Maximal overlap discrete wavelet packet transform details
the maximal overlap discrete wavelet packet transform (MODWPT) details
for the 1-D real-valued signal,
w = modwptdetails(
x. The MODWPT
details provide zero-phase filtering of the signal. By default,
only the terminal nodes, which are at level 4 or at level
whichever is smaller.
To decide whether to use
modwpt, consider the type of data analysis
you need to perform. For applications that require time alignment,
such as nonparametric regression analysis, use
For applications where you want to analyze the energy levels in different
modwpt. For more information, see Algorithms
[___] = modwptdetails(___,'FullTree',,
false, returns details about
only the terminal (final-level) wavelet packet nodes. If you specify
details about the full wavelet packet tree down to the default or specified
level. The default for
Obtain the MODWPT of an electrocardiogram (ECG) signal using the default length 18 Fejer-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 , which demonstrates perfect reconstruction.
xrec = sum(wptdetails); max(abs(wecg-xrec'))
ans = 1.7903e-11
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);
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')
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 Fejer-Korovkin (
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 , where 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')
Obtain the MODWPT details of an ECG waveform using the length 18 Fejer-Korovkin scaling and wavelet filters.
load wecg; [lo,hi] = wfilters('fk18'); wpt = modwptdetails(wecg,lo,hi);
Obtain the MODWPT details for the full wavelet packet tree of an ECG waveform. Use the default length 18 Fejer-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')
x— Input signal
Input signal, specified as a real-valued row or column vector.
have at least two elements.
wname— Analyzing wavelet filter
Analyzing wavelet, specified as one of the following:
'haar' — Haar wavelet
Extremal phase Daubechies wavelet with N
vanishing moments, where N is a positive
integer from 1 to 45.
Symlets wavelet with N vanishing moments,
where N is a positive integer from 2 to
Coiflets wavelet with N vanishing moments,
where N is a positive integer from 1 to
Fejér-Korovkin wavelet with N coefficients,
N = 4, 6, 8, 14, 18
lev— Transform level
Transform level, specified as a positive integer less than or
tf— Return tree option
Return tree option, specified as
returns details about only the terminal (final-level) wavelet packet nodes.
If you specify
modwptdetails returns details about the full
wavelet packet tree down to the default or specified level.
w— Wavelet packet tree details
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
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 2j-by-
matrix. For the full packet table, at level j,
matrix of sequency-ordered wavelet packet coefficients by level and
index. The approximate passband for the nth row
w at level j is cycles
per sample, where n = 1,2,...,2j.
packetlevs— Transform levels
Transform levels, returned as a vector. The levels correspond
to the rows of
only the terminal level coefficients,
a vector of constants equal to the terminal level. If
the full wavelet packet tree of details,
a vector with 2j-1 elements
for each level, j. To select all the MODWPT details
at a particular level, use
packetlevs with logical
cfreq— Center frequencies of passbands
Center frequencies of the approximate passbands in the
returned as a vector. The center frequencies are in cycles per sample.
To convert the units to cycles per unit time, multiply
the sampling frequency.
The MODWPT details (
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 (
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.
 Percival, D. B., and A. T. Walden. Wavelet Methods for Time Series Analysis. Cambridge, UK: Cambridge University Press, 2000.
 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 A. Vol. 454, Issue 1976, 1998, pp. 2243-2266.