## 3 Positioning Based on DVB-T
### Range Measurement
#### 3.2.3 Delay Estimate in the Case of Multiple Emitters  
In a scenario with more than one emitter, the correlation is characterized by the presence of several peaks, each one corresponding to a delay, \(m_{l}\) (with \(l=1,\cdots,N_{e}\)), introduced in the correlation by each specific emitter. Moreover, each emitter can generate both the line of sight (the only path of interest for positioning) and other reflected signals. The peak positions can be evaluated by adopting a threshold mechanism applied in the regions of expected delays. The threshold can be calculated as stated in the Neyman-Pearson theorem. However, in our simulation, since we are interested in the performance of the Position, Velocity and Time (PVT) computation and we know that the signals are present, we skip the phase of signal detection, and we simply estimate the propagation delay by considering the position of the peak.  
In our experiments, we have used a mechanism consisting of two stages; the first one, called _acquisition_, provides a coarse delay estimate, which will be refined by a tracking stage in the second stage. The method adopted in the acquisition stage is based on the comparison of the correlation obtained from the measured samples, with the ideal expected correlation, which is, in the ideal case, of no noise, and other interfering signals, of the type:  
\[R_{N_{e}}(m)=\sum_{l=1}^{N_{e}}R(m-m_{l}) \tag{13}\]The criterion adopted in the acquisition stage is the minimization of the mean squared error between the measured correlation, \(R_{M}(m)\), and the ideal correlation, expressed as:  
\[\hat{\mathbf{m}}=\arg\min_{\mathbf{m}}\sum_{m=1}^{K}\left\|R_{M}(m)-\sum_{l=1}^{ N_{e}}R(m-\bar{m}_{l})\right\|^{2} \tag{14}\]  
where \(\|\cdot\|^{2}\) denotes the two-norm of a vector, \(K\) is the number of correlation points, \(\bar{\mathbf{m}}\) is a test vector, \(\bar{\mathbf{m}}=\{\bar{m}_{1}\bar{m}_{2}\cdots\bar{m}_{N_{e}}\}\), whose elements are variable delays, \(\bar{m}_{l}\), introduced for performing the minimization procedure indicated in Equation (14), and \(\hat{\mathbf{m}}\) is the vector of the estimated delays.  
The minimization can be done by adopting the Matching Pursuit (MP) algorithm [19], which estimates the delays by searching for the correlation peaks in an iterative way. The search space should be the whole range, \(\Delta_{m}\), but in our application, the size of the search interval, \(\Delta_{m}\), can be greatly reduced, thanks to the initialization phase, which allows a prediction of the peak locations. In this way, the estimation of the delays is done only in restricted intervals around the true delays, \(m_{l}\). This allows a hot start of the DVB mode, which presents another important advantage: the majority of peaks due to multipaths are discarded, as they fall outside the search space identified during the initialization phase.  
Once the coarse delay estimation has been achieved by the acquisition stage, the tracking stage can be used to refine this estimation by using an early-late delay lock loop (DLL) similar to the one adopted in a GNSS receiver. An example of DLL design can be found in [6,9]. One difference between the tracking used here and that in GNSS is the re-acquisition phase. This function can be used to refine the tracking performance as the correlation peak jumps from one delay point to the adjacent point. This phase is described in Section 3.2.4.