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CORDIC AlgorithmCOordinate Rotation DIgital Computer

Method for Elementary Function Evaluation (e.g.,sin( z ), cos( z ), tan -1(y ))

Originally Used for Real-time Navigation (Volder 1956)

Idea is to Rotate a Vector in Cartesion Plane bySome Angle

Complexity Comparable to Division

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CORDIC AlgorithmKey Ideas

If we have a computationally efficient way of rotating a vector, we can

evaluate cos, sin, and tan

1 functionsRotation by an arbitrary angle is difficult, so we perform psuedorotations

Use special angles to synthesize a desired angle z

z = (1) + (2) + . . . + (m )

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CORDIC AlgorithmKey Ideas

Rotate the vector OE(i )

with end point at ( x (i )

, y (i )

) by(i )

x (i +1) = x (i )cos (i ) y (i ) sin (i ) = (x (i ) y (i ) tan (i ))/(1 + tan 2 (i ))1/2

y (i +1) = y (i ) cos (i ) + x (i ) sin (i ) = (y (i ) + x (i ) tan (i ))/(1 + tan 2 (i ) ) 1/2

z (i +1) = z (i ) (i )

Goal: eliminate the divisions by (1 + tan2 (i )) 1/2 and choose (i ) so thattan (i ) is a power of 2

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Elimination of Division by (1 + tan2 (i ))1/2

Whereas a real rotation does not change the length R (i ) of the vector, a pseudorotation step increases its length to:

R (i +1) = R (i ) (1 + tan 2 (i ))1/2

The coordinates of the new end point E (I+ 1) after

pseudorotation is derived by multiplying the coordinates of

E (i +1) by the expansion factor

x (i +1) = x (i ) y (i ) tan (i )

y (i +1) = y (i ) + x (i ) tan (i ) [Pseudorotation]

z (i +1) = z (i ) (i )

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Elimination of Division by (1 + tan2 (i ))1/2

Assuming x (0)

= x , y (0)

= y , and z (0)

= z , after m realrotations by the angles (1) , (2) , . . . , (m ), we have:

x (m ) = x cos( (i )) y sin( (i ))

y (m )

= y cos((i )

) + x sin((i )

)z (m ) = z ( (i ))

After m pseudorotations by the angles (1) , (2) , . . . , (m ):

x (m ) = K (x cos( (i )) y sin( (i )))y (m ) = K (y cos( (i )) + x sin( (i ))) [*]

z (m ) = z ( (i ))

where K = (1 + tan 2 (i ))1/2

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Basic CORDIC Iterations

Pick(i )

such that tan(i )

= d i 2 i

, d i {

1, 1}x (i +1) = x (i ) d i y (i )2

i

y (i +1) = y (i ) + d i x (i )2 i [CORDIC iteration]

z (i +1) = z (i ) d i tan 1 2 i

If we always pseudorotate by the same set of angles (with

+ or signs), then the expansion factor K is a constant

that can be precomputed

Example : pseudorotation for 30 degrees30.0 45.0 26.6 + 14.0 7.1 + 3.6 + 1.8 0.9

+ 0.4 0.2 + 0.1 = 30.1

e (i ) = tan 1 2 -i

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Basic CORDIC Iteration

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CORDIC Rotation Mode

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CORDIC Rotation Mode

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CORDIC Vectoring Mode

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CORDIC Vectoring Mode

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CORDIC Hardware

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Generalized CORDIC

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Rotation Modes

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Binary Angular Measurement - BAM Angle Accumulator can Represent Angles as BAM

Encode d i ={-1,+1} as Bit Values {0,1}

Example: -1 Represented by 0 and +1 Represented by 1

LSb Represents d 0

Content z =01011

z =+45 +26.6 -14.0 +7.1 -3.6 =61.1

Can Simplify CORDIC Circuitry for Some Modes May Need BAM encode/decode Can Use Lookup Table

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Review - CORDIC - Rotation Mode

Input is Angle, Initialized in Angle Accumulator Vector Initialized to Lie on x -axis Each Iteration d i Chosen by Sign of Angle Attempt to Bring Angle to Zero Result is x Register Contains ~cos Result is y Register Contains ~sin Also Polar to Rectangular if x Register Initialized to

Magnitude

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Review - CORDIC - Vector Mode

Input is (Pre-scaled) Vector in ( x ,y ) Registers Angle, Initialized to Zero

Each Iteration d i Chosen to Move Vector to Lie AlongPositive x -axis (Want to Reduce y Register to Zero)

Result is Original Vector Angle in Angle Accumulator Can be Used for sin -1 and cos -1 Also Rectangular to Polar Conversion Magnitude in x Register

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CORDIC Rotation/Vector Modes

Rotation Mode:

1

1

1

1

2

2

tan (2 )

ii i i i

ii i i i

i

i i i

x x y d

y y x d

z z d

0 0 0 0

0 0 0 0

2

0

cos sin

cos sin0

1 2

1, 0

1, otherwise

n n

n n

n

ni

ni

i

i

x A x z y z

y A y z x z z

A

zd

Vector Mode:

1

1

11

2

2

tan (2 )

ii i i i

ii i i i

ii i i

x x y d

y y x d

z z d

2 2

0 0

00

0

2

0

0

ta n 1

1 2

1, 0

1, o the rw i se

n n

n

n

ni

ni

ii

x A x y

y

y z z x

A

yd

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Rotation Angle Limits

Rotation/Vector Algorithms Limited to 90 Due to Use of = tan(2 0) for First Iteration Several Ways to Extend Range

Can use trig identities to covert the problem to one that is withinthe domain of convergence

One Way is to Use Additional Rotation for Angles Outside Range

This Rotation is Initial 90 Rotation

'

'' 2

1, 0

1, otherwisei

x d y

y d x z z d

yd

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CORDIC Uses

OPERATION MODE INITIALIZE DIRECTIONSine, Cosine Rotation x=1/ A n, y=0, z= Reduce z to ZeroPolar to Rect. Rotation x=(1/ A n) X mag , y=0, z= X phase Reduce z to ZeroGeneral Rotation Rotation x=(1/ A n) x0, y=(1/ A n) y0, z= Reduce z to ZeroArctangent Vector x=(1/ A n) x0, y=(1/ A n) y0, z=0 Reduce y to ZeroVector Magnitude Vector x=(1/ A n) x0, y=(1/ A n) y0, z=0 Reduce y to ZeroRect. to Polar Vector x=(1/ A n) x0, y=(1/ A n) y0, z=0 Reduce y to ZeroArcsine, Arccosine Vector x=(1/ A n), y=0,

arg =sin or cos Reduce y to Valuein arg Register

Can Use CORDIC For Others Also: Linear Functions Hyperbolic Functions Square Rooting Logarithms, Exponentials

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Iterative CORDIC Structure*

*Taken from A Survey of CORDIC Algorithms for FPGABased Computers, R. Andraka, FPGA98

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Bit-serial CORDIC Structure*

*Taken from A Survey of CORDIC Algorithms for FPGA

Based Computers, R. Andraka, FPGA98