# Automatic Modulation Recognition Using Compressive Cyclic Features
## 2 Statistical Characterization of Signal of Interest  
Assuming an intercepted modulated signal \(x(t)\) is oversampled at a rate \(\rho/T\), where \(\rho\) is an integer and \(T\) is the symbol period, the observed discrete-time sequence can be expressed as [3]  
\[x[l]=ae^{i\theta}e^{j\frac{2\pi}{\rho}\Delta/T\Gamma}\sum_{k}s_{k}p[l-k\rho-l_{ 0}]+n[l] \tag{1}\]  
where \(a\) is the signal amplitude, \(\Delta f\) denotes the carrier frequency offset, \(\theta\) is the phase shift, \(p[l]\) is the overall shape pulse sequence, \(l_{0}\) is the propagation delay, and \(n[l]\) is the additive Gaussian noise sequence. The symbol sequence \(\{s_{k}\}\) is a zero-mean independently and identically distributed (i.i.d.) sequence, with values drawn from a finite-alphabet constellation. Without loss of generality, the assumption of unit variance constellation is given; _i.e._\(E[|s_{k}|^{2}]=1\). Due to the cyclostationarity of the linearly modulated signal, the \(n\)th-order/\(q\)-conjugate discrete-time CC can be inferred as [3]  
\[C_{x}^{\beta}(\boldsymbol{\tau})_{n,q} = a^{n}\frac{C_{s,n,q}}{\rho}e^{-j2\pi a\lambda_{0}}e^{j\frac{2 \pi}{\rho}\Delta f\Gamma\sum_{n=1}^{n}(-)_{n}\tau_{u}} \tag{2}\] \[\times e^{i\theta(n-2q)}\sum_{l}\prod_{u=1}^{n}p^{(*)_{u}}[l+ \tau_{u}]e^{-j2\pi a\lambda l}\]  
\[\alpha=k/\rho,\ \beta=\alpha+(n-2q)T\Delta f/\rho,\ k\ \text{integer},\] \[C_{x}^{\beta}(\boldsymbol{\tau})_{n,q}\
eq 0, \tag{3}\]  
where \(\beta\) represents cyclic frequency (CF) given by (3), \(\boldsymbol{\tau}=[\tau_{1},\ldots,\tau_{n-1},\tau_{n}]|_{\tau_{n}=0}\) is the delay-vector, \(C_{s,n,q}\) denotes the \(n\)th-order/\(q\)-conjugate cumulant of transmitted constellation, and \((-)_{u}\) is the optional minus sign associated with the optional conjugation \((*)_{u}\). Because \(n[l]\) is assumed to be a stationary, zero-mean Gaussian process, its cumulants are time independent and non-zero only for the second order. Since our work focuses on the higher-order (\(n\geq 3\)) CCs of \(x[l]\), the noise contribution does not appear in (3). Moreover, we also find that the CFs are determined by the (normalized) carrier frequency offset \(T\Delta f/\rho\) and the (normalized) symbol rate \(1/\rho\). It is also worth noting that for the raised cosine pulse shape, the actual set of CFs is essentially limited by the signal bandwidth \(B=(1+r)/(2T)\), where \(r\) denotes the roll-off factor, \(0\leq r\leq 1\). Thus, to avoid aliasing in the cyclic frequency domain, the oversampling factor should be fixed according to the order \(n\) and the roll-off factor \(r\), and a result is given as follows: For \(n\) even, if \(2m/n<r\leq(2m+2)/n\), the necessary and sufficient condition for the oversampling factor is \(\rho\geq n+2m+1\), if \(r=0\), \(\rho\geq n-1\), \(m=1,\ldots,n/2-1\). Additionally, when \(n\) odd, the CCs are zero and the set of CFs is empty. One can refer to [9] for more details. By inspection of (2), the fact that the CC is directly proportional to the corresponding cumulant of the signal constellation implies the potential as a signature of modulation format. Nevertheless, the rotation caused by \(l_{0}\), \(\theta\), and \(\Delta f\) considerably restricts its application for AMR. In response, one way is to select the \((n,n/2)\) CCs with \(\boldsymbol{\tau}=\boldsymbol{0}\) (i.e., the zero delay vector) [6; 7], and the other way is to utilize the magnitudes of CCs [8; 9] as discriminating features. Although the \((n,n/2)\) CCs have avoided the effect of carrier frequency offset and is suited for the blind AMR, only limited types of modulations can be recognized. In contrast, based on the prior knowledge of carrier frequency offset and symbol rate, the possible combinations of CC magnitudes are more flexible, and can consequently deal with a wider range of modulations. A typical feature vector formulated by CC magnitudes is given by [9]  
\[F=\big{[}|C_{x}^{\beta}(\boldsymbol{0})_{(n,q_{0})}|,\ \ldots,|C_{x}^{\beta}( \boldsymbol{0})_{(n,q_{k})}|\big{]}, \tag{4}\]where \(\beta=1/\rho+(n-q_{i})T\Delta f/\rho\).