Optimization of ACF-DSR-based joint Doppler shift and SNR estimator for Internet of Vehicle system

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 ACF-DSR (autocorrelation function-double 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 ACF-DSR method with better choice of sampling intervals can achieve better estimation and outperform previous methods.

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 decision-making [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 real-time applications. On the other hand, to adapt the communication receiver to wireless propagation, signal-to-noise 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 ACF-DSR 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 ACF-DSR estimator with better sampling interval setting can produce better Doppler shift and SNR estimation. In contrast to previous methods, the optimized ACF-DSR estimator performs best, which consequently produces the most accurate speed estimates.

The paper is organized as follows: the original ACF-DSR estimator is introduced in Sect. 2. Then, in Sect. 3, the optimized estimator is formulated, and the analysis of sampling intervals is achieved. Finally, in Sect. 4, the superiority of the optimized estimator is demonstrated by numerical results. Conclusion is given in Sect. 5.

ACF-DSR estimator

In this paper, it is assumed that a band-limited pilot signal is transmitted over a Rayleigh fading channel. The ACF calculation for Rayleigh fading channel can be expressed as:

$$R(k) = \sigma_{l}^{2} J_{0} (2\pi f_{d} kT_{s} )$$

(1)

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.,

$$\hat{R}(k) = \frac{1}{K - 1}\sum\limits_{i = 0}^{K - 1} {\hat{c}_{l} (i)\hat{c}_{l}^{ * } (i + k)}$$

(2)

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

$$\hat{f}_{d} = \frac{{\sqrt {2\left( {1 - \sqrt {{{\hat{R}(1)} \mathord{\left/ {\vphantom {{\hat{R}(1)} {\hat{R}(0)}}} \right. \kern-\nulldelimiterspace} {\hat{R}(0)}}} } \right)} }}{{\pi T_{s} }}$$

(3)

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.,

$$\hat{f}_{d} = f_{d} \cdot \sqrt {\frac{{1 - \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}J_{0} (2\pi f_{d} T_{s} )} }}{{1 - \sqrt {J_{0} (2\pi f_{d} T_{s} )} }}}$$

(4)

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 high-speed scenarios.

$$\hat{f}_{d1} = \frac{{\sqrt {2 - \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2 - (m\pi f_{d} T_{s} )^{2} } \right)} }}{{m\pi T_{s} }}$$

(5)

$$\hat{f}_{d2} = \frac{{\sqrt {2 - \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2 - (n\pi f_{d} T_{s} )^{2} } \right)} }}{{n\pi T_{s} }}$$

(6)

Solving above equations, the following joint estimation is obtained.

$$\hat{f}_{d} = \frac{1}{\pi }\sqrt {\frac{2(A - 1)}{{An^{2} T_{s}^{2} - m^{2} T_{s}^{2} }}} \quad \hat{\gamma }_{s} = \frac{1}{{B^{2} - 1}}$$

(7)

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 $(m,n)$.

$$f_{m} < \frac{0.5}{{\max (m,n)}}\left( {f_{m} = f_{d} T_{s} } \right)$$

(8)

Thus, for the application of ACF-DSR 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 $(m,n)$.

Analysis and optimization of ACF-DSR estimator

Algorithm model establishment

Because of nonideal channel model or sample number for estimation, the original Doppler estimation makes Eqs. (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. (5) and (6) for joint Doppler shift and SNR estimations in (7) are reformulated as

$$\hat{f}_{d1} = \frac{{\sqrt {2 - \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2 - (m\pi f_{d} T_{s} )^{2} } \right)} }}{{m\pi T_{s} }} + \frac{\Delta }{m}$$

(9)

$$\hat{f}_{d2} = \frac{{\sqrt {2 - \sqrt {\frac{{\gamma_{s} }}{{\gamma_{s} + 1}}} \left( {2 - (n\pi f_{d} T_{s} )^{2} } \right)} }}{{n\pi T_{s} }} + \frac{\Delta }{n}$$

(10)

Choice of sampling intervals

In this part, above two Eqs. (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 pre-specified 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

$$E_{f} = E\left[ {\left| {{{\hat{f}_{d} } \mathord{\left/ {\vphantom {{\hat{f}_{d} } {f_{d} }}} \right. \kern-\nulldelimiterspace} {f_{d} }} - 1} \right|^{2} } \right]$$

(11)

where $E\left[ \cdot \right]$ denotes the expectation operation.

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. 1, in which different values of m and n make the resulted MSE fluctuates. From Fig. 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.

Fig. 1

The MSE of Doppler shift estimation

In Fig. 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. 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.

Fig. 2

The MSE of Doppler shift estimation: n/m = 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. Figure 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/m’s are not consistent with previous performance presented in Fig. 1. Such deviation suggests us to choose the best sampling intervals (namely m and n) carefully. For further analysis, the MSEs are averaged over $\Delta T_{s}$’s, results are shown in Fig. 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.

Fig. 3

The MSE of Doppler shift estimation for different $\Delta T_{s}$’s

Fig. 4

The MSE of Doppler shift estimation for different $\Delta T_{s}$’s

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.

$$E_{f} (n,m) = \frac{1}{M - 1}\sum\limits_{{1 \le m_{t} < n_{t} \le 5}} \quad {(\hat{f}_{d} (n,m) - \hat{f}_{d} (n_{t} ,m_{t} ))^{2} }$$

(12)

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 Least-Square (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. 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.

Fig. 5

The mismatched MSE (mMSE) for Doppler shift estimates

Simulation

In this section, the optimized ACF-DSR 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 ACF-DSR method (${n \mathord{\left/ {\vphantom {n m}} \right. \kern-\nulldelimiterspace} m} = 2$) [23], the ACF method [16], the LCR-DSR method with fitting [24] and the improved LCR-DSR (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 0.2 ms, the carrier frequency is 2.11 GHz. The vehicle speed range is 30–240 km/h and its corresponding Doppler shift is 58.6–468.8 Hz [24].

The simulated results of SNR estimation are presented in Fig. 6. From this figure, it is explicit that SNR can be very precisely estimated by the optimized ACF-DSR 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. Figure 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 ACF-DSR methods perform approximately, and all of them are slightly better than the LCR-DSR method and its improved version. Among these two LCR-DSR methods, fitting operation is not applied in the LCR-DSR, thus its advantage over the LCR-DSR method with fitting is limited. Moreover, the figures are partially enlarged for detail comparison, from which we can see that the optimized ACF-DSR method yields the best performance.

Fig. 6

The SNR estimation performance of the optimized ACF-DSR estimator

Fig. 7

The Doppler shift estimation comparison of the optimized ACF-DSR method and previous methods

Figure 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 ACF-DSR method is robust to AWGN, thus it can outperform the phase difference method, two LCR-DSR methods and the ACF method. Among the ACF-DSR 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 ${n \mathord{\left/ {\vphantom {n m}} \right. \kern-\nulldelimiterspace} m} = 4$.

Fig. 8

The MSE comparison of the optimized ACF-DSR method and previous methods

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} dB) and three actual speeds ({60, 150, 240} 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 LCR-DSR is slightly better than the LCR-DSR, but both are worse than the ACF-DSR methods. Among the ACF-DSR 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 ACF-DSR estimator can utilized to serve speed-related applications in IoV.

Table 1 The comparison of vehicle speed estimation

Conclusion

In this paper, the ACF-DSR 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 code-multiplexed or time-multiplexed pilot signals, and its accurate Doppler shift estimation can be utilized for vehicle speed estimation in IoV.