Keywords

1 Introduction

Accurate determination of the location and speed of moving objects is necessary in many technical applications, including the vehicular communications. To determine the location and radial speed of objects, various principles can be used: ultrasonic, optical lidar, use of video cameras and microwave [1,2,3,4,5,6,7,8,9,10]. The first three listed principles have a significant drawback - a strong dependence on weather conditions (precipitation, fog) and have a small detectable distance to the object (ultrasound and video camera). For this reason, microwave radars have found the greatest use in automotive telecommunications networks and motion control systems. Radar in automotive applications are used to automatically control the distance between moving vehicles, cross traffic alert and lane change assist, parking aid, obstacle, pedestrian and blind spots detection. In addition, radars are used by traffic police to ensure traffic safety using administrative measures.

Microwave car radars for the simultaneous measurement of speed and distance use the formation and emission of signals of a special form, which include: frequency-modulated continuous wave (FMCW), frequency shift keying (FSK) continuous wave. In [1], a waveform was proposed that became classic for FMCW radars. Such radars have high measurement accuracy, high resolution in distance and speed of targets and high performance. The disadvantage of such radars is the small value of the uniquely measured target speed. In [2], for a radar with a sequence of 64 chirps with a duration of 1 ms each, sweep bandwidth of 150 MHz at a radio frequency of 24 GHz, the maximum detectable speed of only 22.5 km/h is determined, which is not enough for use in vehicular communications applications.

In [3,4,5,6], the problem of unambiguously determining high target velocities is solved by using a signal consisting of multiple chirp sequence segments with different chirp repetition intervals (including random). These methods are quite difficult to implement and are accompanied by a decrease in the maximum measured range. In [2, 7] to extend the range of target speed estimation, a signal in the form of a Chirp Sequence Waveform is used with a shift of the initial frequency in the adjacent chirps of the frame.

We have shown [10] that the conclusions of the authors of [2, 7] are incorrect, including when exposed to noise on the receiving radar system. In [10], it was also shown that the use of phase methods to determine the target’s moving speed significantly reduces the noise immunity of the radar and, therefore, the minimum measured value of the target scattering area. From this point of view, methods based on processing the amplitude spectrum of the fast Fourier transform (FFT) of the received radio signal are more preferable. In [11,12,13] various waveforms of the generated signals are proposed with subsequent processing methods for estimating the speed and distance in the multi-target mode.

In this paper, we consider an algorithm for estimating velocities and distances to targets based on an estimate of the amplitude spectrum of the received microwave signals. The additive white Gaussian noise was used as a model to describe the effect of clutter in real radar system. The causes of the occurrence of false targets are determined and the detection efficiency in the additive white Gaussian noise in the multi-target mode is investigated. These issues are of practical interest and were not considered in [11,12,13].

2 Model Derivation and Parameter Estimation Algorithm

Figure 1 shows the chirp sequence waveform of the radar transmitter’s probe signal. The choice of the chirp sequence waveform is determined by the simplicity of the technical implementation and the wide choice of millimeter-wave FMCW radar for automotive and industrial applications serially supplied to the market by various companies [14]. One frame of a transmitter signal with a duration of \( T_{\sum } = T_{0} + T_{1} + T_{2} \) contains a sequence of one pair of periodically repeating signals with linear frequency modulation. In the first interval, the frequency deviation is 0, i.e. the signal is emitted with a constant frequency equal to f0. The duration of the chirp Tk, the deviation of the frequency of the Δfk, is determined by the variable k (k = 0, 1). The initial frequency of the signal f0.

Fig. 1.
figure 1

Proposed waveform

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The waveform of the radar receiver’s input signal (echo-signal waveform) is delayed with respect to the transmitter’s sounding signal by a time equal to τ. For practically important radar applications, the delay τ can be calculated using the formula

$$ \uptau \left( t \right) = 2R\left( t \right)/c = 2R\left( 0 \right)/c \, - \, 2V_{R} t/c = \uptau_{0} - 2V_{R} t/c,\;\;\;\;t \in [0,T_{\sum } ], $$

where c – is the speed of light, R(0) – is the initial coordinate of the target, and the radial component of the velocity is related to the Doppler shift of the fD frequency as follows 2VR/c = fD/f0.

In Fig. 1, a dashed line represents the echo waveform. At the radar receiver, the input analog microwave signal is converted into an intermediate frequency (IF) signal and then converted into digital form.

For further construction of the mathematical model of the considered radar signals and the synthesis of the algorithm for their processing, we use the results of [10]. The signal SI(tL, k) at the output of the mixer in the in-phase channel I of intermediate frequency of the radar receiver can be written as

$$ \begin{aligned} SI(t_{\text{L}} ,k) \;=\;& U_{\text{M}} \cos \left\{ \begin{aligned} 2\uppi \left[ {\frac{{\Delta f_{k} }}{{T_{k} }}\uptau_{0} + f_{D} + \Delta F(k)} \right]t_{\text{L}} \hfill \\ + \;2\uppi \left[ {f_{D} T(k) + f_{0} \uptau {}_{0}} \right] + \Delta \upvarphi (k) \hfill \\ \end{aligned} \right\} \cdot {\text{rect}}\left[ {\frac{{t_{\text{L }} - \uptau }}{{T_{\text{k}} }} - \frac{1}{2}} \right] \\ & + \;n(t_{\text{L }} ,k) \\ \end{aligned} $$
(1)

Formula (1) describes the mathematical model of the IF signal of the odd k chirp in a waveform sequence under the conditions of the additive noise n(tL, k). The rect(x) function is a symmetric function of a rectangular window. The functions ΔF(k) and \( \Delta \upvarphi (k) \) clarify the mathematical model of the intermediate frequency radar signal, their form is given in [10]. Time tL is “fast” (local time) [10].

The additive noise n(tL, k) is described by a white Gaussian noise model with zero mean and dispersion (power in the receiver frequency band) σ2.

The mathematical model of the intermediate frequency signal of the odd k chirp SQ(tL, k) in the quadrature channel Q is described by a similar expression (1) replacing the function cos(x) with sin(x). The noise correlation in quadrature channels depends on the ratio of the levels of external and internal noise. The model of radar signals allows you to develop a strategy for the simultaneous assessment of target distance R and its speed VR in multi-target mode.

According to (1), the beat frequencies fR1 and fR2, i.e. the frequency of the signal at the mixer output in the I and Q channels of the intermediate frequency of the radar receiver for chirps, respectively, with the numbers k = 0 and k = 1 are determined by the expressions

$$ f_{R1} = \frac{{\Delta f_{1} }}{{T_{1} }}\uptau_{0} - \frac{{2V_{R} }}{c}f_{0} , $$
(2)
$$ f_{R2} = \frac{{\Delta f_{2} }}{{T{}_{2}}}\uptau_{0} - \frac{{2V_{R} }}{c}\left( {f_{0} - \frac{{3\Delta f_{2} }}{2}} \right) $$
(3)

The most difficult task in the multi-target mode is the problem of identifying the beat frequency fR belonging to the same target at t ∈ T1 and t ∈ T2, i.e. for a sequence of chirps with numbers k = 0 and k = 1. From relations (2) and (3) in the first approximation for the beat frequencies fR1 and fR2 belonging to the same target, the following condition must be fulfilled (the condition of “pairing”) for the functional Ξ(fR1, fR2)

$$ \varXi \left( {f_{R1} ,f{}_{R2}} \right) = f_{R1} \frac{{T_{1} }}{{\Delta f_{1} }} - f_{R2} \frac{{T{}_{2}}}{{\Delta f_{2} }} - f_{D} \left( {\frac{{T_{1} }}{{\Delta f_{1} }} - \frac{{T_{2} }}{{\Delta f_{2} }}} \right) \to \hbox{min} $$
(4)

The processing algorithm for the received signal includes a one-dimensional fast complex Fourier transform with a Hamming window, an estimate of the frequencies of the maxima of the amplitude spectrum, a calculation of the distances and speeds of the targets based on the frequencies of the maxima and the use of the algorithm for pairing the targets and the speed of the targets. On the interval where the frequency of the emitted signal is constant (T0), the speed of all targets is determined from the maxima of the amplitude spectrum. The resolution of determining the frequency is determined by the duration of the corresponding interval. Thus, the speed resolution δVR and range are respectively equal to

$$ \updelta V_{R} = \frac{c}{{2T_{0} f_{0} }};\;\updelta R = \frac{c}{2\Delta f} $$

Each Doppler target frequency corresponds to a 2D matrix of beat frequencies for T1 and T2 gaps. The choice of the minimum matrix values according to the algorithm (4) determines the Doppler frequency (speed) and two beat frequencies corresponding to the range R to the target.

The results of a computer experiment to estimate the parameters of R and VR in LabVIEW environment based on the developed algorithm are given in Sect. 4.

3 The Genesis of False Targets in Multi-target Mode

The analysis of the target parameter estimation algorithm in the multi-target mode, described in Sect. 2, shows the possibility of false targets with virtual (not corresponding to the actual) values of the radial velocity VR and the distance R. The reason for the occurrence of false targets is the situation when the beat frequencies fR1 and fR2, corresponding to two different speeds VR1 and VR2 of two distinguishable targets, coincide with the beat frequencies fR1 and fR2 of the third target, which has a speed VR3. A false target has a speed corresponding to one of the real targets, but the distance is erroneously determined. The maximum possible number of false targets nF with a total number of observable targets N with distinct speeds is proportional to the number of placements without repetitions of N elements of 2 each.

$$ n_{F} = N \cdot A_{2}^{N} = N \cdot (N - 1) \cdot (N - 2) $$

The value of nF rapidly increases with increasing N. Thus, when the number of observed targets is N = 9 with distinct velocities VR, the maximum possible number of false targets is 504, which can be comparable with the number of resolvable samples in the distance NR, determined by the formula.

$$ N_{R} = \frac{{R_{MAX} }}{\updelta R} $$

To exclude false targets, complex multi-segment waveform types [2, 3] or methods of tracking the trajectory of targets with subsequent filtering algorithms are used. The multiple-input multiple-output (MIMO) design of radars, which is mainly used to increase the angular resolution by increasing the aperture of the antenna array, will also reduce the probability of false targets. The using of MIMO in the proposed multi-target algorithm requires further study.

4 Results of a Computer Experiment

The standard mode of car radar operation is the simultaneous assessment of parameters (distance and speed) of a significant number of targets, which corresponds to the actual situation on the road. As noted earlier, in radar data processing algorithms there is a technical problem of pairing measured distance values with speed. In this process, ghosts can occur - false targets, the occurrence of which is explained by the discrepancy between the distance-speed pair and the true values of the target. Using a computer experiment in LabVIEW, we investigated the efficiency of multi-target detection in additive white Gaussian noise and calculated the probability of detection for various ratios of the signal to noise at the input of the radar receiver. Simulation parameters are shown in Table 1.

Table 1. Simulation parameters
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The choice of relatively extended modulation period (T0,1,2 = 7 ms) is due to the high-resolution requirements for target distances and speeds. The speed resolution for the parameters indicated in Table 1 is 1 km/h, for the distance - 0.42 m at the first chirp and 0.3 m at the second.

Computational experiments are implemented in the LabVIEW design environment. The block diagram contains a modulator unit and a radar intermediate frequency signal processing unit. The modulator unit sets the target parameters (distance, speed) and the technical characteristics of the radar (sampling frequency, modulation period T0,1,2 of waveform segments, carrier frequency and sweep bandwidth). Based on these parameters, a discrete complex intermediate frequency signal is formed in the modulator unit, consisting of samples of in-phase (I) and quadrature signals (Q), to which is added an additive white Gaussian noise with standard deviation σ. Thus, the modulator unit generates an intermediate frequency signal at the output of the radar mixer. The processing unit performs a complex Fourier transform with the Hamming window of the complex intermediate frequency signal, converts the spectrum indices to bring the first Nyquist zone to zero frequency, which is necessary to determine positive and negative speed values, calculates the amplitude and phase spectrum of the intermediate frequency signal, determines the peak frequencies of the amplitude spectrum on all three waveform segments, forms a functional Ξ(fR1, fR2) in accordance with the expression (4) for the speeds and distances of all targets. As a result, each speed corresponds to a 2D distance matrix of size (Ntarget × Ntarget), in which cells with the minimum value of the functional Ξ(fR1, fR2) are searched. The minimum correspond to the speed from the constant frequency segment and to the distances from the first and second chirps, which coincide with the accuracy of the radar resolution in distance. Thus, pairing of speeds and target distances in the multi-target mode is carried out.

To compare the degree of influence of additive noise on the amplitude and phase spectrum of the FFT, the standard deviations of 50 samples of the maximum amplitude spectrum of the target and phase values at the maximum of the amplitude spectrum under Gaussian noise with a signal-to-noise ratio of −18 dB were measured in the radar. The noise was determined by integrating the entire FFT frequency band, the signal amplitude is 1 V, the root-mean-square deviation (standard deviation - SD) of the Gaussian noise is σ = 4. The simulation results are summarized in Table 2.

Table 2. Simulation results
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The normalized SD, defined as the ratio of the root-mean-square deviation (standard deviation) to the maximum level of the amplitude spectrum and phase at the maximum in the absence of noise, for phase noise is more than three times greater than for amplitude noise. Thus, the use of the amplitude spectrum (without the phase spectrum) in the radar processing algorithm leads to an improvement in the sensitivity of the radar to low signal levels. The waveform of the intermediate frequency, amplitude and phase spectrum of the first chirp for 4 targets in the absence of noise a) and at signal-to-noise ratio (SNR) = −18 dB b) is shown in Fig. 2.

Fig. 2.
figure 2

Intermediate frequency waveform on the T1 interval, its amplitude and phase spectrum

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The signal-to-noise ratio (SNR) is determined as the ratio of the signal power at the beat frequency to the Gaussian noise dispersion σ2. The effect of additive noise most significantly affects the shape of the phase spectrum.

The simulation of the work of the automotive radar in the multi-target mode for the 4 targets under the influence of Gaussian noise was carried out according to the following procedure. At a given level of SNR, 100 computational experiments were carried out to determine the parameters (speed and range) of 4 targets. If at least one of the targets was determined in the experiment incorrectly, the result was recorded as negative. If the parameters of all 4 targets are determined without errors, the result was recorded as positive. The target parameters set in the experiments are shown in Fig. 3. As a result, we obtained the dependence of the probability of the velocity and distance true estimation (%) on the signal-to-noise ratio (dB) shown in Fig. 4.

Fig. 3.
figure 3

Parameters of targets in computational experiments

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Fig. 4.
figure 4

The dependency of probability of targets velocity and distance true estimation, % from signal-to-noise ratio.

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The most dramatic slump in the probability of true detection of target parameters is observed when the signal-to-noise ratio decreases from −17 dB to −20 dB. As shown in [10], the operability of phase algorithms [2,3,4,5,6,7] under the influence of noise on the useful signal is determined by the conditions of SNR > 10 dB.

5 Conclusion

Our studies have shown the advantage of signal processing algorithms based on an amplitude spectrum estimate as compared to phase methods for determining the radial velocities VR and the distances R to targets in the automotive FMCW radar of the millimeter range. A computer experiment in the LabVIEW environment showed the efficiency of the proposed algorithm when noise is applied to a useful signal when fulfilling the conditions SNR > −15 dB. A specific feature of signal processing algorithms based on an amplitude spectrum estimate is the occurrence of false targets when estimating the radial velocities VR and distances R. The probability of false targets occurring increases dramatically with an increase in the total number of accompanied targets in the monitoring process.