Page  1 ï~~STOCHASTIC RESONANCE SOUND SYNTHESIS Rodrigo F. Cddiz and Patricio de la Cuadra Centro de Investigaci6n en Tecnologfas de Audio Pontificia Universidad Cat61ica de Chile { rcadiz,pcuadra} ABSTRACT Stochastic resonance is a nonlinear phenomenon that occurs when the addition of noise to a weak signal enhances its detectability or its information content. An optimal amount of added noise results in a maximum enhancement. The addition of noise allows the perception of a signal in contexts where it could not be perceived at all because of its low amplitude. Two sound synthesis techniques inspired on this phenomenon are proposed in this article. First, by artificially attenuating an audio signal below a certain threshold and then adding the proper amount of noise, it is possible to construct a modified version of the original signal, that retains some of its perceptual characteristics but that is indeed a different signal. Second, a variant of this phenomenon that is not restricted to subthreshold signals, known as suprathreshold stochastic resonance, was also explored as a novel synthesis technique. Patches in Max/MSP were created in order to demonstrate these procedures. 1. INTRODUCTION Stochastic resonance is a statistical phenomenon that is observed on both man-made and naturally occurring non linear systems [5]. This phenomenon occurs when noise enhances an external forcing signal in a nonlinear dynamical system, if and only if the system has a nonzero noise optimum [4]. In other words, a dynamical system subject to both periodic forcing and random perturbation may show a resonance, defined as a peak in its power spectrum, which is absent when either the forcing or the perturbation is absent [1]. This phenomenon does not occur in strictly linear systems, where the addition of noise only degrades the measures of signal quality [5]. The simplest paradigm of stochastic resonance is the non-dynamical or threshold theory [6]. In this manifestation, stochastic resonance results from the concurrence of a threshold, a sub-threshold signal and noise, and it is explained in more detail in section 2. Some authors [7] [2] have shown that for this case there is a direct correspondence between stochastic resonance and the dithering effect, very well known in the theory of digital waveform coding. Stochastic resonance also exists in another form, denoted as dynamical stochastic resonance, that only appears in stochastic, nonlinear, dynamical systems [5]. Stochastic resonance also exists in multi-threshold systems in the presence of noise and periodic inputs. This form is denoted as suprathreshold stochastic resonance, because resonance is found even for large input signals. When the noise on each threshold is independent, and sufficiently large, the optimal thresholds are those given by the suprathreshold stochastic resonance effect. In this case, all threshold devices are identical to the signal mean [3]. This phenomenon is detailed in section 3 Stochastic resonance was discovered and proposed for the first time in 1981 to explain the periodic recurrence of ice ages [1]. Experiments have shown the appearance of stochastic resonance in sensory biology, animal behavior, electrophysiological signals, neuronal function as well as in several aspects of human perception, including human audition. For a detailed account of stochastic resonance in sensory information processing and related applications, please refer to [5]. 2. THRESHOLD STOCHASTIC RESONANCE The necessary components for the threshold paradigm of stochastic resonance are: 1. A threshold 2. A sub-threshold signal 3. Additive noise In threshold stochastic resonance, noise is added to a sub-threshold signal with the purpose of marking threshold crossings in the signal. This is shown in figure 1 (a) and (b). In (a) we observe a subthreshold signal with added noise that is compared to a threshold. For low noise intensities, the signal with added noise does not cross the threshold frequently, so only a small portion of the signal is passed through it. For large noise intensities, the output is dominated by the noise. In both cases, a low signal to noise ratio is obtained. For moderate intensities, the noise allows the signal to reach threshold, but the noise intensity is not so large as to totally cover it. All threshold crossings are marked by a pulse, as shown in figure 1 (b). In the visual domain, this phenomenon has been extensively studied and tested. The perception of a subthreshold image is dramatically improved when additive Gaussian noise is added to the image [6].

Page  2 ï~~(a) Signal with added noise is compared to threshold (b) Output: each threshold crossing is marked by a standard pulse 0.5 -0 -0.5 -Figure 1. Threshold paradigm of stochastic resonance 2.1. Method Let x be an audio signal of finite length, a an attenuation factor between 0 and 1, O an arbitrary threshold value and r a noise signal with normal distribution with zero mean and standard deviation cr. Threshold O Low Pass Filter Audio Fe Audio Signal - Â~ - ---L -H KI-- Signal In L t Out Standard deviation Figure 3. Threshold algorithm implemented as a subpatch in Max/MSP At the sample level, the amplitude of the input signal is compared to a threshold. If the amplitude is greater than the threshold the output signal is marked with 1 and 0 otherwise. This procedure results in an output signal that consists of a sequence of pulses, containing a great amount of undesired high frequency content. To minimize this problem, an optional low pass filter is proposed after this stage with the purpose of smoothing the resulting signal. This process produces a new signal that is dependant on the amount of added noise and the threshold O. By changing these parameters it is possible to control the rugosity or granularity of the output signal, similar to the effect of granular synthesis. Surprinsingly, the results of this technique are very similar to those of granular synthesis, although their foundations are completely different. A complete Max/MSP implementation of this algorithm is shown in figure 4. 3. SUPRATHRESHOLD STOCHASTIC RESONANCE Given a noisy multi-threshold system, it has been shown that for suprathreshold signal levels when all threshold values are equal to the signal mean, the mutual information between the input and output signals has a maximum value for a nonzero noise intensity. This phenomenon is know as suprathreshold stochastic resonance, because it is a form of stochastic resonance that is not restricted to subthreshold signals [2] [3]. Figure 5 shows how this phenomenon works. A single input signal is received by N noisy threshold devices that are subject to independent additive noise. The output from each device, is unity if the sum of the signal and noise at its output is greater than the corresponding threshold and Figure 2. Threshold stochastic resonance synthesis method The proposed sound synthesis method is the following: 1. Attenuate x by the factor a so that it completely falls below O and denote this new signal by ze. 2. Addrto ze. 3. Retain only those samples that are higher in amplitude than O. Make the samples that fall below O zero. 4. Apply a low pass filter to the obtained signal, in order to smooth transitions. The proposed method is shown in figure 2. The implementation of the threshold algorithm in Max/MSP is shown in figure 3.

Page  3 ï~~STOCHASTIC RESONANCE SOUND SYNTHESIS DEMO Rodrigo F. Cadiz and Patricio de la Cuadra Centro de Investigaci6n en Tecnologias de Audio Pontificia Universidad Cat61ica de Chile radtsg o nd 41 h u ok i. 1224+e _gra J J rea Original Sianal Time Fourier Attenuated Signal F Figure 4. Max/MSP implementation of threshold stochastic resonance zero otherwise. The total output of the system is obtained by the sum of each individual output. When all threshold are equal to the signal mean the information transfer between the input and ouput signals is maximized. Audio -- - Audio Signal + I - Signal In - Out t Figure 5. Supra threshold stochastic resonance synthesis method 3.1. Method Let xc be an audio signal of finite length, O an arbitrary threshold value and r7a noise signal with normal distribution with zero mean and standard deviation cr. The proposed sound synthesis method is the following: 1. Create N signals xci resulting of the sum of xc and T2j. 2. Pass each xci through N threshold devices using the same 0. 3. Filter out only those samples that are higher in amplitude than 0. Make the samples that fall below 0 zero. 4. Add all the ouput signals together. Max/MSP implementations of both a noisy threshold device and an array of these devices are shown in figures 3 and 6. The same input signals used in section 2 were used. 16 noisy threshold devices were used. The patch allows a flexible manipulation of the relevant parameters: the threshold value, amount of added noise and optional attenuation and low pass filter effects. When all thresholds are equal to the signal mean, the information transfer is maximized and the output sound is almost identical

Page  4 ï~~Figure 6. Max/MSP implementation of suprathreshold stochastic resonance sound synthesis using 16 noisy threshold devices with the input and little resynthesis is obtained. But if these parameters depart from their optimum values more musically appealing results can be obtained. 4. CONCLUSIONS Two sound synthesis algorithms based on threshold and suprathreshold stochastic resonance were proposed theoretically and implemented in Max/MSP. In terms of sound quality, these techniques resemble certain aspects of granular synthesis. By adjusting the standard deviation of the added noise, and threshold levels, it is possible to control the granularity or rugosity of the produced sound textures. In general, the results obtained by the suprathreshold stochastic resonance method were more satisfactory perceptually than those obtained with threshold stochastic resonance. We believe that this is due to the fact that no attenuation is necessary and to a better transfer of information. However, we believe that both proposed synthesis methods are interesting and worth for further studying and exploring. The proposed techniques may not only be applied to the creation of sounds, but also to some other aspects of audio signal processing such as audio restoration. 5. ACKNOWLEDGEMENTS This research was possible due to the support of an academic insertion grant of the Vicerrectorfa Acaddmica of the Pontificia Universidad Cat61ica de Chile, and a research grant from the World Bank, Programa Bicentenario de Ciencia y Tecnologfa, Conicyt, Government of Chile. 6. REFERENCES [1] R. Benzi, A. Sutera, and A. Vulpiani. The mechanism of stochastic resonance. Journal of Physics A: Mathematical and General, 14(11):453-457, 1981. [2] L. Gammaitoni. Stochastic resonance in multithreshold systems. Physics Letters A, 208(4):315 -322, 1995. [3] M. D. McDonnell, N. G. Stocks, C. E. M. Pearce, and D. Abbott. Optimal information transmission in nonlinear arrays through suprathreshold stochastic resonance. Physics Letters A, 352:183-189, 2006. [4] S. Mitaim and B. Kosko. Adaptive stochastic resonance. Proceedings of the IEEE: Special Issue on Intelligent Signal Processing, 86(11):2152-2183, 1998. [5] F. Moss, L. M. Ward, and W. G. Sannita. Stochastic resonance and sensory information processing: a tutorial and review of application. Clinical Neurophysiology, 115(2):267-281, 2004. [6] E. Simonotto, M. Riani, C. Seife, M. Roberts, J. Twitty, and F. Moss. Visual perception of stochastic resonance. Physical Review Letters, 78:1186-1189, 1997. [7] R. A. Wannamaker, S. P. Lipshitz, and J. Vanderkooy. Stochastic resonance as dithering. Physical Review E, 61:233-236, 2000.