Radix-2 DIT FFT

Radix-2 DIT FFT
- SARANG JOSHI

• r is called as radix or base
• t= no. of stages in FFT algorithm.
• E.g. for N=8 then N=
t
rN 
3
2

Radix-r FFT
 N-pt sequence is decimated into r-point sequences.
 For each r-point sequence , r-point DFT is computed.
 From the results of r-point DFT , -point DFT is
computed.
 From the results of -point DFT , -point DFT is
computed and so on until we get -point DFT
2
r
2
r
3
r
m
r

)(nx
)(1 rx
)(2 rx
)(1 kX
)(2 kX
)(kX

Why FFT?
To compute N-pt DFT :
• No. of multiplications required:
• No. of additions required:
2
N
NN 2
N Multiplications Additions
2 4 2
4 16 12
8 64 56
16 256 240
32 1024 992
64 4096 4032

Properties of twiddle factor
1]
2]
3]
Nk
N
k
N WW


k
N
Nk
N WW 
 2/
2
2/ NN WW 

0 2 4 6 1 3 5 7
0 4 2 6 1 5 3 7
0 4 2 6 1 5 3 7
0 1 2 3 4 5 6 7A signal of 8
samples
2 signals of 4
samples
4 signals of 2
samples
Radix-2 DIT- FFT Algorithm
x(n)
x2(r)x1(r)

1
2
,2,1,0,)12()();2()( 21 
N
rrxrxrxrx 
)1
2
~0()12()()(
)1
2
~0()2()()(
1
2
0 2
1
2
0 2
22
1
2
0 2
1
2
0 2
11












N
kWrxWrxkX
N
kWrxWrxkX
N
r
rk
N
N
r
rk
N
N
r
rk
N
N
r
rk
N
divide N-point sequence x(n) into two N/2-point sequence x1(r) and x2(r)
compute the DFT of x1(r) and x2(r)

)1,2,1,0()()(
)()(
)12()2(
)()()()(
21
1
2
0 2
2
1
2
0 2
1
1
2
0
)12(
1
2
0
2
1
)(0
1
)(0
1
0






















NkkXWkX
WrxWWrx
WrxWrx
WnxWnxWnxkX
k
N
N
r
rk
N
k
N
N
r
rk
N
N
r
kr
N
N
r
rk
N
N
oddn
nk
N
N
evenn
nk
N
N
n
nk
N

compute the DFT of N-point sequence x(n)

)1
2
,1,0()()(
)
2
()
2
()
2
(
)1
2
,1,0()()()(
21
2
)
2
(
1
21




N
kkXWkX
N
kXW
N
kX
N
kX
N
kkXWkXkX
k
N
N
k
N
k
N



)1
2
,1,0()()()
2
(
)1
2
,1,0()()()(
21
21


N
kkXWkX
N
kX
N
kkXWkXkX
k
N
k
N


 Butterfly computation flow graph
)(1 kX
)(2 kX
)()( 21 kXWkX k
N
)()( 21 kXWkX k
N
k
NW
1

N/2-
point
DFT
N/2-
point
DFT
)0(1X
)1(1X
)2(1X
)3(1X
)0(2X
)1(2X
)2(2X
)3(2X
0
NW
1
NW
2
NW
3
NW
)0()0(1 xx 
)2()1(1 xx 
)4()2(1 xx 
)6()3(1 xx 
)1()0(2 xx 
)3()1(2 xx 
)5()2(2 xx 
)7()3(2 xx 
)(1 rx
)(2 rx
)4(X1
)5(X1
)6(X1
)7(X1
)0(X
)1(X
)2(X
)3(X
N-point DFT

2nd Stage of Decimation
1
4
,1,0,)12()();2()( 112111 
N
rrxrgrxrg 
)1
4
,1,0()()()
4
(
)1
4
,1,0()()()(
122/111
122/111


N
kkGWkG
N
kX
N
kkGWkGkX
k
N
k
N



N/4
point
DFT
N/4
point
DFT
)0()0()0( 111 gxx 
)1()2()4( 111 gxx 
)0()1()2( 121 gxx 
)1()3()6( 121 gxx 
)(11 rg
)(12 rg
)0(11G
)1(11G
)0(12G
)1(12G
0
NW
2
NW 1
1
)0(1X
)1(1X
)2(1X
)3(1X

1
4
,1,0,)12()();2()( 222221 
N
rrxrgrxrg 
)1
4
,1,0()()()
4
(
)1
4
,1,0()()()(
222/212
222/212


N
kkXWkG
N
kX
N
kkGWkGkX
k
N
k
N



N/4
point
DFT
N/4
point
DFT
)0()0()1( 212 gxx 
)1()2()5( 212 gxx 
)0()1()3( 222 gxx 
)1()3()7( 222 gxx 
)(21 rg
)(22 rg
)0(21G
)1(21G
)0(22G
)1(22G
0
NW
2
NW 1
1
)0(2X
)1(2X
)2(2X
)3(2X

0
NW 1
)0()0(11 xg 
)4()1(11 xg 
)0(11G
)1(11G

1
1
1
1
1
)0(x
)2(x
)4(x
)6(x
)1(x
)3(x
)5(x
)7(x
)0(X
)1(X
)2(X
)3(X
)4(X
)5(X
)6(X
)7(X
0
NW
0
NW
0
NW
0
NW
0
NW
2
NW
0
NW
2
NW
0
NW
1
NW
2
NW
3
NW
1
1
1
1 1
1 1
STAGE -1 STAGE -2 STAGE -3
S1(0)
S1(1)
S1(4)
S1(5)
S1(2)
S1(3)
S1(6)
S1(7)
S2(0)
S2(1)
S2(4)
S2(5)
S2(2)
S2(3)
S2(6)
S2(7)
Radix-2 DIT FFT

N Multiplications Additions
2 1 2
4 4 8
8 12 24
16 32 64
32 80 160
64 192 384
 The total number of complex multiplications :
 The total number of complex additions is: NNtNaF 2log
N
N
t
N
mF 2log
22


FFT DFT
N Multiplications Additions Multiplications Additions
2 1 2 4 2
4 4 8 16 12
8 12 24 64 56
16 32 64 256 240
32 80 160 1024 992
64 192 384 4096 4032

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Radix-2 DIT FFT

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Radix-2 DIT FFT