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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.

*db*X. 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.
*sym*X. 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.
*coif*X. 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.

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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.