Noise Analysis of Radio Frequency Circuits pp 165-176 | Cite as

# Conclusions and Future Directions

Chapter

## Abstract

In this chapter we summarize our contributions and point out to some future directions where this research can proceed.

## Keywords

Stochastic Differential Equation Phase Noise Flicker Noise Noise Analysis Microwave Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]J. F. Parker and D. Ray, “A 1.6-GHz CMOS PLL with on-chip loop filter,”
*IEEE Journal of Solid-State Circuits*, vol. 33, pp. 337–343, Mar. 1998.CrossRefGoogle Scholar - [2]P. R. Gray and R. G. Meyer, “Future directions in silicon ICs for RF personal communications,” in
*Proceedings of the IEEE 1995 Custom Integrated Circuits Conference*, pp. 83–90, 1995.CrossRefGoogle Scholar - [3]K. L. Fong and R. G. Meyer, “Monolithic RF active mixer design,”
*IEEE Transactions on Circuits and Systems—II: Analog and Digital Signal Processing*, vol. 46, pp. 231–239, Mar. 1999.CrossRefGoogle Scholar - [4]J. C. Rudell, J.-J. Ou, R. S. Narayanaswami, G. Chien, J. A. Weldon, L. Lin, K.-C. Tsai, L. Tee, K. Khoo, D. Au, T. Robinson, D. Gerna, M. Otsuka, and P. R. Gray, “Recent developments in high integration multi-standard CMOS transceivers for personal communication systems,” in
*Proceedings 1998 International Symposium on Low Power Electronics and Design*, pp. 149–154, 1998.CrossRefGoogle Scholar - [5]J. C. Rudell, J.-J. Ou, T. B. Cho, G. Chien, F. Brianit, J. A. Weldon, and P. R. Gray, “A 1.9-GHz wide-band IF double conversion CMOS receiver for cordless telephone applications,”
*IEEE Journal of Solid-State Circuits*, vol. 32, pp. 2071–2088, Dec. 1997.CrossRefGoogle Scholar - [6]A. Patterson, “Computer aided analysis of noise effects in GSM,”
*Electronic Engineering*, vol. 68, no. 838, pp. 53–54, 1996.Google Scholar - [7]R. Gharpurey,
*Modeling and analysis of substrate coupling in integrated circuits*. PhD thesis, University of California at Berkeley, 1996.Google Scholar - [8]G. Wei, “Flicker noise process analysis,” in
*Proceedings of the 1993 IEEE International Frequency Control Symposium*, pp. 321–325, 1993.CrossRefGoogle Scholar - [9]K. Motoishi and T. Koga, “Simulation of a noise source with I/f spectrum by means of an RC circuit,”
*Electronics and Communications in Japan*, vol. 65-A, pp. 19–7, Mar. 1982.Google Scholar - [10]M. K. Nezami, “Phase noise analysis. 1. evaluate the impact of phase noise on receiver performance,”
*Microwaves, and RF*, vol. 37, pp. 165–166, May 1998.Google Scholar - [11]L. Tomba, `Analysis of the effect of phase noise in OFDM systems,”
*European Transactions on Telecommunications*, vol. 9, no. 3, pp. 279–287, 1998.CrossRefGoogle Scholar - [12]R. L. Howald, S. Kesler, and M. Kam, “BER performance analysis of OFDM-QAM in phase noise,” in
*Proceedings. 1998 IEEE International Symposium on Information Theory*, p. 256, 1998.Google Scholar - [13]V. C. Vannicola and P. K. Varshney, “Spectral dispersion of modulated signals due to oscillator phase instability: White and random walk phase model,”
*IEEE Transactions on Communications*, vol. COM-31, pp. 886–895, July 1983.Google Scholar - [14]W. P. Robins,
*Phase noise in signal sources: theory and applications*, vol. 9 of*1EE telecommunications series*. London: Peter Peregrinus on behalf of the Institution of Electrical Engineers, 1982.Google Scholar - [15]B. K. Oksendal,
*Stochastic differential equations: an introduction with applications*. Springer—Verlag, 1998.Google Scholar - [16]D. N. Held and A. R. Kerr, “Analysis of noise in room-temperature millimeter-wave mixers,” in 1977
*International Microwave Symposium Digest*, pp. 483–486, 1977.Google Scholar - [17]U. L. Rohde, A. M. Pavio, and R. A. Pucel, “Accurate noise simulation of microwave amplifiers using CAD,”
*Microwave Journal*, vol. 31, pp. 130–141, Dec. 1988.Google Scholar - [18]V. Rizzoli, F. Mastri, and C. Cecchetti, “Computer-aided noise analysis of MESFET and HEMT mixers,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. 37, pp. 1401–1410, Sept. 1989.CrossRefGoogle Scholar - [19]S. Heinen, J. Kunisch, and I. Wolff, “A unified framework for computer-aided noise analysis of linear and nonlinear microwave circuits,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. 39, pp. 2170–2175, Dec. 1991.CrossRefGoogle Scholar - [20]P. Hudec, “Procedures for exact noise analysis of microwave circuits,”
*Microwave Journal*, vol. 35, pp. 165–170, Sept. 1992.Google Scholar - [21]P. Russer and S. Müller, “Noise analysis of microwave circuits with general topology,” in
*1992 IEEE M7T-S International Microwave Symposium Digest*, vol. 3, pp. 1481–1484, 1992.Google Scholar - [22]M. Okumura, H. Tanimoto, and T. Sugawara, “Noise analysis method for nonlinear circuits with two frequency excitations using the computer,”
*Electronics and Communications in Japan Part**3*, vol. 74, no. 4, pp. 41–50, 1991.CrossRefGoogle Scholar - [23]V. Rizzoli, D. Masotti, and F. Mastri, “General-purpose noise analysis of forced nonlinear microwave circuits,” in
*Conference Proceedings MM**92*, pp. 293–298, 1992.Google Scholar - [24]V. Rizzoli, D. Masotti, and E Mastri, “Advanced piecewise-harmonic-balance noise analysis of nonlinear microwave circuits with application to schottky-barrier diodes,” in
*1992 IEEE MTT-S International Microwave Symposium Digest*, pp. 247–250, 1992.CrossRefGoogle Scholar - [25]V. Rizzoli, F. Mastri, and D. Masotti, “General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance techniques,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. 42, pp. 807–819, May 1994.CrossRefGoogle Scholar - [26]V. Rizzoli, D. Masotti, and F. Mastri, “Full nonlinear noise analysis of microwave mixers,” in
*1994 IEEE MTT-S International Microwave Symposium Digest*, vol. 2, pp. 961–964, 1994.Google Scholar - [27]V. Rizzoli, D. Masotti, and F. Mastri, “Computer-aided noise analysis of integrated microwave front-ends,” in
*1995 IEEE MTT-S International Microwave Symposium Digest*, vol. 3, pp. 1561–1564, 1995.Google Scholar - [28]V. Rizzoli, F. Mastri, and C. Cecchetti, “Signal and noise analysis of large microwave front-ends by the inexact-newton harmonic-balance technique,” in
*1998 IEEE M7T-S International Microwave Symposium Digest*, vol. 3, pp. 1599–1602, 1998.Google Scholar - [29]J. S. Roychowdhury, P. Feldmann, and D. E. Long, “Cyclostationary noise analysis of large RF circuits with multi-tone excitations,”
*IEEE Journal of Solid-State Circuits*, vol. 33, pp. 324–36, Mar. 1998.CrossRefGoogle Scholar - [30]C. D. Hull,
*Analysis and Optimization of Monolithic RF Downconversion Receivers*. PhD thesis, University of California, Berkeley, 1992.Google Scholar - [31]C. D. Hull and R. G. Meyer, “A systematic approach to the analysis of noise in mixers,”
*IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, vol. 40, pp. 909–919, Dec. 1993.CrossRefGoogle Scholar - [32]R. Telichevesky, K. S. Kundert, and J. K. White, “Efficient AC and noise analysis for two-tone RF circuits,” in
*Proceedings of the 33rd Design Automation Conference*, pp. 292–297, 1996.Google Scholar - [33]V. Rizzoli, D. Masotti, F. Mastri, and A. Neri, “Noise analysis of a nonlinear microwave circuit driven by a noise sinusoidal source,”
*Microwave and Optical Technology Letters*, vol. 5, pp. 534–537, Sept. 1992.CrossRefGoogle Scholar - [34]A. Demir, E. Liu, and A. Sangiovanni-Vincentelli, “Time-domain non Monte-Carlo noise simulation for nonlinear dynamic circuits with arbitrary excitations,”
*IEEE Transactions for Computer-Aided Design*, vol. 15, pp. 493–505, May 1996.CrossRefGoogle Scholar - [35]O. Schein and G. Denk, “Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits,” Journal
*of Computational and Applied Mathematics*, vol. 100, no. 1, pp. 77–92, 1998.MathSciNetzbMATHCrossRefGoogle Scholar - [36]A. Dobrowolski, “CAD oriented method for noise analysis of microwave circuits described by the nodal admittance matrix,”
*IEE Proceedings H Microwaves*,*Antennas and Propagation*,vol. 140, pp. 321–325, Aug. 1993.Google Scholar - [37]B. Razavi, “A study of phase noise in CMOS oscillators,”
*IEEE Journal of Solid-State Circuits*, vol. 31, pp. 331–43, Mar. 1996.CrossRefGoogle Scholar - [38]D. B. Leeson, “A simple model of feedback oscillator noise spectrum,”
*Proceedings of the IEEE*, vol. 54, pp. 329–330, Feb. 1966.CrossRefGoogle Scholar - [39]G. Sauvage, “Phase noise in oscillators: a mathematical analysis of Leeson’s model,”
*IEEE Transactions on Instrumentation and Measurement*, vol. 1M-26, pp. 408–410, Dec. 1977.Google Scholar - [40]U. L. Rohde and C.-R. Cheng, `Analysis and optimization of oscillators for low phase noise and low power consumption,”
*RF Design*, vol. 18, pp. 70–79, Mar. 1995.Google Scholar - [41]W. Anzill, F. X. Kärtner, and P. Russer, “Simulation of the phase noise of oscillators in the frequency domain,”
*Archiv für Elektronik und Übertragungstechnik*, vol. 48, pp. 45–50, Jan. 1994.Google Scholar - [42]E. Hafner, “The effect of noise in oscillators,”
*Proceedings of the IEEE*, vol. 54, pp. 179198, Feb. 1966.Google Scholar - [43]K. Kurokawa, “Noise in synchronized oscillators,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. MTT-16, no. 4, pp. 234–40, 1968.Google Scholar - [44]J. R. Ashley, T. A. Barley, and G. J. Rast, Jr., “Automated spectral analysis of microwave oscillator noise,” in
*1976 IEEE MIT-S International Microwave Symposium*, pp. 227–229, 1976.Google Scholar - [45]V. Rizzoli, D. Masotti, F. Mastri, and A. Neri, “Full nonlinear analysis of near-carrier noise in discriminator-stabilized microwave oscillators,”
*Microwave and Optical Technology Letters*, vol. 6, no. 16, pp. 907–911, 1993.CrossRefGoogle Scholar - [46]V. Rizzoli, A. Costanzo, E Mastri, and C. Cecchetti, “Harmonic-balance optimization of microwave oscillators for electrical performance, steady-state stability, and near-carrier phase noise,” in
*Proceedings of IEEFIMTT-S International Microwave Symposium*, pp. 1401–4, May 1994.Google Scholar - [47]M. Okumura and H. Tanimoto, “A time-domain method for numerical noise analysis of oscillators,” in
*Proceedings of the Asia South Pacific Design Automation Conference*, pp. 477–82, Jan. 1997.CrossRefGoogle Scholar - [48]B. De Smedth and G. Gielen, “Accurate simulation of phase noise in oscillators,” in
*Proceedings of the 23rd European Solid-State Circuits Conference*, pp. 208–211, 1997.Google Scholar - [49]A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,”
*IEEE Journal of Solid-State Circuits*, vol. 33, pp. 179–94, Feb. 1998.CrossRefGoogle Scholar - [50]T. Ohira, “Higher-order analysis on phase noise generation in varactor-tuned oscillatorsbaseband noise upconversion in GaAs MESFET oscillators,”
*IEICE Transactions on Electronics*, vol. E76-C, pp. 1851–1854, Dec. 1993.Google Scholar - [51]J. Verdier, O. Llopis, R. Plana, and J. Graffeuil, “Analysis of noise upconversion in microwave field-effect transistor oscillators,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. 44, pp. 1478–1483, Aug. 1996.CrossRefGoogle Scholar - [52]R. Poore, “Accurate simulation of mixer noise and oscillator phase noise in large RFICs,” in
*1997Asia-Pacific Microwave Conference Proceedings APMC*‘87, pp. 357–360, 1997.CrossRefGoogle Scholar - [53]T. Felgentreff, W. Anzill, G. Olbrich, and P. Russer, “Analysis of g-r noise upconversion in oscillators,” in
*1995 IEEE MTT-S International Microwave Symposium Digest*, pp. 947–950, 1995.CrossRefGoogle Scholar - [54]H. J. Siweris and B. Schiek, “Analysis of noise upconversion in microwave FET oscillators,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. MTT-33, pp. 233–242, Mar. 1985.Google Scholar - [55]J. Y. C. Cheah, “Analysis of phase noise in oscillators,”
*RF Design*, vol. 14, no. 12, pp. 99–100, 1991.Google Scholar - [56]C.-L. Chen, X.-N. Hong, and B.-X. Gao, “A new and efficient approach to the accurate simulation of phase noise in microwave MESFET oscillators,” in
*1995 SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference Proceedings*, pp. 230234, 1995.Google Scholar - [57]A. Dec, L. Tóth, and K. Suyama, “Noise analysis of a class of oscillators,”
*IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing*, vol. 45, pp. 757–60, June 1998.CrossRefGoogle Scholar - [58]A. Dec, L. Tóth, and K. Suyama, “Noise analysis of an oscillator with an Mtt3-order filter and comparator-type nonlinearity,” in
*Proceedings of the 1998 IEEE International Symposium on Circuits and Systems*, vol. 1, pp. 225–228, 1998.Google Scholar - [59]J. Yilin and X. Lizhi, “Analysis on phase noise for phase-lock radar frequency synthesizer,” in /997
*IEEE International Conference on Intelligent Processing Systems*, vol. 1, pp. 28–31, 1997.Google Scholar - [60]K. Antoszkiewicz and J. Markowski, “Computer aided noise analysis of an oscillator,”
*Bulletin of the Polish Academy of Sciences*,*Technical Sciences*, vol. 45, no. I, pp. 183–195, 1997.Google Scholar - [61]A. A. Sweet, “A general analysis of noise in Gunn oscillators,”
*Proceedings of the IEEE*, vol. 60, pp. 999–1000, Aug. 1972.CrossRefGoogle Scholar - [62]A. Sjölund, “Analysis of large-signal noise in Read oscillators,”
*Solid-State Electronics*, vol. 15, pp. 971–978, Sept. 1972.CrossRefGoogle Scholar - [63]T. C. Weigandt, B. Kim, and P. R. Gray, “Analysis of timing jitter in CMOS ring oscillators,” in
*Proceedings of IEEE International Symposium on Circuits and Systems*, pp. 27–30, 1994.CrossRefGoogle Scholar - [64]J. A. McNeill, “Jitter in ring oscillators,”
*IEEE Journal of Solid-State Circuits*, vol. 32, pp. 177–193, June 1997.CrossRefGoogle Scholar - [65]A. A. Abidi and R. G. Meyer, “Noise in relaxation oscillators,”
*IEEE Journal of Solid-State Circuits*, vol. SC-18, pp. 794–802, Dec. 1983.Google Scholar - [66]A. Demir and A. Sangiovanni-Vincentelli, “Simulation and modeling of phase noise in open-loop oscillators,” in
*Proceedings of the IEEE 1996 Custom Integrated Circuits Conference*, pp. 453–456, 1996.Google Scholar - [67]M. Lax, “Classical noise. v. noise in self-sustained oscillators,”
*Physical Review*, vol. CAS-160, pp. 290–307, 1967.Google Scholar - [68]G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,”
*IEEE Transactions on Information Theory*, vol. 34, pp. 1437–1448, Nov. 1988.CrossRefGoogle Scholar - [69]A. P. Dekker, “Approximate noise analysis of a feedback oscillator using a nonlinear differential amplifier,”
*Archiv für Elektronik und Übertragungstechnik*, vol. 41, no. 3, pp. 129–132, 1987.Google Scholar - [70]J. Goldberg, “Nonlinear analysis of high Q oscillator phase noise,” in
*Proceedings of the**43rd Annual Symposium on Frequency Control 1989*, pp. 63–74, 1989.Google Scholar - [71]F. X. Kärtner, “Analysis of white and f a noise in oscillators,”
*International Journal of Circuit Theory and Applications*, vol. 18, pp. 485–519, Oct. 1990.zbMATHCrossRefGoogle Scholar - [72]V. F. Kroupa and L. Sojdr, “Phase-lock loops of higher orders,” in
*Second International Conference on Frequency Control and Synthesis*, pp. 65–68, 1989.Google Scholar - [73]B. Kim, T. C. Weigandt, and P. R. Gray, “PLL/DLL system noise analysis for low jitter clock synthesizer design,” in
*Proceedings of the 1994 IEEE International Symposium on Circuits and Systems*, vol. 4, pp. 31–34, 1994.Google Scholar - [74]U. L. Rohde,
*Microwave and wireless synthesizers: theory and design*. Wiley, 1997.Google Scholar - [75]J. C. Nallatamby, M. Prigent, J. C. Sarkissian, R. Quere, and J. Obregon, “A new approach to nonlinear analysis of noise behavior of synchronized oscillators and analog-frequency dividers,”
*IEEE Transactions on Microwave Theory and Techniques*, vol. 46, pp. 1168–1171, Aug. 1998.CrossRefGoogle Scholar - [76]L. Lin, L. Tee, and P. R. Gray, “A 1.4GHz differential low-noise CMOS frequency synthesizer using a wideband PLL architecture,” in
*Digest of Technical Papers*,*IEEE International Solid-State Circuits Conference*, pp. 204–205, 2000.Google Scholar - [77]K. Lim, S. Choi, and B. Kim, “Optimal loop bandwidth design for low noise PLL applications,” in
*Proceedings of the ASP-DAC ‘87. Asia and South Pacific Design Automation Conference**1997*, pp. 425–428, 1997.Google Scholar - [78]K. Lim, C.-H. Park, and B. Kim, “Low noise clock synthesizer design using optimal bandwidth,” in
*Proceedings of the 1998 IEEE International Symposium on Circuits and Systems*, vol. 1, pp. 163–166, 1998.Google Scholar - [79]G. Kolumbân, “Frequency domain analysis of sampling phase-locked loops,” in
*Proceedings 1988 IEEE International Symposium on Circuits and Systems*, vol. 1, pp. 611–614, 1988.CrossRefGoogle Scholar - [80]D. Asta and D. N. Green, “Analysis of a hybrid analog/switched-capacitor phase-locked loop,”
*IEEE Transactions on Circuits and Systems*, vol. 37, pp. 183–197, Feb. 1990.MathSciNetCrossRefGoogle Scholar - [81]A. Demir,
*Analysis and simulation of noise in nonlinear electronic circuits and systems*. PhD thesis, UC Berkeley, 1997.zbMATHGoogle Scholar - [82]W. E. Thain, Jr. and J. A. Connelly, “Simulating phase noise in phase-locked loops with a circuit simulator,” in
*Proceedings of the 1995 IEEE International Symposium on Circuits and Systems*, vol. 3, pp. 1760–1763, 1995.Google Scholar - [83]L. Wu, H. Jin, and W. C. Black Jr., “Nonlinear behavioral modeling and simulation of phase-locked and delay-locked systems,” in
*Proceedings of the IEEE 2000 Custom Integrated Circuits Conference*, pp. 447–450, 2000.Google Scholar - [84]P. Heydani and M. Pedram, “Analysis of jitter due to power-supply noise in phase-locked loops,” in
*Proceedings of the IEEE 2000 Custom Integrated Circuits Conference*, pp. 443–446, 2000.CrossRefGoogle Scholar - [85]M. Farkas,
*Periodic Motions*. Applied mathematical sciences, New York: Springer-Verlag, 1994.CrossRefGoogle Scholar - [86]R. Grimshaw,
*Nonlinear ordinary differential equations*. Applied mathematics and engineering science texts, Oxford, Boston: Blackwell Scientific Publications, 1990.Google Scholar - [87]B. van der Pol, “On oscillation hysteresis in a simple triode generator,”
*Philosophical Magazine*, vol. 43, pp. 177–193, 1922.CrossRefGoogle Scholar - [88]P. Hagedorn,
*Nichtlineare Schwingungen*. Oxford University Press, 1981.Google Scholar - [89]A. H. Nayfeh and D. T. Mook,
*Nonlinear Oscillations*. John Wiley, and Sons, 1979.Google Scholar - [90]L. Arnold,
*Stochastische Differentialgleichungen: Theorie und Anwendung*. John Wiley, and Sons, 1974.Google Scholar - [91]C. W. Gardiner,
*Handbook of stochastic methods for physics*,*chemistry*,*and the natural sciences*, vol. 13 of*Springer series in synergetics*. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag, second ed., 1983.CrossRefGoogle Scholar - [92]G. R. Grimmett and D. R. Stirzaker,
*Probability and random processes*. New York: Oxford University Press, second ed., 1992.Google Scholar - [93]Risken,
*The Fokker-Planck equation: methods of solution and applications*, vol. 18 of*Springer series in synergetics*. Berlin; New York: Springer-Verlag, second ed., 1996.Google Scholar - [94]W. A. Gardner,
*Introduction to Random Processes: with Applications to Signals and systems*. New York: McGraw-Hill, second ed., 1990.Google Scholar - [95]K. S. Kundert, J. K. White, and A. L. Sangiovanni-Vincentelli,
*Steady-state methods for simulating analog and microwave circuits*. Boston; Dordrecht: Kluwer Academic Publishers, 1990.Google Scholar - [96]T. J. Aprille, Jr. and T. N. Trick, “A computer algorithm to determine the steady-state response of nonlinear oscillators,”
*IEEE Transactions on Circuit Theory*, vol. CT19, pp. 354–360, July 1972.Google Scholar - [97]T. J. Aprille, Jr. and T. N. Trick, “Steady-state analysis of nonlinear circuits with periodic inputs,”
*Proceedings of the IEEE*, vol. 60, pp. 108–114, Jan. 1972.MathSciNetGoogle Scholar - [98]P. Kinget, “A fully integrated 2.7 V 0.35 pm CMOS VCO for 5 GHz wireless applications,” in
*Digest of Technical Papers IEEE International Solid-State Circuits Conference*, pp. 226–227, 1998.Google Scholar - [99]P. R. Gray and R. G. Meyer,
*Analysis and design of Analog Integrated Circuits*. New York: Wiley, third ed., 1993.Google Scholar - [100]A. Papoulis,
*Probability*,*Random Variables and Stochastic Processes*. McGraw Hill, third ed., 1991.Google Scholar - [101]K. L. Fong, C. D. Hull, and R. G. Meyer, “A class AB monolithic mixer for 900-MHz applications,”
*IEEE Journal of Solid-State Circuits*, vol. 32, pp. 1166–1172, Aug. 1997.CrossRefGoogle Scholar - [102]H.-G. Brachtendorf, G. Welsch, R. Laur, and A. Bunse-Gerstner, “Numerical steady state analysis of electronic circuits driven by multi-tone signals,”
*Electrical Engineering*, vol. 79, pp. 103–112, 1996.CrossRefGoogle Scholar - [103]J. S. Roychowdhury, “Efficient methods for simulating highly nonlinear multi-rate circuits,” in
*Proceedings of the 34th Design Automation Conference*, pp. 269–274, 1997.CrossRefGoogle Scholar - [104]H.-G. Brachtendorf, G. Welsch, and R. Laur, “A novel time-frequency method for the simulation of the steady state of circuits driven by multi-tone signals,” in
*Proceedings of the 1997 IEEE International Symposium on Circuits and Systems*, vol. 3, pp. 1508–1511, 1997.CrossRefGoogle Scholar - [105]K. S. Kundert,
*Steady state methods for simulating analog circuits*. PhD thesis, University of California at Berkeley, 1989.Google Scholar - [106]B. K. oksendal,
*Stochastic differential equations: an introduction with applications*. Springer—Verlag, 1998.Google Scholar - [107]C. W. Gardiner,
*Handbook of stochastic methods for physics*,*chemistry*,*and the natural sciences*, vol. 13 of*Springer series in synergetics*. Berlin, Heidelberg, New York, Tokyo: Springer—Verlag, second ed., 1983.CrossRefGoogle Scholar - [108]P. Dupuis and H. J. Kushner, “Stochastic systems with small noise, analysis and simulation; a phase locked loop example,”
*SIAM Journal on Applied Mathematics*, vol. 47, pp. 643–661, June 1987.MathSciNetzbMATHCrossRefGoogle Scholar - [109]A. Dembo and O. Zeitouni,
*Large deviations techniques and applications*. Boston: Jones and Bartlett, 1993.zbMATHGoogle Scholar - [110]M. I. Freidlin and A. D. Wentzell,
*Random Perturbations of Dynamical Systems*. Springer—Verlag, 1984.Google Scholar - [111]B. Razavi,
*Design of Analog CMOS Integrated Circuits*. McGraw Hill, 2000.Google Scholar - [112]A. Mehrotra,
*Simulation and Modelling Techniques for Noise in Radio Frequency Integrated Circuits*. PhD thesis, University of California, Berkeley, 1999.Google Scholar - [113]A. Ali and J. L. Tham, “A 900MHz frequency symthesizer with integrated LC voltage-controlled oscillator,” in
*Digest of Technical Papers*,*IEEE International Solid-State Circuits Conference*, pp. 390–391, 1996.Google Scholar - [114]J. Craninckx and M. Steyaert, “A fully integrated CMOS DCS-1800 frequency synthesizer,” in
*Digest of Technical Papers*,*IEEE International Solid-State Circuits Conference*, pp. 372–373, 1998.Google Scholar - [115]A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterisation,” in
*Proceedings 1998 Design Automation Conference*, pp. 26–31, 1998.Google Scholar - [116]A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: a unifying theory and numerical methods for characterization,”
*IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, vol. 47, pp. 655–674, May 2000.CrossRefGoogle Scholar - [117]M. I. Freidlin and A. D. Wentzell,
*Random Perturbations of Dynamical Systems*. Springer—Verlag, 1984.Google Scholar - [118]O. Zeitouni. personal communication, 1998.Google Scholar

## Copyright information

© Springer Science+Business Media New York 2004