Spectral Detection of a Weak Frequency Band Signal Based on the Pre-Whitening Scale Transformation of Stochastic Resonance in a Symmetric Bistable System in a Parallel Configuration

Abstract

The spectral detection of weak frequency band signals poses a serious problem in many applications, especially when the target is within a certain frequency band under low signal-to-noise ratio (SNR) conditions. A kind of novel technique based on the pre-whitening scale transformation of stochastic resonance (SR) in a symmetric bistable system in a parallel configuration is proposed to solve the problem. Firstly, pre-whitening can ensure the Gaussian distribution of the receiving signal fits the requirements for SR processing. Secondly, scale transformation can help to effectively utilize the properties of a weak signal, especially under a low-frequency band. Thirdly, the SR in a symmetric bistable system in a parallel configuration can try to smoothly reduce the variances in the clutter and additive noise. Fourthly, by subtracting the steady state response of the SR in the selected symmetric bistable system from the parallel output, the spectral detection of a weak signal can be realized successfully. Experiment results based on actual sea clutter radar data guarantee the effectiveness and applicability of the proposed symmetric bistable PSR processing approach.

1. Introduction

Weak target signal detection is a common problem in different applications, such as radar target detection [1], image object discovery [2], and optical damage detection [3]. Considerable research has been carried out in solving the problems associated with weak target signal detection, especially when the corresponding signal-to-noise ratio (SNR) is relatively very low (for example, below 0 dB) and the target signal possesses a certain bandwidth.

In the literature, many researchers have investigated the problems of weak target detection by using different methods and techniques. A space–time–waveform joint adaptive detection (STWJAD) method was proposed that combines spatial, temporal, and waveform dimensions; it is based on the linearly constrained minimum variance (LCMV) criterion for effective adaptive processing and detection [4]. The theoretical performance is outstanding, but the complexity of realization remains high. In [5], a new weak maneuvering target detection method was studied in a sea environment with radon–fractional Fourier transform (RFrFT) canceller, which could efficiently suppress the sea clutter and detect targets with low computations, but it needs to utilize different representations of the maneuvering target and sea clutter, and it is hard to apply in other areas. To overcome the shortcomings of fractal analysis in the time domain and Fourier transform domain, ref. [6] mainly studied the joint fractal properties of autoregressive (AR) spectral sea clutter and its application in weak target detection. In the study, the box-counting dimension was combined with AR spectrum estimate theory, which considers the correlation properties of the sea clutter series. Moreover, the intercept is regarded as an auxiliary feature for target detection. However, the intercept can only solve the problems of binary classification and detection and not the spectral detection of a weak frequency band signal. The traditional energy detection method and spectrum analysis method also exist, which perform poorly under low SNR conditions.

Based on the above conditions and up-to-date methods, it is necessary to develop a reliable weak signal detection approach under low SNR conditions with low-to-moderate complexity. In previous studies, it has been found that the nonlinear stochastic resonance (SR) system possesses good enough performance to improve the weak signal detection capability [7,8]. But to achieve an ideal weak signal detection performance, an optimization process should be introduced with some prior knowledge of the SR system [9], which may produce new problems in applications. Although there are some other weak signal frequency detection methods that use nonlinear chaos theory [10], they may only fit narrowband signal detection, not wideband signal detection.

In this research, for the purpose of weak signal detection with more reliable performance and acceptable computational complexity under low SNR conditions, a novel spectral detection approach for a weak frequency band signal is proposed, which combines pre-whitening processing, scale transformation operations, and SR in a symmetric bistable system in a parallel configuration with linear response properties. Based on the introduced techniques, the weak frequency band of the target can be enhanced and remain in the middle part of the scale-transformed frequency band of the SR output of the symmetric bistable parallel system. Finally, real experimental performance evaluations based on real radar data are performed to verify the effectiveness and superiority over the traditional weak signal detection method.

The structure of the paper is as follows: Section 1 briefly introduces the weak signal detection problem and the main contributions of the proposed research. In Section 2, the proposed weak signal detection approach based on the pre-whitening scale transformation of the stochastic resonance in a symmetric bistable system in a parallel configuration is described in detail. The block diagram of the proposed approach and the corresponding SR structure and signal processing flow of the parallel symmetric bistable system are included. Section 3 presents some real experiments to evaluate the application performance of the proposed approach compared with some conventional methods. Finally, the conclusions are given in Section 4 to summarize the whole paper.

2. Proposed Weak Signal Detection Based on Pre-Whitening Scale Transformation of the Stochastic Resonance in a Symmetric Bistable System in a Parallel Configuration

To show a clear structure of the proposed approach, Figure 1 presents the block diagram of the spectral detection of a weak frequency band signal based on the pre-whitening and scale transformation of stochastic resonance in a symmetric bistable system in a parallel configuration. The details of each block will be explained in detail below.

Generally, for the weak target signal detection problem, the signal model can be expressed as follows:

$$r\left(t\right)=s\left(t\right)+c\left(t\right)+n\left(t\right)$$

(1)

where $r\left(t\right)$ is the receiving signal, $c\left(t\right)$ is the clutter signal, $n\left(t\right)$ is the additive noise, including the measurement noise, and $s\left(t\right)$ is the weak frequency band signal. $s\left(t\right)$, $c\left(t\right)$ and $n\left(t\right)$ are independent of each other. Thus, (1) can be mathematically regarded as a two-hypotheses detection problem.

In different applications, the clutter signal $c\left(t\right)$ may have different time–frequency domain characteristics, and the additive noise $n\left(t\right)$ may also possess various distributions. Without full prior knowledge, it will be difficult to carry out corresponding analysis and detection. Here, we first consider letting the receiving signal $r\left(t\right)$ pass through a pre-whitening processor or a pre-whitening filter, so that

$$r_W\left(t\right)=\mathrm{W}\left[r\left(t\right)\right]=\mathrm{W}\left[s\left(t\right)\right]+\mathrm{W}\left[c\left(t\right)\right]+\mathrm{W}\left[n\left(t\right)\right]$$

(2)

where $r_W\left(t\right)$ is the receiving signal passing through the pre-whitening processor and $\mathrm{W}\left[·\right]$ is the corresponding pre-whitening processing operator. In this study, the conventional maximum entropy spectral estimation for the ideal whitening method is used to realize the process.

Considering the weak target detection condition that the power of $s\left(t\right)$ is much lower than those of $c\left(t\right)$ and $n\left(t\right)$, the pre-whitening processing may result in a uniform distribution in the frequency domain of $r_W\left(t\right)$ under both hypotheses. Without loss of generality, $\mathrm{W}\left[c\left(t\right)\right]+\mathrm{W}\left[n\left(t\right)\right]$ can be regarded as a united random process denoted by $c_W\left(t\right)$, and by denoting $\mathrm{W}\left[s\left(t\right)\right]$ as $s_W\left(t\right)$, we have

$$r_W\left(t\right)={s_W\left(t\right)+c}_W\left(t\right)$$

(3)

In the weak signal detection problem, among different techniques, it is found that the stochastic resonance (SR) in a nonlinear symmetric bistable system is a good technique that can help improve the weak signal power, especially under low SNR conditions, which has been applied in many applications [11,12,13]. In the weak signal detection problem, the SNR is defined as the ratio of useful signal power within the detected frequency band to signal noise within the whole detected frequency band. To optimally improve the SNR performance, some prior knowledge regarding the distribution and corresponding parameters of strong clutter, additive noise, and weak signals is required, which may be difficult in certain applications. By using the pre-whitening processing introduced above, it is easy to normalize $r_W\left(t\right)$ to a zero-mean unit variance symmetric Gaussian white noise process. Based on the Linear Response Theory of the SR system [14,15], it is found that when the driving signal of the SR system is composed of weak signals and wideband noise signals that are independent of each other, the response of the SR system can also be divided into two parts related to the weak signal and wideband noise signal. At the same time, for the traditional SR system, when the wideband noise possesses the white Gaussian distribution, it may have better SNR improvement performance. Thus, in order to enhance the performance of the SR, the pre-whitening processing is introduced first in the proposed method.

The frequency band of the weak signal can be estimated approximately as $\left[{\widehat{f}}_{min},{\widehat{f}}_{max}\right]$, where ${\widehat{f}}_{min}$ and ${\widehat{f}}_{max}$ represent the estimated minimal and maximal frequencies of the weak signal band. A nonlinear SR system is introduced to enhance the weak frequency band target signal. Without loss of generality, the following scale transformation-based symmetric bistable SR processing method [14] is introduced to enhance the weak signal power as

$${\displaystyle \frac{dz\left(t\right)}{dt}}=z\left(t\right)-z^3\left(t\right)+\sqrt{{\displaystyle \frac{b}{a^3}}}·r_W\left(t\right)+\sqrt{{\displaystyle \frac{b}{a^3}}}·\sqrt{2Da}·\xi \left(t\right)$$

(4)

where $z\left(t\right)$ is the state variable of the symmetric bistable SR system, a and b are the bistable potential well parameters of the SR system, $\xi \left(t\right)$ is the SR noise with a zero-mean- and unit-variance-obeying symmetric white Gaussian distribution, and D is the noise power. Here, the parameters a and b are chosen for the enhanced symmetric bistable SR performance [15] in the scale-transformed bandwidth of the SR in the symmetric bistable system within $\left(0 \mathrm{H}\mathrm{z},10 \mathrm{H}\mathrm{z}\right)$ as

$$a=0.2\left({\widehat{f}}_{max}-{\widehat{f}}_{min}\right)$$

(5)

$$b=0.7\left({\widehat{f}}_{max}-{\widehat{f}}_{min}\right)$$

(6)

The above settings ensure that the weak frequency band signal spectrum lies in the middle part of the scale-transformed bandwidth of the SR in the symmetric bistable system. From (4), it can be seen that the computational complexity of the introduced SR processing may be regarded as the cubic calculation along with the sampling number due to its nonlinearity. In this case, the computational complexity is not high.

Next, although the symmetric bistable SR system in (4) is selected and the scale transformations in (5) and (6) are introduced, the weak signal enhancement of the symmetric bistable SR system is still related to the time and frequency properties of $s\left(t\right)$. The properties are unknown without prior information. To solve this problem, a kind of symmetric parallel SR processing technique is proposed after the previous pre-whitening and scale transformation processing.

Figure 2 shows the processing structure of the SR in the symmetric bistable system in a parallel configuration. The receiving signal $r_W\left(t\right)$ is used as a common driving signal to all K-symmetric bistable parallel systems with SR. While in these symmetric bistable systems with SR, $g\left[·\right]$ represents the scale transformation-based SR processing in a symmetric bistable system in (4), $z_k\left(t\right)$ is the state variable of the k-th $\left(k=\mathrm{1,2},\cdots ,K\right)$ symmetric bistable SR system, ${\xi }_k\left(t\right)$ is the k-th $\left(k=\mathrm{1,2},\cdots ,K\right)$ SR noise process with the same zero mean, unit variance, and symmetric white Gaussian distribution characteristics, and $r_{WP}\left(t\right)$ is the improved output signal of the SR in the symmetric bistable system in a parallel configuration.

According to the Linear Response Theory of the SR system [14,15], it can be found that, under $H_1$, the stable state variable $z\left(t\right)$ can be written as

$${〈z\left(t\right)〉}_{asy}={〈z\left(t\right)〉}_{st}+{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\kappa \left(t-\tau ,a,b\right)\left[\sqrt{{\displaystyle \frac{b}{a^3}}}·r_W\left(\tau \right)+\sqrt{{\displaystyle \frac{b}{a^3}}}·\sqrt{2Da}·{\xi }_k\left(\tau \right)\right]d\tau$$

(7)

where ${〈z\left(t\right)〉}_{asy}$ is the asymptotic response of $z\left(t\right)$, ${〈z\left(t\right)〉}_{st}$ is the mean value of the unperturbed state variable under $r_W\left(t\right)=0$, and $\kappa \left(t-\tau ,a,b\right)$ is the response function.

When the receiving signal $r_W\left(t\right)$ is introduced into all K-symmetric bistable parallel systems with SR, the corresponding response related to $r_W\left(t\right)$ in each system branch will remain the same. When ${\xi }_k\left(t\right)$ $\left(k=\mathrm{1,2},\cdots ,K\right)$ is independent and identically distributed (i.i.d.), the variance in the PSR-improved output signal $r_{WP}\left(t\right)$ in Figure 2 is as follows:

$$var\left[r_{WP}\left(t\right)\right]=var\left[{〈z\left(t\right)〉}_{st}\right]+var\left[{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\kappa \left(t-\tau ,a,b\right)\sqrt{{\displaystyle \frac{b}{a^3}}}·s_W\left(\tau \right)d\tau \right]+{\displaystyle \frac{1}{K}}var\left[{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\kappa \left(t-\tau ,a,b\right)\left(\sqrt{{\displaystyle \frac{b}{a^3}}}·c_W\left(\tau \right)+\sqrt{{\displaystyle \frac{b}{a^3}}}·\sqrt{2Da}·{\xi }_k\left(\tau \right)\right)d\tau \right]$$

(8)

When the initial values of the state variable in each symmetric bistable parallel SR system are chosen randomly, the response of $c_W\left(\tau \right)$ could be also regarded as a random process. And under the assumptions that when the parallel SR processing unit number K is big enough, it can be found that the last term in (8) will approach zero, which may reduce the impacts of $c_W\left(\tau \right)$ and ${\xi }_k\left(\tau \right)$ significantly, and it may also help to exhibit the time and frequency domain characteristics of the weak signal accordingly.

Finally, when the symmetric bistable SR system is chosen and fixed, the first item on the right side of (8), $var\left[{〈z\left(t\right)〉}_{st}\right]$, can also be obtained if there is no driving signal introduced into the SR parameter in the symmetric bistable system. Thus, based on the above analyses, we can estimate the weak frequency band signal $\widehat{s}\left(t\right)$ as

$$\widehat{s}\left(t\right)=r_{WP}\left(t\right)-{〈z\left(t\right)〉}_{st}={\displaystyle \frac{1}{K}}{\sum }_{k=1}^K{z}_k\left(t\right)-{〈z\left(t\right)〉}_{st}$$

(9)

Then, the spectral detection of the weak frequency band signal can be carried out based on the estimated $\widehat{s}\left(t\right)$. According to the nonlinear weak signal enhancement of the SR against the weak driving signal, the corresponding frequency band property of $s\left(t\right)$ can be observed, which may remain the same as the frequency band of the original weak signal.

3. Real Experiments and Performance Evaluation Results

To evaluate the performance of the proposed spectral detection approach for a weak frequency band signal, real experiments using real data are carried out and compared with the results from conventional methods.

In real experiments, real radar data containing real clutter data and weak target data are used as the objective to be detected; the frequency band is within [800 Hz, 1200 Hz]. The SNR of the receiving signal is in the range from −15 dB to 0 dB, which is a relatively low SNR for weak signal detection. The noise mostly comes from real sea clutter data, and the weak target signal comes from a small ship with a Doppler shift due to target translational motion.

Table 1. Parameters and settings in the experiments and performance evaluations.

Parameters or Settings	Values or Descriptions
radar type	SWR
frequency band	VHF
grazing angle	(−60°, 60°)
radar waveform	continuous wave
radar name	decimeter wave radar
sea state	3
sample size	106
sampling frequency	4096 Hz
step size	0.24 ms
software used	MATLAB R2022a
frequency band of the target signal	[800 Hz, 1200 Hz]
SNR of the receiving signal	−15 dB~0 dB

shows the corresponding parameters and settings in the following experiments and performance evaluations.

During pre-whitening processing, the conventional maximum entropy spectrum estimation for the ideal whitening method is used to realize the corresponding process. The order of the pre-whitening filter is set as 10. In the proposed approach, without loss of generality, the following conventional symmetric bistable nonlinear system with SR is used for weak signal enhancement and improvement:

$${\displaystyle \frac{dz\left(t\right)}{dt}}=z\left(t\right)-z^3\left(t\right)+r_W\left(t\right)+\sqrt{2D}·\xi \left(t\right)$$

(10)

The inner SR noise $\xi \left(t\right)$ is chosen as a white Gaussian noise process, and the initial value of the state variable can be selected randomly within [−1, +1]. In addition, the unperturbed steady state ${〈z\left(t\right)〉}_{st}$ can be obtained by setting $r_W\left(t\right)=0$ in (10).

In the proposed SR processing in the symmetric bistable system in a parallel configuration, K = 1000 symmetric bistable parallel SR processing units are used and averaged to obtain the PSR-improved output signal $r_{WP}\left(t\right)$ in Figure 2. In the performance comparison, the traditional energy detection method is used for comparison; this method is widely used in different applications.

To show the intuitive differentiation between the proposed approach and the traditional method, Figure 3 presents the spectra of the weak frequency band signal under SNR = −15 dB. Firstly, Figure 3a shows the original signal within a certain frequency band without additive noise. Secondly, Figure 3b shows a plot of the original signal with channel noise, where the signal merges in the noise spectrum and is hard to detect. Figure 3c shows the spectrum after traditional energy detection, where the signal spectrum is still unclear and difficult to detect. Finally, Figure 3d presents the spectrum after the proposed approach; the original weak signal band is obvious and can be detected easily compared with that shown in Figure 3c.

Figure 4, Figure 5 and Figure 6 present the spectra of the weak frequency band signal before and after the proposed processing method for comparison based on real radar data, while the SNR is changed from 0 dB to −5 dB and to −10 dB. The original signal is the summation of that in Figure 3a and a high-peak single-frequency signal at 1020 Hz. In Figure 4a, Figure 5a, and Figure 6a, it can be seen that the weak band signal is embedded in the background noise and cannot be detected efficiently, except for the peak single-frequency signal. However, from Figure 4b, Figure 5b, and Figure 6b, it can also be observed that the weak frequency band signal under the channel noise can be enhanced effectively with the introduced symmetric bistable PSR processing, which shows a good advantage in the spectral detection performance of the weak frequency band signal over that of the conventional method.

Next, to reasonably compare the weak signal detection performance, the Receiver Operating Characteristic (ROC) curves are plotted under SNR = −15 dB using real sea clutter data with a weak target (Figure 7). The same performance comparison result is given in Figure 8 when the false alarm rate is below 0.01. The performance is compared with that resulting from the traditional energy detection method, while both theoretical and real experimental results are given for both methods. At the same time, the adaptive filtering method and the spectral analysis method are also investigated. A (5,5)-order autoregression moving average (ARMA) model is used in the adaptive filtering method, and the resolution bandwidth of the spectrum analysis method is set to 1/2 of the sampling bandwidth. It can be observed that the performance of the proposed symmetric bistable PSR processing approach overwhelms all the above traditional weak signal detection methods. The real experimental curve matches the theoretical curve well. The results above guarantee the applicability of the proposed symmetric bistable PSR processing approach commendably.

Finally, the real-time computer run times for the proposed algorithm and other standard algorithms are evaluated, as presented in

Table 2. Real-time computer run times comparison.

Methods	Real-Time Computer Run Times (s)
traditional energy detection method	0.3844
adaptive filtering method	1.5244
spectrum analysis method	0.9869
proposed method	0.5648

. In the calculations, the data rate and other parameters of the radar are the same as in Table 1, and MATLAB R2022a is used to realize the calculations for all methods in a Dell desktop computer with the Intel i5 processor. From the results in Table 2, it can be seen that the proposed method has a moderate computational efficiency, which has run times that are lower compared with the adaptive filtering method and the spectrum analysis method and a little higher than the traditional energy detection method.

4. Conclusions

In this study, a novel spectral detection approach for a weak frequency band signal is proposed, which is based on the pre-whitening scale transformation of the SR in a symmetric bistable system in a parallel configuration. The pre-whitening operation helps to provide the receiving signal with a Gaussian distribution. The scale transformation can ensure that the weak frequency band signal spectrum lies in the low-frequency band in the bistable SR system; this process can improve the weak signal detection efficiency as well. Moreover, the introduction of processing into a symmetric and parallel system with SR may reduce the variance in clutter and additive noise in the detection. Real experimental results based on real sea clutter radar data can be used to evaluate the applicability of this proposed approach, especially in improving the detection probability under a constant false alarm rate. Some further research activities may be focused on optimizing the introduced SR system, including the driving parameters and the inner noise and the relationship between the parallel unit number and the final weak signal enhancement performance.