In theory the wavelet coefficient, *W*_{j},_{n}, equals the convolution of *x*(*t*) with *ψ*_{j},_{n}, but as an empirical time series *x*(*t*) is only observed at a discrete set of observations. Empirically, *W*_{j},_{n} is calculated with a two-channel filter bank.^{5} Thus, one never needs an analytical formula for *ψ*. Instead all the calculations are performed in terms of a filter bank of coefficients.^{6}

A two-channel filter bank representation of the wavelet transform consists of the low-pass filter

$$\varphi \mathrm{(}t\mathrm{)}=\sqrt{2}{\displaystyle \sum _{k=0}^{L-1}}{g}_{l}\varphi \mathrm{(}2t-l\mathrm{}\mathrm{)}\mathrm{,}$$(8)

where ${\mathrm{\{}{g}_{l}\mathrm{\}}}_{l=0}^{L-1}$ are non-zero filter coefficients of positive even length *L*. The matching high-bandpass filter to the wavelet transforms two-channel filter is

$$\psi \mathrm{(}t\mathrm{)}={\displaystyle \sum _{l=0}^{L-1}}{h}_{l}\varphi \mathrm{(}2t-l\mathrm{}\mathrm{)}\mathrm{,}$$(9)

where *h*_{l}=(–1)^{l}*g*_{L}–1–_{l}. Daubechies (1988) provides a sufficient set of non-zero values for ${\mathrm{\{}{h}_{l}\mathrm{\}}}_{l=0}^{L-1}$ where *ψ* is compactly supported with the smallest possible support, *L*–1, for a wavelet possessing *L*/2 vanishing moments and whose regularity (number of derivatives and support size) increases linearly with *L*. Wavelets constructed with the filter coefficients of Daubechies are referred to the Daubechies class of wavelets with order *L*.

The low-pass filter coefficients, {*g*_{l}}, are equivalent to a moving average filter that smooth the high frequency traits of the series. For the Daubechies class of wavelets these low-pass filter coefficients have the squared-gain function

$$\mathcal{G}\mathrm{(}\omega \mathrm{)}\equiv {\left|\frac{1}{2\pi}{\displaystyle \sum _{l=0}^{L-1}}{e}^{-i\omega l}{g}_{l}\right|}^{2}=2{\mathrm{cos}}^{L}\omega {\displaystyle \sum _{l=0}^{L\mathrm{/}2\mathrm{-}1}}\mathrm{(}\begin{array}{c}L\mathrm{/}2\mathrm{-}l+l\\ l\end{array}\mathrm{)}{\mathrm{sin}}^{2l}\omega \mathrm{.}$$(10)

As *L* increase 𝒢 converges to the squared-gain function of a ideal low-pass filter supported on [–*π*, *π*] (see Lai 1995).

The high-bandpass filter coefficients {*h*_{l}} constitute a two-stage filter comprised of a first-state *L*/2-differencing operator and a second-stage weighted moving average. These two-stages are better understood in terms of the squared gain function of {*h*_{l}}

$$\begin{array}{c}\mathscr{H}\mathrm{(}\omega \mathrm{)}=\mathrm{(}4{\mathrm{sin}}^{2}\omega {\mathrm{)}}^{L/2}\frac{1}{{2}^{L-1}}{\displaystyle \sum _{l=0}^{L\mathrm{/}2\mathrm{-}1}}\mathrm{(}\begin{array}{c}L\mathrm{/}2\mathrm{-}l+l\\ l\end{array}\mathrm{)}{\mathrm{cos}}^{2l}\omega \mathrm{,}\\ ={\mathcal{D}}^{L/2}\mathrm{(}\omega \mathrm{)}{\mathcal{A}}_{L}\mathrm{(}\omega \mathrm{}\mathrm{)}\mathrm{,}\end{array}$$(11)

where 𝒟^{L/2}(*ω*)=(4sin^{2}*ω*)^{L/2} is the square modulus of the transfer function for the *L*/2-order differencing operator, (1–*B*)^{L/2}, and the squared gain function

${\mathcal{A}}_{L}\mathrm{(}\omega \mathrm{)}=\frac{1}{{2}^{L-1}}{\displaystyle \sum _{l=0}^{L\mathrm{/}2\mathrm{-}1}}\mathrm{(}\begin{array}{c}L\mathrm{/}2\mathrm{-}l+l\\ l\end{array}\mathrm{)}{\mathrm{cos}}^{2l}\omega \mathrm{,}$

is approximately a low-pass filter. As *L* increases ℋ approaches the squared gain function of an ideal, high-pass, filter with support on the octave, (*π*/2, *π*].

The function *ϕ*(·) is referred to as the scaling function since it does just that. Like the ideal, high-bandpass, wavelet, one can imagine a ideal, low-bandpass, scaling function whose frequency response is

$\widehat{\varphi}\mathrm{(}\omega \mathrm{)}=\{\begin{array}{ll}1\hfill & \text{if\hspace{0.17em}}\omega \in \mathrm{[}-\pi \mathrm{,}\text{\hspace{0.17em}}\pi \mathrm{}\mathrm{]}\mathrm{,}\hfill \\ 0\hfill & \text{otherwise}\mathrm{.}\hfill \end{array}$

Note that unlike the wavelet, *ϕ*(·) does not require any vanishing moments. However, we normalize *ϕ*(·) so that $\int}\varphi \mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}dt=1,$ i.e. ${\sum}_{l}}{g}_{l}^{2}=1.$

In a similar manner to the definition of *ψ*_{j},_{n}, define the dilations and translations of *ϕ*(*ω*) to be *ϕ*_{j},_{n}=2^{−j/2}*ϕ*(2^{−j}*t*–*n*), where *j*, *n*∈**Z**. The filter bank definition of *ψ*_{j},_{n} can then be written as:

${2}^{-j/2}\psi {\mathrm{(}2}^{-j}t-n\mathrm{)}={2}^{-j/2}{\displaystyle \sum _{l=0}^{L-1}}{h}_{l}\varphi {\mathrm{(}2}^{-\mathrm{(}j-1\mathrm{)}\mathrm{}}t-2n-l\mathrm{}\mathrm{)}\mathrm{.}$

Using this high-bandpass filter definition of *ψ*_{j},_{n} it follows that:

$${W}_{j\mathrm{,}n}={2}^{-j/2}{\displaystyle \sum _{l=0}^{L-1}}{h}_{l}{\displaystyle \int}x\mathrm{(}t\mathrm{)}\varphi {\mathrm{(}2}^{-\mathrm{(}j-1\mathrm{)}\mathrm{}}t-2n-l\mathrm{)}\text{\hspace{0.17em}}dt={\displaystyle \sum _{l=0}^{L-1}}{h}_{l}{V}_{j-\mathrm{1,2}n+l}\mathrm{,}$$(12)

where ${V}_{j\mathrm{,}n}={2}^{-j/2}{\displaystyle \int}x\mathrm{(}t\mathrm{)}\varphi {\mathrm{(}2}^{-j}-n\mathrm{)}\text{\hspace{0.17em}}dt$ is the scaling coefficient. Thus, computing *W*_{j,n} in this manner requires knowledge of {*V*_{j−1,n}n∈Z}.

Calculation of the scaling coefficient, *V*_{j},_{n}, can be performed by writing *ϕ*_{j},_{n} in terms of the lowpass filter as:

${2}^{-j/2}\varphi {\mathrm{(}2}^{-j}t-n\mathrm{)}={2}^{-\mathrm{(}j-1\mathrm{)}/2}{\displaystyle \sum _{l=0}^{L-1}}{g}_{l}\varphi {\mathrm{(}2}^{-\mathrm{(}j-1\mathrm{)}\mathrm{}}t-2n-l\mathrm{}\mathrm{)}\mathrm{.}$

Convoluting *x*(*t*) with the above equation we find that *V*_{j},_{n} equals:

$${V}_{j\mathrm{,}n}={2}^{-\mathrm{(}j-1\mathrm{)}/2}{\displaystyle \sum _{l=0}^{L-1}}{g}_{l}{\displaystyle \int}x\mathrm{(}t\mathrm{)}\varphi {\mathrm{(}2}^{-\mathrm{(}j-1\mathrm{)}}t-2n-l\mathrm{)}\text{\hspace{0.17em}}dt={\displaystyle \sum _{l=0}^{L-1}}{g}_{l}{V}_{j-\mathrm{1,2}n-l}\mathrm{.}$$(13)

Thus, both *V*_{j},_{n} and *W*_{j},_{n} are calculated recursively from the smallest to the largest scale with the simple multiplication and addition operators of a two-channel filter bank.^{7}

As an example of the two-channel filter wavelet transform suppose we observe the time series *x*(*t*) at *t*=1, 2, …, 2^{max}. Let *V*_{0,n} be the output from applying the low-pass filter to *x*(*t*) at the lowest possible scale, i.e. let *V*_{0,n}=*x*(*n*) for *n*=1, …, 2^{max}.^{8} The value of *W*_{j,n} and *V*_{j},_{n} for *j*=1, 2, …, max and *n*=1, 2, …, 2^{max−m} are recursively calculated from the *x*(*t*) s by applying over and over the filters of Eqs. (12) and (13).

This recursive algorithm known as the Fast Wavelet Transform is illustrated in Figure 1 where $x=\mathrm{(}x\mathrm{(}1\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}x{\mathrm{(}2}^{\mathrm{max}}\mathrm{)}{\mathrm{)}}^{\prime}\mathrm{,}\text{\hspace{0.17em}}{V}_{j}=\mathrm{(}{V}_{j\mathrm{,1}}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}{V}_{j{\mathrm{,2}}^{\text{max}-j}}{\mathrm{)}}^{\prime}\mathrm{,}$ and ${W}_{j}=\mathrm{(}{W}_{j\mathrm{,1}}\mathrm{,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}{W}_{j{\mathrm{,2}}^{\text{max}-j}}{\mathrm{)}}^{\prime}\mathrm{.}$ The wavelet coefficients for any value of *j* represents the information lost when **V**_{j−1} is filtered by {*g*_{l}}, i.e. **W**_{j} contains the details or information needed to obtain **V**_{j−1} from **V***j*. The box

in

Figure 1 represents the decimation of the output from the filter by 2, i.e. discarding the observations with odd time stamps. By their definition the ideal, low, and high-bandpass, filters,

*ϕ*(·) and

*ψ*(·), include this decimation. Because the filters coefficients {

*g*_{l}} and {

*h*_{l}} are both applied to

**V**_{j}, twice as many observations as the length of

**V**_{j} are created. Only half the output from the two-channel filter is needed to completely represent or recover

**V**_{j}.

Figure 1: Schematic representation of the Fast Wavelet Transform.

Because of the orthogonality of the filter banks for the Daubechies wavelet, the down arrows in Figure 1 can be reversed to synthesize **x** from its wavelet coefficients, **W**_{j}, *j*=1, …, max. In the wavelet synthesis, adding the details of **W**_{j} to the smoothed series **V**_{j} provides us with a representation of **x** at the next degree of resolution **V**_{j−1} with the highest resoluteness being the actual series **x**.

Though not as fast the Fast Wavelet Transform in computing the wavelet coefficients, there exists a cleaner definition of *W*_{j},_{n} and *V*_{j},_{n} in terms of the series *x*(*t*) and the filter banks {*g*_{l}} and {*h*_{l}}. They are

$${W}_{j\mathrm{,}n}={2}^{-j/2}{\displaystyle \sum _{l=0}^{{L}_{j}-1}}{h}_{j\mathrm{,}l}x\mathrm{(}{2}^{j}\mathrm{(}t+1\mathrm{)}\mathrm{}-1-l\mathrm{)}\mathrm{,}$$(14)

$${V}_{j\mathrm{,}n}={2}^{-j/2}{\displaystyle \sum _{l=0}^{{L}_{j}-1}}{g}_{j\mathrm{,}l}x\mathrm{(}{2}^{j}\mathrm{(}t+1\mathrm{)}\mathrm{}-1-l\mathrm{)}$$(15)

where {*h*_{j},_{l}} and {*g*_{j},_{l}} are filter coefficients at scale *j* associated with the Daubechies class of wavelets and *L*_{j}=(2^{j}–1)(*L*–1)+1. Note that for *j*=1, *h*_{1,l}=*h*_{l} and *g*_{1,l}=*h*_{l}. From the definition of *W*_{j},_{n} and *V*_{j},_{n} in Eqs. (12) and (13) and the squared-gain functions of Eqs. (10) and (11), if follows that the squared-gain functions of {*g*_{j},_{l}} and {*h*_{j},_{l}} equal

$${\mathcal{G}}_{j}\mathrm{(}\omega \mathrm{)}\equiv {\left|\frac{1}{2\pi}{\displaystyle \sum _{l=0}^{{L}_{j}-1}}{e}^{-i\omega l}{g}_{j\mathrm{,}l}\right|}^{2}={\displaystyle \prod _{l=0}^{j-1}}\mathcal{G}{\mathrm{(}2}^{l}\omega \mathrm{}\mathrm{)}\mathrm{,}$$(16)

$${\mathscr{H}}_{j}\mathrm{(}\omega \mathrm{)}\equiv {\left|\frac{1}{2\pi}{\displaystyle \sum _{l=0}^{{L}_{j}-1}}{e}^{-i\omega l}{h}_{j\mathrm{,}l}\right|}^{2}=\mathscr{H}{\mathrm{(}2}^{j-1}\omega \mathrm{)}{\mathcal{G}}_{j-1}\mathrm{(}\omega \mathrm{)}$$(17)

where ℋ_{1}=ℋ and 𝒢_{1}=𝒢. Since 𝒢_{j} and ℋ_{j} are the product of dilated 𝒢 and ℋ, as *L* increases 𝒢_{j} approaches the ideal low-pass filter with the support [–*π*/2^{j}, *π*/2^{j}] and ℋ_{j} approaches the ideal band-pass filter with support on the intervals (±*π*/2^{j}, ±*π*/2^{j−1}].

These squared-gain functions also have a nice intuitive flavor to them that follows from the FWT schematic in Figure 1. Skipping **V**_{1} and **W**_{1}, because their squared-gain functions follow directly from (10) and (11), the scaling coefficients **V**_{2} and wavelet coefficients **W**_{2} are computed by, respectively filtering **V**_{1} by {*g*_{l}} and {*h*_{l}} and then decimating the filtered output by two. Since the filtered output of **x** has already been decimated by two in order to obtain **V**_{1}, applying {*g*_{l}} and {*h*_{l}} to **V**_{1} is equivalent to applying filters with the squared-gain functions 𝒢(2*ω*) and ℋ(2*ω*). Since **V**_{1} is the output from filtering **x** with coefficients whose squared-gain function is 𝒢(*ω*), the squared-gain function for the filter of **x** with output **V**_{2} is equal to the product of the squared-gain function for the filter **x** to **V**_{1}, and the squared-gain function of the filter **V**_{1} to **V**_{2}; i.e. 𝒢_{2}(*ω*)=𝒢(*ω*)𝒢(2*ω*). Likewise, a filter applied to **x** whose output is **W**_{2} has a squared-gain function equal to 𝒢(*ω*) ℋ(2*ω*); i.e. ℋ_{2}. The squared-gain functions generalizes to **V**_{j} and **W**_{j} being the output from filtering **V**_{j−1} with {*g*_{l}} and {*h*_{l}}, respectively.

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