Angle and Range Unambiguous Estimation with Nested Frequency Diverse Array MIMO Radars

Abstract

This paper proposes an unambiguous method for joint angle and range estimation in colocated multiple-input multiple-output (MIMO) radar using the nested frequency diverse array (NFDA). Unlike a conventional phased array (PA), the transmission beampattern of FDA-MIMO radar depends not only on angle but also on range, which enables the precise identification of ambiguous regions in the two-dimensional frequency space. As a result, we can simultaneously estimate the angle and range of targets using FDA-MIMO radar, even when range ambiguity exists. By employing a nested array configuration, the degrees of freedom (DOFs) of the FDA are expanded. This expansion leads to improved accuracy in parameter estimation and enables a greater number of identifiable targets. In addition, the Cramér–Rao lower bound (CRLB) and the algorithm complexity are obtained to facilitate performance analysis. The simulation outcomes are presented to showcase the superior performance of the suggested approach.

1. Introduction

Parameter estimation is a critical task in radar target detection, with widespread applications in both military and civilian sectors, such as target pursuit, seismic supervision, and traffic surveillance [1,2,3,4]. The features of each target include azimuth, range, speed, and reflection complex amplitude, among others. Conventional phased array (PA) radars can offer high spatial resolution, thus being widely utilized in target angle estimation [5]. However, the beam steering of the PA radar is kept in a single direction across all range bins, which leads to an inability to directly estimate target ranges when range ambiguity exists. In order to tackle this problem, a multiple pulse repetition frequency (PRF) radar was proposed in [6]. Nevertheless, the inherent accompanying errors in range estimation processes can lead to the generation of ghost targets. The utilization of iterative adaptive methods [7] and staggered PRF [8] can also mitigate range ambiguity. Nonetheless, the implementation of these techniques may lead to a deterioration in clutter suppression performance. In this paper, we focus on using a single PRF to mitigate range ambiguity.

The notion of a frequency diversity array (FDA) first appeared in [9], while [10] presented its hardware implementation and a generalized model. In contrast to conventional phased arrays, the most notable feature of FDA is the introduction of linearly increasing frequency offsets among the radiating elements, which enables the FDA antenna to create a beampattern that depends not only on the angle but also on the range [11]. The coupled feature has garnered significant interest due to its potential use in several areas, such as target detection [12], joint angle–range estimation [13], suppression of range ambiguity clutter [14], and countermeasures against deceptive interference [15]. Nevertheless, the far-field beampattern of a uniform linear FDA shows periodic variations, and the range and angle responses are mutually coupled [16]. To deal with these concerns, several methodologies have been introduced. Since the radiation pattern of an FDA is controllable by adjusting the frequency offset, numerous frequency offset design schemes have been proposed, such as logarithmic [17], Taylor-window [18], and random [19] frequency offsets, among others. These schemes aim to create a time-invariant, spatial-focusing, directional radiation pattern. To determine the optimal frequency offset, several optimization algorithms were developed [20,21]. Furthermore, in [22,23], an optimization method for the transmitter and receiver of an FDA was proposed to mitigate signal-dependent clutter. In [24], a novel structure for the PA-FDA dual-mode radar was proposed to improve parameter estimation accuracy.

The angle and range coupling can also be decoupled by configuring the array appropriately. Multiple-input multiple-output (MIMO) technology has been employed to create a beampattern that decouples angle and range [25,26,27]. Furthermore, sparse array configurations, for instance, minimum redundancy arrays (MRAs) [28], nested arrays [29], and coprime arrays [30], have been introduced because they can achieve larger array apertures and overcome the degree of freedom (DOF) constraints imposed by a finite amount to antennas. Various parameter estimation algorithms have been proposed with respect to sparse arrays [31,32,33]. In [34], a multisignal direction of arrival (DOA) measurement method using MRA subarrays was submitted. In [35], a linear nested array was utilized to detect the DOAs and the number of distributed target signals. Moreover, the authors of [36] developed a novel Bayesian learning algorithm aimed at enhancing the accuracy of DOA estimation for nested arrays. The MIMO radar employing coprime FDAs was investigated in [37], enabling more precise estimation of various target parameters. In the absence of a closed-form solution for the positional coefficients of the MRAs, and given the excessive redundancy present in coprime arrays, this paper focuses on the study of nested arrays. For angle–range joint estimation with high accuracy, we proposed a modified multiple signal classification (MUSIC) algorithm for the nested FDA-MIMO radar in [38]. However, the issue of range ambiguity was not considered.

In this article, a colocated NFDA-MIMO radar architecture is proposed that aims to achieve high-precision estimation of target angles and ranges, even in the presence of range ambiguity. Additionally, the architecture aims to enhance the quantity of source estimates. During the model construction phase, the transmitting and receiving arrays are configured as identical nested arrays. Following this, the received signals are transformed into a single snapshot signal through differential processing, which extends the array aperture. This expansion increases the DOF of the array, resulting in better radar resolution, a greater number of target estimations, and improved estimation accuracy. To address the problem of range ambiguity, we propose a new method in terms of angle–range unambiguous estimation. The prior information is first utilized for range compensation, followed by the estimation of the range ambiguity area and DOA where the target is located. Subsequently, the accuracy of range estimation is further enhanced by estimating the principal range difference. Additionally, we deduce the Cramér–Rao lower bound (CRLB) of the NFDA-MIMO radar and the computational complexity of the proposed algorithm to support comparative analysis. In conclusion, the simulation experiments illustrate that the proposed radar architecture and the unambiguous estimation algorithm substantially improve the number of distinguishable sources, radar resolution, and estimated precision.

The rest of the article is organized as follows. In Section 2, we formulate a colocated NFDA-MIMO radar configuration and then present a signal model based on this radar. Section 3 presents the approach for joint angle and range unambiguous estimation. To better analyze performance, Section 4 derives the CRLB and computational complexity. The effectiveness of the proposed methodology and configuration is substantiated by the outcomes obtained from the simulation experiments in Section 5. Finally, the discussion and conclusion are illustrated in Section 6 and Section 7, respectively.

Notations: The bold lowercase letters represent a vector, while the bold uppercase letters represent a matrix. Superscripted characters ${(·)}^{\mathrm{T}}$, ${(·)}^{\ast }$, ${(·)}^{-1}$, and ${(·)}^{\mathrm{H}}$ denote the transpose, conjugate, inverse, and Hermitian transpose operators in sequence. Operator symbols ⊗, ⊚, and ⊙ stand for the Kronecker product, the Hadamard product, and the Khatri–Rao product, in the same order. Additionally, the trace of a matrix is denoted by $\mathrm{tr}[·]$, the expected value is represented as $\mathbb{E}[·]$, the vectorization operation is indicated through $\mathrm{vec}(·)$, and the absolute value is expressed in $\left|·\right|$.

2. Signal Model

The array structure and the echo processing program are outlined in this section.

2.1. Transmitter Array Configuration

As illustrated in Figure 1, a colocated NFDA-MIMO radar system is equipped with M antennas at the transmitter, consisting of two series-connected uniform linear arrays (ULAs). According to [29], the number of antennas of these two ULAs are ${\mathit{M}}_1$ and ${\mathit{M}}_2$, respectively, and $\mathit{M}={\mathit{M}}_1+{\mathit{M}}_2$. The antenna spacing for the first-level array, $d_1$, is equal to one half of the corresponding wavelength of the maximum carrier frequency. Meanwhile, the antenna spacing for the second-level array is ${\mathit{M}}_1$ times greater than that in the first-level array, i.e., $d_2=\left({\mathit{M}}_1+1\right)d_1$. Assuming the virtual reference antenna is located at position 0, the position of each antenna is provided by the position coefficient matrix $\mathcal{D}=\left\{D_1,D_2,\cdots ,D_M\right\}$, where ${\mathit{D}}_m(m=1,2,3,\cdots ,M)$ is an integer, defined as

$$D_m=\left\{\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}m,\hfill & m=1,2,3,\cdots ,M_1\hfill \\ (m-M_1)(M_1+1),\hfill & m=M_1+1,M_1+2,M_1+3,\cdots ,M\hfill \end{array}\right.$$

(1)

The transmitting carrier frequency on the m-th antenna is represented as

$${\mathit{f}}_m={\mathit{f}}_0+D_m\Delta \mathit{f},m=1,2,3,\cdots ,M$$

(2)

where ${\mathit{f}}_0$ indicates the carrier frequency for virtual reference antenna and $\Delta \mathit{f}$ represents the frequency offset, with $\Delta \mathit{f}\ll {\mathit{f}}_0$. The signal emitted from the m-th antenna is

$$x_m\left(t\right)=\sqrt{{\displaystyle \frac{E}{M}}}{\phi }_m\left(t\right)e^{j2\pi f_{m}t},0\le t\le T_p,m=1,2,3,\cdots ,M$$

(3)

where E denotes the transmitted signal power, ${\mathit{T}}_p$ stands for the pulse duration, and ${\phi }_m\left(t\right)$ is the complex envelope that satisfies the unit orthogonal requirement

$$\left\{\begin{array}{cc}{\int }_{T_p}{\phi }_{m_1}\left(t\right){\phi }_{m_2}^{\ast }(t-\tau )dt=0,\hfill & m_1\ne m_2\hfill \\ \hfill {\int }_{T_p}{\phi }_{m_1}\left(t\right){\phi }_{m_2}^{\ast }\left(t\right)dt=1,& m_1=m_2\hfill \end{array}\right.$$

(4)

where $\tau$ denotes the time delay.

2.2. Received Signal

The receiver of the NFDA-MIMO radar system comprises N antennas, similar in structure to the transmitter, but with $N_1$ elements in the first-level array and $N_2$ elements in the second-level array. Figure 2 depicts the signal processing steps corresponding to any receiving antenna unit at the receiver. It can be seen that the echoes after downconversion and sampling are separated by digital mixing and matched filtering.

The return signal from the n-th receiving antenna can be characterized as

$$y_n\left(t\right)=\sqrt{{\displaystyle \frac{E}{M}}}\sum _{m=1}^M{\xi }_m{\phi }_m(t-{\tau }_{m,n})e^{j2\pi f_m(t-{\tau }_{m,n})}$$

(5)

where ${\xi }_m$ is complex reflection coefficient of the far-field point target and ${\tau }_{m,n}$ represents the bidirectional propagation delay from the m-th antenna of the transmitter to the n-th antenna of the receiver, which is stated as

$${\tau }_{m,n}={\displaystyle \frac{2r}{c}}-{\displaystyle \frac{D_m{d}_{1}sin\left(\theta \right)}{c}}-{\displaystyle \frac{D_n{d}_{1}sin\left(\theta \right)}{c}}$$

(6)

where r and $\theta$ separately stand for the target range and angle, and c denotes the electromagnetic wave velocity. The first term in (6) represents the common delay, while the middle and latter terms are caused by the structure of the transmitter and receiver arrays, respectively.

After mixing and matched filtering [26], the signal received by the n-th receiving antenna and transmitted from the m-th transmitting antenna is approximated with

$$\begin{array}{cc}\hfill y_{mn}(r,\theta )\approx & \sqrt{{\displaystyle \frac{E}{M}}}{\xi }_m{e}^{-j4\pi {\displaystyle \frac{f_0}{c}}r}{e}^{-j4\pi {\displaystyle \frac{\Delta f{D}_{m}r}{c}}}{e}^{j2\pi {\displaystyle \frac{f_0{D}_m{d}_{1}sin\left(\theta \right)}{c}}}{e}^{j2\pi {\displaystyle \frac{f_0{D}_n{d}_{1}sin\left(\theta \right)}{c}}}\hfill \\ \hfill =& \sqrt{{\displaystyle \frac{E}{M}}}\xi e^{-j4\pi {\displaystyle \frac{\Delta f{D}_{m}r}{c}}}{e}^{j2\pi {\displaystyle \frac{f_0{D}_m{d}_{1}sin\left(\theta \right)}{c}}}{e}^{j2\pi {\displaystyle \frac{f_0{D}_n{d}_{1}sin\left(\theta \right)}{c}}}\hfill \end{array}$$

(7)

where $\xi ={\xi }_m{e}^{-j4\pi f_{0}r/c}$. The frequency increment quadratic terms in (7) are neglected for approximation. The target signal output snapshot at the receiver can be written as

$$\mathbf{y}={[y_{11},y_{21},\cdots ,y_{N1},y_{12},\cdots ,y_{MN}]}^{\mathrm{T}}=\sqrt{{\displaystyle \frac{E}{M}}}\xi \mathbf{b}\left(\theta \right)\otimes \mathbf{a}(r,\theta )$$

(8)

where $\mathbf{a}(r,\theta )$ and $\mathbf{b}\left(\theta \right)$ represent emission and reception steering vectors, correspondingly, defined as follows:

$$\begin{array}{cc}\hfill \mathbf{a}(r,\theta )& ={\left[e^{j2\pi D_1{f}_T},e^{j2\pi D_2{f}_T},\cdots ,e^{j2\pi D_M{f}_T}\right]}^{\mathrm{T}}\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill \mathbf{b}\left(\theta \right)& ={\left[e^{j2\pi D_1{f}_R},e^{j2\pi D_2{f}_R},\cdots ,e^{j2\pi D_N{f}_R}\right]}^{\mathrm{T}}\hfill \end{array}$$

(9b)

where ${\mathit{f}}_T={\displaystyle \frac{-2\Delta f}{c}}r+{\displaystyle \frac{d_1}{{\lambda }_0}}sin\left(\theta \right)$ denotes the transmit spatial frequency, ${\mathit{f}}_R={\displaystyle \frac{d_1}{{\lambda }_0}}sin\left(\theta \right)$ stands for the receive spatial frequency, and ${\lambda }_0=c/(f_0+D_M\Delta f)$ is the reference signal wavelength corresponding to the maximum transmission frequency.

Assuming there are K independent targets located at $(r_k,{\theta }_k)$ for $k=1,2,3,\cdots ,K$, the total received snapshot is denoted as

$$\mathbf{x}=\sqrt{{\displaystyle \frac{E}{M}}}\sum _{k=1}^K{\xi }_k\mathbf{b}\left({\theta }_k\right)\otimes \mathbf{a}(r_k,{\theta }_k)+{\mathbf{x}}_n$$

(10)

where ${\mathbf{x}}_n$ indicates the additive Gaussian white noise.

3. The Proposed Algorithm

To ensure the validity of the algorithm, we assume the following [39]:

(1)

The target signals are mutually independent and distinguishable.

(2)

The target signals are independent of the additive Gaussian white noise.

(3)

All snapshots are uncorrelated in time.

All subsequent operations in this section are predicated on these three assumptions.

3.1. Differential Equivalence

The covariance matrix of the echo snapshot can be calculated as

$$\mathbf{R}=\mathbb{E}\left[\mathbf{x}{\mathbf{x}}^{\mathrm{H}}\right]=\sum _{k=1}^K{\sigma }_k^2\mathbf{d}(r_k,{\theta }_k){\mathbf{d}}^{\mathrm{H}}(r_k,{\theta }_k)+{\sigma }_n^2{\mathbf{I}}_{MN}=\mathbf{A}\mathbf{P}{\mathbf{A}}^{\mathrm{H}}+{\sigma }_n^2{\mathbf{I}}_{MN}$$

(11)

where $\mathbf{P}=\mathrm{diag}({\sigma }_1^2,{\sigma }_2^2,\cdots ,{\sigma }_K^2)$ is the signal power matrix with ${\sigma }_k^2=\mathbb{E}\left[{\xi }_k{\xi }_k^{\ast }\right]$, $\mathbf{A}=[\mathbf{d}(r_1,{\theta }_1),\mathbf{d}(r_2,{\theta }_2),\cdots ,\mathbf{d}(r_K,{\theta }_K)]$ and denotes the joint transmit–receive steering vector matrix with $\mathbf{d}(r_k,{\theta }_k)=\mathbf{b}\left({\theta }_k\right)\otimes \mathbf{a}(r_k,{\theta }_k)$. The ${\mathbf{I}}_{MN}$ stands for the $MN\times MN$ identity matrix and ${\sigma }_n^2$ represents the noise power.

The virtual snapshot model, derived by vectorizing $\mathbf{R}$, is provided by

$$\tilde{\mathbf{x}}=\mathrm{vec}\left(\mathbf{R}\right)=\tilde{\mathbf{A}}\mathbf{p}+{\sigma }_n^2\mathrm{vec}\left({\mathbf{I}}_{MN}\right)$$

(12)

where $\tilde{\mathbf{A}}={\mathbf{A}}^{\ast }\odot \mathbf{A}$ , and $\mathbf{p}={[{\sigma }_1^2,{\sigma }_2^2,\cdots ,{\sigma }_K^2]}^{\mathrm{T}}$. From [29], it can be seen that $\tilde{\mathbf{A}}$ corresponds to the joint steering vector of a radar system with $(2M_2(M_1+1)-1)$ transmitting and $(2N_2(N_1+1)-1)$ receiving array elements. The positional coefficients of the transmitting and receiving array elements are formulated separately as ${\mathcal{D}}_{vt}=\{D_{m_1}-D_{m_2}\mid m_1,m_2=1,2,3,\cdots ,M\}$ and ${\mathcal{D}}_{vr}=\{D_{n_1}-D_{n_2}\mid n_1,n_2=1,2,3,\cdots ,N\}$.

Selecting appropriate rows in $\tilde{\mathbf{A}}$ for rearrangement, we can reconstruct a new steering vector matrix of a virtual ULA with extended DOF. Due to the symmetry of ${\mathcal{D}}_{vt}$ and ${\mathcal{D}}_{vr}$, the virtual array is centered at the origin where the reference antenna located. The position of the virtual transmitting array is denoted by $\{1-M_2(M_1+1),2-M_2(M_1+1),\cdots ,0,\cdots ,M_2(M_1+1)-2,M_2(M_1+1)-1\}{d}_1$, and the position of the virtual receiving array is denoted by $\{1-N_2(N_1+1),2-N_2(N_1+1),\cdots ,0,\cdots ,N_2(N_1+1)-2,N_2(N_1+1)-1\}{d}_1$. The corresponding signal model can be expressed as

$$\mathbf{z}=\mathbf{F}\tilde{\mathbf{x}}={\mathbf{A}}_d\mathbf{p}+{\sigma }_n^2\mathbf{F}\mathrm{vec}\left({\mathbf{I}}_{MN}\right)$$

(13)

where $\mathbf{F}$ is defined as the selection matrix in [40], and ${\mathbf{A}}_d$ denotes the differential equivalent joint steering vector matrix, which is written as

$${\mathbf{A}}_d=\left[{\mathbf{b}}_d\left({\theta }_1\right)\otimes {\mathbf{a}}_d(r_1,{\theta }_1),{\mathbf{b}}_d\left({\theta }_2\right)\otimes {\mathbf{a}}_d(r_2,{\theta }_2),\cdots ,{\mathbf{b}}_d\left({\theta }_K\right)\otimes {\mathbf{a}}_d(r_K,{\theta }_K)\right]$$

(14)

where ${\mathbf{a}}_d$ and ${\mathbf{b}}_d$ separately indicate the emission and reception steering vectors for the equivalent symmetric ULA, expressed in the specific form of

$$\begin{array}{cc}\hfill {\mathbf{a}}_d(r,\theta )& ={\left[e^{j2\pi (1-M_2(M_1+1))f_T},e^{j2\pi (2-M_2(M_1+1))f_T},\cdots ,e^{j2\pi (M_2(M_1+1)-1)f_T}\right]}^{\mathrm{T}}\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill {\mathbf{b}}_d\left(\theta \right)& ={\left[e^{j2\pi (1-N_2(N_1+1))f_R},e^{j2\pi (2-N_2(N_1+1))f_R},\cdots ,e^{j2\pi (N_2(N_1+1)-1)f_R}\right]}^{\mathrm{T}}\hfill \end{array}$$

(15b)

3.2. Range Compensation

When range ambiguity is present, the actual target range can be rewritten as

$$r=r_a+(q-1)r_{max}$$

(16)

where $r_a$ is the principal range, $q\in {\mathbb{N}}^{\ast }$ represents the number of range ambiguity with ${\mathbb{N}}^{\ast }$ denoting the set of positive integers, and $r_{max}$ is the maximum unambiguous range. As stated in (15), the target range information is contained within emission steering vector. We can rephrase it as

$${\mathbf{a}}_d(r,\theta )={\mathbf{a}}_r\left(r\right)⊚{\mathbf{a}}_{\theta }\left(\theta \right)$$

(17)

where ${\mathbf{a}}_r$ and ${\mathbf{a}}_{\theta }$ denote the range and angle steering vectors in transmitting end, respectively, defined as

$$\begin{array}{cc}\hfill {\mathbf{a}}_{\theta }\left(\theta \right)=& {\left[e^{j2\pi {\displaystyle \frac{(1-M_2(M_1+1))d_{1}sin\left(\theta \right)}{{\lambda }_0}}},e^{j2\pi {\displaystyle \frac{(2-M_2(M_1+1))d_{1}sin\left(\theta \right)}{{\lambda }_0}}},\cdots ,e^{j2\pi {\displaystyle \frac{(M_2(M_1+1)-1)d_{1}sin\left(\theta \right)}{{\lambda }_0}}}\right]}^{\mathrm{T}}\hfill \end{array}$$

(18a)

$$\begin{array}{cc}\hfill {\mathbf{a}}_r\left(r\right)& ={\left[e^{-j4\pi {\displaystyle \frac{(1-M_2(M_1+1))\Delta f}{c}}r},e^{-j4\pi {\displaystyle \frac{(2-M_2(M_1+1))\Delta f}{c}}r},\cdots ,e^{-j4\pi {\displaystyle \frac{(M_2(M_1+1)-1)\Delta f}{c}}r}\right]}^{\mathrm{T}}\hfill \end{array}$$

(18b)

The effect of principal range $r_a$ on range ambiguity estimation can be eliminated through range compensation in ${\mathbf{a}}_r$. According to (13), the compensating vector can be expressed as

$$\mathbf{g}={\mathbf{1}}_{2M_2(M_1+1)-1}\otimes \mathbf{h}\left(r_b\right)$$

(19)

where ${\mathbf{1}}_{2M_2(M_1+1)-1}$ is a column vector of length $2M_2(M_1+1)-1$ where every element is 1, and $\mathbf{h}\left(r_b\right)$ is the compensating vector applied in the transmit domain, which is in the form of

$$\mathbf{h}\left(r_b\right)={\left[e^{j4\pi {\displaystyle \frac{(-M_2(M_1+1)+1)\Delta f}{c}}{r}_b},e^{j4\pi {\displaystyle \frac{(-M_2(M_1+1)+2)\Delta f}{c}}{r}_b},\cdots ,e^{j4\pi {\displaystyle \frac{(M_2(M_1+1)-1)\Delta f}{c}}{r}_b}\right]}^{\mathrm{T}}$$

(20)

where $r_b$ is a prior range estimate of $r_a$, which is calculated from the number of range bins and the range bin size.

After compensating for the targets, the differential equivalent signal can be indicated as

$$\widehat{\mathbf{z}}=\sum _{k=1}^K{\sigma }_k^2{\mathbf{b}}_d\left({\theta }_k\right)\otimes [{\mathbf{a}}_r(\Delta r+(q_k-1)r_{max})⊚{\mathbf{a}}_{\theta }\left({\theta }_k\right)]+{\sigma }_n^2\sum _{k=1}^K\mathbf{F}\mathrm{vec}\left({\mathbf{I}}_{MN}\right)⊚{\mathbf{g}}_k$$

(21)

where $\Delta r=r_b-r_a$ denotes the principal range difference, and ${\mathbf{g}}_k$ is the range compensation vector for the kth target. Based on (21), we can estimate the range ambiguity area to overcome the challenges posed by range ambiguity. Furthermore, evaluating $\Delta r$ enables more accurate range estimation.

3.3. Range and Angle Estimation

As explained in Section 3.2, the observation vector after range compensation corresponds to a single snapshot signal, leading to rank loss. The traditional MUSIC algorithm cannot be directly applied because it requires a full rank covariance matrix. Based on this fact, we propose a two-dimensional spatial smoothing technique for colocated NFDA-MIMO radar to obtain full rank matrix. Finally, we determine the DOA, q, and $\Delta r$ via two MUSIC algorithms. The details are described as follows.

The number of elements in the spatially smooth partition subarray should be fewer than the elements in both the transmitting and receiving arrays. For simplicity, we assume that $M\le N$. The symmetric ULA can be decomposed into $M_{sub}$ superimposed subarrays, each containing $M_{ele}$ elements, where $M_{ele}=2M_2(M_1+1)-M_{sub}$. The output signal sent by the i-th subarray and picked up by the j-th subarray is

$${\widehat{\mathbf{z}}}_{ij}={\mathsf{\Gamma }}_{ij}\widehat{\mathbf{z}},i,j=1,2,\cdots ,M_{sub}$$

(22)

where ${\mathsf{\Gamma }}_{ij}={\mathsf{\Gamma }}_i\otimes {\mathsf{\Gamma }}_j$ denotes the two-dimensional selection matrix. ${\mathsf{\Gamma }}_i$ is used to select the i-th transmitting subarray, while ${\mathsf{\Gamma }}_j$ is used to select the j-th receiving subarray, defined as

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_i& =\left[{\mathbf{0}}_{M_{ele}\times (i-1)}{\mathbf{I}}_{M_{ele}\times M_{ele}}{\mathbf{0}}_{M_{ele}\times (M_{sub}-i)}\right]\hfill \end{array}$$

(23a)

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_j& =\left[{\mathbf{0}}_{M_{ele}\times (j-1)}{\mathbf{I}}_{M_{ele}\times M_{ele}}{\mathbf{0}}_{M_{ele}\times (M_{sub}-j)}\right]\hfill \end{array}$$

(23b)

where $\mathbf{0}$ indicates the zero matrix. Given the output of subarrays, the covariance matrix processed by aforementioned method is generally arranged in the following form [41]:

$${\mathbf{R}}_{ss}={\displaystyle \frac{1}{M_{sub}^2}}\sum _{i=1}^{M_{sub}}\sum _{j=1}^{M_{sub}}{\widehat{\mathbf{z}}}_{ij}{\widehat{\mathbf{z}}}_{ij}^{\mathrm{H}}$$

(24)

Due to the independence of target signals and noise, the ${\mathbf{R}}_{ss}$ can be decomposed into

$${\mathbf{R}}_{ss}=\mathbf{U}\mathbf{\Lambda }{\mathbf{U}}^{\mathrm{H}}={\mathbf{U}}_{ss}{\mathbf{\Lambda }}_{ss}{\mathbf{U}}_{ss}^{\mathrm{H}}+{\mathbf{U}}_n{\mathbf{\Lambda }}_n{\mathbf{U}}_n^{\mathrm{H}}$$

(25)

where ${\mathbf{U}}_{ss}$ and ${\mathbf{U}}_n$ stand for the signal and noise subspaces, respectively, and $\mathbf{\Lambda }$ denotes the eigenvalue matrix. Using the orthogonal properties between ${\mathbf{U}}_{ss}$ and ${\mathbf{U}}_n$, the target angle and number of range ambiguity may be determined through identifying the peak values of the functions below

$$\mathbf{P}(\theta ,q)=\left|{\displaystyle \frac{1}{{\widehat{\mathbf{A}}}_{ss}(\theta ,q){\mathbf{U}}_n{\mathbf{U}}_n^{\mathrm{H}}{\widehat{\mathbf{A}}}_{ss}^{\mathrm{H}}(\theta ,q)}}\right|$$

(26)

where ${\widehat{\mathbf{A}}}_{ss}$ stands for the joint steering vector matrix corresponding to the subarray.

Substituting the target angle $\widehat{\theta }$ and range ambiguity number $\widehat{q}$ obtained from (26) into ${\widehat{\mathbf{A}}}_{ss}$, the principal range difference $\Delta \widehat{r}$ is searched by

$$\mathbf{P}(\widehat{\theta },\widehat{q},\Delta r)=\max \left|{\displaystyle \frac{1}{{\widehat{\mathbf{A}}}_{ss}(\widehat{\theta },\widehat{q},\Delta r){\mathbf{U}}_n{\mathbf{U}}_n^{\mathrm{H}}{\widehat{\mathbf{A}}}_{ss}^{\mathrm{H}}(\widehat{\theta },\widehat{q},\Delta r)}}\right|$$

(27)

The final computed target range is

$$\widehat{r}=r_b+\Delta \widehat{r}+(\widehat{q}-1)r_{max}.$$

(28)

In summary, we address the range ambiguity issue and enhance the estimation precision by estimating q and $\Delta r$. The algorithm steps are outlined in

Table 1. Unambiguous range–angle estimation algorithm for NFDA-MIMO radar.

Step 1: Vectorize the output data covariance matrix to obtain the differential signal in (12).

Step 2: Convert the differential signal to a virtual ULA signal with extended DOF through redundant and rearrangement operations in (13).

Step 3: Construct the compensating vector with $r_b$ to compensate the principal range in transmit spatial frequency domain using (19) and (21).

Step 4: Obtain the covariance matrix after spatial smoothing in (24).

Step 5: Estimate the angle and range ambiguity number of targets by two-dimensional spectral peak search in (26).

Step 6: Estimate the principal range difference and calculate range of targets using (27) and (28).

.

4. Performance Analysis

4.1. CRLB Derivation

In [40], the CRLB for the sparse array model has been well studied, but only the CRLB of angles was derived without deriving the CRLB of ranges. Therefore, it cannot be directly applied to NFDA-MIMO radar. Based on this situation, we derive CRLBs for range and angle to analyze the performance of the NFDA-MIMO radar.

For (10), the unknown parameter vector can be represented as

$$\mathit{\eta }={\left[{\theta }_1,{\theta }_2,\cdots ,{\theta }_K,r_1,r_2,\cdots ,r_K,{\sigma }_1^2,{\sigma }_2^2,\cdots ,{\sigma }_K^2,{\sigma }_n^2\right]}^{\mathrm{T}}={\left[\mathit{\theta },\mathbf{r},{\mathbf{p}}^{\mathrm{T}},{\sigma }_n^2\right]}^{\mathrm{T}}$$

(29)

where $\mathit{\theta }=[{\theta }_1,{\theta }_2,\cdots ,{\theta }_K]$ and $\mathit{r}=[r_1,r_2,\cdots ,r_K]$. The $(m,n)$-th term in the Fisher information matrix (FIM) [42] is expressed as

$${\mathbf{FIM}}_{m,n}=L\mathrm{tr}\left[{\displaystyle \frac{\partial \mathbf{R}}{\partial {\eta }_m}}{\mathbf{R}}^{-1}{\displaystyle \frac{\partial \mathbf{R}}{\partial {\eta }_n}}{\mathbf{R}}^{-1}\right]$$

(30)

where L represents the number of snapshots. Since $\left({\mathbf{AXB}}^{\mathrm{T}}\right)=(\mathbf{B}\otimes \mathbf{A})vec\left(\mathbf{X}\right)$ and $\mathrm{tr}\left(\mathbf{AB}\right)=\mathrm{vec}{\left({\mathbf{A}}^{\mathrm{T}}\right)}^{\mathrm{T}}\mathrm{vec}\left(\mathbf{B}\right)$, (30) can be rewritten as

$${\mathbf{FIM}}_{m,n}=L{\left[{\displaystyle \frac{\partial \tilde{\mathbf{x}}}{\partial {\eta }_m}}\right]}^{\mathrm{H}}{\left({\mathbf{R}}^{\mathrm{T}}\otimes \mathbf{R}\right)}^{-1}{\displaystyle \frac{\partial \tilde{\mathbf{x}}}{\partial {\eta }_n}}$$

(31)

The FIM can be succinctly expressed by

$$\mathbf{FIM}={\left[{\displaystyle \frac{\partial \tilde{\mathbf{x}}}{\partial \mathit{\eta }}}\right]}^{\mathrm{H}}{\left({\mathbf{R}}^{\mathrm{T}}\otimes \mathbf{R}\right)}^{-1}{\displaystyle \frac{\partial \tilde{\mathbf{x}}}{\partial \mathit{\eta }}}$$

(32)

where $\partial \tilde{\mathbf{x}}/\partial \mathit{\eta }$ stands for the partial derivative of $\tilde{\mathbf{x}}$ with respect to $\mathit{\eta }$, which is calculated as

$${\displaystyle \frac{\partial \tilde{\mathbf{x}}}{\partial \mathit{\eta }}}=\left[{\tilde{\mathbf{A}}}_{\theta }^{{}^{\prime }}\mathbf{P}{\tilde{\mathbf{A}}}_r^{{}^{\prime }}\mathbf{P}\tilde{\mathbf{A}}\tilde{\mathbf{I}}\right]$$

(33)

where ${\tilde{\mathbf{A}}}_{\theta }^{{}^{\prime }}$ and ${\tilde{\mathbf{A}}}_r^{{}^{\prime }}$ represent partial derivatives of angle and range, respectively, which can be computed as

$$\begin{array}{cc}\hfill {\tilde{\mathbf{A}}}_{\theta }^{{}^{\prime }}& ={\displaystyle \frac{\partial {\mathbf{A}}^{\ast }}{\partial \mathit{\theta }}}\odot \mathbf{A}+{\mathbf{A}}^{\ast }\odot {\displaystyle \frac{\partial \mathbf{A}}{\partial \mathit{\theta }}}\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill {\tilde{\mathbf{A}}}_r^{{}^{\prime }}& ={\displaystyle \frac{\partial {\mathbf{A}}^{\ast }}{\partial \mathit{r}}}\odot \mathbf{A}+{\mathbf{A}}^{\ast }\odot {\displaystyle \frac{\partial \mathbf{A}}{\partial \mathit{r}}}\hfill \end{array}$$

(34b)

Note that $\mathbf{R}$ is a positive definite matrix, so we can define the following three auxiliary matrices as

$$\begin{array}{cc}\hfill {\mathbf{M}}_{\theta }& ={\left({\mathbf{R}}^{\mathrm{T}}\otimes \mathbf{R}\right)}^{-1/2}{\tilde{\mathbf{A}}}_{\theta }^{{}^{\prime }}\mathbf{P}\hfill \end{array}$$

(35a)

$$\begin{array}{cc}\hfill {\mathbf{M}}_r& ={\left({\mathbf{R}}^{\mathrm{T}}\otimes \mathbf{R}\right)}^{-1/2}{\tilde{\mathbf{A}}}_r^{{}^{\prime }}\mathbf{P}\hfill \end{array}$$

(35b)

$$\begin{array}{cc}\hfill {\mathbf{M}}_{sn}& ={\left({\mathbf{R}}^{\mathrm{T}}\otimes \mathbf{R}\right)}^{-1/2}\left[\tilde{\mathbf{A}}\tilde{\mathbf{I}}\right]\hfill \end{array}$$

(35c)

Then, the FIM can be denoted by

$$\mathbf{FIM}=L\left[\begin{array}{ccc}{\mathbf{M}}_{\theta }^{\mathrm{H}}{\mathbf{M}}_{\theta }& {\mathbf{M}}_{\theta }^{\mathrm{H}}{\mathbf{M}}_r& {\mathbf{M}}_{\theta }^{\mathrm{H}}{\mathbf{M}}_{sn}\\ {\mathbf{M}}_r^{\mathrm{H}}{\mathbf{M}}_{\theta }& {\mathbf{M}}_r^{\mathrm{H}}{\mathbf{M}}_r& {\mathbf{M}}_r^{\mathrm{H}}{\mathbf{M}}_{sn}\\ {\mathbf{M}}_{sn}^{\mathrm{H}}{\mathbf{M}}_{\theta }& {\mathbf{M}}_{sn}^{\mathrm{H}}{\mathbf{M}}_r& {\mathbf{M}}_{sn}^{\mathrm{H}}{\mathbf{M}}_{sn}\end{array}\right]$$

(36)

The CRLBs of NFDA-MIMO radar corresponding to ${\theta }_k$ and $r_k$ can be calculated by

$$\begin{array}{cc}\hfill {\mathrm{CRLB}}_{{\theta }_k}& ={\left[{\mathbf{FIM}}^{-1}\right]}_{k,k}\hfill \end{array}$$

(37a)

$$\begin{array}{cc}\hfill {\mathrm{CRLB}}_{r_k}& ={\left[{\mathbf{FIM}}^{-1}\right]}_{K+k,K+k}\hfill \end{array}$$

(37b)

4.2. Algorithm Complexity

Given the x, the computational complexity of the covariance operation in (11) is $O\left(L{\left(MN\right)}^2\right)$. Let $M_v=2M_2(M_1+1)-1$, $N_v=2N_2(N_1+1)-1$, and the z can be obtained with $O\left(M_v{N}_v{\left(MN\right)}^2\right)$ operations. Then, the Hadamard product for range compensation requires $O\left(M_v{N}_v\right)$ operations. According to Section 3.3, the complexity of spatial smoothing is $O\left(M_{sub}{M}_{ele}^4\right)$. Since the covariance matrix is of size $M_{ele}\times M_{ele}$, the complexity of the feature decomposition in (25) is $O\left(M_{ele}^6\right)$. In the search phase, $O\left({\kappa }_q{\kappa }_{\theta }{M}_{ele}^2\right)$ operations are needed for the ambiguity number and two-dimensional angle estimation, while the principal range difference estimation requires $O\left({\kappa }_r{M}_{ele}^2\right)$ operations. Here, ${\kappa }_q$, ${\kappa }_{\theta }$, and ${\kappa }_r$ denote the number of search grids for q, $\theta$, and $\Delta r$, respectively. Therefore, the total computational complexity of the algorithm in the paper is $O(L{\left(MN\right)}^2+M_v{N}_v{\left(MN\right)}^2+M_v{N}_v+M_{sub}{M}_{ele}^4+M_{ele}^6+{\kappa }_q{\kappa }_{\theta }{M}_{ele}^2+{\kappa }_r{M}_{ele}^2)$.

The estimation of FDA-MIMO radar target parameters using the MUSIC algorithm requires neither signal rearrangement nor spatial smoothing, thus reducing the computational complexity. However, range compensation is performed snap-by-snap, which requires $O\left(LMN\right)$ operations. The comparison of the complexity of these two parameter estimation algorithms is shown in

Table 2. The complexity comparison.

Methods	Computational Complexity
for NFDA-MIMO	$O(L{\left(MN\right)}^2+M_v{N}_v{\left(MN\right)}^2+M_v{N}_v+M_{sub}{M}_{ele}^4+M_{ele}^6+{\kappa }_q{\kappa }_{\theta }{M}_{ele}^2+{\kappa }_r{M}_{ele}^2)$
for FDA-MIMO	$O(L{\left(MN\right)}^2+{\left(MN\right)}^3+LMN+{\kappa }_q{\kappa }_{\theta }MN+{\kappa }_{r}MN)$

.

5. Simulation Results

In the section, experimental results obtained from MATLAB simulation are illustrated to demonstrate the superior performance of NFDA-MIMO radar and the proposed algorithm. The MIMO and FDA-MIMO (referring to linear FDA-MIMO) radar are selected for comparison. The parameters of the simulations are shown in

Table 3. The simulation parameters.

Parameter	Value	Parameter	Value
Reference frequency	10 GHz	Frequency offset	1,001,250 Hz
Transmit element number	4	Receive element number	4
Transmit element position coefficient	[1 2 3 6]	Receive element position coefficient	[1 2 3 6]
Maximum unambiguous range	30 km	Element spacing	0.015 m
Number of pulses	200	Waveform bandwidth	10 MHz
Target principal range	10 km	Range resolution	15 m
Ambiguous number	5	Target number	2, 3, 4, 5

.

5.1. Comparison of Power Spectrum

In this subsection, we examine a scenario with multiple targets characterized by identical principal ranges and angles yet situated within distinct range ambiguity regions. The target angle is set at 0°, and the signal-to-noise ratio (SNR) is configured as 0 dB.

Figure 3a–i display the target power spectral distributions after range compensation for three types of radars in case the goal counts are 3, 4, and 5, respectively. From Figure 3a,d,g, it can be observed that multiple targets overlap in the two-dimensional frequency space. This phenomenon occurs because conventional MIMO radar lacks the capability to resolve range ambiguities, whereas FDA-MIMO radar and NFDA-MIMO radar possess this capability, as shown in Figure 3b,c,e,f,h,i. However, FDA-MIMO radar is constrained by the amount of array sensors, which means it can only distinguish ambiguous regions that are fewer than the actual number of antennas. In contrast, NFDA-MIMO radar overcomes this limitation and is capable of distinguishing ambiguous regions that exceed the number of antennas. Keep in mind that the ability of NFDA-MIMO radar to identify range ambiguity regions is dependent on the number of subarray elements $N_{ele}$ described in Section 3.3. The maximum number of distinguishable ambiguous regions is $N_{ele}-1$.

5.2. Comparison of Range Resolution

To compare the range resolutions of three types of radars, we conduct a comparison of the power spectra for two point targets that are very close to each other in the same range ambiguity region in Figure 4. Both point targets are coming from 0°, with a range of 5 m between them, and the SNR is designed to be 20 dB.

As shown in Figure 4, the MIMO radar is unable to effectively distinguish between two point targets, even in high-SNR scenarios. This is attributed to the lack of resolution in MIMO radar to distinguish two targets within just one range bin. When the range between two point targets is less than the size of the range bin, the radar misidentifies them as a single target. While it has the ability to discriminate between targets within a range bin, the FDA-MIMO radar fails to accurately identify these two targets due to insufficient range resolution. In comparison, the NFDA-MIMO radar demonstrates a significant improvement in range resolution due to the expansion of the array aperture, successfully identifying both point targets. Hence, NFDA-MIMO radar demonstrates superior range resolution compared to the other two radar systems.

5.3. Performance of Range and Angle Estimation

The feasibility of the presented algorithm is assessed in the first case. To reduce randomness, we conduct 200 Monte Carlo simulations. In these experiments, the q is randomly selected from 1 to 4, and the $\Delta r$ is evenly distributed in the range of (−7.5 m, 7.5 m). The SNR is adjusted to 0 dB. Figure 5 shows the estimation results for the q and $\Delta r$. The angle estimation results will be discussed further in future simulations. From Figure 5a, the estimated number of range ambiguities matches the true values exactly. Figure 5b illustrates that the estimated principal range differences are very close to their true counterparts. The experimental results provide strong evidence for the feasibility of the algorithm.

The root mean square errors (RMSEs) for the estimated values corresponding to the SNR are calculated through 200 Monte Carlo simulations. Figure 6 presents the estimation results in relation to the SNR. The CRLBs are plotted for comparison. As shown in Figure 6a, the angle estimation precision for both MIMO and FDA-MIMO radar is similar, while NFDA-MIMO radar achieves higher accuracy. It is worth noting that the estimation precision of NFDA-MIMO radar surpasses the CRLB of FDA-MIMO radar at high SNR, which is attributed to increased DOF. It is evident from Figure 6b that the precision for range estimation improves as SNR increases in both FDA-MIMO and NFDA-MIMO radars. Nevertheless, the range estimation precision in MIMO radar is no longer improved because the measurement precision is dependent on the bandwidth of the transmitted signal, which remains unchanged.

Figure 7 shows the RMSE curve as a function of the amount of snapshots at an SNR of 0 dB. As illustrated in Figure 7a, the angle estimation accuracy improves for all three radar systems as the amount of snapshots increases. Notably, the NFDA-MIMO radar demonstrates the highest level of angle estimation accuracy, thereby indicating its superior performance in scenarios characterized by limited sample sizes. Figure 7b indicates that the range estimation accuracy of MIMO radar cannot improve with an increasing number of snapshots. In addition, the NFDA-MIMO radar achieves the highest range estimation accuracy for the same quantity of snapshots. However, the range estimation accuracy of the FDA radar is closer to its CRLB, while the NFDA-MIMO radar struggles to attain its own CRLB. This is attributed to the sacrifice of the partial effective array aperture and increased sidelobe level [43] when employing the spatial smoothing algorithm.

6. Discussion

Discussion 1: In Equation (3), we hypothesize that the transmitted envelope signals are fully orthogonal, enabling signal separation at the receiver. However, it is infeasible to obtain perfectly orthogonal transmitted waveforms in practical engineering applications. Consequently, there is a need to design more optimized orthogonal waveforms. Additionally, MIMO radar systems face challenges in forming high-gain narrow beams, requiring coherent accumulation to achieve higher target gain. This presents a promising area for further research.

Discussion 2: The proposed algorithm employs spatial smoothing to obtain the signal covariance matrix, thereby eliminating signal correlation. However, spatial smoothing comes at the cost of partial array aperture loss and an increase in sidelobe level. In our future work, we will continue to conduct in-depth research to circumvent this issue.

7. Conclusions

In this paper, we propose a joint angle and range estimation method for NFDA-MIMO radar in the presence of range ambiguity by leveraging the element expansion characteristic of nested arrays and the angle–range correlation of FDA. The echo signal is transformed into a single snapshot signal through differential processing, which is tantamount to enlarging the array aperture. As a result, the accuracy of angle–range estimation is enhanced with the expansion of the array aperture. By employing the proposed spatially smoothed MUSIC algorithm, the range ambiguity region and the target angle are estimated simultaneously, thereby resolving the issue of range ambiguity. The introduction of principal range difference further enhances the precision of the range estimate. Additionally, the CRLB for the NFDA-MIMO radar and the computational complexity of the proposed algorithm are derived. The numerical simulation results validate the superior performance of the proposed method in angle–range estimation.