Analog VLSI implementation of wavelet transform usingswitched-current circuits

Mu Li • Yigang He • Ying Long

Received: 3 October 2010 / Revised: 9 July 2011 / Accepted: 9 July 2011 / Published online: 19 August 2011

� Springer Science+Business Media, LLC 2011

Abstract For applications requiring low-voltage low-

power and real-time processing, a novel scheme for the

VLSI implementation of wavelet transform (WT) using

switched-current (SI) circuits is presented. SI circuits are

well suited for these applications since the dilation constant

across different scales of the transform can be imple-

mented, and controlled by both the aspect-ratio of the

transistors and the clock frequency. The quality of such

implementation depends on the accuracy of the corre-

sponding wavelet approximation. First, an optimized pro-

cedure based on differential evolution algorithm (DE) is

applied to approximate the transfer function of a linear

steady-state system whose impulse response is the required

wavelet. The proposed approach significantly improves the

accuracy of approximation wavelets. Next, the approxi-

mation of time-domain wavelet function is implemented by

the SI analog filters. Finally, the design of the complete SI

filter based on first-order and biquad section as main

building block is detailed. Simulations demonstrate the

performance of the proposed approach to analog WT

implementation.

Keywords Switched-current circuits � Wavelet filter �Analog VLSI � Differential evolution algorithm �Wavelet transform

1 Introduction

The wavelet transform (WT) has found a wide range of

applications in the signal processing such as signal detec-

tion, image processing, feature extraction, data compres-

sion, etc. The WT has been shown to be a very promising

mathematical tool, particularly for local analysis of non-

stationary signals, due to its good estimation of time and

frequency localization [1–3]. Currently, the WT has been

implemented numerically by software or digital hardware.

However, a principal obstacle of these approaches is the

heavy computational cost, high power consumption, and

large chip area associated with the required analog–digital

(A/D) converter for digital processing. Consequently,

analog hardware implementations have been an attractive

alternative option to achieve low-voltage low-power and

real-time performance.

Some approaches for implementing WT with analog

circuits have been proposed in the literatures [4–16]. In

these approaches, the circuits for implementing the WT

consist of analog filters whose impulse response is the

wavelet function. The transfer function of the filter is given

by approximation. The performance of the implementation

of the WT in analog circuits depends largely on the accu-

racy of the approximation. In [4–6], the switched-capaci-

tors (SC) circuits based on voltage-model technique were

used to implement analog wavelet filters. The key feature

of using SC circuits for implementing WT is the dilations

of a given filter may be easily and very precisely con-

trolled. The mechanism of controlling the dilation constant

M. Li (&) � Y. He � Y. Long

College of Electrical and Information Engineering,

Hunan University, Changsha 410082, China

e-mail: mli@hnust.edu.cn

M. Li

College of Information and Electrical Engineering,

Hunan University of Science and Technology,

Xiangtan 410201, China

Y. Long

Department of Electronics and Communication Engineering,

Changsha University, Changsha 410003, China

123

Analog Integr Circ Sig Process (2012) 71:283–291

DOI 10.1007/s10470-011-9705-7

in a SC circuit implementation involves controlling various

clock frequencies and the rations of certain capacitor val-

ues of the circuits with the same system architecture.

However, the SC filters and circuits require a nonstandard

digital CMOS process to realize integrated (usually double

poly) floating capacitors, making the technique not suited

for mixed analog–digital integrated circuits. In [7–16], the

designs of the current-mode WT circuits were presented

because the current-mode circuits offer suitability for high

frequency applications and potential for low-voltage

operation. Among these, Pade approximation [7, 9, 10, 16]

was used to calculate the transfer function of the wavelet

filter, and then to implement the WT using log-domain

filters or SI filters. However, there are some disadvantages

which limit the practical application of Pade approxima-

tion. One important issue is that the stable transfer function

of a wavelet filter does not automatically result from the

Pade approximation technique. Another is that the quality

of the wavelet approximation is not measured directly in

the time domain but in the Laplace domain. The least

square approximation for wavelet function, which is know

as L2 approximation, has been proposed [11–13, 15] and

can improve the approximation accuracy compared to Pade

approach. But the performances of L2 approximation

greatly depend on the selection of the starting point

because of the existence of local optima in the approxi-

mation. Moreover, the WT implementation used in [7, 9,

10, 14] easily leads to instability of the filter frequency

characteristics because the time constant of log-domain

integrator is proportional to the thermal voltage VT in log-

domain filters.

In this paper, a novel switched-current WT analog cir-

cuit is presented. To obtain the transfer function of the

wavelet filter, the Marr wavelet function is approximated in

time domain using the differential evolution (DE) algo-

rithm. This approach improves the approximation accuracy

compared to previous Pade and L2 approximation approa-

ches. The WT circuits design is based on low-voltage low-

power SI filter bank, which is easily to be implemented

because that the time constant of the SI filter is only con-

trolled by the aspect-ratio (W/L) of the transistors and the

clock frequency in the circuits. The different scale wavelets

have been obtained and simulations demonstrate the fea-

sibility to analog WT implementation with SI circuits.

The paper is organized as follows: Sect. 2 treats the

basic theory of the WT implementation. Section 3 deals

with the computation of a transfer function which

describes a certain wavelet base that can be implemented

as an analog filter using DE algorithm. Section 4

describes the complete design of WT analog circuit based

on SI filter. Subsequently, some results provided by

simulations are given in Sect. 5. Finally, Sect. 6 presents

the conclusions.

2 Wavelet transform and filters

The WT was introduced in order to overcome limited time–

frequency localization with Fourier-transform for non-sta-

tionary signals analysis. The continuous wavelet transform

WTf(a, s) of a signal f(t) at the scale a and position s is

defined by

WTf ða; sÞ ¼1ffiffiffi

apZ

R

f ðtÞ~w s� t

a

� �

dt ¼ f ðtÞ � 1ffiffiffi

ap w

t

a

� �

ð1Þ

where wðtÞðwðtÞ 2 L2Þ is the wavelet base, ~wðtÞ denotes the

complex conjugation and ‘‘*’’ denotes convolution. Hence,

the WT is based on the convolution of the signal with a

dilated impulse response of a filter. If the scale parameter

satisfied a ¼ 2 jðj 2 ZÞ; this transform is defined as dyadic

WT. The main characteristic of the wavelet base w (t) is

given by

Z

þ1

�1

wðtÞdt ¼ 0 ð2Þ

This function needs to satisfy the admissibility condition

so that the original signal can be reconstructed by the

inverse WT

Cw ¼Z

þ1

�1

wðxÞ�

�

�

�

�

�

2

xj j dx\1 ð3Þ

where wðxÞ is the Fourier transform of the wavelet base

w(t).The admissible condition implies that the wavelets are

inherently band-pass filters in the Fourier domain. Note

that wavelet transforms usually can not be implemented

exactly in analog electronic hardware. From (1), we can

easy to understand the analog computation of WTf(a, s) can

be realized through the implementation of a linear filter for

which the impulse response satisfies hðtÞ ¼ a�1=2wðt=aÞ.Figure 1 shows a wavelet system with multiple scales in

parallel that can be used to compute the WT in real time. In

this work, we will discuss the implementation of wavelet

filter whose impulse response is the basic wavelet function.

As an example, a Marr WT system has been presented.

3 Wavelet bases approximation

3.1 Approximation model of the Marr wavelet

Analog filters in general are described by either linear dif-

ferential equations of finite order in time domain, or ideal

arbitrary order rational function in Laplace domain. Hence,

in order to implement the wavelet filter one must first derive

284 Analog Integr Circ Sig Process (2012) 71:283–291

123

these respective differential equations. However, a linear

differential equation having a predefined desired impulse

response does not always exist. Hence, a suitable approxi-

mation approach is researched. In this paper, the approxi-

mation issue of the Marr wavelet function in time domain is

discussed. The Marr wavelet is given by

wðtÞ ¼ ð1� t2Þe�t2=2 ð4Þ

For obvious physical reasons only the hardware

implementation of causal stable filters is feasible. But this

system is non-causal, the Marr wavelet filter is generally not

possible to be implemented exactly in analog electronic

circuits. Therefore, the Marr wavelet W(t) must be time-

shifted (t0) to facilitate an accurate approximation of

wavelet transform in the time domain. The time-shifted

Marr wavelet function w(t - t0) is described as

wðt � t0Þ ¼ ½1� ðt � t0Þ2�e�ðt�t0Þ2=2 ð5Þ

If the impulse response h(t) of the linear filter satisfies

hðtÞ � wðt � t0Þ; the output of the system is the

approximate wavelet transform WTf(t - t0) of the input

signal f(t) as

f ðtÞ � hðtÞ � f ðtÞ � wðt � t0Þ ¼ WTf ðt � t0Þ ð6Þ

From (6), the quality of the analog implementation of

the WT depends mainly on the accuracy of the wavelet

approximation to w(t - t0). For the generic situation of

stable systems with distinct poles, the impulse response

function h(t) may typically have the following form

hðtÞ ¼X

N

i¼1

AieBjt

¼X

n

i¼1

aiebit þ

X

u

j¼1

cjedjt sinðpjtÞ þ fje

gjt cosðpjtÞ ð7Þ

where Ai and Bi is real or complex numbers; ai, bi, ci, di, fi,

gi and pi are real numbers. N is the order of the filter, n

and u correspond to the numbers of the real poles. For

instance, if a 7th order approximation is attempted, this

parameterized class of function h(t), involving a real

parameter vector k = {k1, k2, …, k14}, will typically be

expressed by

hðtÞ ¼ k1ek2t þ k3ek4t sinðk5tÞ þ k6ek4t cosðk5tÞþ k7ek8t sinðk9tÞ þ k10ek8t cosðk9tÞþ k11ek12t sinðk13tÞ þ k14ek12t cosðk13tÞ ð8Þ

where the parameters k2, k4, k8 and k12 must be strictly

negative for reasons of stability. The difference between

the wavelet function w(t - t0) and its approximation h(t)

can be defined by

hðtÞ � wðt � t0Þk k2¼Z

1

0

½hðtÞ � wðt � t0Þ�2dt ð9Þ

Given the explicit form of the wavelet w(t) and the

parameterized class of functions hðtÞ � wðt � t0Þk k can

now be minimized in a straightforward way using standard

numerical optimization techniques. The sum of squares

error of the discrete points be given by

EðkÞ ¼X

N�1

m¼0

½hðmDTÞ � wðmDT � t0Þ�2 ð10Þ

where DT is the sampling time interval and m is the

sampling points k is a parameter vector. In order to obtain

the optimal parameters of the approximation h(t), the

optimization model for approximating h(t) in time domain

is described as

min EðkÞ ¼ minP

N�1

m¼0

½hðmDTÞ � wðmDT � t0Þ�2

s:t: ki\0 ði ¼ 2; 4; 8; 12Þ

8

<

:

ð11Þ

It is easy to know that this is a typical nonlinear

optimization problem with nonlinear constrains. The

ordinary optimization algorithms which obtain the

optimal values of the model are difficultly. Hence, global

intelligent optimization algorithms are applied in next

section for gaining the optimal parameters of the

approximation model.

3.2 Parameters optimization using DE algorithm

Differential evolution algorithm [17] was introduced by

Storm and Price, which is generally considered as accurate,

reasonably fast and robust optimization approach. The

main advantages of DE are its simplicity and therefore easy

to use to use in solving optimization problems requiring

minimization process with real valued and multi-modal

objective functions. Assume that the population is P(G) of

feasible solution space, where G is the evolution genera-

tion. Population size is Np, so G generation population can

11

11

1( ) ( )a a

th t

aaψ= −

22

22

1( ) ( )a a

th t

aaψ= −

33

33

1( ) ( )a a

th t

aaψ= −

1( ) ( )

LLa aLL

th t

aaψ= −

01 2a =

12 2a =

23 2a =

2L–1La =

1( , )fWT a τ

2( , )fWT a τ

3( , )fWT a τ

( , )τf LWT a

( )f t

Fig. 1 System architecture of the dyadic wavelet transform

Analog Integr Circ Sig Process (2012) 71:283–291 285

123

be expressed as XG ¼ ½xG1 ; x

G2 ; . . .; xG

Np�; whose ith individ-

ual can be described as xGi ¼ ½k

Gi;1; k

Gi;2; k

Gi;3; . . .; kG

i;14�. The

basic framework of DE is described as follows:

Step 1 Initialize randomly the individuals of the popula-

tion. Specify the population size Np = 10, cross-

over probability constant CR = 0.7, difference

vector scale factor F = 0.85, the sampling time

interval DT = 0.01 and the numbers of the

sampling points N = 800. Set the maximum

evolution generation Gmax = 25000 and the initial

generation G = 0.

Step 2 Let the individuals of the G generation execute the

operation from Step 3 to Step 5, then generate

(G ? 1) generation individual.

Step 3 For each target vector xiG, a mutant vector v is

generated according to

vGþ1i ¼ xG

r1þ F � ðxG

best � xGr2Þ ð12Þ

where r1; r2 2 f1; 2; . . .;Npg are randomly chosen indices

and r1 6¼ r2 6¼ i: xGbest is the best individual in the current

generation. xGi and vGþ1

i are the father individual in G gener-

ation and the mutated individual in (G ? 1) generation,

respectively. F is a real number to control the amplification of

the difference vector ðxbest � xGr2Þ; defined as the scale factor.

Step 4 Following the mutation operation, the crossover

operator is applied on the up-level individuals of

the population. For each mutant vector vGþ1i ; an

index mbr(i) is randomly chosen, and a trial

vector xGþ1i is generated with

xGþ1i ¼

vGþ1i;j ; ðrandðjÞ�CRÞ or ðj ¼ mbrðiÞÞxG

i;j; ðrandðjÞ[ CRÞ or ðj 6¼ mbrðiÞÞ

(

ð13Þ

Where j ¼ 1; 2; . . .;D; randðjÞ 2 ½0; 1� is the j th evalua-

tion of a uniform random generator number. j is the posi-

tion in D dimensional individual, CR [ [0, 1] is the

crossover probability constant, which has to be determined

previously by the user. mbr(i) [ [1, D] is a randomly

chosen index which ensures that xGþ1i gets at least one

element form vGþ1i;j :

Step 5 Selection is the procedure of producing better

offspring. If the trial vector xGþ1i has a lower

value than that of its target vector, xGi replaces the

target vector in the next generation. Otherwise,

the target retains its places in the population. The

selection operator is as follows

xGþ1i ¼

xGþ1i ; uðxGþ1

i Þ\uðxGi Þ

xGi ; uðxGþ1

i Þ�uðxGi Þ

(

ð14Þ

where u is the evaluation function, whose equation is

given by

u ¼X

799

m¼0

½hð0:01mÞ � wð0:01m� 4Þ�2 ð15Þ

Step 6 Set G = G ? 1, return to Step 2 until to the

maximum number of generation.

According to the above steps of the DE algorithm, the

accurate global optimal solution ki of the approximation

h(t) is obtained in Table 1.

The time domain function of h(t) is as follows

hðtÞ ¼ 2:6469e�1:8223t � 3:3089e�0:3756t sinð1:2211tÞ� 0:6069e�0:3756t cosð1:2211tÞ� 4:6507e�0:8241t sinð�1:8743tÞ� 2:2817e�0:8241t cosð�1:8743tÞ� 0:9157e�0:3458t sinð2:6033tÞþ 0:2608e�0:3458t cosð2:6033tÞ ð16Þ

The Marr wavelet approximation h(t) is shown in Fig. 2.

The following 7th wavelet transfer function of the Marr

wavelet filter ðscale a ¼ 1Þ is obtained as

H1ðsÞ ¼ ½0:01915s6 � 0:4190s5 þ 2:5417s4 � 13:9293s3

þ 33:9734s2 � 75:3264sþ 11:6520�=½s7

þ 4:9133s6 þ 21:2513s5 þ 55:7335s4

þ 109:2857s3 þ 153:5921s2 þ 129:2001s

þ 85:9930� ð17Þ

Table 1 The optimal values of Marr wavelet approximation

i ki i ki i ki

1 2.6469 6 -0.6069 12 -0.3458

2 -1.8223 7 -4.6507 11 -0.9157

3 -3.3089 8 -0.8241 13 2.6033

4 -0.3756 9 -1.8743 14 0.2608

5 1.2211 10 -2.2817

0 1 2 3 4 5 6 7 8

-0.5

0

0.5

1

Time(s)

Am

plitu

de

Marr wavelet

7th order approximation using DE algorithm

Fig. 2 The approximation of Marr wavelet using DE algorithm

286 Analog Integr Circ Sig Process (2012) 71:283–291

123

The other transfer function Ha(s) of the wavelet filters at

scales a [ R can be derived from (17) by the theory of

Laplace transfer. In order to implement Marr wavelet filter

using the cascade structure of first order section and

biquads, the (17) is rewritten as

H1ðsÞ ¼0:0191s� 0:3079

sþ 1:822� s� 0:1663

s2 þ 0:6916sþ 6:897

� s2 � 2:362sþ 17:94

s2 þ 1:648sþ 4:192� s2 � 3:286sþ 12:68

s2 þ 0:7512sþ 1:632

ð18Þ

For comparing the approximation performances based on

DE and L2 approximation algorithm, we define an error

criterion based on the mean-squared error (MSE). In this

scheme the error integral, which is the difference between

the time-shifted wavelet w(t - 4) and its approximation

h(t), is defined by

MSE ¼ 1

8

Z

8

0

hðtÞ � wðt � 4Þj j2dt ð19Þ

In [13, 18], the authors argued that the Pade approximation of

wavelet functions is not the most suited approach and an

alternative approach, based on L2 approximation that works

directly in the time domain, was introduced. A drawback of

L2 approximation is that a starting point is required for the

application of an iterative local search algorithm to find an

optimal approximation. In Table 2, the MSE results of the DE

and L2 approximation have been calculated for various order

approximation of the Marr wavelet. As seen from the MSE

comparison, the approximation method using the DE algo-

rithm has much better approximation accuracy than the L2

approximation approach for a wavelet filter of the same order.

Furthermore, increasing the order of the approximation will

make it easier to find a good approximation. However, it will

also result in an increase in power consumption that may not

be acceptable for the intender application.

4 Wavelet filter implementation using SI circuits

There are many possible circuit design techniques for a

certain filter transfer function. The trend toward lower

power consumption and lower supply voltages has moti-

vated the research of new solutions for the filters. The SI

filters have emerged in recent years as a promising

approach to face these challenges [19–21]. The SI tech-

nique is a relatively new analog sampled-data signal pro-

cessing approach aiming to replace SC circuits and being

implemented using a standard VLSI CMOS technique. An

important feature of using SI circuits for implementing

wavelet transform is that dilations of a given filter may be

easily and very precisely controlled. Two distinct mecha-

nisms exist for implementing dilations via SI circuits.

Given a filter H(s), and its dilated version Ha(s), the direct

approach of SI circuits implementation permits to control

the aspect-ratio of the transistors. A second method of

controlling a in a SI circuits implementation involves

controlling the various clock frequencies of the circuits

with the same system architecture. These convenient con-

trol schemes are in general unachievable using conven-

tional analog current-model circuit designs. In synthesis of

SI filter, the SI first-order and biquad section are basic

building block that can easily be cascaded for higher order

filter realization. The modularity and simplicity of cascade

synthesis approach to the realization of high order filter has

led to its wide acceptance in SI filters design.

General SI first-order section is shown in Fig. 3, whose

the continuous-time first-order transfer function is as follows

HðsÞ ¼ k1sþ k0

sþ x0

ð20Þ

The numerator coefficients k0 and k1 determine the type of

filter. x0 is the pole frequency. The z-domain transfer

function of the first order section is

HðzÞ ¼ ða1 þ a2Þz� a2

ð1þ a3Þz� 1ð21Þ

The bilinear z transform s! 2ð1� z�1Þ=ð1þ z�1ÞT is

applied to (20)

HðzÞ ¼ ðð2k1 þ k0TÞ=AÞz� ð2k1 � k0TÞ=A

ðð2þ x0TÞ=AÞz� 1ð22Þ

where A = 2 - x0T, and the circuit parameters are a1 ¼2k0T=A; a2 ¼ ð2k1 � k0TÞ=A; a3 ¼ ð2þ x0TÞ=A� 1:

Table 2 The MSE results of the Marr wavelet approximation

Order DE approximation L2 approximation

5 5.89518 9 10-4 6.92185 9 10-4

6 2.45862 9 10-4 3.58560 9 10-4

7 6.07620 9 10-5 1.06278 9 10-4

8 5.41755 9 10-6 3.71490 9 10-5

9 2.14622 9 10-6 1.67349 9 10-5

2φ 1φ

1φ

2J 3Jα

1iα

1 : 1 : 3α 1:

oi

J2iα1φ

Fig. 3 Switched-current first-order section

Analog Integr Circ Sig Process (2012) 71:283–291 287

123

The second generation SI biquad section is shown in

Fig. 4, whose transfer function in z-domain is given by

HðzÞ ¼ � ða5 þ a6Þz2 þ ða1a3 � a5 � 2a6Þzþ a6

ð1þ a4Þz2 þ ða2a3 � a4 � 2Þzþ 1ð23Þ

when the s-domain biquad transfer function is given by

HðsÞ ¼ � k2s2 þ k1sþ k0

s2 þ ðx0=QÞsþ x20

ð24Þ

The bilinear z-domain function of (24) is

HðzÞ ¼ �½ð4k2 þ 2k1T þ k0T2Þz2 þ ð2k0T2 � 8k2Þzþ ð4k2

� 2k1T þ k0T2Þ�=½ððx0TÞ2 þ ð2x0TÞ=Qþ 4Þz2

� ð8� 2ðx0TÞ2Þzþ ððx0TÞ2 � ð2x0TÞ=Qþ 4Þ�ð25Þ

where x0, Q and T are the pole frequency, the pole quality

factor and the sampling period, respectively. Comparing

the coefficients of (23) and (25), the parameters can be

calculated through the equations as below

a1a3 ¼ 4k0T2=X; a3a2 ¼ 4x20T2=X; a4 ¼ 4x0T=ðQXÞ;

a5 ¼ 4k1T=X; a6 ¼ ð4k2 � 2k1T þ k0T2Þ=X

ð26Þ

where X ¼ x20T2 � 2ðx0=QÞT þ 4: According to the above

proposed approach, we calculate the parameters (a1 - a6)

of seventh order Marr wavelet filter using SI circuits, which

is shown in Table 3. One first order and three biquad

sections in cascade are to be connected to realize the

approximation transfer function. The AC small signal

block diagram of seventh order Marr wavelet filter using SI

circuit is given in Fig. 5.

sI1α 2α 3α

2φ 2φ

1φ

1φ

1α 5α 6α 3α 2α 4α

1φ

2φ 1φ1φ 2φ

2φ1φ

1φ2φ

2φ

1α 5α 6α 3α 2α 4α

1φ

2φ 1φ1φ 2φ

2φ1φ

1φ 2φ 2φ

1α 5α 6α 3α 4α 2α2φ1φ

1φ1φ 2φ

2φ1φ

1φ2φ

2φ r

Fig. 5 Block diagram of seventh order Marr wavelet filter using SI circuits

Table 3 The circuit parameters of seventh order Marr wavelet filter

ai First section Second section Third section Fourth section

a1 0.012782 0.000269 0.029631 0.020584

a2 0.026213 0.011159 0.006924 0.002649

a3 1.075636 1.000000 1.000000 1.000000

a4 - 0.027974 0.068049 0.030486

a5 - 0.040447 0.097531 0.133356

a6 - 0.020291 1.039706 1.086405

5iα 6iα

2φ 1φ

1φ 2J 3Jα 2φ

1φ

1 : 1 : 1 : 2α : 4α

2J J 2Jα 4Jα

oi

4 oiα

1iα

2 oiα

1 : 1 : 3α

2φ 1φ

Fig. 4 Switched-current biquad section

288 Analog Integr Circ Sig Process (2012) 71:283–291

123

5 Simulation results

To validate the performance of the whole wavelet system,

the analog circuits of the wavelet filter are simulated by the

ASIZ simulator. Since the circuits of wavelet filters across

different scales can be implemented and controlled by the

clock frequency, we can implement the Marr wavelet

functions at dyadic scale system. Setting Is = 1 A,

r = 1 X and clock frequency be 125 kHz for the first scale,

62.5 kHz for the second, 31.25 kHz for the third and

15.625 kHz for the fourth, respectively, also ignoring the

parasitic effects, the simulated impulse response of the SI

wavelet filters with four scales (a = 1, 2, 4, 8) is shown as

Fig. 6. Obviously, the impulse response waveforms of the

wavelet filters approximate the Marr wavelet very well.

The simulated impulse response waveforms of the different

scale filters achieve the peak value 4.15 mA at 0.58, 1.18,

2.35 and 4.7 ms, respectively. The excellent approximation

of the Marr wavelet confirms the performance of the SI

filter. Figure 7 shows the frequency response for different

scales and zero-pole plot of specify scale (a = 1), where s

and 9 denote the zero points and pole points, respectively.

In the zero-pole plot, the pole points of the approximation

system are all involved in the unit circle, which indicates

the built system is stable. Furthermore, the required gain or

attenuation of wavelet at different scales can be realized

with current mirrors. For example, setting the aspect-ratio

W/L = 2 of the output current mirror, the impulse

responses of the SI wavelet filter with scale a = 1 is shown

in Fig. 8. The waveform of the impulse response achieves

the peak value 8.26 mA at 0.58 ms. The above simulation

and measurement results indicate that the proposed system

is feasible.

In addition, in order to evaluate the detection perfor-

mance of the approximation wavelet based on the SI cir-

cuits, the testing signal s(t) with two catastrophe points at

t = 500 s and t = 1000 s is analyzed by the WT circuits.

The simulation of WT circuits in Fig. 5 can be imple-

mented by the ASIZ simulator when the input of the

1=

a2

=a

4=

a8

=a

Fig. 6 Simulated impulse response of the SI wavelet filter with four

scales

1a =

2a =

4a =

8a =

(1, )fWT τ1( )H s

2 ( )H s

4 ( )H s

8 ( )H s

(2, )fWT τ

(4, )fWT τ

(8, )fWT τ

( , )fWT a τ

Fig. 9 System model for detecting the signal based on the WT

circuits

20

10

-10

-20

0

0

Cur

rent

(m

A)

2.01.0

Time (ms)

Fig. 8 Simulated impulse response of the SI wavelet filter with scale

a = 1 (setting output current mirror W/L = 2)

2

-2-2.000 3.263

lm

Re

0 1 2 3 4 5

0

-10

-20

-30

-40

10

a=1a=2a=4

Frequency (kHz)

Gai

n (d

B) a=8

Fig. 7 Simulated frequency response with four scales and zero-pole

figure of the SI wavelet filter (a = 1)

Analog Integr Circ Sig Process (2012) 71:283–291 289

123

circuits is the testing signal s(t). The system model for

detecting the signal based on the WT circuits with four

characteristic scales is given in Fig. 9. The analog WT

results of the signal s(t) at various scales are shown in

Fig. 10. The first plot in Fig. 10 shows the testing signal

waveform. Sub-bands 2–5 are the decomposed components

of s(t) obtained by modified Marr WT filter bank. The

decomposed outputs show that the proposed system can

correctly detect the catastrophe points of the signal.

6 Conclusions

A novel approach for implementing the WT in an analog

way by means of SI circuits has been proposed. To achieve

an excellent approximation of the Marr wavelet, the

approximation method based on a DE algorithm is used.

The approach leads to accurate wavelet approximation

compared with the conventional approximation. Simulation

of the SI filters whose impulse response was the Marr

wavelet function demonstrated the feasibility of the pro-

posed architecture. The SI wavelet circuits provided is

suitable for low-voltage low-power, wide dynamic range

application and is compatible with the digital VLSI tech-

nology. These results suggest that the proposed approach

could also be used to approximate other wavelet bases.

Acknowledgments The authors would like to acknowledge the

supports of the National Natural Science Funds of China for Distin-

guished Young Scholar under Grant No. 50925727, National Natural

Science Foundation of China under Grant No. 60876022, High-Tech

Research and Development Program of China under Grant No.

2006AA04A104, Hunan Provincial Science and Technology

Foundation of China under Grant No. 2008GK2022, the cooperation

project in industry, education and research of Guangdong province and

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

References

1. Akansu, A. N., & Haddad, R. A. (2001). Multiresolution signaldecomposition: transforms, subbands, and wavelets. New York:

Academic.

2. Rioul, O., & Vetterli, M. (1991). Wavelet and signal processing.

IEEE Signal Processing Magazine, 8(4), 14–38.

3. Mallat, S. (2008). A wavelet tour of signal processing. New York:

Academic.

4. Edwards, R. T., & Cauwenberghs, G. (1996). A VLSI imple-

mentation of the continuous wavelet transform. In Proceedings ofIEEE ISCAS, Atlanta, USA (pp. 368–371).

5. Edwards, R. T., & Cauwenberghs, G. (1995). Analog VLSI

processor implementing the continuous wavelet transform. In

Advances in NIPS, Denver, CO (pp. 692–698).

6. Lin, J., Ki, W. H., Edwards, T., & Shamma, S. (1994). Analog

VLSI implementations of auditory wavelet transforms using

switched-capacitor circuits. IEEE Transactions on Circuits andSystems, 41(9), 572–583.

7. Haddad, S. A. P., Karel, J. M. H., Peeters, R. L. M., Westra, R. L.,

& Serdijn, W. A. (2005). Analog complex wavelet filters. In

Proceedings of IEEE ISCAS (Vol. 4, pp. 3287–3290), May

23–26, 2005.

8. Haddad, S. A. P., & Serdijn, W. A. (2002). Mapping the wavelet

transform onto silicon: the dynamic translinear approach. In

Proceedings of IEEE ISCAS (Vol. 5, pp. 621–624), May 2002.

9. Haddad, S. A. P., Verwaal, N., Houben, R., & Serdijn, W. A.

(2004). Optimized dynamic translinear implementation of the

gaussian wavelet transform. In Proceedings of IEEE ISCAS (Vol.

I, pp. 145–148), May 2004.

10. Haddad, S. A. P., Sumit, B., & Serdijn, W. A. (2005). Log-

domain wavelet bases. IEEE Transaction on Circuits and Sys-tems, 52(10), 2023–2032.

11. Agostinho, P. R., Haddad, S. A. P., De Lima, J. A., & Serdijn, W.

A. (2008). An ultra low power CMOS pA/V transconductor and

its application to wavelet filters. Analog Integrated Circuits andSignal Processing, 57, 19–27.

12. Karel, J. M. H., Peeters, R. L. M., Westra, R. L., Haddad, S. A. P.,

& Serdijn, W. A. (2005). Wavelet approximation for imple-

mentation in dynamic translinear circuits. In Proceedings ofIFAC World Congress 2005, Prague, July 4–8, 2005.

13. Karel, J. M. H., Peeters, R. L. M., Westra, R. L., Haddad, S. A. P.,

& Serdijn, W. A. (2005). An L2-based approach for wavelet

approximation. In Proceedings of CDC-ECC, Seville (pp.

7882–7887), December 2005.

14. Li, H. M., He, Y. G., & Sun, Y. C. (2008). Detection of cardiac

signal characteristic point using log-domain wavelet transform

circuits. Circuits, Systems and Signal Processing, 27(5),

683–698.

15. Zhao, W. S., He, Y. G., & Sun, Y. C. (2010). Design of switched-

current wavelet filters using signal flow graph. In IEEE Int.Midwest Symp. Circuits and Systems, Seattle, WA (pp.

1129–1132), August 2010.

16. Hu, Q. C., He, Y. G., & Liu, M. R. (2005). Design and imple-

mentation of wavelet filter using switched-current circuits. In

IEEE Region 10 TENCON, Melbourne, QLD, November 1–6,

2005.

17. Storn, R., & Price, K. (1997). Differential evolution—A simple

and efficient heuristic for global optimization over continuous

spaces. Journal of Global Optimization, 11(4), 341–359.

0 500 1000 1500-1

0

1

0 500 1000 1500-2

0

2

0 500 1000 1500-2

0

2

0 500 1000 1500-5

0

5

0 500 1000 1500-2

0

2

Fig. 10 Testing signal and analysis results of the WT at various

scales

290 Analog Integr Circ Sig Process (2012) 71:283–291

123

18. Akansu, A. N., Serdijn, W. A., & Selesnick, L. W. (2010).

Emerging applications of wavelets: A review. Physical Commu-nication, 3(1), 1–18.

19. Hughes, J. B., Macbeth, I. C., & Pattullo, D. M. (1990). Switched

current filter. IEE Proceedings G: Circuits, Devices and Systems,137(4), 156–162.

20. De Queiroz, A. C. M., Pinheriro, P. R. M., & Caloba, L. P.

(1993). Nodal analysis of switched-current filters. IEEE Trans-action on Circuits and Systems, 40(1), 10–18.

21. Wu, J. (1993). Switched-current filter design using cascaded

sections. Journal of Electronics, 10(3), 273–278.

Mu Li received the B.Sc.

degree in Electronic and Infor-

mation Engineering from Hunan

Normal University, Changsha,

China, in 2002, and the M.Sc.

degrees in circuit and system

from Hunan University, Chang-

sha, China, in 2007. He is cur-

rently working toward the Ph.D.

degree at College of Electrical

and Information Engineering,

Hunan University. Since 2007,

he has been a Lecturer of Elec-

trical Engineering with Hunan

University of Science and

Technology. His current interests include the research of intelligent

signal processing, switched current technology, and testing and fault

diagnosis of analog 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 Profes-

sor of electrical engineering with

the College of Electrical and

Information Engineering, Hunan

University. He was a Senior

Visiting Scholar with the Uni-

versity of Hertfordshire, Hat-

field, UK, in 2002. He is

currently the Director of the Institute of Testing Technology for Circuits

and Systems, 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 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 international conferences.

Ying Long received the M.S.

degree in Department of Elec-

trical and Information Engi-

neering from Hunan Normal

University, Changsha, China, in

2006. She is currently working

toward the Ph.D. degree in

College of Electrical and Infor-

mation engineering at Hunan

University. She is also an

instructor in Department of

Electronics and Communication

Engineering at Changsha Uni-

versity. Her current interests

include the research of signal

processing, switched current technology, and pattern recognition.

Analog Integr Circ Sig Process (2012) 71:283–291 291

123