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.
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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