## 5 OFDM Signal Identification Based on Inherent Features
### High-Order Cumulant-Based OFDM Signal Detection  
High-order cumulants provide an effective means for detecting non-Gaussian burst signals in a Gaussian noise environment. In an ideal Gaussian noise setting, the fourth-order cumulant is expected to be zero; hence, a non-zero measured value serves as a reliable indicator of the presence of a non-Gaussian signal. OFDM signals, which are composed of the superposition of multiple subcarriers, can be approximated to have a Gaussian amplitude distribution when the number of subcarriers is sufficiently large. This is in contrast to the amplitude distribution of single-carrier digital modulation signals, which do not exhibit Gaussian characteristics. Consequently, high-order cumulant-based OFDM signal detection algorithms leverage the asymptotic Gaussian distribution property of the OFDM signal's time-domain envelope, as well as the suppressive effect of high-order cumulants on Gaussian random processes, thereby facilitating effective identification of the signal. The characteristic function of a random variable \(\varphi(v)\) can be expressed as  
\[\varphi(v)=E\Big{\{}e^{jx}\Big{\}}=\int_{-\infty}^{\infty}f(x)e^{jx}dx. \tag{30}\]  
In the given formulation, \(E\{\cdot\}\) represents the expectation is being calculated, and \(f(x)\) denotes the probability density function (PDF). The \(k\)-th moment of a random variable can be obtained by taking the k-th derivative of the characteristic function \(\varphi(v)\) at the origin, as expressed in Equation (23). \[M_{k}=\varphi^{k}(0)=E\Big{\{}x^{\hat{k}}\Big{\}}=\frac{d^{\hat{k}}\varphi(v)}{ dv^{\hat{k}}}\Bigg{|}_{v=0}. \tag{31}\]  
The \(k\)-th cumulant of a random variable can be derived by computing the \(k\)-th derivative of the logarithm of the characteristic function \(\varphi(v)\), evaluated at the origin, as expressed by the following equation:  
\[C_{k}=\frac{d^{\hat{k}}\ln(\varphi(v))}{dv^{\hat{k}}}\Bigg{|}_{v=0}. \tag{32}\]  
The \(k\)-th moment and the \(k\)-th cumulant of a random variable satisfy the following relationship:  
\[C_{x}(\mathbf{H})=\sum_{\cup_{k=1}^{d}H_{k}=\mathbf{H}}(-1)^{q-1}(q-1)!\prod_ {k=1}^{q}M_{x}(H_{k}). \tag{33}\]  
In the expression provided, \(\mathbf{H}=\left\{1,2,...,l\right\}\) denotes the set of symbols associated with the random variable \(\mathbf{x}=\left\{x_{1},x_{2},...,x_{l}\right\}\), and \((\cdot)!\) represents the factorial operation. It follows that the high-order moment and high-order cumulant of the complex random variable \(y\) can be expressed as  
\[M_{p+q,p}=E\big{[}y^{p}(y^{*})^{q}\big{]}, \tag{34}\]  
\[C_{p+q,p}=\text{cum}\left[y_{1},y_{2},...,y_{p},y_{1}^{*},y_{2}^{*},...,y_{q}^ {*}\right]. \tag{35}\]  
In the context provided, \(cum(\cdot)\) refers to the operation of calculating cumulants. The second-order cumulant of a zero-mean stationary signal \(x(n)\) can be expressed as  
\[C_{20}(\tau_{1})=E\big{\{}x(n)x(n+\tau_{1})\big{\}}, \tag{36}\]  
\[C_{21}(\tau_{1})=E\big{\{}x^{*}(n)x(n+\tau_{1})\big{\}}. \tag{37}\]  
The fourth-order cumulant of the signal \(x(n)\) can be represented as  
\[ \begin{split} C_{42}(\tau_{1},\tau_{2},\tau_{3})=E\big{\{}x^{*}( n)x(n+\tau_{1})x(n+\tau_{2})x^{*}(n+\tau_{3})\big{\}}\\ -E\big{\{}x^{*}(n)x(n+\tau_{1})\big{\}}E\big{\{}x(n+\tau_{2})x^{*} (n+\tau_{3})\big{\}}\\ -E\big{\{}x^{*}(n)x(n+\tau_{2})\big{\}}E\big{\{}x(n+\tau_{1})x^{* }(n+\tau_{3})\big{\}}\\ -E\big{\{}x^{*}\Big{(}n\Big{)}x\Big{(}n+\tau_{3}\Big{)}\Big{\}}E \Big{\{}x\Big{(}n+\tau_{1}\Big{)}x\Big{(}n+\tau_{2}\Big{)}\Big{\}}.\end{split}  \tag{38}\]  
In the expression provided, \(\tau_{1}\), \(\tau_{2}\), and \(\tau_{3}\) represent time delays, which typically can be set to 0. The OFDM received signal \(r(t)\) can be represented as  
\[ \begin{cases}r(t)=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}s(t),\\ s(t)=s_{n}\exp\big{[}j2\pi(f_{c}+n\Delta f)\big{(}t-T_{\text{sym}}\big{)}+ \varphi_{0}\big{]}*\text{g}\big{(}t-T_{\text{sym}}\big{)}.\end{cases}