1. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU CORDIC (Coordinate rotation digital computer) Ref: Y. H. Hu, “CORDIC based VLSI architecture for digital signal processing,” IEEE Signal Processing Mag., pp.16-35, July 1992. VLSI Signal Processing 台灣大學電機系 吳安宇

2. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 2 Rotation Operation You need: 4 multipliers. 2 adders. or ROM for Table Look-up

3. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 3 What is “CORDIC” ? COordinate Rotation DIgital Computer Why do we use “CORDIC” ? MAC dominates the implementational cost in some DSP functions. The DSP approach, CORDIC, helps to save the hardware cost.

4. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 4 Basic Concept of The CORDIC To decompose the desired rotation angle (θ) into the weighted sum of a set of predefined elementary rotation angles (am(i)) Such that the rotation through each of them can be accomplished with simple shift-and- add operation.

5. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 5 In General Case: In CORDIC Algorithm:

6. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 6 Behavior of CORDIC V(0) V(1) V(3)

7. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 7 CORDIC Algorithm

8. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 8 Initiation:Given x(0),y(0),z(0) For i=0 to n-1,Do /*CORDIC iteration equation */ /*Angle updating equation*/ /*Scaling Operation (required for m=±1 only)*/ End i loop

9. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 9 X(i)Y(i) X-RegY-Reg +/- Barrel shifter X(i+1)Y(i+1) a(n-1) a(1) a(0) Z-reg Z(i+1) Basic Processor for CORDIC

10. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 10 Modes of Operations Vector rotation mode (θ is given) : determined by the set of  The objective is to compute the final vector (Usually, we set z(0)= θ.) θ  = sign of z(i)

11. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 11 Modes of Operations (cont’d) Angle accumulation mode (θ is not given) The objective is to rotate the given initial vector back to x-axis,and the angle can be accrued.(Now, we let z(0)=0.)  = - sign of x(i)·y(i) θ V(0) V(1) X-axis

12. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 12 Scaling Operation

13. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 13 X(n)Y(n) X(n) Y(n) +/- Barrel shifter X-RegY-Reg Scaling Stage

14. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 14 Advantages and Disadvantages Simple Shift-and-add Operation. (2 adders+2 shifters v.s. 4 mul.+2 adder) - It needs n iterations to obtain n-bit precision. -Slow carry-propagate addition. -Low throughput rate -Area consuming shifting operations.

15. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 15 How to Improve CORDIC ? Use Pipelined Architecture Improve the Performance of the Adders (redundant arithmetic, CSA) Reduce Iteration Number High radix CORDIC. (e.g., Radix-4, Radix-8) Find a optimized shift sequence (e.g., AR-CORDIC) Improve the Scaling Operation – Canonical multiplier recoding – Force Km to 2.

16. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 16 Parallel and Pipelined Arrays Basic CORDIC Processor 1 Basic CORDIC Processor 2 Basic CORDIC Processor n+s x(0) y(0) Basic CORDIC Processor 1 Basic CORDIC Processor 2 Basic CORDIC Processor n+s LATCHLATCH LATCHLATCH LATCHLATCH

17. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 17 In General Case: In CORDIC Algorithm:

18. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 18 Generalized CORDIC Algorithm m  0, linear system ; m=1, circular system ; m=-1, hyperbolic system.

19. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 19 Circular Linear V(2) V(4) V(0) V(1) V(3) V(0) V(2) V(1) V(3) Hyperbolic V(0) V(1) V(2) V(3) Different coordinates

20. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 20 Initiation: Given x(0),y(0),z(0) For i=0 to n-1,Do /*CORDIC iteration equation */ /*Angle updating equation*/ /*Scaling Operation (required for m=±1 only)*/ End i loop

21. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 21 Shift Sequence {s(m,i); 0  i  n-1} Determine the convergence of the CORDIC iteration, as well as the magnitude of the scaling factor Km(n). m=0 or 1, s=(m,i)=i m=-1, s(-1,i)=1,2,3,4,4,5,….,12,13,14,14,.. An angle approximation error:

22. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 22 Application to DSP Algorithms Linear transformation: - DFT, Chirp-Z transform, DHT, and FFT. Digital filters: - Orthogonal digital filters, and adaptive lattice filters. Matrix based digital signal processing algorithms: - QR factorization, with applications to Kalman filtering - Linear system solvers, such as Toeplitz and covariance system solvers,……,etc.

23. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 23 FFT application

24. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 24 Butterfly unit + + - - CORDIC processor

25. ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU pp. 25 Conclusions 1.In some cases, CORDIC evaluates rotational functions more efficiently than MAC units. 2.CORDIC saves more hardware cost. 3.By the regularity, the CORDIC based architecture is very suitable for implementation with pipelined VLSI array processors. 4.The utility of the CORDIC based architecture lies in its generality and flexibility.