1. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU CORDIC (COordinate Rotation DIgital Computer) For Advanced VLSI and VLSI Signal Processing Major Reference: Y. H. Hu, “CORDIC-based VLSI architectures for digital signal processing,” IEEE Signal Processing Mag., pp.16-35, July 1992. 2002/9/25

2. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 2 台灣大學 吳安宇 教授 Outline  Introduction  Conventional CORDIC Algorithm  Enhancement of CORDIC  MVR-CORDIC Algorithm  EEAS-Based CORDIC Algorithm  Vector Rotational CORDIC Family

3. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 3 台灣大學 吳安宇 教授 Introduction  Vector rotation is the kernel of various digital signal processing (DSP) applications, including  Digital filters:  Orthogonal digital filters, and adaptive lattice filters.  Linear transformation:  DFT, Chirp-Z transform, DHT, and FFT.  Matrix-based digital signal processing algorithms:  QR factorization, with applications to Kalman filtering and QR decomposition (QRD) based adaptive filtering.  Linear system solvers  such as Toeplitz and covariance system solvers,……,etc.

4. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 4 台灣大學 吳安宇 教授 Rotational Operation (x in,y in ) (x out,y out )  Each vector rotation takes 4 multiplications and 2 additions

5. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 5 台灣大學 吳安宇 教授 Digital Lattice Filter  Low-sensitivity to coefficient quantization error  Smaller dynamic range

6. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 6 台灣大學 吳安宇 教授 Normalized Lattice Section  Givens Rotation

7. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 7 台灣大學 吳安宇 教授 Fast Fourier Transformation

8. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 8 台灣大學 吳安宇 教授 Fast Fourier Transformation  Twiddle factor

9. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 9 台灣大學 吳安宇 教授 Outline  Introduction Conventional CORDIC Algorithm  Enhancement of CORDIC  MVR-CORDIC Algorithm  EEAS-Based CORDIC Algorithm  Vector Rotational CORDIC Family

10. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 10 台灣大學 吳安宇 教授 Basic Concept of CORDIC Operation  To decompose the desired rotation angle (θ) into the weighted sum of a set of predefined elementary rotation angles (θ(i), I=0,1,2,…W)  Such that the rotation through each of them can be accomplished with simple shift-and-add operation.

11. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 11 台灣大學 吳安宇 教授 Conventional CORDIC Algorithm   where

12. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 12 台灣大學 吳安宇 教授 Conventional CORDIC Algorithm  Ease-to-implementation (shift-and-add only)

13. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 13 台灣大學 吳安宇 教授 Elementary Angle Set of Conventional CORDIC Conventional CORDIC with N = W =8

14. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 14 台灣大學 吳安宇 教授 Conventional CORDIC Algorithm  Example:  Rotation angle:  =  /8 = 0.3927 Sequentially performing of sub-angles, a(i) V(2) V(0) V(1) V(3) V(4)

15. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 15 台灣大學 吳安宇 教授 Generalized CORDIC V(2) V(0) V(1) V(3) V(4) Circular Linear V(1) V(3) Hyperbolic V(0) V(1) V(2) m  0, Linear system ; m=1, Circular system ; m=-1, Hyperbolic system.

16. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 16 台灣大學 吳安宇 教授 Summary of CORDIC Algorithm  Both micro-rotation and scaling phases

17. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 17 台灣大學 吳安宇 教授 Modes of CORDIC Operations θ Rotational Sequence = sign of z(i)  Vector rotation mode (θ is given)  The objective is to compute the final vector  Usually, we set z(0)= θ after the N-th iteration

18. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 18 台灣大學 吳安宇 教授 Usage of Vector Rotation Mode  In below situations, one may choose to evaluate angle by computing the rotational sequence in advance (off-line).  For many DSP problem,, is know in advance (e.g., twiddle factor)  The same will be used over and over.  Store the corresponding set of, instead of, in the memory.  A particular advantage is that there will be no need to implement the angle updating formula resulting in a one-third cost saving of hardware.

19. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 19 台灣大學 吳安宇 教授 Modes of Operations (cont’d)  Angle accumulation mode (θ is not given)  Objective is to rotate the given initial vector back to x-axis, and the angle can be accrued.  Set z(0) = 0;  Rotational Sequence  i = - sign x(i)y(i) θ V(0) V(1) X-axis

20. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 20 台灣大學 吳安宇 教授 Rotational Sequence v.s.  The selection criteria for are:  Note that the set of can be regard as an alternative representation of the rotation angle.  Hence, even in the angle accumulation mode, often it is not necessary to explicitly compute

21. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 21 台灣大學 吳安宇 教授 Basic processor for Micro-Rotation X(i)Y(i) X-RegY-Reg +/- Barrel shifter X(i+1) Y(i+1) mux X-Reg Y-Reg Z-reg Z(i+1) a(1) a(0) a(N-1)

22. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 22 台灣大學 吳安宇 教授 Scaling Operation  Scaling operation is used to maintain the “norm” of the original vector

23. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 23 台灣大學 吳安宇 教授 Conventional Scaling Operation

24. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 24 台灣大學 吳安宇 教授 Selective Scaling Operation  Combine Type-I and Type-II scaling operation by featuring  Complementary distribution  Off-line operation

25. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 25 台灣大學 吳安宇 教授 Basic processor for Scaling Operation X(n)Y(n) X(n) Y(n) +/- Barrel shifter X-Reg Y-Reg

26. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 26 台灣大學 吳安宇 教授 Advantages and disadvantages - Simple Shift-and-add Operation. (2 adders+2 shifters v.s. 4 mul.+2 adder) -Small area. -It needs n iterations to obtain n-bit precision. -Slow carry-propagate addition. -Area consuming shifts (Barrel Shifter)

27. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 27 台灣大學 吳安宇 教授 Enhancement of CORDIC  Architecture  Pipelined Architecture  Faster Adder (CSA)  Algorithm  Radix-4 CORDIC  MVR-CORDIC  EEAS-CORDIC

28. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 28 台灣大學 吳安宇 教授 Pipelined architecture  Expand folded CORDIC processor to achieve the pipelined architecture  Shifting can be realized by wiring

29. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 29 台灣大學 吳安宇 教授 Faster Adder (CSA) + + + sign Ripple Adder and its sign calculation carry sum + + + + + + CSAVMA and its sign calculation carry sum sign

30. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 30 台灣大學 吳安宇 教授 Critical Path of CSA  In on-line approach, we want to get sign bit as soon as possible !  Redundant Number System to be used to obtain the sign bit quickly + + + + + + sign CSAVMA + + + sign CSAVMA RNS

31. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 31 台灣大學 吳安宇 教授 Radix-4 CORDIC  Reduce Iteration Numbers  High radix CORDIC.(e.g. Radix-4, Radix-8)  1 stage of Radix-4 = two stages of Radix-2  Faster computation at higher cost  Employ the Radix-4 micro-rotations to  Reduce the stage number.  But K m may not be constant.

32. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 32 台灣大學 吳安宇 教授 Outline  Introduction  Conventional CORDIC Algorithm  Enhancement of CORDIC Modified Vector Rotation (MVR)-CORDIC Algorithm  EEAS-Based CORDIC Algorithm  Vector Rotational CORDIC Family

33. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 33 台灣大學 吳安宇 教授 Modification of CORDIC Algorithm  Skip some micro-rotation angles  For certain angles, we can only only reduce the iteration number but also improve the error performance.  For example,  =  /4 Conventional CORDIC MVR-CORDIC  m = 7.2*10 -3  m = 0

34. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 34 台灣大學 吳安宇 教授 Modification of CORDIC Algorithm  Repeat some micro-rotation angles  Each micro-rotation angle can be performed repeatedly  For example,  =  /2: execute the micro-rotation of a(0) twice  Confine the number of micro-rotations to R m  In conventional CORDIC, number of iteration=W  In the MVR-CORDIC, R m << W  Hardware/timing alignment

35. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 35 台灣大學 吳安宇 教授 Modification of CORDIC Algorithm  With above three modification where  s(i)  {0, 1, 2, …, W} is the rotational sequence that determines the micro-rotation angle in the i th iteration   (i)  {-1, 0,1} is the directional sequence that controls the direction of the i th micro-rotation

36. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 36 台灣大學 吳安宇 教授 Elementary Angle Set of Conventional CORDIC Conventional CORDIC with N = W =8

37. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 37 台灣大學 吳安宇 教授 Constellation of Reachable Angles (a) Conventional CORDIC with N = W =4 (b) MVR-CORDIC with W =4 and R m =3

38. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 38 台灣大學 吳安宇 教授 Summary of MVR-CORDIC Algorithm  Both micro-rotation and scaling phases

39. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 39 台灣大學 吳安宇 教授 VLSI Architecture  Iterative structure

40. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 40 台灣大學 吳安宇 教授 Outline  Introduction  Conventional CORDIC Algorithm  Enhancement of CORDIC  MVR-CORDIC Algorithm EEAS-Based CORDIC Algorithm  Vector Rotational CORDIC Family

41. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 41 台灣大學 吳安宇 教授 Extended Elementary Angle Set (EEAS)  Apply relaxation on EAS of  EAS is comprised of arctangent of single singed- power-of-two (SPT) term  Effective way to extend the EAS is to employ more SPT terms

42. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 42 台灣大學 吳安宇 教授 Example of EAS and EEAS Example of elementary angles of (a) EAS S 1, (b) EEAS S 2, with wordlength W=3.

43. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 43 台灣大學 吳安宇 教授 Constellation of EEAS Constellation of elementary angles of (a) EAS S 1, (b) EEAS S 2, with wordlength W=8.

44. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 44 台灣大學 吳安宇 教授 Summary of EEAS Scheme  Both micro-rotation and scaling phases

45. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 45 台灣大學 吳安宇 教授 VLSI Architecture  Iterative structure for one-stage rotational operation

46. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 46 台灣大學 吳安宇 教授 Comparison of Rotation Schemes Comparison of existing approaches/algorithms performing vector rotation in 2D plane, where the wordlength, W, is 16. Hardware Requirement Full Adder (FA) Count SQNR Performance Direct Implementation Conventional CORDIC Algorithm Angle Recoding (AR) Technique (R m =6, R s =6) EEAS-based CORDIC Algorithm (R m =2, R s =2) 4 Multipliers, 2 Adders About 43 Adders 24 Adders 16 Adders 1,056 688 384 256 98.7dB 97.4dB 93.3dB 95.1dB

47. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 47 台灣大學 吳安宇 教授 Outline  Introduction  Conventional CORDIC Algorithm  Enhancement of CORDIC  MVR-CORDIC Algorithm  EEAS-Based CORDIC Algorithm Vector Rotational CORDIC Family

48. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 48 台灣大學 吳安宇 教授 Family of Vector Rotational CORDIC Algorithm

49. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 49 台灣大學 吳安宇 教授 Set Diagram of VR CORDIC Family  Relationship among members in Vector Rotational (VR) CORDIC family can be represented as

50. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 50 台灣大學 吳安宇 教授 Angle Quantization  Approximation is applied on “ANGLE”  Decompose target angle into sub-angles

51. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 51 台灣大學 吳安宇 教授 Angle Quantization (Cont.)  Two key design issues in AQ process 1)Determine (construct) the sub-angle, and each sub-angle needs to be easy-to-implementation in practical implementation. 2)How to combine sub-angles such that angle quantization can be minimized.

52. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 52 台灣大學 吳安宇 教授 Angle Quantization (Cont.)  AQ process is conceptually similar to Sum-of-Power- ot Two (SPT) quantization

53. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 53 台灣大學 吳安宇 教授 Angle Quantization (Cont.)  Correspondences between AQ process and SPT (CSD) quantization process

54. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 54 台灣大學 吳安宇 教授 Conclusion  Rotational Engine plays an important role in modern DSP systems  The conventional CORDIC is an interesting idea to reduce VLSI hardware cost in implementing the rotation operation (as well as other arithmetic operations).  The idea of Angle Quantization is firstly introduced in Ref. [4,5]. It is analogous to the SPT concept in quantizing floating-point numbers.  Most CORDIC-related algorithms can be explained using the AQ design framework

55. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 55 台灣大學 吳安宇 教授 Reference 1.Y.H. Hu,, “CORDIC-based VLSI architectures for digital signal processing”, IEEE Signal Processing Magazine, Vol. 9, Issue: 3, pp. 16 -35, July 1992. 2.E. Antelo, J. Villalba, J.D. Bruguera, and E.L. Zapata,” High performance rotation architectures based on the radix-4 CORDIC algorithm”, IEEE Transactions on Computers, Vol. 46, Issue: 8, Aug. 1997. 3.C.S Wu and A.Y. Wu, “Modified vector rotational CORDIC (MVR-CORDIC) algorithm and architecture”, IEEE Trans. Circuits and systems-II: analog and digital signal processing, vol. 48, No. 6, June 2001. 4.A.Y. Wu and C.S. Wu, “A unified design framework for vector rotational CORDIC family based on angle quantization process”, In Proc. of 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 2, pp. 1233 -1236. 5.A. Y. Wu and Cheng-Shing Wu, “A Unified View for Vector Rotational CORDIC Algorithms and Architectures based on Angle Quantization Approach,” in IEEE Trans. Circuits and Systems Part-I: Fundamental Theory and Applications, vol. 49, no. 10, pp. 1442-1456, Oct. 2002. 6.C. S. Wu and A. Y. Wu, “A novel trellis-based searching scheme for EEAS- based CORDIC algorithm” In Proc. of 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol. 2, pp. 1229 -1232.