Intra-Pulse Modulation Recognition for Fractional Bandlimited Signals Based on a Modified MWC-Based Digital Receiver

In this paper, we present a modified modulated wideband converter (MWC)-based digital receiver for fractional bandlimited signals, and further propose an intra-pulse modulation recognition method in discrete time fractional Fourier domain (DTFrFD) for the intercepted signals. The proposed digital receiver can move the cross-channel signal to the baseband and since the nonzero part of its available spectrum is narrower in DTFrFD than in discrete time Fourier domain (DTFD), so a better separation can be achieved by the proposed digital receiver than by original MWC- based receiver. Then, with the data acquired from the digital receiver, we propose an intra-pulse modulation recognition method based on the optimal transformation order and the spectral kurtosis (SK) in DTFrFD for the six types of fractional bandlimited signals. In this algorithm, the optimal transformation order is tested to distinguish the encoded signals from the non-encoded signals, and then the SK is tested to determine the intra-pulse modulation type specially. The computational complexity of the proposed method is much lower than search-based methods. Meanwhile, since the SK of Gaussian signals is zero in DTFrFD, the proposed method shows better robustness than the original MWC discrete compressive sampling structure against SNR variation. Simulation results confirm the obtained analytical results.


INDEX TERMS
Fractional bandlimited signals [1]- [4] such as linear frequency modulation (LFM) are widely used in the radar and ultra-wideband communication because of its ease in implementation, its versatility, and its suitability for VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ anti-jamming [5]. And it is an important task to intercept and detect the fractional bandlimited signals by utilizing the wideband digital receiver. However, under the continued evolution of complex electromagnetic environment, conventional wideband digital receivers based on uniform channelization design have encountered increasing bottleneck problems, such as increasingly complex structures, larger numbers of samples and the difficulty of cross-channel signal processing with the increased receiver bandwidth and channelization number [6], [7].
Recently developed compressed sensing (CS) theories [8], [9] have resulted in new ideas for digital receiver [10]- [14]. Mishali and Eldar proposed a novel compressed sampling architecture for multiband signals called the modulated wideband converter (MWC) [15], and several MWC-based theories have been proposed to realize receiver structure [16]- [21]. Reference [16] design a hardware system of wideband digital reconnaissance receiver based on the MWC theory, and the receiver system has the ability of completing multi-signals acquisition, reconstruction and intra-pulse analysis simultaneously. Reference [17] propose a receiver to sense frequency spectrum which is composed of a uniform linear array (ULA), where each sensor contains an analog front-end equivalent to one channel of the MWC. Reference [11] present a MWC-based receiver structure which can be used to determine the location of ground-based emitters from flying detectors with the goal of minimising the required digitisation rate. To solve the cross-channel signal problem, [18] extend MWC theory to discrete-time Fourier domain (DTFD) in order to construct a new wideband digital receiver in which the cross-channel signals can be converted to the baseband data which contain the full information of the original signals. And based on [18], [19] proposed a parameter estimation algorithm for wideband LFM signal. And [20] propose a joint carrier frequency and direction of arrival (DOA) estimation method using a ULA-based MWC discrete compressed sampling structure. And a modified ULA based MWC discrete CS receiver where another branch is added in each antenna of the original ULA based system in order to estimate carrier frequency and DOA jointly is proposed in [21].
Despite the merits, there are some difficult issues when using the original MWC-based receivers [18]- [21] proposed above for fractional bandlimited signals. It is well known that the MWC-based receivers work in DTFD, and they are constructed based on the sparse prior knowledge and shift invariant property of signal in DTFD. Although Fourier transform (FT) has a good effect on the stationary signals, but not on the non-stationary signals, such as LFM signal [22], [23]. And the shift invariant property of the discrete time Fourier transform (DTFT) is not applicable for the discrete time fractional Fourier transform (DTFrFT). As a result, if the MWCbased receivers are directly applied to intercept the fractional bandlimited signals, the performance will deteriorate with the decrease of signal to noise ratio (SNR). (see Section V for a comparison of the performance) Fractional Fourier transform (FrFT) can be interpreted as a signal decomposition in terms of a LFM basis as its kernel is constituted by LFM functions, it is the natural domain of nonstationary signals. Most research have focused on the MWC theorem expansions for the wideband digital receiver in DTFD from different perspectives [11], [16], [17], [18]- [21], but few have focused on the receiver in discrete time fractional Fourier domain (DTFrFD). It is necessary to generalize a MWC-based signal receiver theorem in DTFrFD.
Thus, in this paper, we present a modified MWC-based digital receiver for fractional bandlimited signals by extending the multichannel compressed sampling structure in [27], and further suggest an intra-pulse modulation recognition method in DTFrFD for the fractional bandlimited signals which is intercepted by the new receiver. The proposed digital receiver consists of two mixing steps. At the first mixing, we multiply exp j 2 (nT ) 2 cot α with the original signal to establish the relationship between DTFrFT and DTFT. and then the spectrum of the mixed product is aliased by the random modulation at the second step. The production of the mixing signal is filtered by a low-pass filter, and then it is sampled under a low-sampling rate. The cross-channel signals can be mixed to the baseband and get better separation than by original MWC-based receiver.
Moreover, it is an important task for the proposed wideband digital receiver to recognize the intercepted fractional bandlimited signals, which could be used to choose the optimal algorithms to estimate the parameters of the signal. In practice, the signal waveform usually has lower signal power, which makes it difficult to classify the waveform directly. The conventional feature extraction and classication methods [28]- [38] work in Fourier domain and often perform unsatisfactorily because of its low concentration. Inspired by the property that the non-stationary signals enjoy both high concentration and absence of cross terms in fractional Fourier domain(FrFD), we propose a recognition method in DTFrFD based on high-order cumulants to realize a classification of six types of signals including normal signal (NS), binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), LFM, nonlinear frequency modulation (NLFM), and binary frequency shift keying (BFSK). First, we introduce the normalized second-order central moment (NSOCM) calculation method to directly obtain the optimal order of the received signal in DTFrFD. The strength and advantage of the NSOCM method lies in its non-ergodic search mechanism, which can improve the computing efficiency to a great extent. The obtained optimal order based on NSOCM is able to effectively distinguish the encoded signals(BPSK,QPSK, and BFSK) and non-encoded signals (LFM and NLFM). Second, we use fourth central moment to calculate the spectral kurtosis(SK) in DTFrFD which is used to complete the specific intra-pulse modulation for the received signal.
The main contributions of this paper are summarized as follows. First, we propose a modified MWC-based digital receiver architecture that can not only intercept the fractional bandlimited signals effectively in a low SNR, but also 85068 VOLUME 8, 2020 obtain better separation for cross-channel signals than by original MWC-based receiver. Second, we propose an intrapulse modulation recognition method based on the optimal transformation order and the SK for the fractional bandlimited signals which is intercepted by the new receiver. Our algorithm bears a relatively low complexity compared with search-based methods and its detection performance and recognition performance are higher than existing algorithms.
The remainder of this paper is organized as follows: In Section II, the basic preliminaries is introduced. The MWC compressed sampling receiver for fractional bandlimited signals is proposed in Section III. In Section IV, the intra-pulse modulation recognition method for fractional bandlimited signals is presented. In Section V, the detection performance and recognition performance are simulated and analyzed. Section VI is the conclusion.

A. SIMPLIFIED FRACTIONAL FOURIER TRANSFORM(SFrFT)
The FrFT is an extension of the ordinary FT, which essentially allows the signal in the time-frequency domain to be projected onto a line of arbitrary angle [22]. SFrFT [39] has the same effect as FrFT of order p for filter design, but it is simpler to implement digitally than the original FrFT. And the first type of pth-order SFRFT is defined as [39]: where p is the transformation order, α is the rotation angle, α = pπ 2 . The inverse SFrFT is denoted as follows: The digital simplified fractional Fourier transform (DSFrFT) is given by [39] F α (m) where y (n) = y (n t), F α (m) = F α (m u), m, n = −N , −N + 1, · · · , N , t and u are the sample spacing in temporal domain and simplified fractional Fourier domain(SFrFD) respectively. And t u = 2π/ (2N + 1). We can also write (1) in matrix form, expressed as where unitary matrix whose element O α F mn at the m row, n column has the following form: where · returns the nearest integer towards positive infinity.
With the change of α, the frequency axis of the SFrFT is located in different positions, and more abundant information about the frequency characteristics of the signal can be obtained compared to the FT.

B. FRACTIONAL BANDLIMITED SIGNALS AND ITS SPECTRAL FEATURES IN SFrFD
A fractional bandlimited signal f (t) has finite energy. The where 2u α is the fractional bandwidth of f (t). According to Parseval's theorem, the bandlimited signal can also be expressed as : (7) where u h = u 0 + u α , and u l = u 0 − u α .

III. PROPOSED MWC-BASED DIGITAL RECEIVER IN DTFrFD
In this part, we explain how the receiver work and describe the advantage of the proposed receiver over original MWC-based digital receivers [18] when processing fractional bandlimited signals.
The proposed digital receiver is designed for intercepting the fractional bandlimited signals which are mainly produced by a pulse working system. The complexity of the signal environment where the digital receiver locates is usually described with λ pulses per second, and the Poisson distribution is used to describe the rule that the radar pulses impinge on the digital receiver. The probability for I pulses impinging on the receiver in τ seconds can be expressed as Considering that the frequency interval 2 − 18GHz which the signals usually appear in practical can be divided into several subbands. So it is appropriate to assume that there are λ = 10 5 pulses per second in the surveillance frequency interval whose bandwidth is about 1GHz. We design to detect and acquire the signal pulses in a very short sampling time by using the proposed receiver. According to (8), the probability for receiving single source is 9.05%, the probability for receiving two sources is 0.45%, and the probability for receiving more than two sources is very low in 1µs. If we VOLUME 8, 2020 design 1µs as the processing time unit, there would be only one signal detected and acquired in most cases [21].
The practical signal environment where the digital receiver locates is sparse distribution. Through the above analysis, the probability for the proposed digital receiver to deal with multi-mixed signals arriving simultaneously is very low. So, we can suppose there is only one signal received by the proposed receiver in the short sampling time.

A. SYSTEM DESCRIPTION
The received signal in discrete-time domain can be expressed as is the complex additive white Gaussian noise(AWGN) with zero mean and σ 2 variance. Let the received signal x [n] be fractional bandlimited to the region (u l , u h ). And the Nyquist sampling rate is f NYQ = u h / (π sin α).
The architecture of the proposed MWC-based digital receiver is shown in Fig. 1. Take the mth channel as an example. There are two mixing steps in each branch of the proposed multi-branch receiver. In the first step, the received signal x [n] is multiplied by the signal exp (j/2) n 2 cot α , and the product is mixed with the random sign signalp m [n] in the second mixing step. The pseudo-random sequencẽ p m [n] is a periodic function with period T p , and there are M p = T p f NYQ elements per period. Then the production of the mixing signal is filtered by a low-pass filter h [n] with a cutoff frequency 1/2T s to obtain the filtered signal w m [n], and then w m [n] is sampled at sampling rate f s = 1/T s to obtain the CS data y m [k], where T s denotes the down-sampling period. It is clear that the original MWC-based digital receivers is a special case at α = π/2.

B. DISCRETE TIME SIMPLIFIED FRACTIONAL FOURIER DOMAIN (DTSFrFD) ANALYSIS
We define the mixing rate in DTSFrFD as u p = f p sin α = sin α/T p , and design u p ≥ B to avoid edge effects [17], here B is the bandwidth of the incident signal s [n] in α-th order DTSFrFD. According to the spectrum distribution of the mixing functionp m [n] · exp j 2 n 2 cot α , the coverage fractional ''frequency'' of the proposed receiver can be divided into M p sub-bands with bandwidths of u p . The interval of the baseband in DTSFrFD is U p = 0, u p . The DTFT of the mixing signal in the mth channel x [n]·p m [n]·exp j 2 n 2 cot α is: , and C α,il is the simplified fractional Fourier series coefficient of According to Eq. (9), the two-step mixing produces a scale transformation and a relative l 2π M p sin α shift in DTSFrFD. Then, the mixing product is truncated by a low-pass filter with a cutoff fractional ''frequency'' u s /2, where u s = f s sin α is the fractional sampling rate for each channel. Consider h (n) to be an ideal rectangular function in DTSFrFD and serves as a preceding anti-aliasing filter. The response of lowpass filter h (n) in DTSFrFD is: For simplicity, we assume u s = u p to truncate the baseband signal which contains the full information of the original signal. The mixed product signal and its DTFT can be denoted by: where * denotes the convolution operator, and F is the DTFT operator. Substituting Eq. (9) into Eq. (12), F (w m [n]) can be simplified as: Then, the low-pass filtered signal w m [n] is down-sampled at a rate of f s = u s csc α to obtain the baseband data y m [k], expressed as where {·} ↓M p denotes the down-sampling operation. s m [k] is the signal component of interest for the m-th branch data, η m [k] is the complex-valued AWGN component of the mth branch data with zero mean and σ 2 /M p variance, and K = u NYQ /u p is the number of the data points in one branch. Therefore, the DTFT of the mth branch data y m [k], which is bandlimited to U s = [0, u s ], can be expressed as We can rearrange (16) in matrix form, which is where Therefore, Eq.(17) can be rewritten as Furthermore, Assume the frequency of the signal x (n) only exists in an unknown hth (0 ≤ h ≤ M p − 1) sub-band in each branch of the proposed receiver. Then, the fractional spectrum information of other sub-bands in each branch can be ignored since there is very little fractional spectrum information of the signal of interest in those sub-bands, except for the hth'subband. The DTFT of the mth-branch and m+1th-branch of the CS data can be approximately expressed as It is obviously that the DTFT of the mth-branch and m + 1th-branch of the CS data have the same form, there is not distortion of the phases of the multi-branch CS data in DTS-FrFD, therefore, the multi-branch CS data can be superposed directly without phase correction in the proposed system.
An example is used to further describe the advantage of the proposed receiver over MWC-based receiver [18] when processing the LFM signal. Assume the received signal x [n] is a cross-channel LFM signal with a bandwidth of B. Both the frequency spectrum of x [n] and the fractional frequency spectrum is shown in Fig. 2, which is depicted as a rectangular band. The FT coefficients ofp m [n] and the SFrFT coefficients ofp m [n] · exp jn 2 cot α /2 are also shown in Fig. 2, which are depicted as equidistant discrete spectral red lines. The frequency information and fractional frequency information after mixing are shown in Fig. 3.
Obviously, the LFM signal is sparse in DTSFrFD rather than in DTFD, and even if the signals satisfy the condition of the MWC-based receiver [18], the maximum bandwidth of signals in DTSFrFD is considerably narrower than that in DTFD, which is helpful to eliminate the cross-channel problem. Although the classic MWC-based receiver [18] can intercept such signals that show better sparsity in the DTSFrFD such as the signals in Fig. 2, the probability of successful intercept is much lower even with more hardware resources. It is not economical to use the classic MWC-based receiver [18] to intercept the fractional bandlimited signals. Fig. 4 shows that the amplitude spectrums of the data for the intercepted LFM signal in DTSFrFD are presented with SNR = 15dB. The simulation parameters are shown in Section V-B. From Fig. 4, the baseband data contains the VOLUME 8, 2020  full spectrum information of the received signal, so we can directly process the data to acquire the parameter estimation of the original signal.

C. INTERCEPTION PERFORMANCE ANALYSIS
In this section, we formulate the interception performance of the proposed receiver and derive the interception criterion. Eq.(18) can be rewritten as . . . where According to Eq. (19), the interception model can be expressed as where Assuming that H 0 hypothesis is true, we randomly select one branch of MWC based on Eq. (19),i.e.
where P i denotes the ith row vector of periodic pseudorandom sequence matrix P. Substituting Eq.(22) into Eq. (21) yields where (•) H denotes the conjugate transpose. Let as η i (n) is modulated by the pseudo-random sequence and low-pass filtered, the elements of the vector η i (n) are uncorrelated with each other. As a result, Eq.(24) can be expressed as Since the distribution of the band-limited AWGN η i (n) is Then the distribution of E i is In practice, we can use a known probability of false alarm P f to calculate each branch threshold γ i . Since Therefore, the threshold of each branch γ i can be expressed as And the threshold for the all branches is where M is the number of branches as shown in Fig.1. M p is the number of elements per period for the pseudo-random sequencep m [n].

IV. PROPOSED INTRA-PULSE MODULATION RECOGNITION METHOD AND PROCEDURES A. INTERCEPTED SIGNAL MODEL
The received discrete time signal is composed of a modulated signal and noise. Its model is given by where x (n) and s (n) are received signal and modulated signal, respectively. η (n) is assumed to be AWGN. The modulated signal s (n) is given by where A is the amplitude. f c and φ 0 are the carrier frequency and the initial phase, respectively. f NYQ is the Nyquist sampling rate. φ (n) is the phase function, which determines the modulation type of the signal. For simplicity and without loss of generality, we assume that A is an invariant constant.

VOLUME 8, 2020
The different pulse compression waveforms considered in this paper are: NS, LFM, NLFM, BPSK, QPSK, BFSK. And φ [n] for different modulation types are expressed as follows: where the phase coding function C BPSK [n] alternates between 0 and 1.
where a 0 , a 1 , a 2 , and a 3 are the frequency modulation coefficients.
where f n is the value of the hopping frequency.

B. METHOD DESCRIPTION
Our proposal consists of two steps. In the first step, we propose using the NSOCM to acquire the optimal transformation order of CS data in the DTSFrFD to distinguish the encoded signals and non-encoded signals roughly. The theoretical analysis in the next subsection will indicate that for the encoded signals(PSK and FSK) and NS signal, the optimal transformation order where X p (u) reaches its extremum values is p = 1 while the values of the optimal order for the non-encoded signals (LFM and NLFM) are concentrated far away from 1. Therefore, the optimal order can be used to classify these signals into two categories. And in the second step, the SK of the CS data in DTSFrFD are used to complete the specific intra-pulse modulation for the received signal.

C. THE OPTIMAL TRANSFORMATION ORDER ANALYSIS FOR THE 6 TYPES OF FRACTIONAL BANDLIMITED SIGNALS
In the fractional Fourier domain (FrFD), support of signals change associated with the transform order and there exists an optimum transform order in which the energy of signals are maximally concentrated [4], [5]. When an signal is transformed by FrFT at its optimum order, transform kernel acts as a matched filter. Therefore, the optimal transformation order has the ability to maximize the absolute amplitude. The optimal transformation order p opt corresponding to maximum magnitude obtained from the FrFT is given by u) is the FrFT of the signals in the p-order FrFD.
In this section, we calculate and compare the optimal order of 6 types of signals in DTSFrFD.
For the NS x NS [n], its FT is |F| = 2π sin c (f − f c ) /f NYQ as its spectrum in DTFD is an impulse, its optimal transformation order is p = 1 obviously. For the LFM signal x LFM [n] , when the fractional rotation angle α = arccot −K f [40], the amplitude spectrum of x LFM [n] in SFrFD is , for the received LFM signal, its DTSFrFT at the fractional rotation angle α = arccot −K f is an impulse. Therefore, the fractional transformation order p = 2α/π = arccot −K f × 2/π is the optimal transformation order in which the energy of LFM signals are most concentrated.
For the NLFM signal x NLFM [n], the optimal transformation order of the NLFM signal is [41] p = 1 + arctan (2πa 3 (n 1 + n 2 ) + 2π a 2 ) 2 π where n 1 and n 2 are the initial time and end time, respectively. For the BFSK signal x BFSK [n], The SFrFT of x BFSK [n] can be calculated by When the fractional order p = 1, the amplitude spectrum of x BFSK [n] in DTSFrFD is |F α | = A (j2π ) 1/2 t sin c π m u − f n /f NYQ Therefore, the optimal transformation order for the BFSK signal is p = 1. Similarly, the optimal transformation order for the PSK signal (BPSK or QPSK) is also p = 1.
Increase the fractional order p from 0 to 2 with a step of 0.015, calculate the discrete time simplified fractional Fourier transform (DTSFrFT) X p (u) of the six types of signals above for every p, and extract the max amplitude X p|max (u) in every p-order DTSFrFD. Traversing all the fractional order p, and we can obtain the evolutions of the max normalized DTSFrFT amplitude X p|max (u) of 6 types of modulation signals with respect to the fractional order p as shown in Fig.5 (simulation parameters are shown in Section V).
As shown in Figure 5, the values of the optimal order for the encoded signals in the DTSFrFD are concentrated around 1, while for the non-encoded signals, the fractional spectrum X p (u) has a maximum around fractional order of p = 1.9. Therefore, the optimal order can be used to distinguish the encoded signals from the non-encoded signals.

D. THE ALGORITHM FLOW OF THE OPTIMAL TRANSFORMATION ORDER ANALYSIS BASED ON THE NSOCM
The optimal transformation order in DTSFrFD will directly affect the recognition results, hence the method of searching the optimal order is important. The traditional method to get the optimal order in the DTSFrFD is peak sweeping method [42]- [44], which is an easy method to realize. And obviously, the search-based algorithms above require numerous extra calculations and have the contradiction between estimation performance and complexity.
In this section, we introduce the NSOCM calculation method [45] to directly obtain the optimal transformation order in DTSFrFD. Compared with the search-based algorithms, the NSOCM approach has higher computational efficiency because of its non-ergodic search mechanism. According to [45], the optimal order p opt is normally given by where TBP X p (u) is the time-bandwidth product for X p (u), And the NSOCM p α of X α (u) is defined by is the normalized first-order origin moment of X α (u), and is the normalized second-order origin moment of X α (u). The NSOCM p α , p α+1 represent the timewidth and frequency width of X α (u), respectively. Hence, Eq. (33) becomes The NSOCM product is given by where µ 0 = (ω 0 + ω 1 ) /2+m 0 m 1 −ω 0.5 is the mixed secondorder moment. Setting the first derivative of p α · p α+1 with respect to the order equal zero, we obtain For this case where α is equal to the extreme point α e , the product p α · p α+1 reaches the extremum values. This result demonstrates that when p satisfies Eq. (34) as follows, the product p α · p α+1 reaches its minimum.
Based on the theoretical analysis above, The calculation process of the optimal transformation order can be summarized into the specific procedures as follows: (1) Take the 0.5-th and 1-th order DTSFrFT of signal x [n] to obtain X 0.5 , X 1 , (2) Calculate the normalized first-order origin moments m 0 and m 1 , the normalized second-order origin moments ω 0 ,ω 0.5 and ω 1 , the mixed second-order moment µ 0 , and NSOCM p 0 and p 1 in accordance with the definition, (3) Obtain the optimal order p opt of p by using Eq. (34) in the range of [0,1].

E. THE SK ANALYSIS OF THE CS DATA IN DTSFrFD
The SK is a statistical tool which can indicate the presence of series of transients and their locations in the Fourier domain [46]. SK measures deviation from Gaussian distribution, the distribution has sharp peaks when the distribution has large kurtosis, conversely the distribution is flat. References [45], [47] extend the SK to fractional Fourier domain, and prove that the signal's SK in the FrFD has the same optimal order with signal's FrFT. And according to [48], the SK of Gaussian signals is zero in FrFD which helps suppressing Gaussian noise.
This subsection propose a DTSFrFT based on SK recognition method to complete the specific intra-pulse modulation  for the received signal. The SK of the m th CS data in the optimal p th-order DTSFrFD is given by where X p (m) is the optimal pth-order DTSFrFT, m is the variable in the pth-order DTSFrFD,E (·) stands for the expected value operator, * denotes the transposed conjugate operator. Table 1 gives the SK of CS data for LFM and NLFM signal using (35) with respect to different SNRs. It is obvious that the LFM signal's SK is far greater than the SK of the NLFM signal, therefore, the SK in DTSFrFD can be used to differentiate LFM signal from NLFM signal. Moreover, As seen from these results of Table 1, when the SNR < 0dB, the SK still has a large value. That is because the SK of Gaussian signals is zero in FrFD which helps suppressing Gaussian noise. Table 2 shows the SK of CS data for the NS signal and the encoded signals through simulations (simulation parameters are shown in Section V) using (35) when the SNR varies from −6dB to 24dB with a step of 3dB. It is obvious that the SK of the NS signal is far greater than the SK of other signals, therefore, the SK in DTSFrFD can be used to recognize NS signal. Similarly, since the SK of the BFSK signal is far greater than the SK of QPSK and BPSK signals, the BFSK signal can be differentiated from the QPSK and BPSK signals. In the same way, QPSK and BPSK signals can be distinguished by differences in SK values.

F. RECOGNITION PROCEDURES
The procedures of the proposed intra-pulse modulation recognition for fractional bandlimited signals based on the proposed digital receiver are shown in Fig. 6 which can be described as follows: 1) Rough recognition module for the encoded signals and the non-encoded signals. Calculate the optimal order of the input CS data by using the NSOCM calculation method. Then, compare the optimal order with the designed threshold p TH . If the optimal order is greater than p TH , the received signal can be classified as the encoded signals. Otherwise, the received signal is classified as non-encoded signals.
2) Specific recognition module for the non-encoded signals. Calculate the SK Q of CS data. Then, compare Q with the designed Q TH , If Q is more than Q TH , the received signal can be classified as LFM signal. Otherwise, the received signal is classified as NLFM signal.
3) Specific recognition module for the encoded signals. Calculate the SK Q of CS data. Then, compare Q with the designed Q TH 1 , If Q is more than Q TH 1 , the received signal can be classified as NS signal. Otherwise, Compare Q with the designed Q TH 2 , if Q is more than Q TH 2 , the received signal can be classified as BFSK signal. Finally, Compare Q with the designed Q TH 3 , If Q is more than Q TH 3 , the received signal can be classified as BPSK signal. Otherwise, the received signal is QPSK signal.

V. NUMERICAL SIMULATION A. DESIGN OF SIMULATION
Examples of numerical simulation are presented to evaluate the detection performance of the proposed modified MWC compressed sampling receiver and the intra-pulse modulation recognition performance of the proposed recognition method based on the new receiver.
The parameters of the proposed receiver are configured as follows. The Nyquist sampling rate of the received signal is f NYQ = u NYQ csc α = 2.2GHz, where u NYQ is the Nyquist sampling rate in DTSFrFD. α is the fractional order which varies from −0.80 × 10 −9 to −0.62 × 10 −9 with a step of 0.01 × 10 −9 . The pseudo-random sequence is generated by a Bernoulli random binary±1 sequence with M p = 400, so the bandwidth of the baseband in DTSFrFD is u p = u NYQ /M p . An ideal low-pass filter with cutoff frequency u p /2 in DTS-FrFD is adopted and the down-sampling rate is u s = u p . The total CS sampling rate is M p u s . The sampling time is T = 10µs. The number of the original sampling data is N = 22000. The number of the data for each branch is K = N /M p = 55. The values of the parameters are listed in Table 3.

B. THE DETECTION PERFORMANCE OF THE PROPOSED MODIFIED MWC COMPRESSED SAMPLING RECEIVER
The detection performance of the proposed system is evaluated by the detection probability, and we demonstrate the simulations of the detection probability of the proposed energy  detection method ( section III-C) based on the proposed receiver. The simulations show the detection probability for two typical fractional bandlimited signals: frequency modulation signals and phase modulation signals with different SNRs. For each type of the signals, the simulations are demonstrated in two aspects,including the tradeoff between detection probability and SNR and the balance between detection probability and the number of sampling channels. The original MWC discrete compressive sampling structure [18] is given for comparison. Each simulation has 300 trials to ensure statistically stable results. We use the LFM signal and the BPSK signal as the test subjects, which are typical frequency modulation signals and phase modulation signals. We simulate the system on the test subjects contaminated by AWGN. The original LFM signal in discrete-time domain is denoted by x LFM (n). The noisy signal is x LFM (n) + ω (n), where ω (n) is the AWGN. The SNR is defined by 10 log ||x|| 2 /||ω|| . x LFM (n) is given by the following: x LFM (n) = Eexp j2π K lfm n 2 /f 2 NYQ cos 2π f l n/f NYQ where E is the amplitude of the signal which could be random or fixed. K lfm = 0.200 × 10 9 Hz/s is the signal modulation rate. The signal duration time is T = 10µs. So the bandwidth of x (n) is B = K lfm · T = 10MHz, f NYQ VOLUME 8, 2020 is the Nyquist sampling rate of the signal. f l = 1GHz is the initial frequency. The signal is both frequency bandlimited and fractional bandlimited with different bandwidths in the observation interval.
The original BPSK signal in discrete-time domain can be depicted as where A is the amplitude of the signal, f b = 1GHz is the frequency carrier. C BPSK (n)is the phase coding function which alternates between 0 and 1.
Assume that the false alarm probability P f = 0.01, the proposed detection threshold can be calculated by Eq.(32).     Fig.7, the detection performance from the proposed receiver is preferable than the original method when SNR is under −5dB. And the detection probability approximates 100% when SNR is above −5dB. And it is common that the detection probability increases with increasing SNR and the number of sampling channels in both the proposed method and classic MWC discrete compressive sampling structure. Besides, when the orders α varies from −0.80 × 10 −9 to −0.62 × 10 −9 with a step of 0.01 × 10 −9 , as can be seen from Eq.(32), the value of the intercept threshold is almost unchanged. Therefore, the interception performance is hardly affected by α when α is small enough.
In Fig. 8(a)and (b), the SNR is {−3, −8, −10} dB, the number of channels varies from 4 to 60 with a step of 2. It is observed that the detection probability has the same trend as the sampling channels, increasing the sampling channels leads to an increase of the detection probability and a smaller SNR correspond to more sampling channels. Occasionally, a low SNR may lead to failure of the detection. From the results we can conclude that the proposed method has better detection performance. In this section, the proposed recognition method based on the new receiver is measured by simulation signals. The purpose is to test the computational efficiency of the optimal order p opt , the accurate rate of the identification results in different conditions.

1) CREATE SIMULATION SIGNALS
For all the six fractional bandlimited waveforms discussed above, there are different parameters that need to be set. For LFM, the initial frequency is f c = 1GHz, the bandwidth is B = 20MHz, and the modulated rate is K f = 2MHz/µs. For NLFM, the frequency modulation parameters are a 0 = 0, a 1 = 1.2 × 10 8 , a 2 = 0.7 × 10 12 , a 3 = 2.0 × 10 15 . For NS, The carrier frequency is f c = 1GHz. For BFSK, the code rate varies from f 1 = 1200MHz to f 2 = 1210MHz. For QPSK, the carrier frequency is f c = 1GHz and a Frank code of length 16 is used to modulate the phase. For BPSK, the carrier frequency is f c = 1GHz and a Barker code of length 11 is used to modulate the phase.For more details, see Table 4.
The parameters of the proposed receiver are as described in section V-A.

2) COMPUTATIONAL EFFICIENCY OF THE OPTIMAL ORDER p opt
Firstly, calculate the optimal order p opt of the CS data for the six types of signals above. Table 5 shows the optimal order p opt of the CS data for NS, BPSK, QPSK, LFM, NLFM, and BFSK signals through simulations without noise. In addition, the search-based method [43] is given for comparison with the NSOCM algorithm.
The optimal order calculated by NSOCM is very close to the search-based results, and the computation time of these methods is given in Table 6. It is obvious that the NSOCM calculation greatly reduces the computation time of searchbased method. As seen from these results, the NSOCM algorithm can give a reasonable optimal order for the DTSFrFT and has the advantage of high computational efficiency.
As seen from Table 5, the values of the optimal order for the NS, BPSK, QPSK, BFSK signals are concentrated around 1. Therefore, the optimal order can be used to distinguish these signals from LFM and NLFM signals. And the range of the order threshold p TH can be expressed as [0.97,1.02].
Secondly, according to (35), calculate the SK of the CS data in the optimal order p opt with 500 Monte Carlo experiments for each signal when the SNR varies from -8dB to 14dB with a step of 2dB. The results are given in Table 1 and  Table 2, respectively. From Table 1, the range of the recognize threshold Q TH which is used to differentiate LFM signal from NLFM signal can be set as [20,30].
Thirdly, from Table 2, the range of the recognize threshold Q TH 1 which is used to distinguished the NS signal from the others can be set as [45,55]. Similarly, the recognize threshold Q TH 2 is used to classify the BFSK signal from BPSK and QPSK signals while the range of Q TH 2 is [20,28]. And finally, the range of the threshold Q TH 3 which can recognize the BPSK signal from QPSK signal is [10,10.7].

3) EXPERIMENT WITH SNR
The experiment research shows the relationship between the ratio of successful recognition (RSR) and SNR. The RSR serves as a recognition performance measure, with a higher ratio corresponding to an improved recognition performance. The probabilities are measured by the testing data of each kind of waveforms. For each signal, 100 Monte Carlo experiments are performed to calculate the RSRs for each SNR which is increased from -8 dB to 14 dB with a step of 2 dB. Figure 9. plots the RSR as a function of the SNR, and the overall probabilities are also calculated. The original MWC discrete compressive sampling structure [18] is given for comparison. It is clear that the RSRs for six types of radar signals have the same trend as the SNR. And when SNR ≥ 8 dB, the RSRs of the original MWC discrete compressive sampling structure [18] are 100%, which rapidly decreases when SNR ≤ 8 dB. At SNR > 8 dB, RSR of the proposed system approaches 100%, with the decrease of SNR, the successful ratio can still be kept at a high level. From Fig.9, We can also see that the RSRs of LFM and NLFM are greater than 90% when SNR is above −3dB, the RSRs of BPSK and QPSK can reach 90% when SNR is above −4dB, the RSRs of BFSK are greater than 90% when SNR is above 1dB, and the RSRs of NS can reach 90% when SNR is above −5dB. And all these results above are the under the condition that only one branch CS data of the proposed compressed sampling receiver are processed. Meanwhile, Fig.9 also shows the results under the condition that ten branches CS data of the proposed compressed sampling receiver are analyzed. And it is clear that the RSRs of the latter are significantly higher than the former at the same SNR, that is because the superposed multi-branch CS data can increase the OSNR [18]. And from the analysis in sec.III-B, since there is not distortion of the phases of the multi-branch CS data, therefore, the multi-branch CS data can be superposed directly without phase correction in the proposed system. Fig.9 validates the proposed recognition method based on superposed CS data outperforms the recognition system based on one branch of CS data under low SNRs. And it is obviously that our system performs better on the classification of each kind of waveform than [18], especially at low SNR and has better robustness against SNR variation.

VI. CONCLUSION
This paper introduces a MWC-based digital receiver architecture in DTSFrFD to intercept fractional bandlimited signals and propose an intra-pulse modulation recognition method in DTSFrFD for the fractional bandlimited signals which is intercepted by the new receiver. The original MWCbased receiver structures are confined to stationary signal and require a high SNR, the proposed structure in this paper overcomes the confine by taking advantage of the properties of FrFT, and the original MWC-based receiver structure is shown to be a special case of it. Meanwhile, an intra-pulse modulation recognition method based on the optimal transformation order and the SK in DTSFrFD is presented for such architecture. This algorithm shows better robustness than the original MWC discrete compressive sampling structure [18] against SNR variation, and it bears a relatively low complexity comparing with the search-based method.
HUALI WANG (Member, IEEE) received the Ph.D. degree in electronic engineering from the Nanjing University of Science and Technology, China, in 1997. He is currently a Professor with the College of Communication Engineering, The Army Engineering University of PLA, Nanjing, China. His research interests include satellite communication and signal processing.
HAICHAO LUO received the Ph.D. degree from the College of Environment and Planning, Henan University, China, in 2017. He is currently a Professor with Henan Normal University, Xinxiang, China. His research interests include data processing and visualization.