Confidence evaluation for blind analysis of LFM/BPSK hybrid signals based on extreme value theory

IET Radar, Sonar & NavigationVolume 13, Issue 3 p. 473-482 Research ArticleFree Access Confidence evaluation for blind analysis of LFM/BPSK hybrid signals based on extreme value theory Hu Guobing, Corresponding Author Hu Guobing s0304152@jit.edu.cn Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorZhao Pinjiao, Zhao Pinjiao Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorChen Zhengyu, Chen Zhengyu Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorLuo Ronghua, Luo Ronghua Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this author Hu Guobing, Corresponding Author Hu Guobing s0304152@jit.edu.cn Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorZhao Pinjiao, Zhao Pinjiao Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorChen Zhengyu, Chen Zhengyu Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this authorLuo Ronghua, Luo Ronghua Department of Electronic and Information Engineering, Jinling Institute of Technology, No. 99 Hongjing Road Jiangning District, Nanjing, People's Republic of ChinaSearch for more papers by this author First published: 16 January 2019 https://doi.org/10.1049/iet-rsn.2018.5322Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The confidence evaluation of a blind analysis for a hybrid modulation scheme combining binary phase shift keying (BPSK) with linear frequency modulation (LFM) signals is derived herein to statistically decide whether an individual processing result is correct by using only the observed signal and analysis results, including modulation identification and corresponding parameter estimations. They formulate the problem using a binary hypothesis, where the null hypothesis states that modulation recognition is correct, and the errors of parameter estimates are all less than some particular metrics (i.e. the analysis result is confident). An auxiliary signal is constructed by using the identified modulation model and corresponding estimated parameters. The modified correlation spectrum between the observed and auxiliary signals is computed accordingly. The spectrum does not exhibit any peaks under the null hypothesis (H0) but does under the alternative hypothesis. This provides the basis of a useful discriminating feature for verifying the confidence of the analysis result. They prove that the extreme values of the modified correlation spectrum under H0 have a Gumbel distribution. A goodness-of-fit test based on the Kolmogorov–Smirnov is used to perform a confidence evaluation of the analysis result of the LFM/BPSK signal. The effectiveness of the algorithm is demonstrated in extensive joint simulation experiments. 1 Introduction A blind signal analysis is a major task for intelligent radios in both military and commercial applications, including electronic warfare, radio surveillance and cognitive radio (CR). The task mainly involves the blind modulation format recognition of the received signal as well as the estimations of the signal's main parameters under a low signal-to-noise ratio (SNR) [1, 2]. The performance evaluation of the signal analysis is also a key operation in practical signal processing systems and can be considered from two points of view. On the one hand, algorithm designers focus on the overall statistical performance that can be expressed by the correct rate of modulation recognition or the mean square error of parameter estimation [3]. On the other hand, users consider the confidence of an individual analysis result to be important. In a non-cooperative context, especially at low SNRs, it is important to decide whether an individual identification is correct or parameter estimations are accurate when the true modulation format and the parameters of the observed signal are unknown on the receiver side. This is because verifying the confidence or credibility of the signal analysis result for an individual processing cycle directly affects successive signal processing operations. For instance, in electronic reconnaissance, the performances of the successive tasks after signal identification and demodulation greatly depend on the credibility of the front-end processing. Similarly, for CR signal processing, a reliable spectrum sensing is a prerequisite for effective spectrum management and other processing operations in the cognitive cycle. In [4], the confidence measurement of the signal classification is used as a metric to detect unknown signals. Thus far, only a few works have addressed the confidence evaluation of modulation classification in radar surveillance or CR scenarios from the viewpoint of algorithm designers. Fehske et al. [5] used the half of the difference between the maximum and second maximum outputs of a backpropagation-based classifier as a confidence metric for the modulation identification algorithm. This method requires a large number of training samples, which are difficult to be collected under non-cooperative conditions. Lin [6] proposed a confidence measurement based on the information entropy for modulation recognition results in single-input and single-output (SISO) as well as multiple-input multiple-output (MIMO) CR channels. The confidence of the algorithm is estimated from the normalised likelihood vectors under each modulation assumption. Obviously, the likelihood function is difficult to be obtained without prior knowledge of both the signal and the noise. Hence, this method is not applicable to the confidence evaluation of the modulation recognition in non-cooperative situations. Moreover, these two methods can only be used to evaluate the overall confidence of a specific identification algorithm. From the viewpoint of users, they are not suitable for verifying a given analysis result, including both the modulation recognition and main parameter estimates. So far, a few methods have been proposed to test the confidence of the blind analysis of binary phase shift keying (BPSK) or sinusoidal signals from the viewpoint of users. The cumulative summation (CUSUM) function between the observed and reference signals constructed using the identified modulation and the corresponding estimated parameters is obtained in [7]. The approximate entropy difference (AED) between the CUSUM function and its ascending forms is accordingly used as a discriminating feature. The credibility test for the BPSK analysis result then boils down to detecting whether the AED is zero or greater than zero. A confidence assessment algorithm that depends on a binary hypothesis test for the frequency estimation of a single-tone sinusoid was proposed in [8, 9]. The correlation between the observed and reference signals generated by the frequency estimate is used as a test statistic, and a threshold is determined using Chebyshev's inequality. The proposed method performs well even at low SNRs. A linear frequency modulation (LFM)/BPSK hybrid modulated signal is well known to be a basic classical signal type used in a low-probability interception radar [10]. A significant amount of literature has been published to identify the modulation type and estimate the parameters of LFM/BPSK hybrid signals [11-13]. To the best of our knowledge, this is the first published study on the confidence evaluation for the analysis result of an LFM/BPSK hybrid modulated signal. Its modulation scheme is more complex than that of the single BPSK or single LFM; hence, the algorithms for assessing the identification of single-modulation signals [7-9] are not applicable. In this study, we focus on the confidence evaluation for the blind analysis of the LFM/BPSK hybrid signal from the viewpoint of users (i.e. verifying the correctness or accuracy of an individual analysis result using only the observed signal and the corresponding analysis result itself). In this work, our goal is to investigate the confidence verification method without any knowledge of the signal parameters or noise power. The spectrum of the correlation between the original received signal and the auxiliary signal, which is constructed using the analysis results, including the estimated parameters and the signal model depending on the identified modulation type, is calculated first. Assuming that an analysis result is either confident or not, we show that the extreme value distribution of the correlation spectrum can be efficiently used to verify the confidence of the blind analysis result for LFM/BPSK signals. Based on this result, we propose an extreme value theory (EVT)-based confidence evaluation algorithm that relies on the Kolmogorov–Smirnov (K–S) test to determine whether the extreme value distribution of the correlation spectrum fits a Gumbel distribution. Furthermore, we investigate the performance of the proposed algorithm in the presence of diverse signal parameters, including the initial frequency, frequency modulation slope, initial phase, code width and so on. 2 Confidence evaluation model for the blind analysis of LFM/BPSK hybrid signals 2.1 Analysis algorithm for an LFM/BPSK hybrid signal A sampled complex LFM/BPSK hybrid modulated signal with added noise is given as follows: (1) where is the sampling interval; N is the number of samples; and is an independent and identically distributed (i.i.d.) sequence of complex Gaussian random variables with a zero mean and a variance of . The signal part, , is uncorrelated with the noise part, . Parameters and denote the amplitude, initial frequency, slope of the frequency modulation and initial phase, respectively. is the phase modulation function, where is a binary coding scheme consisting of 0 or 1. In general, the blind analysis of an LFM/BPSK hybrid signal includes modulation recognition [12, 14] and parameter estimation tasks [13, 15, 16]. Even though algorithms on the parameter estimation of the LFM/BPSK signal have been broadly investigated in the literature, the main ideas are mostly inspired by a combination of the discrete polynomial transform (DPT) method [17] for a single LFM signal parameter estimation and the square transformation for a single BPSK signal parameter estimation [18]. We call this method the DPT-S algorithm [11], and it will be used as an example to verify the confidence test algorithm in the simulations. 2.2 Confidence evaluation model for the blind identification of an LFM/BPSK hybrid signal We formulate the confidence evaluation problem as a binary hypothesis test as follows: : The modulation recognition is correct, and the errors of all parameter estimates of the LFM/BPSK signal are all small. : The modulation recognition is incorrect or at least one of the errors of the parameter estimates of the LFM/BPSK signal is large. In practice, for the parameter estimations of the LFM component, we assume that the estimation errors are all small if holds, where is the discrete frequency interval, and is the sampling frequency [19]. For the BPSK signal component, we assume that the estimation errors are relatively small if no decoding error exists for . Fig. 1 shows the typical diagram of the confidence test for the results of the LFM/BPSK signal analysis. The inputs of the diagram consist of two parts: one is the analysis result of the LFM/BPSK signal, and the other is the observed signal. In this study, the auxiliary signal can be generated as by using the estimated phase function of the observed signal (i.e. ). The correlation between the observed and auxiliary signals can be written as follows: (2) where is the error of the phase function. The correlation contains the information on the confidence of the results of the signal analysis because if the function , the analysis results are confident and vice versa. In the next section, the spectrum of the correlation will be used as a statistic to perform the confident evaluation problem. Fig. 1Open in figure viewerPowerPoint Diagram of the confidence evaluation for the results of the LFM/BPSK signal analysis 3 EVD goodness-of-fit-test-based confidence evaluation algorithm We formulate the confidence evaluation problem as a goodness-of-fit test based on EVT [20]. As for the i.i.d. random variables, asymptotic cumulative distribution function (cdf) or probability distribution function (pdf), the maxima are shown to converge to one of the three specific extreme value distribution (EVD) types: Gumbel, Frechet or Weibull. Moreover, the maximum domain of attraction [21] gives the conditions for determining the discrete EVD type for a specific random variable following a parent distribution. We propose herein a method to test the confidence of the blind analysis of an LFM/BPSK hybrid modulated signal by adopting an EVD goodness-of-fit test formulation. 3.1 Correlation spectrum 3.1.1 hypothesis Under the assumption, the modulation recognition is correct, and the parameter estimation errors are all small; hence, the auxiliary signal can be constructed as follows: (3) where is the estimate of the initial frequency; is the estimate of the frequency modulation slope; and is the estimate of the phase function of the LFM/BPSK signal. The correlation between the observed and auxiliary signals is then given as: (4) where denotes the signal part of the correlation function , and is its additive noise part, which is also an independent complex Gaussian random process with a zero mean and a variance of . Under , the parameter estimates are all close to their true values i.e. and the correlation can be approximately expressed as The above equation infers that can be regarded as a direct current component plus a noise part ; therefore, the discrete Fourier transform (DFT) of can be given as (5) where (6) and (7) As for spectrum , it is also a complex independent Gaussian random sequence with a variance of . As shown in Fig. 2a, the amplitude spectrum of has a peak at the position of the direct current . For the convenience of analysis, we define a modified correlation spectrum as (8) where is the position of the maximum value of that belongs to the direct current. By discarding the direct current component , the spectrum is transformed into , namely, a modified correlation spectrum. Fig. 2b shows the amplitude spectrum of . The approximation clearly holds, and has a Rayleigh distribution because both the real and imaginary parts of have a zero-mean Gaussian distribution. Fig. 2Open in figure viewerPowerPoint Amplitude spectra of and (a) and (b) for SNR = 0 dB. The coding scheme is 13-bit Barker 3.1.2 hypothesis Two mismatching cases exist: (1) The modulation recognition is correct, but the parameter estimation errors are large, or the decoding result is erroneous. This case is denoted by . (2) The modulation recognition is false and denoted by . We next analyse the properties of the correlation spectrum for each case. Case 1: : In this case, the auxiliary signal is also constructed using the LFM/BPSK model because the modulation recognition is correct, and the estimated parameter set is . The auxiliary signal can be obtained as (9) and the correlation is calculated by: (10) where and denote the signal and noise parts of the correlation, respectively, and , and are the estimation errors of the initial frequency, frequency modulation slope and phase function, respectively, in the case of . In general, under the assumption of the three abovementioned parameters, at least one parameter estimation error is large. Furthermore, we rewrite the signal part of the correlation as where , and . The above equation shows that is an equivalent LFM/BPSK hybrid modulation signal, which can be regarded as the product of the LFM component and BPSK component . For , the initial frequency is equal to ; the frequency modulation slope is equal to ; and the initial phase is . The phase function for is . Remarks: (i) becomes a narrow-band LFM signal when and are large, but . Therefore, the modified spectrum of is not a pure noise spectrum because it has an LFM-type spectrum with a bandwidth of . In other words, the modified spectrum should have some peaks located in the band. (ii) If both and are small, but , can be simplified to a baseband BPSK signal, whose bandwidth is approximately , where is the code width of the binary phase code. The modified spectrum in this case also clearly has some peaks in the band. (iii) can be regarded as a narrow-band LFM/BPSK hybrid modulated signal when both and are large and . and ; hence, the LFM/BPSK is a narrow-band signal, whose amplitude spectrum is approximated by the sinc function, which is similar to the spectrum of the BPSK signal. The bandwidth of is approximately , indicating that the in-band fluctuation is large, and the out-of-band fluctuation is relatively small because it is determined by the tail of the sinc function. In brief, the modified spectrum is different from , which is a narrow-band signal that consists of at least one peak. Fig. 3 depicts the modified spectra corresponding to these three cases. Note that stands for the bits of the decoding error for . Fig. 3Open in figure viewerPowerPoint Modified spectra of the correlation for the SNR = 0 dB (a) , (b) and (c) Case 2: : The modulation recognition of LFM/BPSK is incorrect in this case. On the one hand, we note that the model mismatch between the observed and auxiliary signals causes some changes in the correlation because the auxiliary signal is constructed according to the model of the incorrectly estimated modulation signal type. On the other hand, the mechanism of parameter estimations under the mismatched signal model will be different from that under the matched signal model, which will affect the correlation spectrum. In the following, we describe the spectrum properties using an example (i.e. an LFM/BPSK is mistakenly identified as an LFM signal). First, the auxiliary signal is constructed based on the LFM signal model, which is given as (11) where and are the estimates of the initial frequency and frequency modulation slope under a mismatched model, respectively. The correlation is then obtained as follows: (12) where and denote the signal and noise parts of the correlation, respectively, and and are the estimation errors of the initial frequency and frequency modulation slope under model mismatching, respectively. The signal part is clearly the LFM/BPSK signal with parameters and and the phase function ; hence, the modified spectrum of will have at least one peak, despite the fact that both and are small because the spectrum has an approximate bandwidth of . However, the estimation errors and should be larger than the estimation errors and that are considered under the matched model. The explanation according to the estimation procedure is as follows: suppose that the LFM/BPSK signal is mistakenly identified as an LFM signal, and the DPT algorithm is adopted to demodulate it. The estimation procedure would then be presented as follows: Step 1: Calculation of the lag product (13) where the signal part is , and the noise part is (14) As it can be seen in (13), is a BPSK signal with the carrier frequency , initial phase and phase function . Furthermore, the spectrum of is a sinc-like function with the bandwidth . Note that when the DPT algorithm is employed to demodulate the LFM signal, is regarded as a single-tone sinusoid signal, and the DFT-based frequency estimate is obtained by searching for the maximum position of the spectrum. The shape of the BPSK spectrum is theoretically different from that of the single-tone sinusoid, whose spectrum is the line spectrum when the length of the samples is infinite. Therefore, in this case, the frequency estimation errors of the DFT-based estimators are larger than those under a matched signal model because of the uncertainty caused by the model mismatch between the signal and the adopted estimation algorithms and the effect of noise as well. Accordingly, the estimate of the frequency modulation slope is . The estimation error of is related to and relatively larger than that under the matched condition. Step 2: Using the estimate , the intermediate signal is given as: (15) where the estimation error is , and the equivalent noise term is . Equation (15) shows that is an LFM/BPSK signal with the initial frequency , frequency modulation slope and phase function . However, because of the model mismatch, is approximated as a single-tone sinusoid, and the parameter is accordingly estimated using the single-tone frequency estimator denoted by . In general, ; hence, the spectrum of the LFM/BPSK signal is determined by the BPSK component, which is a sinc-like function. Its bandwidth is approximated by , which is wider than that of a single-tone sinusoid. The estimation error under model mismatch is relatively larger than that under the matched condition because of the uncertainty in the maximum position affected by the noise. From the above analyses, we conclude that when the LFM/BPSK signal is mistakenly identified to be an LFM signal, model mismatching happens, leading to larger errors in the parameter estimates. Therefore, the modified correlation spectrum is a sinc-like function, whose amplitude has some peaks in the bandwidth (Fig. 4). Similarly, when an LFM/BPSK signal is mistakenly identified as a BPSK signal or other modulation types, the modified spectrum of the correlation has some peaks because of the model mismatching and corresponding estimation under the model mismatch. Fig. 4Open in figure viewerPowerPoint Modified spectrum under the assumption for SNR = 0 dB: the LFM/BPSK signal is mistakenly identified to be an LFM signal The correlation can be expressed as , where is the auxiliary signal under the different assumption . Accordingly, the modified spectrum can be given as where is the position of the maximum value of that belongs to the direct current. We can conclude that (as explained in Appendix 1) (i) Under , the amplitude of the modified correlation spectrum does not exhibit any peaks. In addition, its pdf follows a Rayleigh distribution because the real and imaginary parts of are dual independent Gaussian random variables with zero means. (ii) Under , for the or case, the amplitude of the modified correlation spectrum (i.e. or ) is different from that under , which exhibits some peaks in the bandwidth of the spectrum. Moreover, or is a random variable that consists of two parts. One part comprises some i.i.d. Rayleigh random variables, whereas the other part, comprises some i.i.d. Rician random variables. As shown in Fig. 5, the histograms of and are fitted well by the Rayleigh pdf, whilst those of are fitted well by the Rice pdf when and for SNR = 0 dB and in the case that the signal is mistakenly identified to be an LFM signal. Hence, we can evaluate the confidence level for the blind analysis of an LFM/BPSK signal using the characteristic differences between the and distributions of the correlation spectra. Fig. 5Open in figure viewerPowerPoint Histograms of the samples of the modified correlation spectra under different assumptions and the corresponding fitted pdfs 3.2 EVD of the correlation spectrum From the above discussions, the distributions of the maximum of and are different from each other because the distributions of and are different. For notational simplicity, we denote or using a uniform term . For convenient analysis, we define as a confidence statistic. Therefore, the distribution of can be regarded as a random vector consisting of i.i.d. central chi-squared random variables with two degrees of freedom (DOF). Moreover, their cdf is given as (16) Let denote the maximum value of : (17) where the length of is . According to order statistics theory, the exact cdf of is given as Therefore, the variance should first be estimated to determine . However, to avoid estimating it, we derive the asymptotic distribution of the maxima of the statistic . According to the extremal types theorem, the asymptotic cdf of can be expressed as (18) where constants and a nondegenerate are presented such that the cdf of a normalised version of converges to . Assume that are i.i.d. central chi-squared random variables with 2 DOF. Their cumulative distribution lies in the domain of attraction of a Gumbel-type EVD [22], which is the limit distribution of the maxima of . According to the order statistic theory and extreme types theorem [20, 21], the asymptotic cdf of can be expressed by the Gumbel EVD (19) In addition, according to [22], the normalising constants can be derived as (20) In practical applications of the limit law, we assume that h is sufficiently large such that (21) which implies that (22) where is of the same type as . Thus, the limit distribution of can be fitted directly to a series of observations of , where the normalising constants and are not required. However, under , the random variables can be regarded as a random vector consisting of the i.i.d. central chi-squared variables and the i.i.d. non-central chi-squared variables in a different range. Thus, denotes the non-i.i.d. variables (23) where the sub-collection consists of the i.i.d. central chi-squared distribution random variables, and is a sub-collection of the L i.i.d. non-central chi-squared distribution random variables. Consequently, the cdf of can be expressed as (24) where is a generalised Q-function. Defining we obtain and (25) where and are the asymptotic cdfs of and , respectively, and is given by: In addition, the notation of the arrow with d in (25) denotes the weak convergence in the distribution. The cdf of the non-central chi-squared random variables with 2 DOF and non-central parameter can be expressed as (26) and (27) is the pdf corresponding to the cdf . Next, we will prove that the cdf is a von Mises function. Theorem 1.The cdf of the non-central chi-squared random variable with 2 DOF is a von Mises function. Proof.See Appendix 2. □ Therefore, the cdf of the maximum of the L i.i.d. random variables with the cdf in (26) is , where the normalising constants can be chosen as (28) By comparing the asymptotic cdfs of and , (29) and (30) it can be seen that is not equal to because the former is a Gumbel distribution, whilst the latter is the product of two Gumbel distribution functions with different parameters. In practice, according to the Gumbel method [23], when modelling the actual data using the EVD, the observed signal is uniformly divided into p blocks, each of length m, and each block's maximum that can be given by . Fig. 6 compares the estimated empirical cdfs generated from the amplitude of the correlation spectrum and the theoretical Gumbel-type cdfs under three sets of simulation conditions. The figure shows that the estimated empirical cdf is in accordance with that of the Gumbel distribution under , whilst the empirical cdf estimated under does not fit the Gumbel distribution. Fig. 6Open in figure viewerPowerPoint Cdfs of for SNR = 0 dB, 13-bit Barker code and a block size of five (a) , (b) when the decoding bit error is 1, and (c) when the LFM/BPSK signal is incorrectly identified as an LFM signal 3.3 Algorithm summary As mentioned earlier, we can conclude that the limit of the distribution of under is not identical to the Gumbel cdf , and the confidence test of the LFM/BPSK signal blind identification results can be transformed into an EVD goodness-of-fit test as follows: (31) The one-sample K–S approach is employed herein to perform the above mentioned hypothesis test [24, 25]. A summary of the proposed confidence test based on the EVT algorithm is given in Table 1. Table 1. Confidence evaluation alg