from which C=O. If the coupling factor for the ith mode, k“), is defined as
The authors wish to thank M. Onoe of the Tokyo University and K. Sawamoto of the Electrical CommunicationLaboratoryfor carefully reading the text and offering valuable criticism and suggestions. TOMOAKI YAMADA NOBUKAZU NIIZEKI Elec. Commun. Lab. Nippon Telegraph and Telephone Public Corp. Musasino-si, Tokyo, Japan
=
h,
(26)
where m is the mass of electrode material per unit area Equation (26) is a special case of (14.42)in [3]. The admittance can be formulated as
’ L. Lundauist. “Dimtal PM m t r a bv transform techniques,”Bell Sys. Tech.J., vol. 48, pp. %7231, February1969. .
ቤተ መጻሕፍቲ ባይዱ
T = T mu
at x t
In a recent paper Lundquist’ derived a formula for the calculation of power spectral densities of data signals. His method is based on the double Fourier transform which relates the power spectral density of any signal to its autocorrelation function. An alternative method for the calculation of power spectral densities is the direct way as used by Bennett’ for the binary signaling wave. The direct way, though in a manner slightly different from Bennett’s, has also been used by Anderson and Sak3in their treatment of digital FM signals. Although Lundquist’s method may be easier than the direct way, as for instance in the FM case, an extension of Bennett’s method leads to a very simple formula for a large class of data signals. The extension is possible for all data signals that can be written as a random process:
where n is positive integers. It should be noted that (24) is the same as the formula quoted by Tiersten as obtained by Coquin [3, eq. (9.77)].and that the result is obtained in the present treatment as the direct consequence of the general formula expressing the admittance of the vibrating piezoelectric plate. The present mathematical method can be easily applied to the cases in which boundary conditions different from (4) and ( 5 ) are to be assumed. As examples, the admittance formulas are givenbelowfor the cases in which 1) mass effect of the electrodes is considered, and 2) air gap exists between the electrodes and the plate. 1) Consider a boundary condition given as
4=
1 (eo)i(uo)Jc~l (Dl/ES1)Xl + C i= 1
3
where C is constant. Substitution of (20) into (5)yields
Full account of the derivation of the above formulas will be published elsewhere [4]. ACKNOWLEDGMENT
PROCEEDINGS where To is the bit length. The integer valued subscripts x, are independent identically distributed random variables with Pr (x, = k ) = p b
2) Boundary conditions are given as
T=O
at x 1 = k h
Dl = E ~ E , at x1 = k h
E , d + S_:*Eldxl = V
By integrating both sides of (15) one obtains
where V is the potential difference, and d, E,, and E,, are the thickness of the gap between the electrodes and the plate, electric field in the gap, and dielectric constant of the material in the gap, respectively. Theadmittance for this configuration is given as
Manuscript received December 23,1969; revised March2.1970.
W. R Bennetr “Statisticsof regeneratwe digital transmission,”Bell Sys. Tech. J., vol. 37, pp. 150-1542, November 1958. R. R.Anderson and J. “Spectra of digital FM,”Bell Syr. Tech. J., vol. 44, pp. 1165-1 189, July-August 1965.
the admittance Y is obtained directly from (21):
where t=U is the thickness and A the area of the plate. For the twodimensional case, where i= 1, 5 the above equation is the same as the one derived by Foster et al. [2] in the analysis of thin-film transducers of crystal class 6 mm. From (23) it is easily shown that the resonant frequencies are obtained as the solutions of the equation
942
PROCEEDINGS OF THE IEEE, JUNE 1970
where the time dependence of uo is assumed to be sinusoidal. Taking into account the boundary condition(16A the solution of (17) can be expressed as
REFERENCES
[ I ] H. F. Tiersten, “Thickness vibrations of piezoelearic plates,” 1. Acousr. SOC.Am., vol. 35, pp. 5 S 5 8 , January 1963. [Z] N . F. Foster, G . A. Coquin,G . A. Rozgonyi, and F. A. Vannatta, “Cadmium sulphide and zinc oxide thin-film transducers,” IEEE Tram. Sonics and Ulfrasonics, vol. SU-15, pp. 28-41, January 1968. [3] H. F. Tiersten, Linear Piezoelectric flare Vibrations. New York: Plenum Ress 1969, p. 93. [4] f. Yamada and N . N i k k i ,“A new formulation of thickness vibration of piedectric n plates,” to be published i Rea. Elec. Commun. Lab. N.T.T. (Tokyo),1970.