adsd_fall2011_14_cordic

8/3/2019 ADSD_Fall2011_14_CORDIC

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Dr. Rehan Hafiz Lecture # 14

ADSD Fall 2011


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Course Website for ADSD Fall 2011

http://lms.nust.edu.pk/

2

Lectures: Tuesday @ 5:30-6:20 pm, Friday @ 6:30-7:20 pm

Contact: By appointment/EmailOffice: VISpro Lab above SEECS Library

Acknowledgement: Material from the following sources has been consulted/used in theseslides:1. A survey of CORDIC algorithms for FPGA based computers, Ray Andraka, Andraka

Consulting Group, Inc2. University of Minnesota : EE 5324 - VLSI Design II - Lectures Slides by Kia Bazargan,

Spring 20063. http://www.indovina.us/~mai/paco/Pacos_cordic_algorithm.xl4. [SHO] Digital Design of Signal Processing System by Dr Shoab A Khan

Material/Slides from these slides CAN be used with following citing reference:

Dr. Rehan Hafiz: Advanced Digital System Design 2010

Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/http://creativecommons.org/licenses/by-nc-sa/3.0/


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COordinate Rotation DIgital Computer

(CORDIC)

Iterative Technique

Enormous Applications

Linear Functions / Multiplication

Division

Hyperbolic Functions

Square Rooting

Logarithms, Exponentials Sine, Cosine Generation

Does so much with just SHIFT & ADD


4/48

Rotation of a Vector

V1

x = r cos()

y = r sin()

V2 x'= r cos(+)

x'= r [cos()cos() -sin()sin()]

x'= r [(x/r)cos() (y/r)sin()]

x'= xcos() ysin()

y'= r sin(+)

y'= ycos() + xsin()

V1

(x,y)

V2

(x,y)

rr


5/48

Re-arranging the Rotation

Transformation

Rotates a vector in a

Cartesian Plane by the

angle

Calculations Re-arranging

So, given a rotation angle ;

these equations require the

computation of tan()

cos =

)tan(..tan1

1

'

)tan(..tan1

1

2

2

xyy

yxx

)sin(.)cos('

)sin(.)cos(

xyy

yxx


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tan() is not simple to compute

in hardware

tan()

0.00 0.000

5.00 0.087

10.00 0.176

15.00 0.268

20.00 0.364

25.00 0.466

30.00 0.577

35.00 0.700

40.00 0.839

45.00 1.000

)tan(..tan1

1'

)tan(..tan1

1

2

2

xyy

yxx


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Example: Lets say we want to rotate a vector

(x0,y0) 30-degrees anti-clockwise to (x3,y3)

Requires calculation of tan(30)

Suppose tan (10) is pre-computed & stored in HW

A 30 degree rotation can be modeled by applying 10 degree

rotation three times

& Similarly for y1,y2,y3 (x0,y0)

(x1,y1)

(x2,y2)

(x3,y3)

)10tan(..)10(tan1

1

)10tan(..)10(tan1

1

)10tan(..)10(tan1

1

222

3

112

2

002

1

yxx

yxx

yxx

1


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Putting a HW efficient

constraint over tan()

tan() is not simple to computein hardware

tan()

0.00 0.0005.00 0.087

10.00 0.176

15.00 0.268

20.00 0.364

25.00 0.466

30.00 0.577

35.00 0.700

40.00 0.839

45.00 1.000

Rotation anglesare restricted sothat tan() = 2-i ,

where i = 0,1,2,3,

Multiplication byTangent is reducedto simple shiftoperation

)tan(..tan1

1'

)tan(..tan1

1

2

2

xyy

yxx

i tan(i) i

i 2-i tan-1(2-i)

0 1 45.000

1 0.5 26.565

2 0.25 14.036

3 0.125 7.125

4 0.0625 3.576

5 0.03125 1.790

6 0.015625 0.895

7 0.0078125 0.448

8 0.00390625 0.2249 0.001953125 0.112


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Putting a HW efficient

constraint over tan()

tan() = 2-i

tan() =d.2-i

d=-1,+1 dependingupon the rotationdirection

i-2i-

i-

2i-

2...21

1'

2...21

1

dxyy

dyxx

)tan(..tan1

1'

)tan(..tan1

1

2

2

xyy

yxx


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In CORDIC we use the

pre-computed 2-i table

Principle: A composite rotation of any

arbitrary angle can be represented

by summation() of elementary

rotations

Using elementary angles from the

constrained tan table

Writing 30 in terms of elementary angles

from table

= 30.0 45.0 26.6 + 14.0 7.1 + 3.6

+ 1.8 0.9 + 0.4 0.2 + 0.1

= 30.1

Iter.

Constrain

-ed tan

Rotation for each

iteration

i tan(i) i

i 2-i tan-1(2-i)

0 1 45.000

1 0.5 26.565

2 0.25 14.036

3 0.125 7.125

4 0.0625 3.576

5 0.03125 1.790

6 0.01562 0.895

7 0.00781 0.448

8 0.00390 0.2249 0.00195 0.112


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X

Y

30

Example: Rewriting

Angles in Terms ofai

= 30.0

45.0 26.6 + 14.0 7.1 + 3.6

+ 1.8 0.9 + 0.4 0.2 + 0.1

= 30.1

45

+ 1.10 - (+ 1.8 ) = - 0.70

Rotate the vector in -ve direction

+ 4.70 - (+ 3.6 ) = + 1.10

Rotate the vector in +ve direction

- 2.40 - (- 7.1 ) = + 4.70


+11.60 - (+14.00 ) = - 2.40


-15.00 - (-26.60 ) = +11.60


+30.00 - (+45.00 ) = -15.00


Angle Accumulator Angle(current tan value) = Residual Angle

+ve+ve direction, -ve -ve direction


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The calculation of residual angle results into

another equation in the CORDIC System

Angle Accumulator Angle(current tan value) = Residual Angle

Actually it is

Angle Accumulator = Angle Acc. (Current Value) (Direction)Angle Moved

)2(tan.

2..

2..

i-1

1

i-

1

-i

1

iii

iiii

iiii

dzz

dxyy

dyxx

Iter.Constrain-ed tan

Rotation for eachiteration

i tan(i) =2-i i=tan

-1(2-i)

0 1 45.000

1 0.5 26.565

2 0.25 14.036

3 0.125 7.125

4 0.0625 3.576

5 0.03125 1.790

6 0.01562 0.895

7 0.00781 0.448

8 0.00390 0.2249 0.00195 0.112

1

-i1)]2(.[tan

iii zdz


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CORDIC- Formal

1


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CORDIC

Derivation Summary

)tan(..tan1

1

)tan(..tan1

1

2

21

1 i

i

i

i

i

xyy

yxx

ii

i

i-1

-i

1

2...

2...

iiiii

iiiii

dxyKy

dyxKx

1,

21

1

2i-

dKi

Provides an iterative method

of performing vector

rotations by arbitrary angles

Uses only shifts & adds

Ki is just a scaling factor

multiplied in each iteration &

can be applied as a

cumulative gain An to the

final xn,yn

The precision increases with

each iteration n

nAGain-2i

21/1/1

647.1 i

nA

)2(tan.

2..

2..

i-11

i-

1

-i

1

iii

iiii

iiii

dzz

dxyy

dyxx


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Modes defined by Volder

Rotation - The coordinate components of a

vector and an angle of rotation are given and the

coordinate components of the original vector,

after rotation through the given angle, arecomputed.

Vector - The coordinate components of a vector

are given and the magnitude and angular

argument of the original vector are computed.

CORDIC Th di f d


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CORDIC

Rotation Mode

Rotates the input vector by aspecified angle

Application Calculating the x,y coordinates

at a given angle Method:

Initialize angle accumulator bydesired angle

Initialize x & y

Decision d: based upon thesign of angle accumulator

Goal : Minimize the AngleAccumulator

The coordinate components of a vector and an

angle of rotation are given and the coordinate

components of the original vector, after rotation

through the given angle, are computed


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Rotation Mode

1

0

3

2

5

46

(x, y)z

(x, y) z=0

rotates a vector (x,y) by a given angle (z)

CORDIC Th di t t f t


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CORDIC

Vectoring Mode

Rotates the input vector to x-axiswhile recording the angle requiredto make that rotation

Application

Calculating the angle of a vector

Calculating the magnitude (scaled) Initialize angle accumulator by

ZERO

Goal : Minimize the y componentof the residual vector.

Decision: based on Sign of the

residual y component Angle accumulator will contain

the traversed angle at the end ofthe iterations

The coordinate components of a vector

are given and the magnitude and

angular argument of the original

vector are computed


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Vector Mode

rotates a vector (x,y) to the x axis whilerecording the angle required to make therotation (z)

0 (x,y)

1

2

3

4

5 (x, 0)z


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Saving an addition for Angular Measurements

System --- Vector Mode

Can we avoid computing z in vector mode & still be able to know the final

angle ?

The angle of a composite rotation is uniquely defined by the directions of

elementary rotations.

Can be stored using : Decision Vector LUT

So, also skip Addition in Vector Mode

d9 d8 d7 d06 d5 d4 d3 d2 d1 d0 Angle

0 0 0 0 0 0 0 0 0 0 99.77

0.112 + 0.224 +0.448+0.895+1.790+3.576+7.125+14.036 + 26.565 + 45.000

0 0 0 0 0 0 0 0 0 1 9.77

0.112 + 0.224 +0.448+0.895+1.790+3.576+7.125+14.036 + 26.565 -45.000

0 0 0 0 0 0 0 0 1 0 46.64

0.112 + 0.224 +0.448+0.895+1.790+3.576+7.125+14.036 - 26.565 + 45.000

CORDIC


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CORDIC

Unified Equations

xi+1 = xi yi.m.di.2-i

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

zi+1 = zi di.ei

Coordinate System m ei

Circular 1 tan -1 (2 -i)

Linear 0 2 -i

Hyperbolic -1 tanh -1 (2 -i)

Mode d

Rotation di = -1 if zi < 0, else +1

Vector di = +1 if yi < 0,else -1

J.Walther,A Unified Algorithm for Elementary Functions, Joint ComputerConference Proceedings

CORDIC Li M d


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CORDIC-Linear Mode

Multiplication

Lets say you are multiplying 28 by anynumber z

Cordics HW optimization constraintlimits the possible multiplier to powerof 2 (i.e. 2-i)

Thus possible combinations are 28*(1)

28*(0.5)

28*(0.25),..

We can represent any composite

multiplier using these elementarymultipliers

Example 28*(1+0.5+0.25) = 28*(1.75)

Example 28*(1-0.5+0.25) = 28*(0.75)

CORDIC Li M d


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CORDIC Linear Mode

Decision for d

The iterations work to produce such a combination (d) of elementary multiplierthat reduces the residual multiplier to zero

After every iteration we compare the residual with zero.

If residual is greater than zero (zi>0)

The combination that we have multiplied with till yet; is still less than the desired multiplier

We need to ADD the next elementary multiplier

If residual is less than zero (zi


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CORDIC-Linear Mode

Multiplication

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11

Series1

CORDIC Li M d

1 2


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CORDIC-Linear Mode

Multiplication

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11

1 2


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Division

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11

Minimize this

term for division

in vector mode


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Minimizing yi+1

Means we wish to force y

0, in our iterations Assume we want to find the ratio

Initialize x=4, y=3 & z=0

Minimizing yi+xi.(di.2-I) means

Minimize 3+4( some term)

This term 3+4( some term) can only reach 0;if some term = the ratio i.e. (-3/4)

AND zi+1 is actually storing the updated value of this someterm in every iteration


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CORDIC Uses

OPERATION MODE INITIALIZE

xo, yo, zo

DIRECTION REQ. OUTPUT

Sine, Cosine of an

angle

Rotation x=1/An, y=0, z=a Reduce z to Zero xn=cos zoYn=sin zo

Polar to Rect. Rotation x=(1/An)Xmag, y=0,

z=Xphase

Reduce z to Zero xn=Xmagcos zo

yn=Xmagsin zoGeneral Rotation

of a vector

Rotation x=(1/An)x0,

y=(1/An)y0, z=a

Reduce z to Zero xn,yn are the

rotated

coordinates

Vector Magnitude Vector x=(1/An)x0,

y=(1/An)y0, z=0

Reduce y to Zero

Arctangent Vector x=(1/An)x0,

y=(1/An)y0, z=0

Reduce y to Zero

Rect. to Polar Vector x=(1/An)x0,

y=(1/An)y0, z=0

Reduce y to Zero Xn is magnitude

Zn is magnitude

Arcsine,

Arccosine

Vector x=(1/An), y=0,

arg=sina or cosa

Reduce y to Value

in arg Register

)/(tan

0

001

0

2

0

2

0

xyzz

y

yxAx

n

n

nn


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Output Equations

n

i

n

n

nn

nn

A

z

zxzyAy

zyzxAx

2

0000

0000

21

0

]sincos[

]sincos[

Circular - Rotation

n

i

n

n

n

nn

A

xyzz

y

yxAx

2

00

1

0

2

0

2

0

21

)/(tan

0

Circular - Vector

0

000

0

zn

zxyy

xx

n

n

Linear - Rotation

)/(

0

000

0

xyzz

y

xx

n

n

n

n

i

n

n

nn

nn

A

z

zxzyAy

zyzxAx

2

0000

0000

21

0

]sinhcosh[

]sinhcosh[

n

in

n

n

nn

A

xyzz

y

yxAx

2

00

1

0

2

0

2

0

21

)/(tanh

0

Hyperbolic - Rotation

Linear - Vector

Hyperbolic - Vector


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Micro-Architecture

CORDIC in HW


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CORDIC in HW

Bit Parallel Architecture



zi+1 = zi

di.eiCoordinate System m ei

Circular 1 tan -1 (2 -i)

Linear 0 2 -i

Hyperbolic -1 tanh -1 (2 -i)

Mode d

Rotation di = -1 if zi < 0, else +1

Vector di = +1 if yi < 0,else -1

ROM has the angle list


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Bit-serial CORDIC StructureMUX selection for cross terms depends on the Iteration number

Each iteration takes w cycles

Unrolled


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Unrolled

CORDIC



zi+1 = zi

di.ei


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Discussion on some CORDIC Papers


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Papers(Please go through the paper)

50 Years of CORDIC: Algorithms,

Architectures, and Applications Pramod K. Meher, Senior Member, IEEE, Javier Valls, Member, IEEE, Tso-Bing Juang,

Member, IEEE, K. Sridharan, Senior Member, IEEE, and Koushik Maharatna, Member,

IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: REGULAR PAPERS, VOL. 56, NO. 9, SEPTEMBER 2009

Adaptive CORDIC: Using Parallel Angle

Recoding to Accelerate Rotations Terence K. Rodrigues and Earl E. Swartzlander, Jr., Fellow, IEEE

IEEE TRANSACTIONS ON COMPUTERS, VOL. 59, NO. 4, APRIL 2010


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CORDIC Implementations

The latency of an iterative algorithm like CORDIC is determinedby the product of the number of iterations times the cycle timeof each iteration.

Speeding up CORDIC

Reduce the number of iterations (Solution : e.g. Higher RADIX)

Reduce the execution time for each cycle (Solution : For e.g. UsingFast Adders)

Two bottlenecks in CORDIC

The micro-rotation for any iteration is performed on theintermediate vector computed by the previous iteration(Solution: Unrolling)

The ith iteration could be started only after the completion of theith iteration, since the value of d needs to be estimated (Solution :Predicting Angle Constants in advance)


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Higher Radix CORDIC - Approach

Diverging pseudorotations only occur when


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Control CORDIC

Simple method; Can be applied in runtime

there is an overshoot, so this method avoids

them by ensuring that the trajectory of the

moving vector will never overshoot the final

target position.


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Angle Recoding

Exhaustive algortihm

At every iteration, the angle constant

that will bring the residual angle (Z i

) closest to 0 is chosen from the set

of available angle constants.

HW wise very complicated angle

selection logic !

It is, therefore, most attractive in

static cases where the rotation

angle is fixed and known a priori.

The selection of the angle constants

can be done offline, and the angle

constants saved in a Look-Up Table.


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Angle Recoding

Dynamic Angle Selection for Angle


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Dynamic Angle Selection for Angle

Recoding

A simulation was done for Angle Recoding to observe the pattern Observation:

There are contiguous bands of rotation angles, all of which use a particular angle constant

during an intermediate iteration. If a rotation angle lies within a particular band, the angle

constant corresponding to that band is used.

For example, anyrotation angle in the

range [20.295, 35.78)

will use the angle

constant 26.56 degrees.

This implies that the

dynamic angle selection

for Angle Recoding can

be done by performing

an initial comparison

step, to determine the

bands that contain an

angle.


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Ranges are then defined for every possible combination of Angle constants

Range comparisons increase exponentially with bit width

Parallel Comparators Required in H/W

H/W mapping for


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H/W mapping for

Parallel Angle Recoding


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Select fewer angle constants at a time, in a given

evaluation cycle. All the angle constants which

are evaluated together in the same cycle in the

comparison logic are said to constitute a section. When all the angle constants from the section

have been processed in the CORDIC unit, the

absolute value of the residual angle that remains

at the end, is less than half the value of the

smallest angle constant in the section.

H/W mapping for


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H/W mapping for

Section based Parallel Angle Recoding


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