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cordiccos

CORDIC-based approximation of cosine

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Syntax

y= cordiccos(theta,niters)

Description

y= cordiccos(theta,niters)computes the cosine ofthetausing aCORDICalgorithm approximation.

Input Arguments

theta

thetacan be a signed or unsigned scalar, vector, matrix, orN-dimensional array containing the angle values in radians. All values ofthetamust be real and in the range [–2π 2π).

niters

nitersis the number of iterations the CORDIC algorithm performs. This is an optional argument. When specified,nitersmust be a positive, integer-valued scalar. If you do not specifynitersor 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 oftheta. 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

yis the CORDIC-based approximation of the cosine oftheta. 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 theWordLength–2.

Examples

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Open Live Script

Compare the results produced by various iterations of thecordiccosalgorithm to the results of the double-precisioncosfunction.

% Create 1024 points between [0, 2*pi)stepSize = pi/512; thRadDbl = 0:stepSize:(2*pi - stepSize); thRadFxp = sfi(thRadDbl, 12);% signed, 12-bit fixed-pointcosThRef = cos(double(thRadFxp));% reference results%使用12位量化输入和vary the number% of iterations from 2 to 10.% Compare the fixed-point CORDIC results to the% double-precision trig function results.forniters = 2:2:10 cdcCosTh = cordiccos(thRadFxp, niters); errCdcRef = cosThRef - double(cdcCosTh);endfigure holdonaxis([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 ofthetato within 0.005187 of the double-precision cosine result.

More About

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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 also increases the expense of the computation and adds latency.

Algorithms

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Signal Flow Diagrams

CORDIC Rotation Kernel

Xrepresents the sine,Yrepresents the cosine, andZrepresents theta. The accuracy of the CORDIC rotation kernel depends on the choice of initial values forX,Y, andZ. This algorithm uses the following initial values:

z 0 is initialized to the θ input argument value x 0 is initialized to 1 A N y 0 is initialized to 0

fimath Propagation Rules

CORDIC functions discard any localfimathattached to the input.

The CORDIC functions use their own internalfimathwhen performing calculations:

  • OverflowAction—Wrap

  • RoundingMethod—Floor

The output has no attachedfimath.

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.

Extended Capabilities

See Also

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Topics

Introduced in R2010a

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