## 3 Signal Processing Method for a Precise Velocity Extraction  
This section presents a processing scheme utilizing the Doppler sensitivity, the absence of range-Doppler coupling in noise radar and the fact that for \(BT_{0}\gg 1\) the signal bandwidth \(B\) of a (pseudo-)random signal is not affected by the segment duration \(T_{0}\). This method reads the functional relationship of (19) in a different light. The deep nulls of the sinc-shaped Doppler-dependency allow for a precise evaluation of the Doppler frequency as the zero points of this function occur at positions for which  
\[f_{D}T_{0}=n\ \,n\in\mathbb{N}^{+} \tag{21}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target** & **Velocity** & **Doppler Frequency** & \(f_{D}<f_{3 dB}\) & **Doppler-Tolerant** \\  & & \(f_{D}\) & & **Segment Duration** \\ \hline sailing boat & 7 knots (3.6 m/s) & 240 Hz & yes (\(\times\)) & 1.8 ms \\ slow car & 30 km/h (8.3 m/s) & 553 Hz & no ( ) & 800 μs \\ fast car & 130 km/h (36 m/s) & 2.4 kHz & no ( ) & 184 μs \\ aircraft & mach 1 (343 m/s) & 22.8 kHz & no ( ) & 19 μs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Common radar targets to be detected by an example X-Band noise radar having a Doppler tolerance of \(f_{3 dB}=443\) Hz defined by a required coherent processing interval of \(T_{r}=1\) ms. The Doppler-tolerant segment duration states the interval the matched filter of the radar must not exceed in order to instead assure a Doppler tolerance with respect to the particular targets. holds true. In Section 2, \(T_{0}\) was considered as a fixed value and (19) was computed for different values of \(f_{D}\) in order to characterize the Doppler tolerance of a given matched filter implementation in noise radar. The idea, here, is to find an unknown Doppler frequency by evaluating the output of multiple matched filter processes, each having a different duration \(T\), that are applied to an identical set of data. However, a variation of the matched filter length also influences the processing gain (18) where \(\frac{1}{L(f_{D})}\) has to be evaluated. In noise radars, the bandwidth \(B\) is constant in all those segments of reasonable duration, as discussed above. Thus, it is only the variation in the matched filter duration which has to be compensated for. This implies the appropriately normalized function  
\[R_{sg}^{(\tau_{D})}(T)=\frac{1}{T}\int\limits_{-T/2}^{T/2}s(t)g^{*}(t-\tau_{D })dt. \tag{22}\]  
Equation (22) describes the matched filter procedure of a signal \(s(t)\) with the replica signal \(g(t)\) for a single delay value \(\tau_{D}\) but different matched filter durations \(T\). The delay value \(\tau_{D}=\frac{2D}{c}\) corresponds to the distance \(D\) of a previously detected target whose velocity information shall be extracted. The term \(c\) denotes the speed of light. But a major difference between the functions of (22) and (19) exists for small input values. When \(f_{D}\to 0\), the relation of (19) converges to unity, whereas for \(T\to 0\) the relation of (22) converges to zero. For this reason, the method of Doppler frequency extraction evaluates the distance between subsequent zero points from which the Doppler frequency of the target can be extracted as follows. For any two subsequent zero points with indices \(n\) and \((n+1)\) it is  
\[\Delta T=(n+1)\frac{1}{f_{D}}-n\frac{1}{f_{D}}=\frac{1}{f_{D}} \tag{23}\]  
and therefore,  
\[f_{D}=\frac{1}{\Delta T}. \tag{24}\]  
Figure 4 illustrates the Doppler extraction algorithm at which a section of duration \(T_{max}\) of the continuous noise signals is extracted to a buffer and evaluated by a series of \(K\) matched filters of different durations \(T(k)\). It is assumed that the maximum available matched filter durations is limited to  
\[0<T(k)\leq T_{max}, \tag{25}\]  
with \(T_{max}\) representing the maximum available segment duration of the signals.