# 2D-Unitary ESPRIT Based Multi-Target Joint Range and Velocity Estimation Algorithm for FMCW Radar
## 1 Introduction  
The millimeter-wave radar, known for its high resolution, finds widespread applications in various domains including autonomous driving [1, 2] and small target detection [3]. The radar system exhibits exceptional detection capabilities and a high probability of target recognition, enabling accurate determination of the target's nature. The millimeter-wave radar is mainly divided into pulsed radar and continuous wave radar according to different ways of radiating electromagnetic waves [4]. The pulsed radar is commonly used for range estimation. The constant frequency continuous wave radar is generally adopted for velocity estimation. The frequency shift keying (FSK) [5] or the phase shift keying (PSK) continuous wave radar achieves range-velocity estimation of multi-targets simultaneously, but the estimation is not accurate when the targets are relatively stationary. Frequency Modulated Continuous Wave (FMCW) radar [6, 7], which utilizes frequency modulation such as triangle wave and sawtooth wave, can provide range and velocity estimates of multiple targets simultaneously. The FMCW radar transmits a chirp signal and receives the signals reflected by multiple moving targets. The information about the time delays and Doppler shifts due to targets' range and radial velocity are embedded in the difference frequency between the transmitted and received signals, and can be estimated by utilizing the signal processing techniques. Therefore, highly accurate frequency estimation of intermediate frequency (IF) signal is a critical step to obtain the range-velocity information of the moving targets. The classical 2D Fast Fourier Transform (2D-FFT) algorithm [8; 9] can perform joint range-Doppler estimation, but the estimation accuracy is inefficient due to its fence effect and spectrum leakage. There are existing algorithms to improve the accuracy of one-dimension (1D) frequency estimation, such as the Rife algorithm [10], the modified Rife algorithm [11], and the Chirp Z-Transform (CZT) algorithm [12], which refine the frequency band to improve the frequency resolution. The Zoom-FFT algorithm [13] achieves a higher frequency estimation accuracy by shifting the spectral band and decreasing the sampling rate. However, the frequency estimation accuracy of the methods in [8; 9; 10; 11; 12; 13] are limited by the Rayleigh limit [14; 15]. Super-resolution frequency estimation is achieved by utilizing the subspace techniques of the conventional Multiple Signal Classification (MUSIC) algorithm [16] and Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm [17], which overcome the Rayleigh limit. The orthogonality of the noise subspace and signal subspace in the signal autocorrelation matrix is used by the MUSIC algorithm to construct a spatial-spectral function for spectral peak searching to obtain high-resolution spectral estimation. The rotational invariance of the signal subspace is employed by the ESPRIT algorithm to obtain higher accuracy without searching for the spectral peaks. However, the subspace-based methods [16; 17] suffer from large computational effort and poor estimation performance in noisy environments. Meanwhile, there exists a mismatch pairing of the range-velocity estimates when there are multiple targets. Therefore, researchers have proposed algorithms to improve the accuracy of two-dimension (2D) frequency estimation [18; 19; 20; 21; 22; 23; 24; 25; 26]. The 2D frequencies are estimated by the 2D-CZT algorithm [18] and the pairing of range-velocity estimates is solved by using the correlation of signals [19]. The 2D-MUSIC algorithm [20] achieves 2D frequency estimation and automatic pairing by constructing a linear epidemic matrix of 2D frequencies. The 2D-Root MUSIC algorithm [21] performs two 1D root estimates of frequencies and then performs range-velocity pairing by using the noise subspace orthogonal to the range-velocity steering vector.