On preprocessing techniques for bandlimited parametric loudspeakers

Applied Acoustics 71 (2010) 486–492

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Technical Note

On preprocessing techniques for bandlimited parametric loudspeakers Ee-Leng Tan *, Peifeng Ji, Woon-Seng Gan Digital Signal Processing Laboratory, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e

i n f o

Article history: Received 13 February 2009 Received in revised form 27 October 2009 Accepted 5 November 2009 Available online 2 December 2009 Keywords: Parametric loudspeaker Self-demodulation Preprocessing techniques

a b s t r a c t The self-demodulation property of finite-amplitude ultrasonic waves can be applied with parametric loudspeaker to produce audible sound. A special characteristic of the reproduced sound waves using parametric loudspeaker is its high directivity. However, the demodulated signal from parametric loudspeaker suffers from high distortion. To reduce the distortion in the demodulated signal, preprocessing of the modulating signal is usually employed. To determine the effectiveness of the preprocessing technique, an important practical constraint on the bandwidth of the ultrasonic transducer of the parametric loudspeaker should be accounted. In this paper, we shall discuss a class of preprocessing techniques that is based on quadrature amplitude modulation. As compared to the conventional preprocessing methods used with bandlimited ultrasonic transducer, the demodulated signal from our proposed preprocessing techniques exhibits lower distortion. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Non-linear interaction ultrasound waves can be applied with parametric loudspeaker for sound reproduction. This phenomenon can be explained with the self-demodulation property of finiteamplitude ultrasonic waves. In 1963, Westervelt [1] described how a difference frequency signal is generated from two highfrequency collimated beams of sound. These high-frequency beams of sound are commonly referred as primary waves. The non-linear interaction of primary waves in the medium (such as water and air) gives rise to an end-fire array of virtual sources which is referred as the parametric array. An illustration of the parametric array is given in Fig. 1. The ultrasonic transducer emits a finite-amplitude (very high intensity) modulated sound beam into the medium. This produces an end-fire virtual array along the axis of propagation. The generation of these virtual sources extends to the point where the sound (ultrasonic beam) ceases to be of finite-amplitude. For years, underwater parametric array has been widely used in sonar applications. The possibility of parametric array in air was first verified experimentally by Bennett and Blackstock [2]. Since then, parametric array has been deployed in air. The ultrasonic transducer and the parametric array (in air) are collectively referred as parametric loudspeaker. In 1982, Yoneyama et al. demonstrated the use of parametric loudspeaker to generate broadband audio [3]. In their experiment, 547 piezoelectric transducers (PZT) were used to project amplitude modulated ultra* Corresponding author. Tel.: +65 67906901. E-mail addresses: [email protected] (E.-L. Tan), [email protected] (P. Ji), ewsgan@ ntu.edu.sg (W.-S. Gan). 0003-682X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2009.11.006

sound. Their experiments revealed that the demodulated signal generated by the non-linear acoustic phenomena has a very sharp directivity pattern, but the demodulated signal suffered from high harmonic distortion and poor frequency response. In 1984, Kamakura et al. [4] showed that it is possible to reduce the distortion found with double sideband amplitude modulation (DSBAM) by preprocessing the modulating signal. Similarly, Pompei introduced a practical device in 1998 [5] which adopted the preprocessing technique proposed by Kite et al. [6]. Their approaches involved square-rooting the modulating signal and these approaches are generally referred as the square root amplitude modulation (SRAM). It is observed from their experiments that SRAM yields lower harmonic distortion as compared to DSBAM. SRAM was motivated by Berktay’s approximation [7] which predicts that the demodulated signal is proportional to second time derivative of the squared envelope of the ultrasound. Verification of Berktay’s results can be found in [3,8,9]. However, distortion in the demodulated signal can be totally removed if and only if all the harmonics of the modulating signal introduced by the square root operation in SRAM are reproduced by the ultrasonic transducer. Hence, an ultrasonic transducer having wide bandwidth (>10 kHz) is required but such wide-band ultrasonic transducer is not realizable with current technology [3,10]. This practical limitation is seldom addressed in literature and was first highlighted by Kite et al. [6]. Extended from [11,14], a new class of preprocessing techniques referred as modified amplitude modulation (MAM), which is a special class of quadrature amplitude modulation (AM), is proposed in this paper. This class of preprocessing techniques is motivated by the limited bandwidth of the ultrasonic transducer. In our performance analysis of DSBAM, MAM and SDRAM, we focus

E.-L. Tan et al. / Applied Acoustics 71 (2010) 486–492

487

Fig. 1. Formation of a parametric loudspeaker.

on the effects of bandlimited ultrasonic transducers which are not discussed in our prior work [11–15]. Unlike DSBAM and SRAM, the proposed class of preprocessing techniques (with different order) offers bandwidth flexibility and can be configured to match the bandwidth of the ultrasonic transducer. At the same bandwidth, the proposed class of preprocessing techniques offers superior performance over DSBAM and SRAM. The rest of the paper is structured as follows. In Section 2, audio reproduction using parametric loudspeaker is reviewed. In Section 3, the proposed class of preprocessing techniques is introduced. Simulation results and observation of different preprocessing techniques are discussed in Section 4. Finally, our conclusions are presented in Section 5. 2. Reproducing audio using parametric array In the case of far field sound, Berktay [7] approximates the demodulated wave pressure p2 to be proportional to the second time derivative of the squared envelope of an ultrasonic wave pressure p1. This modulated ultrasonic wave p1 is radiated into air from a circular piston source with radius a and the ultrasonic wave at an axial distance z from the source is expressed as

p1 ðr; z; sÞ ¼ P0 EðsÞeaðsÞz sinðx0 s þ /ðsÞÞHða  rÞ;

ð1Þ

where r is the transverse radial coordinate, s = t  z/c0 is the retarded time, c0 is the small signal sound speed, P0 is the initial pressure of the primary wave, E(s) is the envelope of the primary wave, x0 is the angular frequency of the carrier, u(s) is the time varying phase function, H(x) is the Heaviside unit step function, and a(s) is the time varying absorption coefficient. The time varying absorption coefficient a(s) is computed as

aðsÞ ¼ ½XðsÞ=x0 2 a0 ;

ð2Þ

where a0 is the thermoviscous attenuation coefficient and X(s) is the instantaneous angular frequency of the carrier. The instantaneous angular frequency X(s) is given as

XðsÞ ¼ x0 þ d/ðsÞ=ds:

ð3Þ

Ignoring the effects of absorption of the demodulated wave, the lossless axial solution is found to be [16]

p2 

bP 20 a2 16 0 c40 z

q

2

envelopes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiof DSBAM and SRAM are stated as 1 + mg(s) and 1 þ mgðsÞ, respectively. In the case of DSBAM, the demodulated is given as signal pDSBAM 2

pDSBAM  2 

bP20 a2 @ 2 ½1 þ mgðsÞ2 16q0 c40 az @ s2 bP20 a2 f2mg 00 ðsÞ þ 2m2 ½gðsÞg 00 ðsÞ þ ðg 0 ðsÞÞ2 g; 16q0 c40 az

where g(s) is the modulating signal, g0 (s) and g 00 ðsÞ are the first and second of time derivatives of g(s), respectively. From (6), it is clear that the fundamental frequency component and distortion in the demodulated signal are given by 2mg 00 ðsÞ and 2m2 ½gðsÞg 00 ðsÞþ ðg 0 ðsÞÞ2 , respectively. We can reduce the amount of distortion in the demodulated signal by reducing m, but this also lowers the sound pressure level (SPL) at fundamental frequency. Next, we analyze the demodulated signal from SRAM whichphas been used in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [4,5,17]. The envelope of SRAM is expressed as 1 þ mgðsÞ, hence, the demodulated signal from SRAM is

pSRAM  2

bP20 a2 @ 2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 bP20 a2 s Þ  1 þ mgð mg 00 ðsÞ: 16q0 c40 az @ s2 16q0 c40 az

3.1. Derivation of the proposed preprocessing technique In this section, we consider a quadrature AM scheme as shown in Fig. 2. To analyze the demodulated signal from quadrature AM using (4) and (5), we have to express the modulated carrier from quadrature AM in the form of (1). By projecting sin (x0s) and cos (x0s) to sin (x0s), the modulated ultrasonic wave of the quadrature AM scheme becomes

g T ðsÞ ¼ g 1 ðsÞ sinðx0 sÞ þ g 2 ðsÞ cosðx0 sÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g 21 ðsÞ þ g 22 ðsÞ sin½x0 s þ tan1 ðg 2 ðsÞ=g 1 ðsÞÞ:

ð4Þ

where b is the coefficient of non-linearity and q0 is the ambient density. When there is no frequency modulation (FM) in the ultrasound wave, a(s) becomes a constant a and (4) reduces to the Berktay’s approximation [7] which is given as:

p2 

bP20 a2 @ 2 2 E ðsÞ: 16q0 c40 az @ s2

ð5Þ

3. Proposed class of preprocessing techniques Before the proposed preprocessing technique is introduced, we first analyze the demodulated signals from DSBAM and SRAM. The

ð7Þ

An infinite bandwidth transducer must be used to reproduce the infinite harmonics introduced by the square root operator in (7). However, this is not the case as ultrasonic transducers generally have a 3 dB bandwidth less than 6 kHz [3,10].

2

@ E ðsÞ ; @ s2 aðsÞ

ð6Þ

Fig. 2. Preprocessing with quadrature amplitude modulation.

ð8Þ

E.-L. Tan et al. / Applied Acoustics 71 (2010) 486–492

The advantage of using quadrature AM is the flexibility of introducing pre-distortion term g2(s) via the orthogonal carrier. In the following, we provide an example on how we could use a pre-distortion term to reduce distortion in parametric loudspeaker. If we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi consider g1(s) = 1 + mg(s) and g 2 ðsÞ ¼ 1  m2 g 2 ðsÞ, this quadrature AM scheme can be viewed as DSBAM with an orthogonal carrier modulating a pre-distortion term. The demodulated signal of such an quadrature AM scheme is described as: AM pQuad:  2



bP20 a2 @ 2 16q0 c40 az @ s2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 g 21 ðsÞ þ g 22 ðsÞ

bP20 a2 ½2mg 00 ðsÞ; 16q0 c40 az

60 50

d /d x 1e5

488

i¼0

ð2iÞ! ð1  2iÞi!2 4i

m2i g 2i ðsÞ;

forjm2 g 2 ðsÞj < 1:

Hence, we approximate (8) as

g^T ðq; sÞ ¼ g 1 ðsÞ sinðx0 sÞ þ g^2 ðq; sÞ cosðx0 sÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g 21 ðsÞ þ g^22 ðq; sÞ sin½x0 s þ tan1 ðg^2 ðq; sÞ=g 1 ðsÞÞ:

ð11Þ

g^2 ðq; sÞ is a truncated series of (10) and is given as q X

ð2iÞ!

i¼0

ð1  2iÞi!2 4i

m2i g 2i ðsÞ;

forjm2 g 2 ðsÞj < 1;

0

0

5

15 10 Bandwidth (kHz)

20

Fig. 3. Value of du(s)/ds for MAM1.

and 0 6 m 6 0.95. We have plotted the values of du(s)/ds for MAM1 in Fig. 3 and the maximum value of du(s)/ds is found to be 5.73  106. This value is relatively small as compared to x0, which is typically around 80 kp rad/s (based on ultrasonic transducer having a resonance frequency of 40 kHz). Therefore, we can approximate MAM1 as an AM scheme, and MAM1 can be analyzed using the Berktay’s far field solution. Using the same approach, we have found that the maximum value of du(s)/ds for MAM2 and MAM3 are 4.69  106 and 4.14  106, respectively. Hence, we can approximate the demodulated signal from MAMq, (for q = 1, 2, 3) using the Berktay’s far field solution.

Next, we investigate the SPL at fundamental frequency generated by DSBAM, MAMq and SRAM. In this section, we assume that the ultrasonic transducer has infinite bandwidth to simplify our analysis in this section. From (11), we obtain the envelope of MAM1, MAM2 and MAM3 as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 21 ðsÞ þ g^22 ð1; sÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 ¼ 2 1 þ mgðsÞ þ m4 g 4 ðsÞ; 8

EMAM1 ðsÞ ¼

To analyze the demodulated signal from MAMq, we need to determine if there exists any frequency modulation in (11). MAMq can be approximated as an purely AM scheme if the FM in MAMq is fairly weak. In this case, the demodulated signal from MAMq can be easily derived using (5). However, if there is significant amount of FM in MAMq, we have to use the lossless axial solution (4) to take into account the effects of FM in the generation of the demodulated signal. Using (3), the instantaneous angular frequency of (11) becomes

ð13Þ

where g 01 ðsÞ and g^02 ðq; sÞ are the first time derivatives of g1(s) and g^2 ðq; sÞ, respectively. The simplest case of MAMq is MAM1 (q = 1) and the pre-distortion term is given as

g^2 ð1; sÞ ¼ 1  0:5m2 g 2 ðsÞ:

0.5 Modulation, m

3.3. Sound pressure level of demodulated signal from MAMq

3.2. Modulation analysis of MAMq

g 1 ðsÞg^02 ðq; sÞ  g 01 ðsÞg^2 ðq; sÞ ; g 21 ðsÞ þ g^22 ðq; sÞ

0 1

ð12Þ

where q is the order of the proposed technique. Since the proposed technique is adapted from DSBAM with inclusion of an orthogonal carrier cos (x0s) modulating the pre-distortion term g^2 ðq; sÞ, we referred this special type of quadrature AM scheme as modified AMq (MAMq), where q is the order of the proposed technique.

XðsÞ ¼ x0 þ

20 10

ð10Þ

g^2 ðq; sÞ ¼

30

ð9Þ

which is similar to (7) but with a gain of 2. Hence, quadrature AM produces a demodulated signal having twice of the SPL in the case of DSBAM. This increase of SPL is a direct consequence of the orthogonal path that is found in thep quadrature ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAM. It is clear that by adding the pre-distortion term 1  m2 g 2 ðsÞ into DSBAM, we are able to completely remove the distortion in DSBAM. However, this is only true if and only if the ultrasonic transducer has infinite bandwidth. As this is not the case with practical ultrasonic transpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ducer; we approximate 1  m2 g 2 ðsÞ using its Taylor series which is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 1  m2 g 2 ðsÞ ¼

40

ð14Þ

To determine the maximum value of du(s)/ds (second term) in (13), we substitute g(s) as sin (2pf1s), where 20 Hz < f1 < 20 kHz

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 21 ðsÞ þ g^22 ð2; sÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 6 6 1 ¼ 2 1 þ mgðsÞ þ m g ðsÞ þ m8 g 8 ðsÞ; 16 128

ð15Þ

EMAM2 ðsÞ ¼

ð16Þ

and

EMAM3 ðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g 21 ðsÞ þ g^22 ð3; sÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5 1 1 m8 g 8 ðsÞ þ m10 g 10 ðsÞ þ m12 g 12 ðsÞ; ¼ 2 1 þ mgðsÞ þ 128 128 512 ð17Þ respectively. By substituting (15)–(17) into (1), we observe that pffiffiffi MAMq results in a higher initial pressure (by a factor of 2) as compared to DSBAM and SRAM. By analyzing the first two terms in the

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E.-L. Tan et al. / Applied Acoustics 71 (2010) 486–492 Table 1 Frequency from various preprocessing techniques. Preprocessing technique

Envelope

Amplitude of carrier

SPL of fundamental frequency

DSBAM

1 + mg(s)

P0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ mgðsÞ þ 18 m4 g 4 ðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 m6 g 6 ðsÞ þ 128 m8 g 8 ðsÞ 1 þ mgðsÞ þ 16 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 1 1 m8 g 8 ðsÞ þ 128 m10 g 10 ðsÞ þ 512 m12 g 12 ðsÞ 1 þ mgðsÞ þ 128 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ mgðsÞ

pffiffiffi 2P 0 pffiffiffi 2P 0 pffiffiffi 2P 0

bP20 a2 @ 2 16q0 c40 ar @ s2

MAM1 MAM2 MAM3 SRAM

square root operator of (15)–(17), we can express the SPL at fundamental frequency as

pMAMq 2

pffiffiffi 2 b 2P 0 a2 @ 2  ½1 þ mgðsÞ þ    16q0 c40 az @ s2 

bP 20 a2 ½mg 00 ðsÞ þ   : 8q0 c40 az

ð18Þ

From (18), it is clear that the SPL at fundamental frequency is proportional to 0:125P20 mg 00 ðsÞ. The SPL at fundamental frequency of DSBAM and SRAM are also derived and summarized in Table 1. Interestingly, the SPL at fundamental frequency is found to be the same for DSBAM and MAMq. However, it should be noted that the envelope of DSBAM produces a fundamental frequency component in the demodulated signal which has an amplitude two times higher than those from MAMq and SRAM. This leads to the fundamental frequency component from DSBAM having the same SPL as the one from MAMq. Among the three modulation techniques, SRAM is found to have half the SPL of DSBAM and MAMq. 4. Simulation results and observation The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation [18] accurately describes the self-demodulation process and can be used to perform an accurate analysis of the distortion found in the demodulated signal. However, in this paper, we simplify the distortion analysis by employing Berktay’s far field solution which adequately reveals the effects of the limited bandwidth of ultrasonic transducer. In [6], Kite et al. discussed the effect of the bandwidth of ultrasonic transducer on the performance of SRAM. By adapting their approach in [6], we compare the effects of bandwidth of ultrasonic transducer for the case of DSBAM, MAMq and SRAM using the simulation steps shown in Fig. 4. Fast Fourier transform (FFT) is used to analyze the performance of DSBAM, MAMq and SRAM when these techniques are applied to a bandlimited ultrasonic transducer. To determine the effects of limited bandwidth of ultrasonic transducer, we used a low-pass fil-

P0

½mgðsÞ

bP20 a2 @ 2 8q0 c40 ar @ s2

½mgðsÞ

bP20 a2 @ 2 8q0 c40 ar @ s2

½mgðsÞ

bP20 a2 @ 2 16q0 c40 ar @ s2

THD ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 22 þ T 23 þ    þ T 2n1 þ T 2n T 21 þ T 22 þ T 23 þ    þ T 2n1 þ T 2n

 100%;

Preprocessing

Low Pass Filtering

ð19Þ

where T1 and Ti are the amplitude of the fundamental frequency (x1) component and the higher harmonics at ix1 (for i = 2, 3, . . . , n), respectively. Inter-modulation distortion (IMD) measurement gives a measure of distortion products not harmonically related to a multi-tone input signal. For a dual tone input (x1 and x2), we expect a series of inter-modulation harmonics at ix1 ± jx2, where i and j are positive integers larger than zero. For an input having dual frequencies, IMD is given as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P1 P1 2 i¼1 j¼1 Dix1 jx2 ffi  100%; IMD ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P1 2 D21 þ D22 þ 1 i¼1 j¼1 Dix1 jx2

ð20Þ

where D1, D2, and Dix±jx is the amplitude of x1, x2 and inter-modulation harmonics at ix1 ± jx2, respectively. In the following, we compare the performance of DSBAM, MAMq and SRAM when they are applied to a bandlimited ultrasonic transducer. To ensure DSBAM, MAMq and SRAM produce a fundamental frequency component of the same SPL, we replace m with m0 for the envelope of DSBAM and MAMq, where m0 = m/2. Fig. 5 shows the THD values of DSBAM and SRAM. We have arbitrarily chosen the input signal as a single tone at 0.025f0. During the self-demodulation process in air, the envelope of DSBAM produces a second harmonic in air which is the sole contributing factor of THD. When the relative bandwidth is lower than 5%, the second harmonic is attenuated by the limited bandwidth of the ultrasonic transducer. This leads to lower THD values when the

-3 dB Bandwidth, fc

Modulating Signal

½mgðsÞ

ter (LPF) in Fig. 4 to simulate a bandlimited ultrasonic transducer. Let fc and f0 denote the 3 dB bandwidth of the LPF and frequency of the ultrasonic carrier in (1), respectively. Hence, we can define a relative bandwidth given as fc/f0. For a single tone input, the amount of distortion in the demodulated signal is measured by total harmonic distortion (THD) and is given as

–24 dB/octave

3 dB

½2mgðsÞ

bP20 a2 @ 2 8q0 c40 ar @ s2

Modulation

Demodulation in Air

Fig. 4. Block diagram of the setup of our simulation.

Demodulated Signal

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E.-L. Tan et al. / Applied Acoustics 71 (2010) 486–492

35

40 30

35 30

25

25

THD (%)

THD (%)

m=0.1 m=0.3 m=0.5 m=0.7 m=0.9

20

20

15

15 10

10 5

5 0

4

6

8

10

12

14

16

18

20

22

0

24

4

6

8

Relative Bandwidth (%)

10

12

14

16

18

20

22

24

Relative Bandwidth (%)

(a)

(b)

Fig. 5. THD values obtained using ultrasonic transducer having a relative bandwidth of 2.5–25% with: (a) DSBAM and (b) SRAM. Same legend in (b) applies to all plots in this figure.

35

m=0.1 m=0.3 m=0.5 m=0.7 m=0.9

30

30

25

25

IMD (%)

IMD (%)

20 20

15

15 10

10

5

5 0

4

6

8

10

12

14

16

18

20

22

24

0

4

6

8

10

12

14

16

18

Relative Bandwidth (%)

Relative Bandwidth (%)

(a)

(b)

20

22

24

Fig. 6. IMD values obtained using ultrasonic transducer having relative bandwidth of 2.5–25% with: (a) DSBAM and (b) SRAM. Same legend in (b) applies to all plots in this figure.

relative bandwidth is lower than 5% as observed in Fig. 5a. In the case of SRAM, the THD values monotonically decrease when the relative bandwidth increases. For all values of m, it is found that SRAM exhibits lower THD than DSBAM. Fig. 6 shows the IMD values of DSBAM and SRAM. We have arbitrarily chosen the input signal having two tones at 0.0175f0 and 0.025f0. In the case of DSBAM, the inter-modulation distortion occurs at 0.0175f0 ± 0.025f0. When the relative bandwidth is higher than 5%, the inter-modulation distortion at 0.0175f0 + 0.025f0 resides in the passband of the LPF. This leads to higher IMD values as compared to those values when relative bandwidth is lower than 5%. Similar IMD values are found in DSBAM and SRAM when the relative bandwidth is lower than 5%. As the relative bandwidth increases, SRAM exhibits significantly lower IMD values as compared to DSBAM.

The THD and IMD values of MAMq, where q = 1, 2, 3, are summarized in Fig. 7. Since there is little difference between MAM4 (and higher order MAMq) and MAM3 in terms of THD and IMD values, we will restrict our discussion to MAM1, MAM2 and MAM3. From Fig. 7, it is clear that THD and IMD values of MAMq are dependent on the available bandwidth of ultrasonic transducer and modulation index. Similar to SRAM, THD and IMD values of MAMq reduce with increasing relative bandwidth, and the reduction of THD and IMD values diminishes as the relative bandwidth increases beyond 7%. Interestingly, THD and IMD values of MAMq remain largely similar except when the relative bandwidth is higher than 7%. For m = 0.9, the smallest THD values of MAM1, MAM2 and MAM3 are 3.2%, 0.44% and 0.07%, respectively. It is clear that if higher bandwidth is available from the ultrasonic transducer, MAM2 and MAM3 should be used. Similar observations are found

491

30

30

25

25

20

20

IMD (%)

THD (%)

E.-L. Tan et al. / Applied Acoustics 71 (2010) 486–492

15

15

10

10

5

5

0

4

6

8

10

12

14

16

18

20

22

0

24

4

6

8

Relative Bandwidth (%)

10

12

25

25

20

20

15

10

5

5

6

8

10

12

14

16

18

20

22

0

24

4

6

8

Relative Bandwidth (%)

10

22

24

12

14

16

18

20

22

24

(d)

30

30

25

25

20

20

IMD (%)

THD (%)

20

Relative Bandwidth (%)

(c)

15

10

5

5

4

6

8

10

12

14

16

18

Relative Bandwidth (%)

(e)

20

22

24

m=0.1 m=0.3 m=0.5 m=0.7 m=0.9

15

10

0

18

15

10

4

16

(b) 30

IMD (%)

THD (%)

(a) 30

0

14

Relative Bandwidth (%)

0

4

6

8

10

12

14

16

18

20

22

24

Relative Bandwidth (%)

(f)

Fig. 7. Distortion performance of MAM1, MAM2 and MAM3 using ultrasonic transducer having relative bandwidth of 2.5–25%. (a) THD values of MAM1, (b) IMD values of MAM1, (c) THD values of MAM2, (d) IMD values of MAM2, (e) THD values of MAM3 and (f) IMD values of MAM3. Same legend in (f) applies to all plots in this figure.

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in IMD values of MAMq. As compared to SRAM, THD and IMD values of MAMq decrease at a faster rate when more bandwidth is available. In addition, lower THD and IMD values are found with MAMq at low bandwidth. Next, we compare generation efficiency of DSBAM, MAMq and SRAM. For this comparison, we define the generation efficiency as the ratio of the output SPL (at fundamental frequency) over the initial pressure of the carrier. Based on the observations as shown in Table 1, we compute the generation efficiency of DSBAM, MAMq and SRAM as

gDSBAM

gMAM

bP20 a2 @2 mbP0 a2 @ 2 ¼ 2m0 gðsÞ ¼ gðsÞ; 4 2 16P0 q0 c0 az @ s 16q0 c40 az @ s2

pffiffiffi 2 b 2P0 a2 @ 2 mbP0 a2 @2 ¼ pffiffiffi m0 gðsÞ ¼ pffiffiffi gðsÞ; 2 4 4 @ s @ 16 2P0 q0 c0 az 16 2q0 c0 az s2

ð21Þ

ð22Þ

and

gSRAM ¼

bP20 a2 @2 mbP0 a2 @ 2 mgðsÞ ¼ gðsÞ; 4 2 16P0 q0 c0 az @ s 16q0 c40 az @ s2

ð23Þ

respectively. From (21)–(23), we notice that there is a 29% drop in the efficiency of MAMq compared to DSBAM and SRAM. However, MAMq yields the lowest THD and IMD values among the various preprocessing techniques discussed in this paper, especially in the case of bandlimited ultrasonic transducer. 5. Conclusions This paper proposes a new class of preprocessing techniques (MAMq) with performance that can be tuned to suit the bandwidth of the ultrasonic transducer. Unlike SRAM whose performance degrades significantly if narrow bandwidth ultrasonic transducer is used, MAMq (with different order) offers bandwidth flexibility. For large values of m, our simulations reveal that MAMq, where q = 1, 2, 3, results in lower THD and IMD values as compared to SRAM. For relative bandwidth lesser than 7%, MAMq is found to exhibit significantly lower THD and IMD values as compared to DSBAM and SRAM when m > 0.7. For all bandwidths, our simulation revealed that MAM3 outperforms SRAM and DSBAM in terms of THD and IMD values when m > 0.7. As MAMq yields significantly lower distortion at higher values of m, higher SPL can be achieved

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