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Stochastic Resonance Sound Synthesis STOCHASTIC RESONANCE SOUND SYNTHESIS Rodrigo F. Cadiz´ and Patricio de la Cuadra Centro de Investigacion´ en Tecnolog´ıas de Audio Pontificia Universidad Catolica´ de Chile frcadiz,[email protected] ABSTRACT Stochastic resonance also exists in multi-threshold sys- tems in the presence of noise and periodic inputs. This Stochastic resonance is a nonlinear phenomenon that oc- form is denoted as suprathreshold stochastic resonance, curs when the addition of noise to a weak signal enhances because resonance is found even for large input signals. its detectability or its information content. An optimal When the noise on each threshold is independent, and amount of added noise results in a maximum enhance- sufficiently large, the optimal thresholds are those given ment. The addition of noise allows the perception of a by the suprathreshold stochastic resonance effect. In this signal in contexts where it could not be perceived at all case, all threshold devices are identical to the signal mean because of its low amplitude. Two sound synthesis tech- [3]. This phenomenon is detailed in section 3 niques inspired on this phenomenon are proposed in this Stochastic resonance was discovered and proposed for article. First, by artificially attenuating an audio signal be- the first time in 1981 to explain the periodic recurrence of low a certain threshold and then adding the proper amount ice ages [1]. Experiments have shown the appearance of of noise, it is possible to construct a modified version of stochastic resonance in sensory biology, animal behavior, the original signal, that retains some of its perceptual char- electrophysiological signals, neuronal function as well as acteristics but that is indeed a different signal. Second, a in several aspects of human perception, including human variant of this phenomenon that is not restricted to sub- audition. For a detailed account of stochastic resonance in threshold signals, known as suprathreshold stochastic res- sensory information processing and related applications, onance, was also explored as a novel synthesis technique. please refer to [5]. Patches in Max/MSP were created in order to demonstrate these procedures. 2. THRESHOLD STOCHASTIC RESONANCE 1. INTRODUCTION The necessary components for the threshold paradigm of stochastic resonance are: Stochastic resonance is a statistical phenomenon that is 1. A threshold observed on both man-made and naturally occurring non linear systems [5]. This phenomenon occurs when noise 2. A sub-threshold signal enhances an external forcing signal in a nonlinear dynam- ical system, if and only if the system has a nonzero noise 3. Additive noise optimum [4]. In other words, a dynamical system sub- ject to both periodic forcing and random perturbation may In threshold stochastic resonance, noise is added to a show a resonance, defined as a peak in its power spectrum, sub-threshold signal with the purpose of marking thresh- which is absent when either the forcing or the perturbation old crossings in the signal. This is shown in figure 1 (a) is absent [1]. This phenomenon does not occur in strictly and (b). In (a) we observe a subthreshold signal with linear systems, where the addition of noise only degrades added noise that is compared to a threshold. For low noise the measures of signal quality [5]. intensities, the signal with added noise does not cross the The simplest paradigm of stochastic resonance is the threshold frequently, so only a small portion of the signal non-dynamical or threshold theory [6]. In this manifes- is passed through it. For large noise intensities, the output tation, stochastic resonance results from the concurrence is dominated by the noise. In both cases, a low signal to of a threshold, a sub-threshold signal and noise, and it is noise ratio is obtained. For moderate intensities, the noise explained in more detail in section 2. Some authors [7] allows the signal to reach threshold, but the noise intensity [2] have shown that for this case there is a direct corre- is not so large as to totally cover it. All threshold crossings spondence between stochastic resonance and the dithering are marked by a pulse, as shown in figure 1 (b). effect, very well known in the theory of digital waveform In the visual domain, this phenomenon has been exten- coding. Stochastic resonance also exists in another form, sively studied and tested. The perception of a subthresh- denoted as dynamical stochastic resonance, that only ap- old image is dramatically improved when additive Gaus- pears in stochastic, nonlinear, dynamical systems [5]. sian noise is added to the image [6]. (a) Signal with added noise is compared to threshold (b) Output: each threshold crossing is marked by a standard pulse 1 0.5 0 !0.5 !1 Figure 1. Threshold paradigm of stochastic resonance 2.1. Method Let x be an audio signal of finite length, α an attenuation Figure 3. Threshold algorithm implemented as a subpatch factor between 0 and 1, Θ an arbitrary threshold value and in Max/MSP η a noise signal with normal distribution with zero mean and standard deviation σ. 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 oth- erwise. This procedure results in an output signal that con- Threshold sists of a sequence of pulses, containing a great amount of Θ Low Pass undesired high frequency content. To minimize this prob- Filter Audio Audio lem, an optional low pass filter is proposed after this stage < Signal x + Signal with the purpose of smoothing the resulting signal. In Out This process produces a new signal that is dependant α η on the amount of added noise and the threshold Θ. By Standard deviation changing these parameters it is possible to control the ru- σ gosity 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 synthe- sis, although their foundations are completely different. A complete Max/MSP implementation of this algorithm is shown in figure 4. Figure 2. Threshold stochastic resonance synthesis method 3. SUPRATHRESHOLD STOCHASTIC The proposed sound synthesis method is the following: RESONANCE 1. Attenuate x by the factor α so that it completely Given a noisy multi-threshold system, it has been shown falls below Θ and denote this new signal by xΘ. that for suprathreshold signal levels when all threshold values are equal to the signal mean, the mutual informa- 2. Add η to xΘ. tion between the input and output signals has a maximum 3. Retain only those samples that are higher in ampli- value for a nonzero noise intensity. This phenomenon is tude than Θ. Make the samples that fall below Θ know as suprathreshold stochastic resonance, because it zero. is a form of stochastic resonance that is not restricted to subthreshold signals [2] [3]. 4. Apply a low pass filter to the obtained signal, in or- Figure 5 shows how this phenomenon works. A single der to smooth transitions. input signal is received by N noisy threshold devices that The proposed method is shown in figure 2. The im- are subject to independent additive noise. The output from plementation of the threshold algorithm in Max/MSP is each device, is unity if the sum of the signal and noise at shown in figure 3. its output is greater than the corresponding threshold and Figure 4. Max/MSP implementation of threshold stochastic resonance zero otherwise. The total output of the system is obtained 3.1. Method by the sum of each individual output. When all thresh- x Θ old are equal to the signal mean the information transfer Let be an audio signal of finite length, an arbitrary η between the input and ouput signals is maximized. threshold value and a noise signal with normal distribu- tion with zero mean and standard deviation σ. The pro- posed sound synthesis method is the following: 1. Create N signals xi resulting of the sum of x and ηi. < + Θ1 2. Pass each xi through N threshold devices using the η1 same Θ. Audio < Audio + Θ2 3. Filter out only those samples that are higher in am- Signal + Signal In Out plitude than Θ. Make the samples that fall below Θ η2 zero. ... ... 4. Add all the ouput signals together. < + ΘN Max/MSP implementations of both a noisy threshold device and an array of these devices are shown in figures ηN 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: Figure 5. Supra threshold stochastic resonance synthesis the threshold value, amount of added noise and optional method attenuation and low pass filter effects. When all thresh- olds are equal to the signal mean, the information trans- fer is maximized and the output sound is almost identical 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 6. REFERENCES these parameters depart from their optimum values more musically appealing results can be obtained. [1] R. Benzi, A. Sutera, and A. Vulpiani. The mechanism of stochastic resonance. Journal of Physics A: Math- ematical and General, 14(11):453–457, 1981. 4. CONCLUSIONS [2] L. Gammaitoni. Stochastic resonance in multi- Two sound synthesis algorithms based on threshold and threshold systems. Physics Letters A, 208(4):315– suprathreshold stochastic resonance were proposed theo- 322, 1995. retically and implemented in Max/MSP. In terms of sound quality, these techniques resemble certain aspects of gran- [3] M. D. McDonnell, N.
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