# A Novel Approach for the Estimation of Doubly Spread Acoustic Channels Based on Wavelet Transform
## 3 WT-Based Sparse Channel Estimation
### WT-Based Algorithm  
Firstly, we consider the transmitted HFM signal as \(s(t)\), then the received signal is as in (3), plugging (3) in (9) and let \(\psi(t)=s(t)\), we get  
\[ \begin{split}& W_{r}(a_{W},b)=\frac{1}{\sqrt{a_{W}}}\int_{- \infty}^{+\infty}r(t)\psi^{*}(\frac{t-b}{a_{W}})dt\\ &=\sum_{l=1}^{L}\frac{1}{\sqrt{a_{W}}}\int_{0}^{T}A_{l}s(a_{l}(t -\tau_{l}))s^{*}(\frac{t-b}{a_{W}})dt\end{split}  \tag{11}\]  
CWT displays the distribution of signal amplitude and phase into time scale \(a_{W}\) and time shift \(b\). Thus, with proper \(a_{W}\) and \(b\), which match to the signal from the \(l\)th path, WT coefficients will appear a peak. Therefore, we can get the frequency information of \(r_{l}(t)\) at time shift \(b\). With this, we can set the values of \(a_{W}\) and \(b\) to be a list of possible values and get the WT coefficients matrix. Thus get the time-frequency information of \(r_{l}(t)\). To get better time-frequency analysis, the values of \(a_{W}\) and \(b\) should be as many as possible. The two-dimensional searching requires large computation cost. Considering the good Doppler-invariability of HFM signal as we mentioned before, we can simplify the two-dimensional searching to one-dimension, that is, by using the transmitted HFM signal as the mother wave, the time scale \(a_{W}\) can be set as 1 and the WT coefficients will meet a peak at a time shift for each path as the following:  
\[b_{l}=-\Delta t_{l}=-\frac{(a_{l}-1)t_{0}}{a_{l}}+\tau_{l} \tag{12}\]  
The detail information can be seen in Appendix A. From (12), the time shift is determined by both time delay and Doppler scale factor. So it is difficult to estimate \(a_{l}\) and \(\tau_{l}\) simultaneously when only sending one HFM signal. We consider to send a superimposed signal, which is defined as  
\[x(t)=\cos(-2\pi k\ln(1-\frac{t}{t_{0}}))+\cos(2\pi k\ln(1+\frac{t}{t_{0}})) \tag{13}\]  
The variables in (13) are defined as in (4) and \(f_{1}<f_{2}\). Therefore, the first part in (13) is a HFM up sweep signal and can be defined as \(s^{+}(t)\); the second part is a HFM down sweep signal and can be denoted as \(s^{-}(t)\). The received signal will be  
\[r(t)=\sum_{l=1}^{L}A_{l}(s^{+}(a_{l}(t-\tau_{l}))+s^{-}(a_{l}(t-\tau_{l})))+n(t) \tag{14}\]  
where \(n(t)\) is the additive noise. In addition, at the receiver, WT will be performed twice and the mother wave will be \(s^{+}(t)\) and \(s^{-}(t)\), respectively. We have verified that when use \(s^{+}(t)\) as mother wave, a constant time shift does not exist which will make \(s^{+}(t)\) match to \(s^{-}(a_{l}(t-\tau_{l}))\) in frequency in Appendix B. Thus the superposition of \(s^{+}(t)\) and \(s^{-}(t)\) will not interfere each other, and corresponding time shifts for the \(l\)th path are:  
\[{b_{l}}^{+}=-\frac{(a_{l}-1)t_{0}}{a_{l}}+\tau_{l} \tag{15}\]  
\[{b_{l}}^{-}=\frac{(a_{l}-1)t_{0}}{a_{l}}+\tau_{l} \tag{16}\]  
The Doppler scale factor and time delay can be calculated by  
\[a_{l}=\frac{2t_{0}}{2t_{0}-({b_{l}}^{-}-{b_{l}}^{+})} \tag{17}\]  
\[\tau_{l}=({b_{l}}^{+}+{b_{l}}^{-})/2 \tag{18}\]  
In addition, the path amplitude can be calculated by normalizing (11), that is  
\[A_{l}=\frac{W_{r}(a_{W},b)}{\frac{1}{\sqrt{a_{W}}}\int_{0}^{T}s(\frac{t-b}{a_ {W}})s^{*}(\frac{t-b}{a_{W}})dt} \tag{19}\]  
For MSML acoustic channels, we propose an iterative algorithm to estimate parameters for each path. Specifically, at each iteration, path with the highest energy can be identified and will be separated from the received signal. In addition, parameters \(\{A_{l},a_{l},\tau_{l}\}\) can be estimated at the same time. The whole estimation scheme is given as follows. WT-based iterative channel estimation algorithm:  
**Input:** Transmitted HFM up sweep signal vector \(\mathbf{S}^{+}\), and HFM down sweep signal vector \(\mathbf{S}^{-}\); received signal vector \(\mathbf{r}\); a small positive \(\varepsilon\).