orthfilt (Wavelet Toolbox)

Orthogonal wavelet filter set.

[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

and the wavelet function

. 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

is compactly supported, the sequence (w_n) 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_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)^kLo_R(2N - 1 - k)), 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 -0.0000 
    -0.0000 1.0000
    id = rf*rf'

id =
    1.0000 0.0000 
    0.0000 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')