Feature-Based Digital Modulation Recognition Using Compressive Sampling

Compressive sensing theory can be applied to reconstruct the signal with far fewer measurements than what is usually considered necessary, while in many scenarios, such as spectrum detection and modulation recognition, we only expect to acquire useful characteristics rather than the original signals, where selecting the feature with sparsity becomes the main challenge. With the aim of digital modulation recognition, the paper mainly constructs two features which can be recovered directly from compressive samples. The two features are the spectrum of received data and its nonlinear transformation and the compositional feature of multiple high-ordermoments of the received data; both of them have desired sparsity required for reconstruction from subsamples. Recognition of multiple frequency shift keying, multiple phase shift keying, and multiple quadrature amplitude modulation are considered in our paper and implemented in a unified procedure. Simulation shows that the two identification features can work effectively in the digital modulation recognition, even at a relatively low signal-to-noise ratio.


Introduction
Constantly increasing volume of data transmitted through the mobile communication networks and the needs of users to increase the data rates lead to rapid development of mobile communication systems.The future fifth generation (5G) wireless communication tends to achieve a remarkable breakthrough both in data rate and spectral efficiency [1].With demand for large data size and high data rate, vast spectrum resources are required urgently.However, most spectrum resources below 2G are fixedly occupied by other industries, although they have not been fully utilized.Considering this, the purpose of spectrum sensing in mobile communication networks is to share spectrum resources with other industries, without interfering with their normal operations.Moreover, spectrum sensing can be also applied to coordinate the public resources, which is a revolutionary change of the fixed spectrum allocation system [2].
On account of the rearrangement function needed in spectrum sensing and the fact that modulation recognition can provide reliable parameters for it, digital modulation recognition is of great importance in the whole system [3].The goal of digital modulation recognition is to identify the modulation format of an unknown digital communication signal.For modulation classification, two general classes of classical methods exist: likelihood-based and feature-based methods, respectively [4,5].Based on the likelihood function of the received digital signal, the former method makes the decision by comparing the likelihood ratio with a threshold.In the feature-based method, several features are usually chosen and the decision is made jointly.
However, in traditional sensing process, two approaches are based on Shannon-Nyquist sampling theorem and the data scale to deal with can be enormous with a quite wide band especially in the cooperation networks.These years, researchers have brought compressive sensing (CS) in, which can solve the problem of high sampling rate caused by Shannon-Nyquist sampling theorem.It is declared that if the signal has a sparse representation in a fixed basis, we can reconstruct the sparse domain of the signal by solving an optimization algorithm, using samplings far fewer than dimensions of the original signal, and the original signal can be obtained by a simple matrix operation [6,7].
In many CS conditions, we expect to acquire some signal characteristics rather than recovering the original signal, since reconstructing signals allows for lots of extra operations, which results in higher complexity in both time and space.Researchers have already carried out much related valuable work in reconstructing signal characteristics based on CS [8,9].Inspired by these researches, we are devoted for finding identification features which can be used in the digital modulation recognition and simultaneously have sparsity, meaning they can be reconstructed directly by compressive samples.
In this paper, we propose a feature-based method based on CS for digital modulation recognition.We construct two identification features and use compressive samples to recover them directly, without recovering the original signals.One identification feature is the spectrum of received data and its nonlinear transformation, which is based on the feature proposed by [10], and the other is a compositional feature of multiple high-order moments of the received data.These two features can be used to identify various kinds of modulation modes, and, in this paper, we only focus on multiple frequency shift keying (MFSK), multiple phase shift keying (MPSK), and multiple quadrature amplitude modulation (MQAM).Simulations would be carried out to indicate that the performance of our method can be effective and reliable, with lower complexity and better antinoise property than traditional ones [11,12].
The rest of this paper is organized as follows.Section 2 would present the system model adopted throughout the work, both the signal model and compressive sensing model.In Section 3, we construct two identification features and analysis sparsity of them.In Section 4, we build the linear relationships between identification features and compressive samples and give a brief introduction of the recovery method.Then, the whole recognition flowchart will be shown in Section 5. Simulations and analysis are present in Section 6.And, finally, we draw the conclusion in Section 7.

System Model
2.1.Signal Model.In a spectrum sensing scenario, we assume that a wide band received signal has one of the following modulation modes: where  represents the amplitude of the received signal,  represents the symbol period, () represents the impulse response of pulse shaping low-pass filter, in which we choose rectangular pulse in this paper, and V() stands for additive Gaussian white noise (AWGN).In the whole process, the timing offset and the carrier offset are both assumed to be zero, and the form of () is chosen as follows: MPSK:  () =  (2  +2((−1)/)) MFSK: where   and , respectively, stand for the carrier frequency and order of the chosen modulation mode, Δ is the carrier spacing, and {   } and {   } are a set of discrete levels.It is worth noting that there is no need of pulse shaping for MFSK, while in order to unity the form as (1), we regard ( − ) in MFSK as 1, which would not influence the use of it.

Compressive Sampling Model.
According to the theory of CS, the compressive sampling process can be modeled analytically as where y is the -length sampling vector of the received signal () at a rate no lower than Nyquist sampling rate.z represents the subsampling measurements.A is a realvalue measurement matrix of size  × , which complies with the restricted isometry property (RIP), such as Gaussian matrix, partial Fourier transform matrix, or others.In order to reconstruct the th power of signal in Section 4.1, we adopt a special measurement matrix, with the value of "1" randomly located in each row and other elements being zero.Owing to the randomization of row elements, the matrix satisfies the RIP requirement as well as Gaussian matrix.Furthermore, the two-valued property of the matrix can enormously simplify the matrix operations, which make it possible to establish linear relationship between the nonlinear reconstruction target with the compressive samples.
To classify the modulated signal based on z, we will firstly build the linear relationships between z and the identification features and then reconstruct the features directly with compressive samples z by solving an optimization algorithm, which would then be used to do digital modulation recognition.

Construction of the Identification Features
To achieve the goal of classifying modulation types accurately, identification features should be chosen with distinguishing details for each modulation type firstly.Secondly, identification features should have desired sparsity, in order to be constructed by compressive samples based on the theory of CS.According to these two requirements, we propose and construct the following two identification features.

Feature 1: Spectrum of the Signal's 𝛾th Power Nonlinear
Transformation.Referring to [9], we calculate the th power of the received signal (); that is, where  = 2  ( = 0, 1, 2, . ..).In the following, it will be shown that the spectrum of specific power order of signal level presents the recognizable characters for certain modulation types and orders.V  () is the noise caused by nonlinear transformation of V().
Then, we calculate the spectrum of [()]  , which is represented as y  , with  ranging from 0 to a larger number.We have the following relationship: where F = [ −2/ ] (,) represents the -point IFFT matrix.
For different kinds of modulation modes, the results are quite different, which can be used to do the recognition, and we call this feature the spectrum feature below.
For MFSK, according to ( 1) and ( 2), the spectrum of it can be calculated as follows: where (⋅) stands for the impulse function.Obviously, there is impulse in the spectrum of MFSK, and the number of these impulses just corresponds to order .As a contrast, the spectrum for MPSK can be presented as follows: where * stands for the convolution operation.The spectrum of MPSK comes out to be a monotone decreasing sine function, with no impulse.The calculation process of spectrum of MQAM is similar to MPSK, and their consequences are also similar.Figure 1 shows the results of different modulation types by  = 1.From (a) and (b), we can see that there is apparent impulse for MFSK, just as we analyze in theory, quite different from that in (c) and (d), which represent the results of MPSK and MQAM, respectively.That is to say, we can distinguish MFSK from others by the spectrum of the signal, and the number of pulses indicates the order of MFSK.
For MPSK, when  < , From the expression, we can see that there is no impulse in this condition.However, when  = , For ( − ), we choose rectangle filter.Since  (2  +2((−1)/)) = 1, ∑ ∞ −∞     ( − ) becomes a constant, and the Fourier transform of it is an impulse.Based on it, we compare the results of  = 1,  = 2,  = 4, and  = 8, referring to Figure 2. It can be seen that the impulse firstly appears when  = , with  varying from small to big, which can be used to determine the order of MPSK.
As for MQAM, owing to similarity of the signal constellation, the property of MQAM is similar with QPSK, shown as Figure 3.It can be easily seen that the impulse firstly appears when  = 4, and it has no relationship with the specific order of it.
The sparsity of this feature is in inverse proportion with , which represents length of the signal as well as the IFFT size, on condition that there appears the impulse.
To sum up, the only remaining problem is to distinguish QPSK and MQAM.Therefore, we construct another feature for it.
In our system model, we intend to acquire  2,1 () and  4,0 () as recognition parameters, which we call the highorder moment feature.
With carrier known, symbols in the digital signals can be regarded as points in the signal constellation [13,14].Since points in digital signals of linear modulations are of equal probabilities, when the data size is large enough, we can use points in the signal constellation to calculate the theoretical values of  2,1 and  4,0 , as Table 1 shows.
Referring to Table 1,  4,0 of different modulation formats are of different theoretical times compared to  2 , which is the square value of  2,1 .
We take QPSK and 16QAM as examples.According to (11), the theoretical values  of QPSK and 16QAM, respectively, come out to be 1 and 0.68.If we get the identification characteristic  of a signal, we can then identify the modulation format by comparing  with a suitable decision threshold.
Since high-order moment is a kind of statistics, we need sample several times.Then, to obtain  2,1 () and  4,0 (), we construct matrixes as follows: where (⋅)  represents conjugate transpose, (⋅)  represents transpose, and vec{⋅} stacks all columns of a matrix into a vector.For R 21 , the element of matrix at row ℎ, column , is
It is obvious that R 21 and R 40 in Figure 4 are sparse.For R 21 , all diagonal elements are nonzero, meaning the sparsity degree of it is 1/.For R 40 , the ((ℎ − 1) ×  + ℎ)th elements of vec{R   } are nonzero, meaning the sparsity degree of it is 1/ 3 .

Recovery of the Identification Features with Compressing Samples
In this section, we introduce the approaches of recovering the two identification features based on CS.We firstly build the linear relationships between compressive samples and the defined features and then give a brief introduction of the  reconstruction algorithm and the practical selection strategy for the measurement matrix.

Linear Relationships between Compressive Samples and the Spectrum Feature.
It is obviously that the th power of the signal is a nonlinear transformation.To get linear relationship between compressive samples and the spectrum of the signal's th power nonlinear transformation, we choose the special measurement matrix proposed in Section 2. According to the nature of this certain-form matrix, we can easily get the following relationship based on (3): and A 1 is the measurement matrix for feature 1.Then, referring to (5), we obtain where Θ = A 1 F is the sensing matrix we needed.

Linear Relationships between Compressive Samples and
the High-Order Moment Feature.For this identification feature, the sampling matrix can be chosen as any one as long as it satisfies the restricted isometry property (RIP): (i) R 21 : according to (3) and the nature of transpose, we get the following relationship, and A 2 stands for the measurement matrix for R 21 : Take the average of both sides: We use R 21 to represent (zz  ), simultaneously refer to (12), and then get Next, we apply the property vec{UXV} = (V  ⊗ U)vec{X} to transform (19) to (20).It is worth noticing that A 2  = A 2  , for A is a real-value matrix: where Ψ = A 2 ⊗ A 2 can be regarded as the sensing matrix, with the scale of  2 ×  2 .
(ii) R 40 : since the sparsity degree of R 40 is far fewer than that of R 21 , the dimension of signal needed and scale of measurement can also be very low.We represent the measurement for R 40 as A 3 , while the only difference of it from A 2 is the dimension.
Similar to (17), there is Then, according to vec{UXV} = (V  ⊗ U)vec{X}, we can transform the two-dimensional relationship into one-dimensional relationship: We can obtain Take the average of both sides: Based on (13), we get the relationship: where R 40 denotes (vec{zz  }vec  {zz  }).And then we have where Φ = (A 3 ⊗A 3 )⊗(A 3 ⊗A 3 ) is the sensing matrix.

Reconstruction of Identification
Features.z  , R 21 , and R 40 can be calculated by the sampling value z.With sensing matrixes and measurement vectors known, the reconstruction of the sparse vectors can be regarded as the signal recovery problem by solving the NP-hard puzzle as follows, taking R 21 as an example: This can be transformed into a linear programming problem: which is called 1-norm least square programming problem and is proved to be convex that there exists a unique optimum solution. > 0 is a weighting scalar that balances the sparsity of the solution induced by the 1-norm term and the data reconstruction error reflected by the 2-norm LS term.In Section 4.1, we have mentioned recovering three recognition features by using measurement matrixes A 1 , A 2 , and A 3 , respectively.However, practically, only using A 1 as the compressive measurement may meet the requirement of recovering all of the features.The reason is that A 2 and A 3 differ in the dimension but are both designed with the constraint of RIP property only.From the other aspect, the primary requirement of constructing matrix A 1 is also the RIP condition.

Modulation Recognition with the Identification Features
Given a received communication signal modulated by MFSK, MPSK, or MQAM, we firstly get compressive samples using measurement matrixes present in Section 2. In this process, due to difference of sparsity we have analyzed in Section 3, various features may apply various length of the signal, and this can be decided based on actual situations.According to the approaches proposed above, the identification features can be easily obtained.Then, we can recognize the modulation format effectively referring to the flowchart shown in Figure 5, and specific steps are listed in the following.
Step 1. Reconstruct the spectrum feature when  = 1 with compressive samples.If there is impulse in the recovered spectrum, the modulation mode can be identified as MFSK, and the number of impulses indicates the order of it.However, if there is no impulse in the feature, the communication signal is modulated by MPSK or MQAM, and then Step 2 should be conducted.
Step 2. Reconstruct the spectrum feature when  = 2, 4, 8, . . .with compressive samples and observe value of  when the impulse firstly appears.If  = 4 when the impulse appears, the modulation mode can be regarded as QPSK or MQAM, and then we go to Step 3.However, if  ̸ = 4 when the impulse appears, the signal is modulated by MPSK and this value of  is the order of it.
Step 3. Reconstruct R 21 and R 40 of the signal with compressive samples, get average values of the diagonal as  2,1 () and  4,0 (), respectively, and then calculate  based on (11).Compare  with the calculated boundary values shown in Table 1 and determine the modulation type.

Numerical Results
This section presents the simulation results of our featurebased recognition method.We firstly generate a stream of signals modulated by MPSK, MFSK, or MQAM.All the signals share the same bit rate 1 kbit/s and the carrier frequency 2 kHz, and the carrier spacing for MFSK is 0.25 kHz.For the two proposed features, the observation time is various because data volume needed by the two features are all different.The performance of reconstruction is closely related to the signal-to-noise ratio (SNR), which is set as a variable in our simulation, and simulations at every SNR are carried out for 500 times.
As mentioned above, information we need to capture in feature 1 is whether there are impulses and the number of them, rather than accurate numerical values.Therefore, we apply correct detection rate of pulse to evaluate the performance of reconstruction of spectrum feature, which is shown in Figure 6.We set a decision threshold which equals two-thirds of the biggest reconstructed value, and if there is no other value larger than the threshold, the biggest value would be regarded as the impulse.In this scenario, the compressive ratio is set as 0.3, which means / = 0.3.We calculate the detection rate for MFSK signal on  = 1, BPSK on  = 2, QPSK and MQAM on  = 4, and 8PSK on  = 8, respectively.It is obvious that the detection rate varies a lot with .The reason is that th power of signal is a nonlinear transform, meaning that the uniformly distributed noise is magnified, and the degree of magnification extends as the increasing of .Therefore, detection rate of impulse when  = 8 is the worst one.
Figure 7 shows the mean square error (MSE) of reconstructed feature 2 with respect to the theoretical ones.That is, We give the MSE of reconstructed R 21 and R 40 , respectively, with the compressive ratio chosen as 0.3 and 0.45.From Figure 7, we can see that the performance of reconstruction of R 40 is closely related to the compressive ratio, while the performance of reconstruction of R 21 is relatively perfect  even at a low compressive ratio.Moreover, we can easily get the conclusion that when the compressive ratio is suitable, the precision of feature 2 is high enough as long as the SNR is higher than 10 dB.
Figure 8 shows the correct classification rate of different modulation modes at relatively low SNR.Difference of the correct classification comes from various performance of reconstruction of features, which has been shown in Figures 6 and 7. MFSK has high recognition rate, larger than 0.93 even when SNR = −6 dB.For MPSK, the correct recognition rate declines as  increases.However, for QPSK and MQAM, the performance is quite different, and we give the following analysis.
According to [14], we have the fact that  4,0 of just the signal and mixture of noise and signal are of the same value, so the main cause of the error comes from  2,1 .
As for  2,1 , we have the following proof stating the variation of the value in noisy condition and noiseless condition.To describe this clearly,  2,1 ( 0 ),  2,1 (V), and  2,1 () are, respectively, used to replace  2,1 while being in the following condition of signal only, noise only, and the mixture of noise and signal: (30) V is zero-mean random measure noises with Gaussian distribution, which is independent from .According to the nature of expectation, we know that  ( 0ℎ * V ℎ ) =  ( 0ℎ  0ℎ * ) = 0.
Therefore, we can obtain the following relationship: meaning  2,1 () is the sum of signal power and noise power.From (11) where  V denotes noise power and   0 denotes signal power.
To sum up,  2,1 () is added by the power of noise, and, as a consequence, the identification parameter  becomes

)
ℎ ,   are elements in the signal y.When ℎ = , meaning diagonal elements, the values are equal to  2,1 () based on the definition of high-order moments.However, when

Figure 2 :
Figure 2: Spectrum of th power of signal modulated by MPSK.

Figure 3 :
Figure 3: Spectrum of th power of signal modulated by MQAM.

Figure 4 :
Figure 4: R 21 and R 40 of the signal modulated by 16QAM.

Figure 5 :
Figure 5: The process of digital modulation recognition.

Figure 7 :
Figure 7: MSE of reconstructed R 21 and R 40 with different compressive ratio.

Figure 8 :
Figure 8: Correct classification rate of different modulation modes.
and (27), we can obtain the relationship of the theoretical  and the actual   :