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CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www.cse.psu.edu/~mji)

CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www.cse.psu.edu/~mji). Cordic Algorithms. A convergence method for evaluating trigonometric (and other) functions

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CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www.cse.psu.edu/~mji)

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  1. CSE 575Computer ArithmeticSpring 2003Mary Jane Irwin (www.cse.psu.edu/~mji)

  2. Cordic Algorithms • A convergence method for evaluating trigonometric (and other) functions • if a unit-length vector with end point at (X,Y) = (1,0) is rotated by an angle Z, its new end point will be at (X,Y) = (cos Z, sin Z) • simple hardware - shifters, adders, lookup table • Family of algorithms: rotation, vector mode • circular rotations • linear rotations • hyperbolic rotations

  3. Real Cordic Rotations If vector OEiis rotated about the origin by an angle i, the new vector OEi+1will have the coordinates Ei+1 (Xi+1, Yi+1) Ei Ri+1 Real rotation: Ei+1 (Xi, Yi) i Xi+1= Xi cos i - Yi sin i Yi+1= Yicos i + Xisin i Zi+1= Zi -i Ri O The variable Z allows us to keep track of the total rotation over several steps. If Z0 is the initial rotation goal and if the i angles are selected at each step such that after n iterations Zn tends to 0, then En will be the end point after rotation by angle Z0

  4. Pseudo Cordic Rotations Expansion factor K = (1 + tan2 i)1/2 depends on the rotation angles 1 ,2…, n. If we always rotate by the same angles, K is a constant. Pseudo rotations increase the vector length to Ri+1 = Ri(1 + tan2i)1/2 E’i+1 (Xi+1, Yi+1) Ri+1 Pseudo rotation: E’i+1 Ei (Xi, Yi) Xi+1= Xi - Yi tan i Yi+1= Yi+ Xitan i Zi+1= Zi -i i Ri O

  5. Basic Cordic Rotations • To simplify pseudo rotations, pick isuch that tan i = di 2-i where di {-1,1}. Then Xi+1= Xi- di Yi 2-i Yi+1= Yi+ di Xi2-i Zi+1= Zi- di tan-12-i • Computation of Xi+1and Yi+1 requires an i-bit right shift and an add/subtract; Zi+1only requires an add/subtract and one table lookup • Precompute and store the function tan-12-i

  6. Zi+1 zero Choosing the Angles

  7. (X0, Y0) (X2, Y2) -45 -14 30 O +26.6 (X3, Y3) (X1, Y1) Rotating the Angles • Illustration of the first three rotations for a Z of 30

  8. Circular Rotation Mode • Choosing di = sign(Zi) {-1,1}to force Z to 0 gives the rotation mode Cordic iterations Xi+1 = Xi- di Yi2-i Yi+1 = Yi+ di Xi 2-i Zi+1= Zi- di Eiwhere Ei= tan-12-i • After n iterations, when Znis sufficiently close to 0, then we have Z = i and Xn= K(X cos Z - Y sin Z) where K = 1.646 760 258 … Yn= K(Y cos Z + X sin Z) Zn= 0 Rule: Choose di  {-1,1} such that Z  0 • Computes cos Z and sin Z by starting with X = 1/K = 0.607 252 935 … and Y = 0 • For k bits of precision, run it k iterations since for large i > k, tan-12-i  2-i

  9. Convergence Domain • Convergence of Z to 0 happens because each angle is more than half the previous angle. • The domain of convergence is 0  Z  99.7 which is the sum of all the angles given in slide 6 • Outside this range, we can use trig identities to convert to the range cos (Z  2j) = cos Z sin (Z  2j) = sin Z cos (Z - ) = - cos Z sin (Z - ) = - sin Z

  10. Rotation Example Computing cos 30 (= 0.866 025) and sin 30 (=0.500 000)

  11. Circular Vector Mode • Choosing di = - sign(Xi Yi)  {-1,1} to force Y to 0 gives the vector mode Cordic iterations Xi+1= Xi- di Yi2-i Yi+1= Yi+ di Xi2-i Zi+1= Zi- di Eiwhere Ei= tan-12-i • After n iterations, when Ynis sufficiently close to 0, then we have tan (i) = -Y/Z Xn= K (X2 + Y2)where K = 1.646 760 258 … Yn= 0 Zn= Z + tan-1 (Y/X) Rule: Choose di  {-1,1} such that Y  0 • Computes tan-1 Y by starting with X = 1 and Z = 0 • For k bits of precision, run it k iterations since for large i > k, tan-12-i  2-i

  12. Vector Example Computing tan-1 0.414 210 (= 22.499 830)

  13.   Circular Cordic Hardware X shift Xi+1 = Xi – di Yi 2-i i counter shift Yi+1 = Yi – di Xi 2-i Y Z Zi+1 = Zi – di Ei lookup table values of Ei= tan-12-i di control

  14. Generalized Cordic • Generalized Cordic defined as Xi+1= Xi-  di Yi2-i Yi+1= Yi+ di Xi2-i Zi+1= Zi - di Ei • Defining  and Eigives  = 1 Circular rotations (basic)Ei= tan-12-i  = 0 Linear rotations Ei= 2-i  = -1 Hyperbolic rotationsEi = tanh-12-i

  15. B  = 1 F A W V U  = -1 E  = 0 Generalized Rotations 2(area EOF) angle EOF = --------------- (OW)2 2(area AOB) angle AOB = -------------- (OU)2 O vector OV

  16.   Enhanced Cordic Hardware X X Xi+1 = Xi – di Yi 2-i shift i counter shift Yi+1 = Yi + di Xi 2-i Y Y Zi+1 = Zi - di Ei Z Z Ei lookup table (tan-12-i, 2-i, tanh-12-i)  di control

  17. Computes functions cos Z and sin Z X = 1/K and Y = 0 Computes function tan-1 Y X = 1 and Z = 0 Circular Mode ( = 1) • Circular rotation mode Xn= K(X cos Z - Y sin Z) Yn= K(Y cos Z + X sin Z) Zn= 0 Rule: Choose di  {-1,1} such that Z  0 • Circular vector mode Xn= K(X2 + Y2)1/2 Yn= 0 Zn= Z + tan-1 (Y/X) Rule: Choose di  {-1,1} such that Y  0

  18. Computes functions multiply and multiply-add Y = 0 Computes functions divide and divide-add Z = 0 Linear Mode ( = 0) • Linear rotation mode Xn= X Yn= Y + X * Z Zn= 0 Rule: Choose di  {-1,1} such that Z  0 • Linear vector mode Xn= X Yn= 0 Zn= Z + Y/X Rule: Choose di  {-1,1} such that Y  0

  19. Computes functions hyperbolic sin and cos X = 1/K’ and Y = 0 Computes functions hyperbolic tanh-1 y X = 1 and Z = 0 Hyperbolic Mode ( = -1) • Hyperbolic rotation mode Xn = K’(X cosh Z + Y sinh Z) Yn= K’(Y cosh Z + X sinh Z) Zn= 0 Rule: Choose di  {-1,1} such that Z  0 K’ = 0.828 159 360 … • Hyperbolic vector mode Xn= K’(X2 - Y2)1/2 Yn= 0 Zn= Z + tanh-1 (Y/X) Rule: Choose di  {-1,1} such that Y  0

  20. CORDIC CORDIC CORDIC CORDIC CORDIC CORDIC Cordic Computation Unit

  21. Cordic Summary • Can compute virtually all trig functions of common interest • When the number of iterations is fixed, K and K’ are constants • Thus, we can not stop computing if Z (or Y) becomes 0 (except when  = 0); we always need k iterations for k digits of precision • Cordic can be extended to higher radices (e.g., for base 4, di{-2,-1,1,2} and the number of iterations will be cut in half with essentially the same hardware)

  22. Key References Duprat and Muller, The CORDIC algorithm: New results for fast VLSI implementation, IEEE Trans. on Computers, 42(2):168-178, 1993. Parhami, Computer Arithmetic, Oxford Univ. Press, 1999. Phatak, Double step branching CORDIC, IEEE Trans. on Computers, 47(5)587-603, May 1998. Vachss, The CORDIC magnification function, IEEE Micro, 7(5):83-83, Oct. 1987. Volder, The CORDIC trigonometric computing technique, IRE Trans. Elecronic Computers, Vol 8, pp. 330-335, Sept. 1959. Walther, A unified algorithm for elementary functions, Proc. Spring Joint Computer Conf, pp. 379-385, 1971.

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