# A Novel Approach for the Estimation of Doubly Spread Acoustic Channels Based on Wavelet Transform
## 1 Introduction  
Underwater acoustic (UWA) channels pose grand challenges for reliable high data-rate communications, due to significant Doppler effects and severe delay spread [1; 2; 3]. In underwater communications, acoustic waves propagate at 1500 m/s, much lower than the speed of electromagnetic waves in terrestrial wireless systems (which is \(3\times 10^{8}\) m/s). Thus, the motion of platform will cause more significant Doppler effects, which is expressed as signal compressing or dilating in time domain and can be treated as Doppler scale [4]. In addition, the delay spread results in severe inter-symbol interference (ISI) [5]. To fully understand the channel characteristics and overcome challenges it poses, accurate channel models and estimation methods are essential to investigate. As observed in many experiments [6; 7], signals from different paths will experience different Doppler scales, arrive at different time and have different energies. Therefore, the multi-scale multi-lag (MSML) channel, denoted in [8], can well model acoustic channel and has been adopted in many researches [7; 9]. In MSML channel model, each path can be parameterized by Doppler scale factor, time delay and amplitude. However, for severe delay spread, this model will be too complex to estimate. To overcome such difficulties, many researchers investigated the sparsity of UWA channel, that is, most of the energy is concentrated in some small regions. Hence only a few channel taps in MSML channel model are nonzero and need to be tracked. As a result, the computational complexity has been reduced and many sparse channel estimation algorithms based on compressed-sensing (CS) have been proposed [7; 10; 11; 12; 13; 14; 15]. These algorithms can be generally grouped in two categories: dynamic programming like matching pursuit (MP), and linear programming like basis pursuit (BP) [7]. BP aims at the \(l_{1}\) minimization which needs high computational complexity, thus is less attractive for practical large-scale applications. Therefore, we will mainly focus on MP algorithm and its successors. In [10], matching pursuit (MP) algorithm is applied to estimate Doppler scale factors of different paths. It iteratively selects one column from the dictionary that is most relevant with the residual signal, and subtracts the estimated path component to update the residual signal. Compared with MP, its orthogonal version, orthogonal matching pursuit (OMP) algorithm [11] makes the residual signal be orthogonal with all the selected columns, thus has better convergence speed and accuracy. The traditional subspace methods and CS-based methods are compared in [7], and the author concludes that CS-based methods have better performances in channel estimation. Meanwhile, some improved algorithms, which focus on adaptively estimating path numbers, like sparsity adaptive matching pursuit (SaMP) [12] and adaptive step size SaMP (AS-SaMP) [13] have been applied to sparse channel estimation. Furthermore, some references propose methods to reduce computation: a two-stage OMP algorithm is proposed by [14], which estimates the Doppler scale factor and time delay respectively, rather than simultaneously as OMP does. However, it requires some preprocessings before channel estimation. The author of [15] proposes that fast Fourier transform (FFT) can be utilized in OMP to simplify calculation. However, the reduction is limited as it only focuses on the computing process rather than on the reducing of column dimensions in the dictionary. Therefore, the main limitation of MP algorithm and its successors is that the estimation accuracy depends on the size of the dictionary, which is determined by the product of grid point numbers on the time delay dimension and the Doppler scale dimension. To guarantee a fine resolution, the number of columns in the dictionary could be extremely large. Thus calculating inner products of the received signal and each column lead to extensive calculation, especially for UWA channel, where the delay-scale spread is large.