# Multi-Feature Multi-Sensor Fusion for Emitter Identification Based on a Modified DS Application

Authors: Jie Chen; Kai Xiao; Kai You; Feng Duan; Xianguo Qing
Journal: Journal of Sensors
 (2022)

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## Abstract

Emitter identification is a crucial task in electronic countermeasure technology area, which deeply affects the accuracy of subsequent threat estimation. In emitter identification system, sensors (transmitter and receiver) have inevitable inconsistency and fuzziness, along with possible ambiguity and instability under interference and malfunction. To manage the uncertainty in emitter identification system, we propose a multi-feature multi-sensor fusion algorithm based on a modified DS application. The modified DS application for emitter identification system is accomplished by two parts—multi-feature fusion based on the improved proximity approach to obtain basic probability assignment (BPA) and multi-sensor fusion based on the combination of two revised evidences to solve potential evidence conflicts. Firstly, the multi-feature fusion method based on the improved proximity approach is raised to produce the identification result for each receiver, which simultaneously build BPAs for DS application. Four entropies are extracted to establish the multi-feature description of received signal. Then, in order to solve the inconsistency among different receivers and realize multi-sensor fusion for emitter identification, the multi-sensor fusion method based on the combination of two revised evidences is proposed. Two revised evidences are put forward, respectively, by the introduction of Lance distance function and spectral angle cosine function before applying DS combination. Experiments and analyses comprehensively demonstrate the great uncertainty management performance and favorable emitter identification effect of the proposed algorithm.


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## Body

## 1. Introduction

With the rapid development of information technology and modern signal processing methods, the electronic countermeasure technology has occupied an important status in modern military field. In increasingly complex electromagnetic environment, uncertainty is caused by various factors, e.g., big signal density, complex signal coding, various modulation parameters, and short spectrum resources [1]. The emitter identification is one of the essential missions in the field of electronic countermeasure technology [2]. Undoubtedly, the uncertainty brings difficulties in emitter identification system and the subsequent estimation of threaten level. Therefore, the management of uncertainty in emitter identification system has great value and significance not only at the improvement of the electronic warfare ability, but also at the mastery of the combat initiative.The uncertainty of emitter identification system is mainly embodied in the following two aspects.On the one hand, both the transmitter and receiver have different degree of uncertainty. In the transmitter, due to the complexity of environment and time-variant property of equipment, transmitted signal has slightly different characteristics [3]. Then, in the receiver, the identification effect is restricted by various factors, e.g., receiver’s type and operating conditions. In addition, even the receivers with same type and same operating conditions will give different identification results under different measurement periods. Thus, both the transmitter and receiver have different levels of uncertainty that need to be solved immediately.On the other hand, traditional identification methods are bound by prior knowledge and recognition technology, which further enhances the uncertainty in system. As emitter identification system belongs to non-cooperative communication field, the prior knowledge is very limited, even not available to use in some adverse environment [4]. In this case, the commonly used uncertainty management methods like probability theory and Bayesian reasoning are not suitable anymore [5, 6]. At the same time, restricted by the measurement accuracy of communication equipment and limited validity of communication technology, the study of emitter identification develops slowly. Therefore, it is urgent to adopt an effective and reliable uncertainty management approach to enhance the precise and robustness of emitter identification.According to these analyses, the emitter identification processing is filled with uncertainty as unreliable and inaccurate information. Hence, the key task in emitter identification field is to study the suitable uncertainty description and management algorithm against specific communication environment and identification demands.Relative mature methods including wavelet analysis [7, 8], Fourier transform analysis [9, 10], bi-spectrum analysis [11, 12], and entropy analysis [13, 14] have been raised to realize accurate emitter identification. Especially, entropy analysis needs less computation amount and can precisely describe system’s uncertainty [15]. Thus, we adopt the conception of entropy analysis to achieve the uncertainty description in this paper. Besides, different entropy characteristics have different resolving ability to different signals [16]. Using single entropy characteristic cannot guarantee accurate signal recognition effect. Hence, in this paper, we build a multi-feature fusion method for each receiver to integrate the recognition ability of multiple entropy characteristics. The receiver in the emitter identification system will extract two singular spectrum entropies and two power spectrum entropies of emitter signal for better expression of signal’s characteristics, and then apply the multi-feature fusion method to get the recognition result. It is worth mentioning that singular spectrum entropies describe signal’s fine characteristics from the perspective of signal’s singularity as time-domain features, and power spectrum entropies describe signal’s fine characteristics from the perspective of signal’s frequency structure as frequency-domain features. The introduction of these four entropies comprehensively expounds the features of received emitter signal in different domains.As system’s uncertainty has been precisely described by multiple entropies, the next core is to achieve the uncertainty management. Signal’s modulation becomes diverse because of complex electromagnetic environment. That is, the complexity of communication processing exacerbates system’s uncertainty [17, 18]. Hence, single receiver is hard to keep the error-free and trouble-free state for emitter identification. Obviously. To address this issue, multiple receivers are adopted to a redundant emitter identification system, where Dempster-Shafer evidence theory (DS theory) is utilized to realize the multi-sensor fusion. Receivers in the emitter identification system are regarded as sensors in the multi-sensor fusion system. After receivers obtain the multi-feature fusion results, the multi-sensor fusion method is applied to integrate these multi-feature fusion results to manage the system’s uncertainty.DS theory is a widely used uncertainty reasoning algorithm. It can operate without prior knowledge and conditional probability, which is considered an inexact derivation of probability theory and Bayesian reasoning. Hence, DS theory is applied to various smart areas, e.g., monitoring system [19], pattern recognition [20], fault diagnosis [21], and target tracking [22]. The wide application of DS theory has attracted a lot of scholars. There are two key problems in the application of DS theory, one is the acquisition of basic probability assignment (BPA), and the other is the solution of evidence conflicts.On the one hand, receivers are regarded as sensors in the emitter identification system based on DS application.BPA is the basic unit of DS application, which reflects the uncertainty of corresponding sensor. How to obtain accurate BPA is the premise step of DS application. In this paper, the BPA acquisition is the way to build the multi-feature fusion method for each sensor. [23] uses the interval gray analysis to transform the interval entropy features to BPAs. However, this method needs the true value of measurement, which is difficult to satisfy in actual application. Considering that both the value of BPA in DS theory and the value of the subordinate function in fuzzy set lie in 0,1, [24] takes fuzzy subordinate function to obtain accurate BPA. But this algorithm requires prior knowledge. Moreover, [25, 26] improve the acquisition accuracy of BPA, respectively, by gray relation theory and the closed degree. It is a pity that these two methods both fuse multiple features of emitter signal without considering the distinguishing contributions of different features. Therefore, to overcome the defects of those methods, an improved proximity approach is proposed in this paper to realize the multi-feature fusion for each receiver and simultaneously obtain accurate BPA.On the other hand, the way to solve the evidence conflicts is the key to achieve reasonable and reliable multi-sensor fusion. Typical modified methods of DS application are divided into two categories—the modification of synthetic equations (DS combination rule), and the revision of evidences before combination. The former methods optimize the DS combination rule by allocating conflict to certain subsets with certain proportions [27–29]. However, these methods sometimes violate the theoretical properties of DS combination rule like commutativity and associativity. The latter methods modify evidences before using DS combination rule [30–32]. They not only guarantee the theoretical properties of DS combination rule, but also relax the restriction of independent evidences. Thus, the application of the latter methods is more extensive. With the above discussion, this paper adopts the idea of latter methods, and proposes two revised evidences, respectively, by the introduction of Lance distance function and spectral angle cosine function. Then, the proposed combination method based on these two revised evidences is utilized to build the multi-sensor fusion method for emitter identification system, which can effectively solve the system’s uncertainty.In summary, we adopt the idea of multi-feature multi-sensor fusion based on a modified DS application for emitter identification system. Firstly, we extract the singular spectrum entropies and power spectrum entropies of received emitter signal as multiple features. Then, an improved proximity approach is presented to build the multi-feature fusion method for each receiver, which also gives theBPA for DS application. At last, the combination method based on the Lance distance function and the spectral angle cosine function is adopted to establish the multi-sensor fusion method for emitter identification system, which effectively manages the uncertainty among multiple receivers.The rest of this paper is organized as follows: Section2 derives the signal models and DS theory separately, which are the foundations of this paper. Section 3 elaborates the implementation of the proposed algorithm. Section 4 presents the experimental results and analyses to comprehensively discuss the great performance of the proposed algorithm. At last, Section 5 concludes the paper.


## 2. Theoretical Foundations

### 2.1. Emitter Signals

According to the differences of application scenarios and modulation characteristics, emitter signals are classified into three categories—analog signal, digital signal and radar signal. In this paper,AM/PM/FM, 2ASK/2FSK/BPSK/QPSK, and LFM/SF are adopted as potential emitter signal sources.


#### 2.1.1. Analog Signal

Based on different modulation mode, analog signals are divided into linear modulation—amplitude modulation (AM) and non-linear modulations—phase modulation (PM), frequency modulation (FM).Assuming that the carrier signal and the modulation signal of emitter identification system are, respectively, defined as(1)uct=Accos2πfct,uΩt=AΩcos2πfΩt,where Ac,fc and AΩ,fΩ separately represent the amplitude and the frequency of carrier signal uct and modulation signal uΩt.Then, the modulated signalsAM/PM/FM can be, respectively, derived as
(2)SAMt=Ac+KauΩtcos2πfct,SPMt=Accos2πfct+KpuΩt,SFMt=Accos2πfct+Kf∫0tuΩtdt,where Ka,Kp,Kf separately represent the modulation index of modulated signals AM/PM/FM.


#### 2.1.2. Digital Signal

Common digital signals are amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK). Their modulation principles are similar to analog signals.Assuming that the baseband signal of emitter transmitter is(3)st=∑angt−nTb,where an,Tb separately represent the amplitude and the symbol period of baseband signal st, and gt−iTb is a rectangular pulse.
(4)gt=10≤t≤T0else.The modulated signalsMASK/MFSK/MPSK are, respectively, derived as
(5)SMASKt=∑iaigt−iTbcos2πfct,SMFSK=∑igt−iTbcos2πfct+2πfit,SMPSK=∑igt−iTbcos2πfct+φi,where M implies the multiple mode of digital signals, and ai,fi,φii=1,2,⋯,M separately represent the modulation index of modulated signals MASK/MFSK/MPSK.


#### 2.1.3. Radar Signal

There are many types of radar signals in communication field. In this paper, we introduce the most common signals: linear frequency modulation signal (LFM) and stepped frequency signal (SF).LFM, also known as Chirp signal, is defined as
(6)SLFM=expj2πf0t+12kt2.We can derive from (6) that the instantaneous angular frequency is linear.
(7)ωt=2πf0+kt.UnlikeLFM, the instantaneous frequency of SF changes discretely.
(8)SSF=∑i=0N−1rectt−iTr−τ/2τexp−j2πf0+iΔft,where f0 is the starting frequency, Tr is the pulse repetition period, τ is the pulse width, Δf is the step frequency, N is the step number, and rectt/τ is a rectangular pulse.
(9)recttτ=1−τ/2≤t≤/20else.


### 2.2. DS Theory

DS theory raises the concept of the upper and lower bounds of probability, which can reasonably describe the uncertainty of emitter identification system.


#### 2.2.1. Frame of Discernment

DS theory is built up on a finite and non-empty set named the frame of discernment (frame), which containsN mutually exclusive and exhaustive hypotheses.
(10)Θ=H1,H2,⋯,HM,where M is the number of hypotheses in frame, and Hii=1,2,⋯,M represents the ith hypothesis which reflects the ith possible recognition result in emitter identification system.Based on frame, the power set2Θ can be derived as
(11)2Θ=∅,H1,H2,⋯,HM,H1,H2,⋯,H1,HM,⋯,H1,H2,⋯,HM.


#### 2.2.2. Basic Probability Assignment

In order to describe the support degree for hypotheses, basic probability assignment (BPA) is introduced. BPA is a mass function m:2Θ⟶0,1, which satisfies the following equations.
(12)m∅=0,(13)∑H⊆ΘmH=1,where mH is the basic support degree of evidence m to proposition H∈2Θ.AsBPA lies in 0,1, ((17)-(18)) separately reflect the non-negativity (mH>0 if H≠∅) and normalization of BPA. Due to the lack of further knowledge, mH cannot be subdivided to any proper subset of H.


#### 2.2.3. Uncertainty Description

According to the definition ofBPA, belief function (Bel) and plausibility function (Pl) can be defined separately as
(14)BelH=∑A⊆HmA,PlH=∑A∩H≠∅mA.It is easily seen in ((19)-(20)) that BelH describes the support of direct evidences to H, while PlH expresses the support of potential evidences to H. Therefore, BelH and PlH, respectively, present the lower and upper bounds of probability, which indirectly form the uncertainty interval BelH,PlH (shown in Figure 1).Figure 1
Uncertainty description of propositionH.


#### 2.2.4. DS Combination Rule

Assuming that two independent and reliable evidencesm1,m2 are separately obtained from two sensors. DS combination rule, also called direct sum operation, can be defined as
(15)mH=11−K∑Hi∩Hj=Hm1Hi·m2HjifH≠∅0ifH=∅,where K is the conflict factor that presents the conflict degree of evidences m1,m2.
(16)K=∑Hi∩Hj=∅m1Hi·m2Hj.In (15), 1/1−K is the normalized factor, which makes m in (15) meet the normalization property in (13).Let the combination of two evidencesm1,m2 expressed as m=m1⊕m2, it is well known that DS combination rule satisfies the properties of associativity and commutativity.
(17)m1⊕m2⊕m3=m1⊕m2⊕m3,m1⊕m2=m2⊕m1.According to same principle, the DS combination rule forNN>2 evidences can be deduced. mH=1/1−K∑∩Hi=H∏1≤k≤NmkHiifH≠∅0ifH=∅,(25)where the conflict degree ofN evidences is
(18)K=∑∩Hi=∅∏1≤k≤NmkHi.


## 2.1. Emitter Signals

According to the differences of application scenarios and modulation characteristics, emitter signals are classified into three categories—analog signal, digital signal and radar signal. In this paper,AM/PM/FM, 2ASK/2FSK/BPSK/QPSK, and LFM/SF are adopted as potential emitter signal sources.


### 2.1.1. Analog Signal

Based on different modulation mode, analog signals are divided into linear modulation—amplitude modulation (AM) and non-linear modulations—phase modulation (PM), frequency modulation (FM).Assuming that the carrier signal and the modulation signal of emitter identification system are, respectively, defined as(1)uct=Accos2πfct,uΩt=AΩcos2πfΩt,where Ac,fc and AΩ,fΩ separately represent the amplitude and the frequency of carrier signal uct and modulation signal uΩt.Then, the modulated signalsAM/PM/FM can be, respectively, derived as
(2)SAMt=Ac+KauΩtcos2πfct,SPMt=Accos2πfct+KpuΩt,SFMt=Accos2πfct+Kf∫0tuΩtdt,where Ka,Kp,Kf separately represent the modulation index of modulated signals AM/PM/FM.


### 2.1.2. Digital Signal

Common digital signals are amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK). Their modulation principles are similar to analog signals.Assuming that the baseband signal of emitter transmitter is(3)st=∑angt−nTb,where an,Tb separately represent the amplitude and the symbol period of baseband signal st, and gt−iTb is a rectangular pulse.
(4)gt=10≤t≤T0else.The modulated signalsMASK/MFSK/MPSK are, respectively, derived as
(5)SMASKt=∑iaigt−iTbcos2πfct,SMFSK=∑igt−iTbcos2πfct+2πfit,SMPSK=∑igt−iTbcos2πfct+φi,where M implies the multiple mode of digital signals, and ai,fi,φii=1,2,⋯,M separately represent the modulation index of modulated signals MASK/MFSK/MPSK.


### 2.1.3. Radar Signal

There are many types of radar signals in communication field. In this paper, we introduce the most common signals: linear frequency modulation signal (LFM) and stepped frequency signal (SF).LFM, also known as Chirp signal, is defined as
(6)SLFM=expj2πf0t+12kt2.We can derive from (6) that the instantaneous angular frequency is linear.
(7)ωt=2πf0+kt.UnlikeLFM, the instantaneous frequency of SF changes discretely.
(8)SSF=∑i=0N−1rectt−iTr−τ/2τexp−j2πf0+iΔft,where f0 is the starting frequency, Tr is the pulse repetition period, τ is the pulse width, Δf is the step frequency, N is the step number, and rectt/τ is a rectangular pulse.
(9)recttτ=1−τ/2≤t≤/20else.


## 2.1.1. Analog Signal

Based on different modulation mode, analog signals are divided into linear modulation—amplitude modulation (AM) and non-linear modulations—phase modulation (PM), frequency modulation (FM).Assuming that the carrier signal and the modulation signal of emitter identification system are, respectively, defined as(1)uct=Accos2πfct,uΩt=AΩcos2πfΩt,where Ac,fc and AΩ,fΩ separately represent the amplitude and the frequency of carrier signal uct and modulation signal uΩt.Then, the modulated signalsAM/PM/FM can be, respectively, derived as
(2)SAMt=Ac+KauΩtcos2πfct,SPMt=Accos2πfct+KpuΩt,SFMt=Accos2πfct+Kf∫0tuΩtdt,where Ka,Kp,Kf separately represent the modulation index of modulated signals AM/PM/FM.


## 2.1.2. Digital Signal

Common digital signals are amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK). Their modulation principles are similar to analog signals.Assuming that the baseband signal of emitter transmitter is(3)st=∑angt−nTb,where an,Tb separately represent the amplitude and the symbol period of baseband signal st, and gt−iTb is a rectangular pulse.
(4)gt=10≤t≤T0else.The modulated signalsMASK/MFSK/MPSK are, respectively, derived as
(5)SMASKt=∑iaigt−iTbcos2πfct,SMFSK=∑igt−iTbcos2πfct+2πfit,SMPSK=∑igt−iTbcos2πfct+φi,where M implies the multiple mode of digital signals, and ai,fi,φii=1,2,⋯,M separately represent the modulation index of modulated signals MASK/MFSK/MPSK.


## 2.1.3. Radar Signal

There are many types of radar signals in communication field. In this paper, we introduce the most common signals: linear frequency modulation signal (LFM) and stepped frequency signal (SF).LFM, also known as Chirp signal, is defined as
(6)SLFM=expj2πf0t+12kt2.We can derive from (6) that the instantaneous angular frequency is linear.
(7)ωt=2πf0+kt.UnlikeLFM, the instantaneous frequency of SF changes discretely.
(8)SSF=∑i=0N−1rectt−iTr−τ/2τexp−j2πf0+iΔft,where f0 is the starting frequency, Tr is the pulse repetition period, τ is the pulse width, Δf is the step frequency, N is the step number, and rectt/τ is a rectangular pulse.
(9)recttτ=1−τ/2≤t≤/20else.


## 2.2. DS Theory

DS theory raises the concept of the upper and lower bounds of probability, which can reasonably describe the uncertainty of emitter identification system.


### 2.2.1. Frame of Discernment

DS theory is built up on a finite and non-empty set named the frame of discernment (frame), which containsN mutually exclusive and exhaustive hypotheses.
(10)Θ=H1,H2,⋯,HM,where M is the number of hypotheses in frame, and Hii=1,2,⋯,M represents the ith hypothesis which reflects the ith possible recognition result in emitter identification system.Based on frame, the power set2Θ can be derived as
(11)2Θ=∅,H1,H2,⋯,HM,H1,H2,⋯,H1,HM,⋯,H1,H2,⋯,HM.


### 2.2.2. Basic Probability Assignment

In order to describe the support degree for hypotheses, basic probability assignment (BPA) is introduced. BPA is a mass function m:2Θ⟶0,1, which satisfies the following equations.
(12)m∅=0,(13)∑H⊆ΘmH=1,where mH is the basic support degree of evidence m to proposition H∈2Θ.AsBPA lies in 0,1, ((17)-(18)) separately reflect the non-negativity (mH>0 if H≠∅) and normalization of BPA. Due to the lack of further knowledge, mH cannot be subdivided to any proper subset of H.


### 2.2.3. Uncertainty Description

According to the definition ofBPA, belief function (Bel) and plausibility function (Pl) can be defined separately as
(14)BelH=∑A⊆HmA,PlH=∑A∩H≠∅mA.It is easily seen in ((19)-(20)) that BelH describes the support of direct evidences to H, while PlH expresses the support of potential evidences to H. Therefore, BelH and PlH, respectively, present the lower and upper bounds of probability, which indirectly form the uncertainty interval BelH,PlH (shown in Figure 1).Figure 1
Uncertainty description of propositionH.


### 2.2.4. DS Combination Rule

Assuming that two independent and reliable evidencesm1,m2 are separately obtained from two sensors. DS combination rule, also called direct sum operation, can be defined as
(15)mH=11−K∑Hi∩Hj=Hm1Hi·m2HjifH≠∅0ifH=∅,where K is the conflict factor that presents the conflict degree of evidences m1,m2.
(16)K=∑Hi∩Hj=∅m1Hi·m2Hj.In (15), 1/1−K is the normalized factor, which makes m in (15) meet the normalization property in (13).Let the combination of two evidencesm1,m2 expressed as m=m1⊕m2, it is well known that DS combination rule satisfies the properties of associativity and commutativity.
(17)m1⊕m2⊕m3=m1⊕m2⊕m3,m1⊕m2=m2⊕m1.According to same principle, the DS combination rule forNN>2 evidences can be deduced. mH=1/1−K∑∩Hi=H∏1≤k≤NmkHiifH≠∅0ifH=∅,(25)where the conflict degree ofN evidences is
(18)K=∑∩Hi=∅∏1≤k≤NmkHi.


## 2.2.1. Frame of Discernment

DS theory is built up on a finite and non-empty set named the frame of discernment (frame), which containsN mutually exclusive and exhaustive hypotheses.
(10)Θ=H1,H2,⋯,HM,where M is the number of hypotheses in frame, and Hii=1,2,⋯,M represents the ith hypothesis which reflects the ith possible recognition result in emitter identification system.Based on frame, the power set2Θ can be derived as
(11)2Θ=∅,H1,H2,⋯,HM,H1,H2,⋯,H1,HM,⋯,H1,H2,⋯,HM.


## 2.2.2. Basic Probability Assignment

In order to describe the support degree for hypotheses, basic probability assignment (BPA) is introduced. BPA is a mass function m:2Θ⟶0,1, which satisfies the following equations.
(12)m∅=0,(13)∑H⊆ΘmH=1,where mH is the basic support degree of evidence m to proposition H∈2Θ.AsBPA lies in 0,1, ((17)-(18)) separately reflect the non-negativity (mH>0 if H≠∅) and normalization of BPA. Due to the lack of further knowledge, mH cannot be subdivided to any proper subset of H.


## 2.2.3. Uncertainty Description

According to the definition ofBPA, belief function (Bel) and plausibility function (Pl) can be defined separately as
(14)BelH=∑A⊆HmA,PlH=∑A∩H≠∅mA.It is easily seen in ((19)-(20)) that BelH describes the support of direct evidences to H, while PlH expresses the support of potential evidences to H. Therefore, BelH and PlH, respectively, present the lower and upper bounds of probability, which indirectly form the uncertainty interval BelH,PlH (shown in Figure 1).Figure 1
Uncertainty description of propositionH.


## 2.2.4. DS Combination Rule

Assuming that two independent and reliable evidencesm1,m2 are separately obtained from two sensors. DS combination rule, also called direct sum operation, can be defined as
(15)mH=11−K∑Hi∩Hj=Hm1Hi·m2HjifH≠∅0ifH=∅,where K is the conflict factor that presents the conflict degree of evidences m1,m2.
(16)K=∑Hi∩Hj=∅m1Hi·m2Hj.In (15), 1/1−K is the normalized factor, which makes m in (15) meet the normalization property in (13).Let the combination of two evidencesm1,m2 expressed as m=m1⊕m2, it is well known that DS combination rule satisfies the properties of associativity and commutativity.
(17)m1⊕m2⊕m3=m1⊕m2⊕m3,m1⊕m2=m2⊕m1.According to same principle, the DS combination rule forNN>2 evidences can be deduced. mH=1/1−K∑∩Hi=H∏1≤k≤NmkHiifH≠∅0ifH=∅,(25)where the conflict degree ofN evidences is
(18)K=∑∩Hi=∅∏1≤k≤NmkHi.


## 3. Multi-Feature Multi-Sensor Fusion Algorithm

To improve the validity and reliability of emitter identification system, the multi-feature multi-sensor fusion algorithm based on a modified DS application is put forward. The implementation diagram of the proposed multi-feature multi-sensor fusion algorithm is drawn in Figure2.Figure 2
Multi-feature models of different signals.


### 3.1. Implementation Chart of the Proposed Algorithm

The innovation points of the proposed algorithm are marked by the dashed rectangles in Figure2. Apparently, the proposed algorithm is formed with two parts—the improved proximity approach to build multi-feature fusion for each receiver and the combination of two revised evidences to establish the multi-sensor fusion.After the information acquisition and pre-processing of multiple receivers in emitter identification system, the proposed algorithm firstly extracts two singular spectrum entropies and two power spectrum entropies of received emitter signal and utilizes the improved proximity approach to achieve multi-feature fusion for each receiver. The multi-feature fusion can precisely describe the characteristics of the received signal. This procedure also buildsBPAs for multiple receivers, which is the foundation of DS application. Then, considering the conflict among multiple receivers, the combination of two revised evidences is applied to realize multi-sensor fusion. As two revised evidences based on Lance distance function and spectral angle cosine function have strong consistency that both manage system’s uncertainty reasonably, the multi-sensor fusion can produce accurate emitter identification result.


### 3.2. Multi-Feature Fusion Based on the Improved Proximity Approach

To achieve the multi-feature fusion for each receiver, the feature library of 9 signals presented in Section2.1 should be built firstly. Then, each receiver will extract the multiple features of received emitter signal, and match them with the feature library to give the recognition result.


#### 3.2.1. Multi-Feature Library

Entropy analysis is a kind of description method for system’s uncertainty. As we all know, the singular spectrum entropies represent signal’s feature in time domain transform, while the power spectrum entropies represent signal’s feature in frequency domain transform. Thus, in order to better manage system’s uncertainty, we introduce two singular spectrum entropies and two power spectrum entropies of emitter signal to establish the multi-feature model.Assuming that the sampling sequence of emitter signal isxii=1,2,⋯,n. The singular spectrum pj1≤j≤n−m can be derived as:
(19)pi=δi∑j=1n−mδj,where δj is the singular value of signal’s spatial reconstruction matrix.Then, the Shannon entropy and exponential entropy of singular spectrum are, respectively, defined as(20)E1=∑j=1n−mpjlogpj,E2=∑j=1n−mpje1−pj.Singular spectrum entropiesE1,E2 both reflect signal’s uncertainty in singular spectrum distribution. It is verified that signal’s energy only focuses on few modes in singular spectrum when E1,E2 are small. On the contrary, signal’s energy almost evenly distributes when E1,E2 are large.The power spectrum of the sampling sequencexi∣i=1,2,⋯,n is
(21)Sω=2n∑t=1nxie−jωt2.After the normalization of power spectrum’s densitySk, its distribution function pk is obtained.
(22)pk=Sk∑k=1nSk.Then, the Shannon entropy and exponential entropy of power spectrum are, respectively, defined as(23)E3=−∑k=1npklogpk,E4=−∑k=1npke1−pk.Power spectrum entropiesE3,E4 both reflect the complexity and uncertainty of signal’s spectrum. It is checked that if E3,E4 are small, the compositions of signal’s frequency are simple. In this case, signal’s energy only focuses on few frequencies in power spectrum, and signal’s uncertainty and complexity are small. On the contrary, signal’s uncertainty and complexity become large if E3,E4 are big.According to the introduction of two singular spectrum entropies and two power spectrum entropies, we can build the multi-feature model for emitter signal.(24)E=E1,E2,E3,E4.Accordingly, the multi-feature library of 9 emitter signals (AM/PM/FM/2ASK/2FSK/BPSK/QPSK/LFM/SF) can be established through Monte-Carlo simulations. Referring to [33, 34], the representative simulation parameters of emitter signals are shown in Table 1.Table 1
Simulation parameters of 9 reference signals.
Frequency of carrier signalfc(MHz)100Sampling frequencyfs(MHz)1000Sampling numberN8000SNR range dB[-10,30]Modulation indexes of analog signalsKa =0 : 8; Kp =50; Kf=50Time series of binary digital signals10110010Time series of QPSK0132120321031122LFM’s parametersτ=56μs; fr =30 MHzSF’s parametersΔf =2KHz; N =1 = fsTake 1dB as the step length, the multi-feature models of 9 signals can be collected, which are exhibited in Figure 3.
(25)FeatureE1FeatureE2FeatureE3FeatureE4.Figure 3
Multi-feature models with Gaussian noise.
(a)(b)(c)(d)We can see from Figure3 that:
(a)
None of the four features can distinguish all signals alone. E.g., the curve of featureE1,E2,E3 has some overlaps and cross parts, which cannot separate congeneric signals like AM/PM/FM. Even the curve of E4 has overlaps under low SNRs.(b)
Different feature has different recognition effect. E.g., featureE3,E4 separates LFM and SF well, whereas, feature E1,E2 cannot.As single feature cannot identify signals precisely and different feature has different recognition performance, the scheme of multi-feature fusion can synthesize the redundancy and complementarity of different features. Thus, Figure3 explains the necessity of multi-feature fusion in emitter identification system. Collecting the multi-feature values of 9 signals in Figure 3, we can build the multi-feature library for emitter identification system.


#### 3.2.2. The Improved Proximity Approach

After the establishment of multi-feature library, we can get signal recognition result through the comparison between the multi-feature model of the received signal and the multi-feature library.As 9 signals are introduced in the multi-feature library, we can define the exhaustive frame for emitter identification system.(26)Θ=HAM,HFM,HPM,H2ASK,H2FSK,HBPSK,HQPSK,HLFM,HSF.Assume that the multi-feature model ofjth signal in the multi-feature library is Ej, and the multi-feature model of the received signal in ith receiver is Fi, which are separately defined as
(27)Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,⋯,9,Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,⋯,N.Then, to describe the difference betweenFi and Ej, we introduce a distance function based on a weighted index wkk=1,2,3,4.
(28)dij=diHj=∑w=14Ejk−FikEjk·wk1/2,where the weighted index wkk=1,2,3,4 satisfies the non-negativity and normalization properties.
(29)wk≥0,∑k=14wk=1.The weighted indexwk is depended on kth feature’s contribution to the recognition result. And the value of wk is set based on experimental conditions, which will be ulteriorly discussed in Section 4.1.In (28), the distance degree dij measures the difference between Fi and Ej, which can reasonably expresses the difference between the received signal and signals in multi-feature library.After that, we use the inverse proportional function to define the similarity degree betweenFi and Ej.
(30)sij=siHj=1dij′,where dij′ is the normalized distance degree.
(31)dij′=dij∑j=19dij.Finally, the improved proximity approach is formed.(32)NiHj=sij∑j=19sij.(43) represents the corresponding relation between the signal of ith receiver and jth signal in feature library.Obviously, (32) indicates the support allocation of ith receiver to the frame Θ in (26). Thus, NiHj is then considered BPA of ith receiver in DS application. That is, miHj=NiHj.


### 3.3. Multi-Sensor Fusion Based on the Modified Combination Method

After gettingBPAs of multiple receivers by multi-feature fusion, evidences are collected as m1,m2,⋯,mN. To solve the uncertainty in emitter identification system, we propose a multi-sensor fusion method. Firstly, Lance distance function and spectral angle cosine function are, respectively, introduced to effectively revise the original evidences. Then, the combination of two revised evidences is applied to manage the evidence conflicts and produce final recognition result for emitter identification system.


#### 3.3.1. Revised Evidence Based on Lance Distance Function

Assuming that two evidences on the frameΘ=H1,H2,⋯,HM are separately expressed as mi=mi1,mi2,⋯,miM and mj=mj1,mj2,⋯,mjM,i=1,2,⋯,N, where N is the number of receivers (sensors), and mit=miHt,mjt=mjHt,t=1,2,⋯,M.The evidence conflict is generally eliminated by Minkowski distance function or Mahalanobis distance function [30], whose definitions between mi,mj are separately shown as:
(33)dijMinq=∑t=1Mmit−mjtq1q,dijMaS=mi−mjTS−1mi−mj.Minkowski distance function can be, respectively, simplified to Manhattan distance function, Euclidean distance function, and Chebyshev distance function whenq=1,2,∞, and S=covmi,mj is the covariance matrix of evidences mi,mj.[30] shows that Minkowski distance function requires the same dimension of attributes that loses the consistency of evidences, while Mahalanobis distance function requires the covariance computation of vectors which is unsuitable for large-scale data processing. Therefore, we introduce a new distance measurement as Lance distance function to solve the evidence conflicts.
(34)dijL=1M∑t=1Mmit−mjtmit+mjt.(34) shows that Lance distance function dijL is a dimensionless attribute, which is not sensitive to large singular values. Hence, dijL overcomes the defects of Minkowski distance function dijMin and Mahalanobis distance function dijMa, which is more ideal for evidence revision.Then, we can define the reliability degree of evidencemi.
(35)relmi=1−Di∑i=1NDi,whereDi satisfies
(36)Di=∑j=1,j≠iNdijL2.Finally, the revised evidenceMLanceHj based on Lance distance function is deduced by relmi.
(37)MLanceHj=∑i=1NmiHj·relmi.


#### 3.3.2. Revised Evidence Based on Spectral Angle Cosine Function

At present, the modification of evidence is mainly realized by distance functions [30–32]. In this paper, from the view point of similarity measurement, we further modify the conflict evidences by spectral angle cosine function. For evidences mi,mj, the definition of spectral angle cosine function is
(38)sij=∑t=1Mmit·mjt∑t=1Mmit2·∑t=1Mmjt2.Then, the support degree and credibility degree of evidencemi can be separately derived as
(39)supmi=∑j=1,j≠iNsij,crdmi=supmi∑i=1Nsupmi,where supmi represents the support degree of mi from other evidences.Ascrdmi represents the credibility degree of evidence mi, the revised evidence MSpectralHj based on spectral angle cosine function can be deduced by crdmi.
(40)MSpectralHj=∑i=1NmiHj·crdmi.


#### 3.3.3. Combination of Two Revised Evidences

After the comprehensive modifications of conflict evidences, respectively, by the introduction of Lance distance function and spectral angle cosine function, the modified DS application is achieved through the synthesis ofMLanceHj and MSpectralHj.
(41)M=MLance⊕MSpectral.MLanceHj and MSpectralHj reasonably and precisely manage system’s uncertainty with different cluster analyses. Ulteriorly, (41) achieves the combination of two revised evidences MLanceHj and MSpectralHj when solving the evidence conflicts among multiple receivers.Accordingly, the proposed multi-sensor fusion method can obtain effective and reliable recognition result for emitter identification system.


## 3.1. Implementation Chart of the Proposed Algorithm

The innovation points of the proposed algorithm are marked by the dashed rectangles in Figure2. Apparently, the proposed algorithm is formed with two parts—the improved proximity approach to build multi-feature fusion for each receiver and the combination of two revised evidences to establish the multi-sensor fusion.After the information acquisition and pre-processing of multiple receivers in emitter identification system, the proposed algorithm firstly extracts two singular spectrum entropies and two power spectrum entropies of received emitter signal and utilizes the improved proximity approach to achieve multi-feature fusion for each receiver. The multi-feature fusion can precisely describe the characteristics of the received signal. This procedure also buildsBPAs for multiple receivers, which is the foundation of DS application. Then, considering the conflict among multiple receivers, the combination of two revised evidences is applied to realize multi-sensor fusion. As two revised evidences based on Lance distance function and spectral angle cosine function have strong consistency that both manage system’s uncertainty reasonably, the multi-sensor fusion can produce accurate emitter identification result.


## 3.2. Multi-Feature Fusion Based on the Improved Proximity Approach

To achieve the multi-feature fusion for each receiver, the feature library of 9 signals presented in Section2.1 should be built firstly. Then, each receiver will extract the multiple features of received emitter signal, and match them with the feature library to give the recognition result.


### 3.2.1. Multi-Feature Library

Entropy analysis is a kind of description method for system’s uncertainty. As we all know, the singular spectrum entropies represent signal’s feature in time domain transform, while the power spectrum entropies represent signal’s feature in frequency domain transform. Thus, in order to better manage system’s uncertainty, we introduce two singular spectrum entropies and two power spectrum entropies of emitter signal to establish the multi-feature model.Assuming that the sampling sequence of emitter signal isxii=1,2,⋯,n. The singular spectrum pj1≤j≤n−m can be derived as:
(19)pi=δi∑j=1n−mδj,where δj is the singular value of signal’s spatial reconstruction matrix.Then, the Shannon entropy and exponential entropy of singular spectrum are, respectively, defined as(20)E1=∑j=1n−mpjlogpj,E2=∑j=1n−mpje1−pj.Singular spectrum entropiesE1,E2 both reflect signal’s uncertainty in singular spectrum distribution. It is verified that signal’s energy only focuses on few modes in singular spectrum when E1,E2 are small. On the contrary, signal’s energy almost evenly distributes when E1,E2 are large.The power spectrum of the sampling sequencexi∣i=1,2,⋯,n is
(21)Sω=2n∑t=1nxie−jωt2.After the normalization of power spectrum’s densitySk, its distribution function pk is obtained.
(22)pk=Sk∑k=1nSk.Then, the Shannon entropy and exponential entropy of power spectrum are, respectively, defined as(23)E3=−∑k=1npklogpk,E4=−∑k=1npke1−pk.Power spectrum entropiesE3,E4 both reflect the complexity and uncertainty of signal’s spectrum. It is checked that if E3,E4 are small, the compositions of signal’s frequency are simple. In this case, signal’s energy only focuses on few frequencies in power spectrum, and signal’s uncertainty and complexity are small. On the contrary, signal’s uncertainty and complexity become large if E3,E4 are big.According to the introduction of two singular spectrum entropies and two power spectrum entropies, we can build the multi-feature model for emitter signal.(24)E=E1,E2,E3,E4.Accordingly, the multi-feature library of 9 emitter signals (AM/PM/FM/2ASK/2FSK/BPSK/QPSK/LFM/SF) can be established through Monte-Carlo simulations. Referring to [33, 34], the representative simulation parameters of emitter signals are shown in Table 1.Table 1
Simulation parameters of 9 reference signals.
Frequency of carrier signalfc(MHz)100Sampling frequencyfs(MHz)1000Sampling numberN8000SNR range dB[-10,30]Modulation indexes of analog signalsKa =0 : 8; Kp =50; Kf=50Time series of binary digital signals10110010Time series of QPSK0132120321031122LFM’s parametersτ=56μs; fr =30 MHzSF’s parametersΔf =2KHz; N =1 = fsTake 1dB as the step length, the multi-feature models of 9 signals can be collected, which are exhibited in Figure 3.
(25)FeatureE1FeatureE2FeatureE3FeatureE4.Figure 3
Multi-feature models with Gaussian noise.
(a)(b)(c)(d)We can see from Figure3 that:
(a)
None of the four features can distinguish all signals alone. E.g., the curve of featureE1,E2,E3 has some overlaps and cross parts, which cannot separate congeneric signals like AM/PM/FM. Even the curve of E4 has overlaps under low SNRs.(b)
Different feature has different recognition effect. E.g., featureE3,E4 separates LFM and SF well, whereas, feature E1,E2 cannot.As single feature cannot identify signals precisely and different feature has different recognition performance, the scheme of multi-feature fusion can synthesize the redundancy and complementarity of different features. Thus, Figure3 explains the necessity of multi-feature fusion in emitter identification system. Collecting the multi-feature values of 9 signals in Figure 3, we can build the multi-feature library for emitter identification system.


### 3.2.2. The Improved Proximity Approach

After the establishment of multi-feature library, we can get signal recognition result through the comparison between the multi-feature model of the received signal and the multi-feature library.As 9 signals are introduced in the multi-feature library, we can define the exhaustive frame for emitter identification system.(26)Θ=HAM,HFM,HPM,H2ASK,H2FSK,HBPSK,HQPSK,HLFM,HSF.Assume that the multi-feature model ofjth signal in the multi-feature library is Ej, and the multi-feature model of the received signal in ith receiver is Fi, which are separately defined as
(27)Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,⋯,9,Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,⋯,N.Then, to describe the difference betweenFi and Ej, we introduce a distance function based on a weighted index wkk=1,2,3,4.
(28)dij=diHj=∑w=14Ejk−FikEjk·wk1/2,where the weighted index wkk=1,2,3,4 satisfies the non-negativity and normalization properties.
(29)wk≥0,∑k=14wk=1.The weighted indexwk is depended on kth feature’s contribution to the recognition result. And the value of wk is set based on experimental conditions, which will be ulteriorly discussed in Section 4.1.In (28), the distance degree dij measures the difference between Fi and Ej, which can reasonably expresses the difference between the received signal and signals in multi-feature library.After that, we use the inverse proportional function to define the similarity degree betweenFi and Ej.
(30)sij=siHj=1dij′,where dij′ is the normalized distance degree.
(31)dij′=dij∑j=19dij.Finally, the improved proximity approach is formed.(32)NiHj=sij∑j=19sij.(43) represents the corresponding relation between the signal of ith receiver and jth signal in feature library.Obviously, (32) indicates the support allocation of ith receiver to the frame Θ in (26). Thus, NiHj is then considered BPA of ith receiver in DS application. That is, miHj=NiHj.


## 3.2.1. Multi-Feature Library

Entropy analysis is a kind of description method for system’s uncertainty. As we all know, the singular spectrum entropies represent signal’s feature in time domain transform, while the power spectrum entropies represent signal’s feature in frequency domain transform. Thus, in order to better manage system’s uncertainty, we introduce two singular spectrum entropies and two power spectrum entropies of emitter signal to establish the multi-feature model.Assuming that the sampling sequence of emitter signal isxii=1,2,⋯,n. The singular spectrum pj1≤j≤n−m can be derived as:
(19)pi=δi∑j=1n−mδj,where δj is the singular value of signal’s spatial reconstruction matrix.Then, the Shannon entropy and exponential entropy of singular spectrum are, respectively, defined as(20)E1=∑j=1n−mpjlogpj,E2=∑j=1n−mpje1−pj.Singular spectrum entropiesE1,E2 both reflect signal’s uncertainty in singular spectrum distribution. It is verified that signal’s energy only focuses on few modes in singular spectrum when E1,E2 are small. On the contrary, signal’s energy almost evenly distributes when E1,E2 are large.The power spectrum of the sampling sequencexi∣i=1,2,⋯,n is
(21)Sω=2n∑t=1nxie−jωt2.After the normalization of power spectrum’s densitySk, its distribution function pk is obtained.
(22)pk=Sk∑k=1nSk.Then, the Shannon entropy and exponential entropy of power spectrum are, respectively, defined as(23)E3=−∑k=1npklogpk,E4=−∑k=1npke1−pk.Power spectrum entropiesE3,E4 both reflect the complexity and uncertainty of signal’s spectrum. It is checked that if E3,E4 are small, the compositions of signal’s frequency are simple. In this case, signal’s energy only focuses on few frequencies in power spectrum, and signal’s uncertainty and complexity are small. On the contrary, signal’s uncertainty and complexity become large if E3,E4 are big.According to the introduction of two singular spectrum entropies and two power spectrum entropies, we can build the multi-feature model for emitter signal.(24)E=E1,E2,E3,E4.Accordingly, the multi-feature library of 9 emitter signals (AM/PM/FM/2ASK/2FSK/BPSK/QPSK/LFM/SF) can be established through Monte-Carlo simulations. Referring to [33, 34], the representative simulation parameters of emitter signals are shown in Table 1.Table 1
Simulation parameters of 9 reference signals.
Frequency of carrier signalfc(MHz)100Sampling frequencyfs(MHz)1000Sampling numberN8000SNR range dB[-10,30]Modulation indexes of analog signalsKa =0 : 8; Kp =50; Kf=50Time series of binary digital signals10110010Time series of QPSK0132120321031122LFM’s parametersτ=56μs; fr =30 MHzSF’s parametersΔf =2KHz; N =1 = fsTake 1dB as the step length, the multi-feature models of 9 signals can be collected, which are exhibited in Figure 3.
(25)FeatureE1FeatureE2FeatureE3FeatureE4.Figure 3
Multi-feature models with Gaussian noise.
(a)(b)(c)(d)We can see from Figure3 that:
(a)
None of the four features can distinguish all signals alone. E.g., the curve of featureE1,E2,E3 has some overlaps and cross parts, which cannot separate congeneric signals like AM/PM/FM. Even the curve of E4 has overlaps under low SNRs.(b)
Different feature has different recognition effect. E.g., featureE3,E4 separates LFM and SF well, whereas, feature E1,E2 cannot.As single feature cannot identify signals precisely and different feature has different recognition performance, the scheme of multi-feature fusion can synthesize the redundancy and complementarity of different features. Thus, Figure3 explains the necessity of multi-feature fusion in emitter identification system. Collecting the multi-feature values of 9 signals in Figure 3, we can build the multi-feature library for emitter identification system.


## 3.2.2. The Improved Proximity Approach

After the establishment of multi-feature library, we can get signal recognition result through the comparison between the multi-feature model of the received signal and the multi-feature library.As 9 signals are introduced in the multi-feature library, we can define the exhaustive frame for emitter identification system.(26)Θ=HAM,HFM,HPM,H2ASK,H2FSK,HBPSK,HQPSK,HLFM,HSF.Assume that the multi-feature model ofjth signal in the multi-feature library is Ej, and the multi-feature model of the received signal in ith receiver is Fi, which are separately defined as
(27)Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,⋯,9,Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,⋯,N.Then, to describe the difference betweenFi and Ej, we introduce a distance function based on a weighted index wkk=1,2,3,4.
(28)dij=diHj=∑w=14Ejk−FikEjk·wk1/2,where the weighted index wkk=1,2,3,4 satisfies the non-negativity and normalization properties.
(29)wk≥0,∑k=14wk=1.The weighted indexwk is depended on kth feature’s contribution to the recognition result. And the value of wk is set based on experimental conditions, which will be ulteriorly discussed in Section 4.1.In (28), the distance degree dij measures the difference between Fi and Ej, which can reasonably expresses the difference between the received signal and signals in multi-feature library.After that, we use the inverse proportional function to define the similarity degree betweenFi and Ej.
(30)sij=siHj=1dij′,where dij′ is the normalized distance degree.
(31)dij′=dij∑j=19dij.Finally, the improved proximity approach is formed.(32)NiHj=sij∑j=19sij.(43) represents the corresponding relation between the signal of ith receiver and jth signal in feature library.Obviously, (32) indicates the support allocation of ith receiver to the frame Θ in (26). Thus, NiHj is then considered BPA of ith receiver in DS application. That is, miHj=NiHj.


## 3.3. Multi-Sensor Fusion Based on the Modified Combination Method

After gettingBPAs of multiple receivers by multi-feature fusion, evidences are collected as m1,m2,⋯,mN. To solve the uncertainty in emitter identification system, we propose a multi-sensor fusion method. Firstly, Lance distance function and spectral angle cosine function are, respectively, introduced to effectively revise the original evidences. Then, the combination of two revised evidences is applied to manage the evidence conflicts and produce final recognition result for emitter identification system.


### 3.3.1. Revised Evidence Based on Lance Distance Function

Assuming that two evidences on the frameΘ=H1,H2,⋯,HM are separately expressed as mi=mi1,mi2,⋯,miM and mj=mj1,mj2,⋯,mjM,i=1,2,⋯,N, where N is the number of receivers (sensors), and mit=miHt,mjt=mjHt,t=1,2,⋯,M.The evidence conflict is generally eliminated by Minkowski distance function or Mahalanobis distance function [30], whose definitions between mi,mj are separately shown as:
(33)dijMinq=∑t=1Mmit−mjtq1q,dijMaS=mi−mjTS−1mi−mj.Minkowski distance function can be, respectively, simplified to Manhattan distance function, Euclidean distance function, and Chebyshev distance function whenq=1,2,∞, and S=covmi,mj is the covariance matrix of evidences mi,mj.[30] shows that Minkowski distance function requires the same dimension of attributes that loses the consistency of evidences, while Mahalanobis distance function requires the covariance computation of vectors which is unsuitable for large-scale data processing. Therefore, we introduce a new distance measurement as Lance distance function to solve the evidence conflicts.
(34)dijL=1M∑t=1Mmit−mjtmit+mjt.(34) shows that Lance distance function dijL is a dimensionless attribute, which is not sensitive to large singular values. Hence, dijL overcomes the defects of Minkowski distance function dijMin and Mahalanobis distance function dijMa, which is more ideal for evidence revision.Then, we can define the reliability degree of evidencemi.
(35)relmi=1−Di∑i=1NDi,whereDi satisfies
(36)Di=∑j=1,j≠iNdijL2.Finally, the revised evidenceMLanceHj based on Lance distance function is deduced by relmi.
(37)MLanceHj=∑i=1NmiHj·relmi.


### 3.3.2. Revised Evidence Based on Spectral Angle Cosine Function

At present, the modification of evidence is mainly realized by distance functions [30–32]. In this paper, from the view point of similarity measurement, we further modify the conflict evidences by spectral angle cosine function. For evidences mi,mj, the definition of spectral angle cosine function is
(38)sij=∑t=1Mmit·mjt∑t=1Mmit2·∑t=1Mmjt2.Then, the support degree and credibility degree of evidencemi can be separately derived as
(39)supmi=∑j=1,j≠iNsij,crdmi=supmi∑i=1Nsupmi,where supmi represents the support degree of mi from other evidences.Ascrdmi represents the credibility degree of evidence mi, the revised evidence MSpectralHj based on spectral angle cosine function can be deduced by crdmi.
(40)MSpectralHj=∑i=1NmiHj·crdmi.


### 3.3.3. Combination of Two Revised Evidences

After the comprehensive modifications of conflict evidences, respectively, by the introduction of Lance distance function and spectral angle cosine function, the modified DS application is achieved through the synthesis ofMLanceHj and MSpectralHj.
(41)M=MLance⊕MSpectral.MLanceHj and MSpectralHj reasonably and precisely manage system’s uncertainty with different cluster analyses. Ulteriorly, (41) achieves the combination of two revised evidences MLanceHj and MSpectralHj when solving the evidence conflicts among multiple receivers.Accordingly, the proposed multi-sensor fusion method can obtain effective and reliable recognition result for emitter identification system.


## 3.3.1. Revised Evidence Based on Lance Distance Function

Assuming that two evidences on the frameΘ=H1,H2,⋯,HM are separately expressed as mi=mi1,mi2,⋯,miM and mj=mj1,mj2,⋯,mjM,i=1,2,⋯,N, where N is the number of receivers (sensors), and mit=miHt,mjt=mjHt,t=1,2,⋯,M.The evidence conflict is generally eliminated by Minkowski distance function or Mahalanobis distance function [30], whose definitions between mi,mj are separately shown as:
(33)dijMinq=∑t=1Mmit−mjtq1q,dijMaS=mi−mjTS−1mi−mj.Minkowski distance function can be, respectively, simplified to Manhattan distance function, Euclidean distance function, and Chebyshev distance function whenq=1,2,∞, and S=covmi,mj is the covariance matrix of evidences mi,mj.[30] shows that Minkowski distance function requires the same dimension of attributes that loses the consistency of evidences, while Mahalanobis distance function requires the covariance computation of vectors which is unsuitable for large-scale data processing. Therefore, we introduce a new distance measurement as Lance distance function to solve the evidence conflicts.
(34)dijL=1M∑t=1Mmit−mjtmit+mjt.(34) shows that Lance distance function dijL is a dimensionless attribute, which is not sensitive to large singular values. Hence, dijL overcomes the defects of Minkowski distance function dijMin and Mahalanobis distance function dijMa, which is more ideal for evidence revision.Then, we can define the reliability degree of evidencemi.
(35)relmi=1−Di∑i=1NDi,whereDi satisfies
(36)Di=∑j=1,j≠iNdijL2.Finally, the revised evidenceMLanceHj based on Lance distance function is deduced by relmi.
(37)MLanceHj=∑i=1NmiHj·relmi.


## 3.3.2. Revised Evidence Based on Spectral Angle Cosine Function

At present, the modification of evidence is mainly realized by distance functions [30–32]. In this paper, from the view point of similarity measurement, we further modify the conflict evidences by spectral angle cosine function. For evidences mi,mj, the definition of spectral angle cosine function is
(38)sij=∑t=1Mmit·mjt∑t=1Mmit2·∑t=1Mmjt2.Then, the support degree and credibility degree of evidencemi can be separately derived as
(39)supmi=∑j=1,j≠iNsij,crdmi=supmi∑i=1Nsupmi,where supmi represents the support degree of mi from other evidences.Ascrdmi represents the credibility degree of evidence mi, the revised evidence MSpectralHj based on spectral angle cosine function can be deduced by crdmi.
(40)MSpectralHj=∑i=1NmiHj·crdmi.


## 3.3.3. Combination of Two Revised Evidences

After the comprehensive modifications of conflict evidences, respectively, by the introduction of Lance distance function and spectral angle cosine function, the modified DS application is achieved through the synthesis ofMLanceHj and MSpectralHj.
(41)M=MLance⊕MSpectral.MLanceHj and MSpectralHj reasonably and precisely manage system’s uncertainty with different cluster analyses. Ulteriorly, (41) achieves the combination of two revised evidences MLanceHj and MSpectralHj when solving the evidence conflicts among multiple receivers.Accordingly, the proposed multi-sensor fusion method can obtain effective and reliable recognition result for emitter identification system.


## 4. Experimental Results and Analyses

Several simulations are conducted here to demonstrate the effectiveness of the proposed multi-feature multi-sensor fusion algorithm. To apply the multi-feature fusion based on the improved proximity approach, the discussion of multi-feature weighted indexes is firstly given. Then, a simplified recognition example is adopted to verify the effectiveness of the multi-feature fusion. After that, the superiority of the multi-sensor fusion based on the modified combination method is verified through analyzing its ability to solve different paradoxes. At last, the recognition of 9 typical emitter signals is comprehensively discussed to test the validity and effectiveness of the proposed multi-feature multi-sensor fusion algorithm in emitter identification system.


### 4.1. Verification for the Multi-Feature Fusion

#### 4.1.1. Determination ofwk

The multi-feature fusion based on the improved proximity approach needs to confirm the weighted indexwkk=1,2,3,4 before applying (28). Different entropy reflects different features of received signal from different angles. Hence, for the multi-feature fusion of each receiver, the contribution of different entropy to signal recognition is different. In this paper, the value of weighted index wk will be calculated according to the recognition ability of kth feature.To begin with, a distance functiondkk=1,2,3,4 of signal’s kth feature is defined.
(42)dk=∑j=−1030∑i=19Eik−FjkEik2,where dk represents the identification effect of signal’s kth feature during SNR=−10,30dB.Then, the weighted indexwkk=1,2,3,4 of signal’s kth feature is derived.
(43)wk=1−dk′∑k=141−dk′,where dk′ is the normalized distance function.
(44)dk′=dk∑k=14dk.Using the same simulation parameters in Table1 and taking 0.6dB Gaussian noise as measure error of SNRs, we can get features’ weighted indexes wkk=1,2,3,4 through Monte-Carlo simulations. Figure 2 shows the multi-feature models of 9 received signals during Monte-Carlo simulations, and the values wkk=1,2,3,4 are given in Table 2.Table 2
Multi-feature weighted indexes.
wiw1w2w3w3Value.1642.2665.2543.3150[FeatureE1] [Feature E2] [Feature E3] [Feature E4]Finally, we take two decimal places of features’ weighted indexes in Table2 as ultimate value, namely, w=w1,w2,w3,w4=0.16,0.27,0.25,0.32. It is obvious that wkk=1,2,3,4 satisfy the non-negativity and normalization properties in ((39)-(40)).


#### 4.1.2. Simulation on the Improved Proximity Approach

[25, 26], respectively, present a BPA acquisition methods based on gray relation and close degree. Both of these two methods are typical methods based on feature matching, which have similar principle to the proposed improved proximity approach. Hence, we take the methods in [25, 26] as the comparison methods.Simplifying the frame of recognition system asΘ=H1,H2,H3,H4, and N=5 sensors are included in the recognition system. Accordingly, the received signals of 5 sensors are Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,3,4,5 and the multi-feature models in feature library are Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,3,4. Referring to [25, 26], the simulation data of Fi,Ej are, respectively, shown in Tables 3 and 4.Table 3
Multi-feature models of received signals.
SensorsFi1Fi2Fi3Fi4Sensor 165.474756.178220.432964.8852Sensor 265.497356.379422.409267.6326Sensor 362.707258.592721.375368.9063Sensor 464.038061.671723.701370.2488Sensor 575.418469.334817.471172.3674Table 4
Multi-feature models in the feature library.
Multi-feature modelsEj1Ej2Ej3Ej4E143.582830.885910.680653.7373E274.360572.139317.810774.1857E363.928658.606421.766067.5529E449.885846.818314.499852.6699Comparing the data in Tables3 and 4, it is intuitively seen that the received signals of sensors 1-4 are close to the multi-feature model E3 in the feature library, while the received signal of sensor 5 conforms to the multi-feature model E2 most. Therefore, effective feature matching method should be consistent to above analyses. That is, sensors m1,m2,m3,m4 should assign most support to E3, while sensor m5 should allocate most support to E2.The multi-feature fusion results of the comparison methods and the proposed improved proximity approach are shown in Table5.Table 5
Multi-feature fusion results.
SensorsMethodsSignal’s typeE1E2E3E4m1[25]0.20640.26320.26640.2640[26]0.18830.25140.31690.2434Proposed0.12080.24090.46260.1757m2[25]0.19600.23760.30060.2658[26]0.18480.25640.32240.2364Proposed0.10250.20300.54980.1447m3[25]0.19240.23220.29630.2791[26]0.18090.25810.32590.2351Proposed0.08890.18340.60100.1267m4[25]0.19610.25370.27910.2711[26]0.17910.27420.31690.2298Proposed0.11700.24780.47360.1616m5[25]0.21310.27920.25230.2554[26]0.17580.32300.28350.2177Proposed0.09930.54940.21500.1363Table5 exposes that:
(a)
Comparison methods and the proposed improved proximity approach all produce reasonable multi-feature fusion results, which are similar to the analyses of Tables3 and 4. That is, the obtained m1,m2,m3,m4 of three methods all give E3 the biggest support as miH3>miHji=1,2,3,4,j=1,2,4, while sensor m5 assigns E2 the biggest support as m5H2>m5Hjj=1,3,4.(b)
The proposed improved proximity approach allocates larger support to the correct signal type than comparison methods, and simultaneously increases the value difference between correct signal type and other signal types. E.g., as discussed before,E3 is the correct signal type for sensors m1,m2,m3,m4. Compared to the comparison methods, E3’s support of m1,m2,m3,m4 in the improved proximity approach is much bigger, and so is its numerical gap between the support to E3 and other signal types.In sum, compared to theBPA acquisition methods in [25, 26], the proposed improved proximity approach produces better multi-feature fusion results, and obtains more reliable and accurate BPAs.


### 4.2. Verification for the Multi-Sensor Fusion

The multi-sensor fusion method for the emitter identification system is realized by the proposed DS combination method based two revised evidences. To prove the validity of the proposed DS combination method, we firstly adopted four paradoxes in this paper.In DS application, four paradoxes (complete conflict paradox, 0 trust paradox, 1 trust paradox, and high conflict paradox) are common types of evidence conflicts [28]. Assuming that the correct proposition of recognition system is H1, the DS combinations of paradoxes are shown in Table 6.Table 6
DS combinations of four paradoxes.
ParadoxesDS combinationElementsPropositionsH1H2H3H4H5Complete conflict paradoxEvidencesm1100\\m2010\\m30.80.10.1\\m40.750.150.1\\Combination resultmNaNNaNNaN\\Conflict degreeK10 trust paradoxEvidencesm10.50.20.3\\m20.60.150.25\\m300.90.1\\m40.70.20.1\\Combination resultm00.87800.1220\\Conflict degreeK0.993991 trust paradoxEvidencesm100.10.9\\m20.90.10\\m30.750.150.1\m40.80.10.1\\Combination resultm010\\Conflict degreeK0.9999High conflict paradoxEvidencesm10.70.10.100.1m200.10.20.10.2m30.60.150.1500.15m40.550.10.10.151m50.60.10.200.1Combination resultm00.35710.42860.21430Conflict degreeK0.9999Table6 shows that:
(a)
Complete conflict paradox: According to (18), the conflict degree of evidences m1,m2,m3,m4 is K=1. Under this circumstance, (25) cannot be applied as the denominator 1−K=0. Hence, we call the evidence conflict with K=1 as complete conflict paradox. Also, the DS combination result is invalid.(b)
0 trust paradox: It is apparent that evidencesm1,m2,m4 are relatively consistent, which all assign H1 the biggest support. The reasonable synthetic result should be similar to the consistent evidences, whereas the synthetic result after using (25) ismH1=0. The evidence conflict is caused by m3H1=0. Thus, 0 trust paradox indicates that the support assignment of evidence after combination is limited by the support assignment of each evidence in system.(c)
1 trust paradox: It is overt that all four evidencesm1,m2,m3,m4 give H2 small support. However, the synthetic result after using (25) is mH2=1, which is totally opposite to intuitive analysis. And the evidence conflict is caused by m1H1=0 and m2H3=0.(d)
High conflict paradox: The conflict degree isK=0.9999, which means evidences are highly conflict. In this case, combined evidences after DS application are usually unreasonable.Several methods are raised to solve the evidence conflicts. To verify the effectiveness of the proposed DS combination method based on two revised evidences, the methods in [13, 23, 28–30] are adopted as the comparison methods.According to the paradoxes in Table6, we can get the synthetic results of the comparison methods and the proposed DS combination method, which are shown in Table 7.Table 7
Synthetic results of different methods.
ParadoxesMethodsPropositionsH1H2H3H4H5ΘComplete conflict paradox (K=1)[13]0.63750.31250.050\\0[23]0.65790.06060.0561\\0.2254[28]0.95140.03110.0107\\0.0068[29]0.96240.03080.0068\\0[30]0.90290.06960.0275\\0Proposed0.99070.00930\\00trust paradox (K=0.9939)[13]0.44720.36570.1871\\0[23]0.44970.17750.2152\\0.1576[28]0.75760.12340.0799\\0.0391[29]0.68750.20630.1062\\0[30]0.58180.25290.1653\\0Proposed0.81630.16220.0215\\01trust paradox (K=0.9999)[13]0.61240.11260.2750\\0[23]0.67010.11540.0655\\0.1490[28]0.94620.02490.0345\\0.0117[29]0.92890.02260.0312\\0[30]0.86550.06060.0739\\0Proposed0.99520.00070.0041\\0High conflict paradox (K=0.9999)[13]0.49000.18000.15000.05000.13000[23]0.52430.08880.11940.02220.09730.1480[28]0.83650.06400.04720.02540.01540.0115[29]0.82060.06700.06190.00590.04460[30]0.63200.12270.11710.03150.09670Proposed0.98310.00760.00600.00010.00320For the DS application in the emitter identification system, the synthetic results will be processed by certain decision rules to give the estimated signal type. The multi-sensor fusion method with synthesis results that are more conducive to decision-making process represents a more effective treatment of evidence conflicts. Thus, to discuss the effectiveness of the comparison methods and the proposed DS combination method, we introduce two decision-making related factorsε1,ε2.
(45)ε1=mA,ε2=mA−mB,where mA,mB separately define the largest and second largest support and their corresponding propositions in the synthetic results.
(46)mA=maxmHj,Hj⊆Θ,mB=maxmHj,Hj⊆Θ,Hj≠A.According to (60-61), the fusion method with largerε1,ε2 is more conducive to decision-making process. That is, the fusion method with larger ε1,ε2 is more effective in dealing with evidence conflicts.The values ofε1,ε2 obtained by different methods are shown in Table 8.Table 8
Decision-making related factors of different methods.
ParadoxesFactorsMethods[13][23][28][29][30]ProposedComplete conflict paradox (K=1)ε10.63750.65790.95140.96240.90290.9907ε20.32500.43250.92030.93160.83330.98140 trust paradox (K=0.9939)ε10.44720.44970.75760.68750.58180.8163ε20.08150.23450.63420.48120.32890.65410 trust paradox (K=0.9999)ε10.61240.67010.92890.94620.86550.9952ε20.33740.52110.89440.91500.79160.9911High conflict paradox (K=0.9999)ε10.49000.52430.83650.82060.63200.9831ε20.31000.37650.77250.75360.50930.9755Tables6–8 indicate that:
(a)
The comparison methods and the proposed DS combination method all assign the biggest support to propositionH1, while the proposed combination method allocates the biggest support to the correct proposition H1. It is already known that the correct propositions of four paradoxes are H1. That is, reasonable and effective synthesis methods should support proposition H1 most. Hence, the proposed DS combination method is more conductive to solve the evidence conflicts.(b)
Compared with five comparison methods, the decision-making related factorsε1,ε2 of the proposed DS combination method are the largest. As we all know, the method with larger ε1,ε2 is more conductive to produce effective decision-making result for emitter identification system. Thus, the proposed DS combination method is more conductive to produce effective emitter recognition result.Overall, the superiority of the proposed multi-sensor fusion based on the proposed DS combination method has been checked.


### 4.3. Verification for the Proposed Algorithm

Sections4.1 and 4.2, respectively, verify the rationality of the multi-feature fusion based on the improved proximity approach and the reliability of the multi-sensor fusion based on the modified DS combination method. Hence, the novelty of the proposed DS application is affirmed.Next, we apply the multi-feature multi-sensor fusion algorithm to the emitter identification system to verify its application performance.


#### 4.3.1. Emitter Identification under DifferentSNR Intervals

Assume that the emitter identification system has certain prior knowledge to give the estimatedSNR interval before signal recognition. Take 5dB as the estimated SNR interval, we can get the emitter recognition results under different SNR intervals by Monte-Carlo simulations. The identification results are shown in Table 9, where the methods presented in [13, 23, 33, 34] are adopted as the comparison methods.Table 9
The recognition accuracy under differentSNR intervals.
SignalsMethodsSNR intervals (dB)[-10,-5)[-5,0)[0,5)[5,10)[10,15)[15,20)[20,25)[25,30)AM[13]0.1890.3110.4820.8990.994111[23]0.1620.4940.7410.9701111[33]0.1450.4850.7370.9721111[34]0.2790.5490.7710.9791111Proposed0.5610.8250.9760.9991111FM[13]0.4660.6440.7940.8590.9960.99911[23]0.5880.9060.9820.9711111[33]0.4760.8940.98010.9990.99911[34]0.7590.9360.9880.9991111Proposed0.8680.965111111PM[13]0.3140.4320.5410.6980.978111[23]0.3120.6120.7980.9711111[33]0.2980.5880.7790.9661111[34]0.4370.6610.8120.9791111Proposed0.7080.8721111112ASK[13]0.3310.5140.6930.8610.999111[23]0.2890.5860.8770.9370.995111[33]0.2460.5200.8720.9930.998111[34]0.3250.5130.7080.9240.999111Proposed0.5120.6650.9060.99911112FSK[13]0.4350.5800.8310.9520.999111[23]0.2600.5380.7620.9090.998111[33]0.2220.4980.6790.8980.997111[34]0.3810.6460.7920.9360.999111Proposed0.4760.6240.8560.9991111BPSK[13]0.5920.7950.8500.9950.973111[23]0.4490.9480.9990.9971111[33]0.4410.9410.9930.9951111[34]0.7100.9650.9910.9981111Proposed0.8380.990111111QPSK[13]0.6130.8150.9320.9770.999111[23]0.5040.962111111[33]0.4320.949111111[34]0.7000.9820.98811111Proposed0.8580.998111111LFM[13]0.7180.8680.99111111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9561111111SF[13]0.7060.7780.9260.9671111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9721111111The recognition results in Table9 reveal that:
(a)
Under differentSNR intervals, the proposed algorithm can always get the biggest emitter recognition rate. Hence, under different SNR intervals, the proposed algorithm always has better recognition performance than the comparison methods.(b)
The proposed algorithm already realizes 99.9% recognition accuracy at5,10SNR interval. Besides, when SNR<0, the comparison methods have low recognition rate, while the recognition accuracy of the proposed algorithm reaches around 50%. Thus, the proposed algorithm has better recognition performance even within low SNR intervals.Accordingly, with the prior knowledge of approximateSNR internal, the proposed algorithm identifies the correct signals more accurate under different SNR intervals. Obviously, the multi-feature multi-sensor fusion structure of the proposed algorithm effectively solves system’s uncertainty and improves the recognition accuracy of the emitter identification system.


#### 4.3.2. Emitter Identification under DifferentSNRs

To further verify the recognition effect of the proposed algorithm, we assume that the emitter identification system has limited prior knowledge thatSNR is unknown. Under this assumption, we take 1dB as SNR step and conduct Monte-Carlo simulations. The recognition results of the comparison methods and the proposed algorithm are shown in Figures 4–8.Figure 4
The recognition results of [13].
(a)(b)(c)(d)Figure 5
The recognition results of [23].Figure 6
The recognition results of [33].Figure 7
The recognition results of [34].Figure 8
The recognition results of the proposed algorithm.It can be seen in Figures4–8 that:
(a)
Both the recognition accuracy of the comparison methods and the proposed algorithm rises asSNR’s increasing. Obviously, with the increase of SNR, the signal features can identify the signal type more clearly.(b)
Compared with the comparison methods, the proposed algorithm has better emitter identification accuracy, especially whenSNR is low. E.g., for signal 2FSK, the proposed algorithm already realizes 90% recognition accuracy when SNR=2dB, while the comparison methods [13, 23, 33, 34] separately realize 90% recognition accuracy till SNR=7dB,5dB,5dB,12dB.Accordingly, with unknownSNR, the proposed algorithm has better recognition performance under different SNRs. That is, we further verify that the multi-feature multi-sensor fusion structure of the proposed algorithm can effectively solve the system uncertainty and improve the recognition accuracy of the emitter identification system.Overall, These two simulations demonstrate that the proposed algorithm accomplishes more effective emitter identification performance.


## 4.1. Verification for the Multi-Feature Fusion

### 4.1.1. Determination ofwk

The multi-feature fusion based on the improved proximity approach needs to confirm the weighted indexwkk=1,2,3,4 before applying (28). Different entropy reflects different features of received signal from different angles. Hence, for the multi-feature fusion of each receiver, the contribution of different entropy to signal recognition is different. In this paper, the value of weighted index wk will be calculated according to the recognition ability of kth feature.To begin with, a distance functiondkk=1,2,3,4 of signal’s kth feature is defined.
(42)dk=∑j=−1030∑i=19Eik−FjkEik2,where dk represents the identification effect of signal’s kth feature during SNR=−10,30dB.Then, the weighted indexwkk=1,2,3,4 of signal’s kth feature is derived.
(43)wk=1−dk′∑k=141−dk′,where dk′ is the normalized distance function.
(44)dk′=dk∑k=14dk.Using the same simulation parameters in Table1 and taking 0.6dB Gaussian noise as measure error of SNRs, we can get features’ weighted indexes wkk=1,2,3,4 through Monte-Carlo simulations. Figure 2 shows the multi-feature models of 9 received signals during Monte-Carlo simulations, and the values wkk=1,2,3,4 are given in Table 2.Table 2
Multi-feature weighted indexes.
wiw1w2w3w3Value.1642.2665.2543.3150[FeatureE1] [Feature E2] [Feature E3] [Feature E4]Finally, we take two decimal places of features’ weighted indexes in Table2 as ultimate value, namely, w=w1,w2,w3,w4=0.16,0.27,0.25,0.32. It is obvious that wkk=1,2,3,4 satisfy the non-negativity and normalization properties in ((39)-(40)).


### 4.1.2. Simulation on the Improved Proximity Approach

[25, 26], respectively, present a BPA acquisition methods based on gray relation and close degree. Both of these two methods are typical methods based on feature matching, which have similar principle to the proposed improved proximity approach. Hence, we take the methods in [25, 26] as the comparison methods.Simplifying the frame of recognition system asΘ=H1,H2,H3,H4, and N=5 sensors are included in the recognition system. Accordingly, the received signals of 5 sensors are Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,3,4,5 and the multi-feature models in feature library are Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,3,4. Referring to [25, 26], the simulation data of Fi,Ej are, respectively, shown in Tables 3 and 4.Table 3
Multi-feature models of received signals.
SensorsFi1Fi2Fi3Fi4Sensor 165.474756.178220.432964.8852Sensor 265.497356.379422.409267.6326Sensor 362.707258.592721.375368.9063Sensor 464.038061.671723.701370.2488Sensor 575.418469.334817.471172.3674Table 4
Multi-feature models in the feature library.
Multi-feature modelsEj1Ej2Ej3Ej4E143.582830.885910.680653.7373E274.360572.139317.810774.1857E363.928658.606421.766067.5529E449.885846.818314.499852.6699Comparing the data in Tables3 and 4, it is intuitively seen that the received signals of sensors 1-4 are close to the multi-feature model E3 in the feature library, while the received signal of sensor 5 conforms to the multi-feature model E2 most. Therefore, effective feature matching method should be consistent to above analyses. That is, sensors m1,m2,m3,m4 should assign most support to E3, while sensor m5 should allocate most support to E2.The multi-feature fusion results of the comparison methods and the proposed improved proximity approach are shown in Table5.Table 5
Multi-feature fusion results.
SensorsMethodsSignal’s typeE1E2E3E4m1[25]0.20640.26320.26640.2640[26]0.18830.25140.31690.2434Proposed0.12080.24090.46260.1757m2[25]0.19600.23760.30060.2658[26]0.18480.25640.32240.2364Proposed0.10250.20300.54980.1447m3[25]0.19240.23220.29630.2791[26]0.18090.25810.32590.2351Proposed0.08890.18340.60100.1267m4[25]0.19610.25370.27910.2711[26]0.17910.27420.31690.2298Proposed0.11700.24780.47360.1616m5[25]0.21310.27920.25230.2554[26]0.17580.32300.28350.2177Proposed0.09930.54940.21500.1363Table5 exposes that:
(a)
Comparison methods and the proposed improved proximity approach all produce reasonable multi-feature fusion results, which are similar to the analyses of Tables3 and 4. That is, the obtained m1,m2,m3,m4 of three methods all give E3 the biggest support as miH3>miHji=1,2,3,4,j=1,2,4, while sensor m5 assigns E2 the biggest support as m5H2>m5Hjj=1,3,4.(b)
The proposed improved proximity approach allocates larger support to the correct signal type than comparison methods, and simultaneously increases the value difference between correct signal type and other signal types. E.g., as discussed before,E3 is the correct signal type for sensors m1,m2,m3,m4. Compared to the comparison methods, E3’s support of m1,m2,m3,m4 in the improved proximity approach is much bigger, and so is its numerical gap between the support to E3 and other signal types.In sum, compared to theBPA acquisition methods in [25, 26], the proposed improved proximity approach produces better multi-feature fusion results, and obtains more reliable and accurate BPAs.


## 4.1.1. Determination ofwk

The multi-feature fusion based on the improved proximity approach needs to confirm the weighted indexwkk=1,2,3,4 before applying (28). Different entropy reflects different features of received signal from different angles. Hence, for the multi-feature fusion of each receiver, the contribution of different entropy to signal recognition is different. In this paper, the value of weighted index wk will be calculated according to the recognition ability of kth feature.To begin with, a distance functiondkk=1,2,3,4 of signal’s kth feature is defined.
(42)dk=∑j=−1030∑i=19Eik−FjkEik2,where dk represents the identification effect of signal’s kth feature during SNR=−10,30dB.Then, the weighted indexwkk=1,2,3,4 of signal’s kth feature is derived.
(43)wk=1−dk′∑k=141−dk′,where dk′ is the normalized distance function.
(44)dk′=dk∑k=14dk.Using the same simulation parameters in Table1 and taking 0.6dB Gaussian noise as measure error of SNRs, we can get features’ weighted indexes wkk=1,2,3,4 through Monte-Carlo simulations. Figure 2 shows the multi-feature models of 9 received signals during Monte-Carlo simulations, and the values wkk=1,2,3,4 are given in Table 2.Table 2
Multi-feature weighted indexes.
wiw1w2w3w3Value.1642.2665.2543.3150[FeatureE1] [Feature E2] [Feature E3] [Feature E4]Finally, we take two decimal places of features’ weighted indexes in Table2 as ultimate value, namely, w=w1,w2,w3,w4=0.16,0.27,0.25,0.32. It is obvious that wkk=1,2,3,4 satisfy the non-negativity and normalization properties in ((39)-(40)).


## 4.1.2. Simulation on the Improved Proximity Approach

[25, 26], respectively, present a BPA acquisition methods based on gray relation and close degree. Both of these two methods are typical methods based on feature matching, which have similar principle to the proposed improved proximity approach. Hence, we take the methods in [25, 26] as the comparison methods.Simplifying the frame of recognition system asΘ=H1,H2,H3,H4, and N=5 sensors are included in the recognition system. Accordingly, the received signals of 5 sensors are Fi=Fi1,Fi2,Fi3,Fi4,i=1,2,3,4,5 and the multi-feature models in feature library are Ej=Ej1,Ej2,Ej3,Ej4,j=1,2,3,4. Referring to [25, 26], the simulation data of Fi,Ej are, respectively, shown in Tables 3 and 4.Table 3
Multi-feature models of received signals.
SensorsFi1Fi2Fi3Fi4Sensor 165.474756.178220.432964.8852Sensor 265.497356.379422.409267.6326Sensor 362.707258.592721.375368.9063Sensor 464.038061.671723.701370.2488Sensor 575.418469.334817.471172.3674Table 4
Multi-feature models in the feature library.
Multi-feature modelsEj1Ej2Ej3Ej4E143.582830.885910.680653.7373E274.360572.139317.810774.1857E363.928658.606421.766067.5529E449.885846.818314.499852.6699Comparing the data in Tables3 and 4, it is intuitively seen that the received signals of sensors 1-4 are close to the multi-feature model E3 in the feature library, while the received signal of sensor 5 conforms to the multi-feature model E2 most. Therefore, effective feature matching method should be consistent to above analyses. That is, sensors m1,m2,m3,m4 should assign most support to E3, while sensor m5 should allocate most support to E2.The multi-feature fusion results of the comparison methods and the proposed improved proximity approach are shown in Table5.Table 5
Multi-feature fusion results.
SensorsMethodsSignal’s typeE1E2E3E4m1[25]0.20640.26320.26640.2640[26]0.18830.25140.31690.2434Proposed0.12080.24090.46260.1757m2[25]0.19600.23760.30060.2658[26]0.18480.25640.32240.2364Proposed0.10250.20300.54980.1447m3[25]0.19240.23220.29630.2791[26]0.18090.25810.32590.2351Proposed0.08890.18340.60100.1267m4[25]0.19610.25370.27910.2711[26]0.17910.27420.31690.2298Proposed0.11700.24780.47360.1616m5[25]0.21310.27920.25230.2554[26]0.17580.32300.28350.2177Proposed0.09930.54940.21500.1363Table5 exposes that:
(a)
Comparison methods and the proposed improved proximity approach all produce reasonable multi-feature fusion results, which are similar to the analyses of Tables3 and 4. That is, the obtained m1,m2,m3,m4 of three methods all give E3 the biggest support as miH3>miHji=1,2,3,4,j=1,2,4, while sensor m5 assigns E2 the biggest support as m5H2>m5Hjj=1,3,4.(b)
The proposed improved proximity approach allocates larger support to the correct signal type than comparison methods, and simultaneously increases the value difference between correct signal type and other signal types. E.g., as discussed before,E3 is the correct signal type for sensors m1,m2,m3,m4. Compared to the comparison methods, E3’s support of m1,m2,m3,m4 in the improved proximity approach is much bigger, and so is its numerical gap between the support to E3 and other signal types.In sum, compared to theBPA acquisition methods in [25, 26], the proposed improved proximity approach produces better multi-feature fusion results, and obtains more reliable and accurate BPAs.


## 4.2. Verification for the Multi-Sensor Fusion

The multi-sensor fusion method for the emitter identification system is realized by the proposed DS combination method based two revised evidences. To prove the validity of the proposed DS combination method, we firstly adopted four paradoxes in this paper.In DS application, four paradoxes (complete conflict paradox, 0 trust paradox, 1 trust paradox, and high conflict paradox) are common types of evidence conflicts [28]. Assuming that the correct proposition of recognition system is H1, the DS combinations of paradoxes are shown in Table 6.Table 6
DS combinations of four paradoxes.
ParadoxesDS combinationElementsPropositionsH1H2H3H4H5Complete conflict paradoxEvidencesm1100\\m2010\\m30.80.10.1\\m40.750.150.1\\Combination resultmNaNNaNNaN\\Conflict degreeK10 trust paradoxEvidencesm10.50.20.3\\m20.60.150.25\\m300.90.1\\m40.70.20.1\\Combination resultm00.87800.1220\\Conflict degreeK0.993991 trust paradoxEvidencesm100.10.9\\m20.90.10\\m30.750.150.1\m40.80.10.1\\Combination resultm010\\Conflict degreeK0.9999High conflict paradoxEvidencesm10.70.10.100.1m200.10.20.10.2m30.60.150.1500.15m40.550.10.10.151m50.60.10.200.1Combination resultm00.35710.42860.21430Conflict degreeK0.9999Table6 shows that:
(a)
Complete conflict paradox: According to (18), the conflict degree of evidences m1,m2,m3,m4 is K=1. Under this circumstance, (25) cannot be applied as the denominator 1−K=0. Hence, we call the evidence conflict with K=1 as complete conflict paradox. Also, the DS combination result is invalid.(b)
0 trust paradox: It is apparent that evidencesm1,m2,m4 are relatively consistent, which all assign H1 the biggest support. The reasonable synthetic result should be similar to the consistent evidences, whereas the synthetic result after using (25) ismH1=0. The evidence conflict is caused by m3H1=0. Thus, 0 trust paradox indicates that the support assignment of evidence after combination is limited by the support assignment of each evidence in system.(c)
1 trust paradox: It is overt that all four evidencesm1,m2,m3,m4 give H2 small support. However, the synthetic result after using (25) is mH2=1, which is totally opposite to intuitive analysis. And the evidence conflict is caused by m1H1=0 and m2H3=0.(d)
High conflict paradox: The conflict degree isK=0.9999, which means evidences are highly conflict. In this case, combined evidences after DS application are usually unreasonable.Several methods are raised to solve the evidence conflicts. To verify the effectiveness of the proposed DS combination method based on two revised evidences, the methods in [13, 23, 28–30] are adopted as the comparison methods.According to the paradoxes in Table6, we can get the synthetic results of the comparison methods and the proposed DS combination method, which are shown in Table 7.Table 7
Synthetic results of different methods.
ParadoxesMethodsPropositionsH1H2H3H4H5ΘComplete conflict paradox (K=1)[13]0.63750.31250.050\\0[23]0.65790.06060.0561\\0.2254[28]0.95140.03110.0107\\0.0068[29]0.96240.03080.0068\\0[30]0.90290.06960.0275\\0Proposed0.99070.00930\\00trust paradox (K=0.9939)[13]0.44720.36570.1871\\0[23]0.44970.17750.2152\\0.1576[28]0.75760.12340.0799\\0.0391[29]0.68750.20630.1062\\0[30]0.58180.25290.1653\\0Proposed0.81630.16220.0215\\01trust paradox (K=0.9999)[13]0.61240.11260.2750\\0[23]0.67010.11540.0655\\0.1490[28]0.94620.02490.0345\\0.0117[29]0.92890.02260.0312\\0[30]0.86550.06060.0739\\0Proposed0.99520.00070.0041\\0High conflict paradox (K=0.9999)[13]0.49000.18000.15000.05000.13000[23]0.52430.08880.11940.02220.09730.1480[28]0.83650.06400.04720.02540.01540.0115[29]0.82060.06700.06190.00590.04460[30]0.63200.12270.11710.03150.09670Proposed0.98310.00760.00600.00010.00320For the DS application in the emitter identification system, the synthetic results will be processed by certain decision rules to give the estimated signal type. The multi-sensor fusion method with synthesis results that are more conducive to decision-making process represents a more effective treatment of evidence conflicts. Thus, to discuss the effectiveness of the comparison methods and the proposed DS combination method, we introduce two decision-making related factorsε1,ε2.
(45)ε1=mA,ε2=mA−mB,where mA,mB separately define the largest and second largest support and their corresponding propositions in the synthetic results.
(46)mA=maxmHj,Hj⊆Θ,mB=maxmHj,Hj⊆Θ,Hj≠A.According to (60-61), the fusion method with largerε1,ε2 is more conducive to decision-making process. That is, the fusion method with larger ε1,ε2 is more effective in dealing with evidence conflicts.The values ofε1,ε2 obtained by different methods are shown in Table 8.Table 8
Decision-making related factors of different methods.
ParadoxesFactorsMethods[13][23][28][29][30]ProposedComplete conflict paradox (K=1)ε10.63750.65790.95140.96240.90290.9907ε20.32500.43250.92030.93160.83330.98140 trust paradox (K=0.9939)ε10.44720.44970.75760.68750.58180.8163ε20.08150.23450.63420.48120.32890.65410 trust paradox (K=0.9999)ε10.61240.67010.92890.94620.86550.9952ε20.33740.52110.89440.91500.79160.9911High conflict paradox (K=0.9999)ε10.49000.52430.83650.82060.63200.9831ε20.31000.37650.77250.75360.50930.9755Tables6–8 indicate that:
(a)
The comparison methods and the proposed DS combination method all assign the biggest support to propositionH1, while the proposed combination method allocates the biggest support to the correct proposition H1. It is already known that the correct propositions of four paradoxes are H1. That is, reasonable and effective synthesis methods should support proposition H1 most. Hence, the proposed DS combination method is more conductive to solve the evidence conflicts.(b)
Compared with five comparison methods, the decision-making related factorsε1,ε2 of the proposed DS combination method are the largest. As we all know, the method with larger ε1,ε2 is more conductive to produce effective decision-making result for emitter identification system. Thus, the proposed DS combination method is more conductive to produce effective emitter recognition result.Overall, the superiority of the proposed multi-sensor fusion based on the proposed DS combination method has been checked.


## 4.3. Verification for the Proposed Algorithm

Sections4.1 and 4.2, respectively, verify the rationality of the multi-feature fusion based on the improved proximity approach and the reliability of the multi-sensor fusion based on the modified DS combination method. Hence, the novelty of the proposed DS application is affirmed.Next, we apply the multi-feature multi-sensor fusion algorithm to the emitter identification system to verify its application performance.


### 4.3.1. Emitter Identification under DifferentSNR Intervals

Assume that the emitter identification system has certain prior knowledge to give the estimatedSNR interval before signal recognition. Take 5dB as the estimated SNR interval, we can get the emitter recognition results under different SNR intervals by Monte-Carlo simulations. The identification results are shown in Table 9, where the methods presented in [13, 23, 33, 34] are adopted as the comparison methods.Table 9
The recognition accuracy under differentSNR intervals.
SignalsMethodsSNR intervals (dB)[-10,-5)[-5,0)[0,5)[5,10)[10,15)[15,20)[20,25)[25,30)AM[13]0.1890.3110.4820.8990.994111[23]0.1620.4940.7410.9701111[33]0.1450.4850.7370.9721111[34]0.2790.5490.7710.9791111Proposed0.5610.8250.9760.9991111FM[13]0.4660.6440.7940.8590.9960.99911[23]0.5880.9060.9820.9711111[33]0.4760.8940.98010.9990.99911[34]0.7590.9360.9880.9991111Proposed0.8680.965111111PM[13]0.3140.4320.5410.6980.978111[23]0.3120.6120.7980.9711111[33]0.2980.5880.7790.9661111[34]0.4370.6610.8120.9791111Proposed0.7080.8721111112ASK[13]0.3310.5140.6930.8610.999111[23]0.2890.5860.8770.9370.995111[33]0.2460.5200.8720.9930.998111[34]0.3250.5130.7080.9240.999111Proposed0.5120.6650.9060.99911112FSK[13]0.4350.5800.8310.9520.999111[23]0.2600.5380.7620.9090.998111[33]0.2220.4980.6790.8980.997111[34]0.3810.6460.7920.9360.999111Proposed0.4760.6240.8560.9991111BPSK[13]0.5920.7950.8500.9950.973111[23]0.4490.9480.9990.9971111[33]0.4410.9410.9930.9951111[34]0.7100.9650.9910.9981111Proposed0.8380.990111111QPSK[13]0.6130.8150.9320.9770.999111[23]0.5040.962111111[33]0.4320.949111111[34]0.7000.9820.98811111Proposed0.8580.998111111LFM[13]0.7180.8680.99111111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9561111111SF[13]0.7060.7780.9260.9671111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9721111111The recognition results in Table9 reveal that:
(a)
Under differentSNR intervals, the proposed algorithm can always get the biggest emitter recognition rate. Hence, under different SNR intervals, the proposed algorithm always has better recognition performance than the comparison methods.(b)
The proposed algorithm already realizes 99.9% recognition accuracy at5,10SNR interval. Besides, when SNR<0, the comparison methods have low recognition rate, while the recognition accuracy of the proposed algorithm reaches around 50%. Thus, the proposed algorithm has better recognition performance even within low SNR intervals.Accordingly, with the prior knowledge of approximateSNR internal, the proposed algorithm identifies the correct signals more accurate under different SNR intervals. Obviously, the multi-feature multi-sensor fusion structure of the proposed algorithm effectively solves system’s uncertainty and improves the recognition accuracy of the emitter identification system.


### 4.3.2. Emitter Identification under DifferentSNRs

To further verify the recognition effect of the proposed algorithm, we assume that the emitter identification system has limited prior knowledge thatSNR is unknown. Under this assumption, we take 1dB as SNR step and conduct Monte-Carlo simulations. The recognition results of the comparison methods and the proposed algorithm are shown in Figures 4–8.Figure 4
The recognition results of [13].
(a)(b)(c)(d)Figure 5
The recognition results of [23].Figure 6
The recognition results of [33].Figure 7
The recognition results of [34].Figure 8
The recognition results of the proposed algorithm.It can be seen in Figures4–8 that:
(a)
Both the recognition accuracy of the comparison methods and the proposed algorithm rises asSNR’s increasing. Obviously, with the increase of SNR, the signal features can identify the signal type more clearly.(b)
Compared with the comparison methods, the proposed algorithm has better emitter identification accuracy, especially whenSNR is low. E.g., for signal 2FSK, the proposed algorithm already realizes 90% recognition accuracy when SNR=2dB, while the comparison methods [13, 23, 33, 34] separately realize 90% recognition accuracy till SNR=7dB,5dB,5dB,12dB.Accordingly, with unknownSNR, the proposed algorithm has better recognition performance under different SNRs. That is, we further verify that the multi-feature multi-sensor fusion structure of the proposed algorithm can effectively solve the system uncertainty and improve the recognition accuracy of the emitter identification system.Overall, These two simulations demonstrate that the proposed algorithm accomplishes more effective emitter identification performance.


## 4.3.1. Emitter Identification under DifferentSNR Intervals

Assume that the emitter identification system has certain prior knowledge to give the estimatedSNR interval before signal recognition. Take 5dB as the estimated SNR interval, we can get the emitter recognition results under different SNR intervals by Monte-Carlo simulations. The identification results are shown in Table 9, where the methods presented in [13, 23, 33, 34] are adopted as the comparison methods.Table 9
The recognition accuracy under differentSNR intervals.
SignalsMethodsSNR intervals (dB)[-10,-5)[-5,0)[0,5)[5,10)[10,15)[15,20)[20,25)[25,30)AM[13]0.1890.3110.4820.8990.994111[23]0.1620.4940.7410.9701111[33]0.1450.4850.7370.9721111[34]0.2790.5490.7710.9791111Proposed0.5610.8250.9760.9991111FM[13]0.4660.6440.7940.8590.9960.99911[23]0.5880.9060.9820.9711111[33]0.4760.8940.98010.9990.99911[34]0.7590.9360.9880.9991111Proposed0.8680.965111111PM[13]0.3140.4320.5410.6980.978111[23]0.3120.6120.7980.9711111[33]0.2980.5880.7790.9661111[34]0.4370.6610.8120.9791111Proposed0.7080.8721111112ASK[13]0.3310.5140.6930.8610.999111[23]0.2890.5860.8770.9370.995111[33]0.2460.5200.8720.9930.998111[34]0.3250.5130.7080.9240.999111Proposed0.5120.6650.9060.99911112FSK[13]0.4350.5800.8310.9520.999111[23]0.2600.5380.7620.9090.998111[33]0.2220.4980.6790.8980.997111[34]0.3810.6460.7920.9360.999111Proposed0.4760.6240.8560.9991111BPSK[13]0.5920.7950.8500.9950.973111[23]0.4490.9480.9990.9971111[33]0.4410.9410.9930.9951111[34]0.7100.9650.9910.9981111Proposed0.8380.990111111QPSK[13]0.6130.8150.9320.9770.999111[23]0.5040.962111111[33]0.4320.949111111[34]0.7000.9820.98811111Proposed0.8580.998111111LFM[13]0.7180.8680.99111111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9561111111SF[13]0.7060.7780.9260.9671111[23]0.7930.9650.99511111[33]0.8670.9600.99211111[34]0.8940.9870.99711111Proposed0.9721111111The recognition results in Table9 reveal that:
(a)
Under differentSNR intervals, the proposed algorithm can always get the biggest emitter recognition rate. Hence, under different SNR intervals, the proposed algorithm always has better recognition performance than the comparison methods.(b)
The proposed algorithm already realizes 99.9% recognition accuracy at5,10SNR interval. Besides, when SNR<0, the comparison methods have low recognition rate, while the recognition accuracy of the proposed algorithm reaches around 50%. Thus, the proposed algorithm has better recognition performance even within low SNR intervals.Accordingly, with the prior knowledge of approximateSNR internal, the proposed algorithm identifies the correct signals more accurate under different SNR intervals. Obviously, the multi-feature multi-sensor fusion structure of the proposed algorithm effectively solves system’s uncertainty and improves the recognition accuracy of the emitter identification system.


## 4.3.2. Emitter Identification under DifferentSNRs

To further verify the recognition effect of the proposed algorithm, we assume that the emitter identification system has limited prior knowledge thatSNR is unknown. Under this assumption, we take 1dB as SNR step and conduct Monte-Carlo simulations. The recognition results of the comparison methods and the proposed algorithm are shown in Figures 4–8.Figure 4
The recognition results of [13].
(a)(b)(c)(d)Figure 5
The recognition results of [23].Figure 6
The recognition results of [33].Figure 7
The recognition results of [34].Figure 8
The recognition results of the proposed algorithm.It can be seen in Figures4–8 that:
(a)
Both the recognition accuracy of the comparison methods and the proposed algorithm rises asSNR’s increasing. Obviously, with the increase of SNR, the signal features can identify the signal type more clearly.(b)
Compared with the comparison methods, the proposed algorithm has better emitter identification accuracy, especially whenSNR is low. E.g., for signal 2FSK, the proposed algorithm already realizes 90% recognition accuracy when SNR=2dB, while the comparison methods [13, 23, 33, 34] separately realize 90% recognition accuracy till SNR=7dB,5dB,5dB,12dB.Accordingly, with unknownSNR, the proposed algorithm has better recognition performance under different SNRs. That is, we further verify that the multi-feature multi-sensor fusion structure of the proposed algorithm can effectively solve the system uncertainty and improve the recognition accuracy of the emitter identification system.Overall, These two simulations demonstrate that the proposed algorithm accomplishes more effective emitter identification performance.


## 5. Conclusions

In this paper, an effective multi-feature multi-sensor fusion algorithm based on a modified DS application is proposed to achieve reliable emitter identification. The modified DS application is accomplished through the multi-feature fusion based on the improved proximity approach and the multi-sensor fusion based on the combination of two revised evidences. Firstly, the multi-feature model of received signal is extracted, and then the improved proximity approach is applied to produce multi-feature fusion result for each receiver. After that, the combination of two revised evidences is raised to build the multi-sensor fusion structure for multiple receivers in the emitter identification system. Experimental results and analyses verify the innovations of the proposed multi-feature multi-sensor fusion algorithm and its practicability in emitter identification system.In the further study, there are still some technical researches need to be studied. On the one hand, entropy analysis is a remarkable method with small computation amount for emitter identification system. We can improve the recognition accuracy of the proposed algorithm through adding other suitable entropy to enrich the multi-feature model of signals. E.g., adding time-frequency features like wavelet coefficients will describe the received signal better, and realize a better recognition accuracy. On the other hand, a multi-feature multi-sensor fusion structure is established in this paper. With effective adjustment, the multi-feature multi-sensor fusion structure can be applied to other fields, like pattern recognition, fault diagnosis, and target identification.



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*Source: 4264057-2022-04-20.xml*