## 6 A Compressed Sensing Based Ultra-Wideband Cognitive Radio
### Discussion
\(\int_{0}^{T_{h}}h_{n}^{2}\left(t\right)dt=E_{nh}\). \"\(\otimes\)\" denotes convolution operation. \(n(t)\) is AWGN. \(x_{n}(t)\) is the received noiseless bit-\"1\" waveform defined as  
\[x_{n}(t)=h_{n}(t)\otimes p_{n}(t) \tag{19}\]  
We further assume that \(T_{b}\geq T_{h}+T_{p}\stackrel{{\rm def}}{{=}}T_{x}\), i.e., no existence of ISI. If the waveforms at different transmitter antennas are assumed to be synchronized, the \(k\)-th decision statistic is,  
Figure 31:0 Figure 31:1  
Figure 31:2 System architecture. In order to maximize the system performance, \(\sum\limits_{n=1}^{N}{{x_{n}}\left({{t_{0}}}\right)}\) should be maximized. Thus the optimization problem can be formulated as follows to get the optimal waveforms \(p_{n}(t)\),  
\[ \begin{array}{l}\mathop{\max}\limits_{n=1}^{N}{{x_{n}}\left({{t_{0}}}\right)} \\ s.t.\mathop{\sum}\limits_{n=1}^{N}{{\int_{{{}^{T_{p}}}}^{{p}}{p_{n}^{2}\left(t \right)d\leq E_{p}}}}\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad 0\leq t_{0}\leq T_{b}\end{array}  \tag{22}\]  
An iterative algorithm is proposed here to give the optimal solution to the optimization problem, which is a computationally efficient algorithm. For the simplicity of the following presentation, \(t_{0}\) is assumed to be zero. Meanwhile,  
\[x\left(t\right)=\sum\limits_{n=1}^{N}{{x_{n}}\left(t\right)} \tag{23}\]  
From inverse Fourier transform,  
\[{x_{nf}}\left(f\right)={h_{nf}}\left(f\right){p_{nf}}\left(f\right) \tag{24}\]  
and  
\[{x_{f}}\left(f\right)=\sum\limits_{n=1}^{N}{{h_{nf}}\left(f\right){p_{nf}} \left(f\right)} \tag{25}\]  
where \({x_{nf}}\left(f\right)\), \({h_{nf}}\left(f\right)\) and \({p_{nf}}\left(f\right)\) are the frequency domain representations of \(x_{n}(t)\), \({h_{n}}(t)\) and \({p_{n}}(t)\) respectively. \({x_{f}}(f)\) is frequency domain representation of \(x(t)\). Thus,  
\[x\left(0\right)=\sum\limits_{n=1}^{N}{{x_{n}}\left(0\right)} \tag{26}\]  
and  
\[{x_{n}}\left(0\right)=\int_{-\infty}^{\infty}{{x_{nf}}\left(f\right)d\textit{f}} \tag{27}\]  
If there is no spectral mask constraint, then according to the Cauchy-Schwarz inequality,  
\[x\left(0\right) = \sum\limits_{n=1}^{N}{{\int_{-\infty}^{\infty}{{h_{nf}}\left(f \right)p_{nf}}\left(f\right)d\textit{f}}} \tag{28}\] \[\leq \sum\limits_{n=1}^{N}{{\sqrt{{\int_{-\infty}^{\infty}{{\left|{h_ {nf}}\left(f\right)\right|}^{2}d\textit{f}}}\int_{-\infty}^{\infty}{{\left|{p_ {nf}}\left(f\right)\right|}^{2}d\textit{f}}}}}\] (29) \[\leq \sqrt{\sum\limits_{n=1}^{N}{{\int_{-\infty}^{\infty}{{\left|{h_ {nf}}\left(f\right)\right|}^{2}d\textit{f}}}\sqrt{\sum\limits_{n=1}^{N}{{\int_ {-\infty}^{\infty}{{\left|{p_{nf}}\left(f\right)\right|}^{2}d\textit{f}}}}}}}\] (30) \[= \sqrt{E_{p}\sum\limits_{n=1}^{N}{{E_{nh}}}} \tag{31}\]  
when \({p_{nf}}\left(f\right)=\alpha{h_{nf}}\left(f\right)\) for all \(f\) and \(n\), two equalities are obtained. \[\alpha=\sqrt{\frac{{{E_{p}}}}{{\sum\limits_{n=1}^{N}{{\int_{-\infty}^{\infty} {{\left|{h_{nf}}\left(f\right)\right|}^{2}d\textit{f}}}}}}} \tag{32}\]In this case, \(p_{n}\left(t\right)=\alpha h_{n}\left(-t\right)\), which means the optimal waveform \(p_{n}(t)\) is the corresponding time reversed multipath impulse response \(h_{n}(t)\). If there is spectral mask constraint, then the following optimization problem will become more complicated,  
\[ \begin{array}{l}\max x\left(0\right)\\ s.t.\sum\limits_{n=1}^{N}\int_{0}^{T_{p}}p_{n}^{2}\left(t\right)df\leq E_{p} \\ \left|p_{nf}\left(f\right)\right|^{2}\leq c_{nf}\left(f\right)\end{array}  \tag{33}\]  
where \(c_{nf}(f)\) represents the arbitrary spectral mask constraint at the transmitter antenna \(n\). Because \(p_{nf}(f)\) is the complex value, the phase and the modulus of \(p_{nf}(f)\) should be determined.