Level-crossing time statistics of Gaussian 1/f noises

Transcription

1 Level-crossing time statistics of Gaussian 1/f noises R. Mingesz 1, Z. Gingl and P. Makra Department of Experimental Physics, University of Szeged, Hungary Dom ter 9., Szeged, H-672 Hungary ABSTRACT It has been recently shown that the amplitude truncation of Gaussian1/f noise does not change the shape of the power spectral density under rather general conditions, including the case when a Heaviside transformation results in a dichotomous noise. This invariance of 1/f noise seems to be an important addition to the knowledge about this kind of noise and may be promising in understanding dichotomous 1/f noise, noise-driven switching and stepping. Probably the most important application is the explanation of ion channel currents in biomembranes. In this work we have extended our investigations, especially concerning the level crossing properties of 1/f noises. We determined the level-crossing time statistics for 1/f noises (< <2) and found an empirical formula for the level-crossing time distribution. The correlation properties of successive level crossing intervals are also explored by measurements and numerical simulations and it is shown that the case =1 is unique in the range from to 2. These time structure related additions to the knowledge about 1/f noise further emphasize the uniqueness of this kind of noise. These results may help to understand 1/f noises better and are strongly relevant to 1/f noise driven switching, dichotomous noises such as the case of ion channel current fluctuations. Keywords: 1/f noise, level-crossing, ion channel current fluctuations INTRODUCTION 1/f noise, which was discovered many decades ago, is very common in several natural and even artificial systems. The general occurrence of this phenomenon is still unexplained and exploring the special properties of these fluctuations is among the most challenging problems in noise research even today. Extensive research into the properties of this noise is required to get new results in this field, and it can help to understand 1/f noise in several systems. One example can be the presence of dichotomous noise in ion channel currents in biomembranes, whose power spectral density follows an 1/f law. Understanding the origin of this fluctuation seems to be still an open question, and results in the field of 1/f noise research may help to explain the reasons. One possibly related result is the discovery of a strange invariant property of Gaussian 1/f, which was found experimentally a few years ago: the power spectral density (PSD) remains 1/f if the amplitude of the noise is truncated using two levels under rather general conditions. This nonlinear transformation can easily occur in overdriven measurements, saturating physical systems and quantities with limited range. Results for amplitude truncation were obtained for 1/f noises with < <2 by experimental investigations, numerical simulations and later a theoretical derivation was also found [1-7]. In this work we extend the investigations to the level crossing properties of 1/f noises. We determined the levelcrossing time statistics for 1/f noises (< <2) and show an empirical formula for the level-crossing time distribution. The correlation properties of successive level crossing intervals are also explored by measurements and numerical simulations and it has been shown that the case =1 is unique in the range from to LEVEL CROSSING INVESTIGATIONS Level-crossing properties of 1/f noises are strongly related to the well-established results about the spectral invariance against amplitude truncation, which is defined by the following formula [6]: 1, if x(t) y(t) = (1) 1, if x(t) 1 rmingesz@titan.physx.u-szeged.hu 312 Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems, Sergey M. Bezrukov, Hans Frauenfelder, Frank Moss, Editors, Proceedings of SPIE Vol. 511 (23) 23 SPIE X/3/$15.

2 Here x(t) is the time-dependent amplitude of the zero-mean Gaussian noise, y(t) is the amplitude of the truncated noise. It has been shown both experimentally and theoretically that the dichotomous y(t) has 1/f spectrum if x(t) represents Gaussian 1/f noise. Level-crossing statistics are derived by analyzing the time spent by the signal continuously over or under a certain level, or in other words, the time duration between successive level crossings. Fig 1. illustrates the level crossings of a signal. T L,i-1 T L,i T L,i+1 T U,i-1 T U,i T U,i+1 Figure 1: Level-crossing time intervals of a signal. T U,i and T L,i denote the time the signal spends above and below the level between neighboring level crossings, respectively. In our level-crossing investigations we used 1/f noises (< <2) with different levels, and analyzed the occurrences of time intervals by measurements and numerical simulations. Time intervals above and below the level were separated and the correlation between the length of neighboring intervals were also calculated, since we can expect some kind of inheritance from the long range correlations of 1/f and dichotomous 1/f noises. Gaussian 1/f noises were generated using different numerical and experimental methods. In numerical simulations we used sample length of 2 16 and averaged 1 of such sequences. We used three different noise generators: synthesis of a signal using 1/f /2 weighted amplitude sinusoidal oscillations with random phase (uniform distribution), spectral filtering of white noise by multiplication by 1/f /2 in the frequency domain using FFT and inverse FFT, proper cascading of digital filters to get 1/f noise from a white noise source. All generators were tested against spectral and amplitude statistical properties. Noise sources for experimental investigations included a white noise source filtered by cascaded analog filters to produce 1/f noise and a real 1/f noise source (the amplified noise of a MOSFET transistor). The bandwidth of the noise sources were.2hz-2khz. The signals were digitized into a uninterrupted stream of 256 million samples with a fast and precision analog-to-digital converter (the sampling frequency was 2kHz) and the data were analyzed off-line by computer. We performed our analysis on all of the noise sources and found no significant differences. Note here that both the inherent and introduced upper and lower frequency limits and the sampling frequency may have an impact on the calculated distribution of the level crossing intervals and one should analyze the results very carefully. It is easy to see that high frequency cutoff reduces the weight of short intervals while absence of low frequencies (also considering measurement time) may be responsible for decreased occurrence of long intervals. Fig. 2 shows the effects of different high frequency cutoffs (the cut-off frequency of the filtered noise is one eigth of the original noise) and Fig. 3 illustrates how the measurement time modifies the proportion of the detected long intervals. Proc. of SPIE Vol

3 1,E+4 Averaged interval number 1,E+2 1,E+ 1,E-2 1,E-4 1,E+ 1,E+1 1,E+2 1,E+3 1,E+4. Figure 2: Effect of high frequency cutoff on the level crossing time distribution of 1/f noise. Curve plotted with solid circles represent the normal case, the effect of reducing the bandwidth by an order of magnitude with first order and sharp cutoff low pass filtering are shown by squares and triangles, respectively. 1,E+ 1,E-1 1,E-2 Probabolity 1,E-3 1,E-4 1,E-5 1,E-6 1,E-7 1,E-8 1,E+ 1,E+1 1,E+2 1,E+3 1,E+4 Figure 3: Level-crossing statistics of a 1/f noise sample consisting of 64k samples (filled squares) and 512k samples. The abovementioned facts were taken into account in our measurements and simulations carried out to determine the distribution of level crossing time intervals. 2. LEVEL CROSSING STATISTICS FOR 1/f NOISES The case of white noise ( =) is quite straightforward if we consider time resolution greater than the noise correlation time. In this case the probability p of spending a time unit above the level determines the time spending N time units above the level as p N-1 (1-p), thus for the level-crossing distribution one gets an exponential law [8]. 314 Proc. of SPIE Vol. 511

4 The case of 1/f 2 noise ( =2) can be derived from the statistics of Brownian motion [9]. The level-crossing time distribution can be calculated quite easily if we consider a discrete representation, the random walk, described by the following formula: x 1 = x w (2) i i where x i represents the amplitude and w i is a 1 or +1 with.5 probability. In this case the probability of starting from a level to returning to that level in N steps is: and using the Sterling-formula for large values of N we get i 1 N pn, (3) N 2 ( N 1) N / 2 1 p N. (4) 3/ 2 N Concerning the cases above, for < <2 we may try to use the following formula for the probability distribution of levelcrossing intervals: bt e (, (5) c t p t) where t is the time between level crossings, b and c are parameters. This formula describes well the case of white ( =, c=) and 1/f 2 noises ( =2, b=,c=1.5). Using simulations (< <2) and measurements ( =1) we have determined the values of b and c as functions of (for the level equal to the mean value), and the results are plotted on Fig. 4. b c Figure 4: Dependence of parameters b and c on the exponent of the 1/f noise. The parameters were determined by numerical simulation and non-linear curve fitting. Note that the curves change significantly around =1 (1/f noise). Proc. of SPIE Vol

5 1,E+ 1,E-1 1,E-2 Probability 1,E-3 1,E-4 1,E-5 1,E-6 1,E-7 1,E+ 1,E+1 1,E+2 1,E+3,6,4,2 Relative error 1,E+ -,2 1,E+1 1,E+2 1,E+3 -,4 -,6 -,8-1 Figure 5: The upper plot shows the zero crossing time distribution of 1/f nose (squares) and the fitted curve based on Eq. 5 (solid line). Below the relative error of the fit is presented, showing good fit in the middle range of t. As we mentioned earlier, the inherent high frequency cutoffs of the simulated or measured noises modify the levelcrossing time distribution for short time intervals. This effect can be minimized by adding a multiplicative term to Eq. 5. as follows: bt e t p ( t) (1 e ), (6) c t Fig. 6 shows the fitting of this formula and the relative error of fitting for 1/f noise. Note that although two new parameters are introduced, this does not affect seriously the reliability of determining parameters b and c, just compensates the error introduced at small t values. 316 Proc. of SPIE Vol. 511

6 1,E+ 1,E-1 1,E-2 Probability 1,E-3 1,E-4 1,E-5 1,E-6 1,E-7 1,E+ 1,E+1 1,E+2 1,E+3,5,4,3 Relative error,2,1 1,E+ 1,E+1 1,E+2 1,E+3 -,1 -,2 Figure 6: The upper plot shows the zero crossing time distribution of 1/f nose (squares) and the fitted curve based on Eq. 6 (solid line). Below the relative error of the fit is presented. Note that the fit has significant error only at long intervals, where the probability is very small, about 5 orders of magnitude smaller than for short intervals. Finally, we refer to our previous related result about the correlation between neighboring level-crossing intervals of 1/f noises [1]. Fig. 1. illustrates the level-crossings of a signal and defines the T U and T L time intervals. We investigated the correlation between successive T U -T U, T L -T L and T U -T L /T L -T U level-crossing intervals by numerical simulations. We have found that the correlation is the strongest around close to 1, namely for 1/f noise (see Fig. 7). We have also noticed that the PSD of the time series of T U -T L /T L -T U type level-crossing intervals can be approximated as 1/f and Fig. 6 shows how depends on. Note that the highest correlation can be observed again at close to 1. Proc. of SPIE Vol

7 correlation x = UU LL LU UL correlation x = UU LL LU UL Figure 7: Correlation coefficients as functions of the power exponent of the input noise for T U -T U, T L -T L and T U -T L /T L -T U type level crossings. The two plots show results for levels equal to the mean value (zero) and to the standard deviation of the noise Figure 8: Spectral exponent versus. The strongest correlation is shown at CONCLUSION In this work we have investigated the level-crossing properties of Gaussian 1/f noises, using numerical simulations and measurements. Several different numerical and real noise sources were used and no dependence on the noise generation method was found. The distribution of the level-crossing time intervals were calculated and an empirical formula was introduced showing good fit to experimental data. The formula follows the theoretical results for white and 1/f 2 noises. We have observed a transition between these two different behaviors of the formula close to =1, 1/f noise, and the correlation between neighboring level crossings also peaks at around =1. These results may help to understand processes and systems wherein amplitude clipping related to threshold crossing occurs (like stochastic resonance), random dichotomous noises with 1/f spectra, and switching or stepping driven by 1/f or 1/f noises. One of the most important related and not yet understood phenomenon is the 1/f noise in ion channel currents [11, 12]. Also note, that the amplitude and time-structure properties of 1/f noises show extreme values at 1, which is in accordance with recent results about the prominently constructive role of 1/f noise in some systems showing stochastic resonance [13, 14]. 318 Proc. of SPIE Vol. 511

8 ACKNOWLEDGMENTS The authors thank L.B. Kish for helpful discussion. This work has been supported by OTKA (Hungarian Academy of Sciences), grant no. T REFERENCES 1. Gy. Trefan, L.B. Kiss, Z. Gingl, I. Hevesi and M.I. Torok, "Diffusion noise generators realized by analog circuits", Proc. 1th International Conference on Noise in Physical Systems, ed. A. Ambrózy, Akadémia, Budapest, 199, pp L.B. Kiss, Z. Gingl, Zs. Marton, J. Kertesz, F. Moss, G. Schmera and A. Bulsara, J.Stat.Phys (1993) 3. Z. Gingl and L.B. Kiss, "A Crucial Property of Gaussian 1/f Fluctuations" Proc. Fluctuation Phenomena in Physical Systems, Ed. V. Palenskis, Vilnius university Press, Palanga, Lithuania, 1994, pp Z.Gingl, L.B. Kiss, "Amplitude-saturation nonlinearity of Gaussian 1/f k fluctuations", Proceedings of the First International Conference on Unsolved Problems of Noise, edited by Doering,Ch.R., Kiss, L.B., and Schlesinger,M.F., World Scientific, 1996, pp Sh. Ishioka, Z. Gingl, D. Choi and N. Fuchikami, "Amplitude truncation of Gaussian 1/f^alpha noises", Physics Letters A (2) 6. Gingl Z., Ishioka S.,Choi D., and Fuchikami N., Chaos 11 3, (21). 7. D. Choi and N. Fuchikami, J Phys Soc Jpn 7 1, (21). 8. S. O. Rice, Bell Systems Technology Journal 24, (1945) 9. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys., 17, No.2-3 (1945) Z. Gingl, R. Mingesz and P. Makra, On the amplitude and time-structure properties of 1/f noises, Proc. Third International Conference on Unsolved Problems of Noise and Fluctuation in Physics, Biology, and High Technology, Ed. S. Bezrukov, Washington, USA, 22, in press 11. S. M. Bezrukov, M. Winterhalter, Physical Review Letters (2). 12. Z. Siwy, A. Fulinski, Origins of 1/f Noise in Membrane Channel Currents, Proc. Third International Conference on Unsolved Problems of Noise and Fluctuation in Physics, Biology, and High Technology, Ed. S. Bezrukov, Washington, USA, 22, in press 13. L.B. Kish, S.M. Bezrukov, Physics Letters A 266, (2). 14. D. Nozaki, Y. Yamamoto, Physics Letters A 243, (1998). Proc. of SPIE Vol