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Vector Radix FFT Algorithm• EEE 507 - Lecture 8
Multidimensional rederivation of the 1-D FFT
Divide-and-conquer algorithm
Operates as a recursive procedure
Solves problem by dividing it into smaller problems that are the sameand then combining answers
Examples of divide-and-conquer algorithms
1-D FFT
Eklundh’s transposition procedure
Vector-Radix
Note:
The R-C algorithm divided the problem into column/row 1-D DFTs andnot into multidimensional DFTs => not into similar problems
=> not divide-and-conquer algorithm
Vector Radix FFT Algorithm• EEE 507 - Lecture 8
1-D FFT Assume N is even:where G(K) and H(K) can be calculated by using N/2-point DFTs.
N K W n x K X
N n
nK
N
odd
n
nK
N
even
n
nK
N
W n x W n x K X
K
H
W
K
G
K
N
Vector Radix FFT Algorithm EEE 507 - Lecture 8
2D FFT Decimation-In-Time algorithm
Note:If
even =>
If
odd =>
2
2
1
1
2 1
2
2
1
1
1 2
2
2
1
1
1 2
2
2
1
1
1 2
) , ( ) , (
) , ( ) , ( ) , (
K
n N
K
n N
odd
n
odd
n
K
n N
K
n N
even
n
odd
n
K
n N
K
n N
odd
n
even
n
K
n N
K
n N
even
n
even
n
W W n n x W W n n x
W W n n x W W n n x K K X
∑
∑
∑
∑
=
n
2
m
n
=
1
2
=
m
n
1
n
Vector Radix FFT Algorithm EEE 507 - Lecture 8
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
2
2
1
1
1
2
) 1 2 , 1 2 (
) 2 , 1 2 ( ) 1 2 , 2 (
) 2 , 2 ( ) , (
K m N K m N N
m
N
m
K m N K m N N
m
N
m
K m N K m N N
m
N
m
K m N K m N N
m
N
m
W
W
m
m
x
W
W
m
m
x
W
W
m
m
x
W W m m x K K X
∑ ∑ ∑ ∑ ∑ ∑
∑ ∑
=
=
=
=
2D FFT Decimation-In-Time algorithm
1 1 1 1 1 1 1 1
2 /
K
m N
K
m
N
j K m N j K m N
W
e
e
W
=
=
=
π
π
Note:
Vector Radix FFT Algorithm EEE 507 - Lecture 8
2D FFT Decimation-In-Time algorithm
where
Note: If
i
=1, multiply
S
ij
by
If
j
=1, multiply
S
ij
by
:N/2-point DFTs
Still need to calculate
for
and
between
N/
and
N
This is simplified by the fact that
is periodic with period
N/2 in
K
and
K
Note:
So, when
N/
is added to a variable (
K
or
K
), reverse the sign of
W
N
having that variable as exponent.
1
2
/
0
1
2
/
0
1 2
−
≤
≤
−
≤
≤
N
K
N
K
2
2
1
1
1
2
2 /
2 /
1
2
0
2
1
1
2
0
2
1
K
m N
K
m N
N m
N m
ij
W W j m i m x K K S
∑ ∑
− =
− =
) , ( ) , ( ) , ( ) , ( ) , (
2
2
1
1
2
K K S W K K S W K K S W K K S K K X
K
K
N
K
N
K
N
=
1
K N
W
2
K N
W
)
,
(
2
1
K
K
X
2
1
K
K
S
ij
,
,
,
S
S
S
S
K
K
) 2 / , 2 / ( ) 2 / , ( ) , 2 / ( ) , (
2 1 2 1 2 1 2 1
N K N K S N K K S K N K S K K S
ij
ij
ij
ij
1
−
=
=
j
N
N
e
W
Vector Radix FFT Algorithm EEE 507 - Lecture 8
2D FFT Decimation-In-Time algorithm
Note:
Procedure:
1. 4 smaller DFTs
need to be calculated
2. By doing three different complex multiplications, we obtain 4 samples of
the DFT, for a given
are calculated by repeating same procedure
) , ( ) , ( ) , ( ) , ( ) , 2 / (
2
1
11
)
(
2
1
10
2
1
01
2
1
00
2
1
2
1
1
2
K K S W K K S W K K S W K K S K N K X
K
K
N
K N
K N
2
1
1
2
K K S W K K S W K K S W K K S N K K X
K
K
N
K
N
K
N
2
1
1
2
K K S W K K S W K K S W K K S N K N K X
K
K
N
K
N
K
N
N
K
N
K
) , ( ) , ( ) , ( ) , ( ) , (
2
1
11
)
(
2
1
10
2
1
01
2
1
00
2
1
2
1
1
2
K K S W K K S W K K S W K K S K K X
K
K
N
K N
K N
=
,
,
,
S
S
S
S
),
,
(
2
1
K
K
1
2
/
,
0
2
1
−
≤
≤
N
K
K
S
S
S
S
Vector Radix FFT Algorithm• EEE 507 - Lecture 8
2D FFT Decimation-In-Time algorithm
Computations
"
#CMULTs=3 / Butterfly
"
#CADDs=12 (=3×4) / Butterfly
since 3 CADDs / DFT sample, and 4 DFT samples/ Butterfly
Note: one can reduce #CADDs to 8 (refer to Problem 2.11 in
Dudgeon & Mersereau)
"
#stages required for 2-D N×N-point FFT
, for
"
#Butterfly per stage=Why? we have N×N initial points;I= # of inputs per stage =
; each Butterfly takes care of 4 inputs
O= # of outputs per stage =
Summary:
# CMULTS= (3 CMULTs/Butt) (
Butt/stage) (
stages)
#CADDs= (8 CADDs/Butt) (
Butt/stage) (
stages)
υ
=
=
N
2
log
V
N
2 = 4 / 2 N
2
N
2
N
N
log
4 / 2 N N N
2
2
log
4
3
N
N
2
2
log
2
2
N
N
2
log
M-Dimensional Vector-Radix FFT EEE 507 - Lecture 8
Array of size
Same basic Butterfly but different # of input/output points
Complexity:
. # inputs/outputs per stage=. # inputs per Butterfly=. # Butterflies per stage=. # stages =
, for
. # CMULTs/Butt=(# inputs/Butt-1)=Total # CMULTs =
4
4
3
4
4
2
1
M
N
N
N
N
×
×
×
...
M
N
M
2
M
M
M
N
N
)
2
(
2
=
ν
=
N
log
V
N
1
2
−
M
N
N
N
N
M
M
M
M
M
2
2
log
.
2
1
2
)
.(log
)
2
).(
1
2
(
−
=
−
