Young Won Lim6/19/10

● Each Row of the DFT Matrix●

DFT Analysis (9A)

Young Won Lim6/19/10

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

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Please send corrections (or suggestions) to youngwlim@hotmail.com.

This document was produced by using OpenOffice and Octave.

9A DFT Frequency 3 Young Won Lim6/19/10

9A DFT Frequency 4 Young Won Lim6/19/10

N=8 DFTDFT : The 1st Row of the DFT Matrix

X[0] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

Sampling Time

Sequence Time Length

Sampling Frequency

Zero Frequency

R

I

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

W 8k n

= e− j

28

k n

k = 0, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos − t = cos t

sin − t = −sin t

t = 2 f t

2⋅ 08 ⋅f s⋅t

0 cycle

9A DFT Frequency 5 Young Won Lim6/19/10

N=8 DFTDFT : The 2nd Row of the DFT Matrix

X[1] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 1 =1T

=1

N =

f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

1st Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 1, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos − t = cos t

sin − t = −sin t

t = 2 f t

2⋅ 18 ⋅f s⋅t

1 cycle

9A DFT Frequency 6 Young Won Lim6/19/10

N=8 DFTDFT : The 3rd Row of the DFT Matrix

X[2] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 2 =2T

=2

N =

2 f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

2nd Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 2, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos − t = cos t

sin − t = −sin t

t = 2 f t

2⋅ 28 ⋅f s⋅t

2 cycles

9A DFT Frequency 7 Young Won Lim6/19/10

N=8 DFTDFT : The 4th Row of the DFT Matrix

X[3] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 3 =3T

=3

N =

3 f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

3rd Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 3, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos − t = cos t

sin − t = −sin t

t = 2 f t

2⋅ 38 ⋅f s⋅t

3 cycles

9A DFT Frequency 8 Young Won Lim6/19/10

N=8 DFTDFT : The 5th Row of the DFT Matrix

X[4] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 4 =4T

=4

N =

4 f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

4th Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 4, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos − t = cos t

sin − t = −sin t

t = 2 f t

2⋅ 48 ⋅f s⋅t

4 cycles

9A DFT Frequency 9 Young Won Lim6/19/10

N=8 DFTDFT : The 6th Row of the DFT Matrix

X[5] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 3 =3T

=3

N =

3 f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

3rd Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 5, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos ' t = cos−−t

sin ' t = sin −− t

− t = −2 f t

2⋅ −38 ⋅f s⋅t

3 cycles

9A DFT Frequency 10 Young Won Lim6/19/10

N=8 DFTDFT : The 7th Row of the DFT Matrix

X[6] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 2 =2T

=2

N =

2 f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

2nd Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 2, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos ' t = cos−−t

sin ' t = sin −− t

− t = −2 f t

2⋅ −28 ⋅f s⋅t

2 cycles

9A DFT Frequency 11 Young Won Lim6/19/10

N=8 DFTDFT : The 8th Row of the DFT Matrix

X[7] measures how much of the above signal component is present in x.

T = N

T = N

f s =1

f 1 =1T

=1

N =

f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

1st Harmonic Freq

R

I

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

W 8k n

= e− j

28

k n

k = 7, n = 0, 1, ... , 7

R

I

sampled values of

sampled values of

cos ' t = cos−−t

sin ' t = sin −− t

− t = −2 f t

2⋅ −18 ⋅f s⋅t

1 cycle

9A DFT Frequency 12 Young Won Lim6/19/10

Fundamental Frequency

T = N

T = N

f s =1

f 1 =1T

=1

N =

f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

1st Harmonic Freq

R

I

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

1 cycle

Fundamental Frequency fo

f 0 = f 1 =f s

N

The Lowest Frequency in a harmonic series.

9A DFT Frequency 13 Young Won Lim6/19/10

Normalized Frequency

T = N

T = N

f s =1

f 1 =1T

=1

N =

f s

N

Sampling Time

Sequence Time Length

Sampling Frequency

1st Harmonic Freq

R

I

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

1 cycle

Normalized Frequency

f n =n⋅f s

N

(samples per second)

(cycles  per sample)

f n

f s

=nNn = 0, 1, 2, ... , N−1

9A DFT Frequency 14 Young Won Lim6/19/10

0th row:  samples of cos00 t j⋅sin 00 t1th row:  samples of cos10 t j⋅sin 10 t2th row:  samples of  cos20 t j⋅sin 20 t3th row:  samples of cos30 t j⋅sin30 t4th row:  samples of cos 40 t j⋅sin 40 t5th row:  samples of cos50 t j⋅sin 50 t6th row:  samples of cos60 t j⋅sin 60 t7th row:  samples of cos70 t j⋅sin70 t

(0 cycle)(1 cycle)(2 cycles)(3 cycles)(4 cycles)(5 cycles)(6 cycles)(7 cycles)

0th row:  samples of cos 00 t j⋅sin 00 t1th row:  samples of cos −70t j⋅sin −70t2th row:  samples of  cos −60t j⋅sin −60 t3th row:  samples of cos −50 t j⋅sin −50 t4th row:  samples of cos −40t j⋅sin −40 t5th row:  samples of cos −30 t j⋅sin −30t6th row:  samples of cos −20 t j⋅sin −20 t7th row:  samples of cos −10 t j⋅sin −10t

(0 cycle)(7 cycles)(6 cycles)(5 cycles)(4 cycles)(3 cycles)(2 cycles)(1 cycles)

N=8 DFTDFT : DFT Matrix in + or – Frequencies

0 = 2⋅f s

N

9A DFT Frequency 15 Young Won Lim6/19/10

0th row:  samples of cos00 t j⋅sin 00 t1th row:  samples of cos10 t j⋅sin 10 t2th row:  samples of  cos20 t j⋅sin 20 t3th row:  samples of cos30 t j⋅sin30 t4th row:  samples of cos 40 t j⋅sin 40 t5th row:  samples of cos50 t j⋅sin 50 t6th row:  samples of cos60 t j⋅sin 60 t7th row:  samples of cos70 t j⋅sin70 t

(0 cycle)(1 cycle)(2 cycles)(3 cycles)(4 cycles)(5 cycles)(6 cycles)(7 cycles)

0th row:  samples of cos00 t j⋅sin 00 t1th row:  samples of cos10 t j⋅sin 10 t2th row:  samples of  cos20 t j⋅sin 20 t3th row:  samples of cos30 t j⋅sin30 t4th row:  samples of cos 40 t j⋅sin 40 t5th row:  samples of cos30 t − j⋅sin30 t6th row:  samples of cos20 t − j⋅sin 20 t7th row:  samples of cos10 t − j⋅sin 10 t

(0 cycle)(1 cycle)(2 cycles)(3 cycles)(4 cycles)(3 cycles)(2 cycles)(1 cycles)

N=8 DFTDFT : DFT Matrix in Both Frequencies

0 = 2⋅f s

N

9A DFT Frequency 16 Young Won Lim6/19/10

N2− 1

N2− 1

f = 1∗ f o

f = 2∗ f o

f = 3∗ f o

f = N2 − 1 ∗ f o

f = N2 ∗ f o

f = −1∗ f o

f = −2∗ f o

f = − N2 − 1∗ f o

1cycle

2cycles

3cycles

f = 0

N2 −1 cycles

N2 −1 cycles

N2 cycles

2cycles

1cycle

0cycle

Frequency View of a DFT Matrix

1N2N3N

12 −

1N

12

1− 1N

1− 2N

12

1N

0

Normalized Frequency

9A DFT Frequency 17 Young Won Lim6/19/10

9A DFT Frequency 18 Young Won Lim6/19/10

9A DFT Frequency 19 Young Won Lim6/19/10

9A DFT Frequency 20 Young Won Lim6/19/10

Young Won Lim6/19/10

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