Acoustic Waves part 16 docx

Dissipation of Acoustic Waves in Barium Monochalcogenides 439 (α/f 2 ) th is directly proportional to rate of heat transfer from compressed regions to rarefied regions. In the low temperature range, 50-200 K, heat is transferred at faster rate from compressional regions to the rarefied regions resulting larger rate of thermoelastic loss. The rate of increase of thermoelastic loss is small beyond 200 K. 2.3 Phonon processes and drag on dislocations A dislocation is a linear imperfection in a crystal. In edge dislocation, near the dislocation line, the crystal is severely strained. In a screw dislocation, Burger vectors are parallel to the dislocation line. In general, a dislocation is composed of mixtures of screw and edge dislocations. Another process for which thermal losses due to p-p interaction can produce an appreciable effect is the drag on disclocations as they are moved through a lattice. Leibfried et al. (1954) discussed the mechanism of scattering of phonons by moving disclocations and the results show that the resulting differential produces a drag force which is proportional to the velocity of the disclocation. Mason (1965) proposed a theory to explain the mechanism involved in the drag produced on a dislocation by phonon-viscosity. This was evaluated on the basis of the effect caused by the change in dimensions of phonon modes and their subsequent equilibrium through a thermal relaxation process. Dislocation damping due to screw and edge dislocations also produces appreciable loss due to phonon-phonon interaction. The loss due to this mechanism can be obtained by multiplying dislocation viscosities by square of dislocation velocity. Dislocation damping due to screw and edge dislocations is given by equations (21) and (22). The Phonon-viscosity, which is analogous to shear-viscosity in liquids damps the motion of both type (screw and edge) disclocations and has the value 2 //3 th EDk C V ED ητ =<>= (19) These phonon-viscosities are presented in the form of drag coefficients for the motion of screw and edge type of disclocations. Here the Cortell’s (Cortell, 1963) condition 0 3/4ab= is valid, where 0 a the disclocation core radius and ‘b’ is is the Brugger’s vector. screw B and ed g e B are given by 22 /8Bb a π = (20) substituting 0 3/4ab= the above equation reduces to, 0.071 screw B η = (21) and 22 (0.0532 0.0079( / ) /(1 ) edge BK η μχσ =+ − (22) where σ , μ , K and χ are Poisson’s ratio, shear modulus, bulk modulus and compressional viscosity respectively. These values can be calculated using the relations 11 12 44 ()/3CCC μ = −+ , 11 12 (2)/3KC C = + , and (4/3 ) lS χ ηη = − (23) Acoustic Waves 440 Compound B screw B edge Long. Shear Long. Shear BaS 0.23 0.10 0.45 0.55 BaSe 0.29 0.17 0.60 0.79 BaTe 0.47 1.30 1.07 3.22 Table 2. Phonon viscosity due to screw and edge dislocation at 300K longitudinal (in cp) and shear (in mp.) waves. Debye average velocity and Debye temperature have been calculated using equations (16) and (17) and are presented in Table 2. Square average Gruneisen numbers < γ i j2 > l and < γ i j 2 > s* and average square Gruneisen parameter < γ i j > 2 l and < γ i j > 2 s and < γ i j > 2 s* for longitudinal and shear waves, nonlinearity coupling constants D l , D s , D s * and their ratios D l /D s , and D l /D s * along different directions of propagation are given in Table 3. Results are as expected [Mason (1967), Kor and Singh (1993)]. Compound Direction < γ i j2 > l < γ i j > 2 l < γ i j > 2 s < γ i j > 2 s* D l D s D s* D l /D s D l /D s* 100 0.94 0.17 0.04 7.82 0.37 20.81 BaS 110 1.06 0.26 0.15 1.93 8.63 1.43 17.37 6.03 0.49 100 0.90 0.24 0.04 7.27 0.43 16.79 BaSe 110 1.04 0.36 0.22 1.80 8.04 1.98 16.28 4.06 0.49 100 1.68 1.28 0.30 10.63 2.73 3.88 BaTe 110 2.14 1.75 4.49 1.33 12.93 40.46 12.00 0.31 1.00 Table 3. Square Average and average square Gruneisen number for longitudinal < γ i j2 > l, < γ i j > 2 l and shear < γ i j > 2 s , < γ i j > 2 s* Waves, nonlinearity coupling constants D l , D s and nonlinearity coupling constants ratios D l / D s , D l / D s * at 300K l for longitudinal wave s for shear wave, polarized along [001] s* for shear wave, polarized along [ 110 ] Viscous drag due to screw (B screw ) and edge dislocations have been obtained (B edge ) using equation (21) and (22), as given in Table 2. The phonon mean free path due to phonon-phonon collision is a rapidly changing function of temperature at low temperatures. Fig. 4 shows the th τ vs T plot for barium monochalcogenides Thermal relaxation time is evaluated using equation (6). Temperature variation of thermal relaxation time is shown in Fig. 4 which shows exponential decay according to relation τ = τ o exp (- t/T), where τ o and t are constants. From the values of thermal relaxation time, it can be seen that the condition th ω τ <<1 is satisfied even at GHz range acoustic wave frequency. Dissipation of Acoustic Waves in Barium Monochalcogenides 441 0 2 4 6 8 10 12 0 100 200 300 400 500 600 T (K) BaS BaSe BaTe Fig. 4. Temperature variation of thermal relaxation time (τ). 3. Conclusions Acoustical dissipation and related parameters have been evaluated over a wide temperature range using simple approach and starting from second and third order elastic constants. These values of second and third order elastic constants have been used to obtain acoustical Gruneisen parameters and non-linearity coupling constants. Utilizing values of non- linearity coupling constants, ultrasonic arttenuation due to phonon-phonon interaction, thermoelastic loss and dislocation dampming due to screw and edge dislocations have been obtained over a wide temperature range. In the present approach, Grunesen parameters have been evaluated for longitudinal and shear modes by considering only finite number of modes (39 modes for longitudinal wave while 18 modes for shear waves). However, a more rigorous approach is needed, in which all possible phonon modes can be incorporated. 4. Acknowledgements I am thankful to the University Grants Commission, New Delhi (Government of India) for financial assistance. 5. References Akhiezer, A., Absorption of sound in metals, J. Phys. (USSR), 1 (1939) 289-298. Bouhemadou, A, Khenata, R., Zegrar, F., Sahnoun, M, Baltache, H., Resh, A.H, Computational Material Science 38, 263 (2006) Bommel H. E and Dransfield , K. Excitation and attenuation of hypersonic waves in quartz , Phys. Rev. 177, 145 (1960). Bommel, H.E. and Dransfeld, K., Phys. Rev., 117 (1960) 245. Breazeale, M.A. and Philip, J., J. Phys. (Colloq), 42 (1981) 134. Brugger, K, Phys. Rev. A. 133, 1611 (1964) τ (10 -10 .sec) Acoustic Waves 442 Charifi Z, Baoziz H., Hassan, F El Haj and Bouarissa, N, J. Phys. Condens. Matter 17, 4083 (2005) Cervantes, P., Williams, Q, Cote, M, Rohlfing, M, Cohen, M. L. and .Louie, S G Phys. Rev. B, 58 (15) 9793 (1998) Elmore, P.A. and Breazeale, M.A., Dispersion and frequency dependent nonlinearity parameters in a graphite–epoxy composite, Ultrasonics, 41 (2004) 709-718. Fabian, J. and Allen, P.B., Theory of sound attenuation in Glasses: The role of thermal vibrations, Phys. Rev. Let., 82 (7) (1999) 1478-1481. Ghate, P.B., Third order elastic constants of Alkali halide crystals, Phys. Rev., 139 (1965) A1666-A1674. Ghate , P.B., Phys. Rev. B 139 (5A) A1666. (1965) Hassan, F. El. Haj and Akbarzadeh, H.Computational and Material Science. 38, 362 (2006) Leibfried, G. and Hahn, H., Temperature dependent elastic constants of alkali halides, Z. Physik, 150 (1958) 497-525. Ludwig, W. and Leibfried, G., Theory of anharmonic effects in crystals, Solid State Physics, Academic Press New York, 12 (1967). Mason, W.P., Ultrasonic attenuation due to lattice-electron interaction in normal conducting metals, Phys. Rev., 97 (1955) 557-558. Mason, W.P. and Bateman, T.B., Ultrasonic wave propagation in pure Si and Ge, J. Acoust. Soc. Am., 36 (1964) 645. Mason, W.P., Effect of impurities and phonon processes in the ultrasonic attenuation of germanium crystal, quartz and silicon, Physical Acoustics, Academic Press New York, IIIB (1965) 237. Mason, W.P., Relation between thermal ultrasonic attenuation and third order elastic moduli for waves along <110> axis of a crystal, J. Acoust. Soc. America, 42 (1967) 253. Mason, W.P. and Rosenberg, A., Thermal and electronic attenuations and dislocation drag in the hexagonal crystal Cadmium, J. Acoust. Soc. America, 45 (2) (1969) 470-480. Pippard, A.B., Ultrasonic attenuation in metals, Philos. Mag., 46 (1955) 1104. Singh, R.K., Singh R. P. Singh and Singh M.P , Proc. 19 th International Congress on Acoustics (ICA-2007), Spain (Madrid) 2007. Singh R. K. *, Singh R. P., Singh M. P., and Chaurasia, S. K., Acoustic Wave Propagation in Barium Monochalcogenides in the B1 Phase Acoustical Physics, 2009, Vol. 55, No. 2, pp. 186–191. Woodruff, R.O. and Ehrenreich, H., Phys. Rev., 123 (1962) 1553. 20 Statistical Errors in Remote Passive Wireless SAW Sensing Employing Phase Differences Y.S. Shmaliy, O.Y. Shmaliy, O. Ibarra-Manzano, J. Andrade-Lucio, and G. Cerda-Villafana Electronics Department, Guanajuato University Mexico 1. Introduction Passive remote wireless sensing employing properties of the surface acoustic wave (SAW) has gained currency during a couple of decades to measure different physical quantities such as temperature, force (pressure, torque, and stress), velocity, direction of motion, etc. with a resolution of about 1% [1]. The basic principle utilized in such a technique combines advantages of the precise piezoelectric sensors [2, 3, 4], high SAW sensitivity to the environment, passive (without a power supply) operation, and wireless communication between the sensor element and the reader (interrogator). Several passive wireless SAW devices have been manufactured to measure temperature [1], identify the railway vehicle at high speed [5], and pressure and torque [6]. The information bearer in such sensors is primarily the time delay of the SAW or the central frequency of the SAW device. Most passive SAW sensors are designed as reflective delay lines with M reflectors 1 and operate as sketched in Fig. 1. At some time instant t 0 = 0, the reader transmits the electromagnetic wave as an interrogating radio frequency (RF) pulse (K = 1), pulse burst (K > 1), pulse train, or periodic pulse burst train. The interdigital transducer (IDT) converts the electric signal to SAW, and about half of its energy distributes to the reflector. The SAW propagates on the piezoelectric crystal surface with a velocity v through double distances (2L 1 and 2L 2 ), attenuates (6 dB per µs delay time [5]), reflects partly from the reflectors (R 1 and R 2 ), and returns back to the IDT. Inherently, the SAW undergoes phase delays on the piezoelectric surface. The returned SAW is reconverted by the IDT to the electric signal, and retransmitted to the interrogator. While propagating, the RF pulse decays that can be accompanied with effects of fading. At last, K pairs of RF pulses (Fig. 1b) appear at the coherent receiver, where they are contaminated by noise. In these pulses, each inter distance time delay Δτ (2k)(2k–1) = 2(L 2 – L 1 )/v, k ∈ [1,K], bears information about the measured quantity, i.e., temperature [1], pressure and torque [5], vehicle at high speed [6], etc. To measure Δτ (2k)(2k–1) , a coherent receiver is commonly used [7], implementing the maximum likelihood function approach. Here, the estimate of the RF pulse phase relative to the reference is formed to range either from –π/2 to π/2 or from –π to π by, respectively, 1 Below, we consider the case of M = 2. Acoustic Waves 444 Fig. 1. Operational principle of remote SAW sensing with phase measurement: a) basic design of passive SAW sensors and b) reflected pulses at the coherent receiver detector [25]. ˆ arctan , Q I θ = (1) arctan( / ), 0 ˆ 0 , arctan( / ) , 0, 0 QI I Q QI I Q θ π ≥ ⎧ ⎪ ≥ = ⎧ ⎨ ±< ⎨ ⎪ < ⎩ ⎩ (2) where I and Q are the in-phase and quadrature phase components obtained for the received pulse. With differential phase measurement (DPM), the phase difference in every pair of pulses is calculated by 221 ˆˆ ˆ kkk θθ − Θ= − (3) and used as a current DPM. Here several estimates may be averaged to increase the signal- to-noise ratio (SNR) [7]. Averaging works out efficiently if the mean values are equal. Otherwise, the differential phase diversity is of interest to estimate either the vehicle's velocity (Doppler shift) or the random error via 1 ˆˆˆ . kkk − Ψ=Θ−Θ (4) Statistical Errors in Remote Passive Wireless SAW Sensing Employing Phase Differences 445 An accurate estimate ˆ k Θ is a principal goal of the receiver. To obtain it with a permitted inaccuracy in the presence of noise, the interrogating signal must be transmitted with a sufficient peak power that, however, should not be redundant. The peak power is coupled with the SNR. Therefore, statistical properties of ˆ k Θ and ˆ k Ψ are of prime interest. Knowing these properties and the peak power of the interrogating pulse, one can predict the measurement error and optimize the system. In this Chapter, we discuss limiting and approximate statistical errors in the estimates (3) and (4). 2. Signal model For SAW sensors with identification marks, the readers are often designed to interrogate the sensors with a linear frequency modulated (LFM) RF impulse request signal [8, 9] 2 00 () 2 ()cos 2 , 2 t xt Sat ft α πθ ⎛⎞ =++ ⎜⎟ ⎜⎟ ⎝⎠ (5) where 2 S and θ 0 are the peak-power and initial phase, respectively, f 0 is the initial carrier frequency, and t is the current time. The LFM pulse can have a near rectangular normalized waveform a(t) of duration T such that α = Δ ω /T , where Δ ω is a required angular frequency deviation, overlapping all the sensor responses. It turns out that noise does not perturb x(t) substantially in the sensor. Therefore, assuming Gaussian envelope in the reflected pulses, the induced SAW reflected from the reflectors R 1 and R 2 and then reconverted and retransmitted can be modeled with, respectively, 22 21 121 1 () 21 21 21 1 () () 2 ( ) cos[2 ( ], ) k K k k K btt kkk k st u t Ste ftt βπϑ − − = −− −−− = = =+ ∑ ∑ (6) 22 2 22 1 () 222 1 () () 2() cos[2 ()], k K k k K btt kkk k st u t Ste ftt βπϑ = −− = = =+ ∑ ∑ (7) where 2 (), [1,2 ], i ti K β ∈ is a normalized instantaneous power caused by attenuation and fading. The full phase shifts relative to the carrier and its constituent induced during the SAW propagation are given by, respectively, 21 21 21 0 , kkk ψ ϑ φθ −−− = −+ (8) 2220 , kkk ψ ϑ φθ =−+ (9) where k ∈[1,K], φ 2k–1 and φ 2k are phase shifts caused by various reasons, e.g., RF wave propagation, Doppler effect, frequency shift between the signals, etc. Here, the relevant information bearing phase shifts can be evaluated with, respectively, Acoustic Waves 446 12 21 21 2 2 4and4. kk kk LL ψ f ψ f vv ππ −− == (10) At the receiver, each of the RF pulses u i (t), i ∈[1, 2K], is contaminated by zero mean additive stationary narrowband Gaussian noise n(t) with a known variance σ 2 , so that, at t = t i , we have a mixture () () () ()cos[2 ()], ii i ii y tutntVt ft t π θ = += + (11) where V i ≥ 0 is a positive valued envelope with the Rice distribution and |θ i | ≤ π is the modulo 2π random phase 2 . Although the frequency f i in the reflected pulses can be different, below we often let the frequencies be equal, by setting f k = f 0 . The instantaneous peak SNR in y i (t) (Fig. 1b) is calculated by 2 2 () . i i St β γ σ = (12) Because of noise, the actual phase difference 3 221212 kkk k k ψψ ϑϑ −− Θ= − = − (13) 211 22 4 [] kk f LfL v π − =− (14) 0(2)(21) 2 , kk f π τ − ≅ −Δ (15) where ϑ 2k–1 = ϑ 1 (t 2k–1 ), ϑ 2k = ϑ 2 (t 2k ), ψ 2k–1 = ψ 1 (t 2k–1 ), and ψ 2k = ψ 2 (t 2k ) cannot be measured precisely and are estimated at the coherent receiver via the noisy phase difference θ 2k – θ 2k–1 as (3), using (1) or (2). Similarly, the time drift in k Θ is evaluated by 1 kkk− Ψ=Θ−Θ (16) 2212223 . kk k k ψψ ψ ψ −−− = −+ + − (17) So, instead of the actual angle k Θ , the coherent receiver produces its random estimate ˆ k Θ and instead of k Ψ we have ˆ k Ψ . Note that, in the ideal receiver, Θ k and ˆ k Θ as well as k Ψ and ˆ k Ψ have the same distributions [11]. 3. Probability density of the phase difference Because both the received signal and noise induced by the receiver are essentially narrowband processes, the instantaneous phase θ i in (11) has Bennett's conditional distribution 2 Throughout the paper, we consider the modulo 2π phase and phase difference. 3 For the sake of simplicity, we assume equal phases φ (t 2k ) and φ (t 2k–1 ). It is important that a linearly drifting phase difference φ (t 2k ) – φ (t 2k–1 ) does not affect distribution of Θ k [8] and may be accounted as a regular error. Statistical Errors in Remote Passive Wireless SAW Sensing Employing Phase Differences 447 () 2 sin (|, ) 2 cos cos , 2 i ii i iii i i i e pe γ γθ γ θ γϑ γ θ θ ππ − − =+ Φ   (18) where 2 /2 1 2 and ( ) d x t iii xet π θθϑ − −∞ =− Φ = ∫  is the probability integral. It has been shown in [17, 13] that (18) is fundamental for the interrogating RF pulses of arbitrary waveforms and modulation laws. Employing the maximum likelihood function approach, the coherent receiver produces an estimate ˆ i θ of θ i [11]. Assuming in this paper an ideal receiver, we let ˆ i θ = θ i . Provided (18), the pdf of the information bearing phase difference Θ k can be found for equal and different SNRs in the pulses and we notice that the problem is akin to that in two channel phase systems. 3.1 Different SNRs in the RF pulses Most generally, one can suppose that the SNRs in the reflected pulses are different, γ 2k–1 ≠ γ 2k , owing to design problems and the SAW attenuation with distance. The relevant conditional pdf was originally published by Tsvetnov in 1969 [16]. Independently, in 1981, Pawula presented an alternative formula [21] that soon after appeared in [18] in a simpler form of () sin 212 0 1 (| , ,) cosh sin cosh(cos) d, 22 y kk kk e pyy e y π γ λ γγ γ γ λ γ π − − ⎡ ⎤ ΘΘ=++ × ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∫  (19) where 21 2 2 21 212 ( ) /2, ( ) /2, arctan , cos , kk kk kkk γ λ γγ γ γγ γ ξ λ γγ −− − =+ =− = = Θ    and . kkk Θ=Θ−Θ  An equivalence of the Tsvetnov and Pawula pdfs was shown in [22]. To avoid computational problems, Tsvetnov expended his pdf in [20] to the Fourier series 212 212 1 11 (| , ,) ( , )cos( ), 2 N kk kk nk k k k n pcn γγ γγ ππ −− = ΘΘ=+ Θ−Θ ∑ (20) where N is proportional to the maximum SNR in the pulses, c n ( γ 2k–1 , γ 2k ) = c n ( γ 2k–1 )c n ( γ 2k ), and /2 (1)/2 (1)/2 () , 222 i i ii ni n n ceI I γ πγ γγ γ − +− ⎡ ⎤ ⎛⎞ ⎛⎞ =+ ⎢ ⎥ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎣ ⎦ (21) where I v (x) is the modified Bessel function of the first kind and fractional order v. The mean and mean square values associated with (20) have been found in [24] to be, respectively, 1 21 2 1 (1) 2(,)sin, n N knkkk n cn n γγ + − = − Θ =Θ ∑ (22) 2 2 212 2 1 (1) 4(,)cos. 3 n N knkkk n cn n π γγ − = − Θ =+ Θ ∑ (23) Acoustic Waves 448 3.2 Equal SNRs in the RF pulses In a special case when the SNRs in the pulses ara supposed to be equal, k γ = γ 2k–1 = γ 2k , the phase difference has the conditional Tsvetnov pdf [20] () /2 cos 0 (|,) 1 cos , 2 k k z kkk k k e p zedz π γ λ γγλ π − ⎡ ⎤ ⎢ ⎥ΘΘ= + + ⎢ ⎥ ⎣ ⎦ ∫ (24) where λ k = k γ cos k Θ  . Note that Tsvetnov published his pdf in the functional form. The integral equivalent (24) shown in [11] does not appear in Tsvetnov's works. It can be observed that, by equal SNRs, (19) becomes (24), although indirectly. 3.3 Probability density of the differential phase difference It has been shown in [22] that the pdf of the differential phase difference (DPD) has two equivalent forms. The first form of this pdf appears to be 12 2212223 12 00 (|, , , ,) 1 cosh( cos )cosh( cos ) ( , , )d d , 24 kkk k k k k pp e AxyFxyxy ππ γγ γγ γ γ γγ π Ψ−−− −− ΨΨ ⎡ ⎤ =+ Ψ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∫∫    (25) where 1212 22322 ( )/2, ( )/2, kk kk γ γγ γγγ −−− =+ =+ 12 2 2 12 22 3 0 12322 2212 1 22212223 1 sin sin cos ( ) 2 cos( )sin cos sin ( ) 1 ( ) cos( 2 ), 2 kk k k kk kk kk k k Fxy Ia xyIa Ia γγ γ γ γ γ γγ γ ζ γγ γ ζ γγ γ γ ζ −−− −− − −−− ⎡⎤ =+ Ψ ⎢⎥ ⎣⎦ ⎡⎤ +Ψ++ ⎣⎦ +Ψ+    (26) 2212223 1/2 22 212 2322 (,, ) 2 sin sin cos sin sin , kk k k kk kk axy x y xy γγ γ γ γγ γγ −−− −−− ⎡ Ψ =Ψ ⎣ ⎤ ++ ⎦  (27) arctan( / ), 0 0 (,, ) , arctan( / ) , 0, 0 QI I Q xy QI I Q ζ π ≥ ⎧ ⎪ ≥ Ψ= ⎧ ⎨ ±< ⎨ ⎪ < ⎩ ⎩  (28) 2322 sin , kk Qy γγ −− =− Ψ  (29) 212 2322 sin sin cos . kk kk Ix y γγ γγ −−− = +Ψ  (30) By changing the variables, namely by substituting sin x with x and sin y with y, the pdf transforms to its second equivalent form of [...]... Mises/Thkhonov-based distributions for systems with differential phase measurement, Signal Process., Vol 85, No 4, (Apr 2005)(693-703), ISSN 0165 1684 [15] Tikhonov, V I The effect of noise on phase-lock oscillation operation, Automatika i Telemekhanika, Vol 20, No 9, (Sep 1959)(1188-1196) [16] Tsvetnov, V V Unconditional statistical characteristics of signals and uncorrelated Gaussian noises in two-channel phase systems,... S.; Shkvarko, Y V.; Torres-Cisnerros, M.; Rojas-Laguna, R & IbarraManzano, O A stochastic analysis of an anharmonic sensor phase response, IEEE Sensors J., Vol 3, No 2, (Apr 2003)(158 -163 ), ISSN 1530- 437X 466 Acoustic Waves [18] Pawula, R F.; Rice, S O & Roberts, J H Distribution of the phase angle between two vectors perturbed by Gaussian noise, IEEE Trans Comm., Vol COM-30, No 8 (Aug 1982)(1828-1841),... [21] Pawula, R F On the theory of error rates for narrow-band digital FM, IEEE Trans Comm., Vol COM-29, No 11, (Nov 1981) (163 4 -164 3), ISSN 0090- 6778 [22] Shmaliy Y S & Shmaliy, O.Y Probability density of the differential phase difference in applications to passive wireless surface acoustic wave sensing, Int J Electron Commun., Vol 63, No 1, (Jan 2009)(623{631), ISSN 1434-8411 [23] Karpov, A F Parameter... 〉 = 0, then the estimates Θ are mutually unbiased ˆ ˆ diversity Ψ k k k ˆ 〉 causes a bias in the multiple DPM that may be associated with the sensor Otherwise, 〈 Ψ k movement (Doppler effect) 454 Acoustic Waves Fig 3 Mean errors calculated for equal SNRs by (19) rigorously (bold) with cn (21) and approximately (dashed) by (44) with n0 (48)[25] 5.1 Mean error The mean error (bias) in the estimate can... behavior (61) is dashed [25] It may also be calculated approximately in two important special cases: γ k < 0 dB, then • 2 εk ≅ π2 3 + • 2 + Θ2 k − πγ k e γ k2 e −γ k ⎛ γ k 16 −γ k ⎛ γ γ2⎞ ⎜ 1 + k + k ⎟ cos Θ k + Θ k sin Θ k ⎜ 4 16 ⎟ ⎝ ⎠ 2 ( ) (60) ⎞ ⎜ + 2 ⎟ cos 2 Θ k + 2 Θ k sin 2 Θ k , ⎝ 3 ⎠ ( ) If γ k > 13 dB, then 2 εk ≅ 1 γk (61) Figure 4 sketches the root MSE (RMSE) calculated rigorously, by... asymptotic line given by (61) 5.3 Error variance and Cramér-Rao lower bound A measure of noise in the estimate is the variance calculated for a single DPM by 2 ˆ ˆ σ Θk = 〈Θ 2 〉 − 〈Θ k 〉 2 k (62) 456 Acoustic Waves With multiple DPM, the variance is often substituted with the Cramér-Rao lower bound (CRLB) having approximate, although typically simple representations Supposing that the measurement vector... (69) ζ = e −γ k π sin ζ π /2 ∫ 0 eγ k cos ζ cos t dt 1 − cos ζ cos t (70) A similar formula employed in [18] is known as the conditional SER, PE ( M|γ k ) = 2 π ∫ π /M p( Θ k |γ k )dΘ k , (71) 458 Acoustic Waves where M is an integer but may be arbitrary in a common case By M = π/ζ, (71) becomes (70) and we notice that (71) was performed in [18] in the integral form that, by symmetry of the integrand,... If the SAW reader system measures velocity of a moving object, then D gives a measure of acceleration 2 In applications, of interest are the mean value 〈D〉 and variance σ D of the drift rate 460 Acoustic Waves Fig 6 Error probability for equal SNRs: rigorous (bold), by (70), and approximated by von Mises/Tikhonov's distribution (dashed), by (72) [25] 7.1 Mean drift rate The mean drift rate 〈D〉 can... the estimate Table 1 gives the relevant values for Ψ ranging from 0.1π to 0.9π A simple measure of accuracy used here is when the exact and approximate values become visually indistinguishable 462 Acoustic Waves 2 Fig 8 The standard deviation σ D for different Ψ : actual (bold), by (90); Gaussian approximation (dashed) for Ψ = 0.032 rad; and simulation with Gaussian distribution, using (91) γ dB 0.1π... ≅ 1− ≅ 1− 1 0 2 N I n (α 1 ) I n (α 2 ) ⎤ ζ ⎡ ⎢1 + 2 ∑ ⎥ ,ζ π⎣ n = 1 I 0 (α 1 ) I 0 (α 2 ) ⎦ ≅ 1− ≤1− 0 ζ (1 + 2N ) , ζ π ζ ( 1 + 4 max γ 1,2 ) ,ζ π π, 1 π ,1 π, γ 1,2 , γ 1,2 (96) (97) (98) 464 Acoustic Waves Fig 9 Error probability of the drift rate to exceed a limit ζ with equal SNRs in the pulses: rigorous (dashed), by (37), and approximate (bold), by (95) Figure 9 illustrates the error probability . seen that the condition th ω τ <<1 is satisfied even at GHz range acoustic wave frequency. Dissipation of Acoustic Waves in Barium Monochalcogenides 441 0 2 4 6 8 10 12 0 100 200 300. Philip, J., J. Phys. (Colloq), 42 (1981) 134. Brugger, K, Phys. Rev. A. 133, 161 1 (1964) τ (10 -10 .sec) Acoustic Waves 442 Charifi Z, Baoziz H., Hassan, F El Haj and Bouarissa, N, J elastic constants of Alkali halide crystals, Phys. Rev., 139 (1965) A1666-A1674. Ghate , P.B., Phys. Rev. B 139 (5A) A1666. (1965) Hassan, F. El. Haj and Akbarzadeh, H.Computational and Material