INTRODUCTION
The Wavelet Transform (WT) is a merited technique for analysis of nonstationary
signals like cardiac signals. Being a multiscale analysis technique, it offers
the possibility of selective noise filtering and reliable parameter estimation.
Often WT systems employ the discrete wavelet transform, implemented in a digital
signal processor. However, in ultra lowpower applications such as biomedical
implantable devices, it is not suitable to implement the WT by means of digital
circuitry due to the high power consumption associated with the required A/D
converter. For power consumption considerations it therefore is preferable to
perform WT in the analog domain.
Lowpower analog realization of the wavelet transform with the technique of
analog circuits (Karel et al., 2005, 2012;
GurrolaNavarro et al., 2010; Haddad
et al., 2005) has been introduced. The quality of such implementations
depends on the accuracy of the corresponding wavelet approximations. Previous
approaches reported for wavelet approximations include mainly pade approximation
method (Haddad et al., 2005) and L_{2}
approximation method (Karel et al., 2005; Haddad
et al., 2005). The Laplace transforms of wavelet functions were
approximated by rational functions in the Laplace domain with Pade approximation
(Baker, 1975; Bultheel and Barel,
1986). However, there are some problems which limit the practical applicability
of Pade approximation. One important issue concerns stability. The stabile transfer
function of wavelet filter does not automatically result from the Pade approximation
technique. If the selection of the point s_{0} is improper, the result
of approximation will be unstable. Some poles of the Morlet wavelet transfer
function obtained by this method in Haddad et al.
(2005) lie in the right half of the splane, which indicates the transfer
function is not stable. Another important drawback is: the quality of the approximation
of the wavelet is not measured directly in the time domain but in the Laplace
domain, which results in a larger error of approximation. The performance of
implementing WT in analog domain depends largely on the accuracy of the approximations
involved in this approach. Karel et al. (2005),
an alternative approach, based on L_{2} approximation that works directly
in the time domain, was introduced. A drawback of this approach is that the
numerical optimization of objective function usually ends in local, nonglobal
optimization when a starting point is not exactly selected. To find a good starting
point for L_{2} approximation, a method involving highorder FIR approximation
and balanceandtruncate model reduction is used. However, it is a computational
complex method and limited to approximate such low order wavelet functions as
the Gaussian and Daubechies. This method has also some convergence problems
when one tries to approximate a function with many oscillations (high order
wavelet), such as the Morlet wavelet. So far, there is a lack of the effective
method to approximate various low or high order wavelet functions, which is
an obstacle for implementing WT in analog domain.
In this study, we focus on the wavelet approximation for implementation in
analog circuits. The innovative aspect of the present work is threefold. First,
by extending pioneering work (Karel et al., 2005;
Hongmin et al., 2008; Li
et al., 2010; GurrolaNavarro et al.,
2010), we propose a generalized optimization mathematical model of approximating
various wavelet functions, which is based on the L_{2 }approximation.
Second, the Particle Swarm Optimization (PSO) (Kennedy and
Eberhart, 1995) algorithm is introduced to solve the optimization problem.
The PSO algorithm is one of the most powerful methods for solving global optimization
problems and is effective, efficient and fairly robust to initial conditions.
This method overcomes these shortcomings of approximation technique in Haddad
et al. (2005) and Karel et al. (2005).
Using PSO algorithm, we have successfully approximated various wavelet functions,
especially the Morlet wavelet (a high order wavelet).
WAVELET TRANSFORM
The wavelet transform provides a timefrequency representation of the signal
(Mallat, 1999; Walnut, 2004). It
was developed to overcome the short coming of the Short Time Fourier Transform
(STFT), which can also be used to analyze nonstationary signals. While STFT
gives a constant resolution at all frequencies, the wavelet transform uses multiresolution
technique by which different frequencies are analyzed with different resolutions.
The definition of the Continuous Wavelet Transform (CWT) for a real valued
time signal x (t) is given as (Mallat, 1999):
where, a is scale parameter (a∈ (0, R)) and τ is translation parameter
(τ∈R). The base function Ψ (t) (Ψ (t)∈L(R)^{2})
is called the mother wavelet. The mother wavelet used to generate all the basis
functions is designed based on some desired characteristics associated with
that function. The translation parameter τ relates to the location of the
wavelet function as it is shifted through the signal. Thus, it corresponds to
the time information in the Wavelet Transform. The scale parameter α is
defined as 1/frequency and corresponds to frequency information. Scaling either
dilates (expands) or compresses a signal. Large scales (low frequencies) dilate
the signal and provide detailed information hidden in the signal, while small
scales (high frequencies) compress the signal and provide global information
about the signal. The above equation shows that the wavelet transform performs
the convolution operation of the signal and the basis function.
The mother wavelet must satisfy two restriction conditions. One is:
This ensures the mother wavelet has no DC component and is fast in decaying
rate. The other is the admissibility condition, i.e,:
where, Ψ (w) is the Fourier transform of the mother wavelet ι (t).
The second restriction in Eq. 3 is stronger than the first
one. The reason for requiring this condition is to guarantee that the reconstruction
of the original time signal from the continuous wavelet transform is possible.
Wavelet transforms usually cannot be implemented exactly in analog electronic
hardware. If a time signal x (t) is passed through a linear system, then x (t)
is convolved with the impulse response h (t) of that linear system, producing
the output signal:
On the other hand, from the definition of WT given by Eq. 1,
the analog computation of WT_{x} (a, t) (scale a) can be achieved through
the implementation of a linear filter of which the impulse response satisfies:
For obvious physical reasons only the hardware implementation of (strictly)
causal stable filters is feasible. In other words, a linear filter will have
a (strictly) proper rational transfer function H (s) that has all its poles
in the complex left half plane. Because the h (t) will then be zero for negative
t, any mother wavelet Ψ (t) which does not have this property must be timeshifted
to facilitate an accurate approximation of its (correspondingly timeshifted)
wavelet transform WT_{x} (a, t). This may result in a truncation error
for a wavelet that does not have compact support, such as the Gaussian wavelet.
Note that an approximation error will also be due to the fact that a wavelet
does not usually possess a rational Laplace transform.
L_{2} APPROXIMATION OF WAVELET FUNCTIONS BASED ON PSO ALGORITHM
Generalized optimization model of l_{2} approximation of wavelet
functions: The theory of L_{2 }approximation (Karel
et al., 2005) provides an alternative framework for studying the
problem of wavelet approximation which offers a number of advantages. On the
conceptual level it is quite appropriate to use the L_{2} norm to measure
the quality of an approximation h (t) of the function .
Indeed, the very definition of the wavelet transform itself involves the L_{2}
inner product between the signal x (t) and the mother wavelet Ψ (t). It
is also desirable that the approximation h (t) of
(t) behaves equally well for all time instances t since h (t) is used as a convolution
kernel with any arbitrary shift. This property holds naturally for L_{2}
approximation but it is not supported by the Pade approximation approach. Another
advantage of L_{2} approximation is that it allows for a description
in the time domain as well as in the Laplace domain, so that both frameworks
can be exploited to develop further insight. According to Parseval’s equality
the squared L_{2} norm of the difference between
(t) and h (t) can be expressed as:
Minimization of
is therefore equivalent to minimization of the L_{2} norm of the difference
between the Laplace transforms
(t) and H (s) over the imaginary axis s = iw.
Particularly in the case of low order approximation, the L_{2} approximation
problem can be approached in a simple and straightforward way in the time domain.
As is well known from linear systems theory any strictly causal linear filter
of finite order n can be represented in the time domain by the impulse response
function h (t) (its Laplace transform H (s)). For the generic situation of stable
systems with distinct poles, the impulse response function h (t) is a linear
combination of damped exponentials and exponentially damped harmonics. For low
order systems, this makes it possible to propose an explicitly parameterized
class of impulse response functions among which to search for a good approximation
of
(t). For instance, if a Nth order approximation is attempted, this parameterized
class of functions h (t) may typically have the following form:
where, the parameters b_{i} and d_{j} must be strictly negative
for reasons of stability. When the expression of wavelet functions includes
sine term A cos(Ωt), such as the Morlet Wavelet, the h (t) may be given
by:
Note that wavelets typically are oscillatory functions so that a good fit requires
the contribution of sufficiently many damped harmonics, which further explains
the structure of this class. Given the explicit form of the wavelet
(t) and the parameterized class of functions h (t), the L_{2} norm of
the difference
can now be minimized in a straightforward way using standard numerical optimization
techniques and software. The negativity constraints on b_{i} and d_{j}
which enforce stability are not difficult to handle.
One common property of a wavelet function
(t) that wasn’t discussed so far is that its integral is usually equal
to zero:
If this property is not shared by the approximation h (t), this will cause
an unwanted bias in the approximation of the wavelet transform. So we have that:
This yields the explicit nonlinear condition, if such an extra nonlinear condition
is not conveniently handled by the optimization software, then it can easily
be used to eliminate one of the variables from the problem. Based on the analysis
above, a generalized optimization mathematical model of approximating various
wavelet functions is then given by:
This is a typical nonlinear and constrained optimization question. It is difficult
to obtain the global optimal solution using common numerical optimization techniques,
which in general provide no global optimality guarantee and give different local
optima with different starting points.
Particle swarm optimization (PSO) algorithm: The PSO algorithm is one
of the most powerful methods for solving global optimization problems and is
effective, efficient and fairly robust to initial conditions. In order to optimize
parameters of h (t), we use the Particle Swarm Optimization (PSO) algorithm
to solve the optimization question in (9), search the whole parameters space
effectively and globally.
Particle swarm optimization algorithms (Kennedy and Eberhart,
1995) are evolutionary computation. The particle swarm optimizer algorithms
find optimal regions of complex search space through the interaction of individuals
in a population of particles. The rapid speed of calculation and simple realization
are its excellent performance. Precision is not good. PSO is a populationbased,
bioinspired optimization method. It was originally inspired in the way crowds
of individuals move towards predefined objectives, but it is better viewed using
a social metaphor. PSO is similar to the other evolutionary algorithms in that
the system is initialized with a population of random solutions. However, each
potential solution is also assigned a randomized velocity and the potential
solutions, call particles, corresponding to individuals. Each particle in PSO
flies in the Ddimensional problem space with a velocity which is dynamically
adjusted according to the flying experiences of its own and its colleagues.
The location of the ith particle is represented as X_{i} = (x_{i1},
…, x_{id}, …, x_{iD}), where x_{id} ∈
[l_{d}, u_{d}], d ∈ [1, D], l_{d}, u_{d }are
the lower and upper bounds for the dth dimension, respectively. The best previous
position (which giving the best fitness value) of the ith particle is
recorded and represented as P_{i} = (p_{i1},…, p_{id},
…, p_{iD}), which is also called pbest. The index of the best particle
among all the particles in the population is represented by the symbol g. The
location Pg is also called gbest. The velocity for the ith particle is represented
as V_{i} = (v_{i1}, …, v_{id}, …, v_{iD}),
is clamped to a maximum velocity V_{max} = (v_{max1}, …,
v_{maxd}, …, v_{maxD}), which is specified by the user.
The particle swarm optimization concept consists of, at each time step, changing
the velocity and location of each particle toward its pbest and gbest locations
according to the Eq. 10 and 11, respectively:
x_{id} = x_{id}+v_{id} 
(11) 
where, w is inertia weight, c_{1} and c_{2} are acceleration
constants and rand() is a random function in the range [0, 1]. For Eq.
10, the first part represents the inertia of pervious velocity; the second
part is the “cognition” part, which represents the private thinking
by itself; the third part is the “social” part, which represents the
cooperation among the particles. If the sum of accelerations would cause the
velocity v_{id} on that dimension to exceed v_{maxd}, then v_{id}
is limited to v_{maxd}.
APPROXIMATION OF THE COMMON WAVELET FUNCTIONS
Approximation of gaussian and morlet wavelet: To demonstrate the proposed
method, we first discuss how to approximate Marr wavelet base. Marr wavelet
is a favorite choice in many signal processing applications. The Marr wavelet
Ψ (t) is the second derivative of a Gaussian probability density function:
Select the timeshift t_{0} = 4, get timereversed and timeshifted
Marr wavelet Ψ (4t). Let h (t) be the impulse response of Marr wavelet
filter to be designed and the order of wavelet filter N be 9, then the parameterized
class of functions h (t) given by:
Note that choice of order of wavelet filter involves an important tradeoff
between optimal solution and complexity of filter circuits. If N is chosen too
small, the h (t) may be far away from the versatile wavelet. On the other hand,
if N is chosen too large, a more complex analog IC is demanded to realize wavelet
transform. We define the distance between h (t) and Ψ (t4):
where, a = (a_{1}, a_{2}, a_{3}, …,
a_{18})^{T} is an undetermined parameter vector. To sample D
(a), the fitness function is given:
The optimization model of approximating Marr wavelet function is described
as:
This is a nonlinear constrained optimization question. Using the proposed hybrid
algorithm to solve Eq. 16, the parameters for PSO are set
as: Population size i = 100, Inertia weight factor w_{min} = 0.4, w_{max}
= 0.9, acceleration constant c_{1} = c_{2} = 2, maximum iteration
N = 9000. The position and the velocity of the i th particle and the fitness
function of corresponding sampling point in the nth iteration are denoted by
.
Its local best position and the global best position of the particle swarm are
denoted as P_{i} and P_{g} , respectively. The PSO optimization
program is run in MATLAB 7.1 and the search process of PSO is shown in Fig.
1. Because PSO is a stochastic algorithm, it is difficult to guarantee a
global optimal solution only by a certain experiment. Here, the number of experiments
is set to 25. After finishing many times test, the best global solution are
selected, which is shown in Table 2. To replace the parameters
in Eq. 13 with a in Table 1, the following
Marr wavelet filter transfer function can be obtained:
Table 1: 
Optimum a for Marr wavelet approximation 

Table 2: 
Optimum a for Morlet wavelet approximation 


Fig. 1: 
Search process of PSO for approximating Marr wavelet 

Fig. 2: 
Approximation of the Marr wavelet 
where, H (s) is the wavelet filter to realize Wt_{xA} (1, τ) under
scale 1. By the theory of Laplace transform, the transfer function of analog
wavelet filter under certain scale a is expressed as .
The imedomain waveform of approximated Marr wavelet in Fig. 2
it inherits the excellent qualities of Mar wavelet.
Now, we consider approximation of the Morlet wavelet. Morlet wavelet Ψ
(t) is defined:
Choosing the timeshift t_{0} = 3, this gives rise to the following
timereversed and timeshifted wavelet function:
To obtain a stable 10th order approximation of
(t), the L_{2} approximation technique was applied using the parameterized
class of functions h (t) given by Eq. (7):
where, the parameters a_{2}, a_{4} and a_{8} must be
strictly negative for reasons of stability. Given the explicit form of
(t) and the parameterized class of functions h (t), the squared L_{2}
norm of the difference between
(t) and h (t) is expressed as:
where, a = (a_{1}, a_{2}, a_{3}, …,
a_{10})^{T} is parameter vector. The sum of squares error of
nonlinear functions is expressed by E (a):
where, the sampling time interval equal 0.01 (ΔT = 0.01). The optimization
problem is described as:
From:
this yields the constrained condition:
Using the Particle Swarm Optimization (PSO) algorithm to solve Eq.
26, the parameters for carrying out PSO are: Population size I = 80, Inertia
weight factor w_{min} = 0.4, w_{max} = 0.9, acceleration constants
c_{1} = c_{2} = 2, maximum iteration N = 6000. The optimum parameter
a = (a_{1}, a_{2}, a_{3}, …,
a_{10})^{T} was obtained in Table 2.
The following filter was obtained:
The corresponding wavelet approximation h (t) is shown in Fig.
3, yielding an L_{2} approximation error equal to 0.0015.
APPROXIMATION PERFORMANCE COMPARISON
To determine the quality of the wavelet approximations obtained with the three
kinds of techniques: pade approximation (Haddad et al.,
2005), L_{2} approximation (Karel et al.,
2005) and L_{2} approximation based on PSO, one may resort to the
computation of their L_{2} approximation errors. L_{2} approximation
errors have been calculated for three different wavelet approximations: the
approximation of Gaussian wavelet, the approximation of Marr wavelet and the
approximation of Morlet wavelet.
Table 3: 
Comparison of approximation performance 


Fig. 3: 
Approximation of the Morlet wavelet 
The calculated results are listed in Table 3. From the results
in Table 3, the approximation errors using L_{2} approximation
based on PSO are the least. One major advantage of the L_{2} approximation
based on PSO is that it can approximate excellently various wavelet functions.
In the pade approximation (Haddad et al., 2005),
this improper selection of the point s_{0} = 0 resulted in an unstable
transfer function of Morlet wavelet. With a longer running time, L_{2}
approximation (Karel et al., 2005) algorithm
is capable of finding the global optimal parameters for low order wavelets.
However, this method has some convergence problems when one tries to approximate
a function with many oscillations (high order wavelet), such as the Morlet wavelet.
From the comparison, it is obvious that the L_{2} approximation based
on PSO method is superior to Pade approximation and L_{2} approximation.
CONCLUSION
For implementing wavelet transforms in analog circuits, a novel method to approximate
various wavelet functions is proposed. To approximate a wavelet with this method,
an optimization mathematical model based on the L_{2} approximation
must be given. Then, the Particle Swarm Optimization (PSO) algorithm is used
to solve the optimization problem. Because the PSO algorithm is one of the most
powerful methods for solving global optimization problems and is effective,
efficient and fairly robust to initial conditions, the proposed method overcomes
these shortcomings of approximation technique in Haddad
et al. (2005) and Karel et al. (2005).
With proposed approach it can approximate wavelets that can not be approximated
sufficiently well with an acceptable order in a straightforward way with the
Pade approach or L_{2} approximation.
ACKNOWLEDGMENTS
The authors thank the Natural Science Foundation of Hunan Province of China
(Grant No. 11JJ3078), China Postdoctoral Science Foundation (Grant No. 2012M521215)
and National Natural Science Funds of China for Distinguished Young Scholar
under (Grant No. 50925727).