Range estimation
The range estimation is fundamental to automotive radars. The range R, to a target, is determined based on the roundtrip time delay that the EM waves take to propagate to and from that target: R = (cτ/2), where τ is the roundtrip time delay in seconds and c is the speed of light in meters per second (c = 3 X 10^{8} m/s). Thus, the estimation of x enables the range measurement [1]. The form of the EM waves (signals) that a radar transmits is important for roundtrip time delay estimation. For example, pulsemodulated continuous waves (CWs) consist of periodic and short power pulses and silent periods. Silent periods allow the radar to receive the reflected signals and serve as timing marks for radar to perform range estimation as illustrated in Figure 2. However, unmodulated CW signals (i.e., Cos(2πf_{c}t)) cannot be used for range estimation since they lack such timing marks. Additionally, the signal reflected from a target should arrive before the next pulse starts. Hence, the maximum detectable range of a radar depends on pulse repetition interval T_{PRF}. The transmitted signal from the radar until it is received back undergoes attenuation due to the path loss and imperfect reflection from the target. In addition, received target signals are subject to internal noise in radar electronics and interference that may be a result of reflected signals from objects not of interest and may come from humanmade sources (i.e., jamming). The typical roundtrip time delay estimation problem considers only ambient noise in the form of additive white Gaussian random process. It is assumed that demodulation has already removed the carrier so that a target signal x(t) at baseband can be modeled as
where α is a complex scalar whose magnitude represents attenuation due to antenna gain, path loss, and the RCS of the target and w(t) is additive white Gaussian noise with zero mean and variance σ^{2}. The goal is to estimate τ with the complete knowledge of the transmitted radar waveform s(t). Assuming the signal s(t) has unit amplitude and finite energy E_{s}, the ideal radar receiver can be found using a matched filter with the impulse response h(t) = s*(t), which maximizes signal to noise ratio (SNR =(α^{2}E_{s}/σ^{2}) =(α^{2}T_{s}/σ^{2}) at the output. Thus, the matched filterbased receiver finds the correlation between the transmitted signal and received reflected pulses
The maximum likelihood (ML) estimate of the time delay is the time that the magnitude of the matched filter output peaks at
The presence of the noise can perturb the location of the peak, which will result in the estimation error. Furthermore, the radar needs to decide whether or not a received signal actually contains an echo signal from a target. A good deal of classical radar literature is devoted to developing strategies that provide the most favorable detection performance.
A typical decision strategy can be formulated based on statistical hypothesis testing (a target present or not). This leads to a simple threshold testing at the matched filter output. Range resolution, another key performance measure, denotes the ability to distinguish closely spaced targets. Two targets can be separated in the range domain only if they produce nonoverlapping returns in the time domain. Hence, the range resolution is proportional to the pulsewidth T_{p} . In other words, finer pulses provide higher resolution. However, shorter pulses contain less energy, which implies poor receiver signalto noise ratio (SNR) and detection performance. As explained in the section “Radar Waveforms,” this problem is overcome by the technique called pulse compression, which uses phase or frequency modulated pulses.
Velocity estimation
Estimation of the target velocity is based on the phenomenon called the Doppler effect. Suppose the car displayed in Figure 2 is moving ahead with differential velocity v. With the existence of relative motion between two cars, the reflected waves are delayed by time τ =(2(R±vt)/c). The time dependent delay term causes a frequency shift in the received wave known as the Doppler shift f_{d} =(±2v/λ). The Doppler shift is inversely proportional to wavelength λ, and its sign is positive or negative, depending on whether the target is approaching or moving away from the radar. While this frequency shift can be detected using CW radar, it lacks the ability to measure the targets range. Here, we discuss a pulsed radar configuration that uses frequency modulated (FM) CW pulses and provides simultaneous range velocity estimation in multitarget traffic scenarios.
The FMCW radar transmits periodic wideband FM pulses, whose angular frequency increases linearly during the pulse.
For the carrier frequency, f_{c} and FM modulation constant K, a single FMCW pulse can be written as [see Figure 3(a)]
The signal reflected from a target is conjugately mixed with the transmitted signal to produce a lowfrequency beat signal, whose frequency gives the range of the target. This operation is repeated for P consecutive pulses. Twodimensional (2D) waveforms in Figure 3(c) depict successive reflected pulses arranged across two time indices. The slow time index p simply corresponds to pulse number. On the other hand, the fast time index n assumes that for each pulse, the corresponding continuous beat signal is sampled with frequency fs to collect N samples within the time duration T. Assuming single target and neglecting reflected signal distortions, the FMCW radar receiver output as a function of these two time indices is given by
Therefore, as illustrated in Figure 3(c), discrete Fourier transform across fast time n can be applied to obtain beat frequency f_{b} =(2KR/c) coupled with Doppler frequency f_{d} . This operation is also known as the range transform or range gating, which allows the estimation of Doppler shift corresponding to unique range gate by the application of second Fourier transform across the slow time. A rangeDoppler map can be found efficiently by using 2D fast Fourier transform (FFT) (5). A demonstrative example based on the aforementioned discussion is shown in Figure 3.
Direction estimation
Use of wideband pulses such as FMCW provides discrimination of targets in both distance and velocity. The discrimination in direction can be made by means of an antenna array. Figure 4(a) depicts a realistic traffic scenario with several targets surrounding the radar that collects direct and multipath reflections from them. In such cases, to spatially resolve equidistant targets and deliver a comprehensive representation of the traffic scene, the angular location of targets should be estimated. Therefore, in automotive radars, the location of a target is often described in terms of a spherical coordinate system (R,θ,∅), where (θ,∅) denote azimuthal and elevation angles, respectively. However, in this case, the single antenna radar setup as used in the rangevelocity estimation problems may not be sufficient, since the measured time delay τ =(2(R±vt)/c) lacks the information in terms of angular locations of the targets.
To enable direction estimation, the radar should collect reflected wave data across multiple distinct dimensions. For example, locating a target using EM waves in 2D requires the reflected wave data from the object to be collected in two distinct dimensions. These distinct dimensions can be formed in many ways using combinations of time, frequency, and space. For instance, a linear antenna array and wideband waveforms such as FMCW form two unique dimensions [13], [14]. Additionally, smaller wavelengths in mmwave bands correspond to smaller aperture sizes and, thus, many antenna elements can be densely packed into an antenna array. Hence, the effective radiation beam, which is stronger and sharper, in turn, increases the resolution of angular measurements.
Consider an antenna array located in plane z = 0, and let l be the abscissa corresponding to each receiver antenna position [see Figure 4(b)]. Let (R_{q},i_{q}) be the position of the q^{th} target in spherical coordinates, moving with velocity vq relative to the radar. With the help of farfield approximation [15], for the q^{th} target, the roundtrip time delay between a transmitter located at the origin and the receiver positioned at coordinate l is given by
where d is the distance between antenna elements (usually half the wavelength) arranged in a linear constellation. Combining (5) and (6) gives the threedimensional (3D) FMCW radar output signal, which enables estimation of range, velocity, and angle. For Q number of targets, the signal can be represented as
where α and ω correspond to same quantities as explained in the range estimation problem. The delay term τl_{q} creates uniform phase progression across antenna elements, which permits the estimation of the angle by FFT in the spatial domain, as shown in (7). Thus, 2D location (range and angle) and speed of targets can be jointly estimated by 3D FFT. The target location and velocity estimation problems are revisited later in the section “Advanced Estimation Techniques” with more emphasis on the highresolution algorithms and computational complexity analysis.
Radar waveforms
Various automotive radar classes, summarized in Table 1, have diverse specifications in terms of several fundamental radar system performance metrics, such as range resolution, velocity resolution, angular direction, SNR, and the probability of target detection. The type of waveform employed by a radar is a major factor that affects these metrics. The radar waveforms, as summarized in Table 2, can be characterized whether or not they are CW, pulsed and frequency, or phase modulated. Modulated radar waveforms include FM CW, stepped frequency (SF) CW, orthogonal frequencydivision multiplexing (OFDM), and frequency shift keying (FSK). Each waveform type has a certain advantage in processing, implementation, and performance as follows:

 In the CW radar, a conjugate mixing of a highfrequency transmitted and received signal produces the output signal at the Doppler frequency of the target. The resolution of frequency measurement is inversely proportional to the time duration of the signal capture. The continuous nature of the waveform precludes roundtrip delay measurement, which is necessary for range estimation of the target [see Figure 5(a)]. Hence, apart from ease of implementation and ability to detect target speed, the CW radar cannot provide the range information.
 Pulsed CW radar can estimate the range information as explained previously in the section “Basic Automotive Radar Estimation Problems.” The Doppler frequency can be estimated by making each pulse longer and measuring the frequency difference between the transmitted and received pulses. As shown in Figure 5(b), the pulse duration and pulse repetition frequency (PRF) are the key parameters in designing pulsed CW radar with desired range and velocity resolution.
 FMCW, also known as linear frequency modulation (LFM) or chirp, is used for simultaneous range and velocity estimation (refer to the “Velocity Estimation” section for details). Due to the pulse compression, the range resolution is inversely proportional to the bandwidth of the FMCW signal and is independent of pulsewidth. For example, the shortrange FMCW radar uses ultrawideband (UWB) waveforms to measure small distances with higher resolution. The Doppler resolution is a function of pulse width and the number of pulses used for the estimation. Thus, with the ability to measure both range and speed with high resolution, FMCW radar is widely used in the automotive industry.
The traffic imaging problem can be turned into a classical parameter estimation problem so that superresolution techniques such as MUSIC can be applied.
 In contrast to FMCW waveforms, the frequency of FSK and SFCW varies in a discrete manner [see Figure 5(c)]. In this case, the range profile of the target and the data collected at discrete frequencies form the inverse Fourier transform relationship. Also, hybrid waveform types can be employed to achieve additive performance. FSK waveform can be combined with multislope FMCW waveform to overcome ghost targets in radar processing [16]. Similarly, alternate pulses of CW and FMCW are used to accurately estimate range and Doppler [17].
 OFDM can be viewed as another multifrequency waveform that offers unique features of the joint implementation of automotive radar and vehicletovehicle communications [18], [19]. For the radar operation, the orthogonality between OFDM subcarriers is ensured by choosing carrier spacing more than maximum Doppler shift, and the cyclic prefix duration is selected greater than the longest roundtrip delay [see Figure 5(d)]. The range profile is estimated through frequency domain channel estimation. OFDM radar processing along with simulation results is explained in [20].
Based on the knowledge of target statistics, radar waveforms can be optimized. Radar waveform design is revisited along with multipleinput, multipleoutput (MIMO) radars in the “MIMO Radar” section.