NAG Library Function Document

nag_dwt (c09cac)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_dwt (c09cac) computes the one-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wfilt (c09aac) must be called first to set up the DWT options.

2 Specification

#include <nag.h>

#include <nagc09.h>

void	nag_dwt (Integer n, const double x[], Integer lenc, double ca[], double cd[], Integer icomm[], NagError *fail)

3 Description

nag_dwt (c09cac) 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_wfilt (c09aac).

4 References

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

5 Arguments

1: n – IntegerInput: On entry: 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_wfilt (c09aac).
2: x[n] – const doubleInput: On entry: x contains the input dataset $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: lenc – IntegerInput: On entry: 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_wfilt (c09aac).
Constraint: $lenc \geq n_{c}$ , where $n_{c}$ is the value returned in nwc by the call to the initialization function nag_wfilt (c09aac).
4: ca[lenc] – doubleOutput: On exit: $ca [i - 1]$ contains the $i$ th approximation coefficient, $C_{a} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
5: cd[lenc] – doubleOutput: On exit: $cd [i - 1]$ contains the $i$ th detail coefficient, $C_{d} (i)$ , for $i = 1, 2, \dots, n_{c}$ .
6: icomm[ $100$ ] – IntegerCommunication Array: On entry: contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wfilt (c09aac).

On exit: contains additional information on the computed transform.
7: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_DIM_LEN: On entry, array dimension lenc not large enough: $lenc = ⟨value⟩$ but must be at least $⟨value⟩$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INITIALIZATION: Either the initialization function has not been called first or array icomm has been corrupted.
Either the initialization function was called with $wtrans = Nag_MultiLevel$ or array icomm has been corrupted.
On entry, n is inconsistent with the value passed to the initialization function: $n = ⟨value⟩$ , n should be $⟨value⟩$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example computes the one-dimensional discrete wavelet decomposition for

8

values using the Daubechies wavelet,

wavnam = Nag_Daubechies4

, with zero end extension.

NAG Library Function Documentnag_dwt (c09cac)