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An in-depth exploration of the CORDIC (Coordinate Rotation Digital Computer) algorithm, focusing on its methodology, rotations, vectoring, conversions, and implementations. The CORDIC algorithm is a versatile technique used for the realization of rotations, calculation of trigonometric functions, and inverse trigonometric functions. It is particularly useful in digital signal processing, computer graphics, and other fields where efficient and accurate computation of angles is required.
What you will learn
Typology: Schemes and Mind Maps
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IMPLEMENTATIONS: word-serial and pipelined
11 – CORDIC
tan
−
1
( a/b
√ a 2 + b 2
, etc.
11 – CORDIC
y
x
(x
in
, y
in
)
(x
R
, y
R
)
Θ
β
M
in
Figure 11.1:
VECTOR ROTATION
11 – CORDIC
α
j
θ
:
θ
∑∞
α
j
θ
) =
∏∞
α
j )
α
j
) :
x
R
j
x
R
j
] cos(
α j ) − y R [
j
] sin(
α
j
)
y
R
j
x
R
j
] sin(
α
j ) +
y
R
j
] cos(
α
j )
11 – CORDIC
7
ROTATION-EXTENSION (cont.)
j
]
j
+1] =
j
] M
j
] =
cos
α
j
M
j
] = (1+
σ
j 2
2 − 2 j ) 1 / 2 M
j
] = (1+
− 2 j ) 1 / 2 M
j
]
∏∞
− 2 j ) 1 / 2 ≈ 1.
z
[ j
z
[ j ] − α j = z [
j
]
−
σ
j
tan
−
1 (
−
j )
11 – CORDIC
x
j
x
j ] − σ j 2 − j y [ j ]
y
j
y
[ j
] +
σ j 2 − j x [
j
]
z
j
z
[ j
]
−
σ
j
tan
−
1 (
−
j )
11 – CORDIC
x
in
, y
in
θ
z
[ j
z
[ j
]
−
σ
j
tan
−
1 (
−
j )
z
[0] =
θ
x
x
in
y
y
in
σ
j
if z
j
]
if z
j
]
x f = K ( x
in
cos
θ
y
in
sin
θ
)
y f = K ( x
in
sin
θ
y
in
cos
θ
)
z
f
11 – CORDIC
y
x
(x
in
, y
in
)
Θ
(x
1 ,y
1 )
(x
2 ,y
2 )
(x
3 ,y
3 )
(x
f ,, y
f )
primitive angles
Figure 11.3:
Rotating a vector using microrotations.
11 – CORDIC
EXAMPLE 11.1 (cont.)
x
and
y
[13]
−
12
11 – CORDIC
cos
θ
sin
θ
x
y
a
b
a
cos
θ
b
sin
θ
a
sin
θ
b
cos
θ
x
a/K
y
[0] =
b/K
11 – CORDIC
x
in
, y
in
y
n
x
R
√
x
in 2
y
in 2
y
R
z
f
= tan
−
1 (
0 . 43
0 . 75
11 – CORDIC
j
y
j ] σ j x [
j ] z [ j ]
11 – CORDIC
z
[ i ] | ≤
∑∞
i
tan
−
1
(
−
j )
θ
max
z
[0]
max
∑∞
tan
−
1
(
− j ) ≈ 1.
o )
σ
j
and
z
[ j
]
θ < θ
max
z
i ] | ≤
tan
−
1 (
−
( i −
tan
−
1 (
−
i −
1
)
∑∞
i
tan
−
1
(
−
j
)
tan
−
1
(
−
i )
∑∞
i
tan
−
1 (
−
j )
i
11 – CORDIC
y
x
α
i-
α
i-
α
i
Figure 11.4:
CONVERGENCE CONDITION: THE MAXIMUM NEGATIVE CASE.
11 – CORDIC
