Ultrasonics 52 (2012) 145–150

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Ultrasonics

journal homepage: www.elsevier .com/locate /ul t ras

Solution to the influence of the MSSW propagating velocity on the bandwidthsof the single-scale wavelet-transform processor using MSSW device

Wenke Lu a,⇑, Changchun Zhu b, Lun Kuang c, Ting Zhang c, Jingduan Zhang a

a School of Information Science and Technology, Donghua University, Shanghai 201620, Chinab School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, Chinac Institute No. 23 of the Second Academy CASC, Beijing 100854, China

a r t i c l e i n f o

Article history:Received 24 June 2011Received in revised form 5 July 2011Accepted 27 July 2011Available online 17 August 2011

Keywords:Wavelet transform processorMagnetostatic surface wave (MSSW)Propagating velocity of MSSW

0041-624X/$ - see front matter � 2011 Elsevier B.V.doi:10.1016/j.ultras.2011.07.008

⇑ Corresponding author. Tel./fax: +86 21 67792132E-mail address: luwenkelu@163.com (W. Lu).

a b s t r a c t

The objective of this research was to investigate the possibility of solving the influence of the magneto-static surface wave (MSSW) propagating velocity on the bandwidths of the single-scale wavelet trans-form processor using MSSW device. The motivation for this work was prompted by the processor that�3 dB bandwidth varies as the propagating velocity of MSSW changes.

In this paper, we present the influence of the magnetostatic surface wave (MSSW) propagating velocityon the bandwidths as the key problem of the single-scale wavelet transform processor using MSSWdevice. The solution to the problem is achieved in this study. we derived the function between the prop-agating velocity of MSSW and the �3 dB bandwidth, so we know from the function that �3 dB bandwidthof the single-scale wavelet transform processor using MSSW device varies as the propagating velocity ofMSSW changes. Through adjusting the distance and orientation of the permanent magnet, we can imple-ment the control of the MSSW propagating velocity, so that the influence of the MSSW propagating veloc-ity on the bandwidths of the single-scale wavelet transform processor using MSSW device is solved.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Wavelet transform finds its application in many disciplines andfields such as in pronunciation, picture, communication, radar,water-sound, earthquake detection, biomedicine, mechanical vibra-tion, chemical industry and torrent analysis. It has pushed the infor-mation industry to a new era. However, its algorithms are verycomplicated and difficult to implement in engineering application.In order to solve this problem, a number of methods have beenemployed in implementing a wavelet transform processor. Physicaldevices such as very large scale integration (VLSI) [1–3], opticaldevices [4,5], and surface acoustic wave (SAW) devices [6–9] havebeen developed to support the implantation process.

In the method of implementation of wavelet transform withSAW devices, [6,7] found that, if the electrode-overlap envelopeof the input interdigital transducer (IDT) for a SAW device is de-signed according to the envelope of wavelet function, the inputIDT of a SAW device can implement the convolution of f(t) andwavelet function, i.e. wavelet transform. Moreover, [6,8] proposedthe implementation of wavelet inverse-transform with SAWdevices.

All rights reserved.

.

The wavelet transform processor and wavelet inverse-transform processor that use SAW devices can benefit from theexcellent properties of the SAW devices, namely, passive, smallsize, low cost, excellent temperature stability, high reliability andhigh reproducibility, which overcome the complicated algorithmsand high power for VLSI, and big size and low reproducibility foroptical devices, but in microwave band, the propagation losses ofthe wavelet transform processor and wavelet inverse-transformprocessor using SAW devices are very high, and their transducerfabrication also is difficult. In this research, MSSW devices are usedto implement the wavelet transform processor in microwave band,and the wavelet transform processor using MSSW devices can ben-efit from the excellent properties of the MSSW devices which are oflow propagation loss and simple transducers (as shown in Table 1).In Table 1, the propagation loss of MSSW device is much lowerthan the one of SAW device, and the conducing strip width ofMSSW device is much wider than the one of SAW device (i.e. thetransducers of the wavelet transform processor using MSSWdevices are easily fabricated). When the envelope of the conduct-ing strips of the input transducer for MSSW device is designedaccording to the wavelet-function envelope, the impulse-responsefunction of the input transducer for MSSW device is equal to wave-let function, so single-scale wavelet transform processor usingMSSW device can be fabricated.

The microwave communication equipment of our design needssingle-scale wavelet transform processor of the center frequency

Table 1Comparison of parameters for SAW device AND MSSW device.

SAW device MSSW device

Frequency 10 MHz–3 GHz 0.5–40 GHzSubstrate 128�y/x – LiNbO3

substrateYttrium–iron–garnet(YIG) film

Propagation loss 100 dB/ls (10 GHz) 12 dB/ls (10 GHz)Conducing strip width 0.099 lm (5 GHz) 40 lm (5 GHz)The spacing between the

conducing strips0.099 lm 310 lm

146 W. Lu et al. / Ultrasonics 52 (2012) 145–150

3.830 GHz and bandwidth 48.000 MHz as a band-pass filter. In thisresearch, we found that, as long as the input transducer of MSSWdevice is designed according to a scale wavelet function, thesingle-scale wavelet transform processor using MSSW device canbe implemented. We know from [6–13] that the single-scale wave-let-transform processor using Morlet wavelet function can be usedas a band-pass filter.

In this paper, we present the influence of the MSSW propagat-ing velocity on the bandwidths as the key problem of the single-scale wavelet transform processor using MSSW device.

Through adjusting the distance and orientation of the perma-nent magnet, we can implement the control of the MSSW propa-gating velocity, so that the influence of the MSSW propagatingvelocity on the bandwidths of single-scale wavelet transform pro-cessor using MSSW device is solved.

This paper is organized as follows. After this introductory sec-tion, in Section 2, we discuss the principle of the single-scale wave-let-transform processor using MSSW device. In Section 3, we givethe transducer design of the single-scale wavelet transform proces-sor using MSSW device. The function between the propagatingvelocity v of MSSW and the �3 dB bandwidth is derived in Section4. We present the solution to influence of the MSSW propagatingvelocity on the bandwidths of single-scale wavelet transform pro-cessor using MSSW Device and also give the experimental resultsof the wavelet transform processor in Section 5. In Section 6, thetheory and the experimental results are discussed. Conclusionsare drawn in Section 7.

2. Principle of single-scale wavelet transform processor MSSWdevice

The wavelet function is [6–13]

wsðtÞ ¼1ffiffisp w

ts

� �ð1Þ

where s denotes the scale of wavelet function.The wavelet transform of signal f(t) is [6–13]

WTsðsÞ ¼ f ðtÞ � wsðtÞ ¼Z

Rf ðtÞ 1ffiffi

sp w

s� ts

� �dt

¼ 1ffiffispZ

Rf ðtÞw s� t

s

� �dt ð2Þ

In formula (1), when ws(t) is Morlet wavelet function, formula(1) is converted into

wsðtÞ ¼1ffiffisp e�

12

tsð Þ2 ej2pf0

s t ¼ PsðtÞej2pf0s t ð3Þ

where Ps(t) is the wavelet-envelope function, PsðtÞ ¼ 1ffiffisp e�

12ð

tsÞ

2, f0

s isthe center frequency.

When s = 2k, the wavelet function shown in formula (3) is con-verted into the Morlet dyadic wavelet function [6–13].

The microwave communication equipment of our design needsthe single-scale wavelet transform processor of the center frequency

3.830 GHz and bandwidth 48.000 MHz as a band-pass filter. In thisresearch, we found that, as long as the input transducer of MSSW de-vice is designed according to a scale wavelet function shown in theformula (4), the single-scale wavelet transform processor usingMSSW device can be implemented. We know form [6–13] the sin-gle-scale wavelet-transform processor using Morlet wavelet func-tion can be used as a band-pass filter.

wsðtÞ ¼1ffiffisp e�

12

tsð Þ2 ej2pf0

s t ¼ w1:492k ðtÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1:492kp e

�12

t1:492k

� �2

ej2p f01:492kt

ð4Þ

where the scale s = 1.492k, f01:492k is the center frequency, k is an inter

from �1 to +1.When k is �13, the scale s is equal to 1.492�13, so the wavelet

function for the scale 1.492�13 is

w1:492�13 ðtÞ ¼ 13:474e�16480:083t2ej2p1:49213f0t

¼ 13:474e�16480:083t2ej2pf�13t ð5Þ

where f�13 are the center frequency for the scale 1.492�13, f�13 =1.49213f0 = 3.830 GHz.

The wavelet transform of signal f(t) for the scale 1.492�13 is

WT1:492�13 ¼ f ðtÞ � w1:492�13 ðtÞ ¼Z

Rf ðtÞw1:492�13 ðs� tÞdt

¼Z

Rf ðtÞ13:474e�16480:083ðs�tÞ2 ej2pf�13ðs�tÞdt ð6Þ

We take the single-scale wavelet transform processor for scale1.492�13 as an example to illustrate the design method of the sin-gle-scale wavelet transform processor (as shown in Fig. 1).

Fig. 1 is schematic diagram of the single-scale wavelet trans-form processor using MSSW device, Fig. 1a is the architecture ofthe single-scale wavelet transform processor using MSSW devicefor scale 1.492�13, Fig. 1b is the transducer substrate for Fig. 1a.

The transducer substrate shown in Fig. 1b is fabricated on singlecrystal GGG substrate. The YIG film grew on single crystal GGGsubstrate through liquid phase epitaxy (LPE), and then the trans-ducer substrate shown in Fig. 1b is pasted onto Fig. 1a.

In Fig. 1b, when the conducting-strip envelope of the inputtransducer for scale 1.492�13 is designed according to the enve-lope function (i.e. 13:474e�16480:083t2 ) of the wavelet functionw1:492�13 ðtÞ, the impulse-response function h1:492�13 ðtÞ of the inputtransducer is equal to the wavelet function w1:492�13 ðtÞ, so theinput transducer can implement the wavelet transform shownin formulas (6), i.e. the convolution of f(t) and w1:492�13 ðtÞ. The out-put transducer of scale 1.492�13 only converts the MSSW signalinto electrical signal.

3. Transducer design of the single-scale wavelet transformprocessor using MSSW device

We know from formula (3) that the wavelet-envelope function

PsðtÞ ¼1ffiffisp e�

12

tsð Þ2 ð7Þ

Substituting s = 1.492�13 into formula (7), we obtain

PsðtÞ ¼1ffiffisp e�

12ð

tsÞ

2 ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:492�13

p e�1

2t

1:492�13

� �2

¼ 13:474e�16480:083t2

¼ 13:474e�16480:083 Hvð Þ2 ð8Þ

where t is the propagating time of MSSW, H is the propagating dis-tance of MSSW, v is the propagating velocity of MSSW.

We define the lengths of the conducting strips

Signal inputSignal output (YIG film)

(GGG substrate)Magnetic field

Input transduer Output transduerfor scale 1.492 for scale 1.492

Metal

-13-13

Gd Ga O garnet substrate3 5 12

Yttrium iron garnet film

a b

Envelope of the

InputransduerOutput transducer

a

-13

-13

Signal input Signal output

for scale 1.492conducing strip

for scale 1.492

GGG substrate

b

W W W1 2 3

(a)

(b)Fig. 1. Schematic diagram of the single-scale wavelet transform processor using MSSW device. (a) Architecture of the single-scale wavelet-transform processor using MSSWdevice for scale 1.492�13. (b) Transducer substrate of scale 1.492�13 for (a).

W. Lu et al. / Ultrasonics 52 (2012) 145–150 147

Ln ¼ kPsðtÞ ¼ 148:434PsðtÞ ð9Þ

where n is an inter from 1 to 6, k is constant, k = 148.434.Substituting formula (8) into formula (9), we have

Ln ¼ 148:434� 13:474e�16480:083 Hvð Þ2

¼ 2000:000e�16480:083 Hvð Þ2 ð10Þ

The conducting-strip envelope shown in Fig. 1b is designedaccording to the formula (10).

The calculation of the lengths for the conducting strips is givenas follows.

Fig. 2a is the conducting-strips for the input transducer of scale1.492�13, Fig. 2b is the input transducer of scale 1.492�13. We knowfrom Fig. 2a that a is width of the conducting strips, b is the spacingbetween the conducting strips. When the propagating distance Hof MSSW is respectively (a + b), 2(a + b), 3(a + b), 4(a + b) and5(a + b), we solve from formula (10) that the lengths of the variousconducting strips

L1 ¼ 2000 ðlmÞ ð11Þ

L2 ¼ L20 ¼ 2000:000e�16480:083 aþbvð Þ2 ðlmÞ ð12Þ

L3 ¼ L30 ¼ 2000:000e�16480:083 2 aþbð Þv

� �2

ðlmÞ ð13Þ

L4 ¼ L40 ¼ 2000:000e�16480:083 3 aþbð Þv

� �2

ðlmÞ ð14Þ

L5 ¼ L50 ¼ 2000:000e�16480:083 4 aþbð Þv

� �2

ðlmÞ ð15Þ

L6 ¼ L60 ¼ 2000:000e�16480:083 5 aþbð Þv

� �2

ðlmÞ ð16Þ

We know from Table 2 that a = 40 lm, b = 310 lm, v = 142 km/s. Substituting a = 40 lm, b = 310 lm and v = 142 km/s into formu-las (12)–(16), we obtain the lengths of the various conductingstrips, as shown in Table 3.

Fig. 2a is designed with the design parameters shown in Tables2 and 3. According to Fig. 2a, the input transducer shown in Fig. 2bis easily designed.

4. Deriving the function between the propagating velocity v ofMSSW and the �3 dB bandwidth

The frequency domain expression of Morlet wavelet function is

wsðxÞ ¼1ffiffisp

ffiffiffiffiffiffiffip

2s2

re�

s22 ðx�x0Þ2 ð17Þ

where s is the scale, x0 is the angular center frequency.

Let YðxÞ ¼ wsðxÞ1ffiffisp

ffiffiffiffiffip2s2

p ¼ e�s22 ðx�x0Þ2 ð18Þ

When Y(x) = 0.707945784, the �3 dB bandwidth is

Df�3dB ¼1ps

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln 0:707945784p

ð19Þ

From the formula (19), we obtain

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 ln 0:707945784p

pDf�3dBð20Þ

Substituting formula (7) into formula (9), we have

Ln ¼ kPsðtÞ ¼ 148:434PsðtÞ ¼148:434ffiffi

sp e�

12ð

tsÞ

2 ð21Þ

Substituting t ¼ Hv into formula (21), we obtain

a

b

L2 L3 L4 L5 L6L1L2'L3'L4'

a+b a+b2(a+b) 2(a+b)

3(a+b) 3(a+b)4(a+b) 4(a+b)

)b+a(5)b+a(5

L6' L5'

12 3

45 6

2'3'4'5'

6'

Signal input

(a)

(b)

0

Fig. 2. Input-transducer design of the single-scale wavelet transform processor using MSSW device for scale 1.492�13. (a) Conducting-strips for the input transducer of scale1.492�13. (b) Input transducer of scale 1.492�13.

Table 3Design parameters of the conducting-strip lengths for the input transducer of scale1.492�13.

Conducting-strip lengths of the input transducer (lm)

L60 L50 L40 L30 L20 L1 L2 L3 L4 L5 L6

164 403 812 1340 1809 2000 1809 1340 812 403 164

Table 4The datum between the propagating velocity v of MSSW and the �3 dB bandwidth.

Propagating velocity vof MSSW (km/s)

216.094 161.964 142.000 132.252 126.800 123.722

�3 dB bandwidthDf�3dB (MHz)

42.000 45.000 48.030 51.000 54.000 57.000

Table 2Design parameters of input transducer and output transducer for scale 1.492�13.

Scale Substrate Propagatingvelocity ofMSSW(km/s)

Centerfrequency(GHz)

�3 dBbandwidth(MHz)

Insertionloss (dB)

Width ofconductingstrips (lm)

Spacing betweenthe conductingstrips (lm)

Number of theconducting strips

Envelope functionof conductingstrips

Inputtransducer

Outputtransducer

1.492�13 YIG film(thickness ofYIG is 23.5 lm)

142.000 3. 830 48.030 �17 40 310 11 2 13:474e�16480:083t2

148 W. Lu et al. / Ultrasonics 52 (2012) 145–150

Ln ¼148:434ffiffi

sp e�

12

Hvsð Þ2 ð22Þ

where H is the propagating distance of MSSW, v is the propagatingvelocity of MSSW.

Substituting n = 2, H = a + b = 350 lm, L2 ¼ L20 ¼ 1809 lm (asshown in Tables 2 and 3) into formula (22), we have

1809 ¼ 148:434ffiffisp e�

12

350vsð Þ2 ð23Þ

Substituting formula (20) into formula (23), we obtain

6:270ffiffiffiffiffiffiffiffiffiffiffiffiffiDf�3dB

p ¼ e�875122:589Df�3dB

v

2

ð24Þ

Formula (24) is the derived function between the propagatingvelocity v of MSSW and the �3 dB bandwidth.

We solve from formula (24) the datum between the propagatingvelocity v of MSSW and the �3 dB bandwidth, as shown in Table 4.

Fig. 3. Single-scale wavelet transform processor using MSSW device for scale 1.492�13.

Fig. 4. Frequency characteristic curve of the single-scale wavelet transform processor using MSSW device for scale 1.492�13 (used to measure the network analyzer E838B).

W. Lu et al. / Ultrasonics 52 (2012) 145–150 149

We know from Table 4 that �3 dB bandwidth of the single-scalewavelet transform processor using MSSW device varies as thepropagating velocity v of MSSW changes.

5. Solution to influence of the MSSW propagating velocity onthe bandwidths of single-scale wavelet transform processorusing MSSW Device and its experimental results

According to the design parameters shown in Tables 2 and 3, wehave fabricated the single-scale wavelet-transform processor usingMSSW device for scale 1.492�13 (as shown in Fig. 3).

The influence of the MSSW propagating velocity on the band-widths of the single-scale wavelet transform processor is solvedas follow.

When the distance and orientation of the permanent magnet forthe single-scale wavelet transform processor (Fig. 3) are adjustedto make the �3 dB bandwidth of the frequency characteristic curve(Fig. 4) for the single-scale wavelet transform processor be approx-imately equal to 48.030 MHz (i.e. the propagating velocity v ofMSSW is equal to 142.000 km/s), we stop adjusting the distanceand orientation of the permanent magnet, so that the influenceof the MSSW propagating velocity on the bandwidths of the sin-gle-scale wavelet transform processor is solved, and also measure

Table 5Comparison of the experimental and theoretical parameters for the single-scale wavelet-transform processor using MSSW device for scale 1.492�13.

Scale Experimental insertion loss(dB)

Experimental center frequency(GHz)

Theoretical bandwidth (MHz) Experimental bandwidth (MHz)

�3 dB �6 dB �9 dB �12 dB �3 dB �6 dB �9 dB �12 dB

1.492�13 �15.180 3.843 48.030 67.924 83.190 96.060 48.042 66.357 77.785 104.361

150 W. Lu et al. / Ultrasonics 52 (2012) 145–150

other parameters of the single-scale wavelet transform processorshown in Fig. 3, as shown in Table 5.

The frequency characteristic curve of the single-scale wavelet-transform processor using MSSW device for scale 1.492�13 isshown in Fig. 4.

The theoretical bandwidths shown in Table 5 are the magni-tudes solved from formula (25), i.e.

w1:492�13 ðxÞ ¼ 227:538e�1

65920:332ðx�x�13Þ2 ð25Þ

where x�13 is the center angular frequency.The experimental bandwidths shown in Table 5 are the actual

measurement bandwidths of the frequency characteristic curveshown in Fig. 4.

6. Discussion

We know from Table 4 that �3 dB bandwidth of the single-scalewavelet transform processor using MSSW device varies as thepropagating velocity v of MSSW changes, of course, its other band-widths (e.g., �6 dB, �9 dB, �12 dB, etc.) will also change. We ex-pect that the propagating velocity v of MSSW is constant, so thatits �3 dB bandwidth also is constant.

When the distance and orientation of the permanent magnet forthe single-scale wavelet transform processor (Fig. 3) are adjusted,the propagating velocity v of MSSW changes, so that the�3 dB band-width also changes. Therefore, in the paper, through adjusting thedistance and orientation of the permanent magnet, we can imple-ment the control of the MSSW propagating velocity v, so that theinfluence of the MSSW propagating velocity on the bandwidths ofthe single-scale wavelet transform processor using MSSW deviceis solved.

7. Conclusions

When the envelope of the conducting strips for the input trans-ducer of the magnetostatic surface wave (MSSW) device is de-signed according to the wavelet-function envelope, the impulse-response function of the input transducer for MSSW device is equalto wavelet function, so the single-scale wavelet transform proces-sor using MSSW device is fabricated.

We know from Table 5 that the theoretical bandwidths areconsistent with the experimental bandwidths. Therefore, we

reach the conclusion that through adjusting the distance andorientation of the permanent magnet, we can implement thecontrol of the MSSW propagating velocity v, so that the influenceof the MSSW propagating velocity on the bandwidths of the sin-gle-scale wavelet transform processor using MSSW Device issolved.

Acknowledgment

This work was supported by the National Natural Science Foun-dation of China (Grant No. 60976058).

References

[1] K. Andra, C. Chakrabarti, T. Acharya, A VLSI architecture for lifting-basedforward and inverse wavelet transform, IEEE Trans. Signal Process. 50 (2009)966–977.

[2] C.T. Huang, P.C. Tseng, L.G. Chen, Efficient VLSI architectures of lifting-baseddiscrete wavelet transform by systematic design method, IEEE Int. Symp.Circuits Syst. 5 (2002) 565–568.

[3] M. Vishwanath, R.M. Owens, M.J. Irwin, VLSI architectures for the discretewavelet transform, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 42(1995) 305–316.

[4] D. Roberge, Y. Sheng, Optical wavelet matched filter, Appl. Opt. 33 (1994)5287–5293.

[5] Xiandong Chen, Xianmin Zhang, Kangsheng Chen, Qingguo Li, Optical wavelet-matched filtering with bacteriorhodopsin films, Appl. Opt. 36 (1997) 8413–8416.

[6] Wenke Lu, Changchun Zhu, Junhua Liu, Qinghong Liu, Implementing wavelettransform with SAW elements, Sci. China (Ser. E) 46 (2003) 627–638.

[7] Wenke Lu, Changchun Zhu, Junhua Liu, Qinghong Liu, Wavelet transformelement of SAW type, Chin. Sci. Bull. 50 (2005) 598–602.

[8] Wenke Lu, Changchun Zhu, Junhua Liu, Qinghong Liu, Study onimplementation and reconstruction element, Acta Electron. Sin. 30 (2002)1156–1159 (in Chinese).

[9] Wenke Lu, Changchun Zhu, Solving three key problems of wavelet transformprocessor using surface acoustic wave devices, IEEE Trans. Ind. Electron. 57(2010) 3801–3806.

[10] Changbao Wen, Changchun Zhu, A novel architecture of implementing wavelettransform and reconstruction processor with SAW device based on MSC, Sens.Actuators, A 126 (2006) 148–153.

[11] Changbao Wen, Changchun Zhu, Time synchronous dyadic wavelet processorarray using surface acoustic wave device, Smart Mater. Struct. 15 (2006) 939–945.

[12] Wenke Lu, Changchun Zhu, Study of small size wavelet transform processorand wavelet inverse-transform processor using SAW devices, Measurement 44(2011) 994–999.

[13] Changbao Wen, Changchun Zhu, Yongfeng, Optimal frequency band designscheme of dyadic wavelet processor array using surface acoustic wave devices,IEEE Trans. Ind. Electron. 56 (2009) 949–955.