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NAG Toolbox

NAG Toolbox: nag_wav_1d_sngl_fwd (c09ca)

Purpose

nag_wav_1d_sngl_fwd (c09ca) computes the one-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wav_1d_init (c09aa) must be called first to set up the DWT options.

Syntax

[ca, cd, icomm, ifail] = c09ca(x, lenc, icomm, 'n', n)
[ca, cd, icomm, ifail] = nag_wav_1d_sngl_fwd(x, lenc, icomm, 'n', n)

Description

nag_wav_1d_sngl_fwd (c09ca) computes the one-dimensional DWT of a given input data array, xixi, for i = 1,2,,ni=1,2,,n, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution and downsampling by two to the input, xx. The approximation (or smooth) coefficients, CaCa, are produced by the low pass filter and the detail coefficients, CdCd, by the high pass filter. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension or zero end extension. The number ncnc, of coefficients CaCa or CdCd is returned by the initialization function nag_wav_1d_init (c09aa).

References

Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint .
x contains the input dataset xixi, for i = 1,2,,ni=1,2,,n.
2:     lenc – int64int32nag_int scalar
The dimension of the arrays ca and cd as declared in the (sub)program from which nag_wav_1d_sngl_fwd (c09ca) is called. This must be at least the number, ncnc, of approximation coefficients, CaCa, and detail coefficients, CdCd, of the discrete wavelet transform as returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
Constraint: lencnclencnc, where ncnc is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
3:     icomm(100100) – int64int32nag_int array
Contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wav_1d_init (c09aa).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
The number of elements, nn, in the data array xx.
Constraint: this must be the same as the value n passed to the initialization function nag_wav_1d_init (c09aa).

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     ca(lenc) – double array
ca(i)cai contains the iith approximation coefficient, Ca(i)Ca(i), for i = 1,2,,nci=1,2,,nc.
2:     cd(lenc) – double array
cd(i)cdi contains the iith detail coefficient, Cd(i)Cd(i), for i = 1,2,,nci=1,2,,nc.
3:     icomm(100100) – int64int32nag_int array
Contains additional information on the computed transform.
4:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry, n is inconsistent with the value passed to the initialization function nag_wav_1d_init (c09aa).
  ifail = 2ifail=2
On entry, lenc < nclenc<nc, where ncnc is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
  ifail = 6ifail=6
On entry, the initialization function nag_wav_1d_init (c09aa) has not been called first or it has been called with wtrans = 'M'wtrans='M', or the communication array icomm has become corrupted.

Accuracy

The accuracy of the wavelet transform depends only on the floating point operations used in the convolution and downsampling and should thus be close to machine precision.

Further Comments

None.

Example

function nag_wav_1d_sngl_fwd_example
n = int64(8);
wavnam = 'DB4';
mode = 'zero';
wtrans = 'Single Level';
x = [1; 3; 5; 7; 6; 4; 5; 2];
fprintf('\n Input Data:\n');
fprintf('%8.4f ', x);
fprintf('\n');

% Query wavelet filter dimensions
[nwl, nf, nwc, icomm, ifail] = nag_wav_1d_init(wavnam, wtrans, mode, n);

if ifail == int64(0)
  % Compute the transform
  [ca, cd, icomm, ifail] = nag_wav_1d_sngl_fwd(x, nwc, icomm);

  if ifail == int64(0)
    fprintf(' Approximation coefficients CA :\n');
    fprintf('%8.4f ', ca);
    fprintf('\n');
    fprintf(' Detail coefficients        CD :\n');
    fprintf('%8.4f ', cd);
    fprintf('\n');

    % Reconstruct original data
    [y, ifail] = nag_wav_1d_sngl_inv(ca, cd, n, icomm);

    if ifail == int64(0)
      fprintf(' Reconstruction       Y : \n');
      fprintf('%8.4f ', y);
      fprintf('\n');
    end
  end
end
 

 Input Data:
  1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000 
 Approximation coefficients CA :
  0.0011  -0.0043  -0.0174   4.4778   8.9557   7.3401   2.5816 
 Detail coefficients        CD :
  0.0237   0.0410  -0.5966   1.7763  -0.7517   0.3332  -0.1188 
 Reconstruction       Y : 
  1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000 

function c09ca_example
n = int64(8);
wavnam = 'DB4';
mode = 'zero';
wtrans = 'Single Level';
x = [1; 3; 5; 7; 6; 4; 5; 2];
fprintf('\n Input Data:\n');
fprintf('%8.4f ', x);
fprintf('\n');

% Query wavelet filter dimensions
[nwl, nf, nwc, icomm, ifail] = c09aa(wavnam, wtrans, mode, n);

if ifail == int64(0)
  % Compute the transform
  [ca, cd, icomm, ifail] = c09ca(x, nwc, icomm);

  if ifail == int64(0)
    fprintf(' Approximation coefficients CA :\n');
    fprintf('%8.4f ', ca);
    fprintf('\n');
    fprintf(' Detail coefficients        CD :\n');
    fprintf('%8.4f ', cd);
    fprintf('\n');

    % Reconstruct original data
    [y, ifail] = c09cb(ca, cd, n, icomm);

    if ifail == int64(0)
      fprintf(' Reconstruction       Y : \n');
      fprintf('%8.4f ', y);
      fprintf('\n');
    end
  end
end
 

 Input Data:
  1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000 
 Approximation coefficients CA :
  0.0011  -0.0043  -0.0174   4.4778   8.9557   7.3401   2.5816 
 Detail coefficients        CD :
  0.0237   0.0410  -0.5966   1.7763  -0.7517   0.3332  -0.1188 
 Reconstruction       Y : 
  1.0000   3.0000   5.0000   7.0000   6.0000   4.0000   5.0000   2.0000 


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