## 2 Quantification of the Doppler Tolerance in CE Noise Radar
#### Impact on Noise Radar Implementations  
The interpretation of (20) leads to the general statement that the longer the matched filter duration \(T_{0}\) the narrower is the main peak of the function \(1/L_{r}(f_{D})\), as \(|f_{3 dB}|\to 0\)  
Figure 3: The matched filter of duration \(T_{0}\) applied to noise radar signals creates a sinc-shaped Doppler characteristic in the zero-delay cut of the ambiguity function \(|\chi(\tau=0,f_{D})|^{2}\). The Doppler loss \(L_{r}(f_{D})\) is higher than 3 dB for \(f_{D}>\frac{0.443}{T_{0}}\) as indicated by the dashed line.  
for \(T_{0}\to\infty\). Therefore, all noise radar applications that use long signal segments for the range-correlation have to cope with a resulting limited Doppler tolerance illustrated by the following example.  
An X-band radar is considered whose parameters are roughly compliant with the demonstrator used in Section 4 and comparable to small marine navigation radars such as [35]. The radar emission is assumed to be restricted, for example by regulations or hardware limitations, to a bandwidth of 50 MHz around a center frequency of 10 GHz. Thus, for an LPI application, any reduction of the emitted power whilst maintaining a certain aimed signal-to-noise ratio for a given task can only be accomplished by utilizing a long processing interval. For example, a radar pulse of 1 us duration and 1 kW peak power might be replaced by a noise signal of the same bandwidth but with 1 W peak power and 1 ms duration. Without going into the details of any actual radar task, a coherent processing interval of \(T_{r}=1\) ms shall be considered favourable for the noise radar in this paragraph leading to a maximum coherent processing gain of \(G=BT_{r}\approx 47\) dB. According to (20), this value of \(T_{r}\) would result in a Doppler cutoff frequency of \(f_{3 dB}=443\) Hz.  
Table 1 illustrates the Doppler tolerance of this particular radar implementation with respect to common radar targets moving at typical velocities. The slowest target, a sailing boat given in the first line of Table 1, is the only target that is compliant with the Doppler tolerance requirement of \(f_{D}<f_{3 dB}\). For all the other targets, either a cost-intense Doppler compensation needs to be performed or the coherent processing interval, that is, the segment duration, needs to be shortened to a value \(T_{r}\leq\widehat{T}_{0}\), presented in the last column of the table, with the consequence of a reduced achievable processing gain. The duration \(\widehat{T}_{0}\leq\frac{0.443}{f_{3 dB}}\), derived from (20), is interpreted as the _Doppler-tolerant segment duration_.  
However, for many applications, especially with LPI background, the coherent processing interval needs to have a certain duration according to the requirements on the processing gain. If the maximum expected Doppler frequency exceeds the \(f_{3 dB}\), then a computationally intense Doppler compensation becomes essential in noise radar even in cases at which the extraction of the target velocity is not required.  
This fact might be considered as a drawback of noise radars but for those applications that indeed ask for Doppler information of the targets, the Doppler sensitivity of noise radar provides an advantageous opportunity that is presented in the next sections including a set-up of two road vehicles for an experimental proof of this idea.