Tips about Wavelet

This introduction about Wavelet is summarized from matlab wavelet tutorial, for this tutorial, in my opinion, is concise and easy to understand.

From Fourier to Wavelet

Fourier method is the most well-known method for signal analysis. It breaks down a signal into constituent sinusoids of different frequencies, in other words, it transforms the signals from time-based to frequency-based.

Fig.1 Fourier Transform

Although this useful technique has been wildly used by people for a long time, it has a serious drawback: time information is lost during the transform, thus Fourier is not suited to detecting nonstatinary signals, such as EEG.

In an effort to correct this deficiency, Dennis Gabor (1946) adapted the Fourier transform to analyze only a small section of the signal at a time -- a technique called windowing the signal. Gabor's adaptation, called the Short-Time Fourier Transform (STFT), maps a signal into a two-dimensional function of time and frequency.

Fig.2 Short Time Fourier Transform

However, in STFT a particular size for time window should be chosen, the drawback is that this window can not provide a flexible analysis for all frequencies.

Wavelet analysis represents the next logical step: a windowing technique with variable-sized regions. Wavelet analysis allows the use of long time intervals where we want more precise low-frequency information, and shorter regions where we want high-frequency information.

Fig.2 Wavelet Transform

NOTE: Most of the information in this article comes from MATLAB tutorial, many thanks for MATLAB! The knowledge here are only basic tips, so if you want more details, see relevant books or papers please!