Many wavelet does not have functional form, but defined by the MRA coefficient.

The visualization of wavelet can be done by using wavelet construction.

\displaystyle v_{j+1,k} = \sum_{n} g_0(k-2n)v_{j,n} + g_1(k-2n) w_{j,n}

For scaling function, we can define v_0 = {1} and w_0 = {0} .

\displaystyle v_{1,k} = \sum_{n} g_0(k-2n)v_{0,n} = g_0(k)

Similarly, the wavelet can be started with v_0 = {0} and w_0 = {1} .

\displaystyle v_{1,k} = \sum_{n} g_1(k-2n)w_{0,n} = g_1(k)

Then build by iteration,

\displaystyle v_{j+1,k} = \sum_{n} g_0(k-2n) v_{j,n}

From last post on the scaling coefficient, i calculated and plot the wavelet for m = 4.

Capture.PNG

we can see the wavelet becomes the Haar wavelet as the free parameter goes to 1. In fact, it becomes a shifted Haar wavelet when the free parameter goes to 0, as we can imagine.

When the free parameter is 0.683013, it is the Daubechies-2 wavelet. Notes that some people will absorbed a factor $latex 1/ \sqrt{2} $ into the coefficient, so that their free parameter is 0.683013/\sqrt{2} = 0.482963.