### I. Introduction

### II. CCM Modeling and Estimation

### 1. Sea CCM Modeling

*N*consecutive pulse transmissions and corresponding receptions during period

*T*into a radar scene. To focus on sea CCM modeling, the target signature in the radar scene is neglected in this section.

*c*

*[*

_{j}*n*] denote a clutter return of the

*j*th range cell for the

*n*th pulse. For

*N*consecutive transmissions, we obtain the clutter return vector of the

*j*th range cell, {

*c*

*[*

_{j}*n*]}(

*n*= 0, 1, ···,

*N*− 1). Let

*c*

*[*

_{j}*n*]}, where

*k*is related to Doppler frequency

*C*

*[*

_{j}*k*] is modelled as the product of two random variables: texture and speckle. Using the Wiener–Khinchin theorem, the DFT of the autocorrelation function of

*c*

*[*

_{j}*n*] corresponds to the clutter power spectral density (PSD), which is approximately E[|

*C*

*(*

_{j}*k*)|

^{2}]. According to the DFT theory,

*c*

*[*

_{j}*n*] can be expressed as

*c**∈ ℂ*

_{j}

^{N}^{×1}denotes

*c**≡ [*

_{j}*c*

*[0], …,*

_{j}*c*

*[*

_{j}*N*−1]]

^{T}(superscript T denotes transpose) and for notational simplicity,

*r*

_{j}_{,}

*is used in place of*

_{k}*C*

*[*

_{j}*k*]. Meanwhile,

*u**in Eq. (2) denotes*

_{k}

*u**∈ ℂ*

_{k}

^{N}^{×1}is, in fact, a Doppler steering vector of

*u**| = 1. The overall interference vector*

_{k}

*v**can be expressed as*

_{j}

*n**is a complex additive white Gaussian noise vector. Using Eq. (2) and the statistical independence between*

_{j}

*c**and*

_{j}

*n**, the theoretical covariance matrix for*

_{j}

*v**, i.e.,*

_{j}

*R**∈ ℂ*

_{j}

^{N}^{×}

*, is expressed as*

^{N}

*u**⊥*

_{k}

*u**for*

_{m}*k*≠

*m*and

*u**⊥*

_{k}

*n**. According to [16], a collection of random variables—namely,*

_{j}*γ*

_{j}_{,}

*s—can be assumed to be mutually independent, having zero mean. That is,*

_{k}*δ*

_{k}_{,}

*is the delta function defined as*

_{m}*δ*

_{k}_{,}

*= 1 for*

_{m}*k*=

*m*and

*δ*

_{k}_{,}

*= 0 otherwise.*

_{m}*σ*

^{2}is noise variance, and

**is the identity matrix.**

*I**N*×

*N*matrix whose columns are the eigenvectors of

*R**, and*

_{j}

*Λ**= diag{*

_{j}*λ*

_{j}_{,1}

*λ*

_{j}_{,2}…

*λ*

_{j}_{,}

*} is a diagonal matrix in which*

_{N}*λ*

_{j}_{,}

*=*

_{k}*Nη*

_{j}_{,}

_{k}^{2}+

*σ*

^{2}is an eigenvalue corresponding to eigenvector

*u**∈ ℂ*

_{k}

^{N}^{×1}.

### 2. Proposed Algorithm for CCM Estimation

*R̃**denote an estimate for*

_{j}

*R**obtained by the SCM, which used to be a primary choice for CCM estimation.*

_{j}

*R̃**is then expressed as*

_{j}*L*is the number of valid training vectors adjacent to the CUT and

*v**is the interference vector at the*

_{l}*l*th range cell.

*R̃**cannot be ideally eigen-decomposed, as expressed in Eq. (8).*

_{j}

*z**be the DFT of*

_{l}

*v**.*

_{l}

*z**can then be written as*

_{l}

*c**.*

_{j}

*z**is then*

_{l}*γ*

_{j}_{,}

*s, as mentioned in Eq. (6). Although*

_{k}**. Thus, the proposed CCM estimate, hereafter denoted as**

*I*

*R̂**, becomes*

_{j}

*v**s from the range bins adjacent to the CUT, ii) obtaining*

_{l}

*z**s by taking the DFT of*

_{l}

*v**s, iii) calculating*

_{l}### III. Filter Construction Using Estimated CCM

### 1. Max SINR Filter

*p*

_{1},

*p*

_{2}] in the

*k*domain.

*λ*

_{j}_{,}

*in (8) is then expressed as*

_{k}

*s**with unity intensity, optimum weight*

_{r}

*w*

_{j}_{,}

*becomes [17]*

_{r}*k*≠

*r*,

*s**is orthogonal to*

_{r}

*u**. In this case,*

_{k}

*w*

_{j}_{,}

*is expressed as*

_{r}*f*

*, i.e.,*

_{q}*k*=

*q*. In this case, the received radar return from the

*j*th range cell,

*x**, can be expressed as*

_{j}*ζ*

*is the target reflectivity.*

_{j}*ζ*

*is assumed constant in the following derivation because the change in target reflectivity for short processing intervals might be negligible. To examine the SINR of the filter output at the target bin, that is, the*

_{j}*q*th Doppler bin, we let

*f*

*be located somewhere within the clutter PSD, i.e., −*

_{q}*p*

_{1}≤

*q*≤

*p*

_{2}. The max SINR filter output at

*f*

*, namely,*

_{r}*y*

_{j}_{,}

*, then becomes*

_{r}*β*

*=*

_{r}*Nη*

_{j}_{,}

_{r}^{2}+

*σ*

^{2}. The output SINR at the

*q*th Doppler bin becomes

*γ*

_{j}_{,}

*and*

_{q}*q*th Doppler bin is

*q*th Doppler bin is the same as that of the target return at the

*q*th Doppler bin, except for being scaled by an unknown complex amplitude

*γ*

_{j}_{,}

*. Hence, the filter weight*

_{q}

*w*

_{j}_{,}

*cannot make differences in the SINRs before and after filtering. Moreover, when noise power is negligible,*

_{r}*N*does not contribute to SINR improvement, as will be shown in Section IV.

*r*th Doppler bin be one out of the target-free region, that is,

*r*≠

*q*and −

*p*

_{1}≤

*r*≤

*p*

_{2}. Employing some manipulations, the clutter power at the

*r*th Doppler bin, which is the mean square value of y

_{j,r}, E[|y

_{j,r}|

^{2}], is expressed as

*r*th Doppler bin before filtering is

*η*

_{j}_{,}

_{r}^{2}. This result implies that the bigger

*η*

_{j}_{,}

_{r}^{2}and

*N*are, the larger the clutter suppression ratio becomes.

### 2. Whitening Filter

*f*

*,*

_{r}*y*

_{j}_{,}

*, is*

_{r}*β*

*=*

_{r}*Nη*

_{j}_{,}

_{r}^{2}+

*σ*

^{2}, which is defined the same way it was in Eq. (19). As expected, at the target-free bins where

*r*≠

*q*, the mean square value of

*y*

_{j}_{,}

*in (21) becomes approximately 1, which implies that the interference PSD is whitened. The SINR at the*

_{r}*q*th target Doppler bin is approximately

*r*≠

*q*and −

*p*

_{1}≤

*r*≤

*p*

_{2}, the E[|

*y*

_{j}_{,}

*|*

_{r}^{2}]s of the max SINR filter and the whitening filter are approximately

*r*th Doppler bin is

*η*

_{j}_{,}

_{r}^{2}, their interference suppression ratios are approximately

*Nη*

_{j}_{,}

_{r}^{2}is greater than 1, one filter outperforms the other.

*y*

_{j}_{,}

*|*

_{r}^{2}]s of the max SINR filter and the whitening filter are approximately

*y*

_{j}_{,}

*in Eq. (21). The calculated noise power ratio is*

_{r}### IV. Simulations

*c*

*[*

_{j}*n*] (

*n*= 0, ···, 63), which is the output of the FIR filter. The solid line in Fig. 1 shows an ensemble average of |

*C*

*[*

_{j}*k*]|

^{2}, which is the clutter PSD.

*Lorentzian*function characterized by a small number of mean Doppler frequencies. We arbitrarily chose two mean Doppler frequencies, −150 Hz and 200 Hz. We next obtained the speckle Doppler function by convolving the

*Lorentzian*function with an arbitrarily chosen Gaussian function. Since HH polarization returns are related to scatterers moving faster than Bragg scatterers, the typical function shape of the Doppler spectrum cannot be specified in such a way as to associate it with sea state and radar parameters. Thus, we do not enumerate the associated parameter values. For a given Doppler function, the rest of the procedure to simulate the time-series speckle return data is the same as before. The dashed line and the solid line in Fig. 2, respectively, show the initially chosen Doppler function and the PSD of the simulated HH polarized speckle data.

*R̃**using the SCM shown in Eq. (9) with the VV polarized data, we used 25 training vectors adjacent to the CUT. We next decomposed*

_{j}

*R̃**into eigenvectors and eigenvalues. For all eigenvectors, we took the DFT and obtained their spectra. Fig. 3(a) shows the spectra arranged in such a manner that the spectrum of an eigenvector associated with a larger eigenvalue lay to the left of the*

_{j}*x*axis. We next applied the proposed algorithm to obtain

*R̂**. Fig. 3(b) shows the plots of the spectra of the eigenvectors in*

_{j}

*R̂**, which are arranged as in Fig. 3(a). We note that from the spectra shown in Fig. 3(b), the eigenvectors look to have a form of a Doppler steering vector of a single frequency, which is particularly apparent in the eigenvectors associated with larger eigenvalues. In order to determine the results that could be achieved if there were a sufficient number of training vectors, we arbitrarily generated 250 training vectors, calculated the*

_{j}

*R̃**using the SCM, and finally obtained the eigenvector spectra shown in Fig. 3(c). More eigenvectors with larger eigenvalues appear to have the form of Doppler steering vectors of a single frequency. Fig. 3(d) is the plot of the eigen vector spectra obtained with the HH polarized data by applying the proposed method. Two dominant Doppler frequencies can be observed in the spectra of the eigenvectors associated with the large eigenvalues.*

_{j}

*R̃*

_{j}^{−}

^{1}

*s**,*

_{r}

*R̂*

_{j}^{−}

^{1}

*s**, and*

_{r}

*R̃*

_{j}^{−}

^{1}

*s**using 250 training vectors is called filter D. Fig. 4(a)–(d) show the frequency responses of filters A–D, respectively. The frequency responses of filter B and filter C appeared smoother around the clutter-dominant region than those of filter A, which implies that filter A might not effectively suppress clutter components. As expected, filter D, which was obtained with many training vectors, exhibits better performance than filter A.*

_{r}