Calculating Discrete Fourier Transform using Matrix method

1. Calculating DFT using Matrix
method
- SARANG JOSHI

2. • N-point DFT :
• Twiddle factor:

−
=
−
=
1
0
2
)()(
N
n
N
n
kj
enxkX

For k=0,1,2…..N-1
N
j
N eW
2
−
=

−
=
=
1
0
)()(
N
n
kn
NWnxkX For k=0,1,2…..N-1

3. 























=
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
kn
N
WWWW
WWWW
WWWW
WWWW
W
............
..........
..........
..........
.............
............
.............
n=0 n=1 n=2 ………….. n=N-1
k=0
k=1
k=2
.
.
k=N-1
TWIDDLE FACTOR MATRIX

4. 























=
−−−
−
−
2
)1()1(210
)1(2420
1210
0000
........
..........
..........
..........
.............
............
.............
N
N
N
N
N
NN
N
NNNN
N
NNNN
NNNN
kn
N
WWWW
WWWW
WWWW
WWWW
W
n=0 n=1 n=2 ………….. n=N-1
k=0
k=1
k=2
.
.
k=N-1
TWIDDLE FACTOR MATRIX

5. 







= 1
2
0
2
0
2
0
2
2
WW
WW
W kn






−
=
11
11
2
kn
W
N
j
N eW
2
−
=
2
2
2
j
eW
−
=
j
2
−
= eW
IdentitysEuler'...........)sin()cos( 
je j
−=−

6. 











−−
−−
+−−
=
jj
jj
W kn
11
1111
11
1111
4














=
9
4
6
4
3
4
0
4
6
4
4
4
2
4
0
4
3
4
2
4
1
4
0
4
0
4
0
4
0
4
0
4
4
WWWW
WWWW
WWWW
WWWW
W kn

7. Matrix Relations:
• Where
NNN xWX ][=


























−
=
)1(
.
.
.
)2(
)1(
)0(
NX
X
X
X
X N
DFT :

8. Ex.1] Find DFT of x(n)={1,2,0,1}
Solution: NNN xWX ][=
























−−
−−
+−−
=
1
0
2
1
11
1111
11
1111
4
jj
jj
X

9. }1,2,1,4{4 jjX +−−=

10. RATE, FOLLOW & SHARE
https://unacademy.com/user/jsarang70-7008

11. THANK YOU !