1. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Project Presentation Computing Multiplication & division using CORDIC in Visual DSP++ PROJECT BY Mohammad Waqas Arbab Waseem Abbas

2. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC OUTLINE Abstract Introduction Methodology Results & Discussion Conclusion/summary References

3. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Abstract What is “CORDIC” ? Coordinate Rotation Digital Computer. Three methods of CORDIC algo. linear, circular and hyperbolic For multiplication & div. linear method is used.

4. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC WHY CORDIC ? CORDIC algorithms are efficient in terms of both computation time and hardware resources Dominates the implementation & hardware cost

5. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC HOW ? The CORDIC algorithm makes use of only shifters and adder blocks to compute these functions.

6. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC INTRODUCTION Background ?  COordinate Rotational DIgital Computer  Jack E. Volder (1959)  Primary concern was trig functions  Extended by J. Walther in 1971  Used by most calculators today  Efficient shift add algorithm/ no mulitplies needed

7. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Comparison ? Simple Shift-and-add Operation. (2 adders+2 shifters v.s. 4 mul.+2 adder) CORDIC algorithms are efficient in terms of both computation time and hardware resources - and in most systems, these resources are normally a premium This algorithm uses only minimal hardware (adder and shift) for computation of all trigonometric and other function values. It consumes fewer resources than any other techniques and so the performance is high. Thus, almost all scientific calculators use the CORDIC algorithm in their calculations.

8. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Application ? calculators digital-systems satellite microcontrollers Application to DSP algo.s (below) 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.

9. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC USES ? – Linear Functions – Trigonometric Functions – Hyperbolic Functions – Square Rooting – Logarithms, Exponentials –The functions that can be evaluated using CORDIC methods are sine, cosine, tangent, inverse tangent, hyperbolic sine, hyperbolic cosine, hyperbolic tangent, inverse hyperbolic tangent, natural logarithm, natural exponential, square root, multiplication, division.

10. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC METHODOLOGY Explanation of algo.? Embedding of elementary function evaluation as a generalized rotation operation. Decompose rotation operation into successive basic rotations. Each basic rotation can be realized with shift and add arithmetic operations.

11. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Contd. A CORDIC algorithm for multiplication can be derived using a series representation for x as shown below From this, z is composed of shifted versions of y. The unknown value for z, may be found by driving x to zero 1 bit at a time. If the ith bit of x is nonzero, yi is right shifted by i bits and added to the current value of z. The ith bit is then removed from x by subtracting 2-i from x. If x is negative, the ith bit in the twos complement format would be removed by adding 2-i. In either case, when x has been driven to zero all bits have been examined and z contains the signed product of x and y correct to B bits.

12. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Contd. multiply(x,y) { for (i=1; i=<B; i++) { if (x > 0) x = x - 2^(-i) z = z + y*2^(-i) else x = x + 2^(-i) z = z - y*2^(-i) } return(z) }

13. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC A CORDIC division algorithm is based on rewriting the equation z=x/y into the form x- y*z=0. If z is expanded into its series representation, the second version of the equation takes the form in Figure (a), which, after some manipulation, yields Figure (b). This final form of the equation shows that the quotient z may be estimated 1 bit at a time by driving x to zero using right- shifted versions of y. If the current residual is positive, the ith bit in z is set. Likewise, if the residual is negative the ith bit in z is cleared.

14. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC divide(x,y) { for (i=1; i=<B; i++) { if (x > 0) x = x - y*2^(-i); z = z + 2^(-i); else x = x + y*2^(-i); z = z - 2^(-i); } return(z) }

15. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC X(i)Y(i) X-RegY-Reg +/- Barrel shifter X(i+1)Y(i+1) A Flow chart diagram of CORDIC

16. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC RESULTS Loading: "C:\Documents and Settings\arbab waseem abbas\My Documents\VisualDSP Projects\dsp_project\Debug\dsp_project.dxe"... Load complete. Breakpoint Hit at MULTIPLY 36.000000 DIVIDE 9.000000 Breakpoint Hit at Loading: "C:\Documents and Settings\arbab waseem abbas\My Documents\VisualDSP Projects\dsp_project\Debug\dsp_project.dxe"... Load complete. Breakpoint Hit at MULTIPLY 50.000000 DIVIDE 4.500000 Breakpoint Hit at

17. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC CONCLUSION CORDIC algorithms are an efficient method to compute many different functions Low area, high speed Used in calculators, DSPs, math-coprocessors and supercomputers. CORDIC saves more hardware cost. By the regularity, the CORDIC based architecture is very suitable for implementation with pipelined VLSI array processors. The utility of the CORDIC based architecture lies in its generality and flexibility.

18. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Contd. Using CORDIC algorithms may allow a single chip solution where algorithms using the look-up table method may require a large ROM size or where power series calculations require a separate co-processor because of the computation time required.

19. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC How to Improve ? Use Pipelined Architecture Improve the Performance of the Adder(redundant arithmetic, CSA) Reduce Iteration Number –High radix CORDIC. –Find a optimized shift sequence. –Improve the Scaling Operation.

20. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC REFRENCES 1. Volder, Jack E., “The CORDIC Trignometric Computing Technique”, IRE Transactions Electronic Computers, vol. EC-8, pp. 330-334, September 1959. 2. Specker W. H., “A Class of Algorithms for Ln x, Exp x, Sin x, Cos x, Tan-1x and Cot-1x”, IEEE Transactions Electronic Computers, vol. EC-14, pp. 85-86, 1965. 3. Walther, J. S., “A Unified Algorithm For Elementary Functions”, 1971 Proceedings of the Joint Spring Computer Conference, pp. 379-385, 1971. 4. Jarvis, Pitts, “Implementing CORDIC Algorithms”, Dr. Dobb’s Journal, #169`, pp. 152-156, October 1990. 5. Dr Dobb’s official website.

21. Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP Presentation Computing Multiplication & division using CORDIC Thanks for your patience Any queries?