# A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix
## 5 Conclusions  
The continuous wavelet transform of a signal with finite length for a fixed scale is considered as a linear time-invariant operator. Furthermore, the filter with causal and stability is constructed. The algorithm of linear convolution constitutes a unifying framework to the continuous wavelet transform methods previously published in [17]. The precision of these methods based on this framework is almost the same, no matter what method is used, and higher than other methods that use the approximation of wavelet, for example, the \(cut\) function in wavelet toolbox of Matlab. The edge effect is also easily handled by using Equivalent Condition of Time-domain and Frequency-domain Algorithms of CWT.  
The algorithms of CWT consist of two methods, time-domain method and frequency-domain method. For time-domain method, by constructing the causal filter \(\tilde{\psi}_{a}(t)\), we know the wavelet coefficients are just the middle part of the corresponding linear convolution. For frequency-domain method, by exploring the relationship of \(\tilde{\psi}_{a}(t)\) and \(\tilde{\tilde{\psi}}_{a}(t)\), we know the wavelet coefficients are just the first \(N\) coefficients of the corresponding convolution. Furthermore, by constructing the different filters, the wavelet coefficients can be the first \(N\) coefficients, or the middle \(N\) coefficients, or the last \(N\) coefficients of the corresponding convolution for frequency-domain method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(\sigma_{2}\) & \(\eta\) & CWT & Zhao’s Method & Algorithm 2 & Algorithm 3 \\ \hline
1 & 5 & 9.5190 & 9.4150 & 9.4144 & 9.4534 \\
1 & 6 & 9.5051 & 9.4044 & 9.4042 & 9.4332 \\
1 & 7 & 9.4869 & 9.3897 & 9.3893 & 9.4117 \\
1 & 8 & 9.4677 & 9.3713 & 9.3709 & 9.3892 \\
1.5 & 5 & 9.5012 & 9.4030 & 9.4026 & 9.4304 \\
1.5 & 6 & 9.4797 & 9.3832 & 9.3831 & 9.4037 \\
1.5 & 7 & 9.4545 & 9.3601 & 9.3598 & 9.3762 \\
1.5 & 8 & 9.4323 & 9.3377 & 9.3374 & 9.3513 \\ \hline \end{tabular}
\end{table}
Table 2: The wavelet entropy corresponding to different wavelet transform methods.  
Figure 7: This contour, computed with the case 0 of Algorithm 3 with \(M=N+2 aT\), manifests no edge effect.  
There are three methods for the calculation of linear convolution. The first one is to directly implement the definition of linear convolution. The second one is known as the FFT-based method, for example, Algorithm 1. Lastly, if the length of the signal is far larger than the length of the filter, an overlap-add procedure for the calculation of linear convolution is faster than the previous two methods [23]. How to combine the Frequency-domain method and an overlap-add procedure for the computation of CWT is a question for future research.  
The work is supported by the Natural Science Foundation of Jiangxi Province, China (Grant No. 20161 BAB201017), the Technology Plan Project of Jiangxi Provincial Education Department (No. GJ)160758), and the Doctoral research startup project of Jinggangshan University(No. JZB11002).  
[PERSON], [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] wrote the introduction part of this paper; [PERSON] wrote the experimental part; [PERSON] wrote the rest parts of this paper.  
The authors declare no conflict of interest.