# Multi-Frequency Target Detection Techniques for DVB-T Based Passive Radar Sensors
## 3 Multi-Frequency Detection Schemes  
We assume that each receiving antenna simultaneously collects \(N\) radio frequency channels, all transmitted by the same transmitter. Then we apply to each frequency channel the first two processing stages described in Section 2, thus obtaining \(NK\) different outputs, namely the CAF maps \(\chi_{k,n}\left[l,m\right]\ \left(k=1,..,K,\ n=1,..,N\right)\) evaluated at \(K\) surveillance channels for \(N\) different frequency channels. The analysis conducted in [23] against FM radio-based PR data demonstrated that quite effective and robust MF detection schemes are provided by:  
1. Resorting to a centralized linear integration followed by a CA-CFAR detection technique, or
2. using a binary integration after applying a first-decision threshold separately on each frequency channel output. As is apparent, both approaches exploit a non-coherent integration across frequency channels whose potentialities are well known in the radar literature [31; 32] where typically this is used to: (i) increase the detection performance by exploiting multiple target echoes received with the same SNR on multiple channels; (ii) average over the fluctuation of the target radar cross section (especially when using frequency agility). However, for the MF PR, we also aim to exploit the non-coherent integration to make detection schemes robust with respect to the undesired characteristics of the received waveforms. When FM radio broadcast transmissions are exploited such undesired characteristics are typically due to the time-varying nature of both the signal AF and of the propagation channel EM conditions (i.e., multipath effects, presence of co-/adjacent-channel interfering transmissions). Based on the successful results obtained in the FM radio-based PR case, we selected the above MF approaches for potential application to the DVB-T-based PR. The resulting detection schemes are briefly summarized below followed by a detailed discussion of the modifications required by the exploitation of the DVB-T broadcast transmissions:  
_Centralized MF detection scheme (CEN-MF-N)_: Basically such an approach operates on sin gle surveillance antennas by incoherently adding (after square law detector) the corresponding CAFs obtained at different frequency channels:  
\[\mathrm{z}_{k}\left[l,m\right]=\sum_{n=1}^{N}\left|\chi_{k,n}\left[l,m\right] \right|^{2} \tag{2}\]  
and then comparing the cell under test \(\mathrm{z}_{k}\left[l_{0},m_{0}\right]\) with the average intensity estimated over \(Q\) secondary data surrounding the cell under test (\(\mathrm{z}_{k}\left[l_{q},m_{q}\right],\ q\in I_{\left[l_{0},m_{0}\right]}, \ \left|I_{\left[l_{0},m_{0}\right]}\right|=Q\)):  
\[\mathrm{z}_{k}\left[l_{0},m_{0}\right]\underset{H_{0}}{\overset{H_{1}}{ \gtrgtr}}\mathrm{G}_{C}\cdot\left(\sum_{q\in I_{\left[l_{0},m_{0}\right]}} \mathrm{z}_{k}\left[l_{q},m_{q}\right]\right) \tag{3}\]  
where, under the Gaussian assumption for the disturbance affecting each map, the threshold \(G_{C}\) is readily found by inverting the following expression of the theoretical false alarm probability:  
\[P_{fa}=\sum_{n=0}^{N-1}\left( \begin{array}{c}QN+n-1\\ n\end{array} \right)\left(\frac{G_{C}}{QN}\right)^{n}\left(1+\frac{G_{C}}{QN} \right)^{-QN-n} \tag{4}\]Along with the expected enhancement of the SNR on the resulting integrated maps, with this approach the characteristics of a bad frequency channel are likely to be averaged with the good ones, thus yielding a limited impact on the final performance. Decentralized MF detection scheme L/N (DEC-MF-L/N)In this case, a detection is declared at a given range/velocity location when \(L\) detections out of \(N\) channels are obtained for the considered pixel over the single-channel maps.