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

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XI. Hilbert Huang Transform (HHT)
Proposed by ???? (AD. 1998 )
??????????
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References
1 N. E. Huang, Z. Shen, S. R. Long, M. C. Wu,
H. H. Shih, Q. Zheng, N. C. Yen, C. C. Tung, and
H. H. Liu, The empirical mode decomposition and
the Hilbert spectrum for nonlinear and
non-stationary time series analysis, Proc. R.
Soc. Lond. A, vol. 454, pp. 903-995, 1998. 2
N. E. Huang and S. Shen, Hilbert-Huang Transform
and Its Applications, World Scientific,
Singapore, 2005.
(PS ?? 2007 ???????????????)
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11-A The Origin of the Concept
????? instantaneous frequency ??? Hilbert
transform
? Hilbert transform
or
H(f)
j
f-axis
f 0
-j
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Applications of the Hilbert Transform
? analytic signal
? another way to define the instantaneous
frequency
where
Example
4
Problem of using Hilbert transforms to determine
the instantaneous frequency
This method is only good for cosine and sine
functions with single component.
Not suitable for (1) complex function
(2) non-sinusoid-like function
(3) multiple components
Example
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? Hilbert-Huang transform ?????
?????????? sinusoid-like components trend
(? Fourier analysis ???????,?? sinusoid-like
components ? period ? amplitude ???????)
??? Hilbert transform (? STFT,number of zero
crossings) ????? components ? instantaneous
frequency
?????? Fourier transform
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11-B Intrinsic Mode Function (IMF)
??
(1) The number of extremes and the number of
zero-crossings must either equal or
differ at most by one.
(2) At any point, the mean value of the envelope
defined by the local maxima and the
envelope defined by the local minima is near to
zero.
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11-C Procedure of the Hilbert Huang Transform
Steps 18 are called Empirical Mode Decomposition
(EMD)
(Step 1) Initial y(t) x(t), (x(t) is the
input) n 1, k 1
(Step 2) Find the local peaks
y(t)
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(Step 3) Connect local peaks
???? B-spline,??? cubic B-spline ???
(?????)
9
(Step 4) Find the local dips
(Step 5) Connect the local dips
10
(Step 6-1) Compute the mean
(pink line)
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(Step 6-2) Compute the residue
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(Step 7) Check whether hk(t) is an intrinsic mode
function (IMF)
(1) ???? local maximums ??? 0
local minimums ??? 0
(2) ??? u1(t), ??? u0(t)
????
for all t
If they are satisfied (or k ? K), set cn(t)
hk(t) and continue to Step 8 cn(t) is the nth IMF
of x(t).
If not, set y(t) hk(t), k k
1, and repeat Steps 26
(??????????,??? k ??? K)
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(Step 8) Calculate and check whether x0(t) is a
function with no more than one extreme point.
If not, set n n1, y(t) x0(t) and
repeat Steps 27 If so, the empirical mode
decomposition is completed.
Set y(t) x(t)
Step 8
Step 7
Y
x0(t) has only 0 or 1 extreme?
hk(t) is an IMF?
Y
Step 9
Step 1
Steps 26
N
y(t) hk(t)
trend
N
y(t) x0(t)
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(Step 9) Find the instantaneous frequency for
each IMF cs(t) (s 1, 2, , n). Method 1 Using
the Hilbert transform Method 2 Calculating the
STFT for cs(t). Method 3 Furthermore, we can
also calculate the instantaneous frequency from
the number of zero-crossings directly.
instantaneous frequency Fs(t) of cs(t)
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Technique Problems of the Hilbert Huang Transform
??????? ??????????,?????? (1) ??????? extreme
points (2) ??????????? extreme points (3) ???????
extreme points ??????
Noise ??? ?? pre-filter ???
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????????????? extreme points
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11-D Example
Example 1
After Step 6
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IMF1
IMF2
x0(t)
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Example 2
hum signal
IMF1
IMF2
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IMF3
IMF4
IMF5
IMF6
21
IMF7
IMF8
IMF9
IMF10
22
IMF11
x0(t)
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11-E Comparison
(1) ???????????? (2) ?????? function ????? (3)
??????????,????????????????? (4) ??? Climate
analysis Economical data
Geology
Acoustics Music signal

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? Conclusion
????????? ??????? sinusoid functions
?????,????sinusoid functions ? amplitudes
?????,??? HHT ???
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???? Interpolation and the B-Spline
Suppose that the sampling points are t1, t2, t3,
, tN and we have known the values of x(t) at
these sampling points. There are several ways
for interpolation.
(1) The simplest way Using the straight lines
(i.e., linear interpolation)
t1
t2
t3
t4
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(2) Lagrange interpolation
? ???????,
(3) Polynomial interpolation
solve a1, a2, a3, , aN from
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(4) Lowpass Filter Interpolation
??? sampling interval ?????? tn1 ? tn ?t
for all n
discrete time Fourier transform
lowpass mask
X1(f)
X (f)
x(tn)
inverse discrete time Fourier transform
x(t)
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(5) B-Spline Interpolation
B-spline ??? spline
for tn lt t lt tn1
otherwise
m 1 linear B-spline m 2 quadratic
B-spline m 3 cubic B-spline (????)
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In Matlab,the commandspline can be used for
spline interpolation (Note In the command, the
cubic B-spline is used)
Example Generating a sine-like spline curve
and samples it over a finer mesh x
0110 original sampling points
y sin(x) xx 00.110 new
sampling points yy spline(x,y,xx)
plot(x,y,'o',xx,yy)