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Wavelet transform is one of the most useful tools for working in signal processing and other disciplines like biology, economics, physics.. etc. When you work with Fourier transform you don't know in what time the harmonics for an specific frecuency have been generated. This problem is solved with wavelet transform.
Wavelet transform uses principal bases of linear algebra for having a kind of "mechanism" and be tolerant with Heisenberg's uncertainty principle. This tool allows to have better compresion rates for image compression, sound correction.. image processing etc, etc..
The applications are enormous, that's way all who are involved with electronics and processing of any kind of signal by this discipline MUST learn wavelet transform. I hope this will be usefull for you.