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,

x_{i}

, for

i = 1, 2, \dots, 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,

x

. The approximation (or smooth) coefficients,

C_{a}

, are produced by the low pass filter and the detail coefficients,

C_{d}

, 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

n_{c}

, of coefficients

C_{a}

C_{d}

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: x contains the input dataset $x_{i}$ , for $i = 1, 2, \dots, n$ .
2: $lenc$ – int64int32nag_int scalar: The dimension of the arrays ca and cd. this must be at least the number, $n_{c}$ , of approximation coefficients, $C_{a}$ , and detail coefficients, $C_{d}$ , of the discrete wavelet transform as returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).

Constraint: $lenc \geq n_{c}$ , where $n_{c}$ is the value returned in nwc by the call to the initialization function nag_wav_1d_init (c09aa).
3: $icomm (100)$ – 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, $n$ , in the data array $x$ .

Constraint: this must be the same as the value n passed to the initialization function nag_wav_1d_init (c09aa).

Output Parameters

1: $ca (lenc)$ – double array: $ca (i)$ contains the $i$ th approximation coefficient, $C_{a} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
2: $cd (lenc)$ – double array: $cd (i)$ contains the $i$ th detail coefficient, $C_{d} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
3: $icomm (100)$ – int64int32nag_int array: Contains additional information on the computed transform.
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, n is inconsistent with the value passed to the initialization function.

$ifail = 3$: On entry, array dimension lenc not large enough.

$ifail = 6$: Either the initialization function has not been called first or array icomm has been corrupted.

Either the initialization function was called with $wtrans ='M'$ or array icomm has been corrupted.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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

This example computes the one-dimensional discrete wavelet decomposition for

8

values using the Daubechies wavelet,

wavnam ='DB4'

, with zero end extension.

Open in the MATLAB editor: c09ca_example

function c09ca_example


fprintf('c09ca example results\n\n');

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

c09ca example results


 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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox