## 6 A Compressed Sensing Based Ultra-Wideband Cognitive Radio
### Channel estimation  
After down-sampling, \(y\) is processed at the receiver with \(\Theta\) using BP. In constructing \(\Theta\), \(f(t)\), \(c(t)\) and \(\Psi(t)\) are required. \(f(t)\) and \(\Psi(t)\) are fixed and can be considered as prior knowledge at the receiver. The channel, \(c(t)\), however, needs to be estimated. A CS based channel estimation method is proposed. A 3-8 GHz channel can be estimated by a 500 Msps A/D.  
Similar to Equation 8, the UWB channel can be modeled as:  
\[c\left(t\right)=\sum_{i=0}^{L-1}c_{i}\delta\left(t-iT_{h}\right) \tag{14}\]  
The channel estimation block diagram is illustrated in Figure 29. A UWB probing pulse \(p(t)*f(t)\) is transmitted to \"probe\" the channel, where \(p(t)\) is a UWB pulse and \(f(t)\) is a PN sequence. At the receiver, sub-Nyquist rate A/D collects \(M\) uniform measurements. This process can be represented as \(y=D\downarrow\left(c(t)*f(t)*p(t)\right)\), where \(D\downarrow\) denotes a down-sampling factor of \(\left\lfloor N/M\right\rfloor\) and \(y\) denotes the measurement vector. Since the system is LTI, an alternative block diagram can be drawn as Figure 30. Then, \(y=D\downarrow\left(\left(f(t)*p(t)\right)*c(t)\right)\). In matrix notation, \(y=\Theta c\), where \(\Theta\) is a \(quasi-Toeplitz\) matrix derived from \(f(t)*p(t)\) and \(c=\left[c_{0},c_{1},...,c_{L-1}\right]^{T}\). The channel estimation problem is to get \(\hat{c}\) from measurements \(y\), which is identical to the CS problem described in Equation 4.  
Successful recovery requires \(c\) to be sparse and the incoherent property of measurement matrix \(\Theta\)[21]. Indoor UWB channel is sparse and PN sequence structured \(\Theta\) has the incoherent property. PN chip rate should be the same as the bandwidth of the channel under estimation. We demonstrate an estimation result in the following.  
Figure 28: Spectrum Mask of the transmitted signal. The ’notch’ part is set to cancel the interference at the receiver. Transmitter also has the ’notch’ part because CS requires consistency at the receiver and transmitter. The ’mask’ part is set to avoid interfering primary users.  
First, we need to set up the real channel \(c(t)\) as the estimation target. Vector network analyzer (VNA) is used to get the real indoor channel coefficient \(c\). The 3-8 GHz channel is measured by VNA with 1 MHz frequency step and 128 averages. \(c(t)\) (Figure 31) is derived from the VNA data using CLEAN algorithm with a rectangular window. There are about 50 non-zero entries in \(c\). PN chip rate is 5 GHz and length of \(f(t)\) is 1 \(\mu s\). Baseband Gaussian UWB pulse \(p(t)\) has 5 GHz bandwidth. Since the measured channel is in passband, up-conversion is applied after the PN filter. At the receiver, 500 Msps A/D is used to get measurements. BP is then used to get the estimated vector \(\hat{c}\) with the knowledge of \(f(t)\), \(p(t)\) and \(y\) only. Additive white Gaussian noise (AWGN) is added at the received samples as \(y=\Theta c+w\), where \(w\) is the noise vector. Basis pursuit denoising (BPDN) is used to solve the recovery problem with noise. Figure 32(a) shows the estimation result and Figure 32(b) shows the zoomed in result. It can be seen that although \(\hat{c}\) is a little noisy, all major paths in \(\hat{c}\) perfectly match to \(c\). Only the amplitudes are slightly different.