It has recently been reported that in some systems showing stochastic resonance, the signal-to-noise ratio (SNR) at the output can significantly exceed that at the input; in other words, SNR gain is pos-sible. We took two such systems, the non-dynamical Schmitt trigger and the dynamical double well potential, and using numerical and mixed-signal simulation techniques, we examined what SNR gains these systems can provide. In the non-linear response limit, we obtained SNR gains much greater than unity for both systems. In addition to the classical narrow-band SNR definition, we also measured the ratio of the total power of the signal to the power of the noise part, and it showed even better signal improvement. Here we present a brief review of our results, and scrutinise, for both the Schmitt-trigger and the double well potential, the behaviour of the SNR gain by stochastic resonance for different signal amplitudes and duty cycles. We also discuss the mechanism of providing gains greater than unity.
Keywords: Stochastic resonance; signal-to-noise ratio gain; Schmitt trigger; double well potential.
1. Introduction Stochastic resonance (SR), whose study has grown to be a significant branch of noise research, is a rather general phenomenon wherein increasing the amount of input noise can optimise signal transfer in a system. To be more exact, we speak of stochastic reso-nance when the measure reflecting the quality of the transfer (most often the signal-to-noise ratio, or SNR) shows a maximum at non-zero input noise amplitude. Stochastic resonance has attracted considerable attention from various scientific fields, and the term applies to a wide range of systems and processes: the cycle of ice ages, neural models, physiological phenomena such as baroreflex mechanisms, bistable ring lasers, supercon-
L146 P Makra, Z Gingl & L B Kish
ducting quantum-interference devices (SQUIDs), electronic circuits, etc (see [1–14] and references therein).
Stochastic resonance means that the output signal gets better with a certain increase in input noise, but it does not usually involve the improvement of the output signal com-pared to the input. Yet it is a legitimate question whether systems showing SR can possi-bly function as filters, making the signal passing through them less noisy. A positive an-swer to this question might have great significance: it might establish the theoretical basis for technical applications and might lead us to a better understanding of several (espe-cially biological) systems.
Early investigations aimed at finding SR-based signal improvement were largely un-successful, and the exceptions soon turned out to be artefacts. The search for SNR gain was further discouraged by linear-response theory (LRT) [15], which proved that the out-put SNR cannot exceed that at the input (SNR gain is not possible) in the linear-response (LR) limit, that is, for signals whose amplitude is small compared to that of the noise. This result, however, did not rule out SNR gain in the non-linear response (NLR) range, so subsequent studies mostly assumed the condition of NLR.
The first theoretical finding that predicted significant SNR improvement involved a level-crossing detector (LCD) driven by a random spike train with additional Gaussian white noise [1]. Here the system produced an SNR gain of 5 orders of magnitude when operated in the strongly non-linear limit. Later, the same set-up also yielded SNR im-provement with a deterministic spike train as input [2]. In the studies that followed, sig-nificant SNR gains were observed, for example, in saturating threshold device models [3] and neuronal models [4].
In our brief review, we would like to revisit two of our previous results that elaborate on the issue of SNR improvement in bistable systems: one concerning a non-dynamical system, the Schmitt trigger [16], and the other examining a dynamical model, the double-well potential [17]. Our aim is to explore how the gains obtained depend on the duty cy-cle of the deterministic signal and the non-linearity of the arrangement, and compare the behaviour of the two systems in this respect.
2. Signal-to-noise ratio gain by SR in a Schmitt trigger
The Schmitt trigger is one of the first systems where stochastic resonance was observed. It has two threshold levels and its output is a dichotomous signal with a lower and an upper stage. If the input drops below the lower threshold, the output assumes the lower stage and remains so until the input exceeds the upper threshold, when it switches to the upper stage. In our computer simulations, we considered a model Schmitt trigger with symmetric threshold levels, -AT and AT. The input consisted of a periodic pulse train p(t) with variable amplitude and duty cycle, plus a Gaussian white noise w(t) with variable amplitude σ; the output, x(t), was defined in the following way:
−<+−
>+=
otherwise,)()(if,1)()(if,1
)(valueprevious
AtwtpAtwtp
tx T
T (1)
Fig. 1 illustrates the shape of the deterministic input signal p(t): A is the amplitude ex-pressed as a percentage of the threshold value AT, T stands for the period, τ denotes the width of the pulses and the duty cycle is defined as 2τ/T. The noise w(t) we used was
Signal-to-noise ratio gain in non-dynamical and dynamical bistable stochastic resonators L147
band-limited: its spectrum was flat below a cut-off frequency of 50 kHz and was attenu-ated by 20 dB/decade above this frequency.
A T
τ
Fig. 1. The shape of the periodic input signal p(t).
To obtain the SNR gain as a function of the input noise amplitude σ, we recorded 1000 independent samples with lengths of 32768 data points for each σ value; the input pulse train was periodic by 1024 data points. Then we calculated the averaged power spectral density (PSD) of these samples and from the PSD we obtained the signal-to-noise ratio. We used two classical definitions for the SNR:
(i) The most common SNR definition, which is the ratio of signal power around the signal frequency f0 and the level of the background noise at the same frequency [18]:
)(
)(lim:
0
0
0
0
fS
dffSSNR
N
ff
fff ∫
∆+
∆−→∆
= , (2)
(ii) and a more realistic wide-band definition, which is the total power of the periodic signal divided by the total power of the noise (for a detailed discussion, see [17]):
.)(
)(lim:
0
1 0
0
0
∫
∑ ∫∞
∞
=
∆+
∆−→∆
==dffS
dffS
PPSNR
N
k
fkf
fkff
N
Sw
(3)
In the definitions above, f0 denotes the frequency of the deterministic signal, S(f) stands for the PSD of the signal and SN(f) signifies the PSD of the noise component in the signal. Since the signals we considered were periodic, and the noise had a smooth spectrum, we could easily separate the signal from the noise: at a given harmonic, the PSD of the signal could be obtained as the corresponding data point in the total PSD minus the background noise level, which was interpolated from six neighbouring data points. The SNR gain was simply the ratio of output and input SNRs:
in
out
SNRSNRG =: , and
win
woutw SNR
SNRG
,
,:= . (4a-b)
We should note here that SNR as a measure to quantify stochastic resonance has at-tracted widespread criticism. Several objections have emerged, including the argument that SNR as a spectral measure cannot capture the whole probabilistic structure of SR and general information-theoretic distance measures offer a more comprehensive description [19]. For the narrow-band concept, additional shortcomings have been pointed out: it characterises the response of the system only in a narrow frequency interval, and conse-quently, its applicability is limited to sinusoidal inputs [20]. In spite of these objections,
we stuck to SNR, since it is still the most widely used response measure in the literature of SR. We included the wide-band extension so as to avoid some of the problems with SNR.
The results of our simulations are illustrated in Figs. 2 and 3. These figures demon-strate that both kinds of gains can exceed unity, although the wide-band gain remains greater than one over a wider range of input noise amplitude than its narrow-band coun-terpart. To illustrate the differences between the narrow-band and wide-band SNR, we included Fig. 4, which shows them side by side. Figure 2 shows how narrow-band and wide-band gains depend on the amplitude of the input noise, for three different values of the amplitude of the periodic input signal p(t). Since working in the NLR limit is a pre-requisite of producing SNR gain, it does not surprise us that the gains we obtain get higher as the amplitude (thus the non-linearity) increases.
As Fig. 3 reveals, greater duty cycles result in smaller gains. To understand this fact, one should consider that due to the output levels’ being fixed, the noise can manifest it-self in the output signal only in the form of jitter, that is, switch-time fluctuations. Conse-quently, SNRout does not depend significantly on the duty cycle if the correlation time of the noise is smaller than the width of the pulses in the periodic signal, and the amplitude of the periodic signal is close to the threshold (these requirements are met in our case). On the other hand, the input SNR is considerably higher for greater pulse width: the power of the deterministic component (the numerators in Eqs. (2) and (3)) increases, while the noise level (the denominators in Eqs. (2) and (3)) remains constant. The SNR gain, therefore, will be smaller if we increase the value of the duty cycle.
0.05 0.10 0.15 0.20 0.25 0.30 0.350.1
1
10
100Signal amplitude
70% 80% 90%
G
σ/AT
0.05 0.10 0.15 0.20 0.25 0.30 0.350.1
1
10
100Signal amplitude
70% 80% 90%
Gw
σ/AT
Fig. 2. Narrow-band (G) and wide-band (Gw) SNR gains in a Schmitt trigger as functions of input noise ampli-tude σ, for three different values of periodic signal amplitude. The latter is expressed as a percentage of the threshold value AT, and σ is also scaled by AT. The duty cycle is 10%.
0.05 0.10 0.15 0.20 0.25 0.30 0.350.1
1
10
100Duty cycle
10% 20% 30%
G
σ/AT
0.05 0.10 0.15 0.20 0.25 0.30 0.350.1
1
10
100Duty cycle
10% 20% 30%
Gw
σ/AT
Fig. 3. Narrow-band (G) and wide-band (Gw) SNR gains in a Schmitt trigger as functions of input noise ampli-tude σ, for three different duty cycles. The amplitude of the noise is scaled by the threshold value AT; the ampli-tude of the periodic signal is 90% of the threshold value.
Signal-to-noise ratio gain in non-dynamical and dynamical bistable stochastic resonators L149
0.05 0.10 0.15 0.20 0.25 0.30 0.350.1
1
10
100 G Gw
G
σ/AT
Fig. 4. Narrow-band (G) and wide-band (Gw) SNR gains compared in the Schmitt trigger. The amplitude of the noise is denoted by σ; it is scaled by the threshold value AT. The amplitude of the periodic signal is 90% of the threshold value, and the duty cycle is 10%.
3. Signal-to-noise ratio gain by SR in a double-well potential We considered an archetypal SR model: a particle moving in a double-well potential, excited by a periodic pulse train plus a band-limited Gaussian white noise. The over-damped dynamics of this system can be given by the following Langevin equation:
),()()()( 3 twtptxtxdtdx
++−= (5)
where x(t) denotes the position of the particle, p(t) is the periodic signal and w(t) stands for the noise. Since it has recently been shown that if the periodic excitation is sinusoidal, the SNR gain in the double well is almost always below unity even though the system works in the NLR range [20], we used a symmetric periodic pulse train with variable amplitude and duty cycle as periodic input signal p(t) (see Fig. 1). The noise w(t) we ap-plied was a band-limited Gaussian white noise.
We have developed a mixed-signal – both analogue and digital – circuitry for our in-vestigations. We generated the periodic excitation and the noise in the digital domain, then applied D/A converters to convert them into analogue signals. In order to solve Eq. (5) by analogue methods, we transformed it into an integral equation first:
∫ ′′+′+′−′=t
tdtwtptxtxtx0
3 .)]()()()([)( (6)
We applied analogue circuits to realise the mathematical operations required to solve this integral equation, such as integration, multiplication, addition and subtraction. The whole system was driven by a high-performance digital signal processor (DSP) and the data were acquired by high-resolution A/D converters with fast roll-off anti-aliasing filters. We used the same data acquisition equipment (including filters) for input and output sig-nals, guaranteeing the same measurement method and thus avoiding artefacts due to dif-ferent treatment. A host personal computer running LabVIEW controlled the DSP and performed additional evaluation tasks. The block diagram of the system is shown in Fig. 5.
It is important to note that the analogue realisation introduces a 1/t0 factor in the right side of Eq. (6). Correspondingly, the original Langevin equation (5) is modified in the following way:
In our case, the value of t0 is 1.2⋅10-4 s. This implies that the signal frequency and the bandwidth of the noise (to be given in the next paragraph) must be re-scaled if one would like to use the original Langevin equation (5).
We measured the threshold value of the input signal amplitude (AT) — the minimum value at which switching between the two wells can occur without noise — and set input signal amplitudes that were 70, 80 and 90% of this threshold. For every noise amplitude value σ, we recorded one thousand samples of x(t), each consisting of 2048 data points, and from these records we calculated the averaged power spectral density (PSD) of the signal. Using this averaged PSD, we computed the SNRs according to definitions (2) and (3), then obtained both the narrow-band and the wide-band gains as in (4a-b). The sam-pling frequency was 8 kHz, the frequency of the periodic input signal was set to 31.25 Hz and the bandwidth of the white noise was 50 kHz (these parameters pertain to Eq. (7) and correspond to 3.75·10-3 Hz and 6 Hz in Eq. (5)).
The results of our simulations can be seen in Figs. 6, 7 and 8. The behaviour reflected in these figures is very similar to that of the Schmitt trigger, though here the obtainable gains are smaller and the choice of the SNR definition makes greater difference: while the wide-band gain can exceed unity for all three signal amplitudes, its narrow-band counterpart always remains smaller than one for signals whose amplitude is 70% of the threshold value. Increasing the amplitude and the duty cycle of the input pulse train has the same kind of effect here as in the case of the Schmitt trigger.
Fig. 5. Block diagram of the mixed-signal simulation system. The periodic signal and the noise are generated digitally and fed into an analogue simulation system. The solution x(t) is digitised by an A/D converter and the data are processed by a DSP and the host PC.
Signal-to-noise ratio gain in non-dynamical and dynamical bistable stochastic resonators L151
0.2 0.4 0.6 0.8
100
1000
10000
Signal amplitude OUT, 70% OUT, 80% OUT, 90% IN, 70% IN, 80% IN, 90%
SNR
(au)
σ/AT
0.2 0.4 0.6 0.80.1
1
10
100
Signal amplitude
OUT, 70% OUT, 80% OUT, 90% IN, 70% IN, 80% IN, 90%
SN
Rw
σ/AT Fig. 6. Input (IN) and output (OUT) SNRs in the double well as functions of the input noise amplitude σ, both narrow-band (SNR, left panel) and wide-band (SNRw, right panel). The duty cycle is 10%.
0.2 0.4 0.6 0.8
0.1
1
10
Signal amplitude 70% 80% 90%
G
σ/AT
0.2 0.4 0.6 0.8
0.1
1
10
Signal amplitude 70% 80% 90%
Gw
σ/AT Fig. 7. Narrow-band (G) and wide-band (Gw) SNR gains in the double well as functions of input noise ampli-tude σ, for three different values of periodic signal amplitude. The latter is expressed as a percentage of the threshold value AT, and σ is also scaled by AT. The duty cycle is 10%.
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.1
1
10
Duty cycle
10% 20% 30%
G
σ/AT
0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.1
1
10
Duty cycle
10% 20% 30%
Gw
σ/AT
Fig. 8. Narrow-band (G) and wide-band (Gw) gains as functions of input noise amplitude σ, for three different duty cycles. The amplitude of the noise is scaled by the threshold value AT; the amplitude of the periodic signal is 90% of the threshold value.
L152 P Makra, Z Gingl & L B Kish
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1
1
10
G Gw
G
σ/AT
Fig. 9. Narrow-band (G) and wide-band (Gw) SNR gains compared in the double well. The amplitude of the noise is denoted by σ; it is scaled by the threshold value AT. The amplitude of the periodic signal is 90% of the threshold value, and the duty cycle is 10%.
4. Conclusions On the basis of our previous studies, we have compared two bistable systems — the non-dynamical Schmitt trigger and the double-well potential, which is dynamical — with re-gard to the signal-to-noise ratio gain induced by stochastic resonance. We have demon-strated that these systems can provide significant SNR amplification with a periodic pulse train plus Gaussian white noise as input. We have also shown that the dependence of the SNR gain on the amplitude and the duty cycle of the input signal is very similar in these systems: in both cases, greater gains can be obtained by increasing the amplitude and decreasing the duty cycle of the periodic input. This implies a common mechanism in the background, namely, that the input SNR drops considerably if the duty cycle decreases, while the output SNR is less sensitive to the value of the duty cycle, and the sum of these effects may result in SNR amplification. The main difference between the Schmitt trigger and the double well lies in the fact that the non-linear dynamics of the double well intro-duces more complex signal-transfer effects, which leads to smaller SNR gains and a stronger dependence on the input signal amplitude.
Our findings strengthen the view that SNR gain is significant only for signals wherein the power is concentrated in a narrow, pulse-like segment. These pulse-like signals are not without relevance: the majority of signals encountered in neural systems — which represent one of the main direction for the application of SR-related phenomena — also belong to this class.
We would also like to emphasise that the choice of the SNR definition has consider-able significance: in both cases, SNR gain is greater than unity even over a wider range if we apply a much more realistic wide-band definition based on the total power of signal and noise.
Acknowledgements We would like to convey our special thanks to the anonymous referee for helpful and constructive comments. Our research has been supported by OTKA (Hungary), under grant T037664.
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