Young Won Lim5/14/11

DFT Matrix Properties (3A)

Young Won Lim5/14/11

Copyright (c) 2009, 2010, 2011 Young W. Lim.

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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 5 Young Won Lim5/14/11

Symmetric Matrices

A =

B =

A = AT

B = BT

IDF

DFT

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 11 Young Won Lim5/14/11

Product AB

A B = C

N

N

00=

IDFTDFT

A A* = C

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

3A DFT Matrix Properties 13 Young Won Lim5/14/11

Young Won Lim5/14/11

References

[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] A “graphical interpretation” of the DFT and FFT, by Steve Mann