Acoustic signals are quite fluctuating as their signal strengths are used to represent time-varying information. It always has great advantage if future information could be forecasted via predicting the acoustic signal strengths. In this paper, we try to answer several challenging questions in acoustic signal research: (i) Are acoustic signals predictable? (ii) How are acoustic signals predicted? (iii) Are there any potential applications based on acoustic signal prediction? These studies are made possible with the development of wireless sensor technology. Wireless sensors can sense different modalities such as acoustic, video, image, seismic, and thermal. In this paper, we will use wireless acoustic sensors. The main goal of wireless acoustic sensors is to monitor physical world. Usually, people are more interested in unexpected events. For example, in a battlefield scenario, people are more interested in the appearance of enemies. If a wireless sensor is to monitor power generator motor, unusual increasing of the acoustic sound may indicate a possible machinery failure. Both the appearance of enemies and the unusual increasing of the machine sound can be seen as events, and it will have great benefit if such events could be forecasted on the basis of acoustic signal prediction.
In the existing works on engineering approach for acoustic signal and voice research, nonlinear dynamic analysis was used for reliable measurement of the aperiodic esophageal voice ; jitter, shimmer, signal-to-noise ratio, correlation dimension, and second-order entropy were measured from audio recordings of subjects' voices. In , the local and global changes of fundamental frequency F0 during phonation were examined, and a biomechanical model of predictions of F0 contours based on the mechanics of vibration of vocal fold lamina propria was proposed. In , multidimensional assessment was applied to voice after vertical partial laryngectomy, and the aim of this cross-sectional cohort study was to analyze the acoustic, stroboscopic, and perceptual parameters in patients who had undergone vertical partial laryngectomy and to compare them with normal subjects and total laryngectomy patients. In , laryngeal electroglottography was used as a predictor of laryngeal Electromyography.
In this paper, we investigate the first question via analyzing the self-similarity of acoustic signals. Acoustic signal is a kind of time series, and it is widely recognized that if a time series is self-similar, it is predictable. We use Xbow wireless acoustic sensors professional developer's kit MOTE-Kit  (www.xbow.com) as our testbed to get data sets from different scenarios. We show that the acoustic signal (i.e., its strength) is self-similar and long-range dependent by using variance-time plotting, a common statistical method that has been widely used to verify self-similarity of time series. Because the acoustic signal is self-similar, its characteristics can be captured, so it is predictable. We apply a type-1 fuzzy logic systems (FLS) and an interval type-2 FLS to acoustic signal strength predicting. Furthermore, we apply the acoustic signal prediction to event detection based on the forecasted signal.
The remainder of the paper is organized as follows. Section 2 studies the self-similarity of acoustic signals. Section 3 gives an overview of type-2 fuzzy sets and interval type-2 FLSs. In Section 4, we demonstrate that acoustic signal strength should be modeled as a type-2 membership function (MF), a Gaussian MF with uncertain mean. Hence, we apply this knowledge and design an interval type-2 FLS to predict the acoustic signal strength. A singleton type-1 FLS is also designed for performance comparison. In Section 5, we propose double sliding window scheme to make event detection based on the forecasted signal. Simulation results and discussions are presented in Section 6. Section 7 concludes this paper.
2 Self-Similarity of Acoustic Signals
For a detailed discussion on self-similarity in time series, see [[5, 6]]. Here, we briefly present its definition . Given a zero-mean, stationary time series X = (Xt ; t = 1, 2, 3, …), we define the m-aggregated series by summing the original series X over nonoverlapping blocks of size m. Then, it is said that X is H-self-similar if, for all positive m, X(m) has the same distribution as X rescaled by mH. That is,
If X is H-self-similar, it has the same autocorrelation function r(k) = E[(Xt − μ)(Xt + k − μ)]/σ2 as the series X(m) for all m, which means that the series is distributionally self-similar: the distribution of the aggregated series is the same as that of the original.
Self-similar processes can show long-range dependence. A process with long-range dependence has an autocorrelation function r(k) ~ k− β as k → ∞, where 0 < β < 1. The degree of self-similarity can be expressed using Hurst parameter H = 1 − β/2. For self-similar series with long-range dependence, 1/2 < H < 1. As H → 1, the degree of both self-similarity and long-range dependence increases.
One method that has been widely used to verify self-similarity is the variance-time plot, which relies on the slowly decaying variance of a self-similar series. The variance of X(m) is plotted against m on a log–log plot, a straight line with slope (− β) greater than −1 is indicative of self-similarity, and the parameter H is given by H = 1 − β/2. We use this method to verify the self-similarity of acoustic signal.
In our experiments, eight acoustic sensors were deployed in our lab at the University of Texas at Arlington. The location of each sensor is plotted in Figure 1. We designed two scenarios, one is with a fixed acoustic sound source (a CD player playing music) and the other is without a fixed source (the CD player keeps changing its location). In Figure 2, we plot the variance of X(m) against m on a log–log plot for the acoustic signals collected by the eight sensors in the first scenario and Figure 3 for the second scenario, respectively. From the two figures, it is very clear that the no matter under what kind of condition, the acoustic signals have self-similarity because their traces have slopes much greater than −1. This demonstrates that acoustic signals are self-similar, so they are predictable. We will apply interval type-2 FLS to acoustic signal predicting.
3 Introduction to Interval Type-2 Fuzzy Logic Systems
General type-2 FLSs are computationally intensive because type reduction is very intensive. Things simplify a lot when secondary MFs are interval sets (in this case, the secondary memberships are either 0 or 1). When the secondary MFs are interval sets, the type-2 FLSs are called “interval type-2 FLSs.” In , Liang and Mendel proposed the theory and design of interval type-2 FLSs. They proposed an efficient and simplified method to compute the input and antecedent operations for interval type-2 FLSs, one that is based on a general inference formula for them.
In an interval type-2 nonsingleton FLS with type-2 fuzzification and that meet under minimum or product t-norm, the result of the input and antecedent operations, Fl, is an interval type-1 set, that is, , where and simplify to
where xi (i = 1, …, p) denotes the location of the singleton. In this paper, we use center-of-sets type reduction , which can be expressed as follows:
where Ycos is an interval set determined by two end points, yl and yr; ; , and Yi is the centroid of the type-2 interval consequent set , and i = 1, …, M. We also use the training method proposed in  for designing an interval type-2 FLS in which its parameters are tuned.
Interval type-2 FLS has been applied to a number of very important applications where uncertainties abound, such as fading channel equalization  and co-channel interference elimination , network video traffic modeling and classification , connection admission control for ATM network , sensor network lifetime estimation [[13, 14]], sensed signal forecasting in wireless sensor networks , cross-layer optimization in ad hoc networks [[16, 17]], and situation understanding .
4 Acoustic Singal Strength Forecasting using Interval Type-2 FLS
Acoustic sensor node measures sound amplitude at its microphone. Assuming that the sound source is a point source and sound propagation is lossless and isotropic, a root-mean-squared (RMS) amplitude measurement z is related to the sound source position X as
where a is the RMS amplitude of the sound source, ς is the location of the sensor, and w is RMS measurement noise . According to , w is modeled as a Gaussian with zero mean and variance σ2. The sound source amplitude a is also modeled as a random quantity, which is uniformly distributed in the interval [alo, ahi]. Given the location of the sound source X and the sensor position ς, is uniformly distributed as a is uniformally distributed. Therefore, z should be modeled as a Gaussian primary MF having a fixed standard deviation and an uncertain mean.
Fuzzy logic systems have been extensively used in time-series forecasting (e.g., [[8, 20]]). Because the sensed acoustic signal strength is self-similar as demonstrated in Section 2, its characteristics can be captured, which also means that it can be forecasted. Here, we apply an interval Type-2 FLS to do a multi-step forecasting, the step size is L. We use four antecedents, that is, x(k − 4 × L), x(k − 3 × L), x(k − 2 × L), and x(k − 1 × L), as inputs of the FLS to predict x(k). Similarly, we use x(k − 4 × L + i), x(k − 3 × L + i), x(k − 2 × L + i), and x(k − 1 × L + i) to predict x(k + i), ∀ i < L. If antecedent has two fuzzy sets, the number of rules is 24 = 16. The rules are set up as in the following example:
Rl: IF x(k − 4 × L) is and x(k − 3 × L) is and x(k − 2 × L) is and x(k − 1 × L) is , THEN x(k) is .
We use center-of-sets type reduction and steepest-descent training algorithm  to design this interval type-2 FLS.
For comparison, we also design a type-1 FLS for signal strength forecasting. Antecedents are the same as in the interval type-2 FLS; however, Gaussian MFs are chosen for this type-1 FLS. There are also 16 rules because each of the antecedents has two fuzzy sub-sets as well. The rule is designed as follows:
Rl: IF x(k − 4 × L) is and x(k − 3 × L) is and x(k − 2 × L) is and x(k − 1 × L) is , THEN x(k) is Gl.
We use center-of-sets defuzzifier and steepest-descent training algorithm to design this type-1 FLS.
In this paper, we present a potential application based on the predicted acoustic signals, event detection, and propose a new event-detection algorithm, double sliding window event-detection.
5 An Application: Event Detection
In , the acoustic energy in a fixed period is integrated; when it exceeds a threshold, an event is claimed occurring as follows:
However, this simple method suffers from a significant drawback; namely, the value of the threshold depends on the sensed signal energy. When there is no event occurring in the sensing range, the sensed signal consists of only noise. The level of the noise power is generally unknown and can change when the environment changes or if unwanted interferers go on and off. Therefore, it is quite difficult to set a fixed threshold. We propose a double sliding window algorithm for event detection so as to alleviate the threshold value selection problem.
The double sliding window event-detection algorithm calculates two consecutive sliding windows of the sensed signal energy. The basic principle is to form the decision variable as the ratio of the total energy contained inside the two windows. Figure 4 shows the windows A and B and the response of the ratio mn to the start and end of a sensed event. It can be seen that when only noise is sensed the response is flat, since both windows contain ideally the same amount of noise energy.
The calculation of the window A and window B value is represented as
Then the decision variable mn is
When mn exceeds the threshold Th1, an event is claimed occurring(Figure 4(a)). The advantages of this approach are as follows: first, the decision variable mn does not depend on the sensed signal energy but on the ratio of the energy of two consecutive windows; second, we can predict not only the starting edge of the event but also the ending edge, that is, when Mn is below the threshold Th2, the event is claim ending (Figure 4(a)).
Our simulations were based on N = 480 samples, x(1), x(2), …, x(480). The first 240 data, x(1), x(2), …, x(240), are for training, and the remaining 240 data, x(241), x(242), …, x(480) are for testing. In Figure 5, we plot the sensed data that we used for training and testing, x(1), x(2), …, x(480). A standard 1-kHz audio signal with different volume levels was used to simulate the events. Each sample has a 1024-ms duration.
We applied a type-1 FLS and an interval type-2 FLS for sensed signal forecasting. The initial locations of antecedent MFs were based on the mean, mt, and std, σt, of the training data set. The parameters and number of parameters in the type-1 FLS and interval type-2 FLS are summarized in Table 1. The initial values that we choose for the Guassian MFs are listed in Table 2. Then, we use steepest-descent algorithm to train all the parameters based on the training data. After training, all the parameters and rules are fixed, and we test the interval type-2 FLS based on the remaining 240 samples, x(241), x(242), …, x(480). We set the step size as L = 5 in both the type-1 FLS and the interval type-2 FLS. Meanwhile, the window size M equals to 5 in double sliding window event-detection as well. That makes the sensed signal forecasting meaningful.
Table 1. The parameters and number of parameters in type-1 and interval type-2 fuzzy logic systems (FLSs).
interval type-2 FLS
Parameters in one antecedent
Parameters in one consequent
Total number of parameters
Table 2. Initial values of the parameters in type-1 and interval type-2 fuzzy logic systems (FLSs). Each antecedent is described by two fuzzy sets.
Interval Type-2 FLS
mt − 2σt
[mt − 2.5σt, mt − 1.5σt]
or mt + 2σt
or [mt + 1.5σt, mt + 2.5σt]
We compared the performance of the interval type-2 FLS with that of the type-1 FLS for sensed signal strength forecasting. For each FLS, we ran 100 Monte-Carlo realizations to eliminate the randomness of the consequences, and the two FLSs were tuned using a simple steepest-descent algorithm for five epochs. We used the testing data to see how each FLS performed by evaluating the root-mean-square error (RMSE) between the defuzzified output of the FLS and the actual sensor data (x(k + 1)), that is,
where xk = [x(k − 4 × 5), x(k − 3 × 5), x(k − 2 × 5), x(k − 1 × 5)]T, and T denotes transpose. The RMSE of all simulations are summarized in Figure 6. We can observe in Figure 6 that the interval type-2 FLS outperforms the type-1 FLS in the sensed signal strength forecasting.
We are more interested in the system's capability of forecasting the events, especially the starting point of the events. We used the forecasted data sets to detect the starting point of the events, that is, the time stamp of event occurrence and then compared with the actual time stamp. We evaluated our double sliding window algorithm and compared it against the cumulated signal strength scheme . We chose Th1 = mean + std for the double sliding window event-detection. Because the threshold is hard to choose for cumulated signal strength scheme, we ran simulations for three different thresholds: that is, mean, mean + std/2, and mean + std. We also ran 100 Monte-Carlo simulations so as to get the average absolute error between the forecasted and actual time stamp, , where Di is the detected starting point (based on the forecasted signal) and Pi is the actual starting point. The results are summarized in Table 3.
Table 3. Average absolute error between the forecasted and actual time stamp of the starting edge of events in type-1 fuzzy logic system (FLS) and interval type-2 FLS. Here, 1 stands for one sample or 1024 ms; m and σ stands for the mean and the standard deviation of the cumulated signal strength of the training data, respectively.
Interval type-2 FLS
SS, Signal Strength.
Double sliding window
SS with th = m
SS with th = m + σ/2
SS with th = m + σ
We can observe Table 3 that the performance of event detection based on the forecasted signal from type-2 FLS is much better than that based on the forecasted signal from type-1 FLS. Meanwhile, our double sliding window is more effective than the existing cumulated signal strength scheme. Event forecasting helps us to prepare future events ahead of time.
In this paper, we studied the security assurance in application layer in wireless acoustic sensors via event forecasting and detection. In order to perform even forecasting and detection, we have answered three challenging questions in acoustic signal research using Xbow acoustic sensors:
Acoustic signals are predictable because we have demonstrated that real-world acoustic signals are self-similar.
We applied interval type-2 FLS to acoustic signals prediction. We showed that a type-2 fuzzy MF, that is, a Gaussian MF with uncertain mean, is appropriate to model the acoustic signal strength. Two FLSs, a type-1 FLS, and an interval type-2 FLS were designed for acoustic signal forecasting.
Furthermore, we proposed a double sliding window scheme for event detection based on the forecasted signals. Simulation results showed that the interval type-2 FLS outperforms the type-1 FLS in signal strength forecasting and the performance of event detection based on the forecasted signal from type-2 FLS is much better than that based on type-1 FLS.