DSP (2015 Spring) Computation of DFT NCTU EE 1 Computation of DFT Efficient algorithms for computing DFT – Fast Fourier Transform. (a) Compute only a few points out of all N points (b) Compute all N points What are the efficiency criteria? Number of multiplications Number of additions Chip area in VLSI implementation DFT as a Linear Transformation Matrix representation of DFT Definition of DFT: 1 , , 1 , 0 , ) ( 1 ) ( 1 , , 1 , 0 , ) ( ) ( 1 0 1 0 N n W k X N n x N k W n x k X N k kn N N n kn N where Let , ) 1 ( ) 1 ( ) 0 ( , ) 1 ( ) 1 ( ) 0 ( N X X X N x x x N N X x and ) 1 )( 1 ( ) 1 ( 2 ) 1 ( ) 1 ( 2 4 2 1 2 1 1 1 1 1 1 1 N N N N N N N N N N N N N N N N W W W W W W W W W W Thus, N N N N N N N N N N N X W X W x x W X * 1 1 IDFT point - point DFT - Because the matrix (transformation) N W has a specific structure and because k N W has par- ticular values (for some k and n), we can reduce the number of arithmetic operations for computing this transform.
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DSP (2015 Spring) Computation of DFT
NCTU EE 1
Computation of DFT
Efficient algorithms for computing DFT – Fast Fourier Transform.
(a) Compute only a few points out of all N points
(b) Compute all N points
What are the efficiency criteria?
Number of multiplications
Number of additions
Chip area in VLSI implementation
DFT as a Linear Transformation Matrix representation of DFT
Definition of DFT:
1,,1,0,)(1
)(
1,,1,0,)()(
1
0
1
0
NnWkXN
nx
NkWnxkX
N
k
knN
N
n
knN
where
Let ,
)1(
)1(
)0(
,
)1(
)1(
)0(
NX
X
X
Nx
x
x
NN
Xx
and
)1)(1()1(2)1(
)1(242
12
1
1
1
1111
NNN
NN
NN
NNNN
NNNN
N
WWW
WWW
WWW
W
Thus,
NN
NNN
NNN
N
N
N
XW
XWx
xWX
*
1
1IDFTpoint -
point DFT-
Because the matrix (transformation) NW has a specific structure and because k
NW has par-
ticular values (for some k and n), we can reduce the number of arithmetic operations for
computing this transform.
DSP (2015 Spring) Computation of DFT
NCTU EE 2
Example 3] 2 1 0[][ nx
jj
jj
WWW
WWW
WWW
WWWW
WWWW
WWWW
WWWW
11
1111
11
1111
1
1
1
1111
14
24
34
24
04
24
34
24
14
94
64
34
04
64
44
24
04
34
24
14
04
04
04
04
04
4W
Only additions are needed to compute this specific transform.
(This is a well-known radix-4 FFT)
Thus, the DFT of ][nx is
j
j
22
2
22
6
444 xWX
Fast Fourier Transform -- Highly efficient algorithms for computing DFT
General principle: Divide-and-conquer
Specific properties of kNW
Complex conjugate symmetry: *)( knN
knN WW
Symmetry: kN
Nk
N WW 2
Periodicity: kN
NkN WW
Particular values of k and n: e.g., radix-4 FFT (no multiplications)
Direct computation of DFT
1
0
1
0
Re][ImIm][Re
Im][ImRe][Re
1,,1,0 ,][][
N
nkn
Nkn
N
knN
knN
N
n
knN
WnxWnxj
WnxWnx
NkWnxkX
For each k, we need N complex multiplications and N-1 complex additions. 4N real