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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 by

Some Angle - Complexity Comparable to Division

CORDIC AlgorithmKey Ideas

If we have a computationally efficient way of

rotating a vector, we can evaluate cos, sin, and

tan1 functions Rotation by an arbitrary angle is

difficult, so we perform psuedorotations Use

special angles to synthesize a desired angle z z

a(1) a(2) . . . a(m)

CORDIC AlgorithmKey Ideas

Rotate the vector OE (i) with end point at (x

(i), y (i)) by a (i) x (i1) x (i)cos a (i) y

(i) sin a (i) (x (i) y (i) tan a (i))/(1

tan2a (i))1/2 y (i1) y (i) cos a (i) x (i)

sin a (i) (y (i) x (i) tan a (i))/(1 tan2a

(i) ) 1/2 z (i1) z (i) a (i) Goal eliminate

the divisions by (1 tan2a (i)) 1/2 and choose a

(i) so that tan a(i) is a power of 2

Elimination of Division by (1 tan2a(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(i1) R(i) (1 tan2a

(i))1/2 The coordinates of the new end point

E(I1) after pseudorotation is derived by

multiplying the coordinates of E(i1) by the

expansion factor x (i1) x (i) y (i) tan a

(i) y (i1) y (i) x (i) tan a (i)

Pseudorotation z (i1) z (i) a (i)

Elimination of Division by (1 tan2a(i))1/2

Assuming x(0) x, y (0) y, and z (0) z,

after m real rotations by the angles a(1), a (2),

. . . , a (m), we have x(m) x cos(åa (i)) y

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

(i)) z(m) z (åa (i)) After m pseudorotations

by the angles a(1), a (2), . . . , a (m) x(m)

K(x cos(åa (i)) y sin(åa (i))) y(m) K(y cos(åa

(i)) x sin(åa (i))) z(m) z (åa

(i)) where K P(1 tan2a(i))1/2

Basic CORDIC Iterations

Pick a (i) such that tan a (i) di 2 i, di Î

1, 1 x(i1) x(i) di y(i)2i y (i1) y

(i) di x(i)2iCORDIC iteration z (i1) z

(i) di tan1 2i 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

degrees 30.0 _at_ 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

Basic CORDIC Iteration

CORDIC Rotation Mode

CORDIC Rotation Mode

CORDIC Vectoring Mode

CORDIC Vectoring Mode

CORDIC Hardware

Generalized CORDIC

Rotation Modes

Binary Angular Measurement - BAM

- Angle Accumulator can Represent Angles as BAM
- Encode di-1,1 as Bit Values 0,1
- Example -1 Represented by 0 and 1 Represented

by 1 - LSb Represents d0
- Content z01011
- z45 ? 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

Review - CORDIC - Rotation Mode

- Input is Angle, ? Initialized in Angle

Accumulator - Vector Initialized to Lie on x-axis
- Each Iteration di 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

Review - CORDIC - Vector Mode

- Input is (Pre-scaled) Vector in (x,y) Registers
- Angle, ? Initialized to Zero
- Each Iteration di Chosen to Move Vector to Lie

Along Positive 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

CORDIC Rotation/Vector Modes

- Rotation Mode

- Vector Mode

Rotation Angle Limits

- Rotation/Vector Algorithms Limited to ?90?
- Due to Use of ? tan(20) for First Iteration
- Several Ways to Extend Range
- Can use trig identities to covert the problem to

one that is within the domain of convergence - One Way is to Use Additional Rotation for Angles

Outside Range - This Rotation is Initial ?90? Rotation

CORDIC Uses

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

Iterative CORDIC Structure

Taken from A Survey of CORDIC Algorithms for

FPGA Based Computers, R. Andraka, FPGA98

Bit-serial CORDIC Structure

Taken from A Survey of CORDIC Algorithms for

FPGA Based Computers, R. Andraka, FPGA98

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