2 3 1.0737 0.0737 603.8 1.6873 0.0448 366.8 4 0.8561 0.1440 1179.2 1.8074 0.0753 617.2 5 0.9672 0.0329 269.2 1.7505 0.0185 151.2 6 1.0214 0.0213 174.8 1.7195 0.0126 102.8 7 0.9944 0.0056 46.2 1.7351 0.0031 25.2 8 1.0079 0.0079 64.8 1.7274 0.0046 37.8 9 1.0011 0.0011 8.8 1.7313 0.0007 5.8 10 0.9978 0.0022 18.2 1.7333 0.0012 10.2 11 0.9994 0.0006 5.2 1.7323 0.0003 2.2 12 1.0002 0.0002 1.8 1.7318 0.0002 1.8 13 0.9999 0.0002 1.2 1.7321 0.0000 0.2 14 0.9996 0.0004 3.2 1.7321 0.0000 0.2 15 0.9998 0.0003 2.2 1.7321 0.0000 0.2```

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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. This algorithm takes its initial values for X, Y, and Z from the inputs, `r` 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`.