Scaling function:

Wavelet function:

Filter coefficients:

Approximate a function using Haar wavelet coefficients:

Perform a LiftingWaveletTransform:

Approximate original data by keeping largest coefficients and thresholding everything else:

Compare the different approximations:

Compute the multiresolution representation of a signal containing an impulse:

Compare the cumulative energy in a signal and its wavelet coefficients:

Compute the ordered cumulative energy in the signal:

The energy in the signal is captured by relatively few wavelet coefficients:

Compare range and distribution of wavelet coefficients:

Plot distribution of wavelet coefficients:

Compare with wavelet coefficients plotted along a common axis:

DaubechiesWavelet[1] is equivalent to HaarWavelet:

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Scaling function integrates to unity; :

In particular, :

Haar scaling function is orthogonal to its shift; :

Wavelet function integrates to zero; :

Haar wavelet function is orthogonal to its shift; :

Wavelet function is orthogonal to the scaling function at the same scale; :

The lowpass and highpass filter coefficients are orthogonal; :

HaarWavelet has one vanishing moment; :

This means constant signals are fully represented in the scaling functions part ({0}):

Linear or higher-order signals are not:

satisfies the recursion equation :

Symbolically verify recursion:

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a highpass filter:

Fourier transform of is given by :

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function: