Wavelet Toolbox

Daubechies Wavelets: dbN

In dbN, N is the order. Some authors use 2N instead of N. More about this family can be found in [Dau92] pages 115, 132, 194, 242. By typing waveinfo('db), at the MATLAB command prompt, you can obtain a survey of the main properties of this family.

Figure 6-12: Daubechies Wavelets db4 on the Left and db8 on the Right

This family includes the Haar wavelet, written db1, the simplest wavelet imaginable and certainly the earliest. Using waveinfo('haar'), you can obtain a survey of the main properties of this wavelet.

Haar

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dbN

These wavelets have no explicit expression except for db1, which is the Haar wavelet. However, the square modulus of the transfer function of h is explicit and fairly simple.

• Let , where denotes the binomial coefficients.

• Then

• The support length of and is 2N - 1. The number of vanishing moments of is N.
• Most dbN are not symmetrical. For some, the asymmetry is very pronounced.
• The regularity increases with the order. When N becomes very large, and belong to where µ is approximately equal to 0.2. Certainly, this asymptotic value is too pessimistic for small-order N. Note that the functions are more regular at certain points than at others.
• The analysis is orthogonal.

Wavelet Families: Additional Discussion

Symlet Wavelets: symN

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