As shown in Fig. 1, the radar is mounted on an airplane flying at altitude H with the velocity $Va$. Let the $X$-, $Y$- and $Z$-axes denote the cross-track, along-track, and height directions, respectively. In cross-track, a sparse MIMO array with $MT$ transmitters and $MR$ receivers is adopted to form a uniform virtual array, which has $M=MTMR$ virtual elements and total length $L=(M\u22121)d$, where $d$ represents the element spacing. The position of the $m$’th virtual element is $(xm,yn,H)$, where $yn$ represents the along-track spatial sample coordinate. The illuminated scenario is located at the nadir area of the platform, and for a point target $P$ with reflectivity $\sigma p$ located at $(xp,yp,zp)$ in the 3-D target support band $\Omega $, the instantaneous target-to-$m$’th virtual element distance is Display Formula
$R\u2032(xm,yn;P)=(xm\u2212xp)2+(yn\u2212yp)2+(H\u2212zp)2.$(1)
Assuming that orthogonal pulses with pulse length $Tp$, frequency bandwidth $B$, and carrier frequency $fc$ are transmitted, the echo signal data corresponding to each virtual element channel will transform to the following form after carrier frequency down-conversion, orthogonal demodulation,^{6} and range compression: Display Formula$src(t,xm,yn)=\u222d\Omega rect[xmL]rect[yn\u2212ypLsyn]\sigma pe\u2212j4\pi fccR\u2032sinc[B(t\u22122R\u2032c)]dxpdypdzp,$(2)
where $t$ denotes the fast-time variable, $Lsyn$ denotes the along-track synthetic aperture length, and $c$ is the speed of light. For notational simplicity, we use rectangular window to represent the antenna patterns and signal weighing functions in Eq. (2). In some continuous wave radar, the wavenumber domain form of the range compressed signal $src(t,xm,yn)$ is directly accessible, which is Display Formula$Src(k,xm,yn)=\u222d\Omega rect[k\u2212kcB]rect[xmL]rect[yn\u2212ypLsyn]\xd7\sigma pexp(\u2212jk2R\u2032)dxpdypdzp,$(3)
where $k\u2208[kmin,kmax]$ is the fast-time wavenumber variable, with $kmin=2\pi (fc\u2212B/2)/c$, $kmax=2\pi (fc+B/2)/c$, and $kc=2\pi fc/c$.