Cordic Algorithms in FPGAs - hk.sidviv.com

Cordic Algorithms in Cordic Algorithms in FPGAsFPGAs

Cordic Cordic AlgorithmsAlgorithms

Cordic AlgorithmCordic Algorithm

�� COrdinateCOrdinate Rotation Digital ComputerRotation Digital Computer

�� An iterative method to perform Coordinate Rotations An iterative method to perform Coordinate Rotations using only shifts and addsusing only shifts and adds

�� Hardware Efficient algorithms used for calculating Hardware Efficient algorithms used for calculating trigonometric and transcendental functionstrigonometric and transcendental functions


Vector rotation in Cartesian Vector rotation in Cartesian Coordinate SystemCoordinate System

X

Y

X’

Y’

φ

Px’

Py’

Px

Py

φ

P



X

Y

X’

Y’

φ

Px’

Py’

Px

Py

φ

φ

Py*cos(φ)

Px*sin(φ)Py’=Px*sin(φφφφ)+Py*cos(φφφφ)

P



X

Y

X’

Y’

φ

Px’

Py’

Px

Py

φ

Px*cos(φ)

Py*sin(φ)

Px’=Px*cos(φφφφ)-Py*sin(φφφφ)

P



X

Y

X’

Y’

φ

Px’

Py’

Px

Py

φ

P

Px’=Px*cos(φφφφ)-Py*sin(φφφφ)

Py’=Px*sin(φφφφ)+Py*cos(φφφφ)

Px’= cos(φφφφ)*( Px-Py*tan(φφφφ))

Py’= cos(φφφφ)*(Py+Px*tan(φφφφ))


x x i+1i+1 = = coscos((φφφφφφφφii) * () * ( xxii -- yyii * tan(* tan(φφφφφφφφii) )) )

y y i+1i+1 = = coscos((φφφφφφφφii) * () * ( yyii + x + x I I * tan(* tan(φφφφφφφφii) )) )

z z i+1i+1 == zzii -- φφφφφφφφii

Assume Assume tan(tan(φφφφφφφφii) = d) = di i * 2* 2--ii where where ddi i = +1= +1 or or --11

φφφφφφφφii = = ddii * tan* tan--11(2(2--ii))

KKi i = cos(= cos(φφφφφφφφii) =) = coscos(tan(tan--11(2(2--II)) = 1/)) = 1/

CORDIC EquationsCORDIC Equations

1 2 2i+ −


Factor Out the constantFactor Out the constant KKii from each iterationfrom each iteration

xxii+1+1 = x = x I I -- yyi i * d* di i * 2* 2--ii

yyi+1i+1 = y = y I I + x + x ii* d* di i * 2* 2--ii

zzi+1i+1 = z= zii -- ddi i * tan* tan--11(2(2--ii))

Multiply by Multiply by AAnn after n iterations to account for after n iterations to account for constants constants KKii

AAnn = =


1 2 2i

i 1

n

+ −

=

∏


�� Previous Equations are valid if initial angle Previous Equations are valid if initial angle zz0 0 > > --ππππππππ/2/2and and zz00 < < ππππππππ/2/2. For other angles, give an initial rotation . For other angles, give an initial rotation of of ππππππππ..

�� For initial rotation the equations areFor initial rotation the equations are

xx/ / = d * x= d * x where where d = d = --11 if if x < 0x < 0 else else d = 1d = 1

yy/ / = = --d * yd * y

zz/ / = z= z if if dd is equal to is equal to 11

= z = z -- ππππππππ if if dd is equal to is equal to --11

�� An alternate solution is to rotate by An alternate solution is to rotate by ππππππππ/2/2 or or --ππππππππ/2/2initially.initially.



�� Hardware has three accumulators Hardware has three accumulators xx, , yy and and zz

xxi+1i+1 = = xxii -- yyii*d*dii*2*2--ii

yyi+1i+1 = = yyii + + xxii * * ddii * 2* 2--ii

zzi+1i+1 == zzii -- ddii * tan* tan--11(2(2--ii))

�� Two modes of operationTwo modes of operation

—— Rotation ModeRotation Mode

—— Vector ModeVector Mode

Modes of OperationModes of Operation


xxi+1i+1 = x= xii--yyii*d*dii*2*2--ii

yyi+1i+1 = y= yii+x+xii*d*dii*2*2--ii

zzi+1i+1 = z= zii -- ddii*tan*tan--11(2(2--ii))

Rotation Mode of OperationRotation Mode of Operation

•Initialize three accumulators

by x0, y0 and z0

•Rotate to make the angle

accumulator z0 equal to 0.

•di = -1 if zi < 0

•di = 1 otherwise

•Final accumulator

values are xn, yn

and zn

xn=An*(x0cos(z0) - y0sin(z0))

yn=An*(y0cos(z0) + x0sin(z0))

zn= 0


Rotation ModeRotation Mode

�� General Vector Rotator : General Vector Rotator :

—— Rotates a vector Rotates a vector (x(x00,y,y00)) by an angleby an angle zz00..

�� Sine and Cosine Generator : Sine and Cosine Generator :

—— Initial Values Initial Values xx00=1/A=1/Ann, , yy00=0=0 and and zz00==θθθθθθθθ

—— Final Values Final Values xxnn=cos(=cos(θθθθθθθθ)), , yynn=sin(=sin(θθθθθθθθ)) and and zznn=0=0

�� Polar to Rectangular Coordinate Conversion :Polar to Rectangular Coordinate Conversion :

—— Initial Values Initial Values xx00=r=r, , yy00=0=0 and and zz00==θθθθθθθθ

—— Final Values Final Values xxnn=A=Annr cos(r cos(θθθθθθθθ)), , yynn= A= Annr sin(r sin(θθθθθθθθ)) and and zznn=0=0

xxnn = A= Ann*(x*(x

00cos(zcos(z00)) -- yy00sin(zsin(z

00))))

yynn = A= Ann*(y*(y

00cos(zcos(z00)) + x+ x

00sin(zsin(z00))))

zznn = 0= 0


xxi+1i+1 = x= xii--yyii*d*dii*2*2--ii

yyi+1i+1 = y= yii+x+xii*d*dii*2*2--ii


Vectoring Mode of OperationVectoring Mode of Operation

•Initialize three accumulators

by x0, y0 and z0

•Rotate to make the y

accumulator y0 equal to 0.

•di = -1 if yi > 0

•di = 1 otherwise

•Final accumulator

values are xn, yn

and zn

xn=An

yn = 0

zn = z0 + tan-1(y0/x0)

x y0

2

0

2+


Vectoring ModeVectoring Mode

�� Rectangular to Polar Coordinate ConversionRectangular to Polar Coordinate Conversion

—— xxnn is scaled magnitude and is scaled magnitude and zznn is the angle.is the angle.

�� zznn is is tantan--11(y)(y) if if xx00=1=1 and and zz00=0=0

xn=An

yn = 0

zn = z0 + tan-1(y0/x0)

x y0

2

0

2+


ConvergenceConvergence

X

Y

θ

V0 aligning with X axis.

V1 is incorrect.

Subsequent rotations to

V2, V3, … must correct

the error

Condition

tan-1(yi+1/xi+1) >

0.5*tan-1(yi/xi)

ensures convergence

V0

V1

V2

V3

V4


ConvergenceConvergence

�� Cordic Equations in the Rotation Mode and Vectoring Cordic Equations in the Rotation Mode and Vectoring Mode converge.Mode converge.

�� Error reduces in each iteration.Error reduces in each iteration.

�� Each Rotation makes the accumulators more correct. Each Rotation makes the accumulators more correct. Decide on the number of rotations from your Decide on the number of rotations from your application.application.


Functions covered Functions covered ……..

�� Vector Rotation.Vector Rotation.

�� Rectangular to Polar Coordinate ConversionRectangular to Polar Coordinate Conversion

�� Polar to Rectangular Coordinate ConversionPolar to Rectangular Coordinate Conversion

�� Sine, Cosine and Arctan GeneratorsSine, Cosine and Arctan Generators


Arcsine using Inverse CordicArcsine using Inverse Cordic

X

Y

X’

Y’

θθθθ

c

x0X-Y rotates to

X’-Y’

θθθθ = sin-1(c/x0) x c0

2 2−


ArcsineArcsineRotate the vector so thatRotate the vector so that yynn is equal to is equal to cc, a constant, a constant

xxi+1i+1 = x = x I I -- yyii*d*dii*2*2--ii

yyi+1i+1 = = yyii ++ xxii*d*dii*2*2--ii


wherewhere

ddii = +1= +1 if if y y ii< c< c, , --11 otherwiseotherwise

—— Initial Values : Initial Values : xx00=x=x00, , yy00=0=0 and and zz00=0=0

—— Final Values : Final Values : xxnn== , , yynn=c=c and and zznn=sin=sin--11(c/A(c/Annxx00))

(A x ) cn 0

2−


Arccosine using Inverse CordicArccosine using Inverse Cordic

X

Y

X’

Y’

θθθθ

c

x0X-Y rotates to

X’-Y’

θθθθ = cos-1(c/x0)

x c0

2 2−


ArccosineArccosineRotate the vector so thatRotate the vector so that xxnn is equal to is equal to cc, a constant, a constant

xxi+1i+1 = = xxii -- yyii*d*dii*2*2--ii

yyi+1i+1 = = yyii ++ xxii*d*dii*2*2--ii


wherewhere

ddii = = --11 if if xxii < c< c, , +1+1 otherwiseotherwise

—— Initial Values : Initial Values : xx0 0 = x= x00, , yy0 0 = 0= 0 and and zz0 0 = 0= 0

—— Final Values : Final Values : xxnn== , , yynn = c= c and and zznn==coscos--11(c/A(c/Annxx00))

(A x ) cn 0

2 2−


Convergence of Arcsine and Convergence of Arcsine and ArccosineArccosine

X

Y

c

y1

y3

Gain of 1/Ki in

each iteration

1 2 2i+ −1/Ki =

y2

• Due to gain 1/Ki , (y1-y3)/(y1-c) < 1/2, which does not

guarantee convergence


Convergence of Arcsine and Convergence of Arcsine and ArccosineArccosine

�� Double Iteration AlgorithmDouble Iteration Algorithm

—— For each iteration, rotate the vector twice For each iteration, rotate the vector twice instead of once.instead of once.

�� Double Iteration Algorithm makes the results of Double Iteration Algorithm makes the results of arcsine and arccosine more accurate. arcsine and arccosine more accurate.


General Cordic EquationGeneral Cordic Equation

�� Circular Coordinate Transforms : What has been Circular Coordinate Transforms : What has been presented so far.presented so far.

�� Cordic Equations can be extended to calculate Cordic Equations can be extended to calculate more functionsmore functions

—— Linear FunctionsLinear Functions

—— Hyperbolic Functions Hyperbolic Functions


Linear Cordic EquationsLinear Cordic Equations

xxi+1i+1 = x= xi i -- 0 * 0 * yyii * * ddii * 2* 2--i i = x= xii

yyi+1i+1 = y= yi i + + xxii * * ddii * 2* 2--ii

zzi+1i+1 = z= zii -- ddii * 2* 2--ii

�� Rotation ModeRotation Mode ((ddi i = = --11 if if zzii < 0< 0, , +1+1 otherwise) otherwise)

—— A multiplier with shifts and addsA multiplier with shifts and adds

—— Final Values:Final Values:

xxnn = x= x00

yynn = y= y0 0 + x+ x00zz00

zzn n = 0= 0


Linear Cordic EquationsLinear Cordic Equations

xxii+1+1 == xxii

yyii+1+1 == yyii ++ xxii ** ddii * 2* 2--ii

zzii+1+1 == zzii -- ddii * 2* 2--ii

�� Vectoring Mode (Vectoring Mode (ddi i = +1= +1 if if yyii<0<0, , --11 otherwise) otherwise)

—— Calculates the ratio Calculates the ratio yy00/x/x00 in in zz accumulatoraccumulator

—— xxn n = x= x00

yyn n = 0= 0

zzn n = z= z0 0 -- yy00/x/x00


x x i+1i+1 = cosh(= cosh(φφφφφφφφii) * ( ) * ( xxii + + yyii ** tanhtanh((φφφφφφφφii) )) )

y y i+1i+1 = cosh(= cosh(φφφφφφφφii) * ( ) * ( yyii + + xxii ** tanhtanh((φφφφφφφφii) )) )

z z i+1i+1 = = zzii -- φφφφφφφφii

Assume Assume tan(tan(φφφφφφφφii) = ) = ddii * 2* 2--ii where where ddi i = +1= +1 or or --11

φφφφφφφφii = = ddii ** tanhtanh--11(2(2--ii))

KKi i = cosh(= cosh(φφφφφφφφii) = cosh(tanh) = cosh(tanh--11(2(2--ii)) = 1/)) = 1/

HyperbolicHyperbolic CORDIC EquationsCORDIC Equations

1 2 2i− −


xxi+1i+1 = = xxii + + yyii * * ddii * 2* 2--ii

yyi+1i+1 = = yyii + + xxii * * ddii * 2* 2--ii

zzi+1i+1 = z= zii -- ddii ** tanhtanh--11(2(2--ii))

Multiply by Multiply by AAnn after n iterations to account for after n iterations to account for constants constants KKii

AAn n ==

Hyperbolic CORDIC EquationsHyperbolic CORDIC Equations

1 2 2i

i 1

n

− −

=

∏


Rotation ModeRotation Mode

�� Initialize with Initialize with xx00, , yy00 and and zz00..

�� The Hyperbolic Cordic Equations converge toThe Hyperbolic Cordic Equations converge to

xxnn = A= An n * ( x* ( x00 coshcosh(z(z00)) + y+ y00 sinhsinh(z(z00) )) )

yynn = A= An n * ( y* ( y00 coshcosh(z(z00)) + x+ x00 sinhsinh(z(z00) )) )

zznn = 0= 0

xi+1 = xi + yi * di * 2-i

yi+1 = yi + xi * di * 2-i

zi+1 = zi - di * tanh-1(2-i)

•Converge Angle Accumulator

z to 0.

•di = -1 if zi < 0, +1 otherwise


Vectoring ModeVectoring Mode

�� Initialize with Initialize with xx00, , yy00 and and zz00..

�� The Hyperbolic Cordic Equations converge toThe Hyperbolic Cordic Equations converge to

xxnn = A= Ann

yynn = 0= 0

zznn = z= z00 + + tanhtanh--11(y(y00/x/x00))

xi+1 = xi + yi * di * 2-i

yi+1 = yi + xi * di * 2-i

zi+1 = zi - di * tanh-1(2-i)

•Converge Accumulator y to

0.

•di = +1 if yi < 0, +1 otherwise

x y0

2

0

2−


�� Hyperbolic Equations require double iteration for Hyperbolic Equations require double iteration for convergenceconvergence

�� Hyperbolic functions similar to Trigonometric Hyperbolic functions similar to Trigonometric functions described earlier can be calculatedfunctions described earlier can be calculated

�� For more informationFor more information

—— Walther, J.S., Walther, J.S., ““A Unified algorithm for A Unified algorithm for elementary functionselementary functions””, Spring Joint Computer , Spring Joint Computer Conf, 1971, proc, pp. 379Conf, 1971, proc, pp. 379--385.385.

Hyperbolic Cordic EquationsHyperbolic Cordic Equations


Unified EquationsUnified Equations

xxii+1+1 = = xxi i -- m*m*yyii **ddii*2*2--ii

yyi+1i+1 = y= yi i + x+ xii*d*dii*2*2--ii

zzi+1i+1 = z= zii -- ddii**eeii

Unified Cordic EquationUnified Cordic EquationLinear Functions

m = 0 and ei = 2-i

xi+1 = xi

yi+1 = yi + xi * di * 2-i

zi+1 = zi - di * 2-i

Hyperbolic Functions

m = 1 and ei = tanh-1(2-i)

xi+1 = xi + yi * di * 2-i

yi+1 = yi + xi * di * 2-i

zi+1 = zi - di * tanh-1(2-i)

Trigonometric Functions

m = -1 and ei = tan-1(2-i)

xi+1 = xi - yi*di*2-i

yi+1 = yi + xi*di*2-i

zi+1 = zi - di*tan-1(2-i)


Bit Parallel ImplementationBit Parallel Implementation

>> i+-

register

register

>> i

x0

sign(yi)

y0

-mdi

+-

+-

register

ROM

sign(zi)

z0

-di

xi

yi

zi

Unified EquationsUnified Equations

xxii+1+1 = = xxi i -- m*m*yyii **ddii*2*2--ii

yyi+1i+1 = y= yi i + x+ xii*d*dii*2*2--ii

zzi+1i+1 = z= zii -- ddii**eeii

di


Bit Parallel ImplementationBit Parallel Implementation

+-

register

ROM

sign(zi)

z0

-di

zi

Hardwire for

constant ROM

contents

>> i

High Fanout

Slow in FPGAs


Unrolled Bit Parallel Unrolled Bit Parallel

+| +|

+|

>>0 >>0 const

signx0

y0 z0

+| +|

+|

>>1 >>1 const

sign

More Unrolled Cordic units …..

+| +|

+|

>>n >>n const

sign

xn ynzn

Hardwire if

possible


Bit Serial ImplementationBit Serial Implementation

Serial

Add-Sub

Serial

Add-Sub

X Register

Y Register

Z RegisterSerial

Add-SubSerial ROM

x0

y0

z0

xn

yn

zn

sign

sign


Bit Serial Implementation in VirtexBit Serial Implementation in Virtex

Z Register

Serial ROM

sign

D Q

C

CE

A[3:0]

SRL16E

4Tie to

Vcc

LUT used as

Serial In Serial Out

Shift Register


Bit Serial Implementation in VirtexBit Serial Implementation in Virtex

Serial

Add-Sub

X Register

x0

xn

16x1

Dual Port

RAM

Serial Data Out

Shifted Serial

Data OutModulo 16

Counter_1

Counter_1

lags

Counter_0

for

required

delay

A

DPRA

SPO

DPO

DSerial Data In

Modulo 16

Counter_0


�� Good hardware algorithm Good hardware algorithm

—— calculate various trigonometric and hyperbolic calculate various trigonometric and hyperbolic functions functions

—— Rotate Vectors appropriatelyRotate Vectors appropriately

—— Convert from one coordinate system to anotherConvert from one coordinate system to another

�� Pay attention to the convergence of the algorithm.Pay attention to the convergence of the algorithm.

Conclusion Conclusion


Andraka, R.J., Andraka, R.J., ““A survey of CORDIC algorithms for FPGA based A survey of CORDIC algorithms for FPGA based

computerscomputers””, FPGA , FPGA ‘‘98, Proc. Of the 1998 ACM/SIGDA sixth 98, Proc. Of the 1998 ACM/SIGDA sixth international symposium on Field Programmable Gate Arrays, Feb international symposium on Field Programmable Gate Arrays, Feb 2222--24, 1998, Monterey CA, pp 19124, 1998, Monterey CA, pp 191--200.200.

Volder, J., Volder, J., ““The CORDIC Trigonometric Computing TechniqueThe CORDIC Trigonometric Computing Technique””, IRE , IRE Trans. Electronic Computing, Vol ECTrans. Electronic Computing, Vol EC--8, pp3308, pp330--334, Sept 1959.334, Sept 1959.

Walther, J.S., Walther, J.S., ““A unified algorithm for elementary functionsA unified algorithm for elementary functions””, Spring , Spring Joint Computer Conf., pp. 379Joint Computer Conf., pp. 379--385, proc., 1971.385, proc., 1971.

Volder, J., Volder, J., ““Binary Computation algorithms for coordinate rotation and Binary Computation algorithms for coordinate rotation and function generationfunction generation””, Convair Report IAR, Convair Report IAR--1 148 Aeroelectrics Group, 1 148 Aeroelectrics Group, June 1956.June 1956.

ReferencesReferences

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Cordic Algorithms in FPGAs - hk.sidviv.com