### I. Introduction

### II. Estimation Model

*l*-th antenna (

*l*∈ {1,2,..,

*L*}, where

*L*denotes the number of antennas) is expressed as

*a*

*,*

_{k}*τ*

*, and*

_{k}*θ*

*denote the complex amplitude, time delay, and angle of the*

_{k}*k*-th target (

*k*∈ {1,2,..,

*K*}, where

*K*is the number of targets), respectively.

*f*

*denotes the carrier frequency.*

_{c}*B*and

*T*denote the sweep bandwidth and time for one chirp, respectively.

*l*-th element of the angle steering vector in the uniform linear array (ULA) antenna, where

*λ*represents the wavelength and

*d*the distance between antenna elements.

*w*

*(*

_{l}*t*) denotes additive white Gaussian noise (AWGN). The sampled signal

*y*

*(*

_{n}*l*,

*m*) of

*y*

*(*

_{l}*t*) can be represented as follows:

*m*-th element in the

*n*-th snapshot of the range steering vector (

*m*∈ {1,…,

*M*} and

*n*∈ {1,…,

*N*}, where

*M*is the number of samples and

*N*is the number of snapshots, respectively).

*f*

*denotes the sampling frequency,*

_{s}*c*the speed of light, and

*R*

*defines the range of the*

_{k}*k*-th target.

**d̄**

_{l}_{,}

*= [*

_{n}*y*

*(*

_{n}*l*, 1), K,

*y*

*(*

_{n}*l*,

*M*)]

*. The covariance matrix for*

^{T}**x**

*is defined as follows:*

_{n}*denotes the Hermitian transpose. The corresponding eigenvalue decomposition is as follows:*

^{H}*λ*

*(*

_{i}*λ*

_{1}>

*λ*

_{2}> ⋯ >

*λ*

*) denotes the*

_{LM}*i*-th eigenvalue and

**e**

*the corresponding eigenvector of the matrix R, respectively. E*

_{i}_{s}and E

_{n}represent the signal subspace and noise subspace of matrix R, respectively. Using the noise subspace E

_{n}, we can establish the 2D-MUSIC spectrum for joint range and angle estimation, such as

**v**(R,

*θ*) = [

*a*

_{1}(

*θ*), …,

*a*

*(*

_{L}*θ*)] ⊗ [

*b*

_{1}(

*R*), …,

*b*

*(*

_{M}*R*)], and ⊗ denotes the Kronecker product.

*P*(

*R*,

*θ*) has peaks at the localization of targets through a 2D search.

### III. Proposed Pre-processing Technique

*y*

_{1,}

*(*

_{n}*m*) and

*y*

_{2,}

*(*

_{n}*m*) using

**q**

_{1,}

*and*

_{n}**q**

_{2,}

*, an (*

_{n}*L*× 1)-dimensional vector

**y**

*can be expressed as follows:*

_{n}**b**=[

*b*

_{m}_{,}

*(*

_{n}*R*

_{1}),⋯,

*b*

_{m}_{,}

*(*

_{n}*R*

*)]*

_{K}*.*

^{T}**w**represents an (

*L*×1)-dimensional AWGN vector because the addition and subtraction between AWGN is still AWGN. The (

*l*,

*k*)-th element in matrix

**A**,

**A**(

*l*,

*k*), is determined as follows:

*L*×

*M*/2)-dimensional matrices

**Y**

_{1,}

*and*

_{n}**Y**

_{2,}

*as follows:*

_{n}**Z**

_{1,}

*= (*

_{n}**Y**

_{2,}

*+*

_{n}**Y**

_{1,}

*)/2 and*

_{n}**Z**

_{2,}

*= (*

_{n}**Y**

_{2,}

*−*

_{n}**Y**

_{1,}

*)/2j, the (*

_{n}*L*×

*M*)-dimensional matrix

**Z**

*is determined as follows:*

_{n}**W**

*is also an AWGN matrix. The (*

_{n}*k*,

*m*)-th element of the matrix

**B**,

**B**(

*k*,

*m*) is determined as follows:

*y*

*(*

_{n}*l*,

*m*) in (2) can be transformed into a signal with a real-valued steering vector by using the proposed pre-processing technique and can be expressed as follows:

*l*,

*k*)-th element of the matrix

**A**,

*k*,

*m*)-th element of the matrix

**B**, and

*l*,

*m*)-th element of the matrix

**W**

*. Taking the real part of*

_{n}**Z**

*(*

_{n}*l*,

*m*) can be expressed as follows:

*l*-th element ofthe new angle steering vector, and

*k-*th element of the new range steering vector for 2D-MUSIC, respectively. Accordingly, the steering vector for the 2D-MUSIC spectrum can be defined as follows:

*Q*is the number of iterations. In the spectrum search process,

*a*

*and*

_{s}*r*

*are the angle and range resolutions, respectively.*

_{s}*R*

*denotes the maximum range. As shown in Table 1, the proposed 2D-MUSIC reduces the multiplications by about four times and the additions by about three times in the calculation of the covariance matrix. In addition, both multiplications and additions are reduced by about four times in the eigenvalue decomposition. Moreover, the division and arctangent operations are reduced by two times. In the process of the MUSIC spectrum search, multiplications and additions are reduced by about four times and two times, respectively.*

_{max}### IV. Simulation Results

*R*

_{1},

*θ*

_{1}) = (50 m, 20°) and (

*R*

_{2},

*θ*

_{2}) = (70 m, 60°). There are four or eight receiving antennas, and the number of snapshots is 64, 96, and 128. The range and angle resolution are 0.1 m and 0.1°, respectively. Figs. 2–5 illustrate the variations of the azimuth angle and range estimation root mean square error (RMSE) with signal-to-noise ratio (SNR), respectively, where 1,000 Monte Carlo simulations were performed. In addition, Fig. 6 shows a spatial spectrum when SNR is set to 15 dB.

*timeit*in MATLAB instructions can be used to calculate the runtime as in [17], and the runtime is analyzed against the number of snapshots. We can observe from Table 2 that the runtime of the basic 2D-MUSIC using the proposed pre-processing technique is reduced by a maximum of 65.7% compared to the case without pre-processing. In addition, the runtimes of the 2D-gold-MUSIC and RD-MUSIC with the proposed pre-processing technique are reduced by a maximum 67.2% and 67.9%, respectively.