## 2 Quantification of the Doppler Tolerance in CE Noise Radar
The matched filter output of a radar echo shows a particular signal-to-noise ratio (SNR), evaluated by common detection algorithms as presented in ([32], [ch. 2]). It is  
\[\text{SNR}=\frac{P_{y}}{P_{N}}, \tag{5}\]  
where \(P_{y}\) denotes the signal power of the, in the frame of this paper matched filtered, radar echo and \(P_{N}\) denotes the power of the receiver noise, bounded below by the thermal noise. The current paper proposes a quantification of the Doppler tolerance of CE noise radar by investigating the influence of the Doppler effect on the matched filtered signal power \(P_{y}\). In this approach, the squared ambiguity function \(|\chi(\tau,f_{D})|^{2}\), directly proportional to \(P_{y}\), is used as the main indicator of the loss that a particular Doppler shift of an echo signal has on the resulting correlator output. As the current work focuses on CE noise radars, the following derivations make use of the characteristic thumbtack-like shape of its ambiguity function. These considerations lead to a loss function \(L(\tau,f_{D})\), given by  
\[\frac{1}{L(\tau,f_{D})}=\frac{|\chi(\tau,f_{D})|^{2}}{|\chi(0,0)|^{2}}, \tag{6}\]  
with the maximum value of the ambiguity function always located at \((\tau=0,f_{D}=0)\). The illustration of the ambiguity function in Figure 1 emphasizes the absence of range-Doppler coupling in noise radar [7] why, in this case, (3) may be computed by  
\[|\chi(\tau,f_{D})|=|\chi(\tau,0)|\cdot|\chi(0,f_{D})|. \tag{7}\]  
This independence of delay and Doppler in the ambiguity function of CE noise radar leads to the idea to fully characterize this Doppler tolerance by evaluating the zero-Delay cut (ZDC) of the AF, that is, \(|\chi(0,f_{D})|\). Consequently, for CE noise radar, the indicator of the Doppler tolerance, defined in (6), simplifies to  
\[\frac{1}{L(f_{D})}=\frac{|\chi(0,f_{D})|^{2}}{|\chi(0,0)|^{2}}. \tag{8}\]  
A threshold value of \(L_{th}=2\) is set that represents the \"3 dB point\" of the ambiguity function in the Doppler dimension. Consequently, a _cutoff-frequency_\(f_{3 dB}\) shall be defined as  
\[f_{3 dB}=min(|f_{D}|):L(f_{D})\geq L_{th}\,. \tag{9}\]  
In the following, the cutoff-frequency is investigated by evaluating the respective zero-delay cut of the ambiguity function depending on the matched filter duration. In noise radar technology, the emitted signal \(g_{0}(t)\) is often assumed to be the realization of a Gaussian or at least of an ergodic random process. The echo of a target located in a particular distance \(D\) and moving with a radial velocity \(v_{r}\) forms a received signal  
\[s_{0}(t)=A\cdot g_{0}\bigg{(}t-\frac{2D}{c}\bigg{)}e^{-j2\pi f_{D}t}, \tag{10}\]  
which is a delayed and Doppler-shifted copy of the transmitted signal \(g_{0}(t)\) attenuated by a factor \(A\) that mainly depends on the range and the size of the target. The term \(f_{D}\) was defined in (1). A range measurement can be performed by computing the cross-correlation of the received signal and the transmitted signal, in other words by a matched filter process. But any practical computation of the matched filter has a finite duration which requires a segmentation of the continuous noise signals \(s_{0}(t)\) and \(g_{0}(t)\) to finite signals \(s(t)\) and \(g(t)\)In general, the segmentation procedure of a noise radar waveform can be modelled by an appropriate window function \(w(t)\) from which a signal,  
\[u(t)=n(t)\cdot w(t), \tag{11}\]  
is computed with \(n(t)\) representing a realization of a white Gaussian process filtered to a 3 dB bandwidth equal to \(B\). A specific window function is presented later in this section. From the definition of the ambiguity function, given in (3), it follows that for \(\tau=0\) the phase of the waveform \(u(t)\) has no influence on the zero-delay cut \(\chi(0,f_{D})\).