modwptdetails

Maximal overlap discrete wavelet packet transform details

Syntax

w = modwptdetails(x)

w = modwptdetails(x,wname)

w = modwptdetails(x,lo,hi)

w = modwptdetails(___,lev)

[w,packetlevs] = modwptdetails(___)

[w,packetlevs,cfreq] = modwptdetails(___)

[___] = modwptdetails(___,'FullTree',tf)

Description

w = modwptdetails(x) returns the maximal overlap discrete wavelet packet transform (MODWPT) details for the 1-D real-valued signal, x. The MODWPT details provide zero-phase filtering of the signal. By default, modwptdetails returns only the terminal nodes, which are at level 4 or at level floor(log2(numel(x))), whichever is smaller.

Note

To decide whether to use modwptdetails or modwpt, consider the type of data analysis you need to perform. For applications that require time alignment, such as nonparametric regression analysis, use modwptdetails. For applications where you want to analyze the energy levels in different packets, use modwpt. For more information, see Algorithms.

w = modwptdetails(x,wname) uses the orthogonal wavelet specified by wname.

w = modwptdetails(x,lo,hi) uses the orthogonal scaling filter, lo, and wavelet filter, hi.

w = modwptdetails(___,lev) returns the terminal nodes of the wavelet packet tree at positive integer level lev.

[w,packetlevs] = modwptdetails(___) returns a vector of transform levels corresponding to the rows of w.

[w,packetlevs,cfreq] = modwptdetails(___) returns cfreq, the center frequencies of the approximate passbands corresponding to the MODWPT details in w.

[___] = modwptdetails(___,'FullTree',tf), where tf is false, returns details about only the terminal (final-level) wavelet packet nodes. If you specify true, then modwptdetails returns details about the full wavelet packet tree down to the default or specified level. The default for tf is false.

Examples

MODWPT Details Using Default Wavelet

Obtain the MODWPT of an electrocardiogram (ECG) signal using the default length 18 Fejér-Korovkin ('fk18') wavelet and the default level, 4.

load wecg;
wptdetails = modwptdetails(wecg);

Demonstrate that summing the MODWPT details over each sample reconstructs the signal. The largest absolute difference between the original signal and the reconstruction is on the order of $$10^{-11}$$, which demonstrates perfect reconstruction.

xrec = sum(wptdetails);
max(abs(wecg-xrec'))

ans = 1.7903e-11

MODWPT Details for Two Sine Waves

Obtain the MODWPT details for a signal containing 100 Hz and 450 Hz sine waves. Each row of the modwptdetails output corresponds to a separate frequency band.

dt = 0.001;
fs = 1/dt;
t = 0:dt:1;
x = (sin(2*pi*100*t)+sin(2*pi*450*t));
[lo,hi] = wfilters('fk22');
wptdetails = modwptdetails(x,lo,hi);

Use modwpt to obtain the energy and center frequencies of the signal. Plot the energy in the wavelet packets. The fourth and fifteenth frequency bands contain most of the energy. Other frequency bands have significantly less energy. The frequency ranges of fourth and fifteenth bands are approximately 94-125 Hz and 438-469 Hz, respectively.

[wpt,~,cfreqs,energy] = modwpt(x,lo,hi);
figure
bar(1:16,energy);
xlabel('Packet')
ylabel('Packet Energy')
title('Energy by Wavelet Packet')

Plot the power spectral density of the input signal.

pwelch(x,[],[],[],fs,'onesided');
title('Power Spectral Density of Input Signal')

Show that the MODWPT details have zero-phase shift from the 100 Hz input sine.

p4 = wptdetails(4,:);
plot(t,sin(2*pi*100*t).*(t>0.3 & t<0.7))
hold on
plot(t,p4.*(t>0.3 & t<0.7),'r')
legend('Sine Wave','MODWPT Details')

MODWPT Details for Noisy Sine Wave

Obtain the MODWPT details for a 100 Hz time-localized sine wave in noise. The sampling rate is 1000 Hz. Obtain the MODWPT at level 4 using the length 22 Fejér-Korovkin ('fk22') wavelet.

dt = 0.001;     
t = 0:dt:1;
x = cos(2*pi*100*t).*(t>0.3 & t<0.7)+0.25*randn(size(t));
wptdetails = modwptdetails(x,'fk22');
p4 = wptdetails(4,:);

Plot the MODWPT details for level 4, packet number 4. The MODWPT details represent zero-phase filtering of the input signal with an approximate passband of $$[3Fs/2^5,4Fs/2^5)$$, where $$Fs$$ is the sampling frequency.

plot(t,cos(2*pi*100*t).*(t>0.3 & t<0.7));
hold on
plot(t,p4,'r')
legend('Sine Wave','MODWPT Details')
hold off

MODWPT Details Using Scaling and Wavelet Filters

Obtain the MODWPT details of an ECG waveform using the length 18 Fejér-Korovkin scaling and wavelet filters.

load wecg; 
[lo,hi] = wfilters('fk18');
wpt = modwptdetails(wecg,lo,hi);

MODWPT Details for Full Packet Tree

Obtain the MODWPT details for the full wavelet packet tree of an ECG waveform. Use the default length 18 Fejér-Korovkin ('fk18') wavelet. Extract and plot the node coefficients at level 3, node 2.

load wecg;     
[w,packetlevels] = modwptdetails(wecg,'FullTree',true);
p3 = w(packetlevels==3,:);
plot(p3(3,:))
title('Level 3, Node 2 MODWPT Details')

Input Arguments

x — Input signal

real-valued vector

Input signal, specified as a real-valued row or column vector. x must have at least two elements.

Data Types: single | double

wname — Analyzing wavelet

"fk18" (default) | character vector | string scalar

Analyzing wavelet, specified as a character vector or string scalar. The wavelet must be orthogonal. Orthogonal wavelets are designated as type 1 wavelets in the wavelet manager, wavemngr.

Valid built-in orthogonal wavelet families are: Best-localized Daubechies ("bl"), Beylkin ("beyl"), Coiflets ("coif"), Daubechies ("db"), Fejér-Korovkin ("fk"), Haar ("haar"), Han linear-phase moments ("han"), Morris minimum-bandwidth ("mb"), Symlets ("sym"), and Vaidyanathan ("vaid").

For a list of wavelets in each family, see wfilters. You can also use waveinfo with the wavelet family short name. For example, waveinfo("db"). Use wavemngr("type",wn) to determine if the wavelet wn is orthogonal (returns 1). For example, wavemngr("type","db6") returns 1.

lo,hi — Filters

even-length real-valued vectors

Filters, specified as a pair of even-length real-valued vectors. lo is the orthogonal scaling filter and hi is the orthogonal wavelet filter. The filters must satisfy the conditions for an orthogonal wavelet. For more information, see wfilters and isorthwfb. You cannot specify both wname and a filter pair lo,hi.

Note

By default, the wfilters function returns two pairs of filters associated with an orthogonal or biorthogonal wavelet you specify. To agree with the usual convention in the implementation of MODWPT in numerical packages, when you specify an orthogonal wavelet wname, the modwptdetails function internally uses the second pair of filters returned by wfilters. For example,

wptdetails = modwptdetails(x,"db2");

is equivalent to

[~,~,lo,hi] = wfilters("db2"); wptdetails = modwptdetails(x,lo,hi);

This convention is different from the one followed by most Wavelet Toolbox™ discrete wavelet transform functions when decomposing a signal. Most functions internally use the first pair of filters.

Data Types: single | double

lev — Transform level

positive integer

Transform level, specified as a positive integer less than or equal to floor(log2(numel(x))).

tf — Return tree option

false (default) | true

Return tree option, specified as false or true. If tf is false, then modwptdetails returns details about only the terminal (final-level) wavelet packet nodes. If you specify true, then modwptdetails returns details about the full wavelet packet tree down to the default or specified level.

For the full wavelet packet tree, w is a 2^j+1-2-by-numel(x) matrix. Each level j has 2^j wavelet packet details.

Output Arguments

w — Wavelet packet tree details

matrix

Wavelet packet tree details, returned as a matrix with each row containing the sequency-ordered wavelet packet details for the terminal nodes. The terminal nodes are at level 4 or at level floor(log2(numel(x))), whichever is smaller. The MODWPT details are zero-phase-filtered projections of the signal onto the subspaces corresponding to the wavelet packet nodes. The sum of the MODWPT details over each sample reconstructs the original signal.

For the default terminal nodes, w is a 2^j-by-numel(x) matrix. For the full packet table, at level j, w is a 2^j+1-2-by-numel(x) matrix of sequency-ordered wavelet packet coefficients by level and index. The approximate passband for the nth row of w at level j is $$\left[\frac{n-1}{2^{\left(j+1\right)}},\frac{n}{2^{\left(j+1\right)}}\right)$$ cycles per sample, where n = 1,2,...,2^j.

packetlevs — Transform levels

vector

Transform levels, returned as a vector. The levels correspond to the rows of w. If w contains only the terminal level coefficients, packetlevs is a vector of constants equal to the terminal level. If w contains the full wavelet packet tree of details, packetlevs is a vector with 2^j-1 elements for each level, j. To select all the MODWPT details at a particular level, use packetlevs with logical indexing.

cfreq — Center frequencies of passbands

vector

Center frequencies of the approximate passbands in the w rows, returned as a vector. The center frequencies are in cycles per sample. To convert the units to cycles per unit time, multiply cfreq by the sampling frequency.

Algorithms

The MODWPT details (modwptdetails) are the result of zero-phase filtering of the signal. The features in the MODWPT details align exactly with features in the input signal. For a given level, summing the details for each sample returns the exact original signal.

The output of the MODWPT (modwpt) is time delayed compared to the input signal. Most filters used to obtain the MODWPT have a nonlinear phase response, which makes compensating for the time delay difficult. All orthogonal scaling and wavelet filters have this response, except the Haar wavelet. It is possible to time align the coefficients with the signal features, but the result is an approximation, not an exact alignment with the original signal. The MODWPT partitions the energy among the wavelet packets at each level. The sum of the energy over all the packets equals the total energy of the input signal.

References

[1] Percival, Donald B., and Andrew T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge ; New York: Cambridge University Press, 2000.

[2] Walden, A. T., and A. Contreras Cristan. “The Phase–Corrected Undecimated Discrete Wavelet Packet Transform and Its Application to Interpreting the Timing of Events.” Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, no. 1976 (August 8, 1998): 2243–66. https://doi.org/10.1098/rspa.1998.0257.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• The input wname must be constant.

Version History

Introduced in R2016a

R2023a: Supports single-precision data

The modwptdetails function supports single-precision data.