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CORDIC Algorithm
COordinate Rotation DIgital Computer
- Example: vector rotation:
= ^ −
y
x
y
x
sin( ) cos( )
cos( ) sin( )
φ φ
φ φ
x
y
x'
y'
φ
Implementation cost:
- 2 multiplications
- 2 addition or subtraction
- 2 sin() and cos()
How do we compute sin(φ) or cos(φ)?
- Easy, but expensive (for high accuracy): Direct table look-up:
- Exploit symmetry of the sine function:
φ (^) sin(φ)
sin(φ)=sin(π − φ) sin(-φ)=-sin(φ) also: sin(φ)=cos(π/2 − φ) Only one quadrant needed
Direct Table Look-up (Example)
from Peter Cheung, Imperial College London
Direct Table Look-up (Example)
- Trade-off between memory and the logic for multiplexing and complement computation
- Sine tables are provided in several DSPs
- Note: a BRAM on our Spartan-6 can hold 2K x 9 bit (considering the example, extra logic is not required)
- For higher precision: two level table look-up:
sin(α + β) = sin(α) cos(β) + cos(α) sin(β) (trick: coarse table for α and fine table for β) Requires two multiplications and one addition
CORDIC Algorithm
COordinate Rotation DIgital Computer
- Method for elementary function evaluation
(e.g., sin( z ), cos( z ), tan -1( y ))
- The modern CORDIC algorithm was first described in
1959 by Jack E. Volder. It was developed to replace the analog resolver in the B-58 bomber's navigation computer. (from Wikipedia)
- Used in Intel 80x87 coprocessor and Intel 80486
- Commonly used for FPGAs
- Complexity Comparable to Division
from Wikipedia
CORDIC Algorithm: Key Ideas
- Rather than computing sin(φ) directly, we iteratively rotate β towards φ
- Ideal search within first quadrant: - Step 1: set β = 45° - Step 2: if φ >= β then β = β + (45/2)° else β = β - (45/2)° - Step 3: if φ >= β then β = β + (45/4)° else β = β - (45/4)° - Continue by halving step size of β in each iteration
- Search is stable if b[i]>b[i+1]>=b[i]/
CORDIC Algorithm:
- What does it mean to rotate tan( β ) = ± 2 -i^ rather then rotating ± β · 2 -i^ in each iteration?
- For smaller angles, it converges against the same:
- Important: ± 2 -i^ is a simple shift! (can in some cases be directly implemented within the routing)
- Rotation by an arbitrary angle is difficult, so we perform psuedorotations
from Wikipedia
i 2 -i^ arctan(2 -i )360/2π^ 45 2 -i
- 1 0.5 26.56505118 * 4.065051177 * 22. 0 1 45 * - 2 0.25 14.03624347 0.753717879 11. - 3 0.125 7.125016349 0.106894615 5. - 4 0.0625 3.576334375 0.013826201 2. - 5 0.03125 1.789910608 0.001743421 1. - 6 0.015625 0.89517371 0.000218406 0. - 7 0.0078125 0.447614171 2.73158E-05 0. - 8 0.00390625 0.2238105 3.41494E-06 0. - 9 0.001953125 0.111905677 4.26882E-07 0.
- 10 0.000976563 0.055952892 5.33607E-08 0.
- 11 0.000488281 0.027976453 6.6701E-09 0.
- 12 0.000244141 0.013988227 8.33763E-10 0.
- 13 0.00012207 0.006994114 1.0422E-10 0.
- 14 6.10352E-05 0.003497057 1.30276E-11 0.
- 15 3.05176E-05 0.001748528 1.62844E-12 0.
- 16 1.52588E-05 0.000874264 2.03555E-13 0.
- 17 7.62939E-06 0.000437132 2.54444E-14 0.
- 18 3.8147E-06 0.000218566 3.18056E-15 0.
- 19 1.90735E-06 0.000109283 3.97577E-16 8.58307E-
- 20 9.53674E-07 5.46415E-05 4.96903E-17 4.29153E-
- 4.06505=26.565-45/ CORDIC Rotation Angles
CORDIC Hardware (Bit-parallel, unrolled)
- Implementation cost: three ADD/SUB ALUs per iteration.
- Shift operations: hardwired
- rotate angles ( a (^) i ) are fit into the logic
- Typically with pipeline register after each iteration (results in very high throughput)
- Improvement of the angle resolution by almost one signal bit iteration. http://en.wikibooks.org/wiki/Digital_Circuits/CORDIC
CORDIC Hardware (Bit-parallel, iterative)
- Lower throughput (n times less)
- The barrel-shifter is variable and costs logic
- an is stored in a small register file
http://en.wikibooks.org/wiki/Digital_Circuits/CORDIC
CORDIC Algorithm: Gain
- Cordic uses iterative rotations in steps of tan( β ) = ± 2 -i
x i+1 = cos( tan -1^ (± 2 -i ) ) · [ x i – y i · d i · 2 -i^ ] y i+1 = cos( tan -1^ (± 2 -i ) ) · [ y i + x i · d i · 2 -i^ ]
- How we deal with cos ( tan-1 ( ± 2 -i^ ) )?
- The cosine is symmetric: cos ( tan-1 (2 -i) ) = cos ( tan-1 ( - 2 -i) )
- cos ( tan -1 (2 -i) ) is the gain K i of an iteration
- We can compute K offline for all n iterations:
- The gain approaches 0.6037, if n goes to infinity
= (^) ∏ n
K Ki
i
i K (^) i 2 1 2
1 cos(arctan( 2 − )) −
= =
CORDIC Algorithm: Gain
- In order to compensate the gain, we have to scale the result with the reciprocal value of the gain:
- We can compute A offline for all n iterations
- A approaches 1.647, if n goes to infinity
- No overhead if combined with other scaleing values
∏ = + − n
A 1 2 2 i
CORDIC Uses
OPERATION MODE INITIALIZE DIRECTION Sine, Cosine Rotation (^) x =1/ An , y =0, z = α Reduce z to Zero Polar to Rect. Rotation x =(1/ An ) Xmag , y =0, z = Xphase Reduce z to Zero General Rotation Rotation (^) x =(1/ An ) x 0 , y =(1/ An ) y 0 , z = α Reduce z to Zero Arctangent Vector x =(1/ An ) x 0 , y =(1/ An ) y 0 , z = 0 Reduce y to Zero Vector Magnitude Vector x =(1/ An ) x 0 , y =(1/ An ) y 0 , z = 0 Reduce y to Zero Rect. to Polar Vector x =(1/ An ) x 0 , y =(1/ An ) y 0 , z = 0 Reduce y to Zero Arcsine, Arccosine Vector x =(1/ An ), y =0, arg =sin α or cos α
Reduce y to Value in arg Register
- Can Use CORDIC For Others Also:
- Linear Functions
- Hyperbolic Functions
- Square Rooting
- Logarithms, Exponentials
CORDIC – Rotation/Vector Modes
• Rotation Mode:
1 1 1 1
tan (2 )
i i i i i i i 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 sin 0 1 2 1, 0 1, otherwise
n n n n n n n i i i i
x A x z y z y A y z x z z A
d z
−
= − = + = = +
= −^ < +
∏
• Vector Mode:
1 1 1 1
tan (2 )
i i i i i i i i i i i i i i
x x y d
y y x d
z z d
−
−
− −
0 2 02
0 0 0 2 0
0 tan 1
1 2 1, 0 1, otherwise
n n n n n (^) i n i
i i
x A x y y z z y x A
d y
−
= +
= + −
= +
= +^ < −
∏
