In some applications, the signal cannot be precisely processed in its original resolution. For example when working with signals that their spikes carry the underneath information, then working in the original resolution cannot be interpreted very well and the patterns cannot be extracted well.
For example, a signal can be periodic in several time scales. Mathematically, the largest period must be accepted as the main period, but when there are non-stationary or semi-stationary signals, then there is no global periodicity, but several local periods.
Smoothing, with lowpass filters or Gaussian filters, is one approach to make blurred versions of the original signal and then work with several planes or resolutions of the signal.
Wavelet is another powerful solution. Wavelet decomposes the signal into multiresolution and the signal can be reconstructed from these lower resolutions. The reconstructions power of the wavelet is very useful in the compression methods. Moreover, the filter bank decomposition is almost a kind of wavelet which tries to decompose the signal into lowpass and highpass versions and repeats the same for the lowpass version. Therefore, the signal will be decomposed into fs/2, fs/4, fs/8, fs/16, fs/32, etc.
But there are some considerations when we use the wavelet:
- What wavelets can best describe a typical signal?
- The wavelet decomposition need to work on all samples in the signal, so it is not always causal.
- The wavelets are not extracted from the signal shape, therefore is it the best choice? This question is almost similar to the first question.
There are two other approaches that need to be reviewed:
- Sparse decomposition
The beauty of this method is that it just keeps the necessary points, not more.
- Discrete Geometry
This field discusses the representation of an object in several resolutions. Therefore, it can also be used in multiresolution signal processing.
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