Time Frequency Analysis and Wavelet Transforms ?????????

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Time Frequency Analysis and Wavelet Transforms
?????????

• ??? ? ? ?
• Office???723?, TEL [phone]
• E-mail jjding_at_ntu.edu.tw
• ????http//djj.ee.ntu.edu.tw/TFW.htm
• ???????,???????????!

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? ???? ???? 15 scores
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Homework 60 scores 5 times, ? 3 ???
????,??????? ,?? 60 ???,??????????? 4095
???,????? 8 ?,?????? ??????,?? E-mail
????,??????????   Term paper 25 scores
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Term paper 25 scores ?????,??????? (1) ????
(10???(????),?????,11?12???,?????????????????? ??
????????????????,?? abstract, conclusion, ?
references,???? sections,????subsections?
References ???, ????? IEEE ??????
) ?????????,?????? ?????? (Ctrl-C , Ctrl-V)
???,??? 60 ??? (2) Tutorial ??????,?????????,?
18???(??????? tutorial,???? (2/3)N 13 ??,N ???
tutorial ???),???????,?????????????????????????,??
????????????,?Word ?? ?????????,????? 3?
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(3) ???? ?????,??? 40??,????????????????????????
12?14?(? 12 ?)?????????,??12?1?????,???????? ?????
,??????(?????????,???????????????)????????,??????
2 ?? ?????????,????? 2? (4) ?? Wikipedia
?????????,?? 2 ???,??????????????????80??????????
?,????,??????????? ???? Wikipedia
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???),????????????????,??????????????????? ??????
? Wikipedia,??? 1?18?
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????17 ?
11/16, ? HW3 11/23, 11/30, ? HW3 12/7, ?
HW4 12/14, Oral 12/2, 12/28, ? HW4, 1/4,
? HW5
9/14, 9/21, 9/28 10/5, ? HW1 10/12,
10/19, ? HW1 10/26, ? HW2 11/2, 11/9, ?
HW2
1/18, ? HW5 ? term paper
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???? (1) Introduction (2) Short-Time Fourier
Transform (3) Gabor Transform (4)
Implementation of Time-Frequency Analysis (5)
Wigner Distribution Function (6) Cohens Class
Time-Frequency Distribution (7) S Transforms,
Gabor-Wigner Transforms, Matching Pursuit, and
Other Time Frequency Analysis Methods (8)
Movement in the Time-Frequency Plane and
Fractional Fourier Transforms (9) Filter Design
by Time-Frequency Analysis (10) Modulation,
Multiplexing, Sampling, and Other Applications
(?)
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???? (11) Hilbert Huang Transform (12) From
Haar Transforms to Wavelet Transforms (13)
Continuous Wavelet Transforms (14) Continuous
Wavelet Transform with Discrete Coefficients (15)
Discrete Wavelet Transform (16) Applications of
the Wavelet Transform
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• ????
• (1) ?? (??????,???????????)
• (2) S. Qian and D. Chen, Joint Time-Frequency
  Analysis Methods and
• Applications, Prentice-Hall, 1996.
• (3) L. Cohen, Time-Frequency Analysis,
  Prentice-Hall, New York, 1995.
• (4) K. Grochenig, Foundations of Time-Frequency
  Analysis, Birkhauser,
• Boston, 2001.
• (5) L. Debnath, Wavelet Transforms and
  Time-Frequency Signal Analysis,
• Birkhäuser, Boston, 2001.
• (6) S. Mallat, A Wavelet Tour of Signal
  Processing The Sparse Way,
• Academic Press, 3rd ed., 2009.
• (7) Others

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Matlab Program Download ??????? http//comm.ntu.e
du.tw/matlab/request.php ???? ???,Matlab 7
????,??,???,2010. . (??????) ???,Matlab
???????,???,??,2011. ???,????????
Matlab,??,???,2007. ????,????????-Matlab,??,2005.
?????????? 35 ?
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Tutorial ???????(??????)
?? ????????,????????
(1) Time-Frequency Reassignment (2) Sparse
Time-Frequency Representation (3) Fast Algorithm
for Time-Frequency Analysis (4) Time-Frequency
Analysis for Machine Fault Detection (5)
Orthogonal Matching Pursuit (6) Compressive
Sensing for Signal Reconstruction (7) Compressive
Sensing for Radar Imaging (8) Compressive
Sensing for Communication (9) Compressive
Sensing for Denoising (10) Hilbert-Huang
Transform for EMG Signal Processing (11) Wavelet
Transforms for Image Feature Extraction (12)
Wavelet Transforms for Watermarking
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I. Introduction

• Fourier transform (FT)
• Time-Domain
  ? Frequency Domain
• ? t varies from ?88
• Laplace Transform
• Cosine Transform, Sine Transform, Z Transform.
• Some things make these operations not practical
• Only the case where t0 ? t ? t1 is interested.
• (2) Not all the signals are suitable for
  analyzing in the frequency domain.
• It is hard to observe the variation of spectrum
  with time by these operations

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Example 1 x(t) cos(440? t) when t lt 0.5,
x(t) cos(660? t) when 0.5 ? t lt 1, x(t)
cos(524? t) when t ? 1 The Fourier transform of
x(t)
Frequency
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• Finite-Supporting Fourier Transform
• Short-Time Fourier Transform (STFT)
• w(t)
  window function ? mask function
• STFT ??? windowed Fourier transform ?
• time-dependent Fourier transform
• Ref L. Cohen, Time-Frequency Analysis,
  Prentice-Hall, New York, 1995.
• Ref A. V. Oppenheim and R. W. Schafer,
  Discrete-Time Signal Processing,
• London Prentice-Hall, 3rd ed., 2010.

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?????? w(t) 1 for t ? B,
w(t) 0 otherwise ?? Short-time Fourier
transform ???? ?????
????? exp(- st2) ??? Gaussian
function ? Gabor function ??? Short-Time
Fourier Transform ??? Gabor Transform
t-axis
t-axis
0
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• (C) Gabor Transform
• S. Qian and D. Chen, Joint Time-Frequency
  Analysis Methods and Applications,
• Prentice Hall, N.J., 1996.
• R. L. Allen and D. W. Mills, Signal Analysis
  Time, Frequency, Scale, and
• Structure, Wiley- Interscience.
• Common Features for short-time Fourier transforms
  and Gabor transforms
• (1) The instantaneous frequency can be observed
• (2) Without Cross Term
• (3) Poor clarity

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Example x(t) cos(440? t) when t lt 0.5, x(t)
cos(660? t) when 0.5 ? t lt 1, x(t)
cos(524? t) when t ? 1 The Gabor transform of
x(t) (? 200)
f -axis (Hertz)
taxis (Second)
? Gray level ??? X(t, f) ? amplitude
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Instantaneous Frequency ???? If
around t0
then the instantaneous frequency of x(t) at t0
are
(??? frequency ??)
(???? angular frequency ??)
If the order of
gt 1, then instantaneous frequency varies with
time
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????,?????????????
Frequency Modulation
Music
Speech
Others (Animal voice, Doppler effect, seismic
waves, radar system, optics, rectangular function)
In fact, in addition to sinusoid-like functions,
the instantaneous frequencies of other functions
will inevitably vary with time.
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• Sinusoid Function
• Chirp function
• 
  Instantaneous frequency
• acoustics, wireless communication, radar system,
  optics
• ? ? (F1 900Hz, F2 1200Hz) , ? (F1 300Hz,
  F2 2300Hz)
• F1 ????????, F2 - F1 ????????
• Higher order exponential function

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Example 2 (1)
t ? 0,
3 (2)
t ? 0, 3
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Fourier transform
f (Hz)
f (Hz)
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(1)
(2)
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Example 3
left x1(t) 1 for t ? 6, x1(t) 0
otherwise, right x2(t) cos(6t ? 0.05t2)
Gabor transform
? -axis
t -axis
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Example 4
Data source http//oalib.hlsresearch.com/Whales/i
ndex.html
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Why Time-Frequency Analysis is Important?

• Many digital signal processing applications are
  related to the spectrum or the bandwidth of a
  signal.
• If the spectrum and the bandwidth can be
  determined adaptive, the performance can be
  improved.
• ? modulation, ? signal
  identification,
• ? multiplexing, ? acoustics,
• ? filter design, ? system modeling,
• ? data compression, ? radar system analysis
• ? signal analysis, ? sampling

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Example Generalization for sampling theory
???????, ? The supporting of x(t) is t1 ? t ?
t1 T, x(t) ? 0 otherwise ? The supporting
of X( f ) ? 0 is -B ? f ? B, X( f ) ? 0
otherwise ??????, ?t ? 1/F , F 2B,
B?? ??,???? N ???? N T/?t ? TF
???????????????????,??????????
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Q1Scaling ????????????????? Hint
Q2 How to use time-frequency analysis to reduce
the number of sampling points?
Time-frequency analysis is an efficient tool for
adaptive signal processing.
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????????
spectrogram
square
(1) Short-time Fourier transform (STFT)
Asymmetric STFT
improve
S transform
(rec-STFT, Gabor, )
combine
Gabor-Wigner Transform
windowed WDF
improve
(2) Wigner distribution function (WDF)
Cohens Class Distribution
improve
(Choi-Williams, Cone-Shape, Page, Levin,
Kirkwood, Born-Jordan, )
improve
Pseudo L-Wigner Distribution
Haar and Daubechies
(3) Wavelet transform
Coiflet, Morlet
Directional Wavelet Transform
(4) Time-Variant Basis Expansion
Matching Pursuit
Prolate Spheroidal Wave Function
(5) Hilbert-Huang Transform
(???? Fourier transform ???)
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? Continuous Wavelet Transform forward wavelet
transform
?(t) mother wavelet, a location, b scaling,
inverse wavelet transform

?a,b(t) is dual orthogonal to
?(t).
output
Fourier transform X(f), f frequency
time-frequency analysis X(t, f), t time, f frequency
wavelet transform X(a, b), a time, b scaling
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?? (1)
when a1 a and b1 b,

otherwise (2) ?(t) has a finite
time interval Two parameters, a ????, b
???? ?? adaptive signal analysis
????????????,b ???????
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Wavelet ????? Mexican hat wavelet, Haar
Wavelet, Daubechies wavelet, triangular wavelet,
?? Mexican hat wavelet ? a and b ?????
a 2, b 1
a 6, b 1
a 10, b 1
a 6, b 2
a 6, b 3
a 6, b 0.5
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? Discrete Wavelet Transform (DWT)
The discrete wavelet transform is very different
from the continuous wavelet transform. It is
simpler and more useful than the continuous one.
L-points
down sampling
lowpass filter
xLn
xn ?????
x1,Ln
gn
? 2
N-points
L-points
xn
highpass filter
down sampling
xHn
x1,Hn
? 2
xn ?????
hn
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??2-point Haar wavelet gn 1/2
for n -1, 0 gn 0 otherwise
h0 1/2, h-1 -1/2, hn 0 otherwise
½ ½
½
gn
hn
n
n
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
-½
then
(????)
(????)
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Discrete wavelet transform ???? (discrete Haar
wavelet, discrete Daubechies wavelet, B-spline
DWT, symlet, coilet, ..)
??? wavelet, gn ? hn ????? 2 ? ?? gn ????
lowpass filter ??? hn ???? highpass
filter ???
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2-D ???
gm
? 2
x1,Lm, n
L-points
m ??, n ??
along m
gn
? 2
v1,Lm, n
along n
hm
? 2
x1,H1m, n
M N
m ??, n ??
xm, n
along m
L-points
x1,H2m, n
gm
? 2
hn
? 2
v1,Hm, n
m ??, n ??
along m
along n
? 2
x1,H3m, n
hm
along m
m ??, n ??
xm, n
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??? Pepper.bmp
x1,Lm, n
x1,H2m, n
2-D DWT ???
x1,H3m, n
x1,H1m, n
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3?2-D DWT???
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?? ???? (JPEG 2000) ????edge
detection corner detection
filter design
pattern recognition music
signal processing economical
data temperature analysis
feature extraction
biomedical signal processing
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????????? (by Matlab)
A. ?????

• ???,???????????? .wav ???
• ?? wavread (2015??????? audioread)
• ? x, fs wavread('C\WINDOWS\Media\ringin.wav'
  )
• ??? ringin.wav ????? x ???? fs sampling
  frequency
• ?????? size(x) 9981 1 fs 11025
  
• ?? ??,???????
• ??????????

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??????? time 0length(x)-1/fs
x ???? wavread ?????? plot(time, x)
?? .wav ????????,??? ?1 ? 1
??
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• ?????????,?????????????
• x, fswavread('C\WINDOWS\Media\ringin.wav',
  4001 5000)
• 
  ???4001?5000?
• x, fs, nbits wavread('C\WINDOWS\Media\ringin
  .wav')
• nbits x(n) ?bit ?
• ???bit ???,???bit 2?1,???bit 2?2, ..,
• ? n ?bit 2?nbits 1, ?? x ??2nbits ?1 ?????
  
• ??????, nbits 8,?? x ?? 128????

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• ?????? ??? (Stereo)??? (?????)
• ? x, fswavread('C\WINDOWS\Media\notify.wav')
  
• size(x) 29823 2 fs 22050

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B. ????
X fft(x) plot(abs(X)dt) dt 1/fs
abs(X)dt
ringin.wav ???
m
fft ?? ????? (1) Using normalized
frequency F F m / N. (2) Using
frequency f, f F ? fs m ? (fs / N).
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abs(X)dt
ringin.wav ???
F
abs(X)dt
ringin.wav ???
f
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C. ?????
(1) wavplay(x) ? x ? 11025Hz ?????

(???? 1/11025 9.07 ? 10-5 ?) (2) sound(x)
? x ? 8192Hz ????? (3) wavplay(x,
fs) ? sound(x, fs) ? x ? fs Hz ????? Note
(1)(3) ? x ???1 ?column (?2? columns),? x ????
?? ?1 ? 1 ?? (4) soundsc(x, fs) ??? x ????
-1 ? 1 ?? ???
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D. ? Matlab ?? .wav ? wavwrite
wavwrite(x, fs, waveFile) ??? x ???? .wav
?,????? fs Hz ? x ???1 ?column (?2? columns)
? x ??? ?? ?1 ? 1 ?? ? ?????fs,????fs ?
8000Hz
(2015??????? audiowrite)
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E. ? Matlab ?????
????,??????????,????????? (??? notebooks
??????????)
????
Sec 3 Fs 8000 recorder
audiorecorder(Fs, 16, 1) recordblocking(recorder
, Sec) audioarray getaudiodata(recorder)
???????,????? ????????,sampling frequency ? 8000
Hz ????? audioarray,??? column vector
(??????,???? column vectors)
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???? (?)
wavplay(audioarray, Fs)
??????? t 0length(audioarray)-1./Fs plot
(t, audioarray)
??????????? xlabel('sec','FontSize',16) wavwrite(
audioarray, Fs, test.wav) ???????? .wav ?
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????
recorder audiorecorder(Fs, nb, nch)
(?????????)
Fs sampling frequency, nb using nb bits to
record each data nch number of channels (1 or 2)
recordblocking(recorder, Sec)
(?????)
recorder the parameters obtained by the command
audiorecorder Sec the time length for
recording
audioarray getaudiodata(recorder)
(??????,?? audioarray ?? column vector,??????,?
audioarray ??? column vectors)
???????,???,?????
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F?MP3 ?????
????????? mp3read.m, mp3write.m
??? http//www.mathworks.com/matlabcentral/fileex
change/13852-mp3read-and-mp3write ?????Dan
Ellis mp3read.m ?? mp3 ??? mp3write.m
?? mp3 ???
??? .wav ? (????????),.mp3 ??? MPEG-2 Audio
Layer III??????????
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?? Write an MP3 file by Matlab fs8000
sampling frequency t 1fs3/3
filename test Nbit32 number
of bits per sample x 0.2cos(2pi(500t300(t-1
.5).3)) mp3write(x, fs, Nbit, filename)
make an MP3 file test.mp3 Read an MP3 file by
Matlab x1, fs1mp3read('phase33.mp3') x2x1(57
7end) delete the head sound(x2, fs1)