cordicrotate

CORDIC-based approximation of rotation of input

Syntax

v = cordicrotate(theta,u)

v = cordicrotate(theta,u,niters)

v = cordicrotate(___,ScaleOutput)

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(___,ScaleOutput) specifies whether to scale the output.

Examples

CORDIC-Based Complex Rotation

Run this code to evaluate the accuracy of 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');

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     -4.0229	 0.2563	 1049.8
4       -6.3096   0.2858	 1170.5    -3.2654	 0.5013	 2053.2
5       -6.0935   0.0697	 285.5     -3.6526	 0.1141	 467.2
6       -5.9766   0.0472	 193.5     -3.8413	 0.0746	 305.8
7       -6.0356   0.0118	 48.5      -3.7473	 0.0193	 79.2
8       -6.0063   0.0175	 71.5      -3.7947	 0.0280	 114.8
9       -6.0212   0.0026	 10.5      -3.7710	 0.0043	 17.8
10      -6.0283   0.0045	 18.5      -3.7590	 0.0076	 31.2
11      -6.0247   0.0009	 3.5       -3.7649	 0.0018	 7.2
12      -6.0227   0.0011	 4.5       -3.7678	 0.0012	 4.8
13      -6.0237   0.0001	 0.5       -3.7664	 0.0003	 1.2
14      -6.0244   0.0006	 2.5       -3.7656	 0.0010	 4.2
15      -6.0239   0.0001	 0.5       -3.7661	 0.0005	 2.2

Input Arguments

theta — Angle values in radians

scalar | vector | matrix | multidimensional array

Angle values in radians, specified as a signed or unsigned scalar, vector, matrix, or multidimensional array. All values of theta must be in the range [–2π 2π).

u — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a signed or unsigned scalar, vector, matrix, or multidimensional array. u can be a scalar value or have the same dimensions as theta. u can be real or complex valued.

niters — Number of iterations the CORDIC algorithm performs

positive integer-valued scalar

Number of iterations the CORDIC algorithm performs, specified as a positive integer-valued scalar. This argument is optional. 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.

ScaleOutput — Whether to scale output by inverse CORDIC gain factor

true or 1 (default) | false or 0

Whether to scale the output by the inverse CORDIC gain factor, specified as a Boolean value. 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.

Output Arguments

v — Approximated result of the CORDIC rotation algorithm

scalar | vector | matrix | multidimensional array

The data type of the output v depends on the data type of the input u:

• 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.

Algorithms

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, arcsine, arccosine, arctangent, and vector magnitude. You can also use this algorithm for divide, square root, hyperbolic, and logarithmic functions.

The precision of the CORDIC algorithm is a function of the data type used and the maximum shift value or number of iterations of the CORDIC kernel. Using a data type with a larger word length and performing more iterations of the CORDIC algorithm can reduce the numeric error of the result. However, doing so also increases the latency of the computation and the utilizes more hardware resources. For more information, see How to Set CORDIC Input Word Length and Maximum Shift Value to Achieve Desired Precision.

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, Jack E. “The CORDIC Trigonometric Computing Technique.” IRE Transactions on Electronic Computers. EC-8, no. 3 (Sept. 1959): 330–334.

[2] Andraka, Ray. “A Survey of CORDIC Algorithm for FPGA Based Computers.” In Proceedings of the 1998 ACM/SIGDA Sixth International Symposium on Field Programmable Gate Arrays, 191–200. https://dl.acm.org/doi/10.1145/275107.275139.

[3] Walther, J.S. “A Unified Algorithm for Elementary Functions.” In Proceedings of the May 18-20, 1971 Spring Joint Computer Conference, 379–386. https://dl.acm.org/doi/10.1145/1478786.1478840.

[4] Schelin, Charles W. “Calculator Function Approximation.” The American Mathematical Monthly, no. 5 (May 1983): 317–325. https://doi.org/10.2307/2975781.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Variable-size signals are not supported.

• The number of iterations the CORDIC algorithm performs, niters, must be a constant.

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

The cordicrotate function also supports MATLAB^® to High-Level Synthesis (HLS) code generation.

Version History

Introduced in R2011a