Implementing wavelet inverse-transform processor with surface acoustic wave device

Ultrasonics 53 (2013) 447–454

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Implementing wavelet inverse-transform processor with surface acoustic wave device Wenke Lu a,⇑, Changchun Zhu b, Qinghong Liu a, Jingduan Zhang a a b

School of Information Science and Technology, Donghua University, Shanghai 201620, China School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

Article history: Received 2 April 2012 Received in revised form 14 August 2012 Accepted 15 August 2012 Available online 27 August 2012 Keywords: Wavelet inverse-transform processor Surface acoustic wave (SAW) Length function of the electrodes Load resistance Internal resistance

a b s t r a c t The objective of this research was to investigate the implementation schemes of the wavelet inversetransform processor using surface acoustic wave (SAW) device, the length function of defining the electrodes, and the possibility of solving the load resistance and the internal resistance for the wavelet inverse-transform processor using SAW device. In this paper, we investigate the implementation schemes of the wavelet inverse-transform processor using SAW device. In the implementation scheme that the input interdigital transducer (IDT) and output IDT stand in a line, because the electrode-overlap envelope of the input IDT is identical with the one of the output IDT (i.e. the two transducers are identical), the product of the input IDT’s frequency response and the output IDT’s frequency response can be implemented, so that the wavelet inverse-transform processor can be fabricated. X-1120Y LiTaO3 is used as a substrate material to fabricate the wavelet inversetransform processor. The size of the wavelet inverse-transform processor using this implementation scheme is small, so its cost is low. First, according to the envelope function of the wavelet function, the length function of the electrodes is defined, then, the lengths of the electrodes can be calculated from the length function of the electrodes, finally, the input IDT and output IDT can be designed according to the lengths and widths for the electrodes. In this paper, we also present the load resistance and the internal resistance as the two problems of the wavelet inverse-transform processor using SAW devices. The solutions to these problems are achieved in this study. When the amplifiers are subjected to the input end and output end for the wavelet inversetransform processor, they can eliminate the influence of the load resistance and the internal resistance on the output voltage of the wavelet inverse-transform processor using SAW device. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The application of wavelet transform has been found in a variety of fields, such as communication, pronunciation, picture, radar, water-sound, earthquake, biomedicine, mechanical vibration, chemical industry and torrent analysis. Wavelet transform has pushed the information industry to a new era. However, its algorithms are now complicated, and its programs also are complicated and difficult in engineering application. In order to solve those problems, various methods with physical devices have been reported. Physical devices such as very large scale integration (VLSI) [1,2], optical devices [3,4] and SAW devices [5–9] have been developed to support the implantation process. In the method of implementation of wavelet transform with SAW devices [5] and [6], found that, if the electrode-overlap envelope of the input IDT for ⇑ Corresponding author. Tel./fax: +86 21 67792132. E-mail address: [email protected] (W. Lu). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.08.010

SAW device is designed according to the envelope function of wavelet function, then the input IDT of SAW device can implement the convolution of f(t) and wavelet function, i.e. wavelet transform. Chen et al. [5] proposed the implementation of wavelet inversetransform with two wavelet transform processors using SAW devices. Lu et al. [7] used the implementation of the wavelet inverse-transform with the multistrip coupler (MSC) [11]. The sizes of the wavelet inverse-transform processors using these two implementation schemes are large, so their costs are high. In this paper, the implementation scheme that the input interdigital transducer (IDT) and output IDT stand in a line is used to implement the wavelet inverse-transform processor. In this implementation scheme, because the input IDT is identical with the output IDT, the product of the input IDT’s frequency response and the output IDT’s frequency response can be implemented, so that the wavelet inverse-transform processor can be fabricated. The size of the wavelet inverse-transform processor using this implementation scheme is small, so its cost is low. We also present the load

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resistance and the internal resistance as the two problems of the wavelet inverse-transform processor using SAW devices. The solutions to these problems are achieved in this study. The wavelet transform processor and the wavelet inversetransform processor that use SAW devices can benefit from the excellent properties of the SAW devices, namely, passive, small size, low cost, excellent temperature stability, high reliability and high reproducibility, which overcome the high power for VLSI, and big size and low reproducibility for optical devices. This paper is organized as follows. After this introductory section, in Section 2, we give the implementation schemes of wavelet inverse-transform processor using SAW device. The design and fabrication for multiple-scale wavelet inverse-transform processor using SAW devices are discussed in Section 3. Solutions to the load resistance and the internal resistance are achieved in Section 4. Conclusions are drawn in Section 5. 2. Implementation schemes of wavelet inverse-transform processor using SAW devices The dyadic wavelet is [5–8]

  k t w2k ðtÞ ¼ 22 w k 2

ð1Þ

where 2k is scale of dyadic wavelet, k is an integer form 1 to +1. The dyadic wavelet transform of signal f(t) is [5–8]

WT 2k ðsÞ ¼ f ðtÞ  w2k ðtÞ ¼

Z R

  k st dt f ðtÞ22 w 2k

ð2Þ

We know from formulas (3), (6), and (7) that the wavelet inverse-transform of scale 21 is

Z pffiffiffi 2 2 WT 21 ðsÞ 2e2ðtsÞ ej2pf0 ðtsÞ ds AþB R Z Z 2 ¼ ½ f ðtÞw21 ðs  tÞdtw21 ðt  sÞds AþB R R

y21 ðtÞ ¼

(8), w21 ðs  tÞ ¼ pffiffiffiIn formula 2 2e2ðtsÞ ej2pf0 ðtsÞ . Fourier transform of y21 ðtÞ

Y 21 ðxÞ ¼

ð8Þ

pffiffiffi 2 2e2ðstÞ ej2pf0 ðstÞ , w21 ðt  sÞ ¼

2 FðxÞw21 ðxÞw21 ðxÞ AþB

ð9Þ

There are three implementation schemes for the wavelet inverse-transform processor of scale 2–1: implementation scheme using two wavelet-transform processors, implementation scheme using the multistrip coupler (MSC), and implementation scheme that the input IDT and output IDT stand in a line, where the third implementation scheme is used in the paper. We will discuss the three implementation schemes in detail, as shown below. 2.1. Implementation scheme using two wavelet-transform processors [5] Fig. 1 can implement the product of w21 ðxÞ and w21 ðxÞ shown in formula (9), i.e. when the input signal is f(t), Fig. 1 can implement the wavelet inverse-transform shown in formula (9) or formula (8) [5].

The formula of the dyadic-wavelet inverse-transform is [5–8]

yðtÞ ¼

2 X A þ B k2z

Z R

  k ts ds WT 2K ðsÞ22 w 2k R      Z Z k 2 X st ts 2k dt 2 ds ¼ f ðtÞ22 w w A þ B k2z R R 2k 2k 2 X ¼ A þ B k2z

2.2. Implementation scheme using the MSC [7,9,10]

WT 2K ;ðsÞ w2k ; sðtÞds

Z

ð3Þ

In Fig. 2, when the MSC is inserted between the input IDT and the output IDT, Fig. 2 can implement the product of w21 ðxÞ and w21 ðxÞ shown in formula (9), i.e. when the input signal is f(t), Fig. 2 can implement the wavelet inverse-transform shown in formula (9) or formula (8) [7,9,10].

where y(t) is the dyadic-wavelet inverse-transform signal of f(t). Fourier transform of y(t)

YðxÞ ¼

2 X FðxÞw2k ðxÞw2k ðxÞ A þ B k2z

ð4Þ

When w2k ðtÞ of formula (1) is Morlet wavelet function, formula (1) is converted into k

1

 2 t

w2k ðtÞ ¼ 22 e 2 22k k 2

12

e

j2pf0

Wavelet transform processor for scale 2-1

Wavelet transform processor for scale 2-1

y2−1(t)

Fig. 1. Implementation scheme using two wavelet-transform processors.

t 2k

ð5Þ

t 2k

pffiffiffi 2 2e2t ej2p2f 0 t

ð6Þ

pffiffiffi 2 where 2e2t is the envelope function of wavelet function w21 ðtÞ for scale 21, 2f0 is the center frequency. Therefore, we know from formulas (2) and (6) that the wavelet transform of scale 21 is

Z WT 21 ðsÞ ¼ f ðtÞ  w21 ðtÞ ¼ f ðtÞw21 ðs  tÞdt R Z pffiffiffi 2 f ðtÞ 2e2ðstÞ ej2p2f 0 ðstÞ dt ¼ R

WT2−1

 

where 2 e is the envelope function of Morlet wavelet function. Here we take the wavelet inverse-transform processor for scale 21 as an example to illustrate the implementation scheme of the wavelet inverse-transform processor. When scale is 21, formula (5) is converted into

w21 ðtÞ ¼

f(t)

Envelope of electrode overlap MSC Input signal (Electrical f(t) signal)

Envelope of electrode overlap a

b

The input IDT for scale 2-1

Output signal y2-1(t) (Electrical signal) a

b

The output IDT for scale 2-1 Piezoelectric substrate

ð7Þ Fig. 2. Implementation scheme using the MSC.

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2.3. Implementation scheme that the input IDT and output IDT stand in a line In the implementation scheme that the input IDT and output IDT stand in a line, when the overlap weighting (electrode-overlap envelope) of the input IDT is not identical with the one of the output of IDT, the product of the input IDT’s frequency response and the output IDT’s frequency response can be not implemented, and when the overlap weighting (electrode-overlap envelope) of the input IDT is identical with the one of the output of IDT (i.e. the two transducers are identical), the product of the input IDT’s frequency response and the output IDT’s frequency response can be implemented. In Fig. 3, because the electrode-overlap envelope of the input IDT for scale 2–1 is identical with the one of the output of IDT for scale 2–1 (i.e. the electrode-overlap envelopes of the input IDT and the output IDT for scale 2–1 are designed according to the envelope function of the wavelet function w21 ðtÞ), Fig. 3 can implement the product of w21 ðxÞ and w21 ðxÞ shown in formula (9), i.e. when the input signal is f(t), Fig. 3 can implement the wavelet inverse-transform shown in formula (9) or formula (8).

Envelope of electrode overlap

Input signal (Electrical f(t) signal)

SAW signal

a

b

Output signal y2-1(t) (Electrical signal)

The input IDT for scale 2-1

a

3. Design and fabrication for mutiple-scale wavelet inversetransform processor using SAW devices The third scheme (i.e. implementation scheme that the input IDT and output IDT stand in a line) is applied to fabricate multiple-scale wavelet inverse-transform processor using SAW devices. In Fig. 4, the input IDTs and output IDTs can also be designed with the method shown in Fig. 3. We know from Fig. 4 that the output signals of the output IDTs for scales, 2þ1 ;    ; 20 ;    ; 21 are 2 y2þ1 ðtÞ ¼ AþB 2 ¼ AþB

R

WT 2þ1 ðsÞw2þ1 ;s ðtÞds R   ih þ1  i R hR þ1 t t f ðtÞ2 2 w 2sþ1 dt 2 2 w 2sþ1 ds R R

.. . R 2 WT 20 ðsÞw20 ;s ðtÞds y20 ðtÞ ¼ AþB R   ih 0   i R hR 0 2 s ds ¼ AþB f ðtÞ22 w s2t dt 22 w t 0 R R 20 .. . R 2 WT 21 ðsÞw21 ;s ðtÞds y21 ðtÞ ¼ AþB R

ih 1 ts i R hR 1 2 t ¼ AþB R R f ðtÞ2 2 w 2s1 dt 2 2 w 21 ds

Envelope of electrode overlap

WT2-1

Through the above analysis and research, the sizes of the front two schemes are larger than the one of the third scheme, so their costs are also higher than the one of the third scheme. Therefore, in the paper, the third scheme (i.e. implementation scheme that the input IDT and output IDT stand in a line) is applied to fabricate the wavelet inverse-transform processor using SAW device.

b

The output IDT for scale 2-1

Piezoelectric substrate Fig. 3. Implementation scheme that the input IDT and output IDT stand in a line.

After the output signals, y2þ1 ðtÞ; . . . ; y20 ðtÞ; . . . ; y21 ðtÞ pass the adder and amplifier A4, the output signal of the amplifier A4 is

yðtÞ ¼ y2þ1 ðtÞ þ    þ y20 þ    þ y21 ðtÞ Z 2 X ¼ WT 2k ðsÞw2k ;s ðtÞds A þ B k2Z R   Z Z k k 2 X st ts ¼ f ðtÞ22 wð k Þdt 22 wð k Þ ds A þ B k2z R R 2 2 where

2 AþB

ð10Þ

is the amplification of the amplifier A4 shown in Fig. 4.

Fig. 4. Principle of multiple-scale wavelet inverse-transform processor using SAW devices.

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We know from formula (10) that Fig. 4 can implement the wavelet inverse-transform of multiple scales. Here we take the wavelet inverse-transform processor using SAW devices for scale 21 shown in Fig. 3 as an example to illustrate the design and fabrication for multiple-scale wavelet inverse-transform processor using SAW devices shown in Fig. 4. In Fig. 3, because the electrode-overlap envelope of the input IDT for scale 2–1 is identical with the one of the output IDT for scale 2–1 (i.e. the electrode-overlap envelopes of the input IDT and the output IDT for scale 2–1 are designed according to the envelope function of the wavelet function w21 ðtÞ). We know from formula (5) that the wavelet function of scale 21 is

pffiffiffi 2 w21 ðtÞ ¼ 2e2t ej2p2f 0 t ¼ Ps ðtÞej2p2f 0 t

In Fig. 5a, when the propagating time of SAW t = t1, the propagating time of SAW

t1 ¼

t1 ¼

2

L2 ¼ L20 ¼ 1600e2ð0:014567526Þ ¼ 1599 lm: Substituting 2t1 = 2  0.014567526 ls and n = 3 into formula (12), we have the lengths of the electrodes 3 and 30

ð12Þ

where n is an inter from 1 to 100, k is constant, k = 1131.371. The electrode-overlap envelopes of the input IDT for scale 21 and the output IDT for scale 21 shown in Fig. 3 are designed according to the function shown in formula (12). The calculation of the lengths for the electrodes is given as follows. In Fig. 5a, when the propagating time of SAW t = 0 and n = 1, substituting t = 0 and n = 1 into formula (12) gives the length of the electrode 1

2

L3 ¼ L30 ¼ 1600e2ð20:014567526Þ ¼ 1597 lm Likewise, in Fig. 5a, respectively substituting 3t1, n = 4, 4t1, n = 5, . . . ,99t1, n = m 100 into formula (12), we obtain the lengths of the electrodes, 4, 40 , 5, 50 , . . . , 100 and 1000 , as shown in Table 2. According to Fig. 5a, the input IDT for scale 21 or the output IDT for scale 21 shown in Fig. 5b is easily designed. In Fig. 3, the wavelet inverse-transform processor for scale 21 was fabricated on X-1120Y LiTaO3 substrate. The velocity Vs of the SAW propagating on X-1120Y LiTaO3 substrate is 3295 ms1. The design parameters of the wavelet inverse-transform processors for scale 21 are shown in Tables 1 and 2. The fabricated wave-

pffiffiffi 2 2 L1 ¼ kP s ðtÞ ¼ 1131:371Ps ðtÞ ¼ 1131:371 2e2t ¼ 1600e20 ¼ 1600 lm

(a)

2

L 4'

L 3'

L2'

3t1

4 L3

L2

L1

t1 2t1 99t 1

3

1

4'

......

L100'

b

2'

3'

24 þ 24 ¼ 0:014567526 ls 3295

Substituting t1 = 0.014567526 ls and n = 2 into formula (12), we have the lengths of the electrodes 2 and 20

where is the envelope function of wavelet function, pffiffiffi Ps(t) 2 Ps ¼ 2e2t . We define the lengths of the electrodes

a

0

ð13Þ

m

where a is width of the electrode, b is the spacing between the electrodes, v is the propagating velocity of SAW. We know from Table 1 that the electrode width a = 24 lm, spacing between the electrodes b = 24 lm, and on X-1120Y LiTaO3 substrate, SAW propagating velocity m = 3295 lm/ls. Substituting a = b = 24 lm and m = 3295 lm/ls into formula (13), we obtain

ð11Þ

pffiffiffi 2 Ln ¼ kP s ðtÞ ¼ 1131:371Ps ðtÞ ¼ 1131:371 2e2t

aþb

L4

......

L 100

t1 2t1 3t1 99t 1

(b)

......

......

Fig. 5. Design of the input IDT for scale 21 or the output IDT for scale 21. (a) Electrode lengths for the input IDT for scale 21 or the output IDT for scale 21. (b) The input IDT for scale 21 or the output IDT for scale 21.

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W. Lu et al. / Ultrasonics 53 (2013) 447–454 Table 1 Design parameters of the wavelet inverse-transform processor for scale 21 (fabricated on X-1120Y LiTaO3 substrate). Scale

Center frequency (MHZ)

3 dB bandwidth (MHZ)

3 dB relative bandwidth

21

34.323

0.374

1.1%

Number of electrodes

Input IDT

Output IDT

199

199

Electrode width (lm)

Spacing between the electrodes (lm)

Function of electrode-overlap envelope

24

24

pffiffiffi 2 2e2t

Table 2 Design parameters of the electrode lengths for the input IDT of scale 21. L1000

L40

L30

L20

L1

L2

L3

L4

L100

25

1594

1597

1599

1600

1599

1597

1594

25

let inverse-transform processor for scale 21 is shown in Fig. 6. The wavelet inverse-transform processor for scale 21 shown in Fig. 6 is measured with the network analyzer HP8712ET, with its frequency characteristic curve being the upper curve shown in Fig. 7 (the lower curve shown in Fig. 7 is the part-amplification diagram of the upper curves). The ordinates of the upper curves are 10 dB/ ref, and the ordinates of the lower curves are 1 dB/ref, and its experimental parameters are shown in Table 3. The theoretical bandwidths shown in Table 3 are respectively the magnitudes solved from formula (14), i.e.

pffiffiffiffiffiffi1ðx2x0 Þ2 pffiffiffiffiffiffi18ðx2x0 Þ2 w21 ðxÞw21 ðxÞ ¼ 2 pe 8 2 pe 1

¼ 4pe4ðx2x0 Þ

2

ð14Þ

where 2x0 is the center angular frequency, 2x0 = x1 = 2pf1, f1 is center frequency. The experimental bandwidths shown in Table 3 are respectively the actual measurement bandwidths of the frequency characteristic curve being the upper curve shown in Fig. 7. The experimental results shown in Fig. 7 and Table 3 confirm that the implementation scheme that the input IDT and output IDT stand in a line (as shown in Fig. 3) can implement the wavelet inverse-transform processor using SAW devices. With the above method, the wavelet inverse-transform processors of other scales shown in Fig. 4 are also designed. Here we take the wavelet inverse-transform processor for scale 21 as an example to illustrate the basis of using substrate material. In the wavelet inverse-transform processor for scale 21, if the Y-Z LiNbO3 substrate of larger electromechanical coupling coefficient k2 is used, the wavelet inverse-transform processor for scale

Fig. 7. Wavelet inverse-transform processor for scale 21 (fabricated on X-1120Y LiTaO3 substrate).

21 is measured with the network analyzer HP8712ET, with its frequency characteristic curve being the upper curve shown in Fig. 8. The lower curve shown in Fig. 8 is the part-amplification diagram of the upper curve, the ordinate of the upper curve is 10 dB/ref, and the ordinate of the lower curve is 1 dB/ref. In the wavelet inverse-transform processor for scale 21, if the X-1120Y LiTaO3 substrate of smaller electromechanical coupling coefficient k2 is used, the wavelet inverse-transform processor for scale 21 is measured with the network analyzer HP8712ET, with its frequency characteristic curve being the upper curve shown in Fig. 7. The lower curve shown in Fig. 7 is the part-amplification diagram of the upper curve, the ordinate of the upper curve is 10 dB/ ref, and the ordinate of the lower curve is 1 dB/ref.

Fig. 6. Wavelet inverse-transform processors for scale 21.

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W. Lu et al. / Ultrasonics 53 (2013) 447–454

Table 3 Comparison of theoretical and experimental parameters for wavelet inverse-transform processors for scale 21. Scale

Insertion loss (dB)

Theoretical bandwidth (MHZ)

Experimental bandwidth (MHZ)

3 dB

6 dB

9 dB

12 dB

3 dB

6 dB

9 dB

12 dB

Theory

Experiment

21

12.777

0.374

0.529

0.646

0.748

0.371

0.511

0.617

0.706

34.323

34.281



Y¼

Y 11 Y 12 Y 21 Y 22

Center frequency (MHZ)

 ð15Þ

where Y11 is the input admittance, Y 11 ¼ Vi11 jV 2 ¼0 ; Y 22 is the output   , Y12 is the transfer admittance, Y 12 ¼ admittance;Y 22 ¼ Vi22  V 1 ¼0    i1  ; Y 21 is the transfer admittance;Y 21 ¼ Vi21  . V2 V 1 ¼0

V 2 ¼0

We know from Fig. 10 that [13]

"

0

i1

#

 ¼

0 i2



Y 11 þ jB1

Y 21

Y 12

Y 22 þ jB2

e1 e2

 ð16Þ

where B1 and B2 are the matching susceptances, e1 is the input voltage, e2 is the output voltage [13]. In Fig. 10, Rin is the internal resistance of the signal es, Rout is the 0 0 load resistance, i1 ¼ ðes  e1 Þ=Rin ; i2 ¼ e2 =Rout [13]. 0 0 i1 ¼ ðes  e1 Þ=Rin and i2 ¼ e2 =Rout when substituted in formula (16) gives [13]

"

es Rin

# ¼

0 Fig. 8. Frequency characteristic curve of wavelet inverse-transform processor for scale 21 (fabricated on Y-Z LiNbO3 substrate).

i2

SAW signal

V1

V2

WT2-1

The output IDT for scale 2 -1 Piezoelectric substrate

Fig. 9. Parameter measurement of admittance matrix for wavelet inverse-transform processor of scale 21 [13].

Comparing Fig. 7 with Fig. 8, we know that if the substrate of smaller electromechanical coupling coefficient k2 (such as X1120Y LiTaO3) is used for the wavelet inverse-transform processor, the finger reflection is reduced, so that the frequency characteristic is good (Fig. 7); if the substrate of larger electromechanical coupling coefficient k2 (such as Y-Z LiNbO3) is used for the wavelet inverse-transform processor, the finger reflection is serious, so that the frequency characteristic is poor (Fig. 8). 4. Solution to the load resistance and the internal resistance Fig. 9 is the parameter measurement of the admittance matrix for the wavelet inverse-transform processor of scale 21.The admittance matrix of the wavelet inverse-transform processor using SAW device is

Y 21

Y 12

1 Y 22 þ jB2 þ Rout

½1 þ Rin ðjB1 þ Y 11 Þ

#

e1



e2

ð17Þ



Y 21 es

1 Rout

 þ jB2 þ Y 22  Y 12 Y 21 Rin

ð18Þ

We know from formula (18) that the load resistance Rout and the internal resistance Rin of the signal es have influence on the output voltage e2 of the wavelet inverse-transform processor using SAW device. The solutions of these two problems are achieved as follows. We know form Fig. 11 that the output resistance of the amplifier A1

RA1 ¼

The input IDT for scale 2 -1

Y 11 þ jB1 þ R1in

According to formula (17), we can obtain [13]

e2 ¼

i1

"

R0 A0

ð19Þ

where A0 is the open-loop gain of the amplifier A1, R0 is the inherent output resistance of the amplifier A1. AD840 or AD8009 can be used as the amplifier A1. If the amplifier A1 is AD840, A0 of the amplifier AD840 is 1.3  105, and its R0 is 15 X. Therefore,

RA1 ¼

R0 15 X ¼ ¼ 0:000115 X  0 X A0 1:3  105

ð20Þ

We also know from Fig. 11 that the input resistance of the amplifier 21

 Rif ¼ Ri 1 þ A0

R12 R12 þ R13

 ð21Þ

where Ri is the inherent input resistance of the amplifier 21 (such as Ri of the amplifier AD840 is 30 kX), also A0 ¼ 1:3  105 , so

  R12 Rif ¼ Ri 1 þ A0 R12 þ R13   5 ¼ 8:970  105 kX ¼ 30 1 þ 1:3  105  5 þ 16:739 ¼ 8:970  108 X where Rif is very large.

ð22Þ

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W. Lu et al. / Ultrasonics 53 (2013) 447–454

i1

i'1

i2

i'2

e1 Rin

e2

jB1

SAW signal

V1

eS

V2 jB 2

WT2-1

The input IDT for scale 2-1

Rout

The output IDT for scale 2-1 Piezoelectric substrate

Fig. 10. Influence of the load resistance and the internal resistance on wavelet inverse-transform processor using SAW device for scale 21 [13].

Amplifier A1

RA1 i'1

Rif i1

i2

e1 Rin

jB1

eS

i'2

R1

Amplifier 2-1

e2 WT2-1

V1

V2 jB 2

SAW Signal The input IDT for scale 2-1

The output IDT for scale 2-1

e2 R12

R13 Rout

Piezoelectric substrate

Fig. 11. Solutions of the load resistance and the internal resistance for wavelet inverse-transform processor using SAW device for scale 21.

After we contrast Fig. 10 with Fig. 11, formula (18) is converted into

e2 ¼

½1 þ RA1 ðjB1 þ Y 11 Þ



Y 21 es

1 Rif

 þ jB2 þ Y 22  Y 12 Y 21 RA1

ð23Þ

Formula (20) and formula (22) when substituted in formula (23) gives

e2 ¼

¼

½1 þ 0  ðjB1 þ Y 11 Þ



1 8:970108

Y 21 es



 þ jB2 þ Y 22  Y 12 Y 21  0

ð1=jB2 ÞRif ð1=jB2 ÞRif   1=jB2 ð1=jB2 Þ þ Rif Rif

ð26Þ

We know for formula (26) that when 1/jB2 is smaller, the SER becomes weaker.

Y 21 es jB2 þ Y 22

5. Conclusion

ð24Þ

We know from formula (24) that the output voltage e2 of the wavelet-transform processor using SAW device has nothing to do with the load resistance Rout and the internal resistance Rin of the signal es. Therefore, in Fig. 11, the amplifier A1and amplifier 21 can eliminate the influence of the load resistance Rout and the internal resistance Rin on the output voltage e2 of the wavelet inversetransform processor using SAW device. Because sound-electricity-reclamation (SER) temporarily delays or stores some input energy, it can cause the reflection of the incident sound wave for IDT, and make the IDT performances deteriorate. The SER of the wavelet inverse-transform processor using SAW device for scale 21 is also related to its load impedance. When its load impedance is lower, also SER becomes weaker. The low impedance load makes the SER voltage have a short circuit, so that SER can be eliminated [8,12]. We also know form Fig. 11 that the load impedance of the wavelet inverse-transform processor using SAW device for scale 21 is

ð1=jB2 ÞRif ð1=jB2 Þ þ Rif

Z out ¼

Y 21 es

1:115  109 þ jB2 þ Y 22

Z out ¼

where Rif is very large (Rif = 8.970  108 X), Rif is much larger than 1/jB2. Therefore,

ð25Þ

In this paper, we investigate the implementation schemes of the wavelet inverse-transform processor using surface acoustic wave (SAW) device. In the implementation scheme that the input interdigital transducer (IDT) and output IDT stand in a line, because the input IDT is identical with the output IDT, the product of the input IDT’s frequency response and the output IDT’s frequency response can be implemented, so that the wavelet inverse-transform processor can be fabricated. The size of the wavelet inverse-transform processor using this implementation scheme is small, so its cost is low. The theory and the experimental results confirm that this implementation scheme can implement the wavelet inverse-transform processor using SAW device, In this paper, first, according to the envelope function of the wavelet function, the length function of the electrodes is defined, then, the lengths of the electrodes can be calculated from the length function of the electrodes, finally, the input IDT and output IDT can be designed according to the lengths and widths for the electrodes. In this paper, we give the function relationship among the load resistance, the internal resistance and the output voltage of the wavelet inverse-transform processor using SAW device. This function relationship indicates the influence of the load resistance and the internal resistance on the output voltage of the wavelet inverse-transform processor using SAW device. When the amplifiers

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are subjected to the input end and output end for the wavelet inverse-transform processor, they can eliminate the influence of the load resistance and the internal resistance on the output voltage of the wavelet inverse-transform processor using SAW device. Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant Nos. 60976058, 61274078), Innovation Program of Shanghai Municipal Education Commission (Grant No. 13ZZ049). References [1] Cheng Zhang, Chungyan Wang, M.O. Ahmad, A pipeline VLSI architecture for fast computation of the 2-D discrete wavelet transform, IEEE Transactions on Circuits and Systems I: Regular Papers 99 (1) (2012) 1–11. [2] Xin Tian, Wu Lin, Yi-Hua Tan, Jin-Wen Tian, Efficient multi-input/multi-output VLSI architecture for two-dimensional lifting-based discrete wavelet transform, IEEE Transactions on Computers 60 (8) (2011) 1207–1211. [3] D. Roberge, Y. Sheng, Optical wavelet matched filter, Applied Optics 33 (23) (1994) 5287–5293.

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