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Optimization of ACFDSRbased joint Doppler shift and SNR estimator for Internet of Vehicle system
EURASIP Journal on Advances in Signal Processing volumeÂ 2022, ArticleÂ number:Â 37 (2022)
Abstract
Accurate vehicle speed estimation is required for intelligent internet of vehicles, and it can be realized by Doppler shift estimation in mobile communication. In this paper, the ACFDSR (autocorrelation functiondouble sampling rate) method for joint Doppler shift and SNR estimation is further investigated. Based on the analysis of ACF, an improved algorithm model taking account of estimation deviation is formulated. Then, the effects of sampling intervals in DSR are figured out by mean square error analysis of estimation, and two better choices are obtained. By Monte Carlo simulations, it is demonstrated that the optimized ACFDSR method with better choice of sampling intervals can achieve better estimation and outperform previous methods.
1 Introduction
The Internet of Things (IoT) is recognized as one of the most important areas of future technology and is gaining vast attention from a wide range of industries [1,2,3,4]. For example, IoT is optimized with 5G network to expand spectrum resources and supply large data volume business [5], simultaneous wireless information and power transfer has been proposed for energy efficiency and ubiquitous links [6]. With the development of IoT, Internet of Vehicles (IoV) is evolving as a new theme of research from traditional vehicular Ad hoc networks (VANETs) [7, 8] to enable more intelligent driving experiences such like vehicle localization, behavior analysis even automatic driving [9,10,11]. For these applications, the vehicle speed estimated by perception operation is an essential input. The accurate speed estimation makes the cognitive computing based on it reliable in complex traffic environment, thus providing further support for intelligent decisionmaking [12, 13].
One way to obtain vehicle speed is through Doppler shift estimation in mobile communication and subsequent conversion calculation [14]. Several estimators had been proposed, but most of them suffered from additive white Gaussian noise (AWGN) [15,16,17,18]. Although maximum likelihood methods [19, 20] took account of the noise effects for accuracy enhancement, an exhausted search was required, making them not competitive for realtime applications. On the other hand, to adapt the communication receiver to wireless propagation, signaltonoise ratio (SNR) estimation is also needed [21]. To reduce realization cost, Doppler shift and SNR were jointly estimated [22,23,24]. A nonparametric estimator based on peak search in frequency domain was proposed, but its Doppler shift estimation was not reliable [22]. For improvement, signal processing methods including double sampling rate (DSR), autocorrelation function (ACF) and level crossing rate (LCR) had been applied in estimator, making the estimator also applicable in noisy scenarios [23, 24].
In former research on ACFDSR estimator [23], though the simulation results showed a good performance in a wide range of velocities and SNRs, the analysis should be further refined. Because the fixed integer ratio of two sampling intervals limits estimator performance, and the perfect estimation of original Doppler shift does not work in practice. Therefore, an improved Doppler shift estimator taking account of nonideal deviation is established in this paper, and factional ratio is also applied to analyze its effect on estimator performance under the assumption of large estimation error. By Monte Carlo simulations, two better ratio of sampling intervals are obtained, in which the fractional one can also benefit computation reduction. Simultaneously, it is demonstrated that the optimized ACFDSR estimator with better sampling interval setting can produce better Doppler shift and SNR estimation. In contrast to previous methods, the optimized ACFDSR estimator performs best, which consequently produces the most accurate speed estimates.
The paper is organized as follows: the original ACFDSR estimator is introduced in Sect.Â 2. Then, in Sect.Â 3, the optimized estimator is formulated, and the analysis of sampling intervals is achieved. Finally, in Sect.Â 4, the superiority of the optimized estimator is demonstrated by numerical results. Conclusion is given in Sect.Â 5.
2 ACFDSR estimator
In this paper, it is assumed that a bandlimited pilot signal is transmitted over a Rayleigh fading channel. The ACF calculation for Rayleigh fading channel can be expressed as:
where \(\sigma_{l}^{2}\) and \(f_{d}\) represent the actual channel variance and Doppler shift, \(T_{s}\) is the pilot symbol interval, \(J_{0} ( \cdot )\) denotes the first kind Bessel function [23].
To estimate Doppler shift and SNR jointly, the channel ACF is firstly estimated base on channel estimates, i.e.,
where \(\hat{c}_{l} (i)\) is the channel estimates (i is the discrete time index and l denotes the path index), and k is the index for discrete time delay. Besides, K is the total sample number which should be larger enough to ensure the time length larger than the fading period. Then, the original Doppler shift estimator based on ACF can be given by
Above estimator is known bias when AWGN exists. For improvement, the ACF was adapted for noisy scenarios by setting \(R(0) = \sigma_{l}^{2} + \sigma_{z}^{2}\), and its estimation \(\hat{R}(0)\) was correspondingly modified [23].
Above Doppler shift estimation is original, to enhance accuracy also to obtain the SNR and Doppler shift estimations jointly, the DSR technique was applied [23]. It is realized by establishing equation after getting an original Doppler shift estimation, i.e.,
where \(\gamma_{s} = {{\sigma_{l}^{2} } \mathord{\left/ {\vphantom {{\sigma_{l}^{2} } {\sigma_{z}^{2} }}} \right. \kern\nulldelimiterspace} {\sigma_{z}^{2} }}\) denotes the SNR.
In above equation, there are two unknown parameters, i.e., Doppler shift \(f_{d}\) and SNR \(\gamma_{s}\). For joint estimation through equations solving, two sampling intervals \(T_{s1} = mT_{s}\) and \(T_{s2} = nT_{s}\) (m, n are integers and \(m < n\)) are adopted, then two equations based on two original Doppler shift estimations (\(\hat{f}_{d1}\) and \(\hat{f}_{d2}\)) are formulated as below, in which the fourth order approximation of Bessel function is adopted to enables Doppler shift estimator practicable in highspeed scenarios.
Solving above equations, the following joint estimation is obtained.
where \(A = \frac{{2  (m\pi \hat{f}_{d1} T_{s} )^{2} }}{{2  (n\pi \hat{f}_{d2} T_{s} )^{2} }}\) and \(B = \frac{{(\hat{f}_{d} n)^{2}  (\hat{f}_{d} m)^{2} }}{{(\hat{f}_{d2} n)^{2}  (\hat{f}_{d1} m)^{2} }}\).
For above two sampling intervals \(T_{s1} = mT_{s}\) and \(T_{s2} = nT_{s}\), the increase in \(m(n)\) results the decrease in channel sampling rate, thus the computation cost can be definitely reduced when doing ACF calculation. At the same time, the sampling theory must be held, i.e., the sampling rate must be larger than two times of the channel bandwidth. Correspondingly, the following inequation should be satisfied, which limits the value choice of \((m,n)\).
Thus, for the application of ACFDSR estimator, the maximum value of the normalized Doppler shift \(f_{m}\) should be firstly estimated to determine the integer values for \((m,n)\). It is explicit that the smaller value of \(f_{m}\) enlarges the value range for \((m,n)\).
3 Analysis and optimization of ACFDSR estimator
3.1 Algorithm model establishment
Because of nonideal channel model or sample number for estimation, the original Doppler estimation makes Eqs.Â (5) and (6) untenable. To enhance the authenticity and reliability of the analysis in this section, an estimation deviation \(\Delta\) is considered existing between the values on both sides of the equal sign in (5) and (6). According to our simulations, the following remarks have been concluded.

When the actual Doppler shift \(f_{d}\) and SNR \(\gamma_{s}\) are increasing, the estimation deviation \(\Delta\) is generally decreasing. Besides, the maximum of the observed \(\Delta\) in simulation can nearly be 23% of \(f_{d}\).

The larger sampling interval leads to smaller deviation, i.e., \(T_{s1} < T_{s2} \Rightarrow \Delta_{1} > \Delta_{2}\). Because the noise bandwidth becomes smaller when larger sampling interval is applied, thus SNR is equivalently increased, which results in smaller deviation.

The deviation \(\Delta_{1} (\Delta_{2} )\) is approximately a linear decreasing function of \(m(n)\).
Based on above remarks, Eqs.Â (5) and (6) for joint Doppler shift and SNR estimations in (7) are reformulated as
3.2 Choice of sampling intervals
In this part, above two Eqs.Â (9) and (10) are utilized to analyze the effect of two sampling intervals, i.e., the values of \((m,n)\). In this paper, the maximum value of the normalized Doppler shift \(f_{m}\) is prespecified 0.1 for simulation and analysis. Due to the limit in (8) and the assumption as \(m < n\), the maximum value for n is 5. Thereby, there are two types of value choice for \((m,n)\):

\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is integer: \(m = 1\), and \(n \in \{ 2,3,4,5\}\).

\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is fraction: m and n are relatively prime and \(2 \le m < n \le 5\).
In this paper, the estimation deviation \(\Delta\) is prespecified when doing simulation for analysis. For reliability, it is relaxed to 30% from the observed maximum 23%. The analysis for different \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m}\) is conducted by two cases according to whether \(\Delta T_{s}\) is fixed or changing. In addition, the mean square error (MSE) of the final Doppler shift estimation \(\hat{f}_{d}\) is applied for comparison, which is defined by
where \(E\left[ \cdot \right]\) denotes the expectation operation.
3.2.1 Case I (\(\Delta T_{s}\) is fixed)
In this case, \(f_{m} \in \{ 0.01:0.01:0.1\}\), \(SNR \in \{ 0:3:30\}\) and \(\Delta T_{s}\) is fixed 0.03, which corresponds to the worst case (estimation deviation reaches 30% of \(f_{d}\)). After computer simulation, the MSE of Doppler shift estimation is shown in Fig.Â 1, in which different values of m and n make the resulted MSE fluctuates. From Fig.Â 1a, it can be found that larger n can result in smaller MSE. When using the same n, the fraction n/m outperforms integer n/m, and larger m is better for fraction n/m. Obviously, the originally used \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\) in [23] is the worst choice.
In Fig.Â 1b, the SNR increasing makes the MSE changing like a concave curve. Relatively, the turning point that the MSE changes from decreasing to increasing is latter for lager n. Though in previous remarks, larger \(f_{d}\) and \(\gamma_{s}\) leading to smaller deviation \(\Delta\) is derived, the final estimation \(\hat{f}_{d}\) obtained by equations solving is nonlinearly and comprehensively affected by \(f_{d}\), \(\gamma_{s}\) and \(\Delta\). When negative effects outweigh positive effects, the MSE curve rises. Therefore, in Fig.Â 2, we can also find concave curves of MSE changing for different \(f_{m}\) (\({{f_{d} = f_{m} } \mathord{\left/ {\vphantom {{f_{d} = f_{m} } {T_{s} }}} \right. \kern\nulldelimiterspace} {T_{s} }}\)), in which the negative effects is more evident when \(f_{m}\) is smaller.
3.2.2 Case II (\(\Delta T_{s}\) is changing)
In this case, \(\Delta T_{s}\) is changing from 0.005 to 0.03 by step 0.005. FigureÂ 3 displays the MSE of Doppler shift estimation for different \(\Delta T_{s}\), which increases as \(\Delta T_{s}\) becomes larger. For small \(\Delta T_{s}\), the increase in MSE is rapid and the performances of n/mâ€™s are not consistent with previous performance presented in Fig.Â 1. Such deviation suggests us to choose the best sampling intervals (namely m and n) carefully. For further analysis, the MSEs are averaged over \(\Delta T_{s}\)â€™s, results are shown in Fig.Â 4, and nearly the same conclusion found in Figs. 1 and 2 can be obtained. Hence, the following analysis is carried out with the same parameter setting as that in Case I.
According to all above presentations and analysis, it seems that \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {5 \mathord{\left/ {\vphantom {5 4}} \right. \kern\nulldelimiterspace} 4}\) is the best choice, but it is hard to confirm because the deviation is modeled approximately, and the estimator can be sensitive for two very close sampling intervals. To find a better n/m for practical use, another performance called mismatch MSE (mMSE) is applied.
where M is the total number of \((n,m)\). Actually, \(E_{f} (n,m)\) can be viewed as a measurement for estimation stability under the principle of LeastSquare (LS). The most stable n/m, i.e., the one results in smallest \(E_{f} (n,m)\), can also be robust against nonideal factors.
The mMSE results are presented in Fig.Â 5. It is obvious that \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\) is the worst choice. Moreover, a larger gap between n and m or a smaller n for fraction n/m can lead to higher stability of mMSE. Based on all figures and above comprehensive analysis, \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern\nulldelimiterspace} 3}\) and \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) are two better choices for the scenario in this paper. On the other side, the fraction \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern\nulldelimiterspace} 3}\) has an advantage of computation reduction over \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) for its larger m.
4 Simulation
In this section, the optimized ACFDSR method with better choices of n/m (\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) and \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = {4 \mathord{\left/ {\vphantom {4 3}} \right. \kern\nulldelimiterspace} 3}\)) is further demonstrated in comparison with previous methods including the phase difference method [15], the original ACFDSR method (\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\)) [23], the ACF method [16], the LCRDSR method with fitting [24] and the improved LCRDSR (without fitting) [25]. Computer simulations are executed for estimator comparison, in which the Jakes channel model is adopted. Besides, the pilot symbol interval \(T_{s}\) is 0.2Â ms, the carrier frequency is 2.11Â GHz. The vehicle speed range is 30â€“240Â km/h and its corresponding Doppler shift is 58.6â€“468.8Â Hz [24].
The simulated results of SNR estimation are presented in Fig.Â 6. From this figure, it is explicit that SNR can be very precisely estimated by the optimized ACFDSR estimator. Relatively, the estimations in low SNR scenarios are more accurate than those in high SNR scenarios. For reliable vehicle speed estimation in IoV, the Doppler shift estimation performance is more concerned. FigureÂ 7 shows the comparison of Doppler shift estimations by different methods. Among, the phase difference method and the ACF method performs badly, because they cannot eliminate the influence of AWGN. On the other hand, the three ACFDSR methods perform approximately, and all of them are slightly better than the LCRDSR method and its improved version. Among these two LCRDSR methods, fitting operation is not applied in the LCRDSR, thus its advantage over the LCRDSR method with fitting is limited. Moreover, the figures are partially enlarged for detail comparison, from which we can see that the optimized ACFDSR method yields the best performance.
FigureÂ 8 compares different estimators by averaged MSE of Doppler shift estimation along SNR dimension. Due to that the improved algorithm modeling takes account of estimation deviation, the optimized ACFDSR method is robust to AWGN, thus it can outperform the phase difference method, two LCRDSR methods and the ACF method. Among the ACFDSR methods (the optimized and the original) using different n/m, the optimized with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) or \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4/3\) performs better than the original with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\), and the optimized with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4/3\) is slightly better than the other with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\). Moreover, the optimized with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4/3\) also has an advantage on computation, which can be reduced to nearly one third of the other with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\).
According to the Doppler shift estimates, the vehicle speed estimation can be obtained by linear transformation. The comparison of vehicle speed estimation for two SNRs ({0, 10}Â dB) and three actual speeds ({60, 150, 240}Â km/h) is shown in Table 1. As can be seen from the table, the phase difference method and the ACF method are obviously the two worst estimators. Moreover, the improved LCRDSR is slightly better than the LCRDSR, but both are worse than the ACFDSR methods. Among the ACFDSR methods, the optimized (\({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4\) and \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 4/3\)) perform better than the original with \({n \mathord{\left/ {\vphantom {n m}} \right. \kern\nulldelimiterspace} m} = 2\) for their improved speed estimations. Thus, the optimized ACFDSR estimator can utilized to serve speedrelated applications in IoV.
5 Conclusion
In this paper, the ACFDSR estimator for joint Doppler shift and SNR is optimized by taking estimation deviation into algorithm model and selecting better sampling intervals. Its effectiveness and reliability are confirmed by simulations. Moreover, the estimator is simple to realize in mobile communication with either codemultiplexed or timemultiplexed pilot signals, and its accurate Doppler shift estimation can be utilized for vehicle speed estimation in IoV.
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Abbreviations
 ACF:

Autocorrelation function
 DSR:

Double sampling rate
 IoT:

Internet of things
 IoV:

Internet of Vehicles
 VANETs:

Vehicular Ad hoc networks
 AWGN:

Additive white Gaussian noise
 SNR:

Signaltonoise ratio
 LSR:

Level crossing rate
 WSSUS:

Widesense stationary and mutually uncorrelated scattering
 MSE:

Mean square error
 mMSE:

Mismatch mean square error
 LE:

Logarithmic envelope
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Acknowledgments
The authors would like to thank Zhejiang Gongshang University, Zhejiang Provincial Key Laboratory of New Network Standards and Technologies for their support and anyone who supported the publication of this paper.
Funding
This paper was sponsored by the Natural Science Foundation of Zhejiang Province (Grant No. LQ21F010008), and Zhejiang Gongshang University, Zhejiang Provincial Key Laboratory of New Network Standards and Technologies (Grant No. 2013E10012).
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JYH contributed to the development of ideas. JGW and LY conducted algorithm modeling, numerical analysis, and wrote the paper. XFF also contributed paper writing. ZWN and ZJX proofread and revised this paper. All authors read and approved the final manuscript.
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Wen, J., Yan, L., Feng, X. et al. Optimization of ACFDSRbased joint Doppler shift and SNR estimator for Internet of Vehicle system. EURASIP J. Adv. Signal Process. 2022, 37 (2022). https://doi.org/10.1186/s13634022008707
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DOI: https://doi.org/10.1186/s13634022008707