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Orthogonal wavelet filter set

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

`[Lo_D,Hi_D,Lo_R,Hi_R] = orthfilt(W)` computes
the four filters associated with the scaling filter

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:

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

A low-pass FIR filter

Of length 2

*N*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_D | Hi_D |

Reconstruction | Lo_R | Hi_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) ^{k}`

% 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.

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

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