A Novel Analog VLSI Implementation of Wavelet Transform Based On SI Circuits

Meirong Liu, Qinchun Hu, Yigang He College of Electrical and Information Engineering, Hunan University,

Changsha, 410082, P.R. China Email: liumr12@yahoo.com.cn, hyghnu@yahoo.com.cn

Abstract

A general scheme for the VLSI implementation of wavelet transform using switched-current (SI) circuits is firstly presented in this paper. SI circuits are well suited for this application since the dilation constant across different scales of the transform can be precisely implemented and controlled by both the trans- conductance (W/L) ratios and the clock frequency. The circuits are composed of analog filter bank whose impulse responses are the first derivative of Gaussian and its dilations. The transfer functions of the basic wavelet filter can be given by Padé approximation, which is able to decompose the transfer function into rational form so as to be conveniently implemented by SI circuits. Therefore, it can easily be applied to other filters at different scales with a prescribed impulse response. Simulated results indicated a good performance of wavelet transform.

1. Introduction

Over the last decade or so, wavelet transform, which has received considerable interest and has found widespread use in various signal processing applications, such as signal analysis, signal coding, denoising, image processing and compression etc, has been shown to be an effective tool for the analysis of non-stationary signals[1,2]. Wavelet transform, as representing an alternative to traditional Fourier transform based analysis techniques, decomposes finite energy signals into elementary functions, generated form shifted and scaled versions of a prototype wavelet function, which are well localized both in time and frequency domains. However, a principal obstacle to the wider utilization of wavelet transform is the heavy computational cost. Consequently, analog implementations have been an attractive option to achieve real-time performance. Some researches have been done in analog VLSI implementations of wavelet transform [3-6] .

The switched-current (SI) technique is a relatively new analog sampled-data signal processing technique that aims to replace switched-capacitors (SC) [7-9]. In contrast to the SC technique, which requires a nonstandard digital CMOS process to realize floating linear capacitors, SI can perform accurate signal processing functions in a standard digital CMOS process without the direct use of any capacitor. Moreover, the SI technique does not utilize CMOS op-amps but rather performs its entire analog signal processing with much simpler current mirrors. In a word, The advantages of the SI is its simplicity of implementation, potential for high speed, low-voltage operation and compatibility with digital CMOS processes which is an attractive feature due to the tendency for the integration of large analog/digital systems in a single chip. This paper presents a new analog VLSI implementation of wavelet transform based on SI circuit.

2. Wavelet Transform

The wavelet transform is performed using a basic function called mother wavelet ( )tψ . ( ) )(2 RLt ∈ψ ,

)(2 RL denoting the Hilbert space of measurable, square-integrabel, one-dimensional functions. The function ( )tψ , whose Fourier spectrum being ( )ψ ω , needs to satisfy the admissibility condition [Eq.(1)] so that the original signal can be reconstructed by the inverse wavelet transform.

( ) 2

C dψ

ψ ωω

ω+∞

−∞= < ∞∫ (1)

This admissibility condition implies that the Fourier transform of the wavelet must have a zero component at the zero frequency. Therefore, the wavelets are inherently band-pass filters in the Fourier domain. This means that ( )tψ is oscillatory and has zero-mean value.

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2008 Congress on Image and Signal Processing

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A set of wavelet basis functions ( ){ }ta τψ , , can be generated by translating and scaling the basic wavelet

( )tψ , as

( )

−=

at

ata

τψψ τ1

, (2)

The wavelet transform of a signal ( ) )(2 RLtf ∈ is described as:

( ) ( ) ( ) ( ) ( )dtttfttfaWT aR

af ττ ψψτ ,,,, ∫== (3)

Wavelet transform depends upon two parameters, being scale a and position τ .

Wavelet transform is highly redundant when the parameters a and τ are continuous. Let 2ja= ( )j Z∈ , the dyadic wavelet transform is:

( ) ( ) ( )

( )

,2 2

222

j j

j

jR

WT f t t

tf t dt

τ ψ

τψ−

=

− = ∫

(4)

To cover the whole frequency domain, the Fourier transform of ( )tj2ψ must satisfy the condition:

( ) 12 =∑∞

−∞=j

j ωψ (5)

When satisfied the following condition:

( )2

2 (0 )j

jA B A Bω

+∞

=−∞

≤ Ψ ≤ < ≤ < +∞∑ (6)

Where A and B are constant, and the original signal can be reconstructed by the inverse dyadic wavelet transform when A is equal to B.

( ) ( ) ( )( )∑∫∈

=Zj R

dtWTtf jj τψτ τ，22~ (7)

( )τψ，j2

~ is the dual frame of ( )τψ，j2

, the up and

down limits of its frame are 1B− and 1A− respectively[1].

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2

2 2

2

1

j j

j j

t t A BA B

t t A BA

τ τ

τ τ

ψ ψ

ψ ψ

= ≈ + = =

， ，

， ，

(8)

3. Filter Bank And Wavelet Transform

According to the conclusion in section 2, a wavelet, being satisfy admissible condition, is similar to the transfer function of a band-pass filter. In fact, any band-pass filter impulse response with zero mean that decays to zero fast enough with increasing frequency can serve as a basic wavelet for wavelet transform.

If ( )tψ is a impulse response function of a band-pass filter, whose Fourier spectrum being ( )ψ ω ,

and satisfying the admissibility condition, then ( )tψ can be called basic wavelet.

( )a tψ is defined as the general wavelet basis function at scale a as

( )

=

at

ata ψψ 1 (9)

This is the basic wavelet scaled by a and normalized by 21−a . It defines a set of functions that become broader with increasing a . It also defines

( ) ( )* *1a a

tt taa

ψ ψ ψ = − = −

(10)

Which is the reflected complex conjugate of the scaled wavelet. If ( )tψ is real and even, as is often the cases, the reflection and conjugation have no effect.

Now the wavelet transform [Eq.(3)] can be rewritten as

( ) ( ) ( ) ( ) ( ),f a aR

WT a f t t dt f t tτ ψ τ ψ= − = ∗∫ (11)

For fixed a , then, ( )τ,aWTf is the

convolution of ( )tf with the reflected conjugate wavelet at scale a .

Figure 1. Filter bank analogy for the dyadic wavelet transform of a signal

Figure 1 shows the dyadic wavelet transform as a bank of linear (convolution) filters acting upon ( )tf .

( )tf

( )τ,2 0fW T

( )τ,2 nfW T

( )τ,2 2fW T

( )τ,2 1fW T

02( )tψ

12( )tψ

22( )tψ

2( )n tψ

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Each value of ( )Zja j ∈=2 defines a different band- pass filter, and the outputs of all the filters, taken together, comprise the wavelet. Further the ( )tf can be reconstructed as

( ) [ ]( ) ( )

( ) ( )

20

20

1 *

1 * *

a a

a a

daf t f t dC a

daf t tC a

ψ

ψ

ψ τ ψ τ τ

ψ ψ τ

∞ ∞

−∞

∞

= −

= −

∫ ∫

∫ (12)

which implies that filter outputs, each filtered again by ( )taψ and properly scaled, combine to reconstruct

( )tf .

4. Padé Approximant

The filter bank, whose impulse response being a basic wavelet, play a key role in the method for the analog implementation of wavelet transform proposed in section 3. The transfer function of a filter is usually a rational expression, which is suitable for imple- mentation.

The Padé approximants are a particular type of rational fraction approximation to the value of a function. In the Padé approximation, the coefficients of the approximating rational expression are computed from the Taylor coefficients of the Laplace Transform of ( )tψ .

Suppose we have the truncated Taylor series expansion of a general function ( )A s around some point, e.g., s = 0:

( ) ( )110

+++++= jjj sOsasaasA (13)

The constants 0a to ja are called the Taylor

coefficients. Unfortunately, ( )sA is not a suitable expression to build a filter, since it has only zeros. Then, the Padé approximation ( )sR ML, of a function

( )sA is given by:

( ) ( ) ( )sQsPsR MLML =, (14)

Where ( )sPL is a polynomial of degree at most L and ( )MQ s is a polynomial of degree at most M . Then, ( )sA can be approximated up to order L M+ . The coefficients of ( )sPL and ( )MQ s are deter- mined by the equation:

( ) ( ) ( ) ( )1++=− MLML sOsQsPsA (15)

Since we can obviously multiply the numerator and denominator by any constant and leave [ L/M ] unchanged. We impose the normalization condition

( ) 10 =MQ (16)

Finally we require the coefficients of ( )sPL and

( )sQM as

( )( )

+++=+++=

MMM

LLL

sqsqsQspsppsP

1

10

1 (17)

Then by Eq.16 we may multiply Eq.15 by ( )sQM , which linearizes the coefficient equation. We

can write out Eq.(15) in more detail as

=

−−− MMLLLLL q

qq

aaaa

aaaaa

a

p

ppp

2

1

21

012

01

0

2

1

0 1

000000

(18)

and

=

−+−++

+−++

+−−+

0

00

1

2

1

21

212

111

MLMLMLML

MLLLL

MLLLL

q

qq

aaaa

aaaaaaaa

(19)

Where we define 0≡na if 0<n and

0≡jq if j M> . By solving Eq.18 and Eq.19, we

can get the coefficients of ( )LP s and ( )sQM [10].

The main advantage of the Padé approximation is its computational simplicity and its general applicability. Therefore, it can easily be applied to other filters with a prescribed impulse response, for example to build WT filters with alternative mother wavelets.

On the basis of the time-frequency resolution product, the [3/5] Padé approximation is a suitable candidate for implementation. It is almost symmetrical, it has almost no overshoot and its time-frequency resolution product is only 5% above the Heisenberg limit of 2

1 , while its order 5, is reasonable. We conclusion this section by giving the transfer function

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of the first derivative of Gaussian wavelet.

[ ]( ) 5432

32

5/3 33.80.338.745.943.5275.53.184.92

sssssssssH

++++++−=

(20) 5. SI Circuit Design

Switched-current technique was first reported by Hughes et al in 1989[7]. A key feature of using SI circuits for implementing WT is that dilations of a given filter may be easily and very precisely controlled. Two distinct mechanisms exist for implementing dilations via SI circuits. Given a filter ( )sH , and its dilated version ( )H as , the direct approach of SI circuit implementation permits control of the dilation constant a , since a depends only on the trans- conductance ratios (or W/L ratios). Second, and perhaps more interesting method of controlling a , in a SI circuit implementation involves controlling the various clock frequency ( cf ) of the circuits. Thus, a rational form of the transfer function of a seed filter can be implemented directly as SI filter circuits, with other filers can being simple dilations of it.

The modularity and simplicity of cascade synthesis approach, which decomposes transfer function of filter into biquads, and if order being odd, additional first order circuit is needed, to the realization of high order filter has led to its wide acceptance in active RC, switched capacitor and switched current filter design. Therefore, SI first order circuit and second order circuit (biquads) are the basic building blocks of SI filter. In this section, the SI circuit technique will be applied to the design of wavelet filter for analog implementation of the WT.

5.1. Switched-current Integrator

Integrator is the basic block of filter and used as first order section. Second generation SI integrator overcomes the drawback of the earlier SI integrator. A universal SI integrator configuration (Figure 2), based on Second generation SI technique, is proposed in [9], which is obtained from the superposition of Forward and Backward Euler and feed-forward inputs. The input currents are weighted a1, a2 and a3, accomplished by scaling the weights of the output stages supplying these currents, and the output stage has unit weight.

M 2M 3

M 1 M 4

a 1i1

a 3i3a 2i2 1φ

2φ

2φ

1φ

1φ

1 a 41 1

io

a4J J2J

Figure 2. Universal switched-current integrator

The three different input modes are non-inverting, inverting and amplifying. They are as follows

In Figure 2, 1i is non-invert input, when

2 3 0i i= =

( ) ( )1

1111o

A zi z i zBz

−

−=−

(21)

In Figure 2, 2i is invert input, when 1 3 0i i= = ,

( ) ( )ziBzAzio 21

2

1 −−−= (22)

In Figure 2, 3i is amplify input, when

1 3 0i i= = ,

( ) ( ) ( )1

331

1

1o

A zi z i z

Bz

−

−

−= −

− (23)

Where 11

41aA

a=

+, 2

241

aAa

=+

, 4

33 1 a

aA+

=

and 41

1a

B+

= .

By superposition of equations (21), (22) and (23), the z-domain output current is given by

( ) ( ) ( )

( ) ( )

1 11 2

1 21 1

13

31

1 11

1

oA z A zi z i z i z

Bz BzA z

i zBz

− −

− −

−

−

= −− −

−−

−

(24)

A bilinear-z integrator may be formed by setting ( ) ( ) ( )zizizi =−= 21 , ( )3 0i z = and

1 241

aA A Aa

= = =+

, the z-domain transfer response

then becomes

( ) ( )( )

( )1

1

1 11

−

−

−+==Bz

zAzizizH o (25)

320320

Therefore, we get the bilinear integrators.

5.2. Switched-current Biquad

Consider the continuous-time biquad described by

( ) ( )( ) ( ) 2

002

012

2

/ ωω ++++

==sQs

ksksksisi

sHi

o (26)

Where ( )sii and ( )oi s are the input and

output current-mode signals. 0k , 1k and 2k are

constants. 0ω and Q are the pole frequency and quality factor respectively.

The signal flow graph (SFG) is showed in figure 3, and the transfer function of integrator based biquad [8] circuit given in figure 4 will be

( ) ( )( )

( ) ( )( ) ( )

25 6 1 3 5 6 6

24 2 3 4

21 2 1

o

i

i z a a z a a a a z aH z

i z a z a a a z+ + − − +

= = −+ + − − +

(27)

( )zi1a

1−−

ZZ

13

−− Za

2a

( )165 1 −−+ Zaa

1−−

ZZ

4a

( )zio

Figure 3. SFG representation of SI biquad

Figure 4. SI biquad circuit

Applying the bilinear z-transform of ( )sH

[Eq.(26)] gives

(28)

Where 2 20 02 / 4D T T Qω ω= − + .

Comparing coefficients of Eq. (27) and Eq. (28) gives

DTkaa /4 2031 = (29-a);

DTaa /4 22032 ω= (29-b);

( )QDTa /4 04 ω= (29-c); DTka /4 15 = (29-d);

( ) DTkTkka /24 20126 +−= (29-e)

And filter parameter in terms of circuit parameter will be

2 30

2 a aT B

ω = (30)

2 34

12

Q a a Ba

= (31)

Where, 4 2 32 4B a a a= − +

All type of switched current filter can be implemented with integrator-based biquad. For higher order filter realization, several biquads are used in cascade. The required gain or attenuation is realized with current mirrors.

6. Simulation Results

In order to verify the feasibility of the scheme proposed in this paper, a 5th-order first derivative of Gaussian wavelet filter (FDGWF) was designed and simulated. The FDGWF, whose transfer function [Eq.(20)] is provided in section 4, can be realized by the cascade structure of first order section and biquads. And it can be implemented by SI circuits shown in Figure 5.

The circuit is shown in a schematic form, with the biasing current sources omitted (the biasing currents can be assumed as one for each transistor, with values proportional to the

transistor transconductances). All transistors have unitary transconductances in the normalized design except for some transistors whose transconductance ratios are given in table 1.

To validate the circuit principle, the circuit was simulated by ASIZ [11,12]. The switches to

M7

M2 M4M3 M5

M6 M8

M1

1φ

2φ 1φ

a3

1φ

2φ

2φ 1φ

io

a2 a4

a5i a6i

a1i

a2io a4io

a3J2J 2JJ a2J a4J

( )

2 2 222 1 0 0 2 2 1 0

2 2 2 220 0 0

4 2 2 8 4 2

2 / 4 2 8 1

k k T k T k T k k k T k Tz zD D D

H zT T Q Tz z

D Dω ω ω

+ + − − ++ + = −

+ + −+ +

321321

ground are unnecessary in an actual implementation, but are required by the simulator for some nodes will be connected only to current source in phase 1.

For illustration we give the plots of ideal first derivative of Gaussian and its Padéapproximation, proposed in section 4, shown in figure 6. The impulse response of the FDGWF simulated by ASIZ is shown in figure 7. In order to generate wavelets at different scales, for example

02=a , 12−=a and 22−=a , the clock frequency ( cf ) of figure 7(a), figure 7(b) and figure 7(c) are set to 100KHz, 200KHz and 400KHz respectively, and the W/L of M36 (output

current mirror) are set to 1, 2 and 2 respectively.

Figure 5. The SI realization using Cascade structure for FDGWF

Table 1. Transconductance (W/L) ratios for the FDGWF

M4 M5 M8 M13 M14 M20

0.637 0.319 0.216324 0.05691 0.113817 0.312586

M21 M26 M27 M28 M34 M35

0.07251 0.193855 0.38393 1.446786 0.050732 0.438076

Figure 6. The first derivative of Gaussian

(dashed line) and Padé approximation (solid line)

Simulation results shown that the dilations of basic wavelet can be implemented by changing the clock frequency. The excellent impulse response in figure 7 can be compared with the ideal first derivative of Gaussian and its Padé approximation in figure 6 to confirm the performance of the SI based FDGWF. The required gain or attenuation of wavelet at different scales is realized with current mirrors.

(a) (b) (c)

Figure 7. Simulated impulse response of the FDGWF using ASIZ

(a) 02=a , KHzfc 100= ; (b)

12−=a , KHzfc 200= ; (c) 22−=a , KHzfc 400=

322322

7. Conclusions

A new general analog implementation of wavelet transforms using switched-current circuits are presented in this paper. Compared to digital imple- mentation, analog implementations have been an attractive option to achieve real-time performance because that every signal is not suitable for digital processing. The switched-current techniques are a new analog sampled-data signal processing technique. Design techniques for switched-current filters have been described, and the key feature of SI circuits from the viewpoint of wavelet transform is that changing the dilation constant a can be very precisely controlled by either the clock frequency or the transconductance ratios. The desired transfer functions of wavelet filter is approximated by Padé transform which is able to decompose the transfer function into rational form so as to be conveniently implemented by SI circuits. Therefore, arbitrary wavelet transform filters can be implemented by the method proposed in this paper. Simulated results of FDGWF have demonstrated the effectiveness of the proposed new method for the implementation of wavelet transform.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No.50677014, Doctoral Special Fund of Ministry of Education under grant No. 20060532002, High-Tech Research and Development Program of China（No.2006AA04A104），the Program for New Century Excellent Talents in University of china (NCET-04-

0767), Foundation of Hunan Provincial Science and Technology (06JJ2024).

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