Wavelet functions are powerful mathematical tools used to analyze signals at multiple scales and resolutions. Unlike traditional Fourier analysis which only provides frequency information, wavelets give us both time and frequency localization. This makes them particularly useful for analyzing non-stationary signals where frequency content changes over time. The example shown here is a Morlet wavelet, which combines a Gaussian envelope with a sinusoidal oscillation.
The power of wavelets lies in their ability to be scaled and translated. Scaling changes the width of the wavelet, affecting its frequency content. A smaller scale parameter creates a narrower wavelet that captures high-frequency details, while a larger scale creates a wider wavelet for low-frequency analysis. Translation shifts the wavelet in time, allowing us to analyze different temporal locations. This combination of scaling and translation enables wavelets to provide a multi-resolution analysis of signals.
The key advantage of wavelet analysis over traditional Fourier analysis becomes clear when we compare their representations. Fourier analysis decomposes a signal into its frequency components, but loses all time information. We know which frequencies are present, but not when they occur. Wavelet analysis, on the other hand, provides a time-frequency representation. Each wavelet coefficient tells us not only about frequency content, but also when that frequency occurs in time. This makes wavelets particularly powerful for analyzing signals where frequency content changes over time, such as music, speech, or biomedical signals.