## 6 A Compressed Sensing Based Ultra-Wideband Cognitive Radio
### Discussion
Meanwhile,  
\[x\left(0\right)=\int_{-\infty}^{\infty}x_{f}\left(f\right)df \tag{34}\]  
and  
\[x_{f}\left(f\right)=\sum\limits_{n=1}^{N}\left|h_{nf}\left(f\right)\right| \left|p_{nf}\left(f\right)\right|e^{j2\pi\left(\arg\left(h_{nf}(f)\right)+ \arg\left(p_{nf}(f)\right)\right)} \tag{35}\]  
where the angular component of the complex value is \(\arg\left(\bullet\right)\). For the real value signal \(x(t)\),  
\[x_{f}\left(f\right)=x_{f}^{\ast}\left(-f\right) \tag{36}\]  
where \"\(\ast\)\" denotes conjugate operation. Thus,  
\[x_{f}\left(-f\right)=\sum\limits_{n=1}^{N}\left|h_{nf}\left(f\right)\right| \left|p_{nf}\left(f\right)\right|e^{-j2\pi\left(\arg\left(h_{nf}(f)\right)+ \arg\left(p_{nf}(f)\right)\right)} \tag{37}\]  
and \(x_{f}(f)+x_{f}(-f)\) is equal to  
\[\sum\limits_{n=1}^{N}\left|h_{nf}\left(f\right)\right|\left|p_{nf}\left(f \right)\right|\cos(2\pi\left(\arg\left(h_{nf}\left(f\right)\right)+\arg \left(p_{nf}\left(f\right)\right)\right)) \tag{38}\]  
If \(h_{nf}(f)\) and \(\left|p_{nf}\left(f\right)\right|\) are given for all \(f\) and \(n\), maximization \(x(0)\) is equivalent to setting,  
\[\arg\left(h_{nf}\left(f\right)\right)+\arg\left(p_{nf}\left(f\right)\right)=0 \tag{39}\]  
which means the angular component of \(p_{nf}(f)\) is the negative angular component of \(h_{nf}(f)\). The optimization problem (33) can be simplified as,  
\[ \begin{array}{l}\max\sum\limits_{n=1}^{N}\int_{-\infty}^{\infty}\left|h_{nf }\left(f\right)\right|\left|p_{nf}\left(f\right)\right|df\\ s.t.\sum\limits_{n=1}^{N}\int_{-\infty}^{\infty}\left|p_{nf}\left(f\right) \right|^{2}df\leq E_{p}\\ \left|p_{nf}\left(f\right)\right|^{2}\leq c_{nf}\left(f\right)\end{array}  \tag{40}\]  
Because  
\[\left|h_{nf}\left(f\right)\right|=\left|h_{nf}\left(-f\right)\right| \tag{41}\]\[\left|p_{nf}\left(f\right)\right|=\left|p_{nf}\left(-f\right)\right| \tag{42}\]  
\[\left|c_{nf}\left(f\right)\right|=\left|c_{nf}\left(-f\right)\right| \tag{43}\]  
for all \(f\) and \(n\). Thus uniformly discrete frequency points \(f_{0}\), \(\ldots\), \(f_{M}\) are considered in the optimization problem (40). Meanwhile, \(f_{0}\) corresponds to the DC component and \(f_{1}\), \(\ldots\), \(f_{M}\) correspond to the positive frequency components. Define column vectors \(\mathbf{h}_{f}\), \(\mathbf{h}_{1f}\), \(\ldots\), \(\mathbf{h}_{Nf}\),  
\[\mathbf{h}_{f}=\left[\mathbf{h}_{1f}^{T}\ \mathbf{h}_{2f}^{T}\ \cdots\ \mathbf{h}_{Nf}^{T}\right]^{T} \tag{44}\]  
\[\left(\mathbf{h}_{nf}\right)_{i}=\left\{ \begin{array}{c}\left|h_{nf}\left(f _{i-1}\right)\right|,i=1\\ \sqrt{2}\left|h_{nf}\left(f_{i-1}\right)\right|,i=2,\ldots,M+1\end{array} \right. \tag{45}\]  
where \"\(T\)\" denotes transpose operation. Define column vectors \(\mathbf{p}_{f}\), \(\mathbf{p}_{1f}\), \(\ldots\), \(\mathbf{p}_{Nf}\),  
\[\mathbf{p}_{f}=[\mathbf{p}_{1f}^{T}\ \mathbf{p}_{2f}^{T}\ \cdots\ \mathbf{p}_{Nf}^{T}]^{T} \tag{46}\]  
\[\left(\mathbf{p}_{nf}\right)_{i}=\left\{ \begin{array}{c}\left|p_{nf}\left(f _{i-1}\right)\right|,i=1\\ \sqrt{2}\left|p_{nf}\left(f_{i-1}\right)\right|,i=2,\ldots,M+1\end{array} \right. \tag{47}\]  
Define column vectors \(\mathbf{c}_{f}\), \(\mathbf{c}_{1f}\), \(\ldots\), \(\mathbf{c}_{Nf}\),  
\[\mathbf{c}_{f}=[\mathbf{c}_{1f}^{T}\ \mathbf{c}_{2f}^{T}\ \cdots\ \mathbf{c}_{Nf}^{T}]^{T} \tag{48}\]  
\[\left(\mathbf{c}_{nf}\right)_{i}=\left\{ \begin{array}{c}\sqrt{\left|c_{nf} \left(f_{i-1}\right)\right|},i=1\\ \sqrt{2\left|c_{nf}\left(f_{i-1}\right)\right|},i=2,\ldots,M+1\end{array} \right. \tag{49}\]  
Thus, the discrete version of the optimization problem (40) is shown below,  
\[ \begin{array}{c}\max\mathbf{h}_{f}^{T}\mathbf{p}_{f}\\ s.t.\left\|\mathbf{p}_{f}\right\|_{2}^{2}\leq E_{p}\\ 0\leq\mathbf{p}_{f}\leq\mathbf{c}_{f}\end{array}  \tag{50}\]  
An iterative algorithm is shown as follows. 1. Initialization: \(P=E_{p}\) and \(\mathbf{p}_{f}^{*}\) is set to be all-0 column vector. 2. Solve the following optimization problem to get the optimal \(\mathbf{q}_{f}^{*}\) using Cauchy-Schwarz inequality. \[