cordicatan2

CORDIC-based four quadrant inverse tangent

Syntax

theta = cordicatan2(y,x)

theta = cordicatan2(y,x,niters)

Description

theta = cordicatan2(y,x) computes the four quadrant arctangent of y and x using a CORDIC algorithm approximation.

theta = cordicatan2(y,x,niters) performs niters iterations of the algorithm.

Examples

Compute CORDIC Arctangent

Define floating-point Cartesian coordinates.

y = 0.5;
x = -0.5;

Use cordicatan2 to compute floating-point CORDIC arctangent. Compare the result to the arctangent computed using atan2.

theta_cdat2_float = cordicatan2(y,x)

theta_cdat2_float = 
2.3562

theta_atan2_float = atan2(y,x)

theta_atan2_float = 
2.3562

Define fixed-point Cartesian coordinates.

y = fi(0.5,1,16,15);
x = fi(-0.5,1,16,15);

Use cordicatan2 to compute fixed-point CORDIC arctangent. Compare the result to the arctangent computed using atan2.

theta_cdat2_fixpt = cordicatan2(y,x)

theta_cdat2_fixpt = 
    2.3562

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

theta_atan2_fixpt = atan2(y,x)

theta_atan2_fixpt = 
    2.3562

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

Input Arguments

y — Cartesian y-coordinate

scalar | vector | matrix | multidimensional array

Cartesian y-coordinate, specified as a scalar, vector, matrix, or multidimensional array.

y and x must be the same size. If they are not the same size, at least one value must be a scalar value. y and x must have the same data type.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes

x — Cartesian x-coordinate

scalar | vector | matrix | multidimensional array

Cartesian x-coordinate, specified as a scalar, vector, matrix, or multidimensional array.

y and x must be the same size. If they are not the same size, at least one value must be a scalar value. y and x must have the same data type.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fi
Complex Number Support: Yes

niters — Number of iterations of CORDIC algorithm

positive integer-valued scalar

Number of iterations of CORDIC algorithm, specified as a positive, integer-valued scalar.

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:

• For fixed-point operation, the maximum number of iterations is one less than the word length of y or x.

• For floating-point operation, 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

theta — Arctangent value

scalar | vector | matrix | multidimensional array

Arctangent value in the range [-pi, pi] radians, returned as a scalar, vector, matrix, or multidimensional array.

If y and x are floating-point numbers, then theta has the same data type as y and x. Otherwise, theta is a fixed-point data type with the same word length as y and x and with a best-precision fraction length for the [-pi, pi] range.

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 Diagram

CORDIC Vectoring Kernel

The accuracy of the CORDIC kernel depends on the choice of initial values for X, Y, and Z. This algorithm uses the following initial values:

• X₀ is initialized to the X input value

• Y₀ is initialized to the Y input value

• Z₀ is initialized 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.

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 cordicatan2 function also supports MATLAB^® to High-Level Synthesis (HLS) code generation.

Version History

Introduced in R2011b