EE 5324 – VLSI Design II
Part IX: CORDIC Algorithms
Kia Bazargan
University of Minnesota
Spring 2006
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References and Copyright
• Textbook referenced
 [Par00] B. Parhami
“Computer Arithmetic: Algorithms and Hardware Designs”
Oxford University Press, 2000.
• Slides used(Modified by Kia when necessary)
 [©Oxford U Press] © Oxford University Press,
New York, 2000
Slides for [Par00] (With permission from the author)
http://www•ece•ucsb•edu/Faculty/Parhami/files_n_docs•htm
 [©Wilde] © Prof. Doran Wilde, Lecture notes on the
Computer Arithmetic Course, ECE Dept., Brigham
Young University
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What is CORDIC?
• How to evaluate trigonometric functions?
 Table lookup
 Polynomial approximations
 CORDIC
• CORDIC (COordinate Rotation DIgital Computer)





Introduced in 1959 by Jack E. Volder
Rotate vector (1,0) by f to get (cos f, sin f)
Can evaluate many functions
Rotation reduced to shift-add operations
Convergence method (iterative)
o N iterations for N-bit accuracy
 Delay / hardware costs comparable to division or
square rooting!
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Basic CORDIC Transformations
Y
sin f
• Basic idea
f
 Rotate (1,0) by f degrees
to get (x,y): x=cos(f), y=sin(f)
cos f
Y
• Rotation of any (x,y) vector:
(x’,y’)
x  x. cos(f )  y. sin( f )
y  y. cos(f )  x. sin( f )
f
• Rearrange as:
x  cos(f ).[ x  y. tan( f )]
y  cos(f ).[ y  x. tan( f )]
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X
Note :
EE 5324 - VLSI Design II - © Kia Bazargan
(x,y)
X
sin( f )
 tan(f )
cos(f )
371
Rotation and Magnitude Components
x  cos(f ).[ x  y. tan( f )]
y  cos(f ).[ y  x. tan( f )]
• Two components:
 cos(f)
Y
o Reduces the magnitude of the vector
o If don’t multiply  pseudo rotation
(x’,y’)
 tan(f)
o Rotates the vector
o Break f into a series of successively
O
shrinking angles ai such that:
Shift operation
tan (ai) =2-i
o Should we use all ai ’s?
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EE 5324 - VLSI Design II - © Kia Bazargan
f
(x,y)
X
[© Oxford U Press]
372
Pre-computation of tan(ai)
• Find ai such that tan(ai)=2-i : (or, ai=tan-1(2-i) )
i
ai
tan(ai)
0
1
2
3
4
5
6
7
8
9
45.0°
26.6°
14.0°
7.1°
3.6°
1.8°
0.9°
0.4°
0.2°
0.1°
1
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.0078125
0.00390625
0.001953125
=
=
=
=
=
=
=
=
=
=
2-0
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
• Note: decreasing ai.
 Possible to write any angle f = a0  a1  …  a9
as long as -99.7°  f  99.7° (which covers –90..90)
 Convergence possible: ai ≤ Nj=i+1ai+1 (10-5 deg accuracy)
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Example: Rewriting Angles in Terms of ai
• Example: f=30.0°







Start with a0 = 45.0
45.0 – 26.6 = 18.4
18.4 + 14.0 = 32.4
32.4 – 7.1 = 25.3
25.3 + 3.6 = 28.9
28.9 + 1.8 = 30.7
...
Y
(>
(<
(>
(<
(<
(>
30.0)
30.0)
30.0)
30.0)
30.0)
30.0)
45°
30°
X
 f = 30.0
 45.0 – 26.6 + 14.0 – 7.1 + 3.6
+ 1.8 – 0.9 + 0.4 – 0.2 + 0.1
= 30.1
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Rotation Reduction
x  cos(f ).[ x  y. tan( f )]
y  cos(f ).[ y  x. tan( f )]
• Rewrite in terms of ai: (0  i  n)
xi 1  cos(a i ).[ xi  yi .di . tan(a i )]
yi 1  cos(a i ).[ yi  xi .d i . tan(a i )]
i

xi 1  K i .[ xi  yi .d i .2 ]
i
yi 1  K i .[ yi  xi .d i .2 ]
• Where:
K i  cos(a i )  cos(tan (2 ))
1
i
di  1
Note :
cos(a i )  cos( a i )
• What about Ki’s?
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Taking Care of the Magnitude
xi 1  K i .[ xi  yi .d i .2i ]
yi 1  K i .[ yi  xi .d i .2 i ]
• Observations:
 We choose to always use ALL ai terms, with +/- signs
 Ki = cos(ai) = cos(-ai)
 At each step, we multiply by cos(ai) [constant?]
• Let the multiplications aggregate to:
n
K   Ki
n  , K  0.607 252 935...
i 0
 Multiply this constant ONLY ONCE at the end
Even better: start with (K,0) instead of (1,0)
 no multiplication at all!
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Hardware Realization: CORDIC Rotation Mode
 First rotate by f, then rotate by –di.ai to get 0
(no subtraction to compare f and current angle)
• Algorithm: (z is the current angle)









Spring 2006
Similar to
non-restor’g sqrt
• To simplify the hardware:
Mode: rotation: “at each step, try to make z zero”
Initialize x=0.607253,y=0,z=f
For i=0 n
x0,y0
y
di = 1 when z>0, else -1
x2,y2
xi+1= xi – di . 2-i . yi
-45
30
x10 x
-14
yi+1 = yi + di . 2-i . xi
+26.6
x3,y3
zi+1 = zi – di . ai
x1,y1
Result: xn=cos(f), yn=sin(f)
Precision: n bits (tan-1(2-i)  2-i)
[© Oxford U Press]
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CORDIC Rotation Mode C Code
0.6072529350
((long int)((X) * 65536.0))
// downloaded (and modified by Kia) from
// www.execpc.com/~geezer/embed/cordic.c
#include <stdio.h>
#include <math.h>
#define AG_CONST
#define FXD(X)
typedef long int fixed; /* 16.16 fixed-point */
static const fixed Alpha[]={ FXD(45.0),
FXD(26.565), FXD(14.0362), FXD(7.12502),
FXD(3.57633), FXD(1.78991), FXD(0.895174),
FXD(0.447614),FXD(0.223811), FXD(0.111906),
FXD(0.055953),FXD(0.027977) };
int main(void){
fixed X, Y, CurrAngle;
unsigned i;
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Spring 2006
X=FXD(AG_CONST); /* AG_CONST * cos(0) */
Y=0;
/* AG_CONST * sin(0) */
CurrAngle=FXD(28.027);
for(i=0; i < 12; i++){
fixed NewX;
if(CurrAngle > 0) {
NewX=X - (Y >> i);
Y+=(X >> i);
X=NewX;
CurrAngle -= Alpha[i]; }
else {
NewX=X + (Y >> i);
Y-=(X >> i);
X=NewX;
CurrAngle += Alpha[i];
} // if-else
} // for (i=…
printf(“cos(28.027)=%6.4f, sin()=%6.4f\n”,
x/65536.0, y/65536.0);
} // main
Only used:
>0, >>, +
Integer ops
CORDIC Hardware
x
shift
y
shift
z
±
±
±
Lookup
Table
[© Oxford U Press]
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CORDIC Vectoring Mode
• Difference with rotation mode?
 When choosing di, instead of trying to make z
converge to 0, try to make yi zero
 di = –sign (xi . yi)
• Variables will converge to:



xn = 1/K (x2 + y2)½
yn = 0
Zn = z + tan-1(y/x)
• Application?
 If start with x=1, z=0, the final z is tan-1(y)
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Generalized CORDIC
1
y
• Generalized CORDIC
iteration:
 xi+1= xi –  . di . 2-i .
yi
 yi+1 = yi + di . 2-i . xi
 zi+1 = zi – di . e(i)
• Variations:

1
0
-1
Spring 2006
Function
Circular rotation (basic CORDIC)
Linear rotation
Hyperbolic rotation
EE 5324 - VLSI Design II - © Kia Bazargan
B
A
F
O
0
1
E
W V
U
x
C
D
e(i)
tan-1(2-i)
2-i
tanh-1(2-i)
[© Oxford U Press]
381
Various CORDIC Applications
• Directly computes:
 sin, cos, sinh, cosh
 tan-1, tanh-1
 Division, multiplication
• Also directly computes:





tan-1(y/x)
y + x.z
(x2+y2)1/2
(x2-y2)1/2
ez = sinh(z) + cosh(z)
[© Wilde]
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Various CORDIC Applications (cont.)
• Indirectly computes:
sin z
tan z 
cos z
sinh z
tanh z 
cosh z
ln w  2 tanh
1
w e
t
Spring 2006
t ln w
1
1
1
cos w  tan
sin w  tan
w 1
w 1
log b w  K . ln w
1

w  ln w 
1 w
w
w
1 w
2
2

1 w 
cosh 1 w  ln w  1  w2
sinh
1
2
w  ( w  1 / 4)  ( w  1 / 4)
2
2
[© Wilde]
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Summary of CORDIC Applications
Rotation mode
di=sign(zi), zi  0
= 1
Circular
e(i) =
tan-12-i
= 0
Linear
e(i)= 2-i
x
y
z
C
O
R
D
I
C
K(x cos z - y sin z)
K(y cos z + x sin z)
0
For cos & sin, set x=1/K, y=0
tan z = sin z / cos z
x
y
z
C
O
R
D
I
C
Vectoring mode
di=-sign(xi yi), yi  0
C
K (x2 + y2)1/2
O
R
y
0
D
-1
I
z
z + tan (y/x)
C
For tan-1, set x = 1, z = 0
cos-1w = tan-1[(1-w2)1/2/w]
sin-1w = tan-1[w/(1-w2)1/2]
x
x
x
y + xz
y
0
For multiplication, set y = 0
z
C
O
R
D
I
C
x
0
z + y/x
For division, set z = 0
Note: in linear mode, limited input range (convergence)
Spring 2006
EE 5324 - VLSI Design II - © Kia Bazargan
[© Oxford U Press]
[Par00], p. 371
384
Summary of CORDIC Applications (cont.)
Rotation mode
di=sign(zi), zi  0
Vectoring mode
di=-sign(xi yi), yi  0
C
K’(x2 - y2)1/2
O
R
 = -1 y
K'(y cosh z+x sinh z) y
0
D
-1
I
Hyperz
z
0
z
+
tanh
(y/x)
C
bolic
For tanh-1 set x = 1, z = 0
For cosh & sinh, set x=1/K', y=0
e(i) =
ln w=2 tanh-1 |(w - 1)/(w + 1)|
tanh z = sinh z / cosh z
w1/2=[(w+1/4)2 - (w-1/4)2]1/2
-1
-i
tanh 2
exp(z) = sinh z + cosh z
cosh-1w = ln (w + (1-w2)1/2)
w t = exp(t ln w)
sinh-1w = ln (w + (1+w2)1/2)
In the  = -1 case, steps 4, 13, 40, 121, ... , j , 3j + 1, ... must be repeated
for the method to converge. These repetitions are incorporated in the
constant K' below.
xi+1= xi –  . di . 2-i . yi
  {-1,0,1}, di  {-1,1}
yi+1 = yi + di . 2-i . xi
K = 1.646 760 258 121…
[Par00], p. 371
zi+1 = zi – di . e(i)
K’= 0.828 159 360 960 2…
x
C
O
R
D
I
C
K'(x cosh z-y sinh z) x
[© Oxford U Press]
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To Probe Further…
• Tutorials
 http://cnmat.cnmat.berkeley.edu/~norbert/cordic/node3.html
 http://www.execpc.com/~geezer/embed/cordic.htm
(including C code)
• Papers
 Survey paper on FPGA implementation of CORDIC algorithms:
http://www.andraka.com/files/crdcsrvy.pdf
 http://www.taygeta.com/cordic_refs.html
• Hardware implementations
 http://www.free-ip.com/cordic/
 http://www.stanford.edu/~chet/cordic.html
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