Figure 1a shows a typical radio frequency (RF) transmit pulse train of a pulse Doppler radar in which *τ* is the pulse width. For the received signals, two different scenarios may be considered for extraction of the Doppler shift/frequency, *f*
_{
d
} [32, 33]. In a general scenario shown in Fig. 1b, *τ* and *f*
_{
d
} are large enough to have *f*
_{
d
}
*τ* > 1, for which at least one period of *f*
_{
d
} lies within the receive pulse width. However, in the second scenario shown in Fig. 1c, we have *f*
_{
d
}
*τ* < 1 (usually *f*
_{
d
}
*τ* < < 1), in which case several pulses are required to extract *f*
_{
d
}. This scenario; which we have considered in this work, is commonly encountered in aircraft surveillance radars [32]. In this scenario, the effect of the Doppler frequency on each pulse is negligible and may be viewed as the sampling of the Doppler signal by the radar pulse repetition frequency (PRF), *f*
_{
r
}.

The data model and problem formulation for Fig. 1c are presented as follows. As shown in Fig. 2, the transmit signal is a train of *N*
_{
P
} probing pulses each of which containing *N*
_{
s
} sub-pulses with the bandwidth *B*. We assume that the targets are located behind the maximum unambiguous ranges with no ambiguity in the Doppler frequency interval of interest \( \left[\frac{-{f}_r}{2},\frac{f_r}{2}\right) \). This interval is divided into *N*
_{
D
} Doppler bins as

$$ {f}_d=-\frac{f_r}{2}+\frac{f_r\left(d-1\right)}{N_D},\ d=1,2,\cdots, {N}_D. $$

(1)

Now, we calculate the amount of phase shift over one sub-pulse caused by the Doppler phenomenon. Thus, for the *d*th Doppler bin, the Doppler phase shift over one sub-pulse can defined as

$$ {\omega}_d=\frac{2\pi {f}_d}{B},\ d=1,2,\cdots, {N}_D. $$

(2)

By considering the waveform for sub-pulses \( {\boldsymbol{s}}_i\in {\mathrm{\mathbb{C}}}^{1\times {N}_s}, \)
*i* = 1, …, *M*
_{
t
} as the code sequence of the *i*th transmit antenna, transmit signals are defined by an *M*
_{
t
} × *N*
_{
s
} matrix as

$$ \boldsymbol{S}={\left[{\boldsymbol{s}}_1^T\ {\boldsymbol{s}}_2^T\ \cdots\ {\boldsymbol{s}}_{M_t}^T\right]}^T. $$

(3)

Also, the range dimension of surveillance area is divided into *N*
_{
R
} bins. Accordingly, the largest possible delay between the transmit and receive pulses is *N*
_{
R
} − 1. Then, the transmitted pulse waveforms can be arranged into the matrix \( \tilde{\boldsymbol{S}}, \) so that

$$ \tilde{\boldsymbol{S}}=\left[\boldsymbol{S}\kern1em {\mathbf{0}}_{M_t\times \left({N}_R-1\right)}\right] $$

(4)

where \( {\mathbf{0}}_{M_t\times \left({N}_R-1\right)} \) is an *M*
_{
t
} × (*N*
_{
R
} − 1) matrix of zeros, and we have \( \tilde{\boldsymbol{S}}\in {\mathrm{\mathbb{C}}}^{M_t\times \left({N}_s+{N}_R-1\right)} \). Also, we assume that the angular interval of interest *θ*
_{
a
} is divided into *N*
_{
A
} angular bins (*a* = 1, ⋯, *N*
_{
A
} ). Then, in a uniform linear array, the steering vectors of *M*
_{
t
} transmit and *M*
_{
r
} receive antennas are respectively denoted by

$$ {\boldsymbol{a}}_a={\left[\begin{array}{cc}\hfill 1\hfill & \hfill {e}^{-\frac{j2\pi {\Delta}_t \sin \left({\theta}_a\right)}{\lambda_0}}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {e}^{-\frac{j2\pi \left({M}_t-1\right){\Delta}_t \sin \left({\theta}_a\right)}{\lambda_0}}\hfill \end{array}\right]}^T $$

(5)

and

$$ {\boldsymbol{b}}_a={\left[\begin{array}{cc}\hfill 1\hfill & \hfill {e}^{-\frac{j2\pi {\Delta}_r \sin \left({\theta}_a\right)}{\lambda_0}}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {e}^{-\frac{j2\pi \left({M}_r-1\right){\Delta}_r \sin \left({\theta}_a\right)}{\lambda_0}}\hfill \end{array}\right]}^T $$

(6)

where *λ*
_{0} is the radar carrier wavelength and *Δ*
_{
t
} and *Δ*
_{
r
} show the distances between two adjacent transmit and receive antennas, respectively. Therefore, the *p*th received pulse matrix \( {\boldsymbol{Y}}_p\in {\mathrm{\mathbb{C}}}^{M_r\times \left({N}_s+{N}_R-1\right)} \) can be written as

$$ {\boldsymbol{Y}}_p={\displaystyle \sum_{r=1}^{N_R}}{\displaystyle \sum_{a=1}^{N_A}}{\displaystyle \sum_{d=1}^{N_D}}{\alpha}_{r,a,d}{e}^{j{T}_R\left(p-1\right){\omega}_d}{\boldsymbol{b}}_a{\boldsymbol{a}}_a^T\tilde{\boldsymbol{S}}{\boldsymbol{J}}_r+{\boldsymbol{E}}_p $$

(7)

where *Ε*
_{
p
} is the additive white Gaussian noise matrix for the *p*th pulse (*p* = 1, 2, ⋯, *N*
_{
P
}), *T*
_{
R
} is the ratio of pulse repetition interval (PRI) over the single sub-pulse duration, and

$$ {\boldsymbol{J}}_r=\left[\begin{array}{cccc}\hfill \overset{r-1}{\overbrace{0......0}}\hfill & \hfill 1\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \ddots \hfill & \hfill \hfill \\ {}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill 1\hfill \\ {}\hfill 0\hfill & \hfill \hfill & \hfill \hfill & \hfill 0\hfill \end{array}\right] $$

(8)

is an (*N*
_{
s
} + *N*
_{
R
} − 1) × (*N*
_{
s
} + *N*
_{
R
} − 1) shift matrix for the *r*th range bin. Also, *α*
_{
r,a,d
} for *r* = 1, ⋯, *N*
_{
R
}, *a* = 1, ⋯, *N*
_{
A
}, and *d* = 1, ⋯, *N*
_{
D
} denote the return complex reflection coefficients of targets corresponding to the radar cross-section. In this scenario, since the number of range-angle-Doppler bins, in which actual targets are detected, is much smaller than the total number of radar bins, most of reflection coefficients are zero, and thus, the radar signal model can be assumed sparse. We assume that complex reflection coefficients during pulse repetitions are constant.

We will show that (7) can be converted to a 2D sparse from in which the unknown parameters *α*
_{
r,a,d
} are presented in matrix form. By defining and using

$$ {\boldsymbol{v}}_{r,a}=\mathrm{v}\mathrm{e}\mathrm{c}\left[{\boldsymbol{b}}_a{\boldsymbol{a}}_a^T\tilde{\boldsymbol{S}}{\boldsymbol{J}}_r\right], $$

(9)

$$ \boldsymbol{A}=\left[{\boldsymbol{v}}_{1,1}\ \begin{array}{ccc}\hfill {\boldsymbol{v}}_{1,2}\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{v}}_{N_R,{N}_A}\hfill \end{array}\right], $$

(10)

and

$$ {\boldsymbol{x}}_d={\left[\begin{array}{cc}\hfill {\alpha}_{1,1,d}\hfill & \hfill {\alpha}_{1,2,d}\hfill \end{array}\begin{array}{cc}\hfill \cdots \hfill & \hfill {\alpha}_{N_R,{N}_A,d}\hfill \end{array}\right]}^T,\ d=1,\cdots, {N}_D, $$

(11)

in (7), we obtain

$$ {\boldsymbol{y}}_p=\mathrm{v}\mathrm{e}\mathrm{c}\left({\boldsymbol{Y}}_p\right)={\displaystyle \sum_{d=1}^{N_D}}{e}^{j{T}_R\left(p-1\right){\omega}_d}\boldsymbol{A}{\boldsymbol{x}}_d+{\boldsymbol{e}}_p $$

(12)

where *e*
_{
p
} = vec(*E*
_{
p
}) is a complex Gaussian noise vector with zero mean and covariance matrix *I*. Equivalently, in a more compact form, we get

$$ {\boldsymbol{y}}_p=\boldsymbol{AX}{\boldsymbol{\theta}}_p+{\boldsymbol{e}}_p $$

(13)

where \( \boldsymbol{X}=\left[\begin{array}{ccc}\hfill {\boldsymbol{x}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{x}}_{N_D}\hfill \end{array}\right] \) contains the complex reflection coefficients corresponding to the radar cross-section, and \( {\boldsymbol{\theta}}_p={\left[\begin{array}{ccc}\hfill {e}^{j{T}_R\left(p-1\right){\omega}_1}\hfill & \hfill \cdots \hfill & \hfill {e}^{j{T}_R\left(p-1\right){\omega}_{N_D}}\hfill \end{array}\right]}^T. \)

Next, by defining *y*
_{
p
}, *p* = 1, 2, ⋯, *N*
_{
P
} as the columns of matrix *Y*, we have

$$ \boldsymbol{Y}=\left[\begin{array}{ccc}\hfill {\boldsymbol{y}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{y}}_{N_P}\hfill \end{array}\right]=\boldsymbol{AX}\boldsymbol{\Theta } +\boldsymbol{E} $$

(14)

where \( \boldsymbol{E}=\left[\begin{array}{ccc}\hfill {\boldsymbol{e}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{e}}_{N_P}\hfill \end{array}\right] \) and \( \boldsymbol{\Theta} =\left[\begin{array}{ccc}\hfill {\boldsymbol{\theta}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{\theta}}_{N_P}\hfill \end{array}\right] \).

Equation (14) presents a 2D sparse signal model for pulse Doppler MIMO radars where \( \boldsymbol{Y}\in {\mathrm{\mathbb{C}}}^{M_r\left({N}_s+{N}_R-1\right)\times {N}_P}, \)
\( \boldsymbol{A}\in {\mathrm{\mathbb{C}}}^{\left[{M}_r\left({N}_s+{N}_R-1\right)\right]\times \left[{N}_R{N}_A\right]}, \)
\( \boldsymbol{\Theta} \in {\mathrm{\mathbb{C}}}^{N_D\times {N}_P}, \) and \( \boldsymbol{X}\in {\mathrm{\mathbb{C}}}^{N_R{N}_A\times {N}_D} \). Due to the underdetermined nature of (14) for a sparse model (i.e., *M*
_{
r
}(*N*
_{
s
} + *N*
_{
R
} − 1) < *N*
_{
R
}
*N*
_{
A
} and *N*
_{
P
} < *N*
_{
D
}), it has no unique solution. Our goal is to find the *sparsest* matrix for *X* in which we have as many zero elements as possible.

The 2D sparse signal model given by (14) can be converted to a 1D model by using the following property [34]:

$$ \mathrm{v}\mathrm{e}\mathrm{c}\left(\boldsymbol{AX}\boldsymbol{\Theta } \right)=\left({\boldsymbol{\Theta}}^T\otimes \boldsymbol{A}\right)\;\mathrm{v}\mathrm{e}\mathrm{c}\;\left(\boldsymbol{X}\right). $$

(15)

Therefore, we have

$$ \boldsymbol{y}=\boldsymbol{\Phi} \boldsymbol{x}+\boldsymbol{e}, $$

(16)

where *x* = vec(*X*), *y* = vec(*Y*), *e* = vec(*E*), and **Φ** = **Θ**
^{T} ⊗ *A*. Although *x* can be computed using 1D sparse recovery algorithms such as IAA [35] and SLIM [22], due to the large dimension of \( \boldsymbol{\Phi} \in {\mathrm{\mathbb{C}}}^{\left[{N}_P{M}_r\left({N}_s+{N}_R-1\right)\right]\times \left[{N}_R{N}_A{N}_D\right]}, \) 1D solutions are computationally extremely expensive. Accordingly, due to the smaller number of products appeared in (14) compared to (16), developing the 2D algorithm for direct solution of (14) leads to an extreme reduction of computational load compared to 1D one.