CORDIC

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CORDIC (digit-by-digit method, Volder's algorithm) (for COordinate Rotation DIgital Computer) is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions. It is commonly used when no hardware multiplier is available (e.g., simple microcontrollers and FPGAs) as the only operations it requires are addition, subtraction, bitshift and table lookup.

Origins

The modern CORDIC algorithm was first described in 1959 by Jack E. Volder. It was developed at the aeroelectronics department of Convair to replace the analog resolver in the B-58 bomber's navigation computer.^[1]

Although CORDIC is similar to mathematical techniques published by Henry Briggs as early as 1624, it is optimized for low complexity finite state CPUs.

John Stephen Walther at Hewlett-Packard further generalized the algorithm, allowing it to calculate hyperbolic and exponential functions, logarithms, multiplications, divisions, and square roots.^[2]

Originally, CORDIC was implemented using the binary numeral system. In the 1970s, decimal CORDIC became widely used in pocket calculators, most of which operate in binary-coded-decimal (BCD) rather than binary.

CORDIC is particularly well-suited for handheld calculators, an application for which cost (e.g., chip gate count has to be minimized) is much more important than is speed. Also the CORDIC subroutines for trigonometric and hyperbolic functions can share most of their code.

Applications

Hardware

CORDIC is generally faster than other approaches when a hardware multiplier is unavailable (e.g., in a microcontroller based system), or when the number of gates required to implement the functions it supports should be minimized (e.g., in an FPGA).

On the other hand, when a hardware multiplier is available (e.g., in a DSP microprocessor), table-lookup methods and power series are generally faster than CORDIC. In recent years, the CORDIC algorithm is used extensively for various biomedical applications, especially in FPGA implementations.

Software

Many older systems with integer-only CPUs have implemented CORDIC to varying extents as part of their IEEE Floating Point libraries. As most modern general-purpose CPUs have floating point registers with common operations such as add, subtract, multiply, divide, sin, cos, square root, log10, natural log, the need to implement CORDIC in them with software is nearly non-existent. Only microcontroller or special safety and time-constrained software applications would need to consider using CORDIC.

Mode of operation: rotation mode

CORDIC can be used to calculate a number of different functions. This explanation shows how to use CORDIC in rotation mode to calculate sine and cosine of an angle, and assumes the desired angle is given in radians and represented in a fixed point format. To determine the sine or cosine for an angle β, the y or x coordinate of a point on the unit circle corresponding to the desired angle must be found. Using CORDIC, we would start with the vector v₀:

$v_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

In the first iteration, this vector would be rotated 45° counterclockwise to get the vector v₁. Successive iterations will rotate the vector in one or the other direction by size decreasing steps, until the desired angle has been achieved. Step i size is arctan(1/(2ⁱ⁻¹)) for i = 1, 2, 3, ….

An illustration of the CORDIC algorithm in progress.

More formally, every iteration calculates a rotation, which is performed by multiplying the vector v_{i − 1} with the rotation matrix R_i:

$v_{i} = R_{i}v_{i-1}\$

The rotation matrix R is given by:

$R_{i} = \left( \begin{array}{rr} \cos \gamma_{i} & -\sin \gamma_{i} \\ \sin \gamma_{i} & \cos \gamma _{i}\end{array} \right)$

Using the following two trigonometric identities

$\begin{align} \cos \alpha & = & {1 \over \sqrt{1 + \tan^2 \alpha}} \\ \sin \alpha & = & {{\tan \alpha} \over \sqrt{1 + \tan^2 \alpha}} \end{align}$

the rotation matrix becomes:

$R_i = {1 \over \sqrt{1 + \tan^2 \gamma_i}} \begin{pmatrix} 1 & -\tan \gamma_i \\ \tan \gamma_i & 1 \end{pmatrix}$

The expression for the rotated vector v_i = R_iv_{i − 1} then becomes:

$v_i = {1 \over \sqrt{1 + \tan^2 \gamma_i}} \begin{pmatrix} x_{i-1} - y_{i-1} \tan \gamma_i \\ x_{i-1} \tan \gamma_i + y_{i-1} \end{pmatrix}$

where x_{i − 1} and y_{i − 1} are the components of v_{i − 1}. Restricting the angles γ_i so that tan γ_i takes on the values $\pm 2^{-i}$ the multiplication with the tangent can be replaced by a division by a power of two, which is efficiently done in digital computer hardware using a bit shift. The expression then becomes:

$v_i = K_i \begin{pmatrix} x_{i-1}-\sigma_i 2^{-i} y_{i-1} \\ \sigma_i 2^{-i} x_{i-1} + y_{i-1} \end{pmatrix}$

where

$K_i = {1 \over \sqrt{1 + 2^{-2i}}}$

and σ_i can have the values of −1 or 1 and is used to determine the direction of the rotation: if the angle β_i is positive then σ_i is 1, otherwise it is −1.

We can ignore K_i in the iterative process and then apply it afterward by a scaling factor:

$K(n) = \prod_{i=0}^{n-1} K_i = \prod_{i=0}^{n-1} 1/\sqrt{1 + 2^{-2i}}$

which is calculated in advance and stored in a table, or as a single constant if the number of iterations is fixed. This correction could also be made in advance, by scaling v₀ and hence saving a multiplication. Additionally it can be noted that

$K = \lim_{n \to \infty}K(n) \approx 0.6072529350088812561694$^[3]

to allow further reduction of the algorithm's complexity. After a sufficient number of iterations, the vector's angle will be close to the wanted angle β. For most ordinary purposes, 40 iterations (n = 40) is sufficient to obtain the correct result to the 10th decimal place.

The only task left is to determine if the rotation should be clockwise or counterclockwise at every iteration (choosing the value of σ). This is done by keeping track of how much we rotated at every iteration and subtracting that from the wanted angle, and then checking if β_{n + 1} is positive and we need to rotate clockwise or if it is negative we must rotate counterclockwise in order to get closer to the wanted angle β.

$\beta_{i} = \beta_{i-1} - \sigma_i \gamma_i. \quad \gamma_i = \arctan 2^{-i},$

The values of γ_n must also be precomputed and stored. But for small angles, arctan(γ_n) = γ_n in fixed point representation, reducing table size.

As can be seen in the illustration above, the sine of the angle β is the y coordinate of the final vector v_n, while the x coordinate is the cosine value.

Mode of operation: vectoring mode

The rotation-mode algorithm described above can rotate any vector (not only a unit vector aligned along the x axis) by an angle between -90° and +90°. Decisions on the direction of the rotation depend on β_i being positive or negative.

The vectoring-mode of operation requires a slight modification of the algorithm. It starts with a vector the x coordinate of which is positive and the y coordinate is arbitrary. Successive rotations have the goal of rotating the vector to the x axis (and therefore reducing the y coordinate to zero). At each step, the value of y determines the direction of the rotation. The final value of β_i contains the total angle of rotation. The final value of x will be the magnitude of the original vector scaled by K. So, an obvious use of the vectoring mode is the transformation from rectangular to polar coordinates.

Software implementation

The following is a MATLAB/GNU Octave implementation of CORDIC that does not rely on any transcendental functions except in the precomputation of tables. If the number of iterations n is predetermined, then the second table can be replaced by a single constant. The two-by-two matrix multiplication represents a pair of simple shifts and adds. With MATLAB's standard double-precision arithmetic and "format long" printout, the results increase in accuracy for n up to about 48.

function v = cordic(beta,n)
% This function computes v = [cos(beta), sin(beta)] (beta in radians)
% using n iterations. Increasing n will increase the precision.
 
if beta < -pi/2 || beta > pi/2
    if beta < 0
        v = cordic(beta + pi, n);
    else
        v = cordic(beta - pi, n);
    end
    v = -v; % flip the sign for second or third quadrant
    return
end
 
% Initialization of tables of constants used by CORDIC
% need a table of arctangents of negative powers of two, in radians:
% angles = atan(2.^-(0:27));
angles =  [  ...
    0.78539816339745   0.46364760900081   0.24497866312686   0.12435499454676 ...
    0.06241880999596   0.03123983343027   0.01562372862048   0.00781234106010 ...
    0.00390623013197   0.00195312251648   0.00097656218956   0.00048828121119 ...
    0.00024414062015   0.00012207031189   0.00006103515617   0.00003051757812 ...
    0.00001525878906   0.00000762939453   0.00000381469727   0.00000190734863 ...
    0.00000095367432   0.00000047683716   0.00000023841858   0.00000011920929 ...
    0.00000005960464   0.00000002980232   0.00000001490116   0.00000000745058 ];
% and a table of products of reciprocal lengths of vectors [1, 2^-j]:
Kvalues = [ ...
    0.70710678118655   0.63245553203368   0.61357199107790   0.60883391251775 ...
    0.60764825625617   0.60735177014130   0.60727764409353   0.60725911229889 ...
    0.60725447933256   0.60725332108988   0.60725303152913   0.60725295913894 ...
    0.60725294104140   0.60725293651701   0.60725293538591   0.60725293510314 ...
    0.60725293503245   0.60725293501477   0.60725293501035   0.60725293500925 ...
    0.60725293500897   0.60725293500890   0.60725293500889   0.60725293500888 ];
Kn = Kvalues(min(n, length(Kvalues)));
 
% Initialize loop variables:
v = [1;0]; % start with 2-vector cosine and sine of zero
poweroftwo = 1;
angle = angles(1);
 
% Iterations
for j = 0:n-1;
    if beta < 0
        sigma = -1;
    else
        sigma = 1;
    end
    factor = sigma * poweroftwo;
    R = [1, -factor; factor, 1];
    v = R * v; % 2-by-2 matrix multiply
    beta = beta - sigma * angle; % update the remaining angle
    poweroftwo = poweroftwo / 2;
    % update the angle from table, or eventually by just dividing by two
    if j+2 > length(angles)
        angle = angle / 2;
    else
        angle = angles(j+2);
    end
end
 
% Adjust length of output vector to be [cos(beta), sin(beta)]:
v = v * Kn;
return

Hardware implementation

The number of logic gates for the implementation of a CORDIC is roughly comparable to the number required for a multiplier as both require combinations of shifts and additions. The choice for a multiplier-based or CORDIC-based implementation will depend on the context. The multiplication of two complex numbers represented by their real and imaginary components (rectangular coordinates), for example, requires 4 multiplications, but could be realized by a single CORDIC operating on complex numbers represented by their polar coordinates, especially if the magnitude of the numbers is not relevant (multiplying a complex vector with a vector on the unit circle actually amounts to a rotation). CORDICs are often used in circuits for telecommunications such as digital down converters.

Related algorithms

CORDIC is part of the class of "shift-and-add" algorithms, as are the logarithm and exponential algorithms derived from Henry Briggs' work. Another shift-and-add algorithm which can be used for computing many elementary functions is the BKM algorithm, which is a generalization of the logarithm and exponential algorithms to the complex plane. For instance, BKM can be used to compute the sine and cosine of a real angle x (in radians) by computing the exponential of 0 + ix, which is cos x + isin x. The BKM algorithm is slightly more complex than CORDIC, but has the advantage that it does not need a scaling factor (K).

History

Volder was inspired by the following formula in the 1946 edition of the CRC Handbook of Chemistry and Physics:

$\begin{align} K_n R \sin(\theta\pm\phi) &= R \sin(\theta) \pm 2^{-n} R \cos(\theta)\\ K_n R \cos(\theta\pm\phi) &= R \cos(\theta) \mp 2^{-n} R \sin(\theta) \end{align}$

with $K_n = \sqrt{1+2^{-2n}}, \tan(\phi) = 2^{-n}.$ ^[1]

Some of the prominent early applications of CORDIC were in the Convair navigation computers CORDIC I to CORDIC III,^[1] the Hewlett-Packard HP-9100 and HP-35 calculators,^[4] the Intel 80x87 coprocessor series until Intel 80486, and Motorola 68881.^[5]

Decimal CORDIC was first suggested by Hermann Schmid and Anthony Bogacki.^[6]

Notes

1. ^ ^{a b c} J. E. Volder, "The Birth of CORDIC", J. VLSI Signal Processing 25, 101 (2000).
2. ^ J. S. Walther, "The Story of Unified CORDIC", J. VLSI Signal Processing 25, 107 (2000).
3. ^ J.-M. Muller, Elementary Functions: Algorithms and Implementation, 2nd Edition (Birkhäuser, Boston, 2006), p. 134.
4. ^ D. Cochran, "Algorithms and Accuracy in the HP 35", Hewlett Packard J. 23, 10 (1972).
5. ^ R. Nave, "Implementation of Transcendental Functions on a Numerics Processor", Microprocessing and Microprogramming 11, 221 (1983).
6. ^ H. Schmid and A. Bogacki, "Use Decimal CORDIC for Generation of Many Transcendental Functions", EDN Magazine, February 20, 1973, p. 64.

References

• Jack E. Volder, The CORDIC Trigonometric Computing Technique, IRE Transactions on Electronic Computers, pp330-334, September 1959
• Daggett, D. H., Decimal-Binary conversions in CORDIC, IRE Transactions on Electronic Computers, Vol. EC-8 #5, pp335–339, IRE, September 1959
• John S. Walther, A Unified Algorithm for Elementary Functions, Proc. of Spring Joint Computer Conference, pp379–385, May 1971
• J. E. Meggitt, Pseudo Division and Pseudo Multiplication Processes, IBM Journal, April 1962
• Vladimir Baykov, Problems of Elementary Functions Evaluation Based on Digit by Digit (CORDIC) Technique, PhD thesis, Leningrad State Univ. of Electrical Eng., 1972
• Schmid, Hermann, Decimal computation. New York, Wiley, 1974
• V.D.Baykov,V.B.Smolov, Hardware implementation of elementary functions in computers, Leningrad State University, 1975, 96p.*Full Text
• Senzig, Don, Calculator Algorithms, IEEE Compcon Reader Digest, IEEE Catalog No. 75 CH 0920-9C, pp139–141, IEEE, 1975.
• V.D.Baykov,S.A.Seljutin, Elementary functions evaluation in microcalculators, Moscow, Radio & svjaz,1982,64p.
• Vladimir D.Baykov, Vladimir B.Smolov, Special-purpose processors: iterative algorithms and structures, Moscow, Radio & svjaz, 1985, 288 pages
• M. E. Frerking, Digital Signal Processing in Communication Systems, 1994
• Vitit Kantabutra, On hardware for computing exponential and trigonometric functions, IEEE Trans. Computers 45 (3), 328-339 (1996)
• Andraka, Ray, A survey of CORDIC algorithms for FPGA based computers
• Henry Briggs, Arithmetica Logarithmica. London, 1624, folio
• CORDIC Bibliography Site, Shaoyun Wang, July 2011
• The secret of the algorithms, Jacques Laporte, Paris 1981
• Digit by digit methods, Jacques Laporte, Paris 2006
• Ayan Banerjee, FPGA realization of a CORDIC based FFT processor for biomedical signal processing, Kharagpur, 2001
• CORDIC Architectures: A Survey, B. Lakshmi and A. S. Dhar, Journal: VLSI Design, January 2010
• Implementation of a CORDIC Algorithm in a Digital Down-Converter, C. Cockrum, Fall 2008

External links

• CORDIC Bibliography Site
• Another USENET discussion
• BASIC Stamp, CORDIC math implementation
• CORDIC as implemented in the ROM of the HP-35 - Jacques Laporte (step by step analysis, simulator running the real ROM with breakpoints and trace facility.
• CORDIC implementation in verilog.
• CORDIC information
• CORDIC Vectoring with Arbitrary Target Value
• PicBasic Pro, Pic18 CORDIC math implementation
• Python CORDIC implementation
• Simple C code for fixed-point CORDIC
• The CORDIC Algorithm
• Tutorial and MATLAB Implementation - Using CORDIC to Estimate Phase of a Complex Number
• USENET discussion
• Descriptions of hardware CORDICs in Arx with testbenches in C++ and VHDL