wavedec

Multilevel 1-D discrete wavelet transform

Syntax

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

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

[c,l] = wavedec(___,Mode=extmode)

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.

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

[c,l] = wavedec(___,Mode=extmode) uses the specified discrete wavelet transform (DWT) extension mode extmode. This syntax can be used with either of the previous syntaxes.

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.

tiledlayout(4,1)
nexttile
plot(approx)
title("Approximation Coefficients")
nexttile
plot(cd3)
title("Level 3 Detail Coefficients")
nexttile
plot(cd2)
title("Level 2 Detail Coefficients")
nexttile
plot(cd1)
title("Level 1 Detail Coefficients")

Multilevel Discrete Wavelet Transform on 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 global DWT extension mode.

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

Obtain the three-level DWT of the signal on the GPU using the db4 wavelet. Even though the zpd extension mode is not supported for gpuArray input, specify it. The wavedec function instead uses the sym extension mode.

[czpd,lzpd] = wavedec(noisdoppg,3,"db4",Mode="zpd");

Confirm that the wavedec function did use the sym extension mode. Set the global extension mode to sym and obtain the three-level DWT of noisdoppg without specifying an extension mode. Confirm this result equals the previous result.

dwtmode("sym","nodisp")
[csym,lsym] = wavedec(noisdoppg,3,"db4");
[max(abs(czpd-csym)) max(abs(lzpd-lsym))]

ans =

     0     0

Obtain the three-level DWT of noisdoppg using the per extension mode, which is supported for gpuArray input. Confirm this result is different from the sym result.

[cper,lper] = wavedec(noisdoppg,3,"db4",Mode="per");
[length(csym) ; length(cper)]

ans = 2×1

        1044
        1024

[lsym ; lper]

ans =

         134         134         261         515        1024
         128         128         256         512        1024

Restore the global extension mode to the original setting.

dwtmode(origMode,"nodisp")

Multiresolution Analysis Based on Decimated Discrete Wavelet Transform

This example shows how, starting with a multilevel 1-D discrete wavelet decomposition of a signal, you can obtain projections of the signal onto wavelet subspaces at successive scales and a scaling subspace. These projections are at the same time scale as the original signal. In other words, you can obtain a multiresolution analysis (MRA) based on the decimated discrete wavelet transform (DWT). You can recover the signal by summing the projections. For more information, see Practical Introduction to Multiresolution Analysis.

Load and plot a signal.

load noissin
plot(noissin)
title("Original Signal")

Use the wavedec function to obtain the discrete wavelet decomposition of the signal down to level 3 using the db4 wavelet.

level = 3;
wv = "db4";
[C,L] = wavedec(noissin,level,wv);

Preallocate a matrix to save the MRA. The number of rows is one more than the level of decomposition, and the number of columns equals the length of the signal.

mra = zeros(level+1,numel(noissin));

Use the wrcoef function to obtain the projections of the signal onto the three wavelet (detail) subspaces. Then obtain the projection onto the final scaling (coarse or approximation) subspace.

for k=1:level
    mra(k,:) = wrcoef("d",C,L,wv,k);
end
mra(end,:) = wrcoef("a",C,L,wv,level);

Confirm the sum along the rows of the MRA equals the original signal.

mraSum = sum(mra,1);
max(abs(mraSum-noissin))

ans = 
1.6591e-12

Plot the MRA.

tiledlayout(level+1,1)
for k=1:level
    nexttile
    plot(mra(k,:))
    title("Projection Onto Detail Subspace "+num2str(k))
end
nexttile
plot(mra(end,:))
title("Projection Onto Approximation Subspace")

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. For more information, see wfilters.

Data Types: single | double

extmode — Extension mode

"zpd" | "sp0" | "spd" | ...

Since R2023b

Extension mode to use for the wavelet decomposition, specified as:

extmode	DWT Extension Mode
"zpd"	Zero padding
"sp0"	Smooth extension of order 0
"spd" (or "sp1")	Smooth extension of order 1
"sym" or "symh"	Symmetric extension (half point): boundary value symmetric replication
"symw"	Symmetric extension (whole point): boundary value symmetric replication
"asym" or "asymh"	Antisymmetric extension (half point): boundary value antisymmetric replication
"asymw"	Antisymmetric extension (whole point): boundary value antisymmetric replication
"ppd", "per"	Periodized extension If the signal length is odd and mode is "per", an extra sample equal to the last value is added to the right and the extension is performed in "ppd" mode. If the signal length is even, "per" reduces to "ppd". This rule also applies to images.

The global variable managed by dwtmode specifies the default extension mode.

Output Arguments

c — Wavelet decomposition vector

vector

Wavelet decomposition vector, returned as a vector.

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 by 2), 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 11, no. 7 (July 1989): 674–93. https://doi.org/10.1109/34.192463.

[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™.

The wavedec function supports GPU array input with these usage notes and limitations:

• wavedec supports only the "sym" and "per" extension modes.

• By default, wavedec uses the "sym" extension mode unless the extension mode is "per". For more information, see the example Multilevel Discrete Wavelet Transform on GPU.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2023b: Specify DWT Extension Mode

The wavedec function can use a DWT extension mode you specify which is different from the default global extension mode.