As

Ma

b

a

ARA

KASWSM

1

pabpdTmtHpaac

taEsuo

HT

1h

Int. J. Electron. Commun. (AEÜ) 68 (2014) 388–394

Contents lists available at ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

j ourna l h omepage: www.elsev ier .com/ locate /aeue

nalog wavelet transform using multiple-loop feedbackwitched-current filters and simulated annealing algorithms

u Lia,b,∗, Yigang Heb

College of Information and Electrical Engineering, Hunan University of Science and Technology, Xiangtan 411201, ChinaCollege of Electrical and Information Engineering, Hunan University, Changsha 410082, China

r t i c l e i n f o

rticle history:eceived 28 November 2012ccepted 1 November 2013

a b s t r a c t

A new approach for implementing continuous wavelet transform (CWT) based on multiple-loop feed-back (MLF) switched-current (SI) filters and simulated annealing algorithms (SAA) is presented. First,the approximation function of wavelet bases is performed by employing SAA. This approach allows for

eywords:nalog circuitswitched-current filtersavelet transform

imulated annealing algorithmsultiple-loop feedback

the circuit implementation of any other wavelets. Then the wavelet filter whose impulse response isthe wavelet approximation function is designed using MLF architectures, which is constructed with SIdifferentiators and multi-output cascade current source circuits. Finally, the CWT is implemented bycontrolling the clock frequency of wavelet filter banks. Simulation results of the proposed circuits andthe filter banks show the advantages of such new designs.

© 2013 Elsevier GmbH. All rights reserved.

. Introduction

The continuous wavelet transform (CWT) is a widely used signalrocessing technique, particularly for local analysis of nonstation-ry and fast transient signals [1–3]. In wearable and implantableiomedical devices, such as wearable detector and pacemakers,ower consumption is a critical issue due to the limited energyensity and the lifetime of currently available portable batteries.his problem implies that the design of such devices has to be opti-ized for low-power dissipation. Traditionally, systems employing

he CWT are implemented using digital signal processing devices.owever, it is not suitable to implement the CWT due to the highower consumption and real-time associated with the requirednalog-to-digital (A/D) converters. From the power consumptionnd real-time perspectives, an excellent alternative is to use analogircuits and to implement the CWT instead.

In the past decades, there have been significant advances inhe analog implementations of the CWT using continuous-timend sample-data circuits and their practical applications [4–21].specially, the analog sample-data circuits for the design of CWT

ystem have caused much attention [14–21]. The main reason ofsing sample-data circuits for implementing CWT is that dilationsf a given filter may be easily and very precisely controlled by

∗ Corresponding author at: College of Information and Electrical Engineering,unan University of Science and Technology, Xiangtan 411201, China.el.: +86 13787426799; fax: +86 731 58290114.

E-mail addresses: limuucn@163.com (M. Li), yghe@hnu.edu.cn (Y. He).

434-8411/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.aeue.2013.11.002

both the component parameter ratios and the clock frequency. In[14,15], the switched-capacitors (SC) circuits are used to implementCWT in an analog way. SC circuits are suited for this applicationsince the dilation constant across different scales of the transformcan be implemented and controlled by both the capacitor ratiosand the clock frequency. However, the reduced supply voltage ofadvanced CMOS technologies has imposed new challenges on thedesign of SC circuits especially for high-speed applications. Fur-thermore, SC is not fully compatible with current trends in digitalCMOS process, because they require good linear floating capacitors.Consequently, the design approach for implementing the CWT bymeans of switched-current (SI) circuits is developed [16–21]. In SImethods of CWT realization, the circuits consist of analog SI fil-ters whose impulse response is the approximated wavelet. So theperformance of the implementation of the analog CWT dependslargely on the accuracy of the approximation. Firstly, Padé approx-imation [7,16] is used to approximate the Laplace transform ofthe desired filter transfer function by a suitable rational function.The main advantage of the Padé method is its computational sim-plicity and its general applicability. However, there are also somedisadvantages limiting its practical applicability. Among others,one important issue is stability. The stable transfer function of awavelet filter does not automatically result from this approach.Another drawback is that the choice of the degrees for the numera-tor and denominator polynomials may yield an inconsistent system

of equations. Subsequently, more accurate L2 approach has beenreported [4,6,18–20,22]. The advantage of the L2 method over Padéapproximation is that L2 approximation offers a more accurateapproximation. Also, the approximation can be performed directly

ommu

ieptfpofi[ipisSHcfi

fitisaiTeccn

h3daSt

2

pt

W

wwit

(

FCl

h

Fobatp

M. Li, Y. He / Int. J. Electron. C

n time domain. A major drawback of this approach is the possiblexistence of local optima. Moreover, the choice of a good startingoint becomes very important and difficult. Recently, a differen-ial evolution algorithm has been used to calculate the transferunction of the filter [17,21]. However, in this algorithm settingroper parameters is difficult and takes longer operation time. Thether key issue of CWT circuit realization is the design of waveletlter topology. In general, there are cascade [16,21] and parallel17] architectures for implementing the wavelet filter based on SIntegrators. However, these structures have high sensitivity [23], inarticular as filter order increases. One of the methods for obtain-

ng low sensitivity in the filter design is to use MLF networks. A MLFtructure for implementing CWT with two-input multiple-outputI bilinear integrator as building blocks has been proposed in [20].owever, the feedforward and feedback coefficients in the circuitannot be directly determined by the transfer function. Moreover,or the proposed circuit, it is difficult to adjust the coefficient valuesn order to design new filters.

In this paper, a new CWT circuit is presented using MLF SIlter and SAA method. In order to obtain the transfer func-ion of the wavelet filters, the wavelet base is approximatedn time domain employing SAA method, which overcomes thehortcomings of Padé, L2 approach and other complex intelligentlgorithms. The CWT circuit consists of a wavelet filter bank whosempulse response is the approximation wavelet and its dilations.he wavelet filter design is based on a MLF structure with SI differ-ntiators and cascade current source. The coefficients of the filteran be directly obtained by the transfer function. In addition, byhanging the output currents of the cascade current source, theew filter is easy to be designed.

This paper is structured as follows. In Section 2, it is arguedow the CWT can be implemented with analog filters. In Section, the SAA method is used to compute the transfer function whichescribes a certain wavelet base that can be implemented as annalog filter. Section 4 describes the complete filter design issues.ome results obtained by simulations are given in Section 5. Finally,he conclusions are presented in Section 6.

. Principle of wavelet transform design

The CWT is a linear operation that decomposes a signal into com-onents that appear at different scales. The CWT of a continuousime signal f(t) at scale a and position � is defined as

(�, a) = 1√a

∫ ∞

−∞f (t) ∗

(t − �

a

)dt (1)

here *(t) is the complex conjugation of the given admissibleavelet function, called the wavelet base (t). When the signal f(t)

s passed through a linear time-invariant filter, the filter output ishe convolution of f(t) with the impulse response h(t) of the filter:

f ∗ h)(�) =∫ ∞

−∞f (t)h(� − t)dt (2)

rom (1) and (2), it is well known that analog computation of theWT W (�, a) can be achieved through the implementation of a

inear filter with impulse response

(t) = 1√a

(−ta

)(3)

or obvious physical reasons only the hardware implementationf causal stable filters is feasible. However, for a given wavelet

ase (t), the transfer function H(s) will usually be non-rationalnd non-causal. Therefore, we consider approximations of H(s) byransfer function that have all their poles in the complex left halflane and which are strictly proper rational. Note that h(t) will be

n. (AEÜ) 68 (2014) 388–394 389

zero for negative t, so that the time-reversed wavelet base (− t)which does not have this property must be time-shifted to facili-tate an accurate approximation of its CWT. In the next section, wewill discuss the wavelet approximation that is suitable for analogimplementation.

3. Wavelet function approximation in time domain

3.1. The model of wavelet function approximation

The first stage in analog filter design is the definition of therespective transfer function. However, a linear differential equa-tion having a desired impulse response does not always exist. Thus,we must employ a suitable approximation method. From linearsystems’ theory, it is known that any strictly causal LTI filter offinite order n can be represented as a state-space system (A, B, C,D), corresponding to a system of associated first-order differentialequations:

x(t) = Ax(t) + Bu(t) (4)

y(t) = Cx(t) + Du(t) (5)

where x(t) is the state vector, u(t) is the input signal and y(t) is theoutput of the filter. The matrix D is set to zero to achieve causality.The associated impulse response h(t) and its Laplace transform H(s)of the system are given by

h(t) = CeAtB (6)

H(s) = C(sI − A)−1B (7)

For the generic situation of stable systems with distinct poles, theimpulse response function h(t) is a linear combination of dampedexponentials and exponentially damped harmonics. The impulseresponse function h(t) of N order filter may typically have the fol-lowing form [4,17,21]

h(t) =m∑i=1

aiebit +

n∑j

cjedjt cos(�jt) + fje

gjt sin(�jt) (8)

where the parameters ai, bi, cj, dj, �j, fj and gj are real numbers.m and n correspond to the number of real poles, and m + n = N. Forinstance, a 5th order approximation function may be described as

h(t) = r1er2t + r3e

r4t cos(r5t) + r6er4t sin(r5t) + r7e

r8t cos(r9t)

+r10er8t sin(r9t) (9)

where the parameters r2, r4 and r8 must be strictly negative forreasons of stability.

As an example, the first derivative of Gaussian approximationis considered in the following. The first derivative of a Gaussianwavelet is expressed as

(t) = −2te−t2

(10)

As stated before, if h(t) is used to approximate a time-shifted andtime-reversed wavelet function (t0 − t), the output of the linearfilter is the approximate wavelet transform W (�, a). The selectionof the time-shift t0 is an important process. If t0 is chosen too small,the truncation error will be too large and the overall approximationperformance will decrease. If t0 is chosen too large, the function tobe approximated will become very flat near t = 0. This effectivelyinduces a large time-delay. Fig. 1 shows the waveforms of differ-ent time-shifting in time-reversed wavelet function. In [17,21], theselection problem of time-shift t0 is not discussed and the value

of time-shift is directly provided. For a selected time-shift t0, thecomputation of an accurate approximation can be achieved usingvarious approaches. Any approximation method should be asso-ciated with some measure of error. Therefore, we define an error

390 M. Li, Y. He / Int. J. Electron. Commun. (AEÜ) 68 (2014) 388–394

-3 -2 -1 0 1 2 3 4 5 6 7 8-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1st diff Gauss ian(1-t)(2-t)(3-t)(4-t)

Amplitude

ca

M

Ihd⎧⎪⎨⎪⎩wtlotoHit

3

rppla

1

Table 2The optimal coefficients of the Gaussian wavelet approximation function using SAA(N = 5, t0 = 2).

Coefficients Values Coefficients Values

r1 0.168253 r6 0.268011r2 −0.084074 r7 1.436899r −0.819217 r −0.422216

TA

Time (s)

Fig. 1. The waveforms of different time-shifting in time-reversed wavelet.

riterion based on the mean-square error (MSE) which is defineds

SE =∫ ∞

0

∣∣h(t) − (t0 − t)∣∣2dt (11)

n order to obtain the optimal parameters of the approximation(t), the discrete optimization model for h(t) in the time domain isescribed as

min

{e(r) =

M∑n=0

(h(n �t) − (t0 − n �t))2

}

s.t. rk < 0, (k = 2, 4, 8)

(12)

here e(r) is sum of squares error, r = (r1, r2, r3, . . ., r10)T, �t denoteshe sampling time (�t = 0.01) and M is the total number of samp-ing points. From (12) it is known that this is a typical nonlinearptimization problem with nonlinear constraints. It is very difficulto obtain an accurate optimal solution using traditional numericalptimization techniques in this nonlinear optimization problem.ence, a global intelligent optimization algorithm is used for min-

mizing the squared of the error function under the constraint inhe next section.

.2. Computation of wavelet approximation using SAA

Simulated annealing (SA) [24,25] is one of the optimal algo-ithms which attempts to solve hard combinational optimizationroblems through a controlled randomization. It is one of the mostowerful and popular heuristics to solve many optimization prob-

ems for its ease of use and capabilities. The basic steps of SAlgorithms for a minimization problem are the following:

. Initialize the SA control parameter: initial temperature T0, num-ber of repetitions L and an initial random solution S0. Set T = T0,current solution S = S0, best solution Sbest = S0, repetition countern = 1. Calculate object function value e(S0).

able 1pproximation performance (MSE) of various orders of systems and various time-shifts u

Order Shift 2.0 Shift 2.5

5 7.7841e−4 1.0368e−3

6 2.9929e−4 8.4681e−4

7 2.3716e−4 6.6049e−4

8 1.7161e−4 4.4521e−4

3 8

r4 −0.441334 r9 −1.413499r5 −2.726476 r10 −0.603626

2. Generate new solution Sn in the neighborhood of S0 and calculate�e = e(Sn) − e(S0).

3. If �e ≤ 0, the generated solution replaces the current solutionS = Sn. Otherwise, the solution is accepted with a criterion prob-ability, i.e., generate a random number q ∈ (0, 1), if q < p = e−�e/T,S = Sn and n = n + 1.

4. If e(S) < e(Sbest), Sbest = Sn.5. Repeat steps 2–4 for L cycles.6. Reduce the temperature T, The temperature-reducing expres-

sion is described as Ti+1 = ˇTi, in which < 1 and constant.7. Repeat steps 2–6 till the stop criterion is met.

The most important feature of SAA is that the performanceof SAA depends on the definition of several control parameters.Obviously, each of these control parameters is chosen according tothe specific problem. In this work, the initial solution is generatedrandomly. The minimization problem between the approximationfunction h(t) and the time-shifted and time-reversed wavelet func-tion (t0 − t) included 2N parameter values. The vector R = [r1, r2,r3, . . ., r10] shows an alternative solution. Setting T = 100, L = 100,

= 0.95 and M = 600. According to the above steps and the givenparameters of SAA, the search for the optimization problem in (12)can be solved. The approximation performance (MSE) of variousorders of systems and various time-shifts using SAA are illustratedin Table 1. Note that Table 1 shows that the approximation errorof high order systems (e.g., order 8) is smaller than the approxi-mation error of low order systems (e.g., order 5). However, highorder systems are more complex than low order systems and theenergy consumption of the circuit increases with the order of thefilter. Moreover, we can see that for the time-shift t0 = 2 the approx-imation is better than for other time-shifts. For a comprehensiveconsideration of the performance on the approximation and thecomplexity in the design, we select a 5th order filter and time-shift t0 = 2 for implementing analog CWT. Using SAA in the previousstates, we can obtain the coefficients of the first derivative of theGaussian wavelet approximation function (i.e., (9)). The situationis given in Table 2. The 5th order transfer function of the Gaussianwavelet filter (scale a = 1) can be obtained:

H(s) = −0.1674s4 + 0.0218s3 − 2.2983s2 − 11.2535s + 1.7574s5 + 1.8112s4 + 10.6953s3 + 9.2496s2 + 17.3045s + 1.3958

(13)

The other scale transfer functions can be derived from (13) usingtheory of Laplace transforms. In Fig. 2, the Gaussian wavelet hasbeen approximated using different approximation methods in timedomain. The MSE results related to wavelet approximations are

sing SAA.

Shift 3.0 Shift 3.5 Shift 4.0

1.3616e−3 5.0837e−3 3.6120e−31.1560e−3 4.2903e−3 2.5908e−31.1089e−3 1.1223e−3 1.2299e−35.3361e−4 6.8854e−4 7.3170e−4

M. Li, Y. He / Int. J. Electron. Commun. (AEÜ) 68 (2014) 388–394 391

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.81st diff GaussianPade approximationL2 approximationSAA approximation

Time (s)

Am

plitu

de

Fig. 2. Approximation of the first derivative of Gaussian.

Table 3The performance comparison of the 5th order Gaussian wavelet function with dif-ferent approximation approach.

Methods Mean-square error (MSE)

Padé 2.3426e−3

siG

r

4

iwAtT

a

H

Tgmtd

2

J

1

J J

1ii oi

a0 = −0.2034; a1 = 2.0136; a2 = 3.1877;a3 = 3.0657; a4 = 1.4813; a5 = 0.3862;b0 = −0.2562; b1 = 2.2806; b2 = 3.5860;

L2 1.0758e−3SAA 7.7841e−4

hown in Table 3. As seen from the MSE comparison, the approx-mation performance of SAA is better than other methods for theaussian wavelet filter of the same order.

In the next section, we will design the SI filter whose impulseesponse is the approximated wavelet and its dilations.

. Switched-current analog wavelet filter design

The wavelet filter performances not only depend on the approx-mation of the wavelet base but also on the filter structure. Most

avelet filter structures used are cascade and parallel topologies.s it is well known, the common shortcoming of these topologies is

hat the sensitivity is higher, in particular as filter order increases.herefore, MLF structure is developed for obtaining low sensitivity.

A z-domain general transfer function implemented with z−1 − 1s the basic elements can be written as [26,27]

(z) = −b0 +∑N

n=1bn(z−1 − 1)n

a0 −∑N

n=1an(z−1 − 1)n(14)

he network function of (14) can be realized by the signal flowraph (SFG) shown in Fig. 3. This SFG consists of adders, coefficient

ultipliers, and the differentiator type elements, z−1 − 1. These

hree basic building blocks implemented with SI technique will beescribed in the following.

0a

1ana 1na

1 1z

ini

oi1 1z 1 1z

1nbnb 1b 0b

Fig. 3. SFG of the MLF filter.

1 1: :

Fig. 4. SI differentiator circuit.

First, a simple SI differentiator is shown in Fig. 4. The transferfunction of this SI differentiator is

H(z) = ˛(z−1 − 1) (15)

This function is obtained by backward Euler mapping[s → (1 − z−1)/T].

Second, the coefficients an and bn in (14) can be realized bymulti-output cascade current source whose circuit is shown inFig. 5. Positive and negative coefficients can be gained by duplicat-ing the output stage with the required mk and nk output weightedtransistors.

Third, the addition operation can be accomplished by connect-ing all the input currents since signals are represented by currents.So the designing circuit according to the SFG in Fig. 3 does not needaddition current adders.

In order to demonstrate the design methodology, the 5thtransfer function of the Gaussian wavelet filter is consideredas an illustrating example. By applying the bilinear transform(s → 2(1 − z−1)/T(1 + z−1)) and the mathematical technique to thetransfer function in (13), it can be transformed into the desiredform as

H(z) = [0.05565(z−1 − 1)5 + 0.62136(z−1 − 1)

4

+2.27492(z−1 − 1)3 + 3.58592(z−1 − 1)

2

+2.28065(z−1 − 1) − (−0.25615)]/[−0.38624(z−1 − 1)5

−1.48131(z−1 − 1)4 − 3.06567(z−1 − 1)

3

−3.18767(z−1 − 1)2 − 2.01362(z−1 − 1) + (−0.20345)]

(16)

Matching the coefficients of (14) to those of (16), we can obtainthe coefficients an and bn as follows:

Fig. 5. Multi-output cascade current source circuit.

392 M. Li, Y. He / Int. J. Electron. Commun. (AEÜ) 68 (2014) 388–394

the SI differentiator as a basic element.

ca

5

aalGcrtfaf“vi

fiuS“gSI

TT

Fig. 7. Ideal frequency response of Gaussian wavelet filter (a = 1).

Fig. 6. Gaussian wavelet filter with

b3 = 2.2750; b4 = 0.6214; b5 = 0.05565.

Based on the above principles and analysis, the Gaussian waveletircuit with the SI differentiator as a basic element can be realizeds shown in Fig. 6.

. Simulation results

In order to verify the feasibility of the proposed method in thebove sections, the wavelet filter has been simulated using Matlabnd ASIZ software. According to the coefficients an and bn calcu-ated in Section 4, we will set the transistor aspect ratios (W/L) ofaussian wavelet circuit. The transistor aspect ratios of the cascadeurrent source circuit are listed in Table 4. Other unlisted aspectatios of the transistors in the circuit are all set as 1. Meanwhile, sethe resistor R = 1 � and the current source Is = 1 A and the samplingrequency 10 Hz. The ideal frequency response of the wavelet filtert a = 1 is shown in Fig. 7. The peak value 4.674 dB is achieved at

0 = 0.221 Hz. Fig. 8 shows the simulated frequency response (curvea”) of the wavelet filter obtained using ASIZ software. The peakalue is achieved at f0 = 0.226 Hz, which is in agreement with thedeal value.

Next, to confirm the low sensitivity of the proposed SI waveletlter, we used ASIZ software to analyze the sensitivity. Consideringncorrelated 5% random errors in all the transconductances of theI circuit, the error margins are plotted as the dotted lines (curvesb”) above and below the nominal gain curve in Fig. 8. The error

ain curves show the lower sensitivity of the proposed filter withI differentiators, due to the MLF structure and the differentiators.n addition, the poles and zeros of the Gaussian filter are also shown

able 4he transistor aspect ratios (W/L) of the multi-output cascade current source circuit.

Coefficients W/L (transistor) Coefficients W/L (transistor)

a0 0.2034 (N0) b0 0.2562 (N1)a1 2.0136 (M1) b1 2.2806 (M6)a2 3.1877 (M2) b2 3.5860 (M7)a3 3.0657 (M3) b3 2.2750 (M8)a4 1.4813 (M4) b4 0.6214 (M9)a5 0.3862 (M5) b5 0.0557 (M10)

Fig. 8. Simulated frequency response and the pole-zero plot of Gaussian waveletfilter (a = 1).

M. Li, Y. He / Int. J. Electron. Commun. (AEÜ) 68 (2014) 388–394 393

im

i1itftwcoFpiwtnFdsrtwdm

voltage, low power signal processing.

Fig. 9. Simulated impulse response of Gaussian wavelet filter (a = 1).

n Fig. 8. All the poles are contained within the unit circle, whicheans that the designed system is stable.The impulse response of the wavelet filter at a = 1 is shown

n Fig. 9. The negative and positive peak values are achieved at.5 s and 3.5 s, respectively, which is slightly different from the

deal value 1.28 s and 2.79 s. In order to implement a waveletransform, we need to be able to scale and shift in time Gaussianunction. By controlling the various sampling clock frequencies ofhe circuit with the same system architecture, the different scaleavelet can be obtained. Adjusting the sampling clock frequen-

ies to 5 Hz, 2.5 Hz and 1.25 Hz, respectively, the impulse responsesf the wavelet filter with four scales (a = 2,4,8) can be seen inig. 10. The impulse responses of different scale filters achieve theositive peak values at 0.7 s, 1.4 s and 2.8 s, respectively. Fig. 11

llustrates the frequency responses of the wavelet filters associatedith 3 dyadic scales with center frequencies ranging from 0.11 Hz

o 0.0275 Hz, respectively. In fact, the transfer function (13) can beormalized to any desired center frequency for some applications.ig. 12 shows the frequency responses of the wavelet filters usingifferent methods. As seen from the frequency response compari-on, the performance of SAA is better than the other methods. Theeason is that the wavelet function approximation performance ofhe SAA method is better than Padé and L2 method for the Gaussianavelet filter of the same order. Obviously, the performance of the

esigned circuit with the same system architecture using the SAAethod can be better than the other methods.

Fig. 10. The impulse response by changing the sampling frequency.

Fig. 11. The frequency response for 3 dyadic scales.

In practice, the transistor is a non-ideal component, whichaffects the performance of the circuit. Therefore, another simula-tion is made, with the circuit simulated in the basic form, withoutenhancement circuits in order to test the effect of finite Gm/Gdsratio in the transistors and parasitic Cgd capacitances. The circuit issimulated in the basic form, assuming Gm/Gds and Cgs/Cgd ratios of1000, with the biasing and signal current sources assumed as ideal.Fig. 13 shows the frequency response of the wavelet filter withparasitic elements considered by different methods. The result bySAA method is curve “c”. Obviously, the designed wavelet filter bySAA method has little effect in the sensitivity to errors and imper-fections. For the practice critical filter, some additional circuits forreducing these effects are definitely necessary.

In order to obtain the power consumption of the proposed CWTSI circuit, the circuit has been simulated by using 0.35 �m CMOSprocess technology with 1.2 V supply voltage in Hspice software.The power consumption of this circuit is only 58 nW per scale,which readily satisfies the low power requirement of the CWTcircuit.

Overall, it is clear that the proposed method here has excel-lent performance for implementing CWT and can be used for low

Fig. 12. The frequency responses of wavelet filters using different methods (a = 1).

394 M. Li, Y. He / Int. J. Electron. Commu

Fm

6

matw(wfbCmmiaueatc

A

F5GFtP2UFXN

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

author of a great number of papers on his research results. His teaching and research

ig. 13. The frequency responses with parasitic elements considered by differentethods (a = 1).

. Conclusions

This paper has presented a low power MLF SI filter for imple-enting the CWT in the analog domain. The wavelet base is

pproximated using universal approximation model based on theheory of linear system. The SAA approach for approximating theavelet performs markedly better than earlier common methods

e.g., Padé and L2), taking the Gaussian wavelet as an example. Theavelet filter has been designed based on MLF structure with SI dif-

erentiators and multi-output cascade current source as the basiclocks. Simulation results confirm that the various scales of theWT can be implemented in an analog circuit. Comparing reportedethods in the literature for implementing the CWT, the proposedethod in this article has the following advantages: (i) the approx-

mation methods of the wavelet function based on SAA have thedvantages at algorithm complexity, approximation accuracy andniversality. (ii) Wavelet filter with MLF structure using SI differ-ntiator building blocks has low sensitivity compared with cascadend parallel structure filter. (iii) The feedback coefficients of the fil-er are easy to be realized and adjusted using multi-output cascadeurrent source.

cknowledgments

This work was supported by the National Natural Scienceunds of China for Distinguished Young Scholar under Grant No.0925727, National Natural Science Foundation of China underrant No. 60876022, Hunan Provincial Science and Technologyoundation of China under Grant No. 2010J4, the Coopera-ion Project in Industry, Education and Research of Guangdongrovince and Ministry of Education of China under Grant No.009B090300196, the Fundamental Research Funds for the Centralniversities, Hunan University, the project supported by Scientificund of Hunan Provincial Education Department No. 10C0672, andiangtan Science and Technology Foundation of China under Granto. CG20121006.

eferences

[1] Haddad SAP, Serdijn WA. Ultra low-power biomedical signal processing: ananalog wavelet filter approach for pacemakers. Springer; 2009.

[2] Martin V, Cormac H. Wavelets and filter banks: theory and design. IEEE Trans-actions on Signal Processing 1992;40(9):2207–31.

[3] Mallat S. A wavelet tour of signal processing. New York: Academic; 1999.

n. (AEÜ) 68 (2014) 388–394

[4] Karel JMH, Haddad SAP, Hiseni S, Westra RL, Serdijn WA, Peeters RLM. Imple-menting wavelets in continuous-time analog circuits with dynamic rangeoptimization. IEEE Transactions on Circuits and Systems 2012;59(1):2023–32.

[5] Gurrola-Navarro MA, Espinosa-Flores-Verdad G. Analogue wavelet transformwith single biquad stage per scale. Electronics Letters 2010;46(9):616–8.

[6] Agostinho PR, Haddad SAP, Lima DEJA, Serdijn WA. An ultra low power CMOSPA/V transconductor and its application to wavelet filters. Analog IntegratedCircuits and Signal Processing 2008;57(1–2):1–14.

[7] Haddad SAP, Sumit B, Serdijn WA. Log-domain wavelet bases. IEEE Transactionson Circuits and Systems 2005;52(10):2023–32.

[8] Li H, He Y, Sun YC. Detection of cardiac signal characteristic point using log-domain wavelet transform circuits. Circuits, Systems and Signal Processing2008;27(5):683–98.

[9] Akansu AN, Serdijn WA, Selesnick W. Emerging applications of wavelets: areview. Physical Communication 2010;3(1):1–18.

10] Zhao W, Sun Y, HE Y. Minimum component high frequency Gm-C wavelet fil-ters based on Maclaurin series and multiple loop feedback. Electronic Letters2010;46(1):34–6.

11] Casson AJ, Rodriguez-Villegas E. A 60pW gm-C continuous wavelet trans-form circuit for portable EEG systems. IEEE Journal of Solid-State Circuits2011;46(6):1406–15.

12] Karel JMH, Peeters RLM, Westra RL, Haddad SAP, Serdijn WA. Wavelet approx-imation for implementation in dynamic translinear circuits. In: Proceedings ofthe 16th IFAC World Congress, IFAC. 2005. p. 1101–6.

13] Huang QX, He YG. Low-voltage/low-power instantaneous companding circuitsfor continuous wavelet transform. In: Proceedings of the IEEE InternationalConference on Neural Networks and Signal Processing, ICNNSP. 2003. p. 331–6.

14] Edwards RT, Cauwenberghs G. A VLSI implementation of the continuouswavelet transform. In: Proceedings of IEEE International Symposium on Circuitsand Systems, ISCAS. 1996. p. 368–71.

15] Lin J, Ki WH, Edwards T, Shamma S. Analog VLSI implementations of audi-tory wavelet transforms using switched-capacitor circuits. IEEE Transactionson Circuits and Systems 1994;41(9):572–83.

16] Hu QC, He YG, Liu MR. Design and implementation of wavelet filter usingswitched-current circuits. In: Proceedings of the IEEE TENCON. 2005. p. 1–6.

17] Li M, He YG. Analogue implementation of wavelet transform using discrete timeswitched-current filters. In: Proceeding of International Conference on Electricand Electronics, EEIC. 2011. p. 677–82.

18] Zhao WS, He YG, Sun Y. SFG realization of wavelet filter using switched-currentcircuits. In: Proceedings of the IEEE ASICON. 2009. p. 37–40.

19] Zhao WS, He YG, Sun Y. Design of switched-current wavelet filters using signalflow graph. In: Proceedings of the IEEE MWSCAS. 2010. p. 1129–32.

20] Zhao WS, He YG. Realization of wavelet transform using switched-current fil-ters. Analog Integrated Circuits and Signal Processing 2012;71(3):571–81.

21] Li M, He YG. Analog VLSI implementation of wavelet transform usingswitched-current circuits. Analog Integrated Circuits and Signal Processing2012;71(2):283–91.

22] Karel JMH, Peeters RLM, Westra RL, Haddad SAP, Serdijn WA. An L2-basedapproach for wavelet approximation. In: Proceedings of the 44th IEEE Confer-ence on Decision and Control, and the European Control Conference, CDC-ECC.2005. p. 7882–7.

23] Schaumann R, Ghausi MS, Laker KR. Design of analog filters: passive, active RC,and switched capacitor. New Jersey: Prentice Hall; 1990.

24] Arkat J, Saidi M, Abbasi B. Applying simulated annealing to cellular man-ufacturing system design. International Journal of Advanced ManufacturingTechnology 2007;32(56):531–6.

25] Kirkpatrick S, Gelatt Jr CD, Vecchi MP. Optimization by simulated annealing.Science 1983;220(4598):671–80.

26] Yu TC, Wu CY, Chang SS. Realization of IIR/FIR and N-path filters using anovel switched-capacitor technique. IEEE Transactions on Circuits and Systems1990;37(1):91–106.

27] Liu SI, Chen CH, Tsao HW, Wu JS. Switched-current differentiator-based IIR andFIR filters. International Journal of Electronics 1991;71(1):81–91.

Mu Li received the M.Sc. degrees in circuit and system from Hunan University,Changsha, China, in 2007. He is currently working toward the Ph.D. degree at Col-lege of Electrical and Information Engineering, Hunan University. Since 2007, he hasbeen a Lecturer of Electrical Engineering with Hunan University of Science and Tech-nology. His current interests include the research of intelligent signal processing,switched current technology, analog filter design, and testing and fault diagnosis ofanalog and mixed-signal circuits.

Yigang He received the M.Sc. degree in Electrical Engineering from Hunan Uni-versity, Changsha, China, in 1992 and the Ph.D. degree in Electrical Engineeringfrom Xi’an Jiaotong University, Xi’an, China, in 1996. Since 1999, he has been a FullProfessor of electrical engineering with the College of Electrical and InformationEngineering, Hunan University. He was a Senior Visiting Scholar with the Universityof Hertfordshire, Hatfield, U.K., in 2002. He is currently the Director of the Insti-tute of Testing Technology for Circuits and Systems, Hunan University. He is the

interests are in the areas of circuit theory and its applications, testing and fault diag-nosis of analog and mixed-signal circuits, RFID, and intelligent signal processing. Dr.He has been on the Technical Program Committees of a number of internationalconferences.