A D* orthogonal matching pursuit algorithm for time-varying channel estimation

Orthogonal frequency-division multiplexing (OFDM) is widely used to achieve high-speed underwater acoustic (UWA) communication. The time-varying UWA channel is a challenging wireless communication environment because of the large delay spreads, significant Doppler effects, and limited bandwidth. Therefore, accurate channel estimation is an important factor in the performance of OFDM systems. This paper briefly describes the use of orthogonal matching pursuit (OMP) with the A* search algorithm (A*OMP) to enhance the channel recovery capabilities. This multipath strategy replaces the traditional single-path OMP scheme.¹ We then apply a dynamic version of A*OMP, named D*OMP, for the dynamic estimation of multi-block joint channels.

Figure 1 shows a system model of UWA-OFDM communication. Each vertical column in a frame represents an OFDM block, and each cell in a vertical column represents a subcarrier frequency: The blue cells indicate the position of the inserted pilots, and the white cells indicate the position of the transmitted data. The comb-shaped pilots are integrated into each OFDM block at the same interval in the frequency domain, enabling better estimation and compensation for UWA channels in OFDM communication.

The reconstruction of sparse signals requires the solution with the minimum l₀ norm. Solving the l₀ minimization problem directly is impractical, so an indirect solution is typically sought using a suitable algorithm.² The semi-greedy A*OMP approach³ searches for a solution on several paths of a search tree. The paths of the tree are evaluated and extended according to a novel dynamic heuristic cost function.

The D* algorithm, a dynamic version of A*, was first introduced by Stentz in 1994.⁴ D* is a reverse heuristic search algorithm that, unlike the progressive forward search of A* from the starting point to the target point, begins at the target point and concludes at the starting point. With each node searched, its corresponding $f(x)$ is calculated, representing the distance metric for that node. The incremental nature of D* means that when sudden obstacles in the dynamic environment disrupt the original search path, the algorithm no longer requires replanning. Instead, it leverages the acquired $f(x)$ from each node to adaptively replan the path in the dynamic environment. This makes D* well-suited to real-time systems, especially those requiring adaptability to unpredictable, dynamic environments, such as joint channel estimation across multiple blocks.

Throughout this paper, bold letters denote vectors or matrices, uppercase letters represent vectors or matrices in the time domain, and lowercase letters correspond to frequency-domain quantities. $A$ and $A2$ denote the absolute value and the l₂ norm of $A$⁠, respectively. $I$ represents the number of initialization paths, $B$ is the number of extension paths, $P$ is the limiting number of paths, and $K$ is the maximum number of paths (also the nonzero number of channels). $M$ is the number of frequency samples in the channel frequency response, and $N$ is the number of time samples in the channel impulse response. $D$ represents the number of OFDM symbol blocks per frame.

The multipath strategy of A*OMP improves reconstruction, especially at sparsity levels where the exact reconstruction probability of a single-path algorithm such as OMP starts falling.⁵ However, A*OMP is only applicable to static channel estimation and is unsuitable for dynamic scenarios. Experimental results show that channel estimation is a dynamic process. The delays of the channels for several consecutive blocks per frame are similar, and the path gains exhibit temporal correlation.⁶ The D*OMP algorithm employs a reverse heuristic function that results in a low computational cost and high efficiency in dynamic environments, making it highly practical for joint channel estimation across multiple blocks. To date, few articles have discussed the application of D*OMP. Therefore, there is significant potential for further research and application of the D*OMP algorithm.

The D*OMP algorithm primarily comprises two stages: First, the A*OMP algorithm is used to estimate the optimal channel for the initial OFDM block; second, the channel is dynamically estimated for subsequent OFDM blocks. By leveraging the information retained from the first stage, the efficiency of channel estimation is significantly enhanced during dynamic estimation.

The core idea of OMP^7–10 is to leverage the fundamental principles of linear algebra to identify the column vectors in matrix $Φ$ that significantly contribute to the received signal $y$ and subsequently determine the positions of nonzero elements in the sparse channel $H$⁠. Specifically, the received signal $y$ can be conceptualized as a linear combination of the column vectors of a matrix $Φ$⁠, where these vectors are weighted and summed by $H$⁠. The sparsity of the vector $H$ means that only a limited number of column vectors in $Φ$ play a crucial role in shaping the received signal $y$⁠. OMP aims to identify these key column vectors and infer the specific positions of nonzero elements in sparse channels $H$ based on their corresponding positions in matrix $Φ$⁠. In Euclidean space, we commonly utilize the inner product to define the similarity between two vectors. Indeed, we can project the received signal $y$ onto the direction of a column vector in matrix $Φ$ to assess the contribution of that particular column vector. This process elucidates the mechanism of signal generation and functions as an efficient method for sparse signal reconstruction.

We initialize $r0=y$⁠. The residue is updated as $rk=y−ŷ$⁠. The vectors in $S$ are not orthogonal to each other, and orthogonal projection ensures that the residue is orthogonal to $S$⁠. Hence, this step is an extremely important extension to OMP. After $K$ iterations, $K$ columns of $Φ$ have been identified. The vectors in $S$ specify the locations of nonzero entries in $H$⁠, and the values of these nonzero entries are the least-squares (LS) solution to Eq. (2). OMP estimates the sparse channel taps one by one after $K$ iterations; ultimately, the residue should be very small. The channel estimation process of OMP is described in Algorithm I:

To limit the number of search paths, a pruning strategy is applied. Despite the use of path pruning, the maximum number of search paths may become excessive in many applications. Therefore, A*OMP limits the number of paths in the stack to $P$⁠. If the number of paths in the tree exceeds $P$ after an iteration, those paths with the highest costs are removed until $P$ paths remain.

Each selected path is compared to the paths already in the tree and is only added to the tree if no equivalent path is found. For each new path, the residue is computed by the orthogonal projection of $y$ onto the set of selected vectors and stored for the next iteration, together with the cost. The selection and expansion of the best path are repeated until either some path $k$ satisfies $frkλ<εfrk−1λ$ or all $K$ paths are complete.

A*OMP prevents solutions from becoming trapped around local optima. Figure 2 illustrates how, at step 16, the OMP algorithm selects the path with the smallest residue from among the multiple paths selected by A*OMP. However, the $r2$ value of that path rises in the 17th and subsequent steps, indicating that OMP has become trapped around a local optimum. The use of multiple paths allows the A*OMP algorithm to choose a better path, thus preventing the estimated channel from being distorted and ultimately leading to communication decoding failure. Figure 2 illustrates that the OMP algorithm and A*OMP select the same optimal path in the first 15 iterations. However, in the 16th iteration, the greedy single-path OMP algorithm lacks the breadth of A*OMP, leading to an increasingly larger discrepancy between the selected path and the original path in subsequent iterations. In contrast, A*OMP balances the search depth and breadth, allowing the global optimum to be obtained. Figure 3 displays the channel impulse response outcomes of the 30th iteration of the entire iteration process for $K=50$⁠. The A*OMP channel is significantly closer to the original channel than that given by OMP.

The sea trial was conducted under level 4–5 sea-breeze conditions, characterized by harsh environmental conditions and relatively strong signal noise. The extensive range of the shallow seawater means that the received signal exhibits less-severe multipath effects than in the lake test. The noise consists primarily of impulse noise and is significantly greater than that found in lake water. Figure 4(a) clearly shows several sparse auxiliary channels besides the main channel, a scenario referred to as block sparse CIR. Figure 4(b) shows the CCTM of the channels for the comb-like pilot structure after Doppler compensation. The sea-trial channels should be regarded as fast time-varying channels because they experience faster changes between different frames, resulting in a channel correlation coefficient that is mostly below 0.5.

The received signal in the lake trial is heavily affected by the short transmission distance and the shallow water, resulting in the generation of numerous small paths alongside the main path in the CIR. The underwater acoustic channel does not show significant changes over time in Fig. 5(a), so we consider this to be a slow time-varying channel. Figure 5(b) illustrates that the channels have some degree of correlation.

There is a relatively stable and obvious path within 0–2 ms in Fig. 6(a). After 3 ms, the multipath effects attenuate to almost zero and the multipath energy is weak. As can be seen from Fig. 6(b), the correlation coefficient between continuous channels in the lake test and the sea trial under calm conditions basically remains above 0.8, indicating a strong correlation. In most cases, the channels are very sparse and can be regarded as slow time-varying coherent channels. The delays of the channels for several consecutive blocks are similar, and the path gains exhibit temporal correlation. Therefore, the correlation for UWA channels can be exploited.

The D* algorithm is extensively applied in the field of dynamic path planning. In the case of path planning, the A* algorithm finds the path that follows the required order from the starting point to the end point. When encountering a new environment with similar characteristics, such as the temporary introduction of moving obstacles along the planned route, it is essential to rerun the algorithm to generate a new path. The D* path planning algorithm operates by planning the route from the end point to the starting point. Once the path planning has been completed, the D* algorithm can swiftly replan the path in response to any blockages encountered in subsequent scenarios. This capability enables real-time updates of the path within a dynamic environment, eliminating the necessity to recalculate the entire route.¹⁴ 

We can consider the process of solving sparse channels as that of finding the optimal paths for the search trees. A strong correlation between channels indicates that the paths of multiple search trees are similar. Let us initialize the preceding estimated channel as a subsequent channel. This enhances both the accuracy and efficiency of channel estimation, as the initial estimate is informed by prior channel conditions, thus minimizing the time and computational complexity associated with searching for the optimal channel. When the next channel is different from the previous channel, we say that the area is obstructed. For example, there are only a few different channel taps in Fig. 6, whereas Fig. 4 has more different channel taps to be replanned. First, A* requires the iteration process for channel estimation to be repeated, which becomes tedious and complex. In contrast, D* considers the correlation between the channels of two OFDM blocks. If the channels are similar, the computation time can be significantly reduced. Even if the channels differ (i.e., when there are obstacles), D* does not search through each column of $Φ$⁠, as in A*, but instead looks at the surrounding taps and the remaining taps in the previous channel, which significantly reduces the computation time.

Therefore, D*OMP is adopted to solve the multi-block joint channel estimation problem. The entire process of D*OMP is presented in Algorithm II.

In addressing the joint channel estimation problem for multiple consecutive OFDM blocks, we first employ the A*OMP algorithm to estimate the channel $H1$ for the first OFDM block. $H1$ is then sorted in descending order from largest to smallest, and the $I$ largest taps in $H1$ are redefined as the initial input for the second block's channel. The selection of $I$ depends on the correlation among channels. If the cost function increases, we consider the main taps between the two channels to be different, requiring backward iteration (from end point to start point). We choose the largest $B$ inner products, denoted as $argmaxB{·}$⁠, instead of considering only one maximum. If the path number $p$ after extension is greater than $P$⁠, we prune the paths with the highest cost functions. The improved A* algorithm is then used to estimate the $p$ channels simultaneously. When the current channel tap encounters an obstacle, searching near to the current tap within the previous OFDM block is prioritized. If this search is unsuccessful, the search continues in the database of the previous OFDM block. If the obstacle is still not found in the database, a final search is conducted across all columns of $Φ$⁠. The loop terminates when $k$ reaches the channel sparsity $K$ or the residual norm falls below the specified threshold $ε$ before reaching $K$⁠. Finally, the optimal estimated channel is selected based on the path that yields the minimum cost function value. Thus, when dynamic changes occur, we can promptly adjust to the new channel using the computed channel, rather than replanning. This makes D*OMP highly practical for joint channel estimation across multiple blocks. Figure 7 shows an example of one iteration of the algorithm.

Simulations were conducted with a sampling frequency of 48 kHz, with each frame containing eight OFDM blocks. The number of subcarriers in each OFDM block was set to $N=1024$⁠, and the pilot interval was set to 4. The numbers of data subcarriers and pilot subcarriers were set to $Nd=682$ and $Np=341$⁠, respectively. The low-density parity check encoding method was employed with a bit rate of $12$⁠, and quadrature phase shift keying (QPSK) modulation was applied. The center frequency of the upconversion was $fc=12 kHz$⁠, and the bandwidth was $B=6 kHz$⁠. The signals passed through a CIR simulated by the bellhop model and a time-varying Gaussian channel with a signal-to-noise ratio (SNR) of $[0, 9]$ dB. The main bellhop^15,16 parameters are listed in Table I. The sound velocity profile had a negative gradient.

The results given by the LS, OMP, and A*OMP models were compared with those of D*OMP to verify the channel estimation performance. The parameter settings of the A*OMP algorithm included the number of initialization paths $I=4$⁠, number of extension paths $B=2$⁠, and limit number of paths $P=4$⁠. The channel sparsity was set to $K=50$ and $ε=0.001$⁠. The measurement indicators are the time consumption, residual convergence speed, constellation diagram, and bit error rate (BER).

We recorded the computation time of each algorithm using the “tic-toc” function in matlab. The average runtimes of the algorithms over 1000 Monte Carlo simulations are listed in Table II.

The LS algorithm has very high computational efficiency (Table II), but gives worse channel estimation performance than the OMP algorithms in the CS field. The accuracy of LS channel estimation decreases rapidly as the SNR decreases. In the field of CS, the D*OMP algorithm achieves the shortest runtime. Through 1000 consecutive Monte Carlo experiments, although D*OMP has a channel estimation time of 123.455 ms for the first OFDM block (similar to A*OMP), the average channel estimation time for D*OMP from the second OFDM block on is 29.225 ms, resulting in an overall average estimation time of 41.4675 ms, which is much lower than that of the OMP algorithm. This reflects the superiority of D*OMP for large-scale UWA communication applications.

Figure 8 compares the convergence speeds of the three algorithms in the first four OFDM symbol blocks in one frame. The OMP algorithm generally fails to attain the global optimum reached by A*OMP and D*OMP. By combining path planning algorithms, it is possible to balance the search depth and breadth to find the global optimum with lower residuals than traditional OMP. In the channel estimation of the first OFDM block, the convergence effects of A*OMP and D*OMP are almost identical. However, from OFDM blocks 2–4, the convergence speed of the D*OMP algorithm is significantly faster than that of A*OMP, while the final convergence errors are approximately the same. Consequently, the channel estimation effect of the D*OMP algorithm is comparable to that of the A*OMP algorithm, but the computation time is much faster.

Figure 9 shows that the QPSK modulated signal exhibits significant random noise after passing through the channel. The four quadrant coordinates are not very clear. The OMP algorithm produces the least-detailed constellation diagram, with aliasing in all four quadrants. The points in the constellation diagrams produced by A*OMP and D*OMP are more tightly packed. This indicates that algorithms with combined path planning are beneficial for improving the channel estimation performance.

Figure 10 shows the BERs of the simulations. For each algorithm, the BER decreases as the SNR increases in both slow and fast time-varying channels. In slow time-varying channels, the D*OMP algorithm can fully utilize the characteristics of slow time-varying channels, resulting in the best channel estimation performance. In fast time-varying channels, the estimation performance of D*OMP is not as good because of the significant channel changes between blocks in each frame. Despite its faster runtime, the D*OMP algorithm still achieves almost the same level of performance as the A*OMP algorithm. Both A*OMP and D*OMP are robust because they perform better in fast and slow time-varying channels.

To validate the algorithm in real noisy marine environments, experimental data were collected from a campus lake and the shallow waters near Qingdao, China (Fig. 11). Table III lists the environmental parameters of these experiments.

Figure 12 shows the probability distribution of the BERs estimated by the three algorithms.

The noise in each environment was acquired from the signal transmission gaps and rounded to the nearest whole number. Figure 12 compares the BERs of channel estimation between the three algorithms for both the lake and sea trials across 100 experimental data. In the lake trial, the probability distribution of BERs gradually increases as the transmission amplitude rises. In the sea trial, however, the BER decreases with increasing distance. Under these two scenarios, the A*OMP and D*OMP algorithms achieve lower BERs.

This study explored the application of D*OMP for channel estimation. A novel acceleration strategy was introduced to reduce the time consumption of conventional A*OMP. Simulations showed that the runtime of the proposed algorithm is equivalent to that of LS. Moreover, the proposed method outperforms other methods in both slow and fast time-varying channels, showing a certain degree of robustness under different UWA environments. Finally, the proposed channel estimator achieves faster parameter convergence and better sparsity control than conventional OMP. Our experimental results demonstrate that the proposed algorithm has a lower probability of error in lake and sea environments, indicating a certain applicability to UWA communication for joint channel estimation.

This work was supported by National Key R&D Program of China under Grant No. 2021YFC3100900. We express our gratitude to all the researchers and staff who participated in the sea trial. Their hard work ensures the smooth collection of valuable and reliable experimental data for this study.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.