For the discrete wavelet transform (DWT), one computes recursively, starting with the coefficient sequence ${\displaystyle s^{(J)}}$  and counting down from k = J - 1 to some M < J,

${\displaystyle s_{n}^{(k)}:={\frac {1}{2}}\sum _{m=-N}^{N}a_{m}s_{2n+m}^{(k+1)}}$  or ${\displaystyle s^{(k)}(z):=(\downarrow 2)(a^{*}(z)\cdot s^{(k+1)}(z))}$ 

and

${\displaystyle d_{n}^{(k)}:={\frac {1}{2}}\sum _{m=-N}^{N}b_{m}s_{2n+m}^{(k+1)}}$  or ${\displaystyle d^{(k)}(z):=(\downarrow 2)(b^{*}(z)\cdot s^{(k+1)}(z))}$ ,

for k=J-1,J-2,...,M and all ${\displaystyle n\in \mathbb {Z} }$ . In the Z-transform notation:

• The downsampling operator ${\displaystyle (\downarrow 2)}$  reduces an infinite sequence, given by its Z-transform, which is simply a Laurent series, to the sequence of the coefficients with even indices, ${\displaystyle (\downarrow 2)(c(z))=\sum _{k\in \mathbb {Z} }c_{2k}z^{-k}}$ .
• The starred Laurent-polynomial ${\displaystyle a^{*}(z)}$  denotes the adjoint filter, it has time-reversed adjoint coefficients, ${\displaystyle a^{*}(z)=\sum _{n=-N}^{N}a_{-n}^{*}z^{-n}}$ . (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

${\displaystyle P_{k}[f](x):=\sum _{n\in \mathbb {Z} }s_{n}^{(k)}\,\varphi (2^{k}x-n)}$ 

is the orthogonal projection of the original signal f or at least of the first approximation ${\displaystyle P_{J}[f](x)}$  onto the subspace ${\displaystyle V_{k}}$ , that is, with sampling rate of 2^k per unit interval. The difference to the first approximation is given by

${\displaystyle P_{J}[f](x)=P_{k}[f](x)+D_{k}[f](x)+\dots +D_{J-1}[f](x),}$ 

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

${\displaystyle D_{k}[f](x):=\sum _{n\in \mathbb {Z} }d_{n}^{(k)}\,\psi (2^{k}x-n),}$ 

with ${\displaystyle \psi }$  denoting the mother wavelet of the wavelet transform.

Given the coefficient sequence ${\displaystyle s^{(M)}}$  for some M < J and all the difference sequences ${\displaystyle d^{(k)}}$ , k = M,...,J − 1, one computes recursively

${\displaystyle s_{n}^{(k+1)}:=\sum _{k=-N}^{N}a_{k}s_{2n-k}^{(k)}+\sum _{k=-N}^{N}b_{k}d_{2n-k}^{(k)}}$  or ${\displaystyle s^{(k+1)}(z)=a(z)\cdot (\uparrow 2)(s^{(k)}(z))+b(z)\cdot (\uparrow 2)(d^{(k)}(z))}$ 

for k = J − 1,J − 2,...,M and all ${\displaystyle n\in \mathbb {Z} }$ . In the Z-transform notation:

• The upsampling operator ${\displaystyle (\uparrow 2)}$  creates zero-filled holes inside a given sequence. That is, every second element of the resulting sequence is an element of the given sequence, every other second element is zero or ${\displaystyle (\uparrow 2)(c(z)):=\sum _{n\in \mathbb {Z} }c_{n}z^{-2n}}$ . This linear operator is, in the Hilbert space ${\displaystyle \ell ^{2}(\mathbb {Z} ,\mathbb {R} )}$ , the adjoint to the downsampling operator ${\displaystyle (\downarrow 2)}$ .

• Lifting scheme
• Fast Fourier transform

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