## 6 A Compressed Sensing Based Ultra-Wideband Cognitive Radio
### Discussion  
Our proposed approach is to exploit the projection matrix with channel itself and a waveform-based pre-coding at the transmitter. Taking the channel as part of CS results in a very simple receiver design, with only one low-rate A/D. The pre-coding is implemented in a natural way using an FIR filter. The  
Figure 30: An equivalent block diagram of channel estimation. Figure 31: Time domain channel derived from VNA measurement. The sparsity of this channel is 50. Figure 29: Block diagram of channel estimation. concept has been demonstrated, through simulations, using real-world measurements. Realistic channel estimation is also considered. The philosophy is to trade computation complexity for hardware complexity, and move receiver complexity to the transmitter. This work is just the beginning of the pre-coded CS. Future work includes reduction of algorithm complexity. Much quicker algorithms are required for real-time applications such as UWB communications. Wideband Waveform Optimization for Multiple Input Single Output Cognitive Radio Using Time Reversal  
Waveform design or optimization is a key research issue in the current wireless communication system. Waveform should be designed according to the different requirements and objectives of system performance. For example, the waveform should be designed to carry more information to the receiver in terms of capacity. For navigation and geo-location, the ultra short waveform should be used to increase the resolution. If the energy detector is employed at the receiver, the waveform should be optimized such that the energy of the signal in the integration window at the receiver should be maximized. In the context of cognitive radio, waveform design or optimization give us more flexibilities to design radio, which can coexist with other cognitive radios and primary radios. From cognitive radio's point of view, spectral mask constraint at the transmitter and the influence of Arbitrary Notch Filter at the receiver should be seriously considered for waveform design or optimization, except for the consideration of the traditional communication objectives. Spectral mask constraint is imposed on the transmitted waveform such that cognitive radio has no interference to primary radio, whereas Arbitrary Notch Filter at the receiver is used to cancel the interference from primary radio to cognitive radio. This section deals with wideband waveform optimization for multiple input single output (MISO) cognitive radio using time reversal. The system architecture is shown in Figure 33. We limit our discussion to a single user scenario. There are \(N\) antennas at the transmitter and one antenna at the receiver. OOK modulation is used for transmission. Thus the transmitted signal at the transmitter antenna \(n\) is,  
Figure 32: (a) Channel estimation result. (b) Zoomed in version of the result. \[s_{n}\left(t\right)=\sum_{j=-\infty}^{\infty}d_{j}p_{n}\left(t-jT_{b}\right) \tag{15}\]  
where \(T_{b}\) is the bit duration, \(p_{n}(t)\) is the transmitted bit waveform defined over \(\left[0,T_{p}\right]\) at the transmitter antenna \(n\) and \(d_{j}\in\left\{0,1\right\}\) is \(j\)-th transmitted bit. Without loss of generality, the minimal propagation delay is assumed to be zero. The energy of transmitted waveforms is \(E_{p}\),  
\[\sum_{n=1}^{N}\int_{0}^{T_{p}}p_{n}^{2}\left(t\right)df=E_{p} \tag{16}\]  
The received noise-polluted signal at the output of low noise amplifier (LNA) is,  
\[r(t) = \sum_{n=1}^{N}h_{n}\left(t\right)\otimes s_{n}\left(t\right)+n \left(t\right) \tag{17}\] \[= \sum_{j=-\infty}^{\infty}d_{j}\sum_{n=1}^{N}x_{n}\left(t-jT_{b} \right)+n\left(t\right) \tag{18}\]  
where \(h_{n}\left(t\right),t\in\left[0,T_{h}\right]\) is the multipath impulse response that takes into account the effect of channel impulse response, the RF front-ends in the transceivers such as Power Amplifier, LNA and Arbitrary Notch Filter as well as antennas between the transmitter antenna \(n\) and the receiver antenna. \(h_{n}(t)\) is available at the transmitter.