|
|
| Line 1: |
Line 1: |
| {{Trigonometry}}
| | == Hollister Australia he says == |
| '''CORDIC''' (for '''CO'''ordinate '''R'''otation '''DI'''gital '''C'''omputer), also known as the '''digit-by-digit method''' and '''Volder's algorithm''', is a simple and efficient [[algorithm]] to calculate [[hyperbolic function|hyperbolic]] and [[trigonometric function]]s. It is commonly used when no [[hardware multiplier]] is available (e.g., simple [[microcontroller]]s and [[FPGA]]s) as the only operations it requires are addition, subtraction, [[bitshift]] and [[lookup table|table lookup]].
| |
|
| |
|
| == Origins ==
| | So while it very cool that the artist that has foundfinancialsuccess with the chart topping Gnarls Barkley, could put out music that challenges, it also a sad reminder of whatintellectual property is treated like; [http://www.outerrimtradingco.com.au/supersecret/affiliate.asp Hollister Australia] a commodity. "_A special edition book of Lynch's visuals for the album has been released and it is available here. <br><br>We are inside a catch 22, where many health professionals feel they cannot recommend Tai-chi because too much of [http://www.matesinconstruction.com.au/flux/plugins/activation/inc_activation.asp Nike Free 3.0] the preliminary research is anecdotal. However, when Tai Chi is jockeying for position to get a crumb of the .5% of total NIH money going to ALL complimentary and alternative medical therapies . <br><br>3) The court system: The whole can do anything a man can do, empowered thing reminds me of a [http://www.romagnolabeefgenetics.com.au/common/standard.asp Nike Air Max 90] tactic John Stewart likes to use. He makes serious commentary that he wants to affect how people think, but when he gets called onto it, he says, I just a comedian. <br><br>The reality of the matter is that Councils can't solve this issue on their own. This is something we can and should do for ourselves.. In case your site is down or takes to long to open up when visitors are looking to get in, it may lead them to click out of your site and move on to the next. They will also more than likely not visit later on since they will remember their bad experience. <br><br>Calm and collected: Whenever you bump into your ex attempt to control the urge to beg these to come home. Act calm and picked up. It has been 20c here at night. The old dog's chain had frozen in the ice and she couldn't reach her food as well as their water was frozen solid. <br><br>NOTES: Only Sacramento and Washington have fewer wins on the road than Philly. Wilkins finished with 13 points. Pekovic stayed in Minnesota for the AllStar break rather than seek out the sun's rays and the sand as so many of his teammates did. He went ice fishing on a nearby lake, and Collins increased to him before the game and playfully asked him why he wasn't invited. <br><br>We all need these things, but they don't need to pay top dollar for it. It would do well for corporations to realize that hiring a web development firm is not like buying a suit. Both IBS and lactose malabsorption have similar symptoms. Both conditions are quite common, at least 10% of the general population. <br><br>They want to take the crown for the pens, that fine. They may be king of the pens. Asian folks are like the wonderful thing about the land itself. You are able to tour and explore different natural spots in japan wherein it is easy to meet and also be friends [http://www.taoist.org.au/custom/emailsend.asp Cheap Polo Shirts] with Asian beauties.<ul> |
| 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 electronics|analog]] [[resolver (electrical)|resolver]] in the [[B-58 Hustler|B-58]] bomber's [[navigation]] computer.<ref name="volder00">[http://dx.doi.org/10.1023/A:1008110704586 J. E. Volder, "The Birth of CORDIC", J. VLSI Signal Processing '''25''', 101 (2000).]</ref>
| | |
| | <li>[http://www.tztea.com.cn/news/html/?209602.html http://www.tztea.com.cn/news/html/?209602.html]</li> |
| | |
| | <li>[http://forum.joshsisto.com/profile.php?id=60835 http://forum.joshsisto.com/profile.php?id=60835]</li> |
| | |
| | <li>[http://281091.free.fr/index.php?site=forum_topic&topic=4663 http://281091.free.fr/index.php?site=forum_topic&topic=4663]</li> |
| | |
| | <li>[http://www.qzjmych.com/news/html/?83672.html http://www.qzjmych.com/news/html/?83672.html]</li> |
| | |
| | <li>[http://www.heatherbrewer.com/board/read.php?15,132951 http://www.heatherbrewer.com/board/read.php?15,132951]</li> |
| | |
| | </ul> |
|
| |
|
| Although CORDIC is similar to mathematical techniques published by [[Henry Briggs (mathematician)|Henry Briggs]] as early as 1624, it is optimized for low complexity finite state CPUs.
| | == Hollister Australia Especially after the senate vote == |
|
| |
|
| [[John Stephen Walther]] at [[Hewlett-Packard]] further generalized the algorithm, allowing it to calculate [[hyperbolic function|hyperbolic]] and [[exponential function]]s, [[logarithm]]s, [[multiplication]]s, [[Division (mathematics)|division]]s, and [[square root]]s.<ref>[http://dx.doi.org/10.1023/A:1008162721424 J. S. Walther, "The Story of Unified CORDIC", J. VLSI Signal Processing '''25''', 107 (2000).]</ref>
| | Justice Ronan Keane concluded in November 1981 the fire was [http://www.outerrimtradingco.com.au/supersecret/affiliate.asp Hollister Australia] probably brought on by arson. This finding, which has been disputed ever since, legally exonerated the owners from responsibility. He rises to inform you compassion. For the Lord is really a God of justice. <br><br>Hear most dating site and obey that top command, Thy Son blest errand from above? The idea in Witchcraft the most horrid free chat and internet dating sites Form of Superstition. I go no step further, answered most dating site Hugh sullenly? What is tennessee dating site there for the reason that word.. <br><br>Especially after the senate vote, we called constituents of democrats that voted against the bill (with the exception of Reid who voted No for tactical purposes) asking them to call in and express their displeasure about this. And people are responsive. <br><br>There's scope to expand and build new beds as well. This is a great project linking local stakeholders, where the community works in union with school pupils and native corporate volunteers, rather than individuals having their own plot. Others believe that the peony is secreting a sugary substance that attracts ants to feed. In any case, the ant is enjoying a symbiotic relationship with the peony and is not doing any harm when you are on the flower bud. <br><br>Before you take to the road in a private car, however experienced your accompanying driver may be, have some lessons with an Approved Driving Instructor first. The advantage of this is that the instructor will be able to use the dual controls and help you get out of trouble. <br><br>You listened closely, you would have likely heard a sizable sigh of relief come from the Buffalo dressing room following a win. While the [http://www.gregoryburgessarchitects.com.au/Mobile/Css/style.asp Timberland Boots Australia] team was 33 entering the tilt, the make an impression on the KHawks was their most convincing W of the season and it couldn have come at a better time.. <br><br>The presence of refulgent lakes and verdant forests make Mount Abu a warm tourist destination. The Mount Abu hotels will also be topclass and come with all facilities.. While many movie download sites offer a lot of the same things, you will find those that stand above the rest. The issue with mainstream sites, like Netflix, Blockbuster and others, is that they just don't compare with regards to selection. <br><br>I have ultra rapid cycling bipolar disorder type 1 and was diagnosed when I was 14, I am 27 now. This illness is [http://www.glendavishotel.com.au/ximages/class.asp Vibram Five Fingers Sydney] very genetic for me in that my father has type 1 bipolar as well as my grandpa on my mom's side from the family. After your web hosting account has been created on [http://www.romavilla.com/graphics/products/large/constants.asp Nike Store Australia] our side we will mail a hosting welcome letter. You can then upload your files, using the IP address to get it all working, set up email addresses, etc.<ul> |
| | | |
| Originally, CORDIC was implemented using the [[binary numeral system]]. In the 1970s, [[decimal]] CORDIC became widely used in pocket [[calculator]]s, most of which operate in [[Binary-coded decimal|binary-coded-decimal]] (BCD) rather than binary.
| | <li>[http://www.txfa.cn/bbs/forum.php?mod=viewthread&tid=145506 http://www.txfa.cn/bbs/forum.php?mod=viewthread&tid=145506]</li> |
| | | |
| CORDIC is particularly well-suited for handheld [[calculator]]s, an application for which cost is much more important than speed (e.g., chip gate count has to be minimized). Also the CORDIC [[subroutines]] for trigonometric and hyperbolic functions can share most of their code.
| | <li>[http://torncraft.tk/forum/viewtopic.php?f=6&t=386773 http://torncraft.tk/forum/viewtopic.php?f=6&t=386773]</li> |
| | | |
| == Applications ==
| | <li>[http://www.ebookabc.net/bbs/forum.php?mod=viewthread&tid=48331&fromuid=4394 http://www.ebookabc.net/bbs/forum.php?mod=viewthread&tid=48331&fromuid=4394]</li> |
| CORDIC uses simple shift-add operations for several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, square-root calculation, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been utilized for applications in diverse areas such as signal and image processing, communication systems, robotics and 3-D graphics apart from general scientific and technical computation.<ref>P. K. Meher, J. Valls, T-B Juang, K. Sridharan, and K. Maharatna, ‘50 Years of CORDIC: Algorithms, Architectures and Applications,’ IEEE Transactions on Circuits & Systems-I: RegularPapers, vol.56, no.9, pp.1893- 1907, September 2009</ref><ref>P. K. Meher and S. Y. Park, ‘CORDIC Designs for Fixed Angle of Rotation,’ IEEE Transactions on
| | |
| VLSI Systems, vol.21, no.2, pp.217-228, February 2013.</ref>
| | <li>[http://51blx.com/bbs/showtopic-7440.aspx http://51blx.com/bbs/showtopic-7440.aspx]</li> |
| | | |
| === Hardware ===
| | <li>[http://daili.xw68.com/in2039/news/html/?27466.html http://daili.xw68.com/in2039/news/html/?27466.html]</li> |
| CORDIC is generally faster than other approaches when a hardware multiplier is not available (''e.g.'', a microcontroller), or when the number of gates required to implement the functions it supports should be minimized (''e.g.'', in an [[FPGA]]).
| | |
| | | </ul> |
| 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 has been 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 the sine and cosine of an angle, and assumes the desired angle is given in radians and represented in a [[Fixed-point arithmetic|fixed point]] format. To determine the sine or cosine for an angle <math> \beta </math>, 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 <math> v_0 </math>:
| |
| | |
| : <math> v_0 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} </math>
| |
| | |
| [[Image:CORDIC-illustration.png|thumb|300px|An illustration of the CORDIC algorithm in progress.]]
| |
| In the first iteration, this vector is rotated 45° counterclockwise to get the vector <math> v_1 </math>. Successive iterations 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/(2<sup>''i''−1</sup>)) for ''i'' = 1, 2, 3, ….
| |
| | |
| More formally, every iteration calculates a rotation, which is performed by multiplying the vector <math>v_{i-1}</math> with the [[rotation matrix]] <math>R_{i}</math>:
| |
| | |
| : <math> v_{i} = R_{i}v_{i-1}\ </math>
| |
| | |
| The rotation matrix is given by:
| |
| | |
| : <math> R_{i} = \begin{bmatrix} \cos \gamma_{i} & -\sin \gamma_{i} \\ \sin \gamma_{i} & \cos \gamma _{i}\end{bmatrix} </math>
| |
| | |
| Using the following two [[List of trigonometric identities#Trigonometric conversions|trigonometric identities]]:
| |
| | |
| : <math>\begin{align} \cos \alpha & = & {1 \over \sqrt{1 + \tan^2 \alpha}} \\ \sin \alpha & = & {{\tan \alpha} \over \sqrt{1 + \tan^2 \alpha}} \end{align} </math>
| |
| | |
| the rotation matrix becomes:
| |
| | |
| : <math> R_i = {1 \over \sqrt{1 + \tan^2 \gamma_i}} \begin{bmatrix} 1 & -\tan \gamma_i \\ \tan \gamma_i & 1 \end{bmatrix} </math>
| |
| | |
| The expression for the rotated vector <math>v_i = R_i v_{i-1}</math> then becomes:
| |
| | |
| : <math> v_i = {1 \over \sqrt{1 + \tan^2 \gamma_i}} \begin{bmatrix} 1 & -\tan \gamma_i \\ \tan \gamma_i & 1 \end{bmatrix} \begin{bmatrix} x_{i-1} \\ y_{i-1} \end{bmatrix} </math>
| |
| | |
| where <math>x_{i-1}</math> and <math>y_{i-1}</math> are the components of <math>v_{i-1}</math>. Restricting the angles <math>\gamma_{i}</math> so that <math> \tan \gamma_{i} </math> takes on the values <math> \pm 2^{-i} </math>, 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:
| |
| | |
| : <math> v_i = K_i \begin{bmatrix} 1 & -\sigma_i 2^{-i} \\ \sigma_i 2^{-i} & 1 \end{bmatrix} \begin{bmatrix} x_{i-1} \\ y_{i-1} \end{bmatrix} </math>
| |
| | |
| where | |
| | |
| : <math> K_i = {1 \over \sqrt{1 + 2^{-2i}}} </math>
| |
| | |
| and <math>\sigma_i</math> can have the values of −1 or 1, and is used to determine the direction of the rotation; if the angle <math>\beta_i</math> is positive then <math>\sigma_i</math> is +1, otherwise it is −1.
| |
| | |
| <math> K_i </math> can be ignored in the iterative process and then applied afterward with a scaling factor:
| |
| | |
| : <math> K(n) = \prod_{i=0}^{n-1} K_i = \prod_{i=0}^{n-1} 1/\sqrt{1 + 2^{-2i}} </math>
| |
| | |
| 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 <math> v_0 </math> and hence saving a multiplication. Additionally it can be noted that:
| |
| | |
| : <math> K = \lim_{n \to \infty}K(n) \approx 0.6072529350088812561694 </math><ref>[http://perso.ens-lyon.fr/jean-michel.muller/SecondEdition.html J.-M. Muller, Elementary Functions: Algorithms and Implementation, 2nd Edition (Birkhäuser, Boston, 2006), p. 134.]</ref>
| |
| | |
| 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 <math>\beta</math>. 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 each iteration (choosing the value of <math>\sigma</math>). This is done by keeping track of how much the angle was rotated at each iteration and subtracting that from the wanted angle; then in order to get closer to the wanted angle <math>\beta</math>, if <math>\beta_{n+1}</math> is positive, the rotation is clockwise, otherwise it is negative and the rotation is counterclockwise.
| |
| | |
| : <math> \beta_{i} = \beta_{i-1} - \sigma_i \gamma_i. \quad \gamma_i = \arctan 2^{-i},</math>
| |
| | |
| The values of <math>\gamma_n</math> must also be precomputed and stored. But for small angles, <math> \arctan(\gamma_n) = \gamma_n</math> in fixed point representation, reducing table size.
| |
| | |
| As can be seen in the illustration above, the sine of the angle <math>\beta</math> is the ''y'' coordinate of the final vector <math>v_n</math>, 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 <math>\beta_i</math> 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 <math>\beta_i</math> 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.
| |
| | |
| <syntaxhighlight lang="MATLAB">
| |
| 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^-2j]:
| |
| 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
| |
| </syntaxhighlight> | |
| | |
| ==Hardware implementation==
| |
| The number of [[logic gate]]s 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 number]]s 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 converter]]s. | |
| | |
| ==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 <math>x</math> (in radians) by computing the exponential of <math>0+ix,</math> which is <math>\cos x + i \sin x.</math> 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]]:
| |
| :<math>\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}</math>
| |
| with <math>K_n = \sqrt{1+2^{-2n}}, \tan(\phi) = 2^{-n}.</math> <ref name="volder00"/> | |
| | |
| Some of the prominent early applications of CORDIC were in the Convair navigation computers CORDIC I to CORDIC III,<ref name="volder00"/> the Hewlett-Packard [[HP-9100]] and [[HP-35]] calculators,<ref>D. Cochran, "Algorithms and Accuracy in the HP 35", Hewlett Packard J. '''23''', 10 (1972).</ref> the [[Intel]] [[80x87]] coprocessor series until [[Intel 80486]], and [[Motorola 68881]].<ref>R. Nave, "Implementation of Transcendental Functions on a Numerics Processor", Microprocessing and Microprogramming '''11''', 221 (1983).</ref>
| |
| | |
| Decimal CORDIC was first suggested by Hermann Schmid and Anthony Bogacki.<ref>H. Schmid and A. Bogacki, "Use Decimal CORDIC for Generation of Many Transcendental Functions", EDN Magazine, February 20, 1973, p. 64.</ref>
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| *[http://www.jacques-laporte.org/Volder_CORDIC.pdf 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
| |
| *[http://www.jacques-laporte.org/Welther-Unified%20Algorithm.pdf John S. Walther, ''A Unified Algorithm for Elementary Functions'', Proc. of Spring Joint Computer Conference, pp379–385, May 1971]
| |
| *[http://www1.i2r.a-star.edu.sg/~pkmeher/papers/CORDIC-TUT-TACS-I.pdf P. K. Meher, J. Valls, T-B Juang, K. Sridharan, and K. Maharatna, ‘50 Years of CORDIC: Algorithms, Architectures and Applications,’ IEEE Transactions on Circuits & Systems-I: Regular Papers, vol.56, no.9, pp.1893- 1907, September 2009.]
| |
| *[http://www1.i2r.a-star.edu.sg/~pkmeher/papers/FRC-TCAS-I.pdf P. K. Meher and S. Y. Park, ‘CORDIC Designs for Fixed Angle of Rotation,’ IEEE Transactions on VLSI Systems, vol.21, no.2, pp.217-228, February 2013.]
| |
| *[http://www.jacques-laporte.org/Meggitt_62.pdf J. E. Meggitt, ''Pseudo Division and Pseudo Multiplication Processes'', IBM Journal, April 1962]
| |
| *[http://baykov.de/cordic1972.htm 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
| |
| *[http://baykov.de/cordic1975.htm V.D.Baykov,V.B.Smolov, ''Hardware implementation of elementary functions in computers'', Leningrad State University, 1975, 96p.]*[http://www.umup.narod.ru/1115.zip 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.''
| |
| *[http://baykov.de/cordic1985.htm 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 function|exponential]] and [[trigonometric function]]s'', IEEE Trans. Computers 45 (3), 328-339 (1996)
| |
| *[http://www.andraka.com/files/crdcsrvy.pdf Andraka, Ray, ''A survey of CORDIC algorithms for FPGA based computers'']
| |
| *Henry Briggs, ''Arithmetica Logarithmica.'' London, 1624, folio
| |
| *[http://cordic-bibliography.blogspot.com/2011/07/cordic-bibliography-site-revive.html ''CORDIC Bibliography Site''], Shaoyun Wang, July 2011
| |
| *[http://www.jacques-laporte.org/TheSecretOfTheAlgorithms.htm ''The secret of the algorithms''], Jacques Laporte, Paris 1981
| |
| *[http://www.jacques-laporte.org/digit_by_digit.htm ''Digit by digit methods''], Jacques Laporte, Paris 2006
| |
| *Ayan Banerjee, [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0X-4313PR1-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c85888a82db1f3b515b4a8d2a529624b FPGA realization of a CORDIC based FFT processor for biomedical signal processing], Kharagpur, 2001
| |
| *[http://www.hindawi.com/journals/vlsi/2010/794891.html ''CORDIC Architectures: A Survey''], B. Lakshmi and A. S. Dhar, Journal: VLSI Design, January 2010
| |
| *[http://cockrum.net/Cockrum_Fall_2008_Final_Paper.pdf ''Implementation of a CORDIC Algorithm in a Digital Down-Converter''], C. Cockrum, Fall 2008
| |
| | |
| ==External links== | |
| *[http://cordic-bibliography.blogspot.com/2011/07/cordic-bibliography-site-revive.html CORDIC Bibliography Site]
| |
| *[http://www.math.ucl.ac.be/~magnus/num1a/cordic.txt Another USENET discussion]
| |
| *[http://www.emesystems.com/BS2mathC.htm BASIC Stamp, CORDIC math implementation]
| |
| *[http://www.jacques-laporte.org/Trigonometry.htm 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)]
| |
| *[http://srohit.googlepages.com CORDIC implementation in verilog]
| |
| *[http://www.math.niu.edu/~rusin/known-math/94/cordic CORDIC information]
| |
| *[http://portal.acm.org/citation.cfm?id=626526.627179 CORDIC Vectoring with Arbitrary Target Value]
| |
| *[http://www.picbasic.co.uk/forum/showthread.php?p=70269#post70269 PicBasic Pro, Pic18 CORDIC math implementation]
| |
| *[http://code.activestate.com/recipes/576792 Python CORDIC implementation]
| |
| *[http://www.dcs.gla.ac.uk/~jhw/cordic/ Simple C code for fixed-point CORDIC]
| |
| *[http://www.andraka.com/cordic.htm The CORDIC Algorithm]
| |
| *[http://luminouslogic.com/dsp-simple-phase-estimation-approximation-cordic-matlab.htm Tutorial and MATLAB Implementation - Using CORDIC to Estimate Phase of a Complex Number]
| |
| *[http://www.eecg.toronto.edu/~bach/HOME/CORDIC/net.refs USENET discussion]
| |
| *[http://bibix.nl/index.php?menu1=arx_ip Descriptions of hardware CORDICs in Arx with testbenches in C++ and VHDL]
| |
| *[http://www.usenetmessages.com/view.php?c=computer&g=2333&id=468984&p=0 Usenet discussion about CORDIC]
| |
| | |
| {{DEFAULTSORT:Cordic}}
| |
| [[Category:Numerical analysis]]
| |
| [[Category:Trigonometry]]
| |