# Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions +
## 1 Introduction  
Random linear time-varying (LTV) system theory was first comprehensively described by [PERSON] [2] and has widely been used in the wireless communications field ever since, particularly to model the multipath fading of mobile radio channels [3; 4]. In particular, wide-sense stationary uncorrelated scattering (WSSUS) models, which are a subset of the category of random LTV systems, are the most common in the literature. As well, more recent work in this field has focused on nonstationary LTV communications channels [5; 6; 7], of which vehicle-to-vehicle (V2V) communication is a prime example [8]. A separate set of models that are distinct, but can be related to [PERSON]'s LTV theory are the [PERSON]/Jakes Doppler spectrum class of models [9; 10], originally only applicable for flat fading (i.e., where the symbol time is much larger than the multipath delay spread). This model is also ubiquitous in the wireless communications literature, and many extensions have been proposed, such as for varying geometries [11] and more accurate fading statistics [12]. It is also commonly coupled with [PERSON]'s LTV theory (which is general enough to model frequency-selective fading where the multipath delay spread is much larger than the symbol duration) to model the Doppler component of multipath scattering [13]. Empirical [14] and analytic [15] expressions characterizing range-Doppler spreading of communication channels using LTV theory are commonplace. Outside of communications applications, random LTV system models have been primarily used for sonar target detection and waveform design problems [16; 17; 18], although a few recent papers haveapplied this method to radar target detection problems as well [19; 20]. Much of the literature on the subject treats the problem agnostically, treating radar and sonar as the same problem [21; 22; 23; 24; 25; 26]. In both radar and sonar, the goal is to detect (usually) man-made targets of interest while suppressing thermal noise, which is internal to the receiver, and clutter, which are unwanted signal-dependent returns from the natural environment. Waveform design problems expressed in this form require characterization of the delay-Doppler \"scattering functions\" of the target and clutter, and the result is an optimized transmit waveform/receive filter pair. In addition to waveform design problems, clutter Doppler spectrum characterization is useful for optimized moving target indication (MTI) and space-time adaptive processing (STAP) filtering, which optimize the receiver only. Specifically, the range-Doppler scattering function can be used to predict pulse-to-pulse correlations in the clutter return to suppress it using eigenfilter techniques [27; 28]. An up-to-date summary of recent work on Doppler spectrum modeling can be found in [29]. Most papers on Doppler spectrum modeling do not use [PERSON]'s LTV formulation; the effect of the transmit waveform, measurement system, and clutter are usually combined. This is appropriate if the goal is to generate realistic clutter samples for simulation purposes or if the goal is to design an MTI/STAP filter because the processing is done on receive. However, waveform design requires the clutter to be partitioned separately from the transmit signal, which creates a motivation to \"translate\" these Doppler spectrum results to the \"language\" of random LTV system theory so that these Doppler spectrum models can be utilized in other problem domains. Because scattering function estimation is a fundamental component of waveform design problems, it is a topic that has been studied for decades [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. However, analytic expressions for scattering functions of sea clutter for airborne radars have not been reported in the literature except for some initial work by the author [1; 42; 43]. In this work we derive such analytic expressions for the case when localized internal clutter motion (ICM) is small relative to radar platform motion using a frequency-selective version of the Jakes model.