dft matrix properties (3a)
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Young Won Lim5/14/11
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3A DFT Matrix Properties 3 Young Won Lim5/14/11
A
N=8 DFTDFT Matrix
x [0 ]
x [1]
x [2]
x [3 ]
x [4 ]
x [5 ]
x [6 ]
x [7 ]
X [0]
X [1 ]
X [2 ]
X [3 ]
X [4 ]
X [5 ]
X [6]
X [7 ]
=
X [k ] = ∑n = 0
7
W 8k n x [n] W 8
k n = e− j 2
8k n
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅0
e− j⋅
4⋅1
e− j⋅
4⋅2
e− j⋅
4⋅3
e− j⋅
4⋅4
e− j⋅
4⋅5
e− j⋅
4⋅6
e− j⋅
4⋅7
e− j⋅
4⋅0
e− j⋅
4⋅2
e− j⋅
4⋅4
e− j⋅
4⋅6
e− j⋅
4⋅0
e− j⋅
4⋅2
e− j⋅
4⋅4
e− j⋅
4⋅6
e− j⋅
4⋅0
e− j⋅
4⋅3
e− j⋅
4⋅6
e− j⋅
4⋅1
e− j⋅
4⋅4
e− j⋅
4⋅7
e− j⋅
4⋅2
e− j⋅
4⋅5
e− j⋅
4⋅0
e− j⋅
4⋅4
e− j⋅
4⋅0
e− j⋅
4⋅4
e− j⋅
4⋅0
e− j⋅
4⋅4
e− j⋅
4⋅0
e− j⋅
4⋅4
e− j⋅
4⋅0
e− j⋅
4⋅5
e− j⋅
4⋅2
e− j⋅
4⋅7
e− j⋅
4⋅4
e− j⋅
4⋅1
e− j⋅
4⋅6
e− j⋅
4⋅3
e− j⋅
4⋅0
e− j⋅
4⋅6
e− j⋅
4⋅4
e− j⋅
4⋅2
e− j⋅
4⋅0
e− j⋅
4⋅6
e− j⋅
4⋅4
e− j⋅
4⋅2
e− j⋅
4⋅0
e− j⋅
4⋅7
e− j⋅
4⋅6
e− j⋅
4⋅5
e− j⋅
4⋅4
e− j⋅
4⋅3
e− j⋅
4⋅2
e− j⋅
4⋅1
3A DFT Matrix Properties 4 Young Won Lim5/14/11
B
N=8 IDFTIDFT Matrix
X [0 ]N
X [1]N
X [2]N
X [3]N
X [4 ]N
X [5]N
X [6 ]N
X [7 ]N
x [0 ]
x [1]
x [2]
x [3 ]
x [4 ]
x [5 ]
x [6 ]
x [7 ]
=
W 8−k n = e
j 28
k nx [n ] = 1
N ∑k = 0
7
W 8−k n X [k ]
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅0
e j⋅
4⋅1
e j⋅
4⋅2
e j⋅
4⋅3
e j⋅
4⋅4
e j⋅
4⋅5
e j⋅
4⋅6
e j⋅
4⋅7
e j⋅
4⋅0
e j⋅
4⋅2
e j⋅
4⋅4
e j⋅
4⋅6
e j⋅
4⋅0
e j⋅
4⋅2
e j⋅
4⋅4
e j⋅
4⋅6
e j⋅
4⋅0
e j⋅
4⋅3
e j⋅
4⋅6
e j⋅
4⋅1
e j⋅
4⋅4
e j⋅
4⋅7
e j⋅
4⋅2
e j⋅
4⋅5
e j⋅
4⋅0
e j⋅
4⋅4
e j⋅
4⋅0
e j⋅
4⋅4
e j⋅
4⋅0
e j⋅
4⋅4
e j⋅
4⋅0
e j⋅
4⋅4
e j⋅
4⋅0
e j⋅
4⋅5
e j⋅
4⋅2
e j⋅
4⋅7
e j⋅
4⋅4
e j⋅
4⋅1
e j⋅
4⋅6
e j⋅
4⋅3
e j⋅
4⋅0
e j⋅
4⋅6
e j⋅
4⋅4
e j⋅
4⋅2
e j⋅
4⋅0
e j⋅
4⋅6
e j⋅
4⋅4
e j⋅
4⋅2
e j⋅
4⋅0
e j⋅
4⋅7
e j⋅
4⋅6
e j⋅
4⋅5
e j⋅
4⋅4
e j⋅
4⋅3
e j⋅
4⋅2
e j⋅
4⋅1
3A DFT Matrix Properties 6 Young Won Lim5/14/11
Conjugate Transpose Matrices
A=
B=
=
= B = A H
H
H
A = B H
IDFT
DFT
B = A∗
A = B∗
3A DFT Matrix Properties 7 Young Won Lim5/14/11
Product AB
A B = CN
N
00=
IDFTDFT
A ⋅ B ⇒ A ⋅ A H ⇒ A ⋅ A∗ ⇒ N I
3A DFT Matrix Properties 8 Young Won Lim5/14/11
Unitary Matrix
C = A ⋅ B = A ⋅ A H
C = N
N
00
U ⋅ U H = I Unitary Matrix
= A ⋅ A∗ = N I
3A DFT Matrix Properties 9 Young Won Lim5/14/11
Symmetric Matrices
A = AT
B = BT
DFT Matrix in the row-wise view DFT Matrix in the column-wise view
IDFT Matrix in the row-wise view IDFT Matrix in the column-wise view
3A DFT Matrix Properties 10 Young Won Lim5/14/11
Conjugate Transpose Matrices
A=
B=
=
= B = A H
H
H
A = B H
IDFT
DFT
Real Imaginary
Real ImaginaryB = A∗
A = B∗
3A DFT Matrix Properties 12 Young Won Lim5/14/11
0 cycle-1 cycles-2 cycles
+1 cycles+2 cycles
-3 cycles-4 cycles+3 cycles
{ e− j 2π
8k n}
{e− j 2π8
k n}
= cos(−2π8 k n)
= sin(−2π8 k n)
R e
I m
X [0]X [1 ]X [2 ]
=X [3]X [4]X [5]X [6]X [7]
x [0 ]x [1]x [2 ]x [3]x [4]x [5]x [6]x [7]
X[0] measures “0 cycle” component in x X[1] measures “+1 cycle” component in x X[2] measures “+2 cycle” component in x X[3] measures “+3 cycle” component in x X[4] measures “+4 cycle” component in x X[5] measures “-3 cycle” component in x X[6] measures “-2 cycle” component in x X[7] measures “-1 cycle” component in x
X[k] measures frequency