Visualization of 1-D wavelets

This applet gives a visual representation of various 1-D mother wavelets. The plot on the top shows the time-domain representation, the plot on the bottom shows the magnitude responses of a dyadic discrete wavelet filter bank. The graph on the top is generated using the technique of successive approximation.

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Wavelet families

• dbX. The Daubechies wavelets (named after Ingrid Daubechies) are a family of orthogonal wavelets. The 'X' denotes the number of vanishing moments. These wavelets have a number of vanishing moments that is maximal for a given wavelet filter support size. The filter coefficients are computed using a technique called "spectral factorization". The Daubechies mother wavelet also has a fractal structure: by zooming into a region of the wavelet, one can recognize the wavelet itself.
• symX. Symlet wavelets are also a family of orthogonal wavelets, very similar to the Daubechies wavelets. The difference is that the spectral factorization is performed using minimal phase selection instead of extremal phase selection as for the Daubechies wavelets. This results in wavelets with the least asymmetry.
• coifX. Coiflets are another famility of orthogonal wavelets, but are based on a different construction (see [1]).
• dualfilt1. This is a complex wavelet, that is based on the 6-tap Q-Shift wavelet filter from Nick Kingsbury [2]. The plot shows the real part, imaginary part and the magnitude (envelope) of the complex wavelet. The imaginary part is approximately the Hilbert transform of the real part of the wavelet.  
• CW_Selesnick_X_Y. This family of complex wavelets is constructed using the Thiran-based design technique from Ivan Selesnick [3]. X is the order of the allpass filter being used (the higher, the better the analyticity of the wavelets) and Y is the number of vanishing moments.

References

• [1]  I. Daubechies, "Ten Lectures on Wavelets," CBMS-NSF Lecture Notes nr. 61, SIAM, 1992.
• [2]  N G Kingsbury, "Complex wavelets for shift invariant analysis and filtering of signals,"
  Journal of Applied and Computational Harmonic Analysis, vol 10, no 3, May 2001, pp. 234-253.
• [3]  I. W. Selesnick. "The design of approximate Hilbert transform pairs of wavelet bases,"
  IEEE Trans. on Signal Processing, 50(5):1144-1152, May 2002.