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Orthogonal wavelet filters

`[`

computes the four filters associated with the scaling filter `Lo_D`

,`Hi_D`

,`Lo_R`

,`Hi_R`

] = orthfilt(`W`

)`W`

corresponding to a wavelet.

The four filters the function computes are decomposition lowpass filter
`Lo_D`

, decomposition highpass filter
`Hi_D`

, reconstruction lowpass filter
`Lo_R`

, and reconstruction highpass filter
`Hi_R`

.

For more information on how the function computes the filters, see Algorithms.

For an orthogonal wavelet in the multiresolution framework, start with the scaling function ϕ and the wavelet function ψ. One of the fundamental relations is the twin-scale relation:

$$\frac{1}{2}\varphi \left(\frac{x}{2}\right)={\displaystyle \sum _{n\in Z}{w}_{n}}\varphi (x-n)$$

All the filters used in the `dwt`

and `idwt`

functions are intimately related to the sequence $${({w}_{n})}_{n\in Z}$$. if ϕ is compactly supported, the sequence
(*w _{n}*) is finite and can be viewed as an
FIR filter. The scaling filter

`W`

is a lowpass FIR filter of length
2N, with the sum 1, and with the norm of 1/√2."For example, for a `db3`

scaling filter,

w = dbwavf('db3') w = 0.2352 0.5706 0.3252 -0.0955 -0.0604 0.0249 sum(w) = 1.000 norm(w) = 0.7071

Define four FIR filters from filter `W`

of length 2N and norm
1.

The function computes the four filters using the following scheme.

The algorithm defines `qmf`

is such that `Hi_R`

and
`Lo_R`

are quadrature mirror filters (that is ```
Hi_R(k) =
(-1)
```

^{k}`Lo_R(2N + 1 - k)`

,
for `k = 1, 2, Ä, 2N`

) and `wrev`

such that it flips
the filter coefficients. Therefore, `Hi_D`

and `Lo_D`

are also quadrature mirror filters.

[1] Daubechies, I. (1992).
*Ten lectures on wavelets*. CBMS-NSF conference series in applied
mathematics, SIAM Ed. pp. 117–119, 137, 152.