# Automatic Modulation Recognition Using Compressive Cyclic Features
## 3 CS-based AMR
### CS-Based Cyclic Characteristic Analysis  
In this subsection, we attempt to exploit the sparsity structure of CC function, and investigate the relationship between the compressive measurements and CCs. Assuming \(\mathbf{x}=\{x[l]\}_{l=0}^{N-1}\) is the set of uniform samples at the oversampling rate \(\rho/T\), the \(n\)th-order/q-conjugate temporal cumulant (TC) can be expressed as [12]  
\[C_{x,n,q}[l,\mathbf{\tau}]=Cumulant\{x^{(*)_{n}}[l+\tau_{u}]\}_{u=1}^{n}, \tag{5}\]  
where \(Cumulant\) is the cumulant operator, and \(\tau_{n}=0\). TC is a periodic function for cyclostationary signals, with its Fourier components given by  
\[C_{x}^{\beta}(\mathbf{\tau})_{n,q}=\frac{1}{N}\sum_{l=0}^{N-1}C_{x,n,q}[l,\mathbf{ \tau}]e^{-j2\pi\beta l}. \tag{6}\]  
From (3), when \(C_{x}^{\beta}(\mathbf{\tau})_{n,q}\
eq 0\), the expression (6) is the CC at CF \(\beta\). Assuming that the vector \(\mathbf{c}\) represents the quantities \(\{C_{x,n,q}[l,\mathbf{\tau}]\}_{l=0}^{N-1}\), \(\mathbf{\eta}\) is the Fourier coefficient vector, and \(\mathbf{\Phi}\) is the \(N\)-point discrete Fourier transform (DFT) matrix, it holds evidently from (6) that \(\mathbf{\eta}=\mathbf{\Phi}\mathbf{c}\). Equivalently, \(\mathbf{c}=\mathbf{\Phi}^{\rm H}\mathbf{\eta}\), where \((\cdot)^{\rm H}\) denotes the conjugate transpose operator. Clearly, the \(n\)th-order/\(q\)-conjugate CCs can be viewed as the set of nonzero elements of vector \(\mathbf{\eta}\), and the location of nonzero element implies the corresponding CF. Thus, the calculation of CCs is equivalent to locating the peaks in \(\mathbf{\eta}\). Since the CFs belong to a finite set depending on the signal bandwidth and order \(n\)[9], the CCs can be treated as sparse in the Fourier domain under the assumption of no cyclic leakage. Sparsity of CCs motivates the application of CS. CS argues that sparse vector \(\mathbf{\eta}\) can be recovered from \(M\ll N\) compressive measurements \(\mathbf{\epsilon}_{\mathbf{z}}\), obtained by \(\mathbf{\epsilon}_{\mathbf{z}}=\mathbf{\
abla}\mathbf{c}\), where \(\mathbf{\
abla}\in\mathbb{R}^{M\times N}\) is the so-called measurement matrix. If \(\mathbf{\
abla}\) satisfies the condition known as the restricted isometry property (RIP), the exact recovery of \(\mathbf{\eta}\) can be achieved by solving an optimization problem, as follows:  
\[\text{minimize}\|\mathbf{\hat{\mathbf{\eta}}}\|_{0}\text{ subject to }\mathbf{ \epsilon}_{\mathbf{z}}=\mathbf{A}\mathbf{\hat{\mathbf{\eta}}} \tag{7}\]  
where \(\mathbf{A}=\mathbf{\
abla}\mathbf{\Phi}^{\rm H}\). In previous studies, a variety of greedy iterative algorithms and convex solvers have been adopted for signal recovery [13; 14]. However, there still remains the issue that the relationship between TCs and the compressive measurements has not been established. Before proceeding, it should be pointed out that in practical implementations with finite data samples, the estimators of involved statistics are defined on the basis of averaging of the sample values taken through the observation interval. Assume that the received signal sequence \(x_{r}[s]\) with length \(L\) is divided into \(W\) frames with length \(N\), where \(W=\lfloor L/N\rfloor\) (\(\lfloor\cdot\rfloor\) denotes floor operation). Then, \(x_{r}[s]\) can be recognized as a mixture of \(W\) sequences \(\{x^{(k)}[l]\}_{k=1}^{W}\)[15]. We define the estimator of higher-order temporal moment (TM) as  
\[R_{x,n,q}[l,\tau]=\frac{1}{W}\sum_{k=1}^{W}\prod_{u=1}^{n}x^{(k)_{u}}[l+\tau_{ u}]. \tag{8}\]  
By substituting (8) into the moment-cumulant (M-C) formula [9], we obtain the estimator of TC  
\[C_{x,n,q}[l,\tau]=\sum_{\{p_{n}\}}(-1)^{I-1}(I-1)!\prod_{i=1}^{I}R_{x,n_{i};q_ {i}}[l,\tau], \tag{9}\]  
where \(\{p_{n}\}\) is the set of distinct partitions of \(\{1,2,\ldots,n\}\), \(I\) is the number of subsets in each partition, \(n_{i}\) and \(q_{i}\) correspond to the number of elements and the number of conjugated terms in the \(i\)th subset.