wavedec

Multilevel 1-D discrete wavelet transform

Syntax

[c,l] = wavedec(x,n,wname)

[c,l] = wavedec(x,n,LoD,HiD)

Description

[c,l] = wavedec(x,n,wname) returns the wavelet decomposition of the 1-D signal x at level n using the wavelet wname. The output decomposition structure consists of the wavelet decomposition vector c and the bookkeeping vector l, which is used to parse c.

Note

For gpuArray inputs, the supported modes are 'symh' ('sym') and 'per'. If the input is a gpuArray, the discrete wavelet transform extension mode used by wavedec defaults to 'symh' unless the current extension mode is 'per'. See the example Multilevel Discrete Wavelet Transform on a GPU.

[c,l] = wavedec(x,n,LoD,HiD) returns the wavelet decomposition using the specified lowpass and highpass wavelet decomposition filters LoD and HiD, respectively.

Examples

Multilevel One-Dimensional Wavelet Analysis

Load and plot a one-dimensional signal.

load sumsin 
plot(sumsin)
title('Signal')

Perform a 3-level wavelet decomposition of the signal using the order 2 Daubechies wavelet. Extract the coarse scale approximation coefficients and the detail coefficients from the decomposition.

[c,l] = wavedec(sumsin,3,'db2');
approx = appcoef(c,l,'db2');
[cd1,cd2,cd3] = detcoef(c,l,[1 2 3]);

Plot the coefficients.

subplot(4,1,1)
plot(approx)
title('Approximation Coefficients')
subplot(4,1,2)
plot(cd3)
title('Level 3 Detail Coefficients')
subplot(4,1,3)
plot(cd2)
title('Level 2 Detail Coefficients')
subplot(4,1,4)
plot(cd1)
title('Level 1 Detail Coefficients')

Multilevel Discrete Wavelet Transform on a GPU

Refer to GPU Computing Requirements (Parallel Computing Toolbox) to see what GPUs are supported.

Load the noisy Doppler signal. Put the signal on the GPU using gpuArray. Save the current extension mode.

load noisdopp
noisdoppg = gpuArray(noisdopp);
origMode = dwtmode('status','nodisp');

Use dwtmode to change the extension mode to zero-padding. Obtain the three-level DWT of the signal on the GPU using the db4 wavelet.

dwtmode('zpd','nodisp')
[c,l] = wavedec(noisdoppg,3,'db4');

The current extension mode zpd is not supported for gpuArray input. Therefore, the DWT is instead performed using the sym extension mode. To confirm this, set the extension mode to sym and take DWT of noisdoppg, then compare with the previous result.

dwtmode('sym','nodisp')
[csym,lsym] = wavedec(noisdoppg,3,'db4');
[max(abs(c-csym)) max(abs(l-lsym))]

ans =

     0     0

Set the current extension mode to per and obtain the three-level DWT of noisdopp. The extension mode per is supported for gpuArray input. Confirm the result is different from the sym results.

dwtmode('per','nodisp')
[cper,lper] = wavedec(noisdoppg,3,'db4');
[length(csym) ; length(cper)]

ans = 2×1

        1044
        1024

[lsym ; lper]

ans = 2×5

         134         134         261         515        1024
         128         128         256         512        1024

Restore the extension mode to the original setting.

dwtmode(origMode,'nodisp')

Input Arguments

x — Input signal

vector

Input signal, specified as a vector.

Data Types: single | double
Complex Number Support: Yes

n — Level of decomposition

positive integer

Level of decomposition, specified as a positive integer. wavedec does not enforce a maximum level restriction. Use wmaxlev to ensure that the wavelet coefficients are free from boundary effects. If boundary effects are not a concern in your application, a good rule is to set n less than or equal to fix(log2(length(x))).

Data Types: single | double

wname — Analyzing wavelet

character vector | string scalar

Analyzing wavelet, specified as a character vector or string scalar.

Note

wavedec supports only Type 1 (orthogonal) or Type 2 (biorthogonal) wavelets. See wfilters for a list of orthogonal and biorthogonal wavelets.

LoD,HiD — Wavelet decomposition filters

even-length real-valued vectors

Wavelet decomposition filters, specified as a pair of even-length real-valued vectors. LoD is the lowpass decomposition filter, and HiD is the highpass decomposition filter. The lengths of LoD and HiD must be equal. See wfilters for additional information.

Data Types: single | double

Output Arguments

c — Wavelet decomposition vector

vector

Wavelet decomposition vector, returned as a vector. The bookkeeping vector l is used to parse the coefficients in the wavelet decomposition vector by level.

Data Types: single | double

l — Bookkeeping vector

vector of positive integers

Bookkeeping vector, returned as a vector of positive integers. The vector contains the number of coefficients by level and the length of the original signal.

The bookkeeping vector is used to parse the coefficients in the wavelet decomposition vector c by level. The decomposition vector and bookkeeping vector are organized as in this level-3 decomposition diagram.

Data Types: single | double

Algorithms

Given a signal s of length N, the DWT consists of at most log₂ N steps. Starting from s, the first step produces two sets of coefficients: approximation coefficients cA₁ and detail coefficients cD₁. Convolving s with the lowpass filter LoD and the highpass filter HiD, followed by dyadic decimation (downsampling), results in the approximation and detail coefficients respectively.

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 2n. If N = length(s), the signals F and G are of length N + 2n −1 and the coefficients cA₁ and cD₁ are of length

floor$$\left(\frac{N-1}{2}\right)+n$$.

The next step splits the approximation coefficients cA₁ in two parts using the same scheme, replacing s by cA₁, and producing cA₂ and cD₂, and so on.

The wavelet decomposition of the signal s analyzed at level j has the following structure: [cA_j, cD_j, ..., cD₁].

This structure contains, for j = 3, the terminal nodes of the following tree:

References

[1] Daubechies, I. Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: SIAM Ed, 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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The input wname must be constant.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• The input wname must be constant.

• The level of decomposition, n must be a compile-time constant.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• Only 'sym' and 'per' extension modes are supported. See dwtmode.

Version History

Introduced before R2006a