cordiccart2pol

CORDIC-based approximation of Cartesian-to-polar conversion

Syntax

[theta,r] = cordiccart2pol(x,y)
[theta,r] = cordiccart2pol(x,y, niters)
[theta,r] = cordiccart2pol(x,y, niters,'ScaleOutput',b)
[theta,r] = cordiccart2pol(x,y, 'ScaleOutput',b)

Description

[theta,r] = cordiccart2pol(x,y) using a CORDIC algorithm approximation, returns the polar coordinates, angle theta and radius r, of the Cartesian coordinates, x and y.

[theta,r] = cordiccart2pol(x,y, niters) performs niters iterations of the algorithm.

[theta,r] = cordiccart2pol(x,y, niters,'ScaleOutput',b) specifies both the number of iterations and, depending on the Boolean value of b, whether to scale the r output by the inverse CORDIC gain value.

[theta,r] = cordiccart2pol(x,y, 'ScaleOutput',b) scales the r output by the inverse CORDIC gain value, depending on the Boolean value of b.

Input Arguments

x,y

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

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 Pair 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

theta

theta contains the polar coordinates angle values, which are in the range [–pi, pi] radians. If x and y are floating-point, then theta has the same data type as x and y. Otherwise, theta is a fixed-point data type with the same word length as x and y and with a best-precision fraction length for the [-pi, pi] range.

r

r contains the polar coordinates radius magnitude values. r is real-valued and can be a scalar value or have the same dimensions as theta If the inputs x,y are fixed-point values, r is also fixed point (and is always signed, with binary point scaling). Both x,y input values must have the same data type. If the inputs are signed, then the word length of r is the input word length + 2. If the inputs are unsigned, then the word length of r is the input word length + 3. The fraction length of r is always the same as the fraction length of the x,y inputs.

Examples

Convert fixed-point Cartesian coordinates to polar coordinates.

[thPos,r]=cordiccart2pol(sfi([0.75:-0.25:-1.0],16,15),sfi(0.5,16,15))

thPos =

    0.5881  0.7854  1.1072  1.5708  2.0344  2.3562  2.5535  2.6780

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

r =

    0.9014  0.7071  0.5591  0.5000  0.5591  0.7071  0.9014  1.1180

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 18
        FractionLength: 15

[thNeg,r]=...
  cordiccart2pol(sfi([0.75:-0.25:-1.0],16,15),sfi(-0.5,16,15))

thNeg =

 -0.5881 -0.7854 -1.1072 -1.5708 -2.0344 -2.3562 -2.5535 -2.6780

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

r =

 0.9014  0.7071  0.5591  0.5000  0.5591  0.7071  0.9014  1.1180

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 18
        FractionLength: 15

More About

expand all

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 also increases the expense of the computation and adds latency.

Algorithms

Signal Flow Diagrams

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 valuey0 is initialized to the y input valuez0 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:

  • OverflowActionWrap

  • RoundingMethodFloor

The output has no attached fimath.

References

[1] Volder, JE. "The CORDIC Trigonometric Computing Technique." IRE Transactions on Electronic Computers. Vol. EC-8, September 1959, pp. 330–334.

[2] Andraka, R. "A survey of CORDIC algorithm for FPGA based computers." Proceedings of the 1998 ACM/SIGDA sixth international symposium on Field programmable gate arrays. Feb. 22–24, 1998, pp. 191–200.

[3] Walther, J.S. "A Unified Algorithm for Elementary Functions." Hewlett-Packard Company, Palo Alto. Spring Joint Computer Conference, 1971, pp. 379–386. (from the collection of the Computer History Museum). www.computer.org/csdl/proceedings/afips/1971/5077/00/50770379.pdf

[4] Schelin, Charles W. "Calculator Function Approximation." The American Mathematical Monthly. Vol. 90, No. 5, May 1983, pp. 317–325.

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