Realization of wavelet transform using switched-current filters

Realization of wavelet transform using switched-current filters

Wenshan Zhao • Yigang He

Received: 3 November 2010 / Revised: 29 July 2011 / Accepted: 6 August 2011 / Published online: 19 August 2011

� Springer Science+Business Media, LLC 2011

Abstract A method for realizing wavelet transform (WT)

is presented, in which the WT is synthesized by a bank of

switched-current (SI) filters whose impulse responses are the

basic wavelet function and its dilations. SI circuits are well

suitable for this application since the dilation constant across

different scales of the transform can be precisely imple-

mented and controlled by the sampling frequency. In this

article, the wavelet base is approximated by a systematic

algorithm with all the involved approximation parameters

taken into account. Also, the SI filter employing the follow-

the-leader feedback (FLF) multiple-loop feedback (MLF)

structure is proposed to synthesize the approximation

function. The Gaussian wavelet is selected as an example to

illustrate the design procedure. Simulation results indicate

that the proposed method has the merits of high approxi-

mation accuracy, strong stability and low sensitivity.

Keywords Wavelet transform � Switched-current filter �Network function approximation �Multiple-loop feedback �Bilinear transform

1 Introduction

The wavelet transform (WT) has been proven to be an

effective tool in signal processing thanks to its multi-scale

analysis characteristics [1, 2]. Due to the heavy

computational burden, conventional implementations of

WTs using software are not suitable for real-time signal

processing. To resolve this problem, hardware implemen-

tations have been exploited over the past few years, mostly

focusing on digital circuit implementations [3–6]. Since

analogue circuits have the characteristics of low power

dissipation compared with digital circuits, analogue circuit

implementations of WT have been investigated to meet the

demand for low-power operation [7–17]. So far, there have

been many methods for the design of analogue WT system

in audio-frequency applications, among which the imple-

mentations using analogue sampled-data circuits have

attracted much attention [12–17]. This approach can realize

the WT at various scales by controlling the sampling fre-

quency precisely. Typically, Lin proposed a method to

implement auditory WTs employing switched-capacitor

(SC) circuits [12]. The presented scheme resolves the

problem of heavy computation when dealing with the

auditory WTs and has been applied to the cochlear fabri-

cation successfully. However, as VLSI feature sizes shrink

into deep sub-micron region, the limitations of SC tech-

nique degrade its attraction. For example, SC circuit is not

compatible with standard digital VLSI processing since the

need for linear floating capacitors leads to the extra fabri-

cation step, e.g. double polysilicon [18–20].

To alleviate the difficulties associated with such SC

designs, the implementation of WT using switched-current

(SI) filters have been proposed [13–17]. Unlike SCs, SI

circuits can be integrated in a standard digital VLSI process

(CMOS) and ideally suitable for mixed analogue/digital

ICs [20]. To date, the prevalent design procedure for SI

WT circuits mainly involves two steps: mathematically

approximate the wavelet bases with realizable rational

expressions and then implement them using SI bandpass

filters (i.e. wavelet filters). In [13], Pade transformation is

W. Zhao (&)

School of Electronics and Information Engineering,

Beijing Jiaotong University, Beijing 100044, China

e-mail: [email protected]

W. Zhao � Y. He

College of Electrical and Information Engineering,

Hunan University, Changsha 410082, Hunan, China

123

Analog Integr Circ Sig Process (2012) 71:571–581

DOI 10.1007/s10470-011-9743-1

utilized to approximate the wavelet base, and the cascade

architecture is employed to synthesize the approximation

function obtained. Although successful in many aspects,

this method still has some shortcomings as below: first, the

approximation accuracy of Pade transformation is rela-

tively low. Second, the stability of wavelet filter cannot be

guaranteed automatically by Pade approximation [21].

Finally, the magnitude sensitivity of cascade connection is

relatively high, especially as the filter order increases [22].

For overcoming the disadvantages with [13], several

approaches have been proposed. In [15], a systematic

algorithm is employed to guarantee the stability of

approximation system. However, since having not dis-

cussed the selection of parameter time shift, the proposed

method cannot achieve the optimal approximation. In fact,

as elaborated in this article, the selection of time shift

would strongly affect the approximation accuracy. Mean-

while, the parallel architecture in [15] still has the high

magnitude sensitivity. Instead of cascade/parallel archi-

tecture, [16, 17] use the signal flow graph (SFG) to syn-

thesize rational approximation functions. Although the

feasibility is verified, these two articles have not given the

canonical SI filter structure, explicit construct procedure

and straightforward design formulas.

Thus, with regard to all the aforementioned problems, this

article presents a method that approximates the wavelet

bases using an improved systematic algorithm, and then

employs the multiple-loop feedback (MLF) structure to

synthesize approximation functions. Based on [15], the

approximation algorithm proposed in this article takes all the

key parameters into account, which results in higher

approximation accuracy both in time domain and frequency

domain. In addition, the MLF approach has the merits of

simple structure and low magnitude sensitivity [22]. How-

ever, the existing MLF structures are mostly applicable to

the design of continuous-time filter. Under this background,

this article constructs a canonical follow-the-leader feed-

back (FLF) MLF SI filter structure by using the bilinear

transform, in which the SI bilinear mapping lossless inte-

grator is employed as the building block. The explicit design

formulas are also presented. And the Gaussian wavelet is

selected as the example to elaborate the design procedure.

2 Design procedure for analogue wavelet filter

The WT consists of expanding input signal f tð Þ over

wavelets which are constructed from the mother wavelet

w tð Þ by means of dilations a and translations b [1]:

WTf a; bð Þ ¼ 1ffiffiffi

ap

Z

1

�1

f tð Þw t � b

a

� �

dt ð1Þ

As can be seen from Eq. 1, the signal’s WT at scale a

can be realized simply by convolving input signal with a

filter whose impulse response is the dilated and time

reversed wavelet base wað�tÞ ¼ 1=ffiffiffi

ap

ð Þw �t=að Þ:Therefore, a WT can be implemented by designing of a

bank of filters whose impulse responses are the basic

wavelet function and its dilated versions. Figure 1

illustrates the wavelet filter bank with multiple scales in

parallel which can be used in computing the WT

coefficients in real-time.

SI circuits are well suited for this implementation since

the dilation constant across different scales of the transform

can be precisely implemented and controlled by the clock

frequency. Given a filter H sð Þ and its dilated version H asð Þ;the direct approach of controlling a in the SI circuit

implementation involves controlling the various clock

frequencies of the circuit with the same system architec-

ture. Such accuracy is in general unachievable using con-

ventional analogue designs. Therefore, this article

concentrates on the SI circuit implementation of basic

wavelet function filters.

Usually, WTs cannot be implemented exactly in ana-

logue circuits, since the wavelet bases do not have the

suitable expressions that can be synthesized directly by the

filter architectures. Hence, it is necessary to find a realiz-

able rational function to approximate the wavelet bases. As

of now, the popular design procedure for analogue imple-

mentation of WT can be summarized as: mathematically

approximate the wavelet bases with rational function and

then implement them using suitable circuit topologies

which act as bandpass filters (i.e. wavelet filters).

3 Construction principle of approximation network

As clarified in Sect. 2, the first step for wavelet filter design

is to determine the rational approximate function of

wavelet base. Meanwhile, for making the analogue wavelet

filter physically realizable, the desired approximation

Fig. 1 Diagram of wavelet transform system using filter bank

572 Analog Integr Circ Sig Process (2012) 71:571–581

123

network should be stable. Owing to having the capability of

building the bridge between Taylor series expansion and

rational expression, Pade transformation has been widely

used to achieve the best possible approximation [10, 13].

Nevertheless, Pade approximation is not convenient for the

design of WT circuits, since it should be operated in

Laplace domain. It is well known that the information for

wavelet is often provided in time domain [1]. Therefore,

the wavelet base should be transformed to Laplace domain

first so as to carry out Pade approximation. In many cases,

this particular feature would lead to special design proce-

dures, e.g. interpolation and envelope determination [10].

These extra steps would thus make the approximation

method complex and have the potential to degrade the

approximation accuracy. To resolve this problem, this

article introduces an approach to construct the approxi-

mation network by using the time-domain information

directly. Also, to achieve the optimal approximation, the

selections of all the involved parameters are discussed

during approximation procedure.

3.1 Systematic algorithm of approximation

Network theory [23] shows that a network function H sð Þwhich has only single real poles or pairs of conjugate poles

can be written as:

H sð Þ ¼ An�1sn�1 þ � � � þ A1s1 þ A0

sn þ Bn�1sn�1 þ � � � þ B1s1 þ B0

¼ K1

s� p1

þ K2

s� p2

þ � � � þ KN

s� pN¼X

N

j¼1

Kj

s� pjð2Þ

Taking the inverse Laplace transform of H sð Þ; the

impulse response h tð Þ is:

h tð Þ ¼X

N

j¼1

Kjepjt; t� 0 ð3Þ

If a desired network function g tð Þ is to be approximated

by the network described by Eq. 2, we must obtain 2N

parameters, i.e. Kj and pj. Here we sample g tð Þ to get 2N

scattered points in order to obtain the approximation

function, where t ¼ 0; T ; 2T ; 3T; . . .; 2N � 1ð ÞT :The 2N points can be expressed by:

g 0ð Þ ¼P

N

j¼1

Kj ¼P

N

j¼1

Kjk0j

g Tð Þ ¼P

N

j¼1

KjepjT ¼

P

N

j¼1

Kjk1j

..

.

g 2N � 1ð ÞT½ � ¼P

N

j¼1

Kje2N�1ð ÞpjT ¼

P

N

j¼1

Kjk2N�1j

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

ð4Þ

in which

kj � epjT ð5Þ

and g 0ð Þ; g Tð Þ; g 2Tð Þ; . . .; g 2N � 1ð ÞT½ � are the ordinates

of the data points.

From Eq. 4, we have:

B1g N � 1ð ÞT½ � þ � � � þBNg 0ð Þ ¼ �g NTð ÞB1g NTð Þþ � � � þBNg Tð Þ ¼ �g N þ 1ð ÞT½ �

..

.

B1g 2N � 2ð ÞT½ � þ � � � þBNg N� 1ð ÞT½ � ¼ �g 2N � 1ð ÞT½ �

9

>

>

>

=

>

>

>

;

ð6Þ

Then using B1 obtained from Eq. 6, we can determine

the zeros (i.e. k1;k2; . . .;kN) of the characteristic

polynomial as below:

kN þ B1kN�1 þ B2k

N�2 þ � � � þ BN ð7Þ

As shown in Eq. 5, pj is given by pj ¼ ln kj

� �

=T , for

j ¼ 1; 2; . . .;N:

Note that, in order to ensure the system is stable, the

computation procedure mentioned above should be repe-

ated by changing T if the real part of pj is positive.

Using the least-squares method, we can determine Kj by

substituting kj into Eq. 4. It is evident that the approxi-

mation network will be obtained by substituting Kj and pj

into Eqs. 2 and 3.

3.2 Selection of parameters for optimal approximation

Based on the method given in Sect. 3.1, we can obtain the

approximation network of the wavelet base w tð Þ: Accord-

ing to the construction principle, it is necessary to select the

suitable parameters, i.e. time shift t0; filter order N and

sampling period T, in order to achieve the optimal

approximation.

Physical hardware can only implement causal filters.

However, some wavelet bases such as Gaussian wavelet is

non-causal since it is not zero at t \ 0. For the purpose of

making wavelet causal, let:

w1 tð Þ ¼ w � t � t0ð Þ½ � ð8Þ

where t0 is the time shift.

Assuming w1 tð Þ is the impulse response of a filter, the

output of the filter is:

f tð Þ � w1 tð Þ ¼ f tð Þ � w � t � t0ð Þ½ � ¼WTf t � t0ð Þ ð9Þ

which suggests that there is a phase translation e�jxt0 ;

without change in amplitude, if the WT has a translation t0.

Obviously, decreasing t0 will truncate part of the wavelet

and make the approximation function not satisfying the

admissible condition. Increasing t0 will lead to a greater

translation. Also, large t0 may reduce the approximation

precision. Hence, a trade-off should be made.

Analog Integr Circ Sig Process (2012) 71:571–581 573

123

Normally, N should be made large in order to reduce the

error in the approximation procedure. However, the greater

the N, the larger the area of the WT circuit. In practice, the

wavelet filter which has a low order and reasonable

approximation accuracy is desired.

To get the 2N sampled data, the sampling period

T should be determined. As for the fixed N, the parameters

T and t0 are partly inter-related. For different t0, there exists

a different optimum Tm resulting in the lowest approxi-

mation error.

4 Design of SI wavelet filter

4.1 Filter architecture and synthesis

The wavelet filter’s performances not only rely on the

approximation accuracy of wavelet bases, but also strongly

on the filter structure. The general rational approximation

functions with arbitrary finite transmission zeros can be

defined as Eq. 2. In view of filter synthesis, many filter

architectures such as cascade, parallel and LC ladder

simulation can be utilized in analogue wavelet filters. For

instance, cascade and parallel structure have been

employed in the design of SI wavelet filter [13, 15], but

suffering from the effect of higher magnitude sensitivity.

Although LC ladder simulation method can alleviate this

problem, it cannot realize real zeros directly. It has been

demonstrated that the MLF approach does not have these

problems [22]. Therefore, to obtain a high-quality SI

wavelet filter, the FLF MLF configuration shown in Fig. 2

is used for the filter design.

The overall transfer function of the filter can be derived

as [22]:

H sð Þ ¼ Iout

Iin

¼c1

s1sn�1 þ � � � þ cn�1

s1s2...sn�1sþ cn

s1s2...sn

sn þ d1

s1sn�1 þ d2

s1s2sn�2 þ � � � þ dn�1

s1s2...sn�1sþ dn

s1s2...sn

ð10Þ

Using the coefficient matching between Eqs. 2 and 10,

the parameter values can be calculated as:

diQi

j¼1sj

¼ Bn�i; i ¼ 1; . . .; n

ciQi

j¼1sj

¼ An�i; i ¼ 1; . . .; nð11Þ

It should be noted that the SFG in Fig. 2 is represented

in s domain, and can only be used for continuous-time filter

synthesis. Since SI circuit belongs to the analogue

sampled-data signal processing technique, the s-domain

FLF topology as shown in Fig. 2 should be mapped into z

domain so as to construct the MLF SI filter.

Till now, several s–z mapping approaches have been

proposed to achieve the best performance, among which

the bilinear transform has been proven to have many

advantages. The bilinear time-discretization method has

the capability to maintain the magnitude and stability. This

feature is very suitable for the wavelet filter design, since

the approximation accuracy (magnitude) and stability of

the wavelet approximation network will be not degraded

during the s–z transform.

The bilinear transform can be defined as

s ¼ 2

Ts

z� 1ð Þzþ 1ð Þ ð12Þ

where Ts is the sampling period of the SI circuit.

On the basis of bilinear transform, Fig. 3 shows the FLF

MLF filter structure in z domain derived from Fig. 2.

Fig. 2 FLF MLF structure in s domain


123

4.2 SI building cells

To realize the integrators in Fig. 3, the bilinear mapping

lossless integrator based on the second-generation SI cir-

cuit [19] is employed in this article. Figure 4(a) shows the

two-input two-output (TITO) SI bilinear integrator, in

which the negative output is achieved by simply inverting

the output current of the integrator with a simple current

mirror circuit. Figure 4(b) is the simplified version of

Fig. 4(a).

The transfer functions of the bilinear integrator in z

domain and s domain can be expressed as Eqs. 13 and 14,

respectively.

H zð Þ ¼ azþ 1

z� 1ð13Þ

H sð Þ ¼ 1=ss ð14Þ

Using the coefficient matching, the parameter value in

Fig. 4 can be determined by substituting Eq. 12 into Eq. 14

as

a ¼ Ts=2s ð15Þ

To realize the feedforward and feedback coefficients as

shown in Fig. 3, we can easily extend the structure of the

TITO SI bilinear integrator by adding more current mirrors.

Figure 5 shows the structure of two-input multiple-output

(TIMO) SI bilinear integrator.

Using the TIMO SI bilinear integrator as building block,

the FLF SI filter described as Fig. 3 can be obtained as

illustrated in Fig. 6. The parameters ai; afi; aci can be used

to realize the time-constant of SI integrator, feedback

coefficients and feedforward coefficients, respectively. It is

worth noting that the parameters aci are considered to be

positive in Fig. 6. The negative aci can be realized simply

by the same way at the negative output of the TIMO SI

bilinear integrator.

From Eq. 15, the parameters ai can be determined as

Fig. 3 FLF MLF structure in z domain based on bilinear transform

(a)

(b)

Fig. 4 Two-input two-output SI bilinear integrator Fig. 5 Two-input multiple-output SI bilinear integrator


123

ai ¼ Ts=2si; i ¼ 1; 2; . . .; n� 1 ð16Þ

Then, afi and aci can be given as

afi ¼ diai ¼ di Ts=2sið Þ; i ¼ 1; 2; . . .; n ð17Þ

aci ¼ ciai ¼ ci Ts=2sið Þ; i ¼ 1; 2; . . .; n ð18Þ

Substituting Eq. 11 into Eqs. 17 and 18, we can obtain

the expressions for afi and aci:

afi ¼ TsBn�i

2

Q

i�1

j¼1

sj; i ¼ 1; . . .; n

aci ¼ TsAn�i

2

Q

i�1

j¼1

sj; i ¼ 1; . . .; n

ð19Þ

5 An example of SI wavelet filter design

We can realize any WT with the systematic algorithm

elaborated in Sect. 3 and the SI filter structure depicted in

Sect. 4. But, for brevity, here we use only the Gaussian

wavelet as an example to illustrate the design procedure for

the approximation network.

5.1 Approximation of Gaussian wavelet

An expression for Gaussian wavelet in the time domain can

be written as:

w tð Þ ¼ �1:7864te�t2 ð20Þ

The time reverse of Eq. 20 can be expressed as

w �tð Þ ¼ 1:7864te�t2 ð21Þ

To use the method introduced in Sect. 3, the 2N sampled

data should be obtained. It is obvious that Gaussian wavelet

has much redundancy in the time–frequency support

domain [1]. In order to facilitate the calculation, here the

time support domain is defined as [-2, 2], which means the

minimum time-shift value is t0 ¼ 2: Table 1 illustrates the

optimal approximation in time domain for various time-

shift and filter order values, in which the mean square error

(MSE) [10] is listed to measure the approximation

accuracy. As seen from Table 1, the parameters N and t0are inter-related. For example, t0 = 2 is the optimum

selection for N = 5 and 7; while t0 = 3.5 is the best choice

for the others.

As discussed in Sect. 3.2, the selection of N needs to

make a balance between the approximation accuracy and

circuit complexity. Herein, we let N = 5 and t0 = 2. For

this case, the optimal sampling period Tm equals 0.5970,

leading to the MSE of 1.9139e-3.

According to the method given in Sect. 3, we can obtain

the parameters of Gaussian wavelet approximation network

by using Matlab, as shown in Table 2. Obviously, the

approximation network has two conjugate pole pairs and a

single real pole. It is worth noting that all the poles have the

negative real part which results in a stable approximation

system.

Substituting Kj and pj shown in Table 2 into Eq. 2, we

can obtain a fifth-order approximation function of Gaussian

wavelet, i.e.

Fig. 6 SI filter based on FLF structure

Table 1 Approximation MSE in time domain for various t0 and N

N t0 = 2 t0 = 2.5 t0 = 3 t0 = 3.5 t0 = 4

5 1.914e-3 3.259e-3 8.729e-3 1.029e-2 4.481e-3

6 3.490e-4 6.707e-4 1.872e-4 6.801e-5 2.476e-4

7 6.110e-5 1.375e-4 2.435e-4 9.801e-5 2.122e-4

8 8.774e-6 2.031e-5 5.190e-6 8.342e-7 5.352e-6

9 1.110e-6 2.654e-6 2.918e-6 7.499e-7 4.976e-6


123

Figure 7 plots the impulse response of the approxima-

tion network compared with ideal Gaussian wavelet. Also,

the fifth-order approximation function of Gaussian wavelet

obtained by Pade transform [10, 13] is given in this figure.

As seen from this figure, the proposed method is better

than Pade approximation at the time-domain approxima-

tion precision, especially around the negative and positive

peak.

Although derived from the time-domain information,

the approximation precision of the proposed method is still

relatively high in frequency domain. Figure 8 depicts the

frequency response of the approximation network com-

pared with Pade approximation function. Obviously, the

proposed method is closer to the ideal Gaussian wavelet.

Table 3 shows the approximation MSE of the proposed

method and Pade transform.

5.2 Design of Gaussian wavelet filter

As seen from Eq. 22, the coefficient of term s4 in the

numerator is small compared with other terms in this

function. Thus, we omit the term s4 so as to ease the

practical circuit design. Figure 9 plots the impulse response

of the simplified function in comparison to the original one.

Apparently, the approximation precision is maintained very

well with the circuit complexity reduced.

Rewriting Eq. 22 without the s4 term, it gives

Based on the filter structure illustrated as Fig. 6, we can

synthesize arbitrary approximation function using SI cir-

cuits. The FLF SI filter structure for Eq. 23 is shown as

Fig. 10, including five bilinear SI integrators, five feedback

branches and four feedforward branches.

According to the application requirement, Eq. 23 can be

denormalized to any desired centre frequency. Herein, the

centre frequency is selected to be 10 kHz as an example.

Figure 11 shows the denormalized frequency response of

Eq. 23, with the amplitude scale in decibels. Meanwhile,

the SI circuit is a sampled-data system in which the min-

imum sampling frequency can be determined by the sam-

pling theorem. However, the greater the sampling

frequency fs, the smaller the W/L of the current mirror; in

this case we let fs = 100 kHz. In addition, to avoid the

nonlinear frequency distortion brought by bilinear trans-

form, the frequency pre-warping is used, which can be

expressed as:

fp ¼fs

ptan

pf0

fs

� �

ð24Þ

where f0 and fp are the frequencies in the z domain and

s domain, respectively. For this case, the centre frequency

is pre-warped to 10,343 Hz.

Letting s1 ¼ s3 ¼ s4 ¼ 1; s2 ¼ s5 ¼ 1=2; the parame-

ters for the wavelet filter as shown in Fig. 10 can be

determined by Eqs. 16, 19 and 23:

Table 2 Optimal parameters

for the approximation networkj pj Kj

1 -0.4100 0.5912

2 -0.6760 ? i1.6111 0.1865 ? i1.4567

3 -0.6760 - i1.6111 0.1865 - i1.4567

4 -0.7811 ? i3.1442 -0.5148 - i0.3728

5 -0.7811 - i3.1442 -0.5148 ? i0.3728

H sð Þ ¼ �0:0654s4 � 2:2574s3 þ 3:6520s2 � 32:3840sþ 1:7552

s5 þ 3:3242s4 þ 16:8560s3 þ 25:3810s2 þ 39:8150sþ 13:1380ð22Þ

H sð Þ ¼ �2:2574s3 þ 3:6520s2 � 32:3840sþ 1:7552

s5 þ 3:3242s4 þ 16:8560s3 þ 25:3810s2 þ 39:8150sþ 13:1380ð23Þ


123

a1 ¼ 0:2069; a2 ¼ 0:4137; a3 ¼ 0:2069;

a4 ¼ 0:2069; af 1 ¼ 0:6876; af 2 ¼ 3:4867;

af 3 ¼ 2:6250; af 4 ¼ 4:1179; af 5 ¼ 1:3588;

ac2 ¼ �0:4669; ac3 ¼ 0:3777; ac4 ¼ �3:3493;

ac5 ¼ 0:1815:

ð25Þ

5.3 Simulation results

In this article, the program ASIZ [24] is employed to

simulate the SI wavelet filter. Setting the sampling fre-

quency 100 kHz, the frequency response of the wavelet

filter at a = 20 is plotted in Fig. 12. The peak value

3.23 dB is achieved at f0 = 9.96 kHz, which is slightly

different from the ideal value 10 kHz. The -3 dB fre-

quencies of this filter are 5 and 13.8 kHz, respectively,

which are close to the ideal values (4.9 and 14.1 kHz).

The proposed SI wavelet filter has the merit of low

sensitivity. To testify this characteristic, we use ASIZ to

calculate the errors introduced by the component toler-

ances. ASIZ can compute the sensitivity in relation to the

variation of a selected group of parameters. As for this

case, the circuit parameters of the proposed Gaussian

wavelet filter can be classified into three typical groups,

that is, ai; afi and aci: Selecting one group, ASIZ can plot

the sensitivity curve by counting all the selected parameters

with a random variability.

For example, Fig. 12 shows the gain sensitivity along

with the gain curve in frequency response window,

counting all the afi with ±5% random variability. Appar-

ently, due to the utilization of MLF structure, the sensi-

tivity of the proposed wavelet filter is very low.

In addition, the poles and zeros of the Gaussian wavelet

filter are plotted in the sub-window of Fig. 12. Obviously,

all the poles are listed within the unit circle, which means

the proposed SI wavelet filter is stable.

The impulse response of the Gaussian wavelet filter at

a = 20 is shown in Fig. 13. The positive peak value is

achieved at around t = 0.07 ms, which is slightly different

from the ideal value 0.068 ms.

By adjusting the sampling frequency, the wavelet filter

at different scales can be realized. Changing the sampling

frequency to 50 and 25 kHz for example, one can realize

the Gaussian wavelet filter at dyadic scale values a = 21

and a = 22, respectively. The frequency responses are

shown in Fig. 14, achieving the peak value at f = 4.98 and

2.49 kHz, respectively. The impulse responses are shown

in Fig. 15, achieving the positive peak value at t = 0.14

and 0.28 ms, respectively. Observed from these figures,

Fig. 7 Impulse response of approximation network

Fig. 8 Frequency response of approximation network

Table 3 Comparison of approximation MSE for Gaussian wavelet

Pade transform [10] This work

Time domain 6.7994e-3 1.9139e-3

Frequency domain 4.6327e-3 1.4081e-3

Fig. 9 Impulse response of approximation function compared with

simplified version


123

simulation results are consistent with theoretical analysis

that the presented approach can easily realize the WT at

different dilations by controlling the sampling frequency.

6 Conclusions

Hardware implementation is an optimum approach to the

real-time application of WTs. In this article, we use a

systematic algorithm to construct an approximation net-

work to implement the WT, taking the Gaussian wavelet as

an example. We then realize the approximation network

with SI filter based on FLF MLF structure, in which the SI

bilinear integrator is employed. The simulation result

suggests that the proposed approach is feasible. Some

useful features of the method are listed below:

(1) Compared to Pade approximation, the introduced

approximation method has the advantages at

Fig. 10 SI circuit of FLF Gaussian wavelet filter

Fig. 11 Denormalized frequency response of Eq. 23 (f0 = 10 kHz)

Fig. 12 Simulated frequency response of Gaussian wavelet filter

(a = 20)

Fig. 13 Simulated impulse response of Gaussian wavelet filter

(a = 20)


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approximation accuracy and stability. Also, it is very

convenient for the wavelet filter design, since it can

use the time-domain information of wavelet directly.

(2) Compared to the cascade/parallel architecture, the

proposed MLF SI filter structure has the merit of low

sensitivity.

(3) Compared to the continuous-time analogue circuit

implementations which often require on-chip tuning

circuitry, the various dilations of wavelet filters

realized with SI circuits can be implemented by

controlling the sampling frequency precisely and

easily.

Acknowledgments This work was supported by the National Nat-

ural Science Funds of China for Distinguished Young Scholar under

Grant No. 50925727, National Natural Science Foundation of China

under Grant No. 60876022, Hunan Provincial Science and Technology

Foundation of China under Grant No. 2010J4, the cooperation project

in industry, education and research of Guangdong province and

Ministry of Education of China under Grant No. 2009B090300196,

and the Fundamental Research Funds for the Central Universities,

Hunan University.

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Fig. 14 Simulated frequency response of Gaussian wavelet filter at

different scales

Fig. 15 Simulated impulse response of Gaussian wavelet filter at

different scales


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Wenshan Zhao received the

M.Sc. and doctor degree in

Electrical Engineering from

Hunan University, Changsha,

China in 2006 and 2011, respec-

tively. Currently, he is working

in School of Electronics and

Information Engineering, Bei-

jing Jiaotong University, Bei-

jing, China. From 2007 to 2009,

he was a visiting Ph.D. student

with University of Hertfordshire,

Hatfield, UK. His research

interests are in analogue filters

and signal processing, analogue

and mixed-signal circuits.

Yigang He received the M.Sc.

degree in Electrical Engineering

from Hunan University, Chang-

sha, China, in 1992 and the Ph.D.

degree in Electrical Engineering

from Xi’an Jiaotong University,

Xi’an, China, in 1996. Since

1999, he has been a full professor

of Electrical Engineering with

the College of Electrical and

Information Engineering, Hunan

University. He was a senior vis-

iting scholar with the University

of Hertfordshire, Hatfield, U.K.,

in 2002. He is currently the

Director of the Institute of Testing Technology for Circuits and Sys-

tems, Hunan University. He is the author of a great number of papers on

his research results. His teaching and research interests are in the areas

of circuit theory and its applications, testing and fault diagnosis of

analog and mixed-signal circuits, RFID, and intelligent signal pro-

cessing. Dr. He has been on the Technical Program Committees of a

number of international conferences.


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Realization of wavelet transform using switched-current filters

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