A novel framework for extracting moment-based fingerprint features in specific emitter identification

Abstract

Extensive experiments illustrate that moments and their derivations can act as effective fingerprint features for specific emitter identification. Nevertheless, the lack of mechanistic explanation restricts the moment-based fingerprint features to a trial-based and data-driven technique. To make up for theoretical weakness and enhance generalization ability, we analytically investigate how intentional modulation and unintentional modulation affect moments. A framework for extracting moment-based fingerprint features is proposed through fine-segmenting slices. Fingerprint features are extracted, followed by segmenting signals into a combination of sinewaves and calculating their moments. The proposed framework shows advantages in mechanism interpretability and generalizing ability. Simulations and experiments verified the correctness and effectiveness of the proposed framework.

1 Introduction

Nowadays, information security is facing challenges caused by false identity attacks. In such non-cooperative communication scenarios, specific emitter identification (SEI) can extract the external features of the received signal to identify their sources, which is widely used in both military and civil applications such as real-time monitoring, supervision of wireless spectrum, wireless network security, and specific target tracking tasks [1, 2].

Limitations in manufacturing processes result in hardware imperfections. Such imperfect structures bring vibrations to electronic parameters, presented as external features. Due to uniqueness, external features can be regarded as identification marks, named fingerprint features [3]. The process, where hardware imperfections influence the transmitted signal, is referred to as unintentional modulation (UIM). Besides UIM, intentional modulation (IM) also exists in the transmitted signal for transmitting information. Hence, the SEI core is to extract fingerprint features, which describe UIM characteristics but neglect IM effects.

Current SEI methods can be divided into two categories depending on the analysis object, i.e., transient signal-based and steady-state signal-based methods. Transient signals are short intervals during which the signal evolves quickly in nontrivial or relatively unpredictable ways. Transient signals are usually generated when switching emitters on/off. Differently, steady-state signals are generated in normal working conditions, which can be regarded as a combination of a finite number of sinusoids. For transient signals, fingerprint features can be extracted directly, such as the fractal dimension [4], natural measures [5], and the energy trajectory [6, 7]. For steady-state signals, transformation is needed before fingerprint feature extraction to mitigate the IM effects. Common transforms include time-frequency spectrum (TFS) [8], wavelet transform (WT) [9], empirical mode decomposition (EMD) [10], and phase space reconstruction (PSR) [11]. Although transient signals are easier to extract fingerprint features, steady-state signals are more approachable in most practical cases. Hence, we select steady-state signals as the analysis object in this paper.

From the statistics perspective, transmitted signals are sampling series following a probability distribution. IM follows a specific subjective distribution, which enables accurate estimation of such signals based on their corresponding communication parameters. In contrast, unpredictable UIM introduces random variations, altering the sampling series’ distribution. This provides an opportunity for accomplishing SEI by comparing the corresponding probability distributions. In the existing SEI techniques, moments, as a quantitative method of measuring probability distributions, often act as fingerprint features.

The first four-order moments are widely used, containing mean, variance, skewness, and kurtosis. Wu et al. [12] and Liu et al. [13] utilized these four moments of the signal as fingerprints to recognize radar emitter identities. He et al. [14] proposed a framework based on signal decomposition methods to extract fingerprint features using skewness and kurtosis. Klein et al. [7, 15] further applied this approach to IEEE 802.11a OFDM signals by segmenting signals according to their frame structures. An alternative approach was adopted by Suski et al. [16], where IEEE 802.11a signals were sliced according to different communication protocols.

Most previous works concentrate on extracting practical fingerprint features for a specific situation, which causes features to suffer weak interpretability and generalization. Therefore, it is crucial to answer the following fundamental questions: (1) Why can moments act as fingerprint features? (2) How to design fingerprint features based on the signal moments?

To answer those questions, we first derive mathematical models from investigating how IM and UIM affect moments. Then, based on the theoretical conclusions, we propose a framework for extracting fingerprint features. The framework contains the following steps: fine-segmenting signals, calculating moments, and designing features. Based on mechanism analysis, this framework provides theoretical support for the moment-based SEI method and strategies to extract effective fingerprint features.

Contributions of this paper are as follows:

• Analyze the mechanism of IM and UIM theoretically and deduce how they affect signal moments via mathematic formulas. Conclusions are summarized in Corollary 1 and Corollary 2. Those two corollaries answer why moments can act as fingerprint features, make up for the lack of mechanistic explanation, and provide a theoretical basis for extracting moment-based fingerprint features.

• Propose a framework for extracting signal fingerprint features based on moments. To the best of our knowledge, this is the first time to give an exact and overall framework for designing moment-based fingerprint features. This provides a well-founded and systematic strategy for designing fingerprint features. This also makes the process of designing fingerprint features to be directive and purposeful instead of relying on extensive trial experiments.

• Facilitate improving generalization abilities of SEI methods. Improvement in generalization contains two aspects. One is that moment-based methods can adapt to various modulations, and the other is to identify emitters using modulations different from training datasets. To our knowledge, it is the first time emitter identification is recognized under different modulation circumstances.

The rest of the paper is organized as follows. Section 2 develops mathematical models for intentional modulation and hardware imperfections. Then, the impacts of the phase noise in high-frequency oscillators and the nonlinearity of the power amplifier on signal moments are investigated, respectively. Section 3 then proposes a novel framework for extracting fingerprint features. In Sect. 4, experiments are implemented to verify the theoretical conclusions and assess our proposed framework’s validity. Results and discussion are presented in detail. Section 5 concludes the whole paper.

2 Theoretical analysis

2.1 Notation and definition

Definitions of notations used in the paper are shown in Table 1.

Table 1 Notation and definition

2.2 Intentional modulation

Modulation can be expressed as a combination of three fundamental modulation methods containing amplitude-shift keying (ASK), frequency-shift keying (FSK), and phase-shift keying (PSK). In these methods, symbols are assigned to either amplitude, frequency, or phase of the signals. For instance, quadrature amplitude modulation (QAM) combines 2ASK and 2PSK. Assume that the transferred information is encoded as

$$\begin{aligned} B(n) = ({B_1},{B_2}, \ldots ,{B_{{N_B}}}), \end{aligned}$$

(1)

where ${B_i} \in \{ 0,1\}$ is the ith bit. $N_B$ is the bit length. Usually, a symbol encodes several bits. For M symbol patterns, we have either multiple amplitude-shift keying (MASK), multiple frequency-shift keying (MFSK), or multiple phase-shift keying (MPSK). In an M-array signal system, there are M alternative symbols. Each symbol represents K bits, hence,

$$\begin{aligned} M=2^K. \end{aligned}$$

(2)

Take QPSK as an example. The system generates four alternative symbols 00, 01, 10, 11, respectively, representing two bits, that are $K=2$, $M=4$. The corresponding symbol sequence is, therefore,

$$\begin{aligned} s(n) = ({s_1},{s_2}, \ldots ,{s_{{N_s}}}), \end{aligned}$$

(3)

where $s_j$ is the jth symbol, and $N_s$ is the number of symbols. Structures of direct up-conversion transmitters for different modulations are elaborated below.

2.2.1 MASK

As a form of amplitude modulation, an MASK system represents symbols as variations in the amplitude of a carrier wave (see Fig. 1).

Fig. 1

Structure of an MASK communication emitter

For each symbol, the impulse generator generates a fixed-voltage impulse, expressed as

$$\begin{aligned} {x_a}(t) = \sum \limits _{j = 1}^{{N_s}} {{a_j}g(t - (j - 1) \cdot {T_s})}, \end{aligned}$$

(4)

where $a_j$ is the amplitude for the jth symbol, $T_s$ represents the symbol period, and g(t) stands for the rectangular window. To avoid inter-symbol interference, a shaping filter h(t) is introduced to eliminate the extra frequency components. Then, the signal is

$$\begin{aligned} {x_h}(t) = \sum \limits _{j = 1}^{{N_s}} {{a_j}h(t - (j - 1){T_s})}. \end{aligned}$$

(5)

Thus, we have

$$\begin{aligned} h(t) = \frac{{\sin \left( {\pi t/{T_h}} \right) }}{{\pi t/{T_h}}} \cdot \frac{{\cos \left( {\alpha \pi t/{T_h}} \right) }}{{1 - {{\left( {2\alpha t/{T_h}} \right) }^2}}}, \end{aligned}$$

(6)

where $\alpha$ and $T_h$ are the roll-off factor and cutoff time, respectively. A high-frequency oscillator (HFO) provides carrier waves for the transmitted signals with the signal’s amplitude amplified before transmission via a power amplifier (PA). The transmitted MASK signals are, therefore,

$$\begin{aligned} {x_\mathrm{tr}}(t) = A\sum \limits _{j = 1}^{{N_s}} {{a_j}h(t - (j - 1){T_s})} \cdot \cos (2\pi {f_\mathrm{c}}t), \end{aligned}$$

(7)

where A represents the amplitude gain and $f_\mathrm{c}$ is the carrier frequency.

2.2.2 MFSK

MFSK is a frequency modulation scheme where symbols are transmitted through discrete frequency changes (as shown in Fig. 2).

Fig. 2

Structure of an MFSK communication emitter

The frequency shifting can be implemented in the FSK systems by switching independent oscillators at the beginning of each symbol period. The MFSK signals are

$$\begin{aligned} {x_\mathrm{tr}}(t) = A\sum \limits _{j = 1}^{{N_s}} {g(t - (j - 1){T_s}) \cdot \cos (2\pi {f_j}t)}, \end{aligned}$$

(8)

where $f_j$ is the carrier frequency corresponding to the jth symbol.

2.2.3 MPSK

MPSK is a phase modulation scheme in which symbols are transmitted as discrete additional phases, as shown in Fig. 3.

Fig. 3

Structure of an MPSK communication emitter

Each symbol is modulated into in-phase and quadrature-phase components. Then, fixed amplitude pulses are generated through the impulse generator, expressed as

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{x_{Ia}}(t) = \sum \limits _{j = 1}^{{N_s}} {{a_{Ij}}g(t - (j - 1) \cdot {T_s})} }\\ {{x_{Qa}}(t) = \sum \limits _{j = 1}^{{N_s}} {{a_{Qj}}g(t - (j - 1) \cdot {T_s})} } \end{array}} \right. . \end{aligned}$$

(9)

These are then separately modulated into two orthogonal basis functions. The two signals are then superimposed as

$$\begin{aligned}{x_\mathrm{tr}}(t) &= {x_{Ia}}(t) + {x_{Qa}}(t)\\ &= A \sum \limits _{j = 1}^{{N_s}} {{a_{Qj}}h(t - (j - 1){T_s})\cos (2\pi {f_\mathrm{c}}t) + {a_{Ij}}h(t - (j - 1){T_s})\sin (2\pi {f_\mathrm{c}}t)} \\ &= A\sum \limits _{j = 1}^{{N_s}} {h(t - (j - 1){T_s})\left( {{a_{Qj}}\cos (2\pi {f_\mathrm{c}}t) + {a_{Ij}}\sin (2\pi {f_\mathrm{c}}t)} \right) } . \end{aligned}$$

(10)

For simplifying (10), set the jth additive phase as $\phi _{j} = - \hbox {arctan}{{a_{Ij}}/{a_{Qj}}}$. Then, the transmitted signal can also be written as

$$\begin{aligned} {x_\mathrm{tr}}(t) &= A\sum \limits _{j = 1}^{{N_s}} {h(t - j{T_s})} \left( {\cos (2\pi {f_\mathrm{c}}t)\cos ({\phi _{j}}) - \sin (2\pi {f_\mathrm{c}}t)\sin ({\phi _{j}})} \right) \\ &= A\sum \limits _{j = 1}^{{N_s}} {h(t - j{T_s})} \cdot \cos (2\pi {f_\mathrm{c}}t + {\phi _{j}}), \end{aligned}$$

(11)

where the jth additive phase ${\phi _j}$ is determined by the M-arry modulation. Possible values are ${\phi _{j}} \in \left\{ {0,{{2\pi }/M}, \ldots ,2\pi } \right\}$.

Shaping filters work on each symbol in the MASK, MFSK, and MPSK schemes. We consider rectangular shaping filters in this paper for simplicity. Comparing (7), (8), and (11), here we present Corollary 1.

Corollary 1

Modulation signals are the summed sinewave signal segments with a certain amplitude, frequency, or phase. Table 2 shows the representation of modulated signals.

Table 2 Representation of modulated signals

2.3 Unintentional modulation

2.3.1 High-frequency oscillator

The HFO is an electronic circuit that produces a periodic oscillating electronic signal, as shown in Fig. 4.

Fig. 4

Structure of HFO

In an RF oscillator, there exist two frequency dividers. Let $f_\mathrm{ref}$ be the input reference frequency. The output frequency is therefore

$$\begin{aligned} {f_\mathrm{c}} = \frac{{(m + 1)n}}{m}{f_\mathrm{ref}}. \end{aligned}$$

(12)

A free-running oscillator generates the reference signal. Imperfect manufacturing processes bring phase noise to the transmitted signals. Kim and Park [17] demonstrated that the phase noise equals adding the sinusoid modulation to the signal phase. For instance, the phase noise for a single-tone signal is

$$\begin{aligned} {x_\mathrm{os}}(t) = F\left[ {x(t)} \right] = x(t) \times \exp [j\lambda \sin (2\pi {f_\mathrm{os}}t)], \end{aligned}$$

(13)

where $x_\mathrm{os}(t)$ denotes the transmitted signal with the HFO phase noise. $F\left( \cdot \right)$ describes the UIM effect of the RF oscillator. $f_\mathrm{os}$ and $\gamma$ are the frequency and amplitude of the phase noise, respectively. Expanding (13) with the Bessel function, phase noise modulation can be transferred into additional monophonic signals, expressed as

$$\begin{aligned} {x_\mathrm{os}}(t) = x\left( t \right) \times \sum \limits _{n = - \infty }^\infty {{j^n}{J_n}(\lambda )\exp \left[ {jn2\pi {f_\mathrm{os}}t} \right] }, \end{aligned}$$

(14)

where $J_n(\lambda )$ is the Bessel function of the first kind, and $\lambda \ll 1$. For the simplest case where only the first item in the summation is considered, we can write

$$\begin{aligned} {x_\mathrm{os}}(t) = x(t) \pm \frac{\mathrm{{1}}}{\mathrm{{2}}}\lambda \sin \left( {2\pi \left( {{f_\mathrm{c}} \pm {f_\mathrm{os}}} \right) t} \right) . \end{aligned}$$

(15)

2.3.2 Power amplifier

PA amplifies the amplitude of the modulated signal ensuring the required transmission power (Fig. 5).

Fig. 5

Structure of PA

Two modulation processes exist in the PA, i.e., AM–AM and AM–PM. Regardless of the system delay, amplitude distortion mainly appears in AM–AM signals. Liu argued that the Taylor series could describe the nonlinearity [18]. Then, we have

$$\begin{aligned} {x_\mathrm{am}}(t) = G\left[ {x(t)} \right] = \sum \limits _{i = 1}^\infty {{g_i} \cdot {{\left[ {x(t)} \right] }^i}}, \end{aligned}$$

(16)

where x(t) and $x_\mathrm{am}(t)$ are the input and output signals of the power amplifier, respectively. $G(\cdot )$ is the nonlinearity function of PA and $g_i$ is the ith coefficient of the Taylor function.

Note that only the first three terms can bring subtle effects. Hence, the high-order terms can be neglected. Furthermore, the components brought by even terms are out of the passband in the spectrum domain. Simplifying (16), we have

$$\begin{aligned} {x_\mathrm{am}}(t) = G\left[ {x(t)} \right] \approx {g_1}x(t) + {g_3}{x^3}(t). \end{aligned}$$

(17)

2.4 Effect on moments

From the statistics perspective, the transmitted signals can be regarded as a probability distribution. Moments such as mean, variance, skewness, kurtosis, and higher moments are often used to characterize the probability density function’s shape quantitatively.

In communication systems, IM and UIM together contribute to signal distribution and affect moments. Considering that modulation can be regarded as the sum of sinusoid signals (shown in Corollary 1), sinusoid signals are selected as analysis items, written as

$$\begin{aligned} X&= A\cos (2\pi {f_\mathrm{c}}t) \\&\quad 0< t < {z_c}{T_c}, \end{aligned}$$

(18)

where $T_c$ is the period of the carrier wave, and $z_c$ is the number of periods. To investigate the impact of UIM on the signal, here we first analyze moments of ideal sinusoid signals. We then investigate the impact of phase noise on high-frequency oscillators and the nonlinearity of power amplifiers. Results are concluded in Table 3.

Table 3 Statistics of sinewave signals

Table 3 illustrates that IM guarantees moments to be a certain constant value regardless of sampling time and hardware imperfection. In contrast, UIM brings vibrations to moments. In the case of imperfect HFO, moments are decided by IM, sampling time, and phase noise parameters. When PA is imperfect, moments rely on IM and nonlinearity parameters. Hence, Corollary 2 is concluded as follows.

Corollary 2

Moments of the ideal sinewave signals are constant values. Phase noise of HFO and amplitude distortion of PA cause fluctuation in moments. Such fluctuation depends on hardware imperfection parameters, which means that moments of signals can act as fingerprint features.

Corollaries 1 and 2 indicate that by fine segmenting the signals into sinewave segments, one can perform the SEI task using conclusions of sinewave signals.

The question of why moments can act as fingerprint features is answered. To accomplish the SEI task, we still have two outstanding issues. The first issue is how to fine-segment the modulated signals into a sum of sinewave signals, and the second issue is how to recognize emitter identities based on moments. Hence, we propose a framework for extracting fingerprint features based on the moments to address those two issues.

3 Methodology

3.1 The proposed framework

The complete process of SEI is presented in Fig. 6.

Fig. 6

The process of SEI

As shown in Fig. 6, three main steps exist in the SEI, including pre-processing, fingerprint extracting, and classifying [19]. Fingerprint extraction is the crucial step that guarantees the efficiency and validity of the proposed method. Our framework further split fingerprint extraction into three steps, i.e., fine-segmenting, moment calculating, and feature designing.

The process of extracting fingerprints can be written as,

$$\begin{aligned} \mathrm{Feature} = \Im \left[ {{M}\left[ {{T}\left[ {{x_\mathrm{tr}}(t)} \right] } \right] } \right] , \end{aligned}$$

(19)

where ${T}\left( \cdot \right)$ is the fine-segment function that slices the transmitted modulated signals into a series of segments. Slicing segments can be expressed as

$$\begin{aligned} {T}\left[ {{x_\mathrm{tr}}(t)} \right] = \sum \limits _{i = 1}^{{z_t}} {{x_\mathrm{tr}}(t){g^{\{ i\} }}(t)}, \end{aligned}$$

(20)

where $z_t$ is the number of segments, and ${g^{\{ i\} }}(t)$ is the ith slicing window. Further, ${M}( \cdot )$ calculates the moments for each component, expressed as

$$\begin{aligned} {M}\left[ {{T}\left[ {{x_\mathrm{tr}}\left( t \right) } \right] } \right] &= {M}\left[ {{T}\left[ {\sum \limits _{i = 0}^{{z_t}} {{a_i}\cos \left( {2\pi {f_i}t + {\phi _i}} \right) } } \right] } \right] \\ &= {\left[ {\begin{array}{*{20}{l}} {\varvec{\mu }}&{{\varvec{\sigma }}_{}^{\varvec{2}}}&{\varvec{\gamma }}&{\begin{array}{*{20}{l}} {\varvec{\kappa }}& \cdots &{{{{\varvec{\mu }}_{\varvec{K}}}}/{{\varvec{\sigma }}_{}^{\varvec{K}}}} \end{array}} \end{array}} \right] ^{{{T}}}} , \end{aligned}$$

(21)

where $\varvec{\mu }$, ${{\varvec{\sigma }}_{}^{\varvec{2}}}$, ${\varvec{\gamma }}$, ${\varvec{\kappa }}$, and ${{{{\varvec{\mu }}_{\varvec{K}}}}/ {{\varvec{\sigma }}_{}^{\varvec{K}}}}$ are row vectors, expressed as

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{\varvec{\mu }} = \left[ {\begin{array}{*{20}{l}} {{\mu ^{\{ 1\} }}}&{{\mu ^{\{ 2\} }}}& \cdots &{{\mu ^{\{ {z_t}\} }}} \end{array}} \right] }\\ {{\varvec{\sigma }}_{}^{\varvec{2}} = \left[ {\begin{array}{*{20}{l}} {\sigma {{_{}^2}^{^{\{ 1\} }}}}&{\sigma {{_{}^2}^{^{\{ 1\} }}}}& \cdots &{\sigma {{_{}^2}^{^{\{ {z_t}\} }}}} \end{array}} \right] }\\ {{\varvec{\gamma }} = \left[ {\begin{array}{*{20}{l}} {{\gamma ^{\{ 1\} }}}&{{\gamma ^{\{ 2\} }}}& \cdots &{{\gamma ^{\{ {z_t}\} }}} \end{array}} \right] }\\ {{\varvec{\kappa }} = \left[ {\begin{array}{*{20}{l}} {{\kappa ^{\{ 1\} }}}&{{\kappa ^{\{ 2\} }}}& \cdots &{{\kappa ^{\{ {z_t}\} }}} \end{array}} \right] }\\ {{{{{\varvec{\mu }}_{\varvec{K}}}} / {{\varvec{\sigma }}_{}^{\varvec{K}}}} = [ {\begin{array}{*{20}{l}} {{{{{{\mu _K}} / {\sigma _{}^K}}}^{\{ 1\} }}}&{{{{{{\mu _K}} / {\sigma _{}^K}}}^{\{ 2\} }}}& \cdots &{{{{{{\mu _K}} / {\sigma _{}^K}}}^{\{ {z_t}\} }}} \end{array}} ]} \end{array}} \right. . \end{aligned}$$

(22)

Besides, $\Im \left( \cdot \right)$ defines the transformation and selection of moments to obtain the fingerprint features depending on the application. Different operations result in different SEI methods. In some cases, $\Im \left( \cdot \right)$ is the slicing operation to select several moments as fingerprint features. In some cases, $\Im \left( \cdot \right)$ is the dimension reduction algorithm, which extracts more effective information. Others may use the vector operation $\Im \left( \cdot \right)$ to obtain a one-dimension feature.

3.2 Fine-segment

When segmenting signals, we consider two special cases. The first approach considers segmenting according to the period of the carrier wave, expressed as

$$\begin{aligned} {g^{\{ i\} }}\left( t \right) = g\left( {t - \left( {i - 1} \right) {T_c}} \right) . \end{aligned}$$

(23)

However, noise may challenge the moment estimation with few sampling points. An alternative approach is to segment signals according to the symbol, i.e.,

$$\begin{aligned} {g^{\{ i\} }}\left( t \right) = g\left( {t - \left( {i - 1} \right) {T_s}} \right) . \end{aligned}$$

(24)

In this approach, the starting and ending times of the symbols are required, i.e., symbol synchronization is required. This is, however, not pratical in most cases.

When designing the fine-segmenting method, we have to balance two demands. We need to ensure that the segment is similar to sinewave signals, which guarantees the characteristics of staying constant in a segment. We further need to acquire sufficient sampling points to ensure an accurate estimation. We analyze how to segment methods that affect moments for MASK, MFSK, and MPSK. Balancing those two segment methods, we propose the fine-segmenting strategy as in Table 4.

Table 4 Representation of modulated signals

3.3 Design features

Fingerprint features are needed for classifying different modulations and emitters. Hence, we propose four criteria for designing features based on the above analysis. In most cases, we must balance efficiency and accuracy when designing fingerprint features. For different cases, we provide 4 methods for designing fingerprint features. Considering efficiency, we have Case 1>Case 2>Case 3>Case 4. Regarding accuracy, we have Case 4>Case 3>Case 2>Case 1.

Case 1 Choose constant values of moments as identification marks. This method needs to guarantee that moments are regardless of IM.

Case 2 Choose a set of moments from different symbols as fingerprint features. Considering the effect of sampling time, one certain value may fail in identification. Hence, it needs to compare sets of moments for emitter authentication.

Case 3 Regard moment variation as fingerprint features. As Corollary 2 illustrates, UIM of hardware imperfection causes vibration in moments. Hence, such fingerprint features can describe hardware characteristics.

Case 4 Combination of those above.

For specific modulation and hardware imperfection, we can choose different design methods for different moments, as presented in Table 5. Notice that we give out the most efficient method in Table 5, meaning that more complex methods are also of capability.

Table 5 Design fingerprint features for different cases

4 Results and discussion

4.1 Data collection

In this section, simulations are performed to verify the analytical results and test the proposed framework’s identification capacity.

In the experiment, we simulate nine emitters for identification. For the UIM in hardware, we consider the UIM of HFO and PA, as mentioned in (15) and (17). Parameters for the phase noise in HFO and the nonlinearity in PA are shown in Table 6. We consider four fundamental modulation signals in the simulations, including CW, 2ASK, 2FSK, and BPSK. The bit rate is $f_b=1$ MHz. Also, we set the sampling frequency $f_s=100$ MHz and the carrier frequency $f_\mathrm{c}=10$ MHz. Besides, additive Gaussian Noise (AWGN) is added to the intercepted signal. For each emitter, we collect 200 samples for different SNR values from 10 to 30 dB.

Table 6 Parameters of the simulated emitters

The experiment is run on a PC equipped with an Intel Xeon Gold 6230R CPU at 2.10 GHz and 128 GB of RAM.

4.2 Theoretical verification

4.2.1 Uniqueness

Here, we choose sinewave signals without noise to show the uniqueness of each emitter (Fig. 7).

Fig. 7

Plots of moments for various emitters

Figure 7 indicates that phase noise of HFO causes variations in the moments, while nonlinearity of PA results in amplitude increase in even order moments. In the experiments, we use various parameters, which are distinguishable for moments. This experiment confirms that moments can be used as fingerprint features.

4.2.2 Effect of segmenting

In this section, we discuss the importance of fine-segmenting. We use ideal signals with multiple modulations, including CW, 2ASK, 2FSK, and BPSK. The results are presented in Fig. 8.

Fig. 8

Moments using different segmenting methods

Subplots in Fig. 8 illustrate that any kind of segmenting method is appropriate for CW signals. S1: Fine-segmenting according to symbol period. S2: Fine-segmenting containing two half-symbol periods. S3: Fine-segmenting of half carrier wave period. For 2ASK, it is necessary to segment signals according to their symbol, as amplitude modulation will affect moments. For 2FSK and BPSK, as long as segmenting is according to carrier period, moments can sufficiently reflect hardware imperfection.

4.3 Emitter identification

4.3.1 Classification

After choosing an appropriate segmenting method, we examine different classifiers in several circumstances. We choose nine common classifiers to recognize emitter identification, containing kernel support vector machine (KSVM) [20], probabilistic neural network (PNN) [12], k-nearest neighbor (KNN) [14], discriminant analysis classifier (DAC) [21], and their variants. Results are presented in Fig. 9.

Fig. 9

Accuracy using different classifiers

Figure 9 indicates that KNN is the most suitable classifier in all four conditions. It can be seen that an appropriate dimension reduction improves accuracy. Moreover, results also reveal that voting of different moments plays an important role in accuracy. The possible reason for the lack of apparent increase is that a hard vote cannot well characterize the weights for each moment and causes weak classifiers suppressing great ones.

4.3.2 Modulation generalization

In this section, we discuss modulation generalization. Here, we choose KNN as the classifier and change the training and test datasets to examine the framework’s capacity. The results are shown in Fig. 10.

Fig. 10

Accuracy of different modulated signals

Comparing the accuracy of cross-modulation shows that higher accuracy is achieved in cases where the training dataset uses complex modulated signals and the test dataset includes simple signals. Note that in the PSK experiment, “Train: QPSK Test: BPSK” performs better than “Train: BPSK Test: BPSK”. This is most likely because, in the MPSK system, UIM is independent of IM. Hence, moments remain the same with different modulation complexity. A slight increase may result from the greater robustness of complex modulation. For ASK and FSK, the UIM depends on the IM. The frequency of phase noise is also related to the carrier frequency in the FSK system. Increasing nonlinearity is related to the amplitude in the ASK system. This suggests that choosing complex modulation as the training dataset makes the classifiers contain more cases and achieve higher accuracy.

5 Conclusion

This paper answers two problems, i.e., why and how moments act as fingerprint features. Firstly, we illustrated how HFO and PA imperfection affected moments for communication signals by using theoretical derivation and experimental verification. Based on our theoretical results, we further proposed a framework for extracting fingerprint features based on moments, containing fine-segmenting signals, calculating moments, and designing features. Extensive experiments verified the effectiveness and efficiency of these methods in both common SEI and modulation generalization scenarios.

Actually, this paper is an initial trial of investigating hardware imperfections. We only take HFO phase noise and PA nonlinearity as an example here. In future works, we would like to investigate the impact of other hardware imperfections, together with their coupling effects. Besides, efficient architectures of neural networks can be designed for the SEI task based on theoretical analysis in this paper.