On the Simulation of Oscillator Phase Noise Workshop at Chair of Communications Theory, May 2008 Christian Müller Communications Laboratory Department of Electrical Engineering and Information Technology Dresden University of Technology, Dresden, Germany E-mail: muellerc@ifn.et.tu-dresden.de May 29, 2008
Objective Motivation impact of oscillator phase noise in communication systems often theoretical results are hard to obtain (system complexity) Objective provide a model for the simulation of oscillator phase noise take care on precision, efficiency, and numerical stability Application verification of theoretical results by simulation simulation instead of theoretical investigations Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 1 of 11
Outline 1 Phase Noise Model and Characterization 2 Phase Noise Simulation Model 3 Discrete Time IIR Filter Design 4 Discrete Time IIR Filter Application 5 Conclusion Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 2 of 11
Phase Noise Model and Characterization
Phase Noise Model and Characterization Oscillator Phase Noise Model a huge class of oscillators can be modeled by Van der Pol s Eq. 1 Đx x 2 1µ x x 0 for 0 1 and initial phase ³ 0 it is shown to have solutions x t µ a cos t ³ 0 O µ a ¾ 2 2 if in addition orbital fluctuations t µ are present (mean square continuous, stationary processes), then it further holds (L 2 ) x t µ a cos t ³ t O µ ³ t ³ 0 1 Miklos Farkas, Periodic motions, Springer, New York, 1994. t 0 s ds Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 3 of 11
Phase Noise Model and Characterization Oscillator Phase Noise Characterization (precision) the model corresponds only to random sources of interference 2 (e.g. acceleration effects, temperature, pressure, fields,...) no systematic disturbances are taken into account 2 assume t µ, and hence ³ t µ, to be a Gaußian process with zero mean 2, 3 specification of t µ by data sheet 3 time domain: Allan variance, total variance frequency domain: power spectral density S of t µ (attention: S ³ of ³ t µ does not necessarily exists) 2 IEEE Guide for Measurement of Environmental Sensitivities of Standard Frequency Generators, IEEE Std 1193-2003, 2004. 3 IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology Random Instabilities, IEEE Std 1139-1999, 1999. Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 4 of 11
Phase Noise Model and Characterization Power Spectral Density Scheme (IEEE Std 1139-1999) ÐÓ Ë Ûµ Û ¾ Û ½ Û ¼ Û ½ Û ¾ ÐÓ Ë ³ Ûµ Û (i) (ii) (iii) (iv) (v) ÐÓ Û Û (i) random walk frequency modulation (ii) (iii) (iv) flicker frequency modulation white frequency modulation flicker phase modulation Û ¾ Û ½ Û ¼ (v) white phase modulation (i) (ii) (iii) (iv) (v) ÐÓ Û Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 5 of 11
Phase Noise Simulation Model
Phase Noise Simulation Model Modeling Approach given the initial oscillator phase noise equation ³ t ³ 0 t 0 s ds the following phase noise simulation model is proposed Ñ-stage Ê ³ ؼ Ø IIR filter ³¼ approximate t µ as stationary ARMA m mµ process 4 design an m-stage IIR filter to approximate S filter a band-limited white Gaußian noise process t µ 4 M. B. Priestley, Spectral Analysis and Time Series, Academic Press, London, 1996. Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 6 of 11
Discrete Time IIR Filter Design
Discrete Time IIR Filter Design Discrete Time IIR Filter Transfer Function convert S to discrete time IIR filter transfer function H from sampling theorem (L 2 ) and uniqueness of S it follows S w µ 2 w s S e iw µ where w 2 w w s as the transfer function H is linear we further have S e iw µ H e iw µ 2 S e iw µ implement discrete time IIR filter as cascade (numerical stability) Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 7 of 11
Discrete Time IIR Filter Design Power Spectral Density Approximation Idea Ò ¼ ½ ÐÓ Ë Ûµ Û ¾ Û ½ ¼ Ò Û ½ ¾ ÐÓ Ë Ûµ Û ¾ Û ½ pol zero ÐÓ Û ÐÓ Û Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 8 of 11
Discrete Time IIR Filter Design Discrete Time IIR Filter Scheme Ò ¼ Ò stage 1 Ò ½ Ò ½ Ò ½ stage k Ò Ò Ñ ½ stage m Ò Ñ Ò Ñ Ò Ò ½ ¼ ½ filter state vector process Ò µ with Ò Ò Ò ½ Ò Ñ µ ¼ Ò Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 9 of 11
Discrete Time IIR Filter Application
Discrete Time IIR Filter Application Probability Distribution of n µ n µ is a discrete time Gaußian process, if n µ it is and 0 is a Gaußian vector independent of n µ n µ has zero mean, if n µ, and 0 have zero mean Stationarity of n µ n µ is stationary, if n µ it is n µ is stationary with covariance matrix function R, if 0 has covariance matrix R 0µ Application Requirement the discrete time IIR filter must be randomly initialized (efficiency) Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 10 of 11
Conclusion
Conclusion Presented Results provided a phase noise simulation model according to data sheets (precision) Ñ-stage ÒÈ ³ Ò IIR filter ³¼ ¼ designed a filter cascade to generate n µ (numerical stability) provided condition for stationarity of n µ (efficiency) Further Results derived a closed form expression for R 0µ provided an algorithm to initialize the discrete time IIR filter Christian Müller, On the Simulation of Oscillator Phase Noise, Workshop, May 2008. Page 11 of 11
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