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M. Lustig, EECS UC Berkeley
EE123Digital Signal Processing
Lecture 9
1
M. Lustig, EECS UC Berkeley
Discrete Transforms (Finite)
• DFT is only one out of a LARGE class of transforms
• Used for:–Analysis–Compression–Denoising–Detection–Recognition–Approximation (Sparse)
Sparse representation has been one of the hottest research topics in the last 15 years in sp
2
• Spectrum of a bird chirping– Interesting,.... but...– Does not tell the whole story– No temporal information!
M. Lustig, EECS UC Berkeley
Example: Bird Chirp
Play Sound!
0 0.5 1 1.5 2 2.5
x 104
0
100
200
300
400
500
600
Hz
Spectrum of a bird chirp
No temporal information!
Miki Lustig UCB. Based on Course Notes by J.M Kahn Fall 2011, EE123 Digital Signal Processing
Example of spectral analysis
x[n]
n
3
• To get temporal information, use part of the signal around every time point
• Mapping from 1D ⇒ 2D, n discrete, w cont.
• Simply slide a window and compute DTFTM. Lustig, EECS UC Berkeley
Time Dependent Fourier Transform
X[n,!) =1X
m=�1x[n+m]w[m]e�j!m
*Also called Short-time Fourier Transform (STFT)
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• To get temporal information, use part of the signal around every time point
M. Lustig, EECS UC Berkeley
Time Dependent Fourier Transform
X[n,!) =1X
m=�1x[n+m]w[m]e�j!m
*Also called Short-time Fourier Transform (STFT)
5
M. Lustig, EECS UC Berkeley
Spectrogram
Time, s
Fre
qu
en
cy,
Hz
2 4 6 8 10 12 14 16 180
1000
2000
3000
4000
0 500 1000 1500 2000 2500 3000 3500 4000 45000
5
10
15
20
25
30
Frequency, Hz
0 500 1000 1500 2000 2500 3000 3500 4000 45000
2
4
6
8
10
12
Frequency, Hz
0 500 1000 1500 2000 2500 3000 3500 4000 45000
5
10
15
20
25
30
Frequency, Hz
0 500 1000 1500 2000 2500 3000 3500 4000 45000
2
4
6
8
10
12
Frequency, Hz
6
Xr[k] =L�1X
m=0
x[rR+m]w[m]e�j2⇡km/N
M. Lustig, EECS UC Berkeley
Discrete Time Dependent FT
• L - Window length• R - Jump of samples • N - DFT length
• Tradeoff between time and frequency resolution
7
M. Lustig, EECS UC Berkeley
Discrete Transforms (Finite)
• Today:– Start with DFT ⇒ Frequency only
– Short-time DFT ⇒ Time-Frequency
– Wavelets ⇒ More flexible/better Time-frequency
–Wavelets ⇒ Sparsity ⇒ ⇒ Compression ⇒ denoising ⇒ approximation
8
t
!
�t · �! � 1
2
�t
�!
M. Lustig, EECS UC Berkeley
Heisenberg Boxes
• Time-Frequency uncertainty principle http://www.jonasclaesson.com
9
�! =2⇡
N
�t = N
�! ·�t = 2⇡
M. Lustig, EECS UC Berkeley
DFT
X[k] =N�1X
n=0
x[n]e�j2⇡kn/N
!
tone DFT coefficient
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X[r, k] =L�1X
m=0
x[rR+m]w[m]e�j2⇡km/N
�! =2⇡
L
�t = L
M. Lustig, EECS UC Berkeley
Discrete STFT
optional
!
tone STFT coefficient
11
x[rR+m]wL[m] =1
N
N�1X
k=0
X[n, k]ej2⇡km/N
M. Lustig, EECS UC Berkeley
STFT Reconstruction
• For non-overlapping windows, R=L :
• What is the problem?
x[n] =x[n� rL]
wL[n� rL]
rL n (r + 1)R� 1
12
x[rR+m]wL[m] =1
N
N�1X
k=0
X[n, k]ej2⇡km/N
M. Lustig, EECS UC Berkeley
STFT Reconstruction
• For non-overlapping windows, R=L :
• For stable reconstruction must overlap window 50% (at least)
x[n] =x[n� rL]
wL[n� rL]
rL n (r + 1)R� 1
13
M. Lustig, EECS UC Berkeley
STFT Reconstruction
• For stable reconstruction must overlap window 50% (at least)
• For Hann, Bartlett reconstruct with overlap and add. No division!
14
M. Lustig, EECS UC Berkeley
Applications
• Time Frequency Analysis
Spectrogram of FM radio
t=0 t=2sec
88.5
88.6
88.4
15
M. Lustig, EECS UC Berkeley
Applications
• Time Frequency Analysis
Spectrogram of Demodulated FM radio (Adele on 96.5 MHz)
0
19KHz
38KHz
57KHz
16
M. Lustig, EECS UC Berkeley
Applications
• Time Frequency Analysis
Spectrogram of digital communications - Frequency Shift Keying
t=0 t=1sec
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Time, s
Fre
quency, H
z
1 2 3 4 5 6 7 80
1000
2000
3000
4000
M. Lustig, EECS UC Berkeley
Applications
• Time Frequency Analysis
Spectrogram of Orca whale
18
M. Lustig, EECS UC Berkeley
Applications
• Noise removal
• Recall bird chirp x[n]
n
Example: Bird Chirp
Play Sound!
0 0.5 1 1.5 2 2.5
x 104
0
100
200
300
400
500
600
Hz
Spectrum of a bird chirp
No temporal information!
Miki Lustig UCB. Based on Course Notes by J.M Kahn Fall 2011, EE123 Digital Signal Processing
19
M. Lustig, EECS UC Berkeley
Application
• Denoising of Sparse spectrograms
• Spectrum is sparse! can implement adaptive filter, or just threshold!
Time, s
Fre
quency, H
z
2 4 6 8 10 12 14 16 180
1000
2000
3000
4000
20
M. Lustig, EECS UC Berkeley
Limitations of Discrete STFT
• Need overlapping ⇒ Not orthogonal
• Computationally intensive O(MN log N)
• Same size Heisenberg boxes
21
M. Lustig, EECS UC Berkeley
From STFT to Wavelets
• Basic Idea:–low-freq changes slowly - fast tracking unimportant–Fast tracking of high-freq is important in many apps.–Must adapt Heisenberg box to frequency
• Back to continuous time for a bit.....
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�t
�!Sf(u,⌦) =
Z 1
�1f(t)w(t� u)e�j⌦tdt
Wf(u, s) =
Z 1
�1f(t)
1ps ⇤(
t� u
s)dt
u
⌦
M. Lustig, EECS UC Berkeley
From STFT to Wavelets
• Continuous time�t
�!
u
⌦
�t
�!
*Morlet - Grossmann
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Z 1
�1| (t)|2dt = 1
Z 1
�1 (t)dt = 0
M. Lustig, EECS UC Berkeley
From STFT to Wavelets
• The function is called a mother wavelet–Must satisfy:
Wf(u, s) =
Z 1
�1f(t)
1ps ⇤(
t� u
s)dt
⇒ Band-Pass
⇒ unit norm
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w(t� u)ej⌦t1ps (
t� u
s)
s = 1
⌦lo
s = 3
M. Lustig, EECS UC Berkeley
STFT and Wavelets “Atoms”
STFT Atoms Wavelet Atoms
u u
⌦hi
u u
25
• Mexican Hat
• Haar
M. Lustig, EECS UC Berkeley
Examples of Wavelets
(t) = (1� t2)e�t2/2
(t) =
8<
:
�1 0 t < 12
1
12 t < 1
0 otherwise
26
M. Lustig, EECS UC Berkeley
Example: Mexican Hat
log(s)
u
SombreroWavelet
27
Wf(u, s) =1ps
Z 1
�1f(t) ⇤(
t� u
s)dt
=�f(t) ⇤ s(t)
(u)
s =1ps (
t
s)
M. Lustig, EECS UC Berkeley
Wavelets Transform
• Can be written as linear filtering
• Wavelet coefficients are a result of bandpass filtering
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i = [1, 2, 3, · · · ]
M. Lustig, EECS UC Berkeley
Wavelet Transform
• Many different constructions for different signals–Haar good for piece-wise constant signals–Battle-Lemarie’ : Spline polynomials
• Can construct Orthogonal wavelets– For example: dyadic Haar is orthonormal
i,n(t) =1p2i (
t� 2in
2i)
29
M. Lustig, EECS UC Berkeley
Orthonormal Haar
Same scalenon-overlapping
Orthogonal between scales
30
M. Lustig, EECS UC Berkeley
Orthonormal Haar
Same scalenon-overlapping
Orthogonal between scales
31
⌦
M. Lustig, EECS UC Berkeley
Scaling function
i,n(t) =1p2i (
t� 2in
2i)
• Problem: –Every stretch only covers half remaining
bandwidth–Need Infinite functions
• Solution:–Plug low-pass spectrum with a scaling function
i=mi=m+1i=m+2i=m+3
32
⌦
�M. Lustig, EECS UC Berkeley
Scaling function
i,n(t) =1p2i (
t� 2in
2i)
• Problem: –Every stretch only covers half remaining
bandwidth–Need Infinite functions
• Solution:–Plug low-pass spectrum with a scaling function
i=mi=m+1
33
�(t) =
⇢1 0 t < 1
0 otherwise
M. Lustig, EECS UC Berkeley
Haar Scaling function
(t) =
8<
:
�1 0 t < 12
1
12 t < 1
0 otherwise
34
M. Lustig, EECS UC Berkeley
Back to Discrete
• Early 80’s, theoretical work by Morlett, Grossman and Meyer (math, geophysics)
• Late 80’s link to DSP by Daubechies and Mallat.
• From CWT to DWT not so trivial!• Must take care to maintain properties
35
!
n
W [k] =N�1X
n=0
x[n] k[n]
M. Lustig, EECS UC Berkeley
Discrete Wavelet Transform
36
1p2
!
t
k = 1k = 0
M. Lustig, EECS UC Berkeley
Example: Discrete Haar WaveletHaar for n=2
1p2
mother wavelet
scalingfunction
detail
approximation
37
M. Lustig, EECS UC Berkeley
Example: Discrete Haar WaveletHaar for n=8 1p
2
!
t
1p2
1p2
1p2
1p2
1p2
1p2
1p2
k=0
k=1
k=2
k=3
k=4
k=5
k=6
k=7
s=22
s=21
s=20
scaling
38
M. Lustig, EECS UC Berkeley
Fast DWT
• Fast Pyramidal Decomposition
• Complexity is O(N), less than FFT
n . --[@j-{f} -0-c_
0.1 WL __, 1
) •y • J . \ • I
'
' 1'
l ..J > f ' -......
is CJ{J/) Ass !kJIJ FAT! fl:_f;(Q#sJ )( U cTZ 0)/ ·-
i -
h[n] g[n]
39
M. Lustig, EECS UC Berkeley
Fast DWT
n . --[@j-{f} -0-c_
0.1 WL __, 1
) •y • J . \ • I
'
' 1'
l ..J > f ' -......
is CJ{J/) Ass !kJIJ FAT! fl:_f;(Q#sJ )( U cTZ 0)/ ·-
i -
40
M. Lustig, EECS UC Berkeley
x[n]
Haar
Haar DWT Example
41
M. Lustig, EECS UC Berkeley
DFT
Haar
Approximation from 25/256 coefficients
42
M. Lustig, EECS UC Berkeley
Example: Denoising Noisy Signals
Haar
43
M. Lustig, EECS UC Berkeley
Example: Denoising by Thresholding
noisy
denoisedlargest 25 coefficients
44