Signal Detection Based on Power-Spectrum Sub-Band Energy Ratio

: The power-spectrum sub-band energy ratio (PSER) has been applied in a variety of fields, but reports on its statistical properties and application in signal detection have been limited. Therefore, the statistical characteristics of the PSER were investigated and a signal detection method based on the PSER was created in this paper. By analyzing the probability and independence of power spectrum bins, as well as the relationship between F and beta distributions, we developed a probability distribution for the PSER. Our results showed that in a case of pure noise, the PSER follows beta distribution. In addition, the probability density function exhibited no relationship with the noise variance—only with the number of bins in the power spectrum. When Gaussian white noise was mixed with the signal, the resulting PSER followed a doubly non-central beta distribution. In this case, the probability density and cumulative distribution functions were represented by infinite double series. Under the constant false alarm strategy, we established a signal detector based on the PSER and derived the false alarm probability and detection probability of the PSER. The main ad-vantage of this detector is that it did not need to estimate noise variance. Compared with time-domain energy detection and local spectral energy detection, we found that the PSER had better robustness under noise uncertainty. Finally, the results in the simulation and real signal showed that this detection method was valid.


Introduction
The power-spectrum sub-band energy ratio (PSER) is a common metric used to represent the proportion of signal energy in a single spectral line. The PSER is derived from spectral analysis and has a direct relation with the amplitude and power spectra. The signal mixed with additive Gaussian white noise (GWN) can be expressed as where , , ,  0 1 1 nN ; () ( (2) where N is the number of spectral bins, j is the imaginary unit, and the arrow superscript denotes a complex function. The k th line in the power spectrum of () sn can be expressed as ( ) | ( ) |  2 1 .
The PSER , () dN Bk is defined as the ratio of the sum of adjacent d bins from the k th bin in the power spectrum to the entire spectrum energy, i.e.,  The PSER is consistent with the spectrum in the waveform, and the data dimension can be greatly reduced by combining spectral bins. Therefore, it has been extensively applied in the fields of remote communication [1,2], earthquake modeling [3], machine design [4], and geological engineering [5]. For example, Preisig et al. used the PSER to study signal detection in underwater acoustic environments [6]. Wang et al. described the distribution of acoustic emission signals in the fracturing of fine sandstone [7]. Xu and Guan extracted the characteristics of underground micro-seismic signals collected by a distributed optical fiber using the PSER [8]. Huang et al. investigated the effects of various dualpeak spectral parameters on dispersive coefficients [9]. Yang et al. used the PSER as a signal feature in coal gangue recognition [10]. Lu et al. studied micro-seismic signals in dual-layer igneous strata [11]. Kong et al. described the feature of electromagnetic radiation under coal oxidation [12]. Though the PSER is widely used, research on its statistical characteristics and detection performance is not sufficient and must be further improved.
Statistical characteristics are the theoretical basics of signal detection. The statistical characteristics of frequency and power spectra have been extensively studied for GWN. The real and imaginary parts of the spectrum follow a Gaussian distribution [2], and the amplitude conforms to a Rayleigh distribution [13]. Groth analyzed the statistical characteristics of the power spectrum with white noise [14]. Johnson et al. studied the distribution characteristics of power spectra acquired by the periodogram averaging technique [15]. Martinez et al. presented a detector based on the Welch's periodogram [16], and the test statistic of this method was approximated by the Gaussian distribution. Pei-Han proposed a power spectral density split cancellation with the same definition as the PSER and derived its probability density function (PDF) and cumulative distribution function (CDF); however, its probability distribution was not specified [2,17]. Bomfin et al. proposed the cooperative power spectral density split cancellation method [18], which was more robust against noise uncertainty than the method of Pei-Han. Bomfin et al. proposed circular folding cooperative power spectral density split cancellation and derived a closed form expression for false alarm threshold [19]. However, he did not provide the expression for CDF when a signal presented. Gurugopinath et al. proposed the fast Fourier transform average ratio (FAR) algorithm that was the same as the PSER. The CDF for the FAR was expressed by a hypergeometric function [20], which was more complicated than our method. In this study, we qualitatively analyzed the probability distribution of the PSER and obtained the accurate PDF and CDF of different existing formulas.
The purpose of signal detection is to detect whether there is a signal on the sampling sequence under a noisy environment. Common signal detection methods [21,22] can be classified as matched filter detection [23], energy-based detection [24], and cyclostation-ary-based detection [25], among others. Energy detection can be classified into two categories: time domain energy detection and frequency domain energy detection. The PSER belongs to a frequency-domain energy detection technology. According to the statistical characteristics of the PSER, we deduced the signal detection method based on the PSER under the constant false alarm (CFAR) strategy.
The remainder of this paper is organized as follows. Section 2 discusses the statistical characteristics of the PSER for GWN. Section 3 develops the signal detection method based on the PSER under the CFAR. In Section 4, simulation experiments with the narrowband and wideband signals are separately used to verify the accuracy of the PSER detection method. The application in vibration signals suggests that the PSER detector can work well in real environments. Section 5 provides additional details concerning the research process.

Statistical Characteristics of GWN
The GWN spectrum, calculated using a discrete Fourier transform (DFT), is given by Let the real and imaginary parts of Zk are mutually independent, and they all follow a Gaussian distribution [14]; that is, The power spectrum for the term () Zk is defined as Pk is also called a power spectrum bin. Any two white noise power spectral bins are independent [2], and () 2 2 z Pk σ follows a chi-squared distribution with two degrees of freedom [2].

Statistical Characteristics of One Power Spectrum Bin
Under 1 H , the discrete Fourier transform of () sn can be expressed as The power spectra for The power spectrum for Xk represents the spectrum of a known signal, the real and imaginary parts of the spectrum can be assumed to be constant (i.e., ()  kR a X k and ()  kI b X k ) [14]. As a result,    (17) According to the PDF of the F distribution, the PDF of Substituting Equation (18) The probability density plot for , () dN Bk is shown in Figure 1.   (22) The CDF for where ( , ) p I a b is the incomplete beta function.
It can be shown that the PDF and CDF of the PSER for GWN have no relationship to the noise variance and are only related to the number of power spectrum bins in a subband d and the number of bins in total power spectrum N.
and its numerator follows a non-central chi-square distribution with 2d degrees of freedom and a non-centrality parameter  k λ , and its denominator follows a non-central chisquare distribution with  22 Nd degrees of freedom and a non-centrality parameter Bk is a ratio of two non-central chi-square distributions, which is referred to a G distribution. Let According to the definition of the G distribution [26], we deduce that the PDF of , The CDF of According to Equation (16), the CDF of , () dN Bk can be derived from the CDF of The derivative of Equation (30) is the PDF of , () dN Bk :

Relationship between
 12 δ δ and Signal-to-Noise Ratio (SNR) 1 δ is the ratio of the energy of valid signal contained in the selected band to the noise energy of one spectral bin. 2 δ is the ratio of the energy of the valid signal not contained in the selected band to the noise energy of one spectral bin.
The total power spectrum energy for the signal and noise can be represented, respectively, as As such, the relationship between , i.e., *  10 12 10 . An indicator to measure the signal quality of the local frequency band, which is used in later text, is the local SNR. It is defined as the ratio of local frequency energy to local noise energy, i.e.,

Principle
The signal detection based on the PSER takes , () The common performance measures are the probability of false alarm Pf, and the probability of detection Pd. Under the Neyman-Pearson criteria (i.e., CFAR criterion), likelihood ratio yields the optimal hypothesis testing solution and performance are measured by a resulting pair of detection and false alarm probabilities (Pd and Pf). Each pair is associated with the particular threshold PSER  that tests the decision statistic. If the test statistic is larger than , the signal is deemed to be present, and it is absent otherwise, i.e., The false alarm probability Pf can be expressed as follows: be solved by referring to the method provided in [27,28]. According to Equation (30), the detection probability d P can be expressed as follows: The decision domain of the PSER is shown in Figure 3. In Figure 3, and the SNR is −9.03 dB. The dotted line is the probability density function under H0, and the solid line is the probability density function under H1. R0 is the decision domain of signal absence, the range of which is

Performance Comparison of the PSER with Other Energy Detection Methods
Under the same conditions, the PSER will compare the detection probability with the commonly used time-domain energy detection (TDED) [29] and local spectral energy detection (LSED) methods to evaluate the detection performance of the PSER.

Local-Spectrum Energy Detection
The detection object of LSED is also the sub-band of interest; therefore, LSED is the method closest to the PSER. If the sub-band is composed of P(i), P(I + 1), …, P(i + d − 1) and total d spectral bins, the detection statistic can be defined as follows: According to the central limit theorem (CLT), the distribution of LFD T is approxi- where  γ is the local SNR, as denoted in Equation (36). The false probability f P is Then, LFD η can be derived from Equation (46), The detection probability d P can be expressed as follows:

Theoretical Detection Probabilities Comparison
In this section, the detection performance of the PSER, LSED, and TDED is compared, and the effects of sub-band energy ratio coefficient a, local bin number d, and full spectrum bin number N on the detection performance of the PSER are discussed.
When N = 512, d = 20, Pf = 0.01, and a = 1 (i.e., the sub-band contains all the energy of signal), the theoretical detection probabilities of the three methods are shown in Figure 4. It can be seen from Figure 4 that the detection probability of the PSER is better than that of TDED, but it is slightly worse than that of LSED.
When the sub-band energy ratio coefficient changes, the detection probabilities of the PSER and LSED are as follows.
In Figure 5, the smaller the sub-band energy ratio coefficient is, the worse the detection probabilities of the PSER and LSED are. When a = 1, the detection probabilities of LSED and the PSER were the best. When a = 0.2, the detection probabilities of LSED and the PSER decrease, and even the detection probability of the PSER is inferior to that of TDED. When the sub-band bin number d changes, the detection probabilities of the PSER and LSED are as follows.
In Figure 6, the smaller the sub-band bin number is, the better the detection probabilities of the PSER and LSED are. However, the detection probability of the PSER is always worse than that of LSED. When the full spectrum bin number N changes, the detection probabilities of the PSER and LSED are as follows.
In Figure 7, the greater the full spectrum bin number N is, the better the detection probabilities of the PSER and LSED are. However, the detection probability of the PSER is slightly worse than that of LSED. The detection probability of the PSER is influenced by the sub-band energy ratio coefficient a, local bin number d, and full spectrum bin number N. When d decreases, N increases, or a increases, the detection performance of the PSER will improve. In general, the detection performance of the PSER is lower than that of LSED but is generally higher than that of TDED. However, the detection performance of the PSER will be lower than that of TDED when the sub-band contains less signal energy.

Detection Performance of the PSER under Noise Uncertainty
In an actual environment, there are many electromagnetic disturbances, including GWN, quantized noise, and thermal noise. Therefore, the noise intensity is time-varying, the noise is only approximately Gaussian distribution, and the noise power fluctuates within a certain range, which is called the noise power uncertainty (NPU) interval. The NPU interval can be expressed as ,    To make the system achieve the ideal detection performance, 2 σ should then max-    In Figures 8 and 9, the solid line is the detection probability under certain noise (i.e., =0 ρ ), and the dotted line is the detection probability under noise uncertainty, i.e.,  0 ρ . Under certain noise, the detection probability of LSED is the best, followed by that of the PSER, while that of TDED is the worst. When =0.5 ρ , the detection probability of TDED decreases the most, followed by that of LSED, while that of the PSER is decreased only slightly. When =1 ρ , the detection probabilities of three methods decrease more significantly than that of =0.5 ρ , but the decrease amplitude of the PSER is the smallest. Obviously, in the case of noise power uncertainty, the PSER is less affected by noise uncertainty, and its robustness is better than that of LSED and TDED, which is the largest advantage of the PSER in signal detection.

Experiments
For this section, we verified and compared the detection performances of the PSER, LSED, and TDED methods discussed in Section 3 through Monte Carlo simulations and the vibration signal collected by an optical time domain reflectometer (OTDR).

Simulations
We used cosine signals to simulate narrow-band signals, which could ensure all the energy contained in the sub-band by using narrow-band signals. In this case, the energy proportion coefficient was 1.
We used Ricker wavelet signals to simulate broadband signals because the energy occupied by the local frequency band is easy to control. In this case, the energy proportion coefficient was less than 1.
We performed all Monte Carlo simulations for at least 10 5 independent trials. We set Pf to 0.01 and N to 512. We used mean-square error (MSE) to measure the deviation between the theoretical values and actual statistical results.

Narrow-Band Signal
We used the cosine signal mixed with GWN as experimental data. The amplitude of the sinusoidal signal was 0.5, the frequency was 30 Hz, the phase was 0, and the duration was 1 s. For convenience of calculation, we set the frequency resolution as 1 Hz. We used a bilateral spectrum in the experiment. Because the energy in the power spectrum of the signal was concentrated at approximately 30 Hz, the selected frequency band was 21-42 Hz; that is, it contained 22 spectral bins. When the SNR was −10 dB, one cosine signal and its power spectral density (PSD), under noise absence and presence, respectively, are shown in Figure 10.   The average false alarm probability of the PSER was 0.0091, that of LSED was 0.0181, and that of the time-domain energy method was 0.0117. Therefore, as shown in Figure 11, the actual false alarm probability of the PSER was below 0.01, that of TDED was above 0.01, and that of LSED was close to 0.02.
As seen in Figure 12, when the sub-band energy ratio coefficient was good, e.g., a = 1, the detection performance of the PSER was better than that of TDED, while the detection performance of LSED was better than that of the PSER.
The MSEs of the actual and theoretical probabilities of these experiments are given in Table 1. It can be seen from Table 1 that the MSE between the actual and theoretical false alarm probabilities of the PSER was the lowest, followed by that of TDED and LSED. At the same time, the MSE between the actual and theoretical detection probabilities of TDED was the smallest, that of the PSER was slightly higher than that of TDED, and that of LSED was significantly higher. The deviation between the PSER actual and theoretical detection probabilities was very small, which indicated that the PSER method is accurate.

Broadband Signal
The PSER is often used for vibration monitoring, especially in microseismic detection, while the Ricker wavelet is often used to simulate seismic or microseismic signals in underground vibration simulation experiments. Therefore, the Ricker wavelet was selected as the simulation broadband signal in this paper. The expression for its time domain is The experiment was focused on the two cases of high-local SNR and poor-local SNR to highlight the influences of 1 δ and 2 δ on the PSER.

High-Local SNR
When we set the sub-band of interest at 36-65 Hz, the frequency band occupied approximately 61.5% of the total energy (i.e., a = 0.615). The theoretical and actual detection probabilities of the three methods are shown in Figures 14 and 15.  In Figure 15, it is shown that the detection performance of the PSER was better than that of TDED, while the detection performance of LSED was better than that of the PSER.

Low-Local SNR
We set the sub-band of interest at 21-40 Hz, and the frequency band occupied only approximately 21% of the total energy (i.e., a = 0.21). The theoretical and actual detection probabilities of the five methods are shown in Figures 16 and 17.   Figure 17 shows that the detection probabilities of the PSER and LSED were lower than that of TDED when the energy proportion coefficient was low. Therefore, the subband energy ratio coefficient should not be too low to ensure the good detection performance of the PSER.

Comparison of Mean Square Error
The MSEs of the actual and theoretical probabilities of these experiments are given in Table 2. As shown in Table 2, whether under high or low local SNR, the MSE between the actual and theoretical false alarm probabilities of the PSER was the smallest and that of TDED was second. The MSE between the actual and theoretical detection probabilities of TDED was the biggest, that of the PSER was the second, and that of LSED was the smallest. Therefore, the difference between the actual and theoretical detection probabilities of PSER was not large, which indicates that the detection method based on PSER was accurate for broadband signals.

Vibration Signal Detection
The vibration signals were collected by an NBX-S3000 instrument (Nebrex Co., Hyogo, Japan). The vibration signals were generated by a tapping machine, which is a standard piece of vibratory equipment. There were five percussion hammers on the tapping machine, and the tapping interval was 0.1 s. Photographs of the NBX-S3000 instrument and tapping machine are shown in Figure 18. The OTDR technique monitors the vibration sensed by an optical fiber. The shape of an optical fiber is a line. One fiber can be divided into many segments of equal length. Each segment is equivalent to a sensor, and the length is denoted as spatial resolution, i.e., the minimum space distance that an OTDR instrument can discern.
The state of one fiber with one vibration source is shown in Figure 19. According to the vibration attenuation law, the vertical intersection point between the vibration source and optical fiber is the strongest point of vibration. In this experiment, the fiber was buried approximately 10 cm below ground. The spatial resolution was 1 m, the sampling rate 4000 Hz, and the monitoring range was from 66 to 126 m from the start side. The tapping machine was fixed almost 97 m from the start side and 1 m from the fiber.

Background Noise
Before detecting the vibration signal, the background noise collected by the NBX-S3000 instrument, totaling 600,000 records, were analyzed. We found that the noise at different distances had different statistical properties, such as mean, variance, and power spectrum. Figure 20 shows the mean and variance at different distances. The average PSD values at 97 m and 85 m, which were calculated by the Welch method, are shown in Figure 21.  It can be seen that the background noise was not true GWN; rather, it was more like pink noise.

Tapping Signal
The recording time of the tapping signal was 10 s, and the tapping interval was 0.1 s; therefore, 100 tapping samples were taken at one distance. The time-domain waveform and PSD of the tapping signal at 97 and 85 m are shown in Figures 22 and 23, respectively.  The tapping signal at 97 m was easier to discern than that at 85 m in the time domain. The bins at 97 m between 50 Hz and 100 Hz were higher than the other bins. Therefore, the characteristic frequency of tapping was in the range of 50-100 Hz. The bins at 85 m between 50 Hz and 100 Hz seemed to be higher than other bins, but it was not obvious.
The approximate SNRs from 84 m to 108 m are shown in Figure 24. The signal at 97 m was closest to the vibration source, so it had a high SNR. The signal at 85 m or 108 m was far from the vibration source, so it had a low SNR.

Detection
Since the noise in the tapping signal was not GWN, we had to change the colored noise into white noise before using LSED and the PSER. We obtained a new PSD by subtracting the average PSD of noise from the PSD of the tapping signal. The new PSD was used in the LSED and the PSER.
When the sub-band was 66-105 Hz, N was 1024, Pf was 0.05, and d was 11. The false alarm and detection probabilities of the PSER, LSED, and TDED are shown in Figures 25  and 26.  The false alarm probability of TDED was approximately 60%, which indicates that indicated that the detection result of TDED was incorrect.
As seen in Figure 26, the detection probability of LSED was greater than that of the PSER, and the false alarm probabilities of LSED and the PSER were close to a, which indicated that the detection results of LSED and the PSER were correct.
Under noise uncertainty, when ρ was 5, the change can be seen in Figure 27. In Figure 27, the thick lines are the detection probabilities of three methods under original noise and the thin lines are the detection probabilities under noise uncertainty. The detection probabilities of LSED and TDED were both decreasing; however, that of the PSER did not change markedly; that is to say, the robustness of the PSER was better than that of LSED and TDED.

Rationale for Rectangular Window Function Selection
In spectral analysis, it is necessary to add a symmetric window function to a sampling sequence in order to suppress side-lobe interference and improve accuracy. This symmetric window function can be expressed as A value of a = 1 corresponds to a rectangular window, a = 0.5 indicates a Hanning window, and a = 0.54 implies a Hamming window. In this study, a rectangular window was used to calculate the spectrum with Equation (2). A rectangular window exhibits inherent limitations, but unlike other window functions, it can ensure that spectral bins remain mutually independent after a discrete Fourier transform of the signal. The use of other window functions often produces a correlation between the power spectrum bins, which makes any subsequent calculation of probability distributions extremely difficult.

Rationality of CDF for the PSER
The expression of CDF for the PSER appearing in the previous literature [2] was greatly improved in this paper, especially when the number of spectral bins in the subband was relatively small, such as d < 10. In [2], the numerator and denominator of the random variable This approximation is reasonable if the number of bins in the numerator and denominator is large enough. However, if the number of bins in the numerator is small, this approximation is subject to a large deviation. In this paper, the numerator and denominator of , ()  dN Bk were regarded as non-central chi-square distributions. When the number of spectral bins in the numerator was relatively small, the cumulative distribution function was still very accurate.

Calculation of CDF for the PSER
The CDF for the PSER derived in this paper was expressed by infinite double series, and its value could only be obtained by numerical calculation. Since the number of calculation terms was set to be large, it took a significant amount of calculation time. However, the CDF for the PSER is only used in theoretical analysis rather than being required in actual detection. How to improve the calculation speed of CDF needs further research.

Detection Performance of the PSER Depends on Sub-Band Energy Ratio Coefficient
The experiment in Section 4 showed that the detection performance of the PSER depends on the sub-band energy ratio coefficient when the SNR is fixed, and the higher the sub-band energy ratio coefficient is, the better the detection performance will be. However, further study is needed to determine how much the sub-band energy ratio coefficient can guarantee that the detection performance of the PSER is better than that of TDED.

Advantages of the PSER
The PDF and CDF of the PSER under GWN have nothing to do with the variance of noise; rather, they are only related to the number of spectral bins of the entire spectrum and sub-band. This means that a detector based on the PSER does not need to estimate noise variance. However, TDED and LSED need to estimate noise variance, and if the variance is estimated incorrectly, the detection performances of TDED and LSED will decrease greatly. Therefore, the detection performance of the PSER is better than that of TDED and LSED under noise uncertainty.

Conclusions
In this paper, a systematic investigation of the statistical characteristics for powerspectrum sub-band energy ratios in the cases of signal absence and signal presence is presented. The statistical characteristics of the PSER provided a theoretical foundation for the use of the PSER in signal detection. The results demonstrated that the PSER of signal absence and signal presence followed beta and doubly non-central beta distributions, respectively. According to the statistical characteristics of the PSER, a signal detection method based on the PSER was established. We found that the detection performance of this method was inferior to that of LSED and generally superior to that of TDED. However, the robustness of PSER detection performance under noise uncertainty was the best among the three methods.  Data Availability Statement: Restrictions apply to the availability of these data. Data sharing is not applicable to this article.

Conflicts of Interest:
The authors declare no conflicts of interest.