0.1 Introduction to wavelets  (Page 4/5)

Haar [link] showed this result in 1910, and we now know that wavelets are a generalization of his work. An example of a Haar system and expansion isgiven at the end of Chapter: A multiresolution formulation of Wavelet Systems .

What do wavelets look like?

All Fourier basis functions look alike. A high-frequency sine wave looks like a compressed low-frequency sine wave. A cosine wave is a sine wavetranslated by 90 ^o or $\pi /2$ radians. It takes a

large number of Fourier components to represent a discontinuity or a sharp corner. In contrast,there are many different wavelets and some have sharp corners themselves.

To appreciate the special character of wavelets you should recognize that it was not until the late 1980's that some of the most useful basicwavelets were ever seen. [link] illustrates four different scaling functions, each being zero outside of $0<t<6$ and each generating an orthogonal wavelet basisfor all square integrable functions. This figure is also shown on the cover to this book.

Several more scaling functions and their associated wavelets are illustrated in later chapters, and the Haar wavelet is shown in [link] and in detail at the end of Chapter: A multiresolution formulation of Wavelet Systems .

Why is wavelet analysis effective?

Wavelet expansions and wavelet transforms have proven to be very efficient and effective in analyzing a very wide class of signals and phenomena.Why is this true? What are the properties that give this effectiveness?

1. The size of the wavelet expansion coefficients $a_{j,k}$ in [link] or $d_{j,k}$ in [link] drop off rapidly with $j$ and $k$ for a large class of signals. This property is called being an unconditional basis and it is why wavelets are so effective insignal and image compression, denoising, and detection. Donoho [link] , [link] showed that wavelets are near optimal for a wide class of signals for compression, denoising, and detection.
2. The wavelet expansion allows a more accurate local description and separation of signal characteristics. A Fourier coefficient represents acomponent that lasts for all time and, therefore, temporary events must be described by a phase characteristic that allows cancellation orreinforcement over large time periods. A wavelet expansion coefficient represents a component that is itself local and is easier to interpret.The wavelet expansion may allow a separation of components of a signal whose Fourier description overlap in both time and frequency.
3. Wavelets are adjustable and adaptable. Because there is not just one wavelet, they can be designed to fit individual applications. They areideal for adaptive systems that adjust themselves to suit the signal.
4. The generation of wavelets and the calculation of the discrete wavelet transform is well matched to the digital computer. We will later see thatthe defining equation for a wavelet uses no calculus. There are no derivatives or integrals, just multiplications and additions—operationsthat are basic to a digital computer.