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# orthfilt

Orthogonal wavelet filter set

## Syntax

[Lo_D,Hi_D,Lo_R,Hi_R] = orthfilt(W)

## Description

[Lo_D,Hi_D,Lo_R,Hi_R] = orthfilt(W) computes the four filters associated with the scaling filter W corresponding to a wavelet:

 Lo_D Decomposition low-pass filter Hi_D Decomposition high-pass filter Lo_R Reconstruction low-pass filter Hi_R Reconstruction high-pass filter

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

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

All the filters used in dwt and idwt are intimately related to the sequence ${\left({w}_{n}\right)}_{n\in Z}$. Clearly if ϕ is compactly supported, the sequence (wn) is finite and can be viewed as a FIR filter. The scaling filter W is

• A low-pass FIR filter

• Of length 2N

• Of sum 1

• Of norm

For example, for the db3 scaling filter,

```load db3
db3
db3 =
0.2352 0.5706 0.3252 -0.0955 -0.0604 0.0249

sum(db3)
ans =
1.000
norm(db3)
ans =
0.7071
```

From filter W, we define four FIR filters, of length 2N and norm 1, organized as follows:

Filters

Low-Pass

High-Pass

Decomposition

Lo_DHi_D

Reconstruction

Lo_RHi_R

The four filters are computed using the following scheme:

where qmf is such that Hi_R and Lo_R are quadrature mirror filters (i.e., Hi_R(k) = (-1)kLo_R(2N + 1 - k), for k = 1, 2, Ä, 2N), and where wrev flips the filter coefficients. So Hi_D and Lo_D are also quadrature mirror filters. The computation of these filters is performed using orthfilt.

## Examples

```% Load scaling filter.
load db8; w = db8;
subplot(421); stem(w);
title('Original scaling filter');

% Compute the four filters.
[Lo_D,Hi_D,Lo_R,Hi_R] = orthfilt(w);
subplot(423); stem(Lo_D);
title('Decomposition low-pass filter');
subplot(424); stem(Hi_D);
title('Decomposition high-pass filter');
subplot(425); stem(Lo_R);
title('Reconstruction low-pass filter');
subplot(426); stem(Hi_R);
title('Reconstruction high-pass filter');

% Check for orthonormality.
df = [Lo_D;Hi_D];
rf = [Lo_R;Hi_R];
id = df*df'

id =
1.0000         0
0    1.0000

id = rf*rf'

id =
1.0000         0
0    1.0000

% Check for orthogonality by dyadic translation, for example:
df = [Lo_D 0 0;Hi_D 0 0];
dft = [0 0 Lo_D; 0 0 Hi_D];
zer = df*dft'

zer =

1.0e-12 *
-0.1883 0.0000
-0.0000 -0.1883

% High- and low-frequency illustration.
fftld = fft(Lo_D); ffthd = fft(Hi_D);
freq = [1:length(Lo_D)]/length(Lo_D);
subplot(427); plot(freq,abs(fftld));
title('Transfer modulus: low-pass');
subplot(428); plot(freq,abs(ffthd));
title('Transfer modulus: high-pass')
% Editing some graphical properties,
% the following figure is generated.
```

## References

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

## See Also

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