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Intrapulse modulation recognition using shorttime ramanujan Fourier transform spectrogram
EURASIP Journal on Advances in Signal Processing volume 2017, Article number: 42 (2017)
Abstract
Intrapulse modulation recognition under negative signaltonoise ratio (SNR) environment is a research challenge. This article presents a robust algorithm for the recognition of 5 types of radar signals with large variation range in the signal parameters in low SNR using the combination of the Shorttime Ramanujan Fourier transform (STRFT) and pseudoZernike moments invariant features. The STRFT provides the timefrequency distribution features for 5 modulations. The pseudoZernike moments provide invariance properties that are able to recognize different modulation schemes on different parameter variation conditions from the STRFT spectrograms. Simulation results demonstrate that the proposed algorithm achieves the probability of successful recognition (PSR) of over 90% when SNR is above 5 dB with large variation range in the signal parameters: carrier frequency (CF) for all considered signals, hop size (HS) for frequency shift keying (FSK) signals, and the timebandwidth product for Linear Frequency Modulation (LFM) signals.
Introduction
Intrapulse modulation recognition aiming at recognizing the intentional intrapulse modulation type of radar signals plays a critical role in modern intercept receivers, which could be used to recognize the signal threat level and choose the optimal algorithm to estimate parameters of the detected signal [1].
In intrapulse modulation recognition context, more interest has been focused on the study of the feature based (FB) algorithms [2,3,4,5]. Thereinto, as a significant means to FB, the timefrequency analysis has been developed because it allows description of the instantaneous characteristics of a signal in the twodimensional (2D) timefrequency space [6,7,8,9,10,11,12,13,14,15]. The authors in [9] proposed a robust method for radar emitter recognition based on the Wigner–Ville distribution (WVD) and transfer learning, the average recognition rate (ARR) reaches more than 90% when signaltonoise ratio (SNR) is 10 dB. In [10], Gustavo LopezRisueno et al. proposed an algorithm based on Shorttime Fourier transform (STFT) to distinguish No Modulation, phase shift keying (PSK), frequency shift keying (FSK) and linear frequency modulation (LFM) sweeping a narrow band, it performs well when SNR is around 10 dB. In [11], a morphological operation based method had been exploited for a recognition of constant hop size (HS), constant timefrequency product, and carrier frequency (CF) ranging from 500 MHz to 1GHz intrapulse modulations, the accuracy can reach more than 95% for SNRs above 4 dB. In [12], Deguo Zeng et al. proposed an approach based on the ambiguity function to recognize six types of modulations, and it suitable for a recognition of LFM signals with bandwidth sweeping from 2 MHz to 15 MHz, pulsewidth (PW) equaling 3,5 and 7 μs when SNRs above 1 dB. In [13], the authors utilized the Rihaczek distribution and the Hough transform (HT) to discriminate Monopulse (MP) and binary phase shift keying (BPSK) signals with limited CF, binary frequency shift keying (2FSK) and 4ary frequency shift keying (4FSK) signals with limited HS, and LFM with large timebandwidth product ranging from 17.5 to 65, their simulation results show that the probability of successful recognition (PSR) is greater than 90% when the SNR is above 4 dB. However, these approaches suffer from low PSR under negative SNR environment, especially have certain limitations for recognizing radar signals with large variation range on CF, HS and timebandwidth product in the complicated noise condition. Therefore, it is paramount to explore new robust algorithms to obtain high PSR under conditions of low SNR and to recognize signals in a large variation range of signal parameters.
Recently, the concept of Ramanujan Fourier Transform(RFT) based timefrequency transform, namely Shorttime Ramanujan Fourier transform(STRFT) has been investigated owing to the good immunity to noise interference of RFT functions [16,17,18]. Following this, the timefrequency analysis of signals based on RFT was considered in a letter by Sugavaneswaran [19]. Their research indicates that in the presence of noise this class of transforms has lower effect in comparison to Discrete Fourier Transform (DFT) based timefrequency transforms. Consequently, regarding the noise robustness, the STRFT is more efficient than the traditional DFT based timefrequency transform, and is a promising solution for intrapulse modulation recognition under low SNR.
Nonetheless, how to realize an efficient recognition procedure for radar signals with large parameter variation range is still a challenging problem. The pseudoZernike moments have opened a wider set of applications for radar signal recognition in recent years, because the moments can provide potentially useful invariance properties such as translation, scale, and rotational invariance [20, 21]. In [22], Jarmo Lundén, et al. examined the suitability of pseudoZernike moments as features for radar waveform recognition, In [23], a new radar classification algorithm based on STFT and pseudoZernike moments features is proposed. Inspired by the aforesaid background, the pseudoZernike moments are beneficial to realize intrapulse modulation recognition in scenarios with large variation in the signal parameters.
The objective in this article is to develop a novel method which contains a “STRFT spectrogram computation”, a “moments feature computation” and a “recognition” to realize a classification of MP, LFM, BPSK, 2FSK and 4FSK signals with large variation range in the signal parameters: CF, HS and timefrequency product under negative SNR. The first part is used to obtain the STRFT spectrograms, which can represent features of intrapulse modulation signals even when the SNR is low. In the second part, the pseudoZernike moments features are used to extract information on spectrograms, which can provide invariance properties that are able to recognize different modulations when parameters change. The last part is used to classify 5 modulations in detail. Simulation results have showed that the recognition algorithm achieves very reliable performance: over 90% PSR when SNR is above 5 dB with CF ranges from 800 MHz to 1600 MHz, HS ranges from 60 MHz to 1000 MHz, and the timebandwidth product ranges from 8 to 500.
The rest of the article is organized as follows. Section 2 proposes an intrapulse modulation recognition model. Section 3 defines the mathematical model of the STRFT spectrogram and presents the spectrogram features for all the modulation schemes under consideration. Section 4 focuses on the mathematical model of pseudoZernike moments computation and describes the process of moments feature selection. Section 5 presents the proposed recognition algorithm. Simulation results are presented and discussed in Section 6. Finally, conclusions are presented in Section 7.
System model
An intrapulse modulation recognition approach based on ShortTime Ramanujan Fourier Transform (STRFT) and pseudoZernike moments feature is proposed in this paper. The system model of the proposed approach is shown in Fig. 1.
Three parts are included in this research: STRFT spectrogram computation, moments feature computation, and recognition. The STRFT analysis is a preprocess of moments feature computation, which could be used to obtain the STRFT spectrograms so as to represent features of intrapulse modulation signals under negative SNR.
In the moments feature computation part, pseudoZernike moments features selected based on the degree of overlapping between each pair of classes of the signal data set are extracted from the spectrograms for its good invariance properties, which consist of ψ _{3,3} feature and ψ _{2,0}, ψ _{5,1} features.
After ψ _{3,3} feature computation, intrapulse modulation signals are described by vectors. These describing vectors are used for recognition by using threshold decision. Furthermore, after ψ _{2,0} and ψ _{5,1} features computation, intrapulse modulation signals are described by matrices, which are used for recognition by using KNN classifier [22].
Mathematical model of STRFT spectrogram and STRFT spectrogram features
Ramanujan Fourier transform (RFT)
In the classical DFT, the basis functions e _{ p }(n) are defined as [16]
It is clear from (1), e _{ p }(n) are obtained as multiples of a basis frequency (1/q).
In the RFT, Ramanujan sums(RS) c _{ q }(n) are sums defined as the nth powers of qth primitive roots of unity [23]
It can be observed that c _{ q }(n) are the sums over the primitive characters e _{ p }(n). In other words, the basis functions are built by summing up components which are multiples of the same periodicity q, and only components satisfying (p, q) = 1 contribute to the sum.
The sums were introduced by Ramanujan to play the role of base functions over which typical arithmetical functions s(n) may be projected
It is obvious that an arithmetical function s(n) is an infinite sequence defined for 1 ≤ n ≤ ∞ for RFT, rather than that for DFT which is taken with a finite n shown in [24].
The s _{ q } is referred to as the RFT coefficient given by [25]
which is what we called the RFT.
Meanwhile, one can write the WienerKhintchine formula according to [26], and the linear property and the frequency multiplication property of RFT can be readily obtained.
STRFT spectrogram computation
In this paper, the STRFT is used to extract the necessary features of 5 modulations for intrapulse modulation recognition. The reasons for this choice is that as a windowed RFT function, the STRFT transform allows simultaneous description of a signal in time and frequency so that the temporal evolution of the signal spectrum can be analyzed in the timefrequency space.
For an arbitrary discretetime signal s(n) of length N, the STRFT of the signal is defined as
where φ(k) is the Rectangular window function of length H, and φ(0) = 1.
Then the STRFT spectrogram S _{ s }(k, q) defined as the squared absolute value of the STRFT of s(n) is given by
In the present work here, we take MP signal as an example to illustrate the deduction of the STRFT spectrogram expression of 5 modulations: MP signal, LFM signal, BPSK signal, 2FSKsignal and 4FSK signal.
Considering the following continuoustime MP signal
where f _{ c } is the carrier frequency(CF), \( T=\frac{1}{f_c} \) is the period of the continuoustime signal, A and φ _{0} are the amplitude and the initial phase of MP separately.
For a sampling interval of T _{ s } (the sampling frequency (SF) is \( {f}_s=\frac{1}{T_s} \)), the discrete representation of signal (7) then becomes
Let us represent \( \frac{T_s}{T} \) as \( \frac{1}{T_0} \), the expression of (8) can be given as
where T _{0} represents the number of samples in one cycle can be written as \( {T}_0=\frac{f_s}{f_c}. \)
Using Eq. (5), the STRFT of MP can be expressed as
in the case of q = T _{0}.
Substituting Eq. (10) into Eq. (6), the STRFT spectrogram of MP becomes
STRFT spectrogram features
Analysis of STRFT spectrogram features
In practice, there exists a tradeoff between time and frequency resolution when determining the window length (the duration of window), that is to say, a long duration of window will provide a poor frequency resolution and vice versa. Through a series of simulation experiments, a Rectangular window of length \( H=\frac{4000}{10}=400 \) is selected, which can provide the best frequency resolutiontime resolution tradeoff for 5 modulations abovementioned. Examples of amplitude normalized STRFT spectrograms P _{ s }(k, q) (a normalization with respect to its maximum value of each STRFT spectrogram S _{ s }(k, q))of 5 modulations computed from a sample of length N = 4000 with a Rectangular window of length H = 400 are shown in Fig. 2ae. The contours on the plot represent relative magnitude with the horizontal axis as q and the vertical axis as k(μs).
Figure 2 shows the amplitude normalized STRFT spectrograms P _{ s }(k, q) reflecting timefrequency distribution features of 5 types of modulation signals. Fig.2a shows the P _{ MP }(k, q) for MP signal. Ideally, there would be a straight line centred about T _{0} in kq plane as the Eq. (11) implied. By contrast, Fig. 2a shows the line to be spread out in q direction at the expense of reduced frequency resolution, and the peak energy is mainly concentrated in the location of T _{0}. The P _{ LFM }(k, q) for LFM signal with chirp rate u = 300 as depicted in Fig.2b. Based on the observation of the spectrogram, the spectrum line can be approximated by a piecewise line starting at T _{0} and finishing at T _{0} − i,where i = 1, 2, … T _{0} − 1, and each segment reflects the change of its frequency and phase. The P _{ BPSK }(k, q) for BPSK signal shown in Fig. 2c illustrates that the amplitude of spectrum obtains the minimum at instant of time of phase conversion, and in the duration of intercode, the P _{ BPSK }(k, q) is the same as the P _{ MP }(k, q). The P _{2FSK }(k, q) for 2FSK signal can be seen in Fig. 2d, which has five vertical line segments centered about T _{0} and T _{1} in kq plane embodying the number of frequency points, while for the P _{4FSK }(k, q) of 4FSK signal has 5 ones centered about T _{0}, T _{2}, T _{1}, T _{0}, T _{3} in kq plane as shown in Fig. 2e.
In summary, the contours on the plot show different spectrogram features of 5 modulations. Hence, the STRFT spectrograms can serve as a discriminating feature.
Analysis of discriminability
The parameter R giving the similarity degree between two amplitude normalized spectrogram P _{ s1}(k, q) and P _{ s2}(k, q) is defined as
The similarity degree is bounded, 0 ≤ R ≤ 1.
The similarity between any two amplitude normalized STRFT spectrograms computed by Eq. (12) with respect to different modulations in the absence of noise is depicted in Table 1. Intuitively, the MP and BPSK signals are difficult to distinguish from each other due to the fact that the spectrograms of the two modulations are similar enough with a similarity degree of 0.9341. In addition, for MP and LFM signals as well as LFM and BPSK signals, the corresponding R are 0.4312 and 0.4544 respectively that means this feature is considered not reliable to provide an effective method of signal differentiation.
Furthermore, the theoretical analysis in Section 3.2 indicates that the location of the STRFT spectral peak will be shifted induced by the variation of CFs of the input signals, then alter the feature value P _{ s }(k, q) and will finally influence the recognition results.
To tackle these problems, we propose a novel signal recognition method that is based on the combination of the STRFT spectrogram and the pseudoZernike moments.
Mathematical model of pseudoZernike moments and moments feature selection
Mathematical model of pseudoZernike moments
Moments have been widely used in image processing for pattern recognition due to its useful invariance properties such as translation, scale, and rotational invariance [21, 27]. Such features capture global information about the image and do not require closed boundaries as boundarybased methods such as Fourier descriptors [27].
The formation of polar coordinates of the pseudoZernike moments for f(x, y) can be obtained by projecting f(x, y) onto orthogonal pseudoZernike polynomials R _{ e,m }(ρ)e ^{ieθ}, by the integral [28].
where \( \rho =\sqrt{x^2+{y}^2} \) represents the distance from the origin to a point in the x − y plane, and \( \theta = \arctan \frac{y}{x} \) is a counterclockwise angular displacement in radians from the positive x axis. R _{ e,m }(ρ) are the radial polynomials expressed as
where e = 0, 1, 2,...., ∞ is the degree of the polynomial, m represents its angular dependence, which takes on positive and negative integer values subject to e ≥ m only.
The translation invariance of pseudoZernike moments
The translation invariance [27] of the pseudoZernike moments is suitable to be applied in illustrating effects of the variation of CFs and is utilized as timefrequency spectrogram features in radar signal classification.
For the amplitude normalized spectrogram P _{ s }(k, q) of the 5 modulations, the translation invariance is done by transforming the original timefrequency spectrogram P _{ s }(k, q) into another one which is \( {P}_s\left( k+\overline{k}, q+\overline{q}\right) \), where \( \overline{k} \) and \( \overline{q} \) are the centroid location of P _{ s }(k, q) computed from
where m _{00} is the zero order moment defined as \( {m}_{00}={\displaystyle \sum_k{\displaystyle \sum_q{P}_s\left( k, q\right)}} \), m _{01} and m _{10} are first order moments, given by \( {m}_{10}={\displaystyle \sum_k{\displaystyle \sum_q k{P}_s\left( k, q\right)}} \) and \( {m}_{01}={\displaystyle \sum_k{\displaystyle \sum_q q{P}_s\left( k, q\right)}}. \)
In other words, the origin is moved to centroid before moment comoutation.
In the present work here, a parameter δ is presented to illustrate the translation invariance of the pseudoZernike moments is suitable to be applied in signal recognition when CFs change.
As Fig. 3 shows, the maximum variations of δ in ψ _{2,0} for MP and LFM signals are 2.31 × 10^{− 6} and 2 × 10^{− 6}, and the maximum variations of δ in ψ _{4,2} for MP and LFM signals are 3.5 × 10^{− 6} and 3.48 × 10^{− 6}, all the values are very small. Consequently, the features are nearly invariant to CFs changing and are feasible for signal classification with the variation in the signal CFs.
PseudoZernike moments feature selection
The overlap measure indicates the degree of overlapping between two clusters, which can be quantified by computing an intercluster overlap [29]. A definition of the overlap rate(OLR) was proposed in [30], which is utilized as representative of the degree of overlap between the given two clusters C _{ i } and C _{ j }. The OLR is determined by the ratio of the number of the overlap points to that the number of small cluster’s points.
where N _{ Over_Region } represents the number of the overlap points, N _{min} is the minimum value of N _{ i } and N _{ j }, which stands for the number of points in each cluster separately. The OLR(C _{ i }, C _{ j }) varies from 0 to1, the closer the OLR(C _{ i }, C _{ j }) is to 0, the better the cluster separation is. Conversely, the closer the OLR(C _{ i }, C _{ j }) is to 1, the two clusters become more strongly overlapped.
In the following, three pseudoZernike moment features based on the average value of OLR(OLR′)are proposed for signal recognition. Here the signal data projected onto the 2D/4D feature space is obtained by testing all features of the pseudoZernike moments ranging from order 1 to order 6.
The algorithm of the moments feature selection for distinguishing LFM with the timebandwidth between 8 and 500 in the case of SNR varying from 5 dB to 5 dB from the rest of signals is summarized as follows:
Algorithm moments feature selection for LFM signal distinction  
8 ≤ uτ
^{2} ≤ 500 Input: s{ MP, BPSK, LFM, 2FSK, 4FSK} 1. Repeat for L = 1, 2,...50 (update the simulation times) 2. Update \( {P}_{s_i, d, u{\tau}^2}^{(L)}\left( k+\overline{k}, q+\overline{q}\right) \) by using Eq. (6) for each d and uτ ^{2} 3. Update \( {\psi_{e, m}}^{(L)}\left({P}_{s_i, d, u{\tau}^2}\right) \) by using Eqs. (13) and (14) for each e 4. if uτ ^{2} > = 8 and uτ ^{2} < = 40 5. Update the \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)}\left( imag\left({\psi}_{e, m}\right)\right) \) by using Eq. (17) and \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)}\left( imag\left({\psi}_{e, m}\right)\right)=\frac{OL{R_{d, u{\tau}^2}}^{(L)}\left( imag\left({\psi}_{e, m}\right)\right)}{4} \) for each e and d 6. Jointly update the minimum of \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)} \) and the corresponding imag ^{(L)}(ψ _{ e,m }) 7. else update the \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)}\left( real\left({\psi}_{e, m}\right)\right) \) by using Eq. (17) and \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)}\left( real\left({\psi}_{e, m}\right)\right)=\frac{OL{R_{d, u{\tau}^2}}^{(L)}\left( real\left({\psi}_{e, m}\right)\right)}{4} \) for each e and d 8. Jointly update the minimum of \( O L{R_{d, u{\tau}^2}^{\hbox{'}}}^{(L)} \) and the corresponding real ^{(L)}(ψ _{ e,m }) 9. end if 10. Until s, e, d, uτ ^{2} do not satisfy the variation range given. 11. Output: {imag(ψ _{ e,m }), real(ψ _{ e,m })} 
step 4: Combining the advantages of imag(ψ _{3,3}) and real(ψ _{3,3}) to discriminate LFM signals with large variation range in the timebandwidth product from other modulations.
The ψ _{3,3} is computed as
For other signals classification, we tested all combinations of two features ranging from order 1 to order 6 and measured the \( O L{R}^{\prime }=\frac{{\displaystyle \sum O L R}}{6} \) which is defined as the average value of OLR between different classes taken in the 4D feature space and find the minimum. Following the foresaid algorithm, the 8th order moments of index 5 versus index27 for pseudoZernike moments referring to ψ _{2,0} and ψ _{5,1} are experimentally selected as features, which specifically suitable for signal classification with the exception of LFM.
The ψ _{2,0} and ψ _{5,1} are given by
Recognition algorithm
Steps of the proposed algorithm
The proposed modulation signal recognition algorithm is shown in Fig. 4.
The starting point are the modulation signals \( \tilde{s}(n), n=0,1,\dots, N1 \) to which Gaussian white noise is added. And the following steps of classifying various modulation types of signals are shown as follows:
Step 1: STRFT spectrogram Computation.
Step 1.1 Computing the amplitude normalized STRFT spectrogram \( {P}_{\tilde{s}}\left( k, q\right) \) of 5 modulations mentionedabove.
Step 1.2 Computing the centroid moved amplitude normalized STRFT spectrogram \( {P}_{\tilde{s}}\left( k+\overline{k}, q+\overline{q}\right) \).
Step 2: LFM signal classification.
Measuring imag(ψ _{3,3}) and real(ψ _{3,3}) respectively. If imag(ψ _{3,3}) > th _{ LFM_1} or real(ψ _{3,3}) < th _{ LFM_2}, the signal is regarded as LFM, else go to step 3.
Step 3: Other signals classification.
Step 3.1 PseudoZernike moments ψ _{2,0} and ψ _{5,1} computation.
Step 3.2 Constructing the 2D feature space by using ψ _{2,0} and ψ _{5,1} and determining the optimal distribution range of the spectrogram features of different modulations from the feature space.
Step 4: Use a Knearest neighbour(KNN)classifier to assign each element to a class for the input radar signals, to perform the classification procedure.
The thresholds for LFM signals recognition
The thresholds th _{ LFM_1} and th _{ LFM_2} are utilized to distinguish LFM with the timebandwidth product between 8 and 40 and to distinguish LFM with the timebandwidth product between 41 and 500 from other signals. They could be obtained by the iterative thresholding algorithm [20] and lots of simulations.
As in Fig. 5a, the minimum of the average value of imag(ψ _{3,3}) for LFM signal class is obtained when uτ ^{2} = 40 at SNR = − 5dB which is close to 0.40 × 10^{− 4} and for the rest of other signal classes the maximum of the average value of imag(ψ _{3,3}) obtained at SNR = 5dB is close to 0.21 × 10^{− 4} in general as shown in Fig. 5b. Finally, we set \( t{h}_{LFM\_1}=\frac{0.40\times {10}^{4}+0.21\times {10}^{4}}{2}=0.31\times {10}^{4} \) as the optimal threshold for LFM classification. Meanwhile, as in Fig. 5c, the average value of real(ψ _{3,3}) for LFM signal class obtains the maximum at uτ ^{2} = 41 for SNR = 5 dB and the maximum is close to − 0.73 × 10^{− 4}, and for other signal classes, the minimum of the average values of real(ψ _{3,3}) is close to − 0.50 × 10^{− 4} obtained at SNR = − 5dB from Fig. 5d. Thus the threshold th _{ LFM_2} can be set to \( \frac{0.73\times {10}^{4}+0.50\times {10}^{4}}{2}=0.62\times {10}^{4} \) to guarantee the correct classification of the LFM signals with the timebandwidth product between 41 and 500.
Results and discussion
Choice of the modulation signal parameters for the Clustering
The parameters used for the clustering are shown in Table 2. CR, PW and HS stand for code rate, pulsewidth and frequency hop size, respectively. Meanwhile, for BPSK, we use 5 bit Barker codes, the 2FSK and 4FSK are encoded by deterministic codes in order to lower the effect of deficiency of some codes. Codes are defined as [0 1 0 1 0] for 2FSK and [0 1 2 3 0]for 4FSK. And the length of Rectangular window is set to be 400. In addition, the SNR values from 5 to 5 dB for most conditions.
Choice of the modulation signal parameters for test
In order to verify that the proposed method can achieve better performance than the algorithm based STFT, we did the following simulations. The code parameters for BPSK、2FSK and 4FSK are same as the parameters set in 6.1 for both algorithms. The CFs are 800 MHz, 1000 MHz and 1600 MHz. The HSs are 60 MHz, 100 MHz and 1000 MHz. The chirp rates for LFM are 40 MHz/ μs,80 MHz/ μs,100 MHz/ μs,1200 MHz/ μs.And the length of rectangular window is set to be 400 and the SNR values from 5 to 5 dB.
Simulation results analysis
To estimate the classifier performance, 50 signals are used for the clustering, and each simulation was run 100 times, evaluating the average recognition rate (ARR).
The effects of CFs and HSs variation
Figure 6 is used to get indications how much the CFs and HSs variation affect the performance at SNR = 5 dB and SNR = 0 dB respectively. The MP、BPSK and LFM signals are simulated with CFs ranging from 800 MHz to 1600 MHz, the FSK signals are simulated with HSs ranging from 60 MHz to 1000 MHz. As expected, the variation in CFs and HSs would not affect the performance much. And the property makes the pseudoZernike moments very suitable to spectrogram features recognization with random variation in the signal parameters: CFs and HSs.
The performance of the proposed algorithm
Figure 7 reports the scatter plots related to ψ _{2,0} and ψ _{5,1} for pseudoZernike for all the data of the 4 types modulation signals for different SNR. Both the STFT and STRFT based algorithms have been considered. The scatter plots shown in Fig. 7a and c demonstrate that for SNR = 5 dB, the extraction of the pseudoZernike moments from STRFT give a certain degree of separation within the class. As shown in Fig. 7b and d, for SNR = 5 dB the 4 classes could be considered to be classified more accurate by the proposed algorithms in comparison to the STFT based algorithm. Consequently, in the presence of noise, the proposed algorithm performs well especially for MP and BPSK signals recognition.
These behaviors are also confirmed by the results illustrated in Fig. 8, where the ARR is plotted versus different SNRs ranging from 5 dB to 5 dB both for STFT, STRFT based algorithms and the algorithm presented in [9]. Thereinto, for STFT and STRFT algorithms, the modulations used to obtain the ARR are MP, LFM, BPSK, 2FSK and 4FSK signals satisfying the signal parameters for test discussed in the paper. And for [9], the modulations used to obtain the ARR are the signals of their own choosing. It is obvious that, an increment in the SNR leads to a higher performance. Obviously, in the case of SNR = 5 dB, the proposed algorithm reaches a ARR of 90%, while the ARR of STFT algorithm reaches 70%,and the STRFT based algorithms assure a higher level of ARR than the STFT counterpart. Consequently, as conclusion to these analyses, it can be claimed that the performance of our algorithm based on the combination of the STRFT and the pseudoZernike moments is preferred to STFT based algorithm. Meanwhile, comparison of the work to the techniques presented in [9] shows that the approach proposed in this paper has better robustness against SNR variation.
Conclusions
In this paper, we have presented a new method for intrapulse modulation recognition under low SNR environment. In this method, the STRFT spectrograms for 5 modulation schemes are firstly calculated. Then the pseudoZernike moments are applied to the STRFT spectrogram to uniquely discriminate the spectrogram features for different modulations when parameters change. Simulation results demonstrate a robust recognition performance over a wide range of SNRs, CFs, HSs, and timebandwidth product. Meanwhile, based on the simulation results analysis, our method is better comparison of the work to the STFT based technique and the technique presented in [9].
However, our work only on few modulation schemes, a discussion on the technique applied to other modulation schemes, such as NLFM, PWM, PPM will be developed.
Abbreviations
 2D:

Twodimensional
 2FSK:

Binary frequency shift keying
 4FSK:

4ary frequency shift keying
 ASK:

Amplitude shift keying
 ARR:

Average recognition rate
 BPSK:

Binary phase shift keying
 CF:

Carrier frequency
 CR:

Code rate
 DFT:

Discrete Fourier Transform
 FB:

Feature based
 FSK:

Frequency shift keying
 GCD:

Greatest common divisor
 HS:

Hop size
 HT:

Hough transform
 KNN:

Knearest neighbour
 LFM:

Linear frequency modulation
 MP:

Monopulse
 MPSK:

Mary phase shift keying
 OLR:

Overlap rate
 PW:

Pulsewidth
 PSK:

Phase shift keying
 PSR:

Probability of successful recognition
 QPSK:

Quadrature phase shift keying
 RS:

Ramanujan sums
 RFT:

Ramanujan Fourier Transform
 STFT:

Shorttime Fourier transform
 STRFT:

ShortTime Ramanujan Fourier Transform
 SF:

Sampling frequency
 SNR:

Signaltonoise ratio
 WVD:

Wigner–Ville distribution
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Acknowledgements
The authors would like to thank the reviewers for their time and effort spent in carefully reviewing the manuscript, and for their valuable comments that have greatly contributed to the enhancement of article’s quality.
Funding
This work is partially supported by Tianjin Research Program Application Foundation and Advanced Technology (15JCQNJC01100).
Authors’ contributions
XM conceived the approach; XM, DL and YS designed the experiments; DL and YS performed the experiments. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Ma, X., Liu, D. & Shan, Y. Intrapulse modulation recognition using shorttime ramanujan Fourier transform spectrogram. EURASIP J. Adv. Signal Process. 2017, 42 (2017). https://doi.org/10.1186/s1363401704699
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Keywords
 Intrapulse modulation recognition
 PseudoZernike moments
 ShortTime Ramanujan Fourier Transform
 Probability of successful recognition