## 6 A Compressed Sensing Based Ultra-Wideband Cognitive Radio
### Compressed Sensing Based UWB Communication System  
With the knowledge of Sections 6.1. and 6.2., we propose a CS-based UWB communication system which is able to reduce the sampling rate to 1.25% of the Nyquist rate. The system architecture is illustrated in Figure 26. A UWB signal is transmitted by feeding a sparse bit sequence through a UWB pulse generator and an pre-coding filter. Then, the received signal is directly sampled after the channel, using a low-rate A/D and then processed by a recovery algorithm. \(\Phi\) is the projection matrix consisting of the pre-coding filter and the channel. It can be noticed that channel itself is part of the projection matrix in CS, so the receiver is very simple, with only one low-rate A/D to collect measurement samples. For example, a 3-8 GHz UWB signals can be successfully recovered by a 125 Msps A/D.  
\(K\)-pulse position modulation (PPM) is used to modulate sparse bit sequence. Each PPM symbol is \(K\)-sparse: there are \(N\) positions and only \(K<<N\) pulses in each symbol, as illustrated in Figure 27. The output of the UWB pulse generator can be written using the notations in Equations 5 and 6, with \(\Psi_{n}\left(t\right)=p\left(t-nT_{p}\right)\), where \(p(t)\) is the function of the UWB pulse and \(T_{p}\) is the period of the pulse. Pre-coding filter and channel are modeled as FIR filters, with combined impulse response \(h(t)=f(t)*c(t)\), where \(f(t)\) and \(c(t)\) are the impulse response for the pre-coding filter and the channel, respectively. Here \(h(t)\) is equivalent to the \(h(t)\) in Equation 8. The received signal \(y(t)=h(t)*x(t)\) is then uniformly sampled by an A/D with sampling period \(T_{s}\). Similar to Equations 9 and 12, the down-sampled measurements form the \(M\times 1\) vector \(y=\Phi\Psi\theta=\Theta\theta\), where \(\Phi\) is a \(quasi-Toeplitz\) matrix. Now, the communication problem becomes a problem of estimating \(\hat{\theta}\) from \(M<<N\) measurements, which is again identical to the problem described as Equation 4.  
The success of recovery relies on the sparsity \(K\) and the incoherence between \(\Psi\) and \(\Phi\). Sparsity is easily met by controlling the transmitted sequence. In a simple case, we set \(K=1\), which means that there is only one pulse in PPM symbol. The incoherence property can be met by proper selection of the pre-coding filter \(f(t)\). If \(f(t)\) is a PN sequence whose chip rate is equal to the bandwidth of the UWB pulse \(p(t)\), then \(\theta\) can be successfully recovered using recovery algorithms. So far the discussion is in baseband. If the transmitted UWB is passband, then up-conversion is applied after the pre-coding filter. PN chip rate and the receiver structure remain the same. No down-conversion is required at the receiver. For example, a 3-8 GHz UWB pulse requires a 5 GHz PN chip rate, which is the same as the signal bandwidth, not the Nyquist rate of the maximum signal frequency, as required by the AIC system. A/D at the receiver directly samples the received signal, without doing down-conversion.  
The number of measurements \(M\) and sampling rate are related and determined by the length of the combined filter \(h(t)\). If \(h(t)\) is long, the received signal is \"spread out\" in the time domain, therefore sufficient measurements can be made under a lower sampling rate.