Wavelet Transform - Other Practical Applications

Other Practical Applications

The wavelet transform can provide us with the frequency of the signals and the time associated to those frequencies, making it very convenient for its application in numerous fields. For instance, signal processing of accelerations for gait analysis, and for fault detection. Ethiopian Highlands rainfall spatial and temporal variability. Also, Sea Surface Temperature spatio-temporal variations.

(1) Discretizing of the c-τ-axis

Applied the following discretization of frequency and time:

\begin{align} c_n &= c_0^n \\ \tau_m &= m \cdot T \cdot c_0^n \end{align}

Leading to wavelets of the form, the discrete formula for the basis wavelet:

Such discrete wavelets can be used for the transformation:

(2) Implementation via the FFT (fast Fourier transform)

As apparent from wavelet-transformation representation (shown below)

where c is scaling factor, τ represents time shift factor

and as already mentioned in this context, the wavelet-transformation corresponds to a convolution of a function y(t) and a wavelet-function. A convolution can be implemented as a multiplication in the frequency domain. With this the following approach of implementation results into:

  • Fourier-transformation of signal y(k) with the FFT
  • Selection of a discrete scaling factor
  • Scaling of the wavelet-basis-function by this factor and subsequent FFT of this function
  • Multiplication with the transformed signal YFFT of the first step
  • Inverse transformation of the product into the time domain results in YW for different discrete values of τ and a discrete value of
  • Back to the second step, until all discrete scaling values for are processed

There are large different types of wavelet transforms for specific purposes. See also a full list of wavelet-related transforms but the common ones are listed below: Mexican hat wavelet, Haar Wavelet, Daubechies wavelet, triangular wavelet.

Read more about this topic:  Wavelet Transform

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