2. • 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
3. 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
8. 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)
13. 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
19. 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