# 0.9 Calculation of the discrete wavelet transform  (Page 4/5)

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If we assume the length of the sequence of the signal is $L$ and the length of the sequence of scaling filter coefficients $h\left(n\right)$ is $N$ , then the number of multiplications necessary to calculate each scaling functionand wavelet expansion coefficient at the next scale, $c\left({J}_{1}-1,k\right)$ and $d\left({J}_{1}-1,k\right)$ , from the samples of the signal, $f\left(Tk\right)\approx c\left({J}_{1},k\right)$ , is $LN$ . Because of the downsampling, only half are needed to calculate thecoefficients at the next lower scale, $c\left({J}_{2}-1,k\right)$ and $d\left({J}_{2}-1,k\right)$ , and repeats until there is only one coefficient at a scale of $j={J}_{0}$ . The total number of multiplications is, therefore,

$\text{Mult}\phantom{\rule{4.pt}{0ex}}=LN+LN/2+LN/4+\cdots +N$
$=LN\left(1+1/2+1/4+\cdots +1/L\right)=2NL-N$

which is linear in $L$ and in $N$ . The number of required additions is essentially the same.

If the length of the signal is very long, essentially infinity, the coarsest scale ${J}_{0}$ must be determined from the goals of the particular signal processing problem being addressed. For this case, the number ofmultiplications required per DWT coefficient or per input signal sample is

$\text{Mult/sample}=N\left(2-{2}^{-{J}_{0}}\right)$

Because of the relationship of the scaling function filter $h\left(n\right)$ and the wavelet filter ${h}_{1}\left(n\right)$ at each scale (they are quadrature mirror filters), operations can be shared between them through the use of alattice filter structure, which will almost halve the computational complexity. That is developed in Chapter: Filter Banks and Transmultiplexers and [link] .

## The periodic case

In many practical applications, the signal is finite in length (finite support) and can be processed as single “block," much as the FourierSeries or discrete Fourier transform (DFT) does. If the signal to be analyzed is finite in length such that

$f\left(t\right)=\left\{\begin{array}{cc}0\hfill & t<0\hfill \\ 0\hfill & t>P\hfill \\ f\left(t\right)\hfill & 0

we can construct a periodic signal $\stackrel{˜}{f}\left(t\right)$ by

$\stackrel{˜}{f}\left(t\right)=\sum _{n}f\left(t+Pn\right)$

and then consider its wavelet expansion or DWT. This creation of a meaningful periodic function can still be done, even if $f\left(t\right)$ does not have finite support, if its energy is concentrated and some overlap isallowed in [link] .

Periodic Property 1: If $\stackrel{˜}{f}\left(t\right)$ is periodic with integer period $P$ such that $\stackrel{˜}{f}\left(t\right)=\stackrel{˜}{f}\left(t+Pn\right)$ , then the scaling function and wavelet expansion coefficients (DWT terms) at scale $J$ are periodic with period ${2}^{J}P$ .

$\text{If}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{˜}{f}\left(t\right)=\stackrel{˜}{f}\left(t+P\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{then}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\stackrel{˜}{d}}_{j}\left(k\right)={\stackrel{˜}{d}}_{j}\left(k+{2}^{j}P\right)$

This is easily seen from

${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{-\infty }^{\infty }\stackrel{˜}{f}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt=\int \stackrel{˜}{f}\left(t+Pn\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$

which, with a change of variables, becomes

$=\int \stackrel{˜}{f}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}\left(x-Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dx=\int \stackrel{˜}{f}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}x-\left({2}^{j}Pn+k\right)\right)\phantom{\rule{0.166667em}{0ex}}dx={\stackrel{˜}{d}}_{j}\left(k+{2}^{j}Pn\right)$

and the same is true for ${\stackrel{˜}{c}}_{j}\left(k\right)$ .

Periodic Property 2: The scaling function and wavelet expansion coefficients (DWT terms) can be calculated from the inner product of $\stackrel{˜}{f}\left(t\right)$ with $\phi \left(t\right)$ and $\psi \left(t\right)$ or, equivalently, from the inner product of $f\left(t\right)$ with the periodized $\stackrel{˜}{\phi }\left(t\right)$ and $\stackrel{˜}{\psi }\left(t\right)$ .

${\stackrel{˜}{c}}_{j}\left(k\right)=〈\stackrel{˜}{f},\left(t\right),,,\phi ,\left(t\right)〉=〈f,\left(t\right),,,\stackrel{˜}{\phi },\left(t\right)〉$

and

${\stackrel{˜}{d}}_{j}\left(k\right)=〈\stackrel{˜}{f},\left(t\right),,,\psi ,\left(t\right)〉=〈f,\left(t\right),,,\stackrel{˜}{\psi },\left(t\right)〉$

where $\stackrel{˜}{\phi }\left(t\right)={\sum }_{n}\phi \left(t+Pn\right)$ and $\stackrel{˜}{\psi }\left(t\right)={\sum }_{n}\psi \left(t+Pn\right)$ .

This is seen from

${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{-\infty }^{\infty }\stackrel{˜}{f}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt=\sum _{n}{\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}\left(t+Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dt={\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\sum _{n}\psi \left({2}^{j}\left(t+Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dt$
${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\psi }\left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$

where $\stackrel{˜}{\psi }\left({2}^{j}t-k\right)={\sum }_{n}\psi \left({2}^{j}\left(t+Pn\right)-k\right)$ is the periodized scaled wavelet.

Periodic Property 3: If $\stackrel{˜}{f}\left(t\right)$ is periodic with period $P$ , then Mallat's algorithm for calculating the DWT coefficients in [link] becomes

${\stackrel{˜}{c}}_{j}\left(k\right)=\sum _{m}h\left(m-2k\right)\phantom{\rule{0.166667em}{0ex}}{\stackrel{˜}{c}}_{j+1}\left(m\right)$

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