Note: On-line weak signal detection via adaptive stochastic resonance Siliang Lu, Qingbo He, and Fanrang Kong Citation: Review of Scientific Instruments 85, 066111 (2014); doi: 10.1063/1.4884715 View online: http://dx.doi.org/10.1063/1.4884715 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Agent-based station for on-line diagnostics by self-adaptive laser Doppler vibrometry Rev. Sci. Instrum. 84, 121703 (2013); 10.1063/1.4845475 A waveguide invariant adaptive matched filter for active sonar target depth classification J. Acoust. Soc. Am. 129, 1813 (2011); 10.1121/1.3557041 Direct detection and time-locked subsampling applied to pulsed electron paramagnetic resonance imaging Rev. Sci. Instrum. 76, 053709 (2005); 10.1063/1.1903163 Adaptive notch filter-based signal processing method and system for vortex flowmeters Rev. Sci. Instrum. 72, 2219 (2001); 10.1063/1.1351832 Digital signal processor-based dc superconducting quantum interference device controller Rev. Sci. Instrum. 72, 2203 (2001); 10.1063/1.1350646
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 066111 (2014)
Note: On-line weak signal detection via adaptive stochastic resonance Siliang Lu, Qingbo He,a) and Fanrang Kong Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
(Received 2 May 2014; accepted 9 June 2014; published online 24 June 2014) We design an instrument with a novel embedded adaptive stochastic resonance (SR) algorithm that consists of a SR module and a digital zero crossing detection module for on-line weak signal detection in digital signal processing applications. The two modules are responsible for noise filtering and adaptive parameter configuration, respectively. The on-line weak signal detection can be stably achieved in seconds. The prototype instrument exhibits an advance of 20 dB averaged signal-to-noise ratio and 5 times averaged adjust R-square as compared to the input noisy signal, in considering different driving frequencies and noise levels. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4884715] Stochastic resonance (SR), as a nonlinear phenomenon that can utilize the noise to increase the output signal-to-noise ratio (SNR) in a proper condition,1 has been widely adopted for weak signal detection (WSD) in a variety of fields, e.g., physical, chemical, biological, engineering systems.2–6 To realize SR, the weak signal, the noise, and the potential should have an optimal matching relation, which can be achieved by tuning the noise intensity and (or) the system parameters.7, 8 By consulting the literatures, two limitations still exist in the present SR-based WSD investigations as: (1) most of the SR studies are concerned with the static signal, namely, the signal to be analyzed has been acquired and stored beforehand; (2) the most commonly used criterion to evaluate whether the SR has occurred is the SNR,9, 10 but the target signal information should be exactly known when applying the SNR-based SR technique, and such a requirement is hardly satisfied in practical application. Motivated by the above two limitations, this Note is committed to investigate a novel adaptive SR technique for realizing on-line WSD. We hope such an attempt could make the SR-based WSD technique simpler and more effective in practical applications. The governing equation corresponding to the bistable SR can be shown as dx dU (x) d 2x −γ + S(t) + N (t) , (1) =− 2 dt dx dt in which N(t) is an additive Gaussian white noise (AGWN), S(t) = Acos(2π fd t) is a periodic signal with A being the signal amplitude and fd being the driving frequency, γ represents the system damping factor, and U(x) = −0.5ax2 + 0.25bx4 is the bistable potential with a and b being positive real values. The schematic model corresponding to Eq. (1) is illustrated in Fig. 1. The computation of SR output x can be regarded as a quadratic integral process, which is also equivalent to a secondary filtering process. Hence, the SR can be regarded as a specific filter that can utilize the noise to enhance the weak signal.11 After equation substitution, and then mathematically let dx/dt = y, Eq. (1) can be separated to two first-order differena) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0034-6748/2014/85(6)/066111/3/$30.00
tial equations as dx/dt = y dy/dt = ax − bx 3 + A cos(2πfd t) + N (t) .
(2)
Equation (2) can be numerically computed via the fourthorder Runge-Kutta equation as ⎫ ⎧ y1 = y[n] x1 = −U (x [n]) − γ y1 + S[n] + N [n] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [n] y = y[n] + x h/2 x = −U (x + y h/2) − γ y + S[n] + N [n] ⎪ ⎪ 2 1 2 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨y3 = y[n]+x2 h/2 x3 = −U (x[n]+y2 h/2)−γ y3 +S[n+1]+N [n+1]⎪ , ⎪ y4 = y[n]+x3 h x4 = −U (x[n]+y3 h)−γ y4 +S[n+1]+N [n+1] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x[n + 1] = x[n] + (y1 + 2y2 + 2y3 + y4 ) h/6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ y[n + 1] = y[n] + (x1 + 2x2 + 2x3 + x4 ) h/6
(3) where h is the calculation step, and S[n], N[n], x[n] represent the discrete forms of S(t), N(t), and x(t), respectively. It can be noted that the SR output x[n] is dominated by the parameters a, b, γ , and h. Without loss of generality, we set a = 1, b = 1, γ = 0.5 for simplify in this study. After that, h becomes the solo parameter that affects the SR realization along with the WSD performance. Subsequently, we numerically evaluate the interaction between the parameter h and the output SNR to reveal the WSD effect. The SNR equals 10log10 (Psignal /Pnoise ) with Psignal and Pnoise being the powers of the signal and the total noise in the spectrum, respectively. The test signals are the sinusoids with A = 10 Vpp , fd = 50 Hz, 75 Hz, and 100 Hz, respectively. The −10 dB AWGNs are injected to the sinusoids separately. The calculated SNRs of the SR output signals with varying h are plotted in Fig. 2. It can be seen that the output SNR has opportunities to surpass the input SNR (denoted by the dashed line), i.e., the noise is suppressed and the weak signal is enhanced, by tuning the parameter h. Moreover, the optimal h values for different driving frequencies are different as marked by the arrows in Fig. 2. This is the basis of the proposed adaptive SR technique for on-line WSD, namely, by adjusting the calculation step h, the frequency of the SR output can be just coincided
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FIG. 1. Schematic model of the bistable SR system.
with that of the driving signal and hence the weak signal can be detected.10 In the following, the proposed technique is implemented in a prototype instrument. The hardware system consists of a computer, an analog-to-digital converter (ADC), a microcontroller unit (MCU), a digital-to-analog converter (DAC), and an oscilloscope, as shown in the right of Fig. 3(a). The detailed types of the instrument components and their purposes in this instrument are also provided in Fig. 3(a). The experimental setup, the instrument wiring diagram, and the pseudo-code are provided in the supplementary material.12 As indicated in Fig. 2, the WSD can be realized by tuning the parameter h; however, a metric is needed to adaptively estimate whether h has been optimized for on-line WSD. This can be achieved via a digital zero crossing detection (DZCD) system with the detailed algorithm flow chart as shown in the left of Fig. 3(a). Here, two explanations for the DZCD are presented as: (1) the “zero point” is the on-line mean of the output signal rather than the fixed “absolute zero,” as we found such a configuration is more effective in practical experiments; and (2) only the rising edge is utilized to trigger the DZCD. A crucial step in the signal processing part is to calculate the difference of the zero crossing point intervals, ∇z, as illustrated in Fig. 3(b) (4 ∇z are used in this study). If all the calculated ∇z are less than a threshold T (T = 10 sampling points in this study), we can conclude that the analyzed signal is periodic and the optimal h has been obtained, and then stop varying h. Moreover, the on-line WSD detection is performed via a running timer embedded in the MCU. Every calculated output point, x[n], can be obtained within a timer interval (100 μs in this study, corresponding to 10 kHz sampling frequency).
FIG. 2. Effect of tuning calculation step h for noisy signals with different driving frequencies.
FIG. 3. (a) Algorithm flow chart (left) and instrument system (right), and (b) illustration of the DZCD system.
Subsequently, a test is conducted to evaluate the practical performance of the prototype instrument. The test includes three parts: (1) fd jumps from 0 to 50 Hz, (2) fd jumps from 50 Hz to 100 Hz, and (3) fd jumps from 100 Hz to 50 Hz. The noise intensity is set as −10 dB in all the parts. The visual on-line WSD effect can be viewed in the multimedia, and four captures in the multimedia are shown in Figs. 4(a)–4(d), respectively. To better examine the periodicity of the output signal, a synchronous zero crossing trigger signal (both the rising edge and the falling edge are considered) corresponding to the output signal is also provided in each subfigure. The parameter h keeps varying until the output signal presents good periodicity; typically, such a process can be finished in seconds. Figs. 4(b) and 4(d) display the stable output signals (h stop varying) driven by noisy 50 Hz and 100 Hz signals, respectively. It can be seen that the output signals are almost noise-free with good periodicities, even though the original sinusoids are corrupted by the heavy injected noise. These results verify the effectiveness of the proposed adaptive SR technique in on-line WSD. Besides the above visual demonstration, the quantitative analyses based on the off-line captured signals are also conducted as follows. We use two metrics to evaluate the performance of the prototype instrument for WSD as: (1) SNR, which reflects the proportions of the periodic signal and
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FIG. 4. Effects of on-line WSD with dynamically varying h in addressing noisy signals with different frequencies (upper: input noisy signal; middle: output signal; lower: synchronously trigger signal, in each subfigure): (a) unstable output (parameter search process) in detecting noisy 50 Hz signal, (b) stable output in detecting noisy 50 Hz signal, (c) unstable output in detecting noisy 100 Hz signal, and (d) stable output in detecting noisy 100 Hz signal. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4884715.1].
the noise in spectrum; and (2) adjust R-square,13 which reflects the waveform fidelity, namely, a higher adjust R-square implies a purer sinusoid with lower distortion. The results in considering different driving frequencies and noise levels are illustrated in Fig. 5. The averaged increased SNR is calculated to be 20 dB and the averaged increased adjust R-square is 500 percent as compared to the input noisy signals. Specially, the adjust R-square of the output signal is very close to 1, which indicates that the output signal is almost a pure sinusoid (also confirmed in Figs. 4(b) and 4(d)). The results indicate that the adaptive SR algorithm has a good anti-noise capability while avoiding signal waveform distortion. SR can be regarded as a specific noise-assisted filter, but this filter is different from the traditional filters that focus on
FIG. 5. Quantitative analyses of WSD for different driving frequencies and different noise levels: (a) SNR, and (b) adjust R-square.
Rev. Sci. Instrum. 85, 066111 (2014)
noise suppression, as it utilizes the noise to enhance the weak signal. Such a distinct merit is plenty useful especially when the target signal frequency is involved in the noise bandwidth. A main contribution of this work is that we propose a novel adaptive SR technique and embed it in a prototype instrument for on-line WSD. Additionally, the proposed algorithm presents low computation complexity and low memory usage features, and hence it can be implemented in a generalpurpose MCU conveniently. Indeed, the algorithm can be more effective if more parameters are tuned jointly (e.g., a, b, γ , h), and more intelligent if the simple linear search algorithm used in this study is replaced by some optimization algorithms. In conclusion, a novel adaptive SR technique is proposed and implemented in a prototype instrument for on-line WSD in this Note. By tuning the calculation step h, the optimal matching relation among the weak signal, the noise and the potential can be obtained, and then a low-noise SR output signal can be presented on line. This work makes the SR-based WSD technique simpler and more effective in practical application. The proposed algorithm can be transplanted to other platforms easily, and the hardware system can be fabricated in a low-power portable device. The designed instrument shows potential applications on on-line signal filtering, signal detection, and waveform restoration. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11274300 and 51075379, and the Program for New Century Excellent Talents in University, China, under Grant No. NCET-13-0539. 1 M.
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