Wavelet transform and cyclic cumulant based modulation classification in wireless network

Abstract

With the development of Internet of things, a large number of embedded devices are interconnected by ad hoc and wireless network. The embedded devices can work correctly, only by ensuring correct communication between them. Identifying modulation scheme is the precondition to ensure the correct communication between embedded devices. However, in the multipath channel, ensuring the correct communication between embedded devices is a great challenge. Multipath channel always exists in the wireless network. However, most of the available modulation classification algorithms are based on ideal channel. It leads to the low-modulation classification probability in multipath channel. To resolve this problem, we propose a novel modulation classification algorithm. The proposed algorithm can classify signal without prior information about multipath channel. We calculate feature by high-order cyclic cumulant and wavelet transform. The feature is robust to multipath channel. The simulation results show that the proposed algorithm can achieve the much better classification accuracy than the available method in multipath channel.

Keywords

Wireless network multipath channel modulation classification wavelet transform cyclic cumulant

Introduction

With the development of Internet of things (IoT), a large number of embedded devices are interconnected by ad hoc and wireless networks.^1,2 Every connected embedded device can communicate with each other. There will be explosive data and massive interconnections in the wireless network.^3,4 In this situation of explosive and massive interconnections, how to ensure the correct communication between embedded devices is a great challenge. Only by ensuring the correct communication, the right information can be received and the cooperation between embedded devices becomes possible.^5,6 Wireless network is most susceptible to multipath fading, and effects of multipath fading always exists in the wireless network.^7,8 In multipath fading channel, it is difficult to identify modulation schemes correctly and we cannot receive the right information.⁹ Therefore, modulation classification in multipath channel is the key technology of signal processing in wireless network.

Modulation classification is widely used in civil and military fields.¹⁰ In the military field, modulation classification is the precondition to interfere and intercept the communication of enemies.¹¹ Modulation classification also is the key to communication intelligence system.¹² Only by correctly identifying the modulation schemes, we can formulate effective reconnaissance and jamming strategies.¹³ In the civil field, modulation classification is mainly applied to wireless spectrum management for signal identity and interference confirmation.¹⁴ The wireless communication department needs to monitor and manage the communication frequency spectrum to prevent illegal user and interference of wireless spectrum. In this way, we can protect legal communication from being affected.^15,16 The use of modulation classification in spectrum monitoring equipment helps us to improve the ability of distinguishing different users, determine the properties of unknown interference signal, and can help manager to solve problems.¹⁷

Modulation classification methods are mainly divided into two categories: maximum likelihood–based method and feature-based method.^18,19 The maximum likelihood–based method is a modulation classification method based on probability theory and hypothesis testing.^20,21 First, we need to know the signal statistical characteristics. Then by theoretical analysis and derivation of the signal statistical characteristics, we can obtain one or more statistical tests. Finally, we can classify modulation schemes based on appropriate threshold.^22,23 Xu et al.²⁴ proposed an automatic modulation classification method based on likelihood function. In their work, they studied the various classification solutions obtained by likelihood ratio test and discussed the detailed features related to all major algorithms. Chavali and Da Silva²⁵ used the expectation maximization (EM) algorithm to solve the maximum likelihood estimation of signal fading factor and Gaussian mixture model parameters. The hybrid likelihood ratio test (HLRT) was used to classify amplitude phase modulation schemes. Headley and Da Silva²⁶ used the quasi HLRT based on moment estimation to classify amplitude phase modulation schemes in non-synchronous fading channel. By oversampling the baseband received signal, Headley derived the moment estimation closed-form solution of received signal amplitude, noise variance, timing deviation, and phase offset. Using the Gibbs sampling, Liu et al.²⁷ estimated the posterior probability distribution of modulation schemes. Then they used posterior probability distribution as test statistic and used test statistic to classify the linear digital modulation schemes. The advantage of this method is that parameter estimation is not needed. But the computational complexity of this method is high.^28,29 The maximum likelihood–based method is optimal in the Bayesian criterion, but its computational complexity is high and is easy to have the problem of model mismatch.^30,31

The feature-based method accomplishes the modulation classification by extracting features from the received signal.^32,33 The key to implement this method is how to find the features that can effectively classify different modulation schemes, that is, selection of feature space.^34,35 Ozdemir et al.³⁶ proposed a modulation classification method based on high-order cumulants. The idea was first to use the independent component analysis to estimate the multiple-input multiple-output (MIMO) channel matrix, second to calculate the estimate of the transmitted signal vector, finally to calculate the fourth-order cumulants of transmitted symbol and fourth-order cumulants formed the feature vector. This article assumed that the feature vector obeyed the joint Gaussian distribution. This article also derived the mean and covariance matrix of the feature vector and used the maximum likelihood criterion to construct the likelihood ratio classifier to classify the modulation schemes. Dan et al.³⁷ proposed a new automatic modulation classification algorithm using wavelet analysis and wavelet support vector machine. The classifier can identify ten modulation schemes, and rate of correct classification is over 96.5% when signal-to-noise ratio (SNR) is not lower than 3 dB. Wang and Wang³⁸ proposed a modulation classification method based on cyclic cumulants. First, the author derived the cyclic cumulant of the first, second, and fourth orders. Second, the author found out the differences between the spectral peaks of different signals under specific cyclic cumulant. Finally, the author used the constant false-alarm rate (CFAR) detection algorithm to complete modulation classification. Zhang et al.³⁹ proposed an algorithm combining high-order cumulants and convolutional self-encoders to achieve modulation classification. The classification performance of feature-based method is suboptimal, but its computational complexity is low and it is more robust to signal model mismatch than maximum likelihood–based method.⁴⁰

All the methods mentioned above are efficient to classify signal in Gaussian channel. However, modulation classification of binary frequency shift keying (2FSK), binary phase shift keying (BPSK), and 16-quadrature amplitude modulation (16QAM) in multipath channel have not been studied. In this article, we propose a novel modulation classification algorithm in multipath channel. The proposed algorithm can avoid the multipath channel effect by combining wavelet transform and higher order cyclic cumulants. The proposed algorithm only need to know the signal carrier frequency and does not require more prior information such as the path number and noise variance.

The remaining parts of this article are organized as follows. In section “Mathematical model and classification algorithm,” we introduce mathematical model and modulation classification algorithm. The results and analysis are presented in section “Simulation results and analysis.” Conclusions are drawn in section “Conclusion.”

Mathematical model and classification algorithm

When the time delay $τ$ and the symbol period $T_{s}$ satisfy $τ << T_{s}$ , we call it flat fading channel. In flat fading channel, the amplitude fading of each frequency point is same, the frequency spectrum is not substantially changed, and signal waveform will not have a relatively large distortion.⁴¹ When the time delay $τ$ and the symbol period $T_{s}$ satisfy $τ >> T_{s}$ , we call it frequency selective fading channel. In frequency selective fading channel, not only the amplitude and phase will be randomly changed, but also the signal waveform will have relatively large distortion, causing inter-symbol interference.⁴² The channel model used in this article is single-input-single-output frequency selective fading channel. The model of the received signal is defined as follows⁴³

\begin{matrix} y (t) & = r (t) + n (t) = \sum_{i = 0}^{L - 1} r_{i} (t) + n (t) \\ = \sum_{i = 0}^{L - 1} h_{i} e^{j θ_{i}} s (t - τ_{i}) + n (t) \end{matrix}

(1)

where $L$ is the number of paths, $h_{i}$ is the attenuation coefficient, $θ_{i}$ is the phase factor, $τ_{i}$ is the time delay of each path, $T_{s}$ is the symbol period, $n (t)$ is Gaussian noise with a mean value of 0, and $n (t)$ is independent with the transmitted signal. $s (t)$ is modulated signal for transmission, and $s (t)$ is defined as follows

s (t) = \tilde{s} (t) e^{j (ω_{c} t + φ_{c})}

(2)

where $ω_{c}$ is the carrier frequency, $φ_{c}$ is the initial phase, and $\tilde{s} (t)$ is the baseband signal.

Wavelet transform is a time–frequency analysis method. Compared with Fourier transform and short-time Fourier transform, wavelet transform is based on the localization analysis of time and frequency.⁴⁴ Wavelet transform can multi-scale refine the signal by calculating flex and transition. It can extract feature information of the signal effectively, and it can find the location and the amplitude of outliers.⁴⁵ Therefore, wavelet transform is known as “mathematical microscope” for analyzing signals. Wavelet transform has the ability to extract the local feature of signals in both time domain and frequency domain. Wavelet transform can detect transient anomalies in normal signals easily. Wavelet transform is more and more widely used in the fields of communication, signal analysis and processing, speech synthesis, pattern recognition, military electronic countermeasures and weapon intelligence, and computer classification and recognition.⁴⁶

The continuous wavelet transform (CWT) of $s (t)$ is defined as follows⁴⁷

\begin{matrix} CWT (a, τ) & = \int s (t) ψ (t) dt \\ = \frac{1}{\sqrt{a}} \int s (t) ψ_{a}^{*} (\frac{t - τ}{a}) dt \end{matrix}

(3)

where $a$ is scale factor, $τ$ is translation factor, * is complex conjugate, and $ψ (t)$ is mother wavelet function and satisfy permissible condition⁴⁸

W_{g} = \int_{- \infty}^{+ \infty} \frac{| \hat{Ψ} (ω) |}{| ω |} d ω < \infty

(4)

where $\hat{Ψ} (ω)$ is Fourier transform of $ψ (t)$ .

The properties of CWT are as follows⁴⁹

If CWT of $s_{1} (t)$ is $CW T_{s_{1}} (a, τ)$ , CWT of $s_{2} (t)$ is $CW T_{s_{2}} (a, τ)$ , and $s (t) = s_{1} (t) + s_{2} (t)$ , then CWT of $s (t)$ satisfies the equation $CW T_{s} (a, τ) = CW T_{s_{1}} (a, τ) + CW T_{s_{2}} (a, τ)$ . According to this property, the wavelet transform of the signal superimposed by modulated signal and noise is equal to the superposition by the wavelet transform of modulated signal and wavelet transform of noise.

If CWT of $s (t)$ is $CW T_{s} (a, τ)$ , then CWT of $s (t - b)$ is $CW T_{s} (a, τ - b)$ .

If CWT of $s (t)$ is $CW T_{s} (a, τ)$ , then CWT of $s (ct)$ is $(1 / \sqrt{c}) CWT (ca, c τ)$ .

The wavelet transform with different scale factors $a$ and different translation factors $τ$ is self similar. According to this property, wavelet transform in multi-scale has similarity. Therefore, the feature can be extracted by recombining the signal after multi-scale wavelet transform.

It is not unique to reconstruct the original signal using CWT. Because of its redundancy, CWT can be discretized, and the discretization results are satisfactory.

Generally, wavelet functions have two characteristics: first, the definition domain of wavelet function is limited, and it has compact support or approximate compact support in the time domain. In principle, any function satisfying the admissibility condition can be wavelet function. But to make the wavelet generating function have better local characteristics in the time–frequency domain, we often choose the real or complex function with compact support or approximate compact support (local in time domain) and regular type (local in frequency domain) as the wavelet function. Second, the wavelet function has a positive and negative undulatory property because its direct current (DC) component must be zero. Because Haar wavelet has good selectivity in time domain and frequency domain and is easy to calculate, we choose Haar wavelet as mother wavelet in this article.⁵⁰

According to the above properties of wavelet transform, we know that in multipath channel, the wavelet transform of received signal equals the sum of wavelet transform on each path signal. The wavelet transform of Gaussian noise approximates to zero. Therefore, without considering Gaussian noise, the wavelet transform of received signal in multipath channel is given by

CW T_{y} (a, τ) = \sum_{i = 0}^{L - 1} h_{i} e^{j θ_{i}} CW T_{s} (a, τ)

(5)

where $CW T_{s} (a, τ)$ is the wavelet transform of transmitted signal.

The properties of cyclic cumulants are as follows⁵¹

The cyclic cumulants of stationary Gauss white noise more than second-order are always zero.

If $λ_{i} (i = 1, 2, \dots, k)$ is constant and $s_{1} = s (t), s_{2} = s (t + τ_{1}), \dots, s_{k} = s (t + τ_{k})$ , then we can get the following equation

\begin{matrix} cu m^{a} (λ_{1} s_{1}, \dots, λ_{k} s_{k}) \\ = (Π_{i = 1}^{k} λ_{k}) cu m^{a} (s_{1}, \dots, s_{k}) \end{matrix}

(6)

If the random variables $x_{i}$ and $y_{i} (i = 1, 2, \dots, k)$ are independent of each other, then we can get the following equation

\begin{matrix} cu m^{a} (x_{1} + y_{1}, \dots, x_{k} + y_{k}) \\ = cu m^{a} (x_{1}, \dots, x_{k}) + cu m^{a} (y_{1}, \dots, y_{k}) \end{matrix}

(7)

Cyclic cumulants have symmetry on variables, and we can get the following equation

cu m^{a} (s_{1}, \dots, s_{k}) = cu m^{a} (s_{i_{1}}, \dots, s_{i_{k}})

(8)

where $(i_{1}, \dots, i_{k})$ is one of permutation of $(1, \dots, k)$ .

In the modulation classification algorithm, cyclic cumulants are computed by cyclic moments. According to the definition of cyclic moments, the kth-order cyclic moments of cyclostationary signal s(i), can be estimated by the sample cyclic moments

{\hat{M}}_{ks}^{a} (τ_{1}, \dots, τ_{k - 1}) = \frac{1}{T} \sum_{i = 0}^{N - 1} s (i) s (i + τ_{1}) (i + τ_{k}) e^{- jat}

(9)

If $\forall m \in Z$ , $Z$ represents integer fields, $s_{n} (i) \in {s_{0} (i), s_{0}^{*} (i) \dots, s_{k} (i), s_{k}^{*} (i)}$ satisfies the following equation

\sum_{τ_{1}, \dots, τ_{m} = - \infty}^{+ \infty} supcum [s_{n_{0}} (t), s_{n_{1}} (t + τ_{1}), \dots, s_{n_{m}} (t + τ_{m})] < \infty

(10)

then we can get the following equations

E {| {\hat{M}}_{ks, N}^{a} (τ) - E {\hat{M}}_{ks, N}^{a} (τ) |}^{2} \leq \frac{C}{N}

(11)

{| {\hat{M}}_{ks, N}^{a} (τ) - E {\hat{M}}_{ks, N}^{a} (τ) |}^{2} \dot{=} o (\frac{1}{N^{1 - γ}})

(12)

{| {\hat{C}}_{ks, N}^{a} (τ) - E {\hat{C}}_{ks, N}^{a} (τ) |}^{2} \dot{=} o (\frac{1}{N^{1 - γ}})

(13)

where $C$ is constant, $γ > 3 / 4$ , and $\dot{=}$ means almost equal. If the kth-order cyclic moment $M_{ks}^{a} (τ)$ and the cyclic cumulant $C_{ks}^{a} (τ)$ exist, then

lim_{N \to \infty} {\hat{M}}_{ks, N}^{a} (τ) \dot{=} M_{ks}^{a} (τ)

(14)

lim_{N \to \infty} {\hat{C}}_{ks, N}^{a} (τ) \dot{=} C_{ks}^{a} (τ)

(15)

The fourth-order cyclic cumulant of signal $s (n)$ is given by the following equation⁵²

C_{40, s}^{a} (0, 0, 0) = {〈 s^{4} (t) e^{j 8 π at} 〉}_{t} - 3 {〈 s^{2} (t) e^{j 4 π at} 〉}_{t}^{2}

(16)

where $a$ is cyclic frequency.

The fourth-order cyclic cumulant of noise $n (t)$ is zero because the cyclic cumulant of stationary white Gaussian noise more than second order is always zero. Since the adjacent symbols of signal can be considered as independent and uncorrelated, signal with time delay more than one symbol period is independent from the original signal. And each path has independent transmission function, so the received signal of different paths can be considered independent. Therefore, we can get the following equation

C_{40, r}^{a} = \sum_{i = 0}^{L - 1} C_{40, r_{i}}^{a}

(17)

\begin{matrix} C_{40, r_{i}}^{a} = {〈 r_{i}^{4} (t) e^{j 8 π at} 〉}_{t} - 3 {〈 r_{i}^{2} (t) e^{j 4 π at} 〉}_{t}^{2} \\ = h_{i}^{4} e^{j 4 θ_{i}} C_{40, s}^{a} \end{matrix}

(18)

When the cyclic frequency satisfies $a = f_{c}$ , we can get the following equation

C_{40, r}^{f_{c}} = C_{40, s}^{f_{c}} \sum_{i = 0}^{L - 1} h_{i}^{4} e^{j 4 θ_{i}}

(19)

Therefore, when the cyclic frequency satisfies $a = f_{c}$ , the cyclic cumulant of the received signal is

C_{40, y}^{f_{c}} = C_{40, r}^{f_{c}} = C_{40, s}^{f_{c}} \sum_{i = 0}^{L - 1} h_{i}^{4} e^{j 4 θ_{i}}

(20)

where $C_{40, s}^{f_{c}}$ is the fourth-order cyclic cumulant of $s (t)$ .

Wavelet transform belongs to linear transformation, satisfies the principle of linear superposition, and has translation invariance. So in multipath channel, the wavelet transform of received signal equals the sum of wavelet transform on each path signal. The signal with time delay for more than one symbol period is independent from the original signal, so each path signal is still independent after wavelet transform. To simplify the mathematical derivation, we make $\bar{y} = CW T_{y} (a, τ)$ and $\bar{s} = CW T_{s} (a, τ)$ . According to equation (5), we can get the following equation

C_{40, \bar{y}}^{f_{c}} = C_{40, \bar{s}}^{f_{c}} \sum_{i = 0}^{L - 1} h_{i}^{4} e^{j 4 θ_{i}}

(21)

We define the feature $f_{40}$ as follow

f_{40} = \frac{C_{40, y}^{f_{c}}}{C_{40, \bar{y}}^{f_{c}}} = \frac{C_{40, s}^{f_{c}}}{C_{40, \bar{s}}^{f_{c}}}

(22)

where $C_{40, \bar{s}}^{f_{c}}$ is the fourth-order cyclic cumulant of $s (t)$ after the processing of wavelet transform, $C_{40, s}^{f_{c}}$ is the fourth-order cyclic cumulant of $s (t)$ .

By the above analysis, the flow chart of the proposed algorithm is shown in Figure 1.

Calculating the wavelet transform of the received signal $y (t)$ to obtain $\bar{y}$ .

Calculating the fourth-order cyclic cumulants of $\bar{y}$ to obtain $C_{40, \bar{y}}^{f_{c}}$ .

Calculating the fourth-order cyclic cumulants of $y (t)$ to obtain $C_{40, y}^{f_{c}}$ .

Calculating the eigenvalue $f_{40}$ by equation (22). Then $f_{40}$ is compared with the threshold. Finally, we can obtain the classification result.

Figure 1.

Algorithm flow chart.

Simulation results and analysis

In this section, we make a lot of experiments to verify the effectiveness and performance of the proposed algorithm. In multipath channel, we test a variety of modulation schemes, including 2FSK, BPSK, minimum-shift keying (MSK), and 16QAM. Sampling frequency is 320 MHz. Carrier frequency is 40 MHz. Symbol transmission rate is 2 Mbps. Number of symbol is 2000. Phase offset is distributed randomly between 0 and $2 π$ . Each path obey Rayleigh fading.

Figure 2 is the eigenvalue curve of signal with SNR when the path number is 3, time delay is distributed randomly between $T_{s} and 2 T_{s}$ ( $T_{s}$ is symbol period).

Figure 2.

Eigenvalue curve with SNR when path number is 3.

Because the eigenvalue of 2FSK is much larger than the other signals, to observe conveniently, Figure 3 is the result of removing 2FSK. From Figures 2 and 3, we can see that eigenvalues of different signals have significant distinction and eigenvalues do not change obviously with SNR. It is shown that eigenvalue is less affected by SNR.

Figure 3.

Eigenvalue curve with SNR when path number is 3 without 2FSK.

To verify the objectivity of the results, the path number is changed to 5 and 8 under the same conditions. The results are shown in Figures 4 –7.

Figure 4.

Eigenvalue curve with SNR when path number is 5.

Figure 5.

Eigenvalue curve with SNR when path number is 5 without 2FSK.

Figure 6.

Eigenvalue curve with SNR when path number is 8.

Figure 7.

Eigenvalue curve with SNR when path number is 8 without 2FSK.

It can be seen from Figures 2 to 7 that the eigenvalues do not change greatly with the path number when SNR is more than −5 dB. It is shown that eigenvalue proposed in this article is less affected by the path number.

Figures 8 –13 are the eigenvalue curve with SNR when the path number is 3, the time delays are $T_{s} ~ 2 T_{s}$ , $T_{s} ~ 3 T_{s}$ , and $T_{s} ~ 5 T_{s}$ , respectively. From the simulation results, we can see that when the time delay is different, eigenvalues do not have obvious change. Combined with Figures 2 –7, it can be proved that the proposed feature can effectively suppress the influence of multipath channel.

Figure 8.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 2 T_{s}$ .

Figure 9.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 2 T_{s}$ without 2FSK.

Figure 10.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 3 T_{s}$ .

Figure 11.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 3 T_{s}$ without 2FSK.

Figure 12.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 5 T_{s}$ .

Figure 13.

Eigenvalue curve with SNR when time delay is $T_{s} ~ 5 T_{s}$ without 2FSK.

Figure 14 shows the probability of correct classification (PCC) of each signal with SNR when the path numbers are 3, 5, and 8. Figure 15 shows the average PCC of all signal with SNR when the path numbers are 3, 5, and 8. From Figure 14, we can see that PCC of each signal are all above 90% at SNR −1 dB. From Figure 15, when SNR is more than −1 dB, we can see that average PCC do not have obvious change even when the path number is different. It is shown that the average PCC is less affected by the path number.

Figure 14.

Different paths numbers, PCC of each signal with SNR.

Figure 15.

Different paths numbers, average PCC of all signal with SNR.

Figure 16 shows the PCC of each signal with SNR when the time delays are $T_{s} ~ 2 T_{s}$ , $T_{s} ~ 3 T_{s}$ , and $T_{s} ~ 5 T_{s}$ . Figure 17 shows the average PCC of all signal with SNR when the time delays are $T_{s} ~ 2 T_{s}$ , $T_{s} ~ 3 T_{s}$ , and $T_{s} ~ 5 T_{s}$ . From Figure 16, we can see that PCC of each signal are all above 90% at SNR > 0 dB. From Figure 17, when SNR is more than −1 dB, we can see that average PCC do not have obvious change even when the time delay is different. It is shown that the average PCC is less affected by the time delay.

Figure 16.

Different time delays, PCC of each signal with SNR.

Figure 17.

Different time delays, average PCC of each signal with SNR.

We compare the performance of the proposed algorithm in this article with the proposed algorithm by Xi and Wu.⁵³ The results are shown in Figures 18 and 19. The dotted line is the algorithm performance curve proposed in this article. The solid line is the algorithm performance curve proposed by Xi. Figure 18 shows the average PCC comparison with SNR when the path numbers are 3, 5, and 8. Figure 19 shows the average PCC comparison with SNR when the time delays are $T_{s} ~ 2 T_{s}$ , $T_{s} ~ 3 T_{s}$ , and $T_{s} ~ 5 T_{s}$ . Table 1 shows the average PCC comparison when the SNR = 0 dB. WH means the proposed algorithm in this article. HOC means the proposed algorithm by Xi. The results show that under the same simulation condition, the average PCC of proposed algorithm in this article is higher than the average PCC of algorithm proposed by Xi. It is shown that the performance of the proposed algorithm in this article outperforms the algorithm proposed by Xi.

Figure 18.

Different paths numbers, average PCC in comparison with SNR.

Figure 19.

Different time delays, average PCC in comparison with SNR.

Table 1.

Average PCC comparison at SNR = 0 dB.

	Path number 3	Path number 5	Path number 8	$T_{s} ~ 2 T_{s}$	$T_{s} ~ 3 T_{s}$	$T_{s} ~ 5 T_{s}$
WH	0.94	0.95	0.96	0.95	0.96	0.97
HOC	0.85	0.76	0.68	0.86	0.77	0.65

PCC: probability of correct classification.

All the above simulation experiments can be run in MATLAB. Length of the proposed algorithm code is less than 3000. For running 100 Monte Carlo experiments, the total time taken is not more than 3 min. When the proposed algorithm is running, not more than 20% of the computational resources are occupied. It shows that the computational complexity of the proposed algorithm is low. It is very suitable for system with high real-time requirement.

Conclusion

In this article, we studied the modulation classification in multipath fading channel and proposed a novel algorithm. Theoretical deduction proves that the proposed algorithm can effectively suppress the effects of multipath channel. We conduct a large number of experiments under different simulation conditions in this article. The results shows, even at low SNR, the proposed algorithm still has high correct classification rate. When SNR is not lower than −1 dB, correct classification rate of 2FSK, BPSK, MSK, and 16QAM are all above 90%. And correct classification rate is less affected by the number of multipath and time delay. Moreover, the computational complexity of the proposed algorithm is low. The proposed algorithm can be used in system with high real-time requirement.

Footnotes

Handling Editor: Daniel Gutierrez-Reina

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper is supported by the National Natural Science Foundation of China (No. 61671167) and the paper is funded by the International Exchange Program of Harbin Engineering University for Innovation-oriented Talents Cultivation.

ORCID iD

Wenwen Li

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