In the Schottky-barrier-gate Gunn-effect digital device [8], this jitter is estimated from (1) to be 4-6 ps, if the rate of rise of the gate input voltage dVG/dt is 1 V/ns.
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PROCEEDINGS LETTERS
IV. JITTER IN GUNK-EFFECT DIGITALDEVICES The front-edge fluctuation of the output current pulse Af is considered as the thermally induced jitter in Gunn-effect digital devices and can be estimated by using (1). I n the Schottky-barrier-gate Gunn-effect digital device [8],this jitter is estimated from ( 1 ) to be 4-6 ps, if the rate of rise of the gate input voltage d V G / d t is 1 V/ns. If a device is triggered by the output of another device of this type, d VG/dt exceeds 1 V/ns and the jitter caused by the thermal noise is estimated to be around 1 ps.
P(s) then
Ckj
= Y t ( j ) ( s k ) / j ! . Applying Leibniz’s rule, the j t h derivative of
P ( s ) is
Substituting in thisequation j ! & ( s t ) , one obtains
[l] G. S. Hobson. ‘Source of FM noise in cavitysontrolled Gunn-eEect oscillators.” Electron. vol. 3. pp. 63-64,Feb. 1967. [2] W.Harth. *Equivalent noise temperature of bulk negativeresistance devices,’ Arch. Ekck. Obatragnmg. vol. 26. pp. 149-150, 1972. [3] K. Kurokawa, ‘Noise in synchronized oscillators,” I E E E Trans. M i n m a o e Theory Tech., vol. MTT-16. pp. 234-240, Apr. 1968. [4] M. Ohtomo. ‘Experimental evaluation of noise parametersin GUM and avalanche oscillators.” I E E E Trams. M i n o w a w TheoryTcch., vol. MTT-20.PP. 42-37, July 1972. [SI A. Ataman and W. Harth. “Intrinsic FM noise of Gunn oscillators.” I E E E Trams. Ekclrom Devices. vol. ED-20, pp. 12-14. Jan. 1973. [6] P. N. Butcher and W. Fawcett. “The intervalley transfer mechanism of negative resistivity inhulk semiconductors.” Proc. Phys. Soc., vol. 86, PP. 1205-1219. 1965. [7] G . S. Hobson. ‘Three-dimensional limitations on space-charge growth in transferredslectron devices,”Brit. J . A p p l . Phys. (J. Phys. D). ser. 2, vol. 2. 1969. [E] T. Sugeta.H. Yanai. and K. Sekido, ‘Schottky-gate bulk effect digital devices.” Proc. I E E E (Lett.), vol. 59. pp. 1629-1630, Nov. 1971.
Leu..
and dividing both sides by
s=st, i
V. CONCLUSION By considering two kinds of fluctuations in the domain formation process, an expression for the thermally induced FM noise in Gunn oscillators has been derived. The result is in good agreement with experimental data. The reduction of the expression will be reported in detail in a later paper. ACKNOWLEDG~NT The authors wish to thank Prof. W. Harth of Institut fur Hochfrequenztechnik, Technische Universitat Braunschweig, Braunschweig, Germany; Prof. Y. Taki of the University of Tokyo, Tokyo, Japan; Dr. K. Sekido, M.Takeuchi,and A. Higashisaka of the Nlppon Electric Company; and Dr. N. Mizushima, M. Yamaguchi, and K. Kurumada of Musashino Electrical Communication Laboratory, Nippon Telegraph and Telephone Public Corporation, for h e l p ful discussions. REFERENCES
yt(S)Qk(S)
=
v t j
ckiutj-i i-0
where
and k = l , 2 , 3; * , n ;j = O , 1, 2, * . , mt-1. From (3), the coefficients Ctj can readily be found successively by settingj = O , 1 , 2 , . into the following recurrence formula:
.
Ctj
= Vw
- Rti
(8)
where
j-0
I t is noted that the evaluation of C t j by ( 8 ) needs a knowledge of Cto, Ctl, . . , Ctj-1, beforehand. In order to have another useful check, one may solve for C k j directly from the mt simultaneous equations given by ( 3 ) for some particular index k . After some mathematical manipulations one finds that
.
j
ctj =
WtiVtj-i i-0
where
Recursive Formulas for the PartialFraction Expansion of a Rational Function with Multiple Poles
In ( l l ) , the symbol 2 indicates summation over all solutions nonnegative integers of the equation
FENG-CHENG CHANG
I
Zi
in
+ +
& = 1, . * +jlj =j. Absfruct-Two simple recursive formulas arederivedforobtaining the coe5cients of the partial fraction expansion with multiple poles. For convenience some of Wtj are listed in the following: The entire process is pure algebraicoperationwithoutanydifferWko = 1 ential calculus andis readily adapted to a digital computer. i-1
Let a rational function of s be F ( s ) =P(s)/Q(s).If deg P ( s )
