Fast Wavelet Transform - Forward DWT

Forward DWT

One computes recursively, starting with the coefficient sequence and counting down from k=J-1 down to some M,

s^{(k)}_n:=\frac12 \sum_{m=-N}^N a_m s^{(k+1)}_{2n+m} or s^{(k)}(z):=(\downarrow 2)(a^*(z)\cdot s^{(k+1)}(z))

and

d^{(k)}_n:=\frac12 \sum_{m=-N}^N b_m s^{(k+1)}_{2n+m} or d^{(k)}(z):=(\downarrow 2)(b^*(z)\cdot s^{(k+1)}(z)) ,

for k=J-1,J-2,...,M and all . In the Z-transform notation:

  • The downsampling operator reduces an infinite sequence, given by its Z-transform, which is simply a Laurent series, to the sequence of the coefficients with even indices, .
  • The starred Laurent-polynomial denotes the adjoint filter, it has time-reversed adjoint coefficients, . (The adjoint of a real number being the number itself, of a complex number its conjugate, of a real matrix the transposed matrix, of a complex matrix its hermitian adjoint).
  • Multiplication is polynomial multiplication, which is equivalent to the convolution of the coefficient sequences.

It follows that

is the orthogonal projection of the original signal f or at least of the first approximation onto the subspace, that is, with sampling rate of 2k per unit interval. The difference to the first approximation is given by

,

where the difference or detail signals are computed from the detail coefficients as

,

with denoting the mother wavelet of the wavelet transform.

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