Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology. DSP...

DSP Presentation Computing Multiplication & division using CORDIC

Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.

Project PresentationComputing Multiplication &

division using CORDIC in Visual DSP++

PROJECT BYMohammad Waqas

Arbab Waseem Abbas



OUTLINE• Abstract• Introduction• Methodology• Results & Discussion• Conclusion/summary• References



Abstract• What is “CORDIC” ?• Coordinate Rotation Digital Computer.• Three methods of CORDIC algo.• linear, circular and hyperbolic• For multiplication & div. linear method is used.



• WHY CORDIC ?

• CORDIC algorithms are efficient in terms of both computation time and hardware resources

• Dominates the implementation & hardware cost



• HOW ?

• The CORDIC algorithm makes use of only shifters and adder blocks to compute these functions.



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



• 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.



• 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.



• 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.



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.



• 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.



• 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) }



• 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.



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) }



X(i) Y(i)

X-Reg Y-Reg

+/- +/-

Barrel shifter

Barrel shifter

X(i+1) Y(i+1)

A Flow chart diagram of 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 <ffa11ba0>•• MULTIPLY 36.000000 DIVIDE 9.000000• Breakpoint Hit at <ffa11d36>• Loading: "C:\Documents and Settings\arbab waseem abbas\My

Documents\VisualDSP Projects\dsp_project\Debug\dsp_project.dxe"...• Load complete.• Breakpoint Hit at <ffa11ba0>•• MULTIPLY 50.000000 DIVIDE 4.500000• Breakpoint Hit at <ffa11d36>



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.



• 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.



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.



• 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.



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