According to the STAP theory, it has been shown that the rank of clutter covariance is far lower than the DOFs of the system.^{22}^{,}^{23} Consequently, some RR-STAP and RD-STAP algorithms have been used to reduce the filter length, i.e., the filter coefficient vector obtained by full-dimension STAP is sparse.^{14} Hence, in the GSC form of the sparsity-based STAP algorithm (see Fig. 2), the filter coefficient vector $\omega b$ can be replaced by $\omega \u02dcb=V\omega b$, where $V=\Delta diag(v)$ and $v\u2208CNK\u22121$ denote a sparse vector. Then, we obtain Display Formula
$z=VHb\u2208C(NK\u22121)\xd71.$(7)
The output of the sparsity-based STAP is Display Formula$yr=d0\u2212\omega bHz=[1\omega bH\u2212\omega bHVH][yb].$(8)
Hence, the output clutter power for the sparsity-based STAP can be computed as Display Formula$Pr=vtHRxvt\u2212rbdHRb\u22121rbd+\u03f5HRb\u03f5,$(9)
where $\u03f5=\omega b\u2212\omega \u02dcb$ is the weight error vector caused by the sparsity constraint. Note that the target signal power is not affected by the sparsity constraint. The output SCNR can be expressed as Display Formula$\xi r=NM|\alpha |2vtHRxvt\u2212rbdHRb\u22121rbd+\u03f5HRb\u03f5.$(10)
Hence, the aim is to minimize the mean-square error $\u03f5HRb\u03f5$. The objective function of the minimization problem can be rewritten as Display Formula$\u03f5HRb\u03f5=rbdHRb\u22121rbd\u2212rbdH\omega \u02dcb\u2212\omega \u02dcbHrbd+\omega \u02dcbHRb\omega \u02dcb.$(11)
$\omega \u02dcb$ is sparse, i.e., most of its elements are considerably smaller than the others. Hence, the minimization problem can be expressed as Display Formula$min\u2009\u2009\u2212rbdH\omega \u02dcb\u2212\omega \u02dcbHrbd+\omega \u02dcbHRb\omega \u02dcb+\lambda \Vert \omega \u02dcb\Vert 0,$(12)
where $\lambda $ is the regularization parameter for regulating the sparseness of $\omega \u02dcb$. However, the $\u21130$-norm problem is nonconvex. Consequently, it is intractable even for optimization problems with a moderate size. Equation (12) can be further programmed as an LASSO algorithm Display Formula$min\u2009\u2009\u2212rbdH\omega \u02dcb\u2212\omega \u02dcbHrbd+\omega \u02dcbHRb\omega \u02dcb+\lambda \Vert \omega \u02dcb\Vert 1.$(13)
In contrast to Eq. (12), Eq. (13) is convex and can be solved by convex optimization algorithms, such as the interior point method (IPM). The complexity of IPM-STAP can be very high when the size of the problem is large, which is not pragmatic in practice.