Microphone arrays

Once upon a time, there was Fourier ...

The well-known Fourier Transform is one of the most useful tools in the domain of signal processing. It has nevertheless some disadvantage, for the modern engineer: it "splashes" fast transients in wide band signals. This comes from the uniform partition of the time-frequency cells. If you choose a high temporal resolution, in order to pick-up transients, you have an bad frequency resolution (bad at low frequencies). If you need a high frequency resolution, the time resolution becomes bad (i.e. fast transients are splashed over a long time interval). 

Multiresolution (Wavelet) Transforms

Starting from Gabor, people are looking for new, better, transforms. First, Gabor introduced non-sinusoidal waveforms as base functions. In this way, elementary functions can contain high frequencies even at slow time-scales. In this case, it is better to say "high or low detail scale", rather than "high or low frequencies" of the base functions.

The idea about wavelet is very straightforward: you can build a transform which can adapt the time resolution to the detail scale: the more detailed (in time) the analysis wave shape, the shorter the time interval of the analysis. In this way, you have a varying time/frequency resolution ratio.

This brings several advantages: fast transients are easily picked-up (as time resolution of the high time-detailed side is high), without affecting the needs of the low detail component of the signal.

If your wavelet (the analyzing wave shape) is very "similar" to your signal, the analysis will lead to a low number of detail bands, in order to describe the signal.

Signal (and image) Compression

From this last idea comes the use of wavelets to compress signals. The problem is to choose (or, to build, if you prefer) a wavelet very "similar" to the signal, in order to analyze the signal in fewer bands. You can thus transmit fewer analysis coefficients, instead of samples of the original signal, and then you can reconstruct the original signal at the receiving end by means of a synthesis algorithm.

In this way you will need a lower bandwidth.

De-noise

This scheme can be made more complex, thinking of some non-linear and time-variant threshold mechanism, in order to suppress (or reduce) the noise embedded in the signal, thus achieving higher S/N ratios. The effectiveness of the method depends on how much your wavelet is good in discriminating  signal from noise (how much it is similar to the signal, and dissimilar from the noise). In any case, the method will preserve the benefit of the multiresolution analysis: fewer bands than in a single-resolution analysis scheme.