## 2 Quantification of the Doppler Tolerance in CE Noise Radar
The relationship of the waveform \(u(t)\) on the ZDC of the coherent AF can be expressed in compact form [14] by  
\[R_{uu}(\tau)=\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)dt. \tag{4}\]0  
with \(\mathcal{F}\{\cdot\}\) denoting the Fourier transform ([33], [ch.2]), which for a signal \(x(t)\) is  
\[R_{uu}(\tau)=\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)dt. \tag{4}\]1  
From (11) and (12) follows that the zero-delay cut of the resulting ambiguity function of \(u(t)\) is computed by a convolution (the symbol is: \(*\) ) to  
\[R_{uu}(\tau)=\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)dt. \tag{4}\]2  
For \(BT\gg 1\), the convolution in (14) may be approximated by  
\[R_{uu}(\tau)=\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)dt. \tag{4}\]3  
A detailed derivation of this approximation can be found in [34]. Equation (15) indicates that the Doppler tolerance in noise radar is mainly influenced by the shape of the window function \(w(t)\) and quantified as follows. For a simple segmentation of the waveform \(n(t)\) into slices of duration \(T_{0}\), a rectangular window,  
\[w_{r}\Big{(}\frac{t}{T_{0}}\Big{)}=\left\{ \begin{array}{ll}1,&\text{for }|t|\leq \frac{T_{0}}{2}\\ 0,&\text{otherwise}\end{array} \right.\, \tag{16}\]  
is implicitly applied by the segmentation procedure leading to a signal  
\[u_{r}(t)=n(t)\cdot w_{r}\Big{(}\frac{t}{T_{0}}\Big{)}\, \tag{17}\]  
for which the Doppler tolerance is computed in this section. Figure 2 illustrates the signal model of \(u_{r}(t)\), created from a noise source \(N_{0}\) that, idealized, is assumed to have a Gaussian distribution. Figure 2: The signal \(u_{r}(t)\) is modeled by limiting the duration, that is, by segmenting, a signal \(n(t)\) that itself represents a band-limited signal extracted from an (idealized) Gaussian noise source \(N_{0}\). For a rectangular window function, as defined in (16), it is \(\int\limits_{-\infty}^{\infty}w_{r}(t)dt=T_{0}\) and, thus, the resulting gain of the respective matched filter process is equal to  
\[G_{MF,r}=\frac{1}{L_{r}(f_{D})}BT_{0}, \tag{18}\]  
with B denoting the bandwidth of the signal \(n(t)\). The equation stated in (15) stresses out that the Doppler behaviour of the ambiguity function of the particular noise waveform defined in (11) is dominated by the shape of the window function \(w(t)\). The normalization in (8) and the fact that \(R_{nn}(0)=const.\) in (15) lead to the following formulation of the Doppler-sensitive loss function of the signal \(u_{r}(t)\). It is  
\[\left|\frac{1}{L_{r}(f_{D})}\right|= \left|\mathcal{F}\left\{\left|w_{r}\left(\frac{t}{T_{0}}\right) \right|^{2}\right\}\right| \tag{19}\] \[= \left|\frac{\sin\left(\pi f_{D}T_{0}\right)}{\pi f_{D}T_{0}} \right|,\]  
where \(T_{0}\) represents the duration of the rectangular window, defined in (16). This Doppler-behaviour is illustrated in Figure 3.  
Equation (19) describes a symmetrical function and, thus, only the positive Doppler axis is drawn in Figure 3. The dotted red line indicates the position of the cutoff Doppler frequency, achieved by applying the rule of (9) to (19) and resulting in  
\[|f_{3 dB,r}|\approx\frac{0.443}{T_{0}}. \tag{20}\]  
The subscript \(r\) indicates the relationship of this Doppler tolerance rule to the particular shape of the rectangular window function of the noise signal \(u_{r}(t)\) as defined in (16) and (17).