Single-level 1-D discrete wavelet transform

`[cA,cD] = dwt(x,wname)`

`[cA,cD] = dwt(x,LoD,HiD)`

`[cA,cD] = dwt(___,'mode',extmode)`

`[`

returns the single-level discrete wavelet transform (DWT) of the vector `cA`

,`cD`

] = dwt(`x`

,`wname`

)`x`

using the wavelet specified by `wname`

. The wavelet must be recognized by
`wavemngr`

. `dwt`

returns the approximation coefficients vector
`cA`

and detail coefficients vector `cD`

of the DWT.

If your application requires a multilevel wavelet decomposition, consider using `wavedec`

.

Starting from a signal *s* of length *N*, two sets of
coefficients are computed: approximation coefficients
*cA*_{1}, and detail coefficients
*cD*_{1}. Convolving *s* with the scaling
filter `LoD`

, followed by dyadic decimation, yields the approximation
coefficients. Similarly, convolving *s* with the wavelet filter
`HiD`

, followed by dyadic decimation, yields the detail coefficients.

where

— Convolve with filter

*X*$$\begin{array}{||}\hline \downarrow 2\\ \hline\end{array}$$ — Downsample (keep the even-indexed elements)

The length of each filter is equal to 2*n*. If *N* =
length(*s*), the signals *F* and *G* are of
length *N* + 2*n* −1 and the coefficients
*cA*_{1} and *cD*_{1}
are of length floor$$\left(\frac{N-1}{2}\right)+n$$.

To deal with signal-end effects resulting from a convolution-based algorithm, a global
variable managed by `dwtmode`

defines the kind of signal extension mode
used. The possible options include zero-padding and symmetric extension, which is the default
mode.

For the same input, this `dwt`

function and the DWT block in the DSP System
Toolbox™ do not produce the same results. The DWT block is designed for real-time
implementation while Wavelet
Toolbox™ software is designed for analysis, so the products handle boundary conditions and
filter states differently.

To make the `dwt`

function output match the DWT block output,
set the function boundary condition to zero-padding by typing `dwtmode('zpd')`

at the MATLAB^{®} command prompt. To match the latency of the DWT block, which is implemented using
FIR filters, add zeros to the input of the `dwt`

function. The number of zeros you add must be equal to half the filter
length.

[1] Daubechies, I. *Ten Lectures on Wavelets*. CBMS-NSF Regional
Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied
Mathematics, 1992.

[2] Mallat, S. G. “A Theory for Multiresolution Signal Decomposition: The Wavelet
Representation.” *IEEE Transactions on Pattern Analysis and Machine
Intelligence*. Vol. 11, Issue 7, July 1989, pp. 674–693.

[3] Meyer, Y. *Wavelets and Operators*. Translated by D. H. Salinger.
Cambridge, UK: Cambridge University Press, 1995.

`dwtfilterbank`

| `dwtmode`

| `idwt`

| `wavedec`

| `waveinfo`