# Antijamming Improvement for Frequency Hopping Using Noise-Jammer Power Estimator
## 2 Proposed Estimator and Detector
### Noise-Jammer Power Estimator  
In order to estimate pure noise-jammer power regardless of transmitted-symbol power, the received symbol was multiplied by its conjugation, and its mean and variance were calculated. With a simple manipulation of mean and variance, the pure noise-jammer power was easily estimated. Figure 1: Block diagram of proposed method. Assume that the data of a slot are copied and repeatedly transmitted to \(L\) slots, the received symbol of the \(l\)-th slot is \(r_{l}\), and \(X_{l}\) is the multiplication of \(r_{l}\) and its conjugation \(r_{l}\). Then, \(X_{l}\) can be written as  
\[ \begin{array}{ll}X_{l}&=r_{l}r_{l}{{}^{\star}}\\ &=(h_{l}s_{l}+\overline{u}_{l})(h_{l}s_{l}+\overline{u}_{l}){{}^{\star}}\\ &=|h_{l}|^{2}|s_{l}|^{2}+\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{\star}}+h_ {l}s_{l}\overline{u}_{l}{{}^{\star}}+\left|\overline{u}_{l}\right|^{2},\end{array}  \tag{1}\]  
where \(h_{l}\) denotes a channel coefficient of the \(l\)-th slot, \(s_{l}\) denotes a transmitted symbol of the \(l\)-th slot, and \({}^{\star}\) denotes a complex conjugate. \(h_{l}\) is uncorrelated to other channel coefficients by frequency-selective fading. \(\overline{u}_{l}\) denotes noise with or without jammer on the \(l\)-th slot, which is defined by  
\[\overline{n}_{l}=\left\{ \begin{array}{l}n_{l},\,w.o.\,jammer\\ n_{l}+j_{l},\,w.\,jammer\end{array} \right., \tag{2}\]  
where \(n_{l}\) and \(j_{l}\) denote pure noise and the jammer on the \(l\)-th slot, respectively. To develop the proposed NJPE, the mean and variance of \(X_{l}\) in Equation (1) were calculated. Assume that noise, transmitted symbol, and channel gain are independent. The PBNJ was modeled as zero-mean Gaussian noise [22; 23]. Therefore, the mean of \(X_{l}\) was obtained as  
\[ \begin{array}{ll}\mathrm{E}[X_{l}]&=\mathrm{E}\Big{[}|h_{l}|^{2}|s_{l}|^{2 }\Big{]}+\mathrm{E}\big{[}\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{\star}} \big{]}+\mathrm{E}\big{[}h_{l}s_{l}\overline{u}_{l}{{}^{\star}}\big{]}+ \mathrm{E}\Big{[}\overline{u}_{l}\big{|}^{2}\Big{]}\\ &=\left\{\begin{array}{l}\mathrm{E}_{s}\mathrm{E}\Big{[}|h_{l}|^{2}\Big{]}+ \sigma_{l,N}{{}^{2}},\,w.o.\,jammer\\ \mathrm{E}_{s}\mathrm{E}\Big{[}|h_{l}|^{2}\Big{]}+\sigma_{l,N}{{}^{2}}+\sigma_ {l,J}{{}^{2}},\,w.\,jammer\end{array}\right.,\end{array} \right.\end{array} \tag{3}\]  
where \(\sigma_{l,N}{{}^{2}}\) and \(\sigma_{l,J}{{}^{2}}\) denote the pure noise power and jammer power on the \(l\)-th slot, respectively, and \(\mathrm{E}_{s}\) denotes a symbol power. Next, the variance of \(X_{l}\) was obtained as  
\[ \begin{array}{ll}\mathrm{Var}[X_{l}]=&\mathrm{Var}\Big{[}|h_{l}|^{2}|s_{l}|^{ 2}\Big{]}+\mathrm{Var}[\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{\star}}]+ \mathrm{Var}\Big{[}|h_{l}s_{l}\overline{u}_{l}{{}^{\star}}\big{]}+\mathrm{Var} \Big{[}|\overline{u}_{l}|^{2}\Big{]}+2\mathrm{Cov}\Big{(}|h_{l}|^{2}|s_{l}|^{ 2},\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{\star}}\Big{)}\\ &\qquad+2\mathrm{Cov}\Big{(}|h_{l}|^{2}|s_{l}|^{2},h_{l}s_{l}\overline{u}_{l}{ {}^{\star}}\Big{)}+2\mathrm{Cov}\Big{(}|h_{l}|^{2}|s_{l}|^{2},\overline{u}_{l} |^{2}\Big{)}+2\mathrm{Cov}\Big{(}\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{ \star}},h_{l}s_{l}\overline{u}_{l}{{}^{\star}}\big{)}\\ &\qquad+2\mathrm{Cov}\Big{(}\overline{u}_{l}s_{l}{{}^{\star}}h_{l}{{}^{\star}},\left|\overline{u}_{l}\right|^{2}\Big{)}+2\mathrm{Cov}\Big{(}|h_{l}s_{l} \overline{u}_{l}{{}^{\star}},\left|\overline{u}_{l}\right|^{2}\Big{)}.\end{array}  \tag{4}\]  
Equation (4) was classified as with- or without-jammer since \(\overline{u}_{l}\) had with- or without-jammer cases in Equation (2). In FHSS, the length of each slot was designed to be shorter than coherence time, and channel gain was assumed as constant in a slot. Then, Equation (4) was simplified as  
\[\mathrm{Var}[X_{l}]=\left\{