An Improved GOMP Sparse Channel Estimation for Vehicle-to-Vehicle Communications

Reliable channel estimation is critical for wireless communication performance, especially in vehicle-to-vehicle (V2V) communication scenarios. Aiming at the major challenges of channel tracking and estimating as the highly dynamic nature of vehicle environments, an improved generalized orthogonal matching pursuit (iGOMP) is proposed for V2V channel estimation. Te iGOMP algorithm transforms the channel estimation problem into a sparse signal recovery problem and replaces the classical inner product criterion with the Dice atom matching criterion. Additionally, the atomic weak progressive selection method is integrated to avoid the suboptimal selection of atoms from the redundant dictionary using the inner product criterion. Te proposed iGOMP method can achieve optimal channel estimation by iterating feedback results. Simulation results demonstrate that the iGOMP method has superior estimation accuracy, mean square error (MSE), and bit error rate (BER) performance compared with traditional channel estimation methods in V2V communications.


Introduction
With the continuous deployment of 6G artifcial intelligence technology in applications such as autonomous driving and the Internet of vehicles (IoV), the application of IoV technology has evolved from initial information entertainment to safety guarantee and efciency improvement under the conditions of vehicle-to-vehicle (V2V) and vehicle-toinfrastructure (V2I) information interaction [1][2][3].In V2V communication systems, both the transmitter (Tx) and the receiver (Rx) are in a moving state, and the antenna height is low, making it easily blocked by scatterers.When switching to diferent scenarios, the wireless channel shows nonwide sense stationary uncorrelated scattering (non-WSSUS) fading characteristics [4][5][6].Moreover, researchers have experimentally demonstrated that V2V channels have sparse properties in V2V environments [7,8].Reliable information transmission is closely related to the transmission characteristics of the wireless channel, and since the wireless channel in V2V environments usually presents highly dynamic characteristics, accurate estimation of the channel impulse response (CIR) is crucial for subsequent equalization and demodulation.Terefore, the accuracy of channel estimation determines reliable information transmission in V2V environments [9,10].Due to the combined efects of multipath fading and Doppler shift, high-speed scenarios exhibit both time/frequency selective fading and timevarying nonstationary channel characteristics in the time domain, which can signifcantly afect the latency communication in IoV scenarios [11][12][13].
In V2V communications, the channel is always sparse, which means that the multipath channel has a large delay spread, but its energy is mainly concentrated in a few paths, with fewer components in other paths [14].In light of this sparsity characteristic, compressed sensing technology can be used to achieve good channel estimation performance through a number of observations, with lower complexity than traditional algorithms.Te authors of [15] investigate the iterative stop criterion for sparsity estimation, as CSbased sparse signal detections require accurate sparsity information.By analyzing the residual energy, the Tx can determine whether active pilot components are included in the residual, preventing an increase in the complexity of JMPA decoding due to excessive false detections.Te greedy algorithm is a common algorithm in compressive sensing reconstruction, with lower complexity compared to convex optimization methods, making it more widely used.Orthogonal matching pursuit (OMP) is a commonly used representative greedy algorithm for estimating sparse channels.However, its iteration time increases when the sparsity level is high [16].In [17], authors proposed a generalized orthogonal matching algorithm (GOMP), which selects a few atoms with the largest product with the residual.Te OMP algorithm is a special GOMP algorithm.Compared with the OMP algorithm, GOMP has a higher computing speed.GOMP uses simple multiatom selection for the OMP algorithm, meaning that after calculating the inner product of the residual and the unselected atoms using the current residual approximation criterion, M atoms with the largest inner product are selected in turn.Tis method of selecting M optimal multiatom selection methods is simple to implement, with less calculation time than that of the OMP algorithm, and improved reconstruction performance compared to the OMP algorithm.In this paper, the algorithms are applied to V2V sparse channel estimation, but the minimum MES and BER are slightly worse.In [18], authors proposed a sequential processing algorithm for unknown sparsity, which increases the sparsity by sampling the signal until the stopping criterion is met.In [19], authors proposed a decision-guided solution to improve the equalization performance by utilizing the sparsity of the arrival angles and reconstructing the channel using the block orthogonal matching pursuit algorithm (BOMP), but at a high computational complexity.Although these solutions improve the accuracy of channel estimation, the sparsity of the channel is usually difcult to obtain, and previous literature often assumes that the channel sparsity is known before performing channel estimation.
Motivated by the aforementioned, an iGOMP algorithm is proposed in this paper, which involves postprocessing the iterative results obtained by the GOMP algorithm and removing redundant atoms.Teoretical analysis and simulation results show that although both the MSE and BER of the GOMP algorithm are slightly worse than those of the OMP algorithm, the former has a shorter running time and lower computational complexity.Compared with the GOMP algorithm, the iGOMP algorithm produces better MSE and BER performance.Te remaining sections of this paper are organized as follows.Section 2 presents the system and channel model.Section 3 describes the proposed improved generalized orthogonal matching pursuit algorithm.Section 4 compares the performance of the proposed method with traditional methods in the V2V channel.Finally, the conclusion is presented in Section V.

System and Channel Model
2.1.System Model.Te rapid movement of the Tx and Rx in V2V scenarios causes a Doppler frequency shift, resulting in a frequency ofset of the subcarriers in the OFDM system.Tis ofset destroys the orthogonality of the subcarriers and signifcantly afects the system's performance.
Assuming that the OFDM system has N subcarriers, of which P subcarriers are allocated for pilot symbol transmission, and each subframe consists of I OFDM symbols, the resource element of the i − th transmitted OFDM symbol on a subcarrier can be denoted as When performing s i OFDM modulation through inverse fast Fourier transform (IFFT), it can be expressed as follows: where S i represents the transmitted time domain signal, F represents the DFT transformation matrix of the point , and (•) H represents the conjugate transpose, and then, the OFDM transmission model is defned as follows: where )] T is the vector of the received signal at i − th OFDM symbol in the time domain, z i is additive complex white Gaussian noise of the channel, and H i denotes the CIR matrix on the i − th OFDM symbol, which could be defned as follows: where h i (k, l) is the k − th CIR sample point on l − th tap at i − th OFDM symbol.
After FFT demodulation at the receiving end, Y is obtained, which is expressed as follows: where ] T are the Tx and Rx signals in the frequency domain, respectively, and Z i is the expression of the additive Gaussian noise vector z i in the frequency domain.
Let Q be a P × N dimensional pilot selection matrix obtained by selecting P rows from an N × N dimensional identity matrix corresponding to the pilot positions.Te received signal at the pilot positions can be expressed as follows: where Y P represents the corresponding signal received at the receiver for the pilot signal, X P � QXQ − 1 represents the signal transmitted for the pilot, and Z P represents the noise in the channel.For the receiver, Y P , X P , and Z P are all known signals.

International Journal of Antennas and Propagation
Te process of recovering h based on (5) can be modeled as a sparse signal reconstruction problem in the presence of noise.Terefore, compressive sensing techniques can be employed to reconstruct the sparse vector h.Subsequently, the frequency-domain channel impulse response samples can be obtained by calculating H p � F p × h p , where H p represents the samples of the channel frequency-domain impulse response.

V2V Channel Model.
Te channel characteristics of V2V communication are scene-dependent.Figure 1 depicts a typical V2V communication scenario in an urban area, where the multipath efect is signifcant due to the presence of near and far scatterers such as buildings and vehicles.
In this paper, the tapped delay line (TDL) model is used as the V2V channel model, and the expression of CIR is derived as follows: where α l (t) � β l (t) • exp(j2πf D,l t) refects the fading characteristics, β l (t) represents the amplitude of the l − th path, f D,l represents the Doppler shift of the l − th path, τ l (t) denotes the time delay of the l − th path, respectively, and L is the number of paths in the V2V channel.In the TDL model, each tap corresponds to a multipath, and its fading amplitude statistics and Doppler spectrum determine the main characteristics of the channel under a specifc scenario.
If a delay tap corresponds to a non-line-of-sight (NLoS) component, the tap follows Rayleigh fading; if a delay tap corresponds to a line-of-sight (LoS) component, the tap is modeled using Rician fading.Te authors of [20] provide specifc channel parameters for six V2V communication scenarios.Terefore, the V2Vurban canyon oncoming (V2V-UCO) scenario is studied, and the specifc channel parameters are shown in Table 1.

Sparse Channel Estimation Algorithm
Based on Improved GOMP

Generalized Orthogonal Matching Algorithm (GOMP).
Te purpose of the GOMP algorithm, based on multiatom selection, is to reduce the computational complexity and running time of the OMP algorithm.Te atomic selection principle of the GOMP algorithm is to choose M atoms with the largest inner product of the residual from all the unselected atoms, which extends the OMP algorithm.Te OMP algorithm is a special case of GOMP (M � l), and the algorithmic pseudocode is as follows.

Atomic Weak Selection.
In V2V communications, it is often difcult to apply the GOMP algorithm since it requires knowledge of the signal sparsity.To address this issue, the atomic weak selection method can be used.Tis method does not select atoms based on their correlation but instead sets a threshold and groups atoms larger than the threshold into atomic sets while discarding the remaining atoms.Te residuals are then updated and stacked until the stopping condition is satisfed.
When the observation matrix is a Gaussian matrix, threshold parameter m is set a value within the range from 2 to 3, where δ t represents the noise power, and the number of columns in the observation matrix is denoted by N.
Te elements in the inner product column that satisfy the selection criterion of the atoms are added to the atom set, and their column index is added to the index set, by computing the inner product between the current residual and the recovery matrix.Te weak selection method of atoms allows the process to be unrestricted by sparsity and can adjust the number of selected atoms according to the inner product value between the current residual and the recovery matrix.Tis efectively avoids the problem of selecting too many or incorrect atoms in advance and improves the stability of the algorithm.

DICE Guidelines.
Te classical channel estimation algorithm based on compressed sensing typically measures the similarity between vectors using the inner product criterion.However, using this method to represent correlation has certain drawbacks.During signal recovery, some similar atoms in the observation matrix can impact the matching of the signal residual, ultimately reducing the accuracy of the signal recovery.
Terefore, selecting an appropriate measurement method to screen the atoms in the support set has become a critical factor that afects the quality of the signal reconstruction algorithm.To address this issue, the measurement criterion has been improved.Specifcally, the Dice coefcient matching criterion is now used in the frst stage of atom screening.Te Dice coefcient matching criterion expresses the similarity between vectors β and λ as follows: Trough a comparative analysis of the mathematical expressions of the dot product criterion and the Dice coefcient matching criterion, it can be inferred that the denominator calculation method used in the dot product criterion can compromise the inherent characteristics of the vectors, making it challenging to diferentiate between similar atoms.Conversely, the calculation method employed in the Dice coefcient matching criterion can efectively address this issue.Terefore, the Dice criterion can identify more appropriate atoms and improve the accuracy of reconstruction.
International Journal of Antennas and Propagation

Improved Generalized Orthogonal Matching Algorithm (iGOMP).
Te GOMP selects a few atoms with the highest product with the residual at each iteration, which can lead to the selection of incorrect atoms.Moreover, the GOMP algorithm can accurately reconstruct the signal only when the sparsity of the channel is known in advance, which is not always feasible in real V2V environments.To solve the technical problems, an improved version of the GOMP algorithm, known as the iGOMP, has been proposed for V2V channels [22,23].Tis algorithm accurately estimates channel information even when sparsity is unpredictable and achieves a low BER as shown in Table 3.

Simulation Results and Analysis
To evaluate the performance of the iGOMP algorithm in V2V scenarios, we conduct a systematic simulation of the proposed iGOMP V2V channel estimation algorithm on the MATLAB R2016a platform, as specifed in Table 4. Specifcally, we compare the performance of the iGOMP algorithm against fve traditional channel estimation algorithms (LS, OMP, SAMP, GOMP, and iGOMP) in the V2V-UCO scenario.BER and MSE are used as performance metrics for channel estimation quality.
Comparing the simulation results of the channel estimation algorithm in the V2V-UCO scenario, it is found that the accuracy of the channel estimation based on the OMP, SAMP, GOMP, and iGOMP algorithms is higher than that of the LS algorithm under the same SNR.Tis is due to the fact that the V2V channel estimation algorithm based on compressed sensing takes into account the sparseness of the channel and can obtain a better estimation efect by using fewer pilots.Since the reconstruction performance of compressive sensing is relatively low SNR, reconstruction is not meaningful in this range, and therefore, the variation range of SNR is set to 5∼30 dB.Te analysis indicates that, under the same number of pilots, the MSE of each method decreases with the increase of SNR.Base channel estimation on the matching pursuit greedy method takes into account the sparse characteristics of the channel, and its MSE performance is signifcantly better than that of LS.For instance, OMP has an SNR peak gain of about 10 dB compared to LS. Figure 2 compares the MSE performance of diferent methods in various speed environments.Te proposed method has an SNR peak gain of about 15 dB compared to OMP and an SNR peak gain of about 5 dB compared to SAMP, which demonstrates that the method proposed in this paper improves the performance based on MSE performance of channel estimation for typical matching pursuit-like greedy methods.Tere are main reasons for this: frst, the method proposed in this paper incorporates the  multiatom selection of GOMP and the backtracking idea of SAMP into OMP, which enables efcient and precise screening of atoms by providing the ability to delete previously selected atoms and ensuring optimality at each iteration.Tis signifcantly improves the performance of the GOMP algorithm.Second, the use of double thresholds addresses the problem of fxed step size in GOMP.At each iteration, the energy diference of the reconstructed signal is evaluated to determine whether it is close to ε 2 .If it approaches ε 2 , a small step size is adopted to gradually approach the estimated value and improve the reconstruction accuracy.Tis leads to an improvement in the performance of channel V2V estimation.
Figure 3 illustrates the BER performance of various channel estimation algorithms at diferent speeds.Te iGOMP channel estimation algorithm proposed in this paper demonstrates superior performance at diferent speeds.As the SNR increases, the channel estimation based on the matching pursuit greedy method and LS shows a downward trend in BER, but the former displays a more pronounced decline.Tis is due to the fact that, at the same SNR, the matching pursuit greedy method has better channel estimation performance than LS.When the SNR increases to 30 dB, the channel estimation based on the matching pursuit greedy method and LS gradually becomes stable since the noise is relatively low at this point, and the performance of each method relies mainly on its own estimation accuracy.Furthermore, the method proposed in this paper has an SNR peak gain of 15 dB relative to OMP and a 5 dB SNR gain relative to SAMP in terms of BER performance.At a velocity of 120 km/h, LS, OMP, GOMP, and SAMP have lower limit as SNR increases, whereas the iGOMP algorithm continues to maintain excellent performance.Te main reason for the signifcant improvement in channel estimation performance is that in low-speed environments, the CIR changes very slowly within a single symbol time unit.However, in high-speed scenarios, the proposed algorithm is able to adaptively estimate the real sparsity of the channel, resulting in greatly improved channel estimation performance.
Figure 4 compares the performance of iGOMP with diferent step lengths in the V2V-UCO scenario.As shown in the fgure, when the step size is set to S � 2, iGOMP achieves better BER and MSE performance compared with International Journal of Antennas and Propagation S � 3 and S � 5.When the MSE is set to 1 × 10 − 3 , there is approximately a 5 dB gain compared with S � 2 and S � 3 about 9 dB gain compared with S � 5. Since diferent step sizes can afect the performance of the algorithm, if the initial step size is large, the support set will quickly expand.Although the signal reconstruction rate will increase, the accuracy of the reconstruction will decrease.If a smaller initial step size is set, the signal can be reconstructed more accurately, thereby improving the accuracy of the algorithm.Table 5 shows a comparison of the runtime performance for the aforementioned methods in a single complete channel estimation at SNR � 30dB.Te simulation tool used

6
International Journal of Antennas and Propagation was MATLAB 2016a, and the computer was equipped with a 2.8 GHz, Intel i5 CPU, 8 GB of memory, and the Windows 10 operating system.Analysis shows that the iGOMP algorithm has a slightly longer average runtime than the other methods due to its more complex and numerous atomic screening steps and the introduction of dual thresholds to set the stopping iteration conditions.Table 6 presents the comparisons of the complexities of diferent algorithms in two aspects: the number of multiplication operations and the time complexity.It can be observed from Table 6 that the LS algorithm has the lowest complexity.Compared with the OMP algorithm, the SAMP algorithm has only one more logarithmic order, which is lg S. Te reason for this is that the SAMP algorithm approximates sparsity for reconstruction, which increases complexity, but its step size is S much smaller than N, so its impact on complexity is limited.Terefore, the complexity of the proposed algorithm is within a tolerable range for research.Te time complexity of the iGOMP algorithm and the SAMP algorithm is the same, and they have the same order of magnitude as a whole.

Conclusion
To further improve the communication quality in the V2V environments of the vehicular communications, this paper proposes an iGOMP algorithm of sparse channel estimation for diverse and dynamic V2V propagation environments.Te algorithm integrates the weak selection method of the Dice criterion to select atoms and improves the reconstruction accuracy of the estimated value by gradually approximating it using a small step size.Tis approach can efectively address the problem in traditional algorithms where the inner product criterion cannot select the optimal atom from a redundant dictionary, thereby enhancing the stability of the algorithm.Simulation results show that the proposed algorithm has higher estimation accuracy compared with LS, OMP, SAMP, and GOMP.At a fxed SNR, the BER and MSE of algorithms were improved, efectively enhancing the performance of V2V channel estimation.Although the proposed method has slightly higher complexity and runtime compared with other methods mentioned aforementioned, it is still within an acceptable range and proves that the algorithm performance is afected by diferent step sizes and V2V communication scenarios.

Data Availability
Te data will be made publicly available upon reasonable request through the corresponding author.

Figure 2 :
Figure 2: MSE comparison of each channel estimation algorithm in V2V-UCO scenario.(a) MSE performance of each algorithm when moving speed is 60 km/h and (b) MSE performance of each algorithm when moving speed is 120 km/h.

Figure 3 :
Figure 3: BER comparison of each channel estimation algorithm in V2V-UCO scenario.(a) BER performance of each algorithm when moving speed is 60 km/h and (b) BER performance of each algorithm when moving speed is 120 km/h.

Figure 4 :
Figure 4: Comparison of MSE and BER performance of iGOMP algorithm under diferent step lengths.(a) MSE comparison of iGOMP algorithm and (b) BER comparison of iGOMP algorithm.

Table 2 :
[21] algorithm pseudocode[21].Input: measure matrix Φ, sample vector Y, channel sparsity K, atoms M; Output: A K-sparse approximation  h of the input signal; Initialization: support sets F 0 � ∅, residual r 0 � Y, iterations t � 1;(1) Calculate u � abs[A T r t−1 ], fnd the u values with the largest measurement matrix and residuals by M, t � t + 1, and get the candidate A t � A t−1 ∪ J 0 ;(2) Calculate  h t � argmin‖Y − Φu‖, take the M mark corresponding to the largest value into a collection F, and calculate residual r t � Y − Φ F Φ +

Table 5 :
Comparisons of running time among diferent algorithms.

Table 6 :
Complexity comparisons among diferent algorithms.International Journal of Antennas and Propagation