# Multi-Frequency Target Detection Techniques for DVB-T Based Passive Radar Sensors
## 3 Multi-Frequency Detection Schemes
### Modifications Required for Application Against DVB-T Signals
#### 3.1.1 CAF Maps Evaluation on Common Bistatic Range/Velocity Grids
Specifically, the maximum relative displacement with respect to the nominal pixel spacing is given by:  
\[E_{rel_{n}}=\frac{1}{\Delta v}\left\lfloor\frac{v_{max}}{\Delta v}\right\rfloor \left|\varepsilon_{v_{n}}\right|=\left\lfloor\frac{v_{max}}{\Delta v}\right\rfloor \frac{|\delta T_{n}|}{T_{int_{a}}+\delta T_{n}} \tag{11}\]  
and assuming that \(v_{max}\) is an integer multiple of \(\Delta v\) and that \(\delta T_{n}\ll T_{int_{a}}\), we obtain:\[E_{rel_{n}}\cong\frac{v_{max}}{\lambda_{n}}\left|\delta T_{n}\right| \tag{12}\]  
As is apparent, the maximum relative displacement \(E_{rel_{n}}\) at the \(n\)-th frequency channel is a function of the carrier frequency and varies with the rounding error affecting the selection of the CPI length and with the CAF extent in the velocity dimension, which in turn depends on the considered surveillance application. The result is shown in Figure 2 assuming, as a worst case, \(\left|\delta T_{n}\right|=\frac{T_{in}}{2}=2.24\cdot 10^{-3}\) s for a 8 k transmission mode, a useful symbol duration \(T_{U}=896\)\(\upmu\)s and a guard interval of duration \(T_{GI}=T_{U}/4\). The maximum relative displacement in Equation (12) is reported as a function of the carrier frequency for two different surveillance applications. Specifically, for the case of a maritime scenario, the maximum expected bistatic velocity for targets of interest is arbitrarily set to 20 m/s; with such position, the maximum relative displacement of the CAF grid in the velocity dimension is negligible across the UHF band. In contrast, taking \(v_{max}=500\) m/s as for aerial surveillance applications, such a displacement might easily exceed the velocity cell thus making the proposed integration scheme completely ineffective. In addition, it is to be noticed that, due to the large diversity among the available carrier frequencies within the DVB-T band, following the approach in Equation (8) would yield non-negligible differences among the jointly exploited frequency channels both in term of coherent integration gain and in term of computational burden. In fact, in the DVB-T case, even limiting the analysis to the UHF band, the largest ratio \(T_{int_{n}}/T_{int_{m}}\) across the band would yield a difference in term of SNR gain approximately equal to 2.6 dB and a possible increase in term of computational load up to 80%. These differences cannot be neglected as they would result in a significant imbalance in the contributions provided by different integrated channels as well as in the computational resources to be assigned to each channel. Taking into account the considerations above, in order to benefit from a MF integration scheme also in a DVB-T-based PR, we propose to modify the CPI length selection strategy with respect to the solution in [23]. Specifically, the nominal \(T_{int}\) is kept constant across different frequency channels in order to guarantee the same coherent integration gain. The value of \(T_{int}\) is selected so as to guarantee the desired velocity resolution at the highest carrier frequency included in the integration scheme, i.e., \(T_{int}=\frac{\lambda_{min}}{\Delta v}\). This would obviously yield under-sampled grids at lower carrier frequencies. Slight differences in the selection of the CPI length are allowed due to the rounding errors in Equation (9). Basically these errors still apply due to the constraints that regulate the selection of  
Figure 2: Maximum relative displacement of the CAF grid in the velocity dimension due to the rounding errors affecting the CPI length selection. the CPI length; in this regard, we observe that different frequency channels might be transmitted using different transmission modes and/or modulation parameters. Consequently we observe that the rounding errors might differ from one channel to another channel even assuming the same desired CPI length.