cordiccos

CORDIC-based approximation of cosine

Syntax

y = cordiccos(theta)

y = cordiccos(theta,niters)

Description

y = cordiccos(theta) computes the cosine of theta using a CORDIC algorithm approximation.

y = cordiccos(theta,niters) specifies niters iterations of the CORDIC algorithm.

Examples

Compare Results of cordiccos and cos Functions

Compare the results produced by various iterations of the cordiccos algorithm to the results of the double-precision cos function.

Create 1024 points between [0,2*pi).

stepSize = pi/512;
thRadDbl = 0:stepSize:(2*pi - stepSize);

Set the fixed-point type to be a signed, 12-bit fixed-point data type. Use the cos function with a double-precision input as reference.

thRadFxp = fi(thRadDbl,1,12);
cosThRef = cos(double(thRadFxp));

Use 12-bit quantized inputs and vary the number of iterations from 2 to 10. Compare the fixed-point CORDIC results to the double-precision trigonometric function results.

for niters = 2:2:10
    cdcCosTh  = cordiccos(thRadFxp,niters);
    errCdcRef = cosThRef - double(cdcCosTh);    
end

Plot the results.

figure
hold on
axis([0 2*pi -1.25 1.25]);
    plot(thRadFxp,cosThRef,'b');
    plot(thRadFxp,cdcCosTh,'g');
    plot(thRadFxp,errCdcRef,'r');
    ylabel('cos(\Theta)');
    gca.XTick = 0:pi/2:2*pi;
    gca.XTickLabel = {'0','pi/2','pi','3*pi/2','2*pi'};
    gca.YTick = -1:0.5:1;
    gca.YTickLabel = {'-1.0','-0.5','0','0.5','1.0'};
    ref_str = 'Reference: cos(double(\Theta))';
    cdc_str = sprintf('12-bit CORDIC cosine; N = %d',niters);
    err_str = sprintf('Error (max = %f)', max(abs(errCdcRef)));
    legend(ref_str,cdc_str,err_str);

After 10 iterations, the CORDIC algorithm has approximated the cosine of theta to within 0.005187 of the double-precision cosine result.

Input Arguments

theta — Input angle in radians

scalar | vector | matrix | multidimensional array

Input angle in radians, 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).

niters — Number of iterations

positive integer-valued scalar

Number of iterations the CORDIC algorithm performs, specified as 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 one less than the word length of theta. 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.

Output Arguments

y — CORDIC-based approximation of cosine of theta

scalar | vector | matrix | multidimensional array

CORDIC-based approximation of the cosine of theta, returned as a scalar, vector, matrix, or multidimensional array.

When the input to the function is floating point, the output data type is the same as the input data type. When the input is fixed point, the output has the same word length as the input, and a fraction length equal to the WordLength – 2.

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 sine, Y represents the cosine, 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:

$$\begin{array}{l}{z}_0\text{ is initialized to the }\theta \text{ input argument value}\\ x_0\text{ is initialized to }\frac{1}{A_N}\\ y_0\text{ is initialized to }0\end{array}$$

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™.

You can generate HDL code for cordiccos function.

Version History

Introduced in R2010a