## Advanced estimation techniques

Advancements in silicon semiconductor technology have had the profound impact on the design of automotive radar systems, providing higher integration and performance at lower cost. This section reviews some sophisticated radar signal processing algorithms, which have become feasible with such advancements, especially for real-time implementation. In this section, most commonly used FMCW radar architecture is assumed and targets are considered to be stationary. Hence, (7) is reduced to a range-azimuth estimation problem with the signal model given by

To elucidate advanced estimation techniques, the dimensionality of the problem is reduced to two dimensions. It should be noted that the discussed techniques can be extended to four-dimensional problems with mobile targets and elevation direction. As discussed previously, the 2-D FFT of (8) can provide joint estimation of distance and angle. The FFT-based estimation has the least complexity of implementation, which is O(LN log LN), where N is the number of time domain samples and L denotes the number of elements in a one-dimensional (1-D) antenna array. However, the resolution of Fourier techniques is dictated by the Rayleigh limit. While the higher range resolution can be obtained with larger FMCW bandwidth, the higher angular resolution requires more antenna elements, adding to the cost of RF front end. Additionally, the radar has to process a larger set of signal samples. However, it is important to reduce the computational load while realizing the desired angular and range resolution. We first visit the ML formulation of joint estimation of range and direction of targets. Then, we review the so-called superresolution techniques as suboptimal and lower complexity alternatives to the ML estimator.

## ML estimation

The complex Gaussian observation noise in (8) is assumed to be temporally and spatially independent. ML estimation of 2-D parameters (R,θ) can be found solving the following equation:

Thus, depending on the granularity of (R,θ) search space, the ML estimator can offer the resolution beyond the Rayleigh limit set by system parameters such as bandwidth and number of antenna elements. However, the complexity of implementing this algorithm depends on the cardinality of the search-space as well as the number of targets. Since (R_{q},i_{q}) are continuous parameters, the computational complexity of ML algorithm O(|(R, θ)|^{Q}) becomes prohibitive. In the subsequent paragraphs, the superresolution techniques that can achieve high resolution at lower computational cost are illustrated.

## Superresolution techniques

Due to their prohibitive computational cost, ML algorithms need to be implemented via suboptimal techniques. These techniques rely on collecting enough signal samples. At a sufficiently high SNR, eigenvalues and associated eigenvectors of sample covariance matrix C (defined in Algorithm 1) represent the ML estimate of their true values. Hence, these eigenvectors can be used to resolve the target with high resolution.

The superresolution algorithms that rely on these techniques include multiple signal classification (MUSIC) [24] and estimation of signal parameters via rotational invariance technique (ESPRIT) [25], [26]. Recalling the 2-D (R,θ) stationary target location estimation problem, the superresolution algorithms can be applied across each dimension separately. However, this approach might lead to the so-called association problem [15]. Since the association of estimated parameters is the key step in interpreting and delivering results to the driver assist system, joint processing can be implemented. As (R,θ) domain is jointly searched for its entire range, the possibility of ghost targets is eliminated and unambiguous results are obtained [27].

For proper transmitter spacing, the colocated MIMO radar can emulate a larger aperture phased array radar.

As discussed previously, the temporal frequency of (8) gives the range, and spatial frequency corresponds to the angular position of the target. Hence, the traffic imaging problem can be turned into a classical parameter estimation problem so that superresolution techniques such as MUSIC can be applied. From (8), a 2-D matrix is formed, which has a Vandermonde structure across each dimension for a uniform linear antenna array. A 2-D joint superresolution was applied in the radar imaging context in [27] and later with FMCW waveforms in [21], which is described in Figure 6 and Algorithm 1. The complexity of the 2-D joint superresolution algorithm lies in the cost of eigenvalue decomposition of covariance matrix C_{LsNs}_{x}_{LsNs} and 2-D exhaustive search over the entire range of (R,θ) domain. Thus, traditional 2-D joint superresolution algorithm has computational complexity of the order of O(L_{s}N_{s})^{3}.

Larger size sampled covariance matrix makes 2-D joint superresolution algorithms difficult in practice. To deal with implementation issue of superresolution algorithms in real time, size of the observation space must be reduced.

## Complexity reduction technique using beamspace projection

FFT-based estimation techniques have a low complexity of implementation. However, its resolution is limited by the radar bandwidth and a number of antenna elements. On the other hand, the superresolution estimation resolves closer targets yet has higher computational complexity. Thus, there exists a trade-off between resolution and complexity. To reduce the computational complexity of superresolution algorithms and maintain their resolution capability, we propose two-stage estimator using a beamspace superresolution algorithm, which breaks the large problem into smaller problems using initial FFT processing [28], [29].

The computational cost of a joint superresolution algorithm lies mainly in the eigenvalue decomposition of large sample covariance matrix. Thus, to reduce the cost, the size of the covariance matrix must be reduced. Hence, as the first stage of a two-stage FFT-based-beamspace algorithm, we obtain the FFT of 2-D matrix D_{LXN} in (8). From the output of this low-resolution 2-D FFT, we can determine temporal and spatial frequencies, which correspond to the approximate location of a target or cluster of targets. Once the frequencies of interest are known, we can project the data from the higher-dimensional subspace of D_{Ls}xN_{s} to the lower subspace of our interest D_{LbXNb} using DFT matrices, which form nonoverlapping beams in range and angular domain. Thus, the superresolution algorithm operates on the smaller data set, and the complexity of a 2-D super resolution imaging can be reduced to O(L_{b}N_{b})^{3}. Moreover, 2-D exhaustive search for the target on a finer grid operates over an area of interest, thereby further reducing the complexity. The performance of the beamspace algorithm is demonstrated in Figure 7. More detailed discussion on the complexity analysis and implementation of radar algorithms can be found in [30].