background (dft.a0)€¦ · 0a dft background 4 young won lim 12/12/2010 power series expansion f x...
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Young Won Lim12/12/2010
Background (DFT.A0)
Young Won Lim12/12/2010
Copyright (c) 2009, 2010 Young W. Lim.
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0A DFT Background 3 Young Won Lim12/12/2010
Taylor Series
f x = f a f a x−a f a
2! x−a2 ⋯
f n a
n! x−an ⋯
f x = f 0 f 0 x f a
2!x2
⋯f n
an!
xn⋯
0A DFT Background 4 Young Won Lim12/12/2010
Power Series Expansion
f x = f 0 f 0 x f a
2!x2
⋯f n
an!
xn⋯
sin x = x −x3
3!
x5
5!−
x7
7!⋯
cos x = 1 −x2
2!
x 4
4!−
x6
6!⋯
e x= 1 x
x2
2!
x3
3!
x4
4!⋯
0A DFT Background 5 Young Won Lim12/12/2010
Complex Number
i = −1
i2= −1
i3= −i
i4= −i
x i y = r cosi r sin
= r cosi sin
x = r cos
y = r sin
0A DFT Background 6 Young Won Lim12/12/2010
Complex Power Series
sin x = x −x3
3!
x5
5!−
x7
7!⋯
cos x = 1 −x2
2!
x 4
4!−
x6
6!⋯e x
= 1 x x2
2!
x3
3!
x4
4!⋯
ei = 1 i
i2
2!
i 3
3!
i 4
4!
i5
5!⋯
= 1 i −
2
2!− i
3
3!
4
4! i
5
5!⋯
= 1 −
2
2!
4
4!⋯ i −
3
3!
5
5!⋯
= cos i sin
0A DFT Background 7 Young Won Lim12/12/2010
Euler Series (1)
ei = cos i sin
ℜ{ei } = cos
ℑ{ei } = sin
∣ei∣ = cos2
sin2 = 1
arg ei = tan−1 sin
cos =
0A DFT Background 8 Young Won Lim12/12/2010
Euler Series (2)
ei = cos i sin
ℜ{ei } = cos =
ei e−i
2
ℑ{ei } = sin =
ei − e−i
2 i
e−i = cos − i sin
0A DFT Background 9 Young Won Lim12/12/2010
DeMoivre’s Theorem
ei = cos i sin
ei
n= cos i sinn
ei n = cos n i sin n
ei
1/n= cos i sin 1/n
ei /n= cos
n i sin
n
0A DFT Background 10 Young Won Lim12/12/2010
Complex Roots
z = r ei
z1/n= r1/ n ei /n
=nr cos
n i sin
n
0A DFT Background 11 Young Won Lim12/12/2010
Phasor
A cos t
A cos t = ℜ{Aei t }
= ℜ{eit⋅Ae i
}
Ae i A∢
0A DFT Background 12 Young Won Lim12/12/2010
Complex Phase Factor (1)
W 40
W 41
W 42
W 43
W 40
W 4−3
W 4−2
W 4−1
W 41= W 4
−3
W 42= W 4
−2
W 43= W 4
−1
W Nk±N
= W Nk
W 4k = e
− j 24
kW 4
−k = e j 2
4k
0A DFT Background 13 Young Won Lim12/12/2010
Complex Phase Factor (2)
W 80
W 81
W 82
W 83
W 84
W 85
W 81= W 8
−7
W 82= W 8
−6
W 83= W 8
−5
W 84= W 8
−4
W 85= W 8
−3
W 86= W 8
−2
W 87= W 8
−1
W 87
W 80
W 8−7
W 8−6
W 8−5
W 8−4
W 8−3
W 8−2
W 8−1
W 86
W Nk±N
= W Nk
W 8k = e
− j 28
kW 8
−k = e j 2
8k
0A DFT Background 14 Young Won Lim12/12/2010
Complex Phase Factor (3)
W Nk−N
= e− j 2
Nk − N
W Nk−N
W Nk =
e− j 2
Nk − N
e−j 2
Nk
= e j 2 = 1
W NkN
= e− j 2
Nk N
W NkN
W Nk =
e− j 2
Nk N
e− j 2
Nk
= e− j2 = 1
W NkN
= e− j 2
NkN
= e− j2k=1
W NkN
= 1
W Nk = e
− j 2N
kW N
−k = e j 2
Nk
W Nk−N
= W Nk W N
kN= W N
k
W NkN
= e j 2
NkN
= e j2k=1
W N−kN
= 1
0A DFT Background 15 Young Won Lim12/12/2010
DFT Symmetry
X∗[k ]
= ∑n = 0
N − 1
x [n] W N−k n
= ∑n = 0
N − 1
x [n] W NnN W N
−k n
= ∑n = 0
N − 1
x [n] W NnN −k
= X [N−k ]
W Nk−N
= W Nk
X∗[k ] = X [N−k ]
X [k ] = ∑n = 0
N − 1
x [n ] W Nk n
W NnN
= 1
0A DFT Background 16 Young Won Lim12/12/2010
● Complex Phase Factors● N=8 DFT
● DFT Matrix● Exponents of W● Common Differences in Exponents of W● Complex Phase Factors in Angles
● N=8 IDFT Matrix● IDFT Matrix● Exponents of W● Common Differences in Exponents of W● Complex Phase Factors in Angles
0A DFT Background 17 Young Won Lim12/12/2010
N=8 DFT DFT Matrix
X [k ] = ∑n= 0
7
W 8k n x [n]
X [0] W 80 W 8
0 W 80 W 8
0 W 80 W 8
0 W 80 W 8
0 x [0 ]
X [1 ] W 80 W 8
1 W 82 W 8
3 W 84 W 8
5 W 86 W 8
7 x [1]
X [2 ] W 80 W 8
2 W 84 W 8
6 W 88 W 8
10 W 812 W 8
14 x [2]
X [3 ] W 80 W 8
3 W 86 W 8
9 W 812 W 8
15 W 818 W 8
21 x [3 ]
X [4 ] W 80 W 8
4 W 88 W 8
12 W 816 W 8
20 W 824 W 8
28 x [4 ]
X [5 ] W 80 W 8
5 W 810 W 8
15 W 820 W 8
25 W 830 W 8
35 x [5 ]
X [6] W 80 W 8
6 W 812 W 8
18 W 824 W 8
30 W 836 W 8
42 x [6 ]
X [7 ] W 80 W 8
7 W 814 W 8
21 W 828 W 8
35 W 842 W 8
49 x [7 ]
=
W 8k n = e
− j 28
k n
0A DFT Background 18 Young Won Lim12/12/2010
N=8 DFTDFT Exponents of W
0 1 2 3 4 5 6 70 0
000
00
00
00
00
00
00
1 00
11
22
33
44
55
66
77
2 00
22
44
66
80
102
124
146
3 00
33
66
91
124
157
182
215
4 00
44
80
124
160
204
240
284
5 00
55
102
157
204
251
306
353
6 00
66
124
182
240
306
364
422
7 00
77
146
215
284
353
422
491
nkn·k mod 8
W Nn k
=
e− j 2/N n k
N = 8
W Nk±N
W Nk =
e− j 2
Nk ± N
e−j 2
Nk
= e∓ j2 = 1
example :−49 mod 8≡ −1 mod 8
0A DFT Background 19 Young Won Lim12/12/2010
N=8 DFTDFT Common Differences in Exponents of W
n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7
k=0
k=1
k=2
k=3
0 0 0 0 0 0 0 00 0 0 0 0 0 0
0 1 2 3 4 5 6 71 1 1 1 1 1 1
0 2 4 6 0 2 4 62 2 2 2 2 2 2
0 3 6 1 4 7 2 53 3 3 3 3 3 3
0 4 0 4 0 4 0 44 4 4 4 4 4 4
0 5 2 7 4 1 6 35 5 5 5 5 5 5
0 6 4 2 0 6 4 26 6 6 6 6 6 6
0 7 6 5 4 3 2 17 7 7 7 7 7 7
k=4
k=5
k=6
k=7
W Nn k
=
e− j 2/N n k
N = 8
W Nk±N
W Nk =
e− j 2
Nk ± N
e−j 2
Nk
= e∓ j2 = 1
n·k mod 8
0A DFT Background 20 Young Won Lim12/12/2010
N=8 DFTDFT Complex Factors in Angles (1)
n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7
k=0
k=1
k=2
k=3
0 0 0 0 0 0 0
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
0A DFT Background 21 Young Won Lim12/12/2010
N=8 DFTDFT Complex Factors in Angles (2)
n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7
k=4
k=5
k=6
k=7
7 7 7 7 7 7 7
4 4 4 4 4 4 4
5 5 5 5 5 5 5
6 6 6 6 6 6 6
0A DFT Background 22 Young Won Lim12/12/2010
● Complex Phase Factors● N=8 DFT
● DFT Matrix● Exponents of W● Common Differences in Exponents of W● Complex Phase Factors in Angles
● N=8 IDFT Matrix● IDFT Matrix● Exponents of W● Common Differences in Exponents of W● Complex Phase Factors in Angles
0A DFT Background 23 Young Won Lim12/12/2010
W 8−k n = e
j 28
k n
N=8 IDFTIDFT
x [n ] =1N ∑
k = 0
7
W 8−k n X [k ]
X [0]W 80 W 8
0 W 80 W 8
0 W 80 W 8
0 W 80 W 8
0x [0 ]
X [1 ]W 80 W 8
−1 W 8−2 W 8
−3 W 8−4 W 8
−5 W 8−6 W 8
−7x [1]
X [2 ]W 80 W 8
−2 W 8−4 W 8
−6 W 8−8 W 8
−10 W 8−12 W 8
−14x [2]
X [3 ]W 80 W 8
−3 W 8−6 W 8
−9 W 8−12 W 8
−15 W 8−18 W 8
−21x [3 ]
X [4 ]W 80 W 8
−4 W 8−8 W 8
−12 W 8−16 W 8
−20 W 8−24 W 8
−28x [4 ]
X [5 ]W 80 W 8
−5 W 8−10 W 8
−15 W 8−20 W 8
−25 W 8−30 W 8
−35x [5 ]
X [6]W 80 W 8
−6 W 8−12 W 8
−18 W 8−24 W 8
−30 W 8−36 W 8
−42x [6 ]
X [7 ]W 80 W 8
−7 W 8−14 W 8
−21 W 8−28 W 8
−35 W 8−42 W 8
−49x [7 ]
= 1N
0A DFT Background 24 Young Won Lim12/12/2010
N=8 IDFTIDFT Exponents of W
0 1 2 3 4 5 6 70 0
000
00
00
00
00
00
00
1 00
11
22
33
44
55
66
77
2 00
22
44
66
80
102
124
146
3 00
33
66
91
124
157
182
215
4 00
44
80
124
160
204
240
284
5 00
55
102
157
204
251
306
353
6 00
66
124
182
240
306
364
422
7 00
77
146
215
284
353
422
491
kn+n·k mod 8
W N−n k
=
e j 2/N n k
N = 8
W N−k±N
W N−k =
e j 2
Nk ± N
e j 2
N k
= e± j2 = 1
example :49 mod 8≡ 1 mod 8
0A DFT Background 25 Young Won Lim12/12/2010
N=8 IDFTIDFT Common Differences in Exponents of W
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
n=0
n=1
n=2
n=3
0 0 0 0 0 0 0 0+0 +0 +0 +0 +0 +0 +0
0 +1 +2 +3 +4 +5 +6 +7+1 +1 +1 +1 +1 +1 +1
0 +2 +4 +6 0 +2 +4 +6+2 +2 +2 +2 +2 +2 +2
0 +3 +6 +1 +4 +7 +2 +5+3 +3 +3 +3 +3 +3 +3
0 +4 0 +4 0 +4 0 +4+4 +4 +4 +4 +4 +4 +4
0 +5 +2 +7 +4 +1 +6 +3+5 +5 +5 +5 +5 +5 +5
0 +6 +4 +2 0 +6 +4 +2+6 +6 +6 +6 +6 +6 +6
0 +7 +6 +5 +4 +3 +2 +1+7 +7 +7 +7 +7 +7 +7
n=4
n=5
n=6
n=7
+n·k mod 8
W N−n k
=
e j 2/N n k
N = 8
W N−k±N
W N−k =
e j 2
Nk ± N
e j 2
N k
= e± j2 = 1
0A DFT Background 26 Young Won Lim12/12/2010
N=8 IDFTIDFT Complex Factors in Angles (1)
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
n=0
n=1
n=2
n=3
+0 +0 +0 +0 +0 +0 +0
+1 +1 +1 +1 +1 +1 +1
+2 +2 +2 +2 +2 +2 +2
+3 +3 +3 +3 +3 +3 +3
0A DFT Background 27 Young Won Lim12/12/2010
N=8 IDFTIDFT Complex Factors in Angles (2)
k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
n=4
n=5
n=6
n=7
+4 +4 +4 +4 +4 +4 +4
+5 +5 +5 +5 +5 +5 +5
+6 +6 +6 +6 +6 +6 +6
+7 +7 +7 +7 +7 +7 +7
Young Won Lim12/12/2010
References
[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003