Digital Signal Processing Reference
In-Depth Information
CE
i
x
i-1
P
P
P+i
5:2
x
i
c
>> i
P+i
0
-
>> i
1
c = r
i
2
-i
y
d = r
i
2
-i
x
i-1
c +1
-
-
-c =
-d =
-
d +1
0
>> i
1
>> i
P+i
d
P
y
i-1
y
i
5:2
P
P+i
1
b [i]
for 2's component
Figure 12.18
A CE with compression tree
Adopting the same procedure - that is, first writing expressions for x
7
and y
7
and then substituting
values of x
6
and y
6
- we get:
x
4
r
5
2
5
y
4
1
r
5
r
6
2
11
r
5
r
7
2
12
þ r
6
r
7
2
13
þ r
6
2
6
þ r
7
2
7
r
5
r
6
r
7
2
18
x
7
¼
x
4
þ
1
r
5
r
6
2
11
y
4
y
7
¼ r
5
2
5
þ r
6
2
6
þ r
7
2
7
r
5
r
6
r
7
2
18
r
5
r
7
2
12
þ r
6
r
7
2
13
It is evident from the above expressions that the terms with 2
x
with x
P will shift the entire
value outside the range of the required precision and thus can simply be ignored. If all these terms are
ignored and we substitute previous expressions into the current iteration, we get the final iteration as
a function of x
4
and y
4
. The final expressions for P
¼
>
16 are:
0
@
1
A
!
N
1
N
1
N
1
r
i
r
j
2
ði þ jÞ
5
r
i
2
i
cos
¼
1
x
4
y
4
i¼
5
j ¼ i þ
1
i¼
ð
i þ j
ÞN
0
@
1
A
!
N
1
N
1
N
1
5
r
i
2
i
r
i
r
j
2
ði þ jÞ
sin
¼
x
4
þ
1
y
4
i¼
i¼
5
j ¼ i þ
1
ð
i þ j
Þ N
Each term in a bracket is reduced to one term. For P
¼
16, the first term in the bracket results in 12
terms and the second bracket also contains 12 terms. These expressions can be implemented as two
compression trees. All the iterations of the CORDIC algorithm are merged into one expression and
the expression can be effectively computed in a single cycle.
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