Term paper A Tutorial of the Morlet Wavelet Transformstudent
R99942036
Outline1. Introduction...p.2 2. Wavelet transform. ...p.2 2.1
Continuous Wavelet Transform (CWT)....p.3 2.2 Continuous Wavelet
Transform with discrete coefficients..p.7 2.3 Discrete Wavelet
Transform (DWT)..p.8 3. Complex Wavelet transform...p.12 4. Some
famous Mother Wavelet....p.14 5. Morlet Waveletp.18 6. Application
of the Morlet wavelet analysis in the ECG...........p.19 7. History
of Wavelet transform.....p.20 8. Conclusion....p.21 9.
Reference..p.21
1
1. IntroductionThe Morlet Wavelet Transform is a kind of Wavelet
transform, and Wavelet transform can be seen a powerful tool in
time-frequency analysis. Above all the tool in time-frequency
analysis, the famous one is the Fourier transform. Fourier
transform is useful to analysis the frequency component over the
whole time, but it can not catch the change in frequency response
with respect to time. Short-Time Frequency transform (STFT) uses a
window function to catch the frequency component in a time
interval. Although STFT can be use to observe the change in
frequency response with respect to time, there is still a problem
that the fixed width of the window function lead to the resolution
fixed. On the other hand, from the Uncertainty Principle, we know
that the product of the time domain resolution and the frequency
resolution is constant, so we can not have high resolution at both
the time domain and frequency domain at the same time. Therefore,
we may have high time resolution when observing the high frequency
signal component and have high frequency resolution when observing
the low frequency signal component. Wavelet transform is one of the
solutions to the above problem: by changing the location and
scaling of the mother wavelet, which is the window function in
Wavelet transform, we can implement the multi-resolution concept
mentioned above. In the way employing a window with variable width,
Wavelet transform can capture both the short duration, high
frequency and the long duration, low frequency information
simultaneously. It is more flexible than STFT and particularly
useful for the analysis of transients, aperiodicity and other
non-stationary signal feature. The Morlet Wavelet transform is the
kind of the Continuous Wavelet Transform, which is one class of the
Wavelet transform. We will force on the Continuous Wavelet
Transform and describe briefly the other classes Wavelet transform.
Besides, the conventional wavelet transform is based on real-valued
wavelet function and scaling function, but the Morlet Wavelet
transform is actually a complex Wavelet transform. In the later
chapter, we will introduce some basic concept of the complex
Wavelet transform.
2. Wavelet transformThere are three distinct classes about
Wavelet transform in use today: Continuous Wavelet Transform (CWT),
continuous Wavelet transform with discrete coefficients and
Discrete Wavelet Transform. With respect to continuous/discrete in
the input/output, we distinguish this three classes following:2
Class Continuous Wavelet Transform Discrete Wavelet
Transform
Input
Output Discrete Discrete
Continuous Continuous Discrete
continuous Wavelet transform with discrete coefficients
Continuous
Now we make the differences of the three classes Wavelet
transform.
2.1 Continuous Wavelet Transform (CWT)2.1.1 Definition a (, ) 1
* t a x(t ) b dt , b [0, } ..(1) b is the complex conjugate of the
mother wavelet (t ) , which is the
X w ( a, b) =
where * (t )
analysis wavelet function, a is the location parameter of the
wavelet and b is the scaling (dilation) parameter of the wavelet.
Note that a is any real number and b is any positive real
number.2.1.2 Inverse Continuous Wavelet Transformx(t ) =
1 C
0
1 b5/ 2
X w (a, b) (
ta )dadb b
where C =
( f ) f
2
0
df < .
2.1.3 Constraints for the mother wavelet
In general, the mother wavelet is not arbitrary function. Here
list three constraints in designing mother wavelet:2.1.3.1 Compact
Support
Support: the region of the mother wavelet where is not equal to
zero Compact support: the width of the support is not
infinite2.1.3.2 Vanishing Moments
We usually need that the mother wavelet is a high frequency
signal, and that could3
make the Wavelet transform sensitive to the high frequency
input. The result of that wavelet transform can be more precise in
time-frequency analysis. Since all signals in nature can be
represented by polynomial, therefore we define the moment: kth
moment: mk = t k (t )dt .
If m0 = m1 = m2 = = mp-1 = 0, we say (t ) has p vanish moments.
At least, werequire the vanishing moment is not less than one,
i.e.
(t )dt = 0The more the order vanishing moment is, the higher
frequency function the mother wavelet is, and the more sensitive
this Wavelet transform to process the high frequency component
is.2.1.3.3 Admissibility Criterion
C =
( f ) f
2
0
df <
where ( f ) is the Fourier transform of the mother wavelet (t )
. This constraint is due to that the inverse wavelet transform can
exist.2.1.4 Modified for digital computer
It is impossible to implement the integral for scaling parameter
b in the infinite interval in digital computer. Since the
constraint Compact Support, which support the mother wavelet exists
in a finite length, we can modify the inverse Wavelet transform by
the scaling function. Define the scaling function (t ) :
(t ) = ( f )e j 2ft df
where ( f ) =
( f ')
2
f
f'
e j 2ft df ' for f > 0 , ( f ) = * ( f ) . The scaling
function
is usually a lowpass filter. Now the modified Wavelet transform
is: X w ( a, b) = LX w (a, b0 ) =
1 * t a x(t ) b dt a (, ) b , 1 b [0, b0 ] * t a x(t ) b0 dt b0
4
The reconstruction is: x(t ) = 1 b0 1 ta 1 t a )dadb + LX w (a,
b0 ) ( )da 0 5 / 2 X w (a, b) ( 3/ 2 b C b b b0 0
2.1.5 Scalogram
Scalogram is the absolute value and square of the output of the
Wavelet transform, i.e., 1 Sc x (a, b) = X w (a, b) = b2 2
t a x(t ) b dt *
The mean of Scalogram is the wavelet energy density function
which is the contribution to the signal energy at the specific
scale parameter a and location parameter b. It is analogous to the
spectrogramthe energy density surface of the STFT.2.1.6 Compare
with Fourier transform and time-frequency analysis Fourier
transform (FT)F ( f ) = x(t )e j 2ft dt (2)
As we know, Fourier transform convert signal in time domain to
frequency domain by integrating over the whole time axis. However,
if the signal is non-stationary, the signal in frequency is
actually a function of time and we can not know when the frequency
component changes. The time-frequency tile allocation of FT is:
Fig.1. From Fig.1, we can know that the time information is
completely lost after Fourier transform. Frequency axis is divided
uniformly, and the frequency resolution may be precise when we
integrate along the whole time axis.5
Short-Time Frequency transform (STFT)
Here we take the Short-Time Frequency transform as the example
of time-frequency analysis:X (t , f ) = w(t )x( )e j 2f d .(3)
where w(t) is the window function, t is the location and f is
the frequency. STFT tries to solve the problem of the FT, which FT
can not catch the change in frequency response with respect to
time, by introducing a window function w(t). The window function is
used to extract a portion of x(t) and then take Fourier transform.
The output of the STFT has two parameters: one is the time
parameter t, which indicating the instant we concern. The other is
the frequency parameter f, which is the same role in the FT.
However, there is another problem: the width of the window is
fixed, which may cause that there is no enough resolution in some
interval. For example, suppose the window size is one, there is a
signal with frequency 0.1Hz. The extracted data from STFT in 1
second look like flat (DC term) in the time domain. The
time-frequency tile allocation of STFT is:
Fig.2. From Fig.2, we can see the time information reserved
after STFT. Besides, both the time axis and frequency axis are
divided uniformly, and there is no one which resolution is
difference with others. By the way, the frequency resolution
depends on the time resolution, which resolution can easily be
understood by the Uncertainty Principle: the product of the time
domain resolution and the frequency resolution is constant.Wavelet
transform6
1 * t a x(t ) b dt (4) b Wavelet transform is one of solutions
to above problem. The mother wavelet (t ) X w ( a, b) = by dilated
(parameter b) and translated (parameter a) is designed to balance
between the time domain and frequency domain resolution. We can see
clearly very low frequency components at large b, which makes the
width of the mother wavelet expansive, and very high frequency
components at small b, which makes the width of the mother wavelet
concentrating. The time-frequency tile allocation of Wavelet
transform is:
Fig.3. the multi-resolution From Fig.3, we can see both the time
axis and frequency axis are not divided in fixed interval, and this
flexibility is very useful in time-frequency analysis. It is worthy
to note that the parameter b is inversely proportional to the
frequency and parameter a is just like the time. This is another
difference between the a-b plot of the Wavelet transform and the
t-f plot of the STFT
2.2 Continuous Wavelet Transform with discrete coefficients2.2.1
Definition
The main difference with Continuous Wavelet Transform is that
the parameter a and b are not chosen arbitrarily. Here we constrain
a to be n2-m and constrain b to be 2-m at the (1): a (, ) 1 * t a X
w (a = n 2 m , b = 2 m ) = x(t ) b dt , b [0, } b X w (a = n2 m , b
= 2 m ) = 1 2 m7
t n2 m x(t ) * 2 m dt
n Z , n (, ) X w (n, m) = 2 m / 2 x(t ) * 2 m t n dt , .(5) m Z
, m (, }
(
)
2.2.2 Inverse Continuous Wavelet Transform with discrete
coefficients
x(t ) =
m = n =
2
m/2
1 (2 m t n) X w (n, m)
where 1 (t ) is the dual function of (t ) , i.e.,
2 m 1 (2 m1 t n1 ) (2 m t1 n)dt = (m m1 ) (n n1 )
should be satisfied. We often desire that 1 (t ) = (t ) , and
then x(t ) =m = n =
2
m/2
(2 m t n) X w (n, m)
The mother wavelet should satisfies
2 m (2 m1 t n1 ) (2 m t n)dt = (m m1 ) (n n1 ) .
2.3 Discrete Wavelet Transform (DWT)2.3.1 Concept
DWT comes from the Continuous Wavelet Transform with discrete
coefficients, but the mathematical part is simplified largely. Even
there is no easy formula to represent the relation between the
input and output.2.3.2 1-D Discrete Wavelet Transform
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Fig.4. one-stage 1-D DWT From Fig.4, we know that x1, L [n] is
the low frequency component, which is the
result after the input passes through the lowpass filter g[n]
and x1, H [n] is the high frequency component, which is the result
after the input passes through the highpass filter h[n] . Above
case of DWT is a one-stage DWT, and if we want to have a more
precise analysis, we can implement more stages in DWT. The
following is the case of two-stage DWT:
Fig.5. two-stage 1-D DWT At the Fig.5, the difference with Fig.4
is that decompose the lowpass component from stage one furthermore.
Sometimes it can also decompose the highpass component from stage
one due to difference goals. It is worthy to note that even if it
pass through many stages decomposing, the output total bits number
after DWT dose not increase many bits when the size of input is
more larger than the size of the filter. On the other hand, we can
observe the effect of more stages DWT from the frequency
domain:
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Fig.6. three-stage 1-D DWT in the frequency domain Fig.6 can
also explain that when we implement more stages in DWT, we may get
a more precise time-frequency analysis.2.3.3 1-D Inverse Discrete
Wavelet Transform
Fig.7. one-stage 1-D inverse DWT In general, we need that the
inverse DWT conforms to the condition Perfect Reconstruction, which
mean that the output of the inverse DWT is fully equal to the input
of the DWT. How to choose the synthesis filter g1[n] and h1[n] is
the other important issue, and here we do not force to this
part.2.3.4 2-D one-stage Discrete Wavelet Transform
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Fig.8. one-stage 2-D DWT From Fig.8, we see that this one-stage
2-D DWT seems to the two-stage 1-D DWT, but they are difference.
The main difference is this 2-D DWT decomposes the low frequency
and high frequency component in two dimensions separately and the
two-stage 1-D DWT decomposes twice in the same (and the only)
dimension. We take an example following for the result of the
one-stage 2-D DWT in the image processing:
Fig.9. an example of the one-stage 2-D DWT11
From Fig.8, it is the result of the one-stage 2-D DWT with the
input is a rectangular. We can see clearly that DWT can be applied
in the edge and corner detection or compression since the lowpass
frequency component have no much difference with the original input
signal but the size, and the other highpass frequency component is
sparse except the edge of the original input reserved.
3. Complex Wavelet transformThe conventional wavelet transform
is based on real-valued wavelet function and scaling function, but
the Morlet Wavelet transform is a complex Wavelet transform. Why do
we consider the complex version? There are some troubles with real
wavelet: Problem 1: Oscillation Since the mother wavelet is a
bandpass filter, the wavelet coefficients will oscillate around
singularities. It complicates wavelet-based processing. Problem 2:
Shift Variance Small shift of the signal will greatly perturb the
wavelet coefficient oscillating around singularities. We can see
the following example:
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Fig. Problem 3: Aliasing The signal after wavelet transform will
result in substantial aliasing. Of course the inverse DWT will
cancel this aliasing, but only when the wavelet and scaling
coefficients are not changed. Any processing about wavelet
coefficient (thresholding, filtering, and quantization) upsets the
delicate balance between the forward and inverse transforms,
leading to artifact errors in the reconstructed signal. Fortunately
there is one solution of above shortcomings: complex wavelet. Now
we start to introduce some basic concept of the complex Wavelet
transform. Before starting, we first note that it seems same
between separable 2-D DWT (here do not describe it) and complex
Wavelet transform. The main difference is that the separable 2D DWT
lacks the directionality. While the separable 2D DWT supports three
directions, the complex Wavelet transform can extend to support six
directions. Since the complex Wavelet transform is based on the
Hilbert transform, we will introduce some concept the Hilbert
transform first.3.1 Hilbert transform
The Hilbert transform of the input signal x(t ) is:
y (t ) = Hilbert{x(t )} = x(t ) * h(t ) ,
where h(t ) =
1 and * is the convolution. The frequency response of h(t ) is:
t H ( w) = j sgn( w) .
We can see the result of the Hilbert transform: Y ( w) = X ( w)
H ( w) = j sgn( w) X ( w)
Fig. the frequency response of the Hilbert transform By the
Hilbert transform, there is an important property: analytic
representation of the signal, which is that the negative frequency
component of the Fourier transform is discarded with no loss of
information:13
xc (t ) = x(t ) + j y (t ) = x(t ) + j Hilbert{x(t )} In the
complex Wavelet transform, it will use this property by the Hilbert
transform.3.2 Complex Wavelet transform
Here first we note the key due to above mention problems: the
Fourier transform does not suffer from these problems. First, the
magnitude of the Fourier transform dose not oscillate positive and
negative but rather provides a smooth positive envelope in the
Fourier domain. Second, the magnitude of the Fourier transform is
perfectly shift invariant. Third, the Fourier coefficients has not
aliasing and do not rely on a complicated aliasing cancellation
property to reconstruct the signal. The main point is the Fourier
transform is based on complex-valued oscillating sinusoids e jwt =
cos( wt ) + j sin( wt ) and the real and imaginary parts of the
oscillating sinusoids is a Hilbert transform pair, i.e. sin( wt ) =
Hilbert{cos( wt )} . Therefore, the oscillating sinusoids e jwt =
cos( wt ) + j sin( wt ) is actually an analytic signal and it is
supported only on one-half of the frequency axis ( w > 0).
Inspired from the Fourier transform, we can imagine the complex
wavelet transform with a complex-valued mother wavelet: c (t ) = r
(t ) + j i (t ) where r (t ) is real and even, i (t ) is imaginary
and odd, and i (t ) is the Hilbert transform of r (t ) . The
complex scaling function is defined similarly. The choice of
complex mother wavelet and complex scaling function is another
important issue, and here we do not force on it. As the Fourier
transform, complex wavelets can be used to analyze both real-valued
signals and complex-valued signals. On the other hand, the complex
WT enables new multi-scale signal processing that exploit the
complex magnitude and phase. In particular, the large magnitude can
indicate the presence of a singularity while the phase indicates
its position within the support of the wavelet.
4. Some famous Mother WaveletThere are many functions chosen to
be the mother wavelet, and here we list some important case.4.1
Haar basis/Haar Wavelet
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Fig.10. Haar Wavelet The Haar Wavelet is the easiest Wavelet
transform, and it is a kind of the Continuous Wavelet Transform
with discrete coefficients. We can see clearly that this mother
wavelet is a highpass filter and the scaling function is a lowpass
filter indeed. There are some advantages of the Haar Wavelet: (1)
Simple (2) Fast algorithm (3) Orthogonal (therefore reversible) (4)
compact, real, odd (5) Vanish moment = 1 The main disadvantage is
both the mother wavelet and the scaling function are not enough
smooth. Since there are less rectangular signals in nature, and in
general it hope the basis seem to the signal we want to analysis in
the signal processing.4.2 Mexican hat function/ Mexican hat
Wavelet2 25 / 4 (1 + 2t 2 )e t 3 The Mexican hat Wavelet is the
second derivative of the Gaussian function. We
(t ) =
can see this Mexican hat function in following plot:
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Fig.11. Mexican hat function In general, the any order
derivative of the Gaussian function can be employed as a mother
wavelet, and it is worthy to note that the p order derivative of
the Gaussian function has p vanishing moment. On the other hand,
since the Gaussian function is a low frequency signal, it is not
suitable to be a mother wavelet (recall the mother wavelet is
usually a high frequency signal). Now we consider the wavelet
transform of an exponential discontinuity, which is a sudden spike
in the signal half way along its length followed by a smooth
exponential decay, at the Fig.12. In the Fig.12 (b), we can see
that the location, where the smallest b is, points to the location
the signal discontinuity in the Fig.12 (a). On the other hand, we
can also observe the effect of the multi-resolution: the resolution
of the wavelet transform is invariant along a (location) but
variant along b (scaling), looked Fig.13.
Fig.12. Point to an exponential discontinuity. (a) A sudden
spike with an exponential tail. (b) The transform plot for the
discontinuity.
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Fig.13. the effect of the multi-resolution Furthermore, here we
can also observe the effect of the vanishing moment: The more the
order vanishing moment is, the more sensitive this Wavelet
transform to process the high frequency component is. Since we know
that observing the high frequency components by observing the
location where the smallest b is, and it points to the location the
high frequency signal (or discontinuity part) from the Fig.12.
Therefore, comparing the red circle part both in the Fig.14 and
Fig.15, we can observe this effect: when the vanishing moment is
higher, the high frequency component of the result after Wavelet
transform is more precise.
Fig.14. the vanishing moment = 1
17
Fig.15. the vanishing moment = 2 A complex version of the
Mexican hat Wavelet can easily be constructed by setting the
analytic version signal of the real-valued mother wavelet. However,
in practice the Morlet wavelet is used when a complex wavelet
function is required.
5. Morlet WaveletThe Morlet wavelet is the most popular complex
wavelet used in practice, which mother wavelet is defined asw 0 1
(t ) = 4 e jw0t e 2 2
t e 2 . 2
where w0 is the central frequency of the mother wavelet. Note
that the term e
2 w0 2
is
used for correcting the non-zero mean of the complex sinusoid,
and it can be negligible when w0 > 5 . Therefore in some
research the mother wavelet definition of the Morlet wavelet is:t2
2
(t ) =where the central frequency w0 > 5 .
14
e
jw0t
e
,
The Morlet wavelet has a form very similar to the Gabor
transform. The important difference is that the window function
dose also be scaled by the scaling parameter, while the size of
window in Gabor transform is fixed.
18
6.Application of the Morlet wavelet analysis in the
electrocardiogramThe application of the Morlet wavelet analysis in
the electrocardiogram (ECG) is mainly to discriminate the abnormal
heartbeat behavior. Since the variation of the abnormal heartbeat
is a non-stationary signal, then this signal is suitable for
wavelet-based analysis. Here we see an example about the wavelet
analysis application in the ECG. Fig.16 shows the pressure in the
aorta and ECG corresponding to an episode of ventricular
fibrillation in a porcine model. We have some note following about
this example: (1) The ECG signal has a typical random or
unstructured appearance. However, the aorta pressure trace reveals
regular low amplitude spikes in the Fig.16 (a). (2) The irregular
activity of the much larger ventricular muscle mass completely
obscured this atrial activity in the standard ECG recording shown
in the Fig.16 (b). (3) The wavelet energy scalogram for this signal
is plotted in the Fig.16 (c). (A Morlet wavelet was used in the
study.) The high amplitude band at around 810 Hz is much more in
this interval than the other traces where no atrial pulsing was
apparent. (4) In the Fig.16 (d), the location of zero wavelet phase
between 1.3 and 1.5 Hz is plotted. The phase plot exhibits a
strikingly regular pattern with the zero phase lines aligning
themselves remarkably well with the atrial pulsing of the pressure
trace. (5) This result suggests that (wavelet) phase information,
which obscure to the traditional methods, may be used to
interrogate the ECG for underlying low-level mechanical activity in
the atria.
19
Fig.16 Simultaneous ECG and pressure recordings. (a)The aorta
pressure trace, with (b) ECG, corresponding (c) wavelet energy plot
obtained using the Morlet wavelet. (d) The zero phase lines of the
Morlet wavelet transform. The plots correspond to the time period
726.23731.31 s after the initiation of VF. (After Addison et al
2002 IEEE Eng. Med. Biol. ( IEEE 2002).)
7. History of Wavelet transform1910, Haar families, which was
proposed by the mathematician Alfrd Haar. Haar wavelet is the first
literature relates to the wavelet transform, but the concept of the
wavelet did not exist at that time. 1981, Morlet, wavelet concept,
which was proposed by the geophysicist Jean Morlet. 1984, Morlet
and Grossman, wavelet. Morlet and the physicist Alex Grossman
invented the term wavelet. 1985, Meyer, orthogonal wavelet. Before
1985, a lot of researchers thought that there was no orthogonal
wavelet except Haar wavelet. The mathematician Yves Meyer
constructed the second orthogonal wavelet called Meyer wavelet in
1985. 1987, International conference in France, which is the 1st
international conference20
about Wavelet transform. 1988, Mallat and Meyer,
multiresolution. Stephane Mallat and Meyer proposed the concept of
multiresolution. 1988, Daubechies, compact support orthogonal
wavelet. Ingrid Daubechies found a systematical method to construct
the compact support orthogonal wavelet. 1989, Mallat, fast wavelet
transform. With the appearance of this fast algorithm, the wavelet
transform had numerous applications in the signal processing
field.
8. ConclusionIn this tutorial, we first compare the difference
for the Fourier transform, Short-Time Frequency transform (STFT),
and Wavelet transform. We mainly introduce the wavelet concept and
force on the two parts Continuous Wavelet Transform and the complex
Wavelet Transform, which two classes does the Morlet Wavelet belong
to. Besides, the complex Wavelet can overcome these problems met in
the real-valued Wavelet: Oscillation, Shift Variance and Aliasing.
In practice the Morlet Wavelet was used in the electrocardiogram
(ECG) study. We also take a briefly example which is the
application of the Morlet wavelet analysis in the ECG. In general,
the mother Wavelet is designed by real-valued function, and however
in this example we can see the application of the phase from the
complex Wavelet transform.
9. Reference[1] C. L. Liu, A Tutorial of the Wavelet Transform.
February 23, 2010. [2] P. S. Addison, Wavelet transforms and the
ECG: a review, in IOP science 8 August 2005 [3] I. W. Selesnick, R.
G. Baraniuk, and N. G. Kingsbury, The Dual-Tree Complex Wavelet
Transform, in IEEE SIGNAL PROCESSING MAGAZINE, NOVEMBER 2005. [4]
W.Wu, Extracting Signal frequency information in time/frequency
domain by means of continuous wavelet transform, in International
Conference on Control, Automation and Systems 2007. [5] P.M.
Bentley and J.T.E. McDonnel, Wavelet transforms: an introduction,
in ELECTRONICS B COMMUNICATION ENGINEERING JOURNAL AUGUST 1994. [6]
]J.J Ding, Slides of time-frequency analysis and wavelet
transform
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