## 1 Introduction
### Background  
Radar signals are sensitive to the motion of the target and to the motion of the radar platform itself. The Doppler effect causes a frequency shift of the involved radio waves in dependency of the relative radial velocity \(v_{r}\) between the radar and the target. In many radar applications, the Doppler frequency \(f_{D}(t)\) is assumed to be constant during the observation time and is approximated, neglecting relativistic effects, by  
\[f_{D}=\frac{-2v_{r}f_{0}}{c}, \tag{1}\]  
with \(f_{0}\) denoting the carrier frequency of the radar signal.  
The historical principle [9] to compensate for the Doppler shift of a radar echo uses a Doppler filter bank, that is, a bank of matched filters which are often tuned to equally spaced Doppler frequencies. In this early implementation, the matched filtered signals then were delivered to a decision network, which in [9] is a \"greatestof\" signal selection. The Doppler filter bank principle was improved over the years leading to sophisticated designs such as [10]. Digital signal processing has opened many occasions for strategies of Doppler analyses, used for the extraction of a target's velocity information, and also for Doppler compensations.  
To efficiently implement a digital Doppler filter bank, a Fast Fourier Transform (FFT) is applied to a series of range measurements. A detailed description can be found in ([11] [ch.17]). The number of consecutive range measurements shall be denoted by \(P\) which, together with the pulse repetition interval \(T_{PRI}\) of a conventional radar, forms the coherent processing interval of the FFT being \(T_{CPI}=P\cdot T_{PRI}\). The digital sampling rate of the resulting FFT is equal to the pulse repetition frequency \(\mathrm{PRF}=\frac{1}{T_{PRI}}\). This signal domain, which represents temporal variations of range measurements, is often referred to as the slow-time scale [12]. A general design criterion of Doppler filters is given in ([13][ch.18]), stating that the bandwidth of each Doppler filter needs to span at least the frequency distance of the filters, a condition which is inherently met when the FFT is applied. The frequency resolution of the Doppler-FFT corresponds to the inverse of the coherent processing interval equal to  
\[\Delta f_{D}=\frac{1}{P\cdot T_{PRI}}. \tag{2}\]  
Moreover, the measurement of the Doppler frequency is ambiguous by multiples of the PRF [14], unaffected by the value of \(P\).  
In order to apply the method of Doppler filter banks to continuous radar signals, either for deterministic continuous wave (CW) radars like the well-known frequency modulated continuous wave (FMCW) or for CE noise radar, a functional equivalent to the PRI needs to be identified. For continuous radar signals, having a duty cycle of 100%, obviously, no pulses exist but the Doppler filter bank principle is directly applicable by segmenting the radar signal properly into slices of duration \(T_{r}\). The choice of \(T_{r}\) depends on the waveform modulation. In an FMCW signal, for example, the segmentation ought to follow the periodic modulation: it practically covers a single frequency ramp in order to maintain the full signal bandwidth during each range measurement. But for the non-repetitive signals in continuous emission noise radar, the segment duration \(T_{r}\) can be chosen freely: The continuous emission of noise radar is understood as an interruption-free non-repetitive modulation based on a random process of limited, constant bandwidth. The randomness of this modulation is non-repetitive and the full signal bandwidth \(B\) is covered in any segment \(T_{r}\) that satisfies the requirement \(B\cdot T_{r}\gg 1\).  
Consequently, in continuous emission radars, the interval \(T_{CPI}\) of an FFT-based Doppler analysis may be formed by \(P\) segments, each of duration \(T_{r}\).