naginterfaces.library.wav.dim1_​multi_​fwd

naginterfaces.library.wav.dim1_multi_fwd(x, nwl, comm)

dim1_multi_fwd computes the one-dimensional multi-level discrete wavelet transform (DWT). The initialization function dim1_init() must be called first to set up the DWT options.

For full information please refer to the NAG Library document for c09cc

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/c09/c09ccf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

$x$ contains the one-dimensional input dataset $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

nwlint

The number of levels, $n_{fwd}$, in the multi-level resolution to be performed.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to dim1_init().

Returns

cfloat, ndarray, shape $\left(\text{lenc}\right)$

Let $q\left(\text{i}\right)$ denote the number of coefficients (of each type) produced by the wavelet transform at level $\text{i}$, for $\text{i}=n_{fwd},\dots ,1$. These values are returned in $dwtlev$. Setting $k_1=q\left(n_{fwd}\right)$ and $k_{\text{j}+1}=k_{\text{j}}+q(n_{fwd}-\text{j}+1)$, for $\text{j}=1,2,\dots ,n_{fwd}$, the coefficients are stored as follows:

$c[\text{i}-1]$, for $\text{i}=1,2,\dots ,k_1$

Contains the level $n_{fwd}$ approximation coefficients, $a_{n_{fwd}}$.

$c[\text{i}-1]$, for $\text{i}=k_1+1,\dots ,k_2$

Contains the level $n_{fwd}$ detail coefficients $d_{n_{fwd}}$.

$c[\text{i}-1]$, for $\text{i}=k_j+1,\dots ,k_{j+1}$

Contains the level $n_{fwd}-\text{j}+1$ detail coefficients, for $\text{j}=2,3,\dots ,n_{fwd}$.

dwtlevint, ndarray, shape $(nwl+1)$

The number of transform coefficients at each level. $dwtlev\left[0\right]$ and $dwtlev\left[1\right]$ contain the number, $q\left(n_{fwd}\right)$, of approximation and detail coefficients respectively, for the final level of resolution (these are equal); $dwtlev[\text{i}-1]$ contains the number of detail coefficients, $q(n_{fwd}-\text{i}+2)$, for the ($n_{fwd}-\text{i}+2$)th level, for $\text{i}=3,4,\dots ,n_{fwd}+1$.

Raises

NagValueError

(errno $1$)

On entry, $\text{n}$ is inconsistent with the value passed to the initialization function: $n=⟨value⟩$, $\text{n}$ should be $⟨value⟩$.

(errno $5$)

On entry, $nwl$ is larger than the maximum number of levels returned by the initialization function: $nwl=⟨value⟩$, maximum $=⟨value⟩$.

(errno $5$)

On entry, $nwl=⟨value⟩$.

Constraint: $nwl\ge 1$.

(errno $7$)

Either the initialization function has not been called first or array $comm$[‘icomm’] has been corrupted.

(errno $7$)

Either the initialization function was called with $wtrans=\text{‘S'}$ or array $comm$[‘icomm’] has been corrupted.

Notes

dim1_multi_fwd computes the multi-level DWT of one-dimensional data. For a given wavelet and end extension method, dim1_multi_fwd will compute a multi-level transform of a data array, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, using a specified number, $n_{fwd}$, of levels. The number of levels specified, $n_{fwd}$, must be no more than the value $l_{max}$ returned in $\text{nwlmax}$ by the initialization function dim1_init() for the given problem. The transform is returned as a set of coefficients for the different levels (packed into a single array) and a representation of the multi-level structure.

The notation used here assigns level $0$ to the input dataset, $x$, with level $1$ being the first set of coefficients computed, with the detail coefficients, $d_1$, being stored while the approximation coefficients, $a_1$, are used as the input to a repeat of the wavelet transform. This process is continued until, at level $n_{fwd}$, both the detail coefficients, $d_{n_{fwd}}$, and the approximation coefficients, $a_{n_{fwd}}$ are retained. The output array, $C$, stores these sets of coefficients in reverse order, starting with $a_{n_{fwd}}$ followed by $d_{n_{fwd}},d_{n_{fwd}-1},\dots ,d_1$.