## 2 Quantification of the Doppler Tolerance in CE Noise Radar  
The general problem of Doppler tolerance in radar is discussed in the literature with different terminology. In 1966, the author of [20] aimed for a waveform design with Doppler invariance by focusing on range-Doppler coupling. As the latter is absent in random waveforms, the approach presented in that paper is not applicable to noise radar technology. The problem of Doppler tolerance is also discussed in [21, 22, 23], but the focus of these papers lies on transmitter code-design which contradicts the idea of randomness in noise radar emissions. The authors of [24, 25] propose a combination of transmission wave design (Golay) combined with expensive phase compensations on receive. An idea, rejected for the same reason. Also in [26], the achievement of Doppler invariance is used as the design criterion for specialized codes, still, that paper does not formally investigate the Doppler tolerance. The most formal approach to Doppler tolerance of radar waveforms can be found in [27] biased by an LFM signal background. But many aspects in [27] can be easily ignored when focusing on noise radar: The particular shape of the ambiguity function of noise signals and the absence of range-Doppler coupling in noise radar allows for a description of the AF by two cuts [7], one cut along the delay axis and a second cut along the Doppler axis. In [28], the Doppler tolerance of OFDM radar waveforms is formulated as a function of the number of carriers in the OFDM signal. The results are not directly applicable to CE noise radar but the paper formulates a general definition of a Doppler characterization by defining a loss-measure based on the ambiguity function. This approach inspired the, still different, procedure for characterizing the Doppler tolerance in CE noise radar of the current paper. The ambiguity function [29], more precisely the auto-ambiguity function, of a radar waveform \(u(t)\) shall be denoted as  
\[|\chi(\tau,f_{D})|=\left|\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)e^{i2 \pi f_{D}t}dt\right|, \tag{3}\]  
which defines a two-dimensional function in the delay-Doppler coordinate system. The ambiguity function for \(f_{D}=0\) forms the auto-correlation function (ACF) of the signal \(u(t)\) with  
\[R_{uu}(\tau)=\int\limits_{-\infty}^{\infty}u(t)u^{*}(t-\tau)dt. \tag{4}\]  
Any concentration of energy located aside the origin of the delay-Doppler coordinate system in (3) might cause ambiguous range-Doppler measurements in radar applications. This might potentially result in the extraction of wrong target parameters or even in false targets. But CE noise radar has an ambiguity function which is close to ideal as is shown in Figure 1 representing an illustration of the thumbtack-shaped ambiguity function of an idealized noise waveform. In real-world applications, restricted to the usage of band-limited and finite noise signals, the resulting ambiguity function still remains similar to the idealized AF shown in Figure 1 but it does not form a two-dimensional Dirac function (also known as Kronecker Delta). Instead, the single peak in the origin of the delay-Doppler coordinate system has a certain spread in both dimensions and is of finite height. The peak height, characterised by the peak-to-average-sidelobe ratio, is determined by the time-bandwidth product of the waveform [30]. The spread of the peak in the delay dimension defines the well-known range resolution and is caused by the finite signal bandwidth ([31], [ch.9]). However, the  
Figure 1: Illustration of the thumbtack-like ambiguity function of a close-to-ideal noise waveform with wide bandwidth and long duration. Neither range nor Doppler ambiguities exist and range-Doppler coupling does not occur. spread in the Doppler dimension, along with its cause, is evaluated here in order to extract a measure for the Doppler tolerance of noise radar technology.