Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
Project Presentation
Computing Multiplication &
division using CORDIC in
Visual DSP++
PROJECT BY
Mohammad Waqas
Arbab Waseem Abbas
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
OUTLINE
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Abstract
Introduction
Methodology
Results & Discussion
Conclusion/summary
References
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
Abstract
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What is “CORDIC” ?
Coordinate Rotation Digital Computer.
Three methods of CORDIC algo.
linear, circular and hyperbolic
For multiplication & div. linear method is used.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• WHY CORDIC ?
• CORDIC algorithms are efficient in terms of both
computation time and hardware resources
• Dominates the implementation & hardware cost
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• HOW ?
• The CORDIC algorithm makes use of only
shifters and adder blocks to compute these
functions.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
INTRODUCTION
• Background ?
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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
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• Comparison ?
• Simple Shift-and-add Operation.
• (2 adders+2 shifters v.s. 4 mul.+2 adder)
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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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• Application ?
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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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• USES ?
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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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• Contd.
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A CORDIC algorithm for multiplication can be
derived using a series representation for x as
shown below
DSP Presentation
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.
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• 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)
}
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
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A CORDIC division algorithm is based on
rewriting the equation z=x/y into the form xy*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).
DSP Presentation
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 rightshifted 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.
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
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)
}
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
X(i)
X-Reg
Barrel
shifter
+/X(i+1)
DSP Presentation
Y(i)
Y-Reg
A Flow chart
diagram of
CORDIC
Barrel
shifter
+/Y(i+1)
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
RESULTS
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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>
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
CONCLUSION
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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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• 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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
• 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.
DSP Presentation
Computing Multiplication & division using CORDIC
Department of Computer Systems Engineering, N-W.F.P. University of Engineering & Technology.
Thanks for your patience
Any queries?
DSP Presentation
Computing Multiplication & division using CORDIC