tqwt
Description
returns the
tunable Q-factor wavelet transform (TQWT) of wt
= tqwt(x
)x
.
The TQWT is computed to the maximum decomposition level with a quality factor of 1. For more information, see TQWT Decomposition Levels.
As implemented, the
tqwt
function uses a redundancy of 3. For more information, see Redundancy.
specifies one or more additional name-value arguments. For example, wt
= tqwt(x
,Name=Value
)wt =
tqwt(x,QualityFactor=2)
specifies a quality factor of 2.
Examples
Tunable Q-factor Wavelet Transform of Multisignal
Load a multichannel EEG signal. The signal has 23 channels.
load Espiga3
size(Espiga3,2)
ans = 23
Obtain the tunable Q-factor wavelet transform of the multisignal to the maximum level using the default quality factor of 1.
wt = tqwt(Espiga3); numel(wt)
ans = 12
For 1 ≤ i ≤ numel(wt)
-1, the ith element of wt
contains the wavelet transform coefficients for the ith subband. The last element of wt
contains the lowpass subband coefficients. Confirm the number of columns of any element of wt
is equal to the number of channels.
k = 7; size(wt{k},2)
ans = 23
Reconstruct the multisignal and demonstrate perfect reconstruction.
xrec = itqwt(wt,size(Espiga3,1)); max(abs(xrec(:)-Espiga3(:)))
ans = 5.1159e-13
Inspect TQWT Information Structure
Load an ECG signal. Obtain the TQWT of the signal down to level 5 with a quality factor of 2. Also obtain information of the TQWT.
load wecg
lvl = 5;
qf = 2;
[wt,info] = tqwt(wecg,Level=lvl,QualityFactor=qf);
Plot the original signal and compare with the lowpass subband coefficients.
subplot(2,1,1) plot(wecg) title("Original Signal") axis tight subplot(2,1,2) plot(wt{end}) title("Lowpass Subband Coefficients") axis tight
Inspect the TQWT information structure. For each subband, confirm that the ratio of the center frequency to the approximate bandwidth equals the quality factor.
info
info = struct with fields:
CenterFrequencies: [0.3333 0.2593 0.2016 0.1568 0.1220]
Bandwidths: [0.1667 0.1296 0.1008 0.0784 0.0610]
Level: 5
Alpha: 0.7778
Beta: 0.6667
info.CenterFrequencies./info.Bandwidths
ans = 1×5
2.0000 2.0000 2.0000 2.0000 2.0000
As implemented, the tqwt
function uses the redundancy . Confirm the highpass and lowpass scaling factors, info.Beta
and info.Alpha
respectively, satisfy the relation .
info.Beta/(1-info.Alpha)
ans = 3
Input Arguments
x
— Input signal
vector | matrix | 3-D array
Input signal, specified as a single- or double-precision vector, matrix, or 3-D
array. If x
is a matrix or 3-D array, the TQWT is computed along
the columns of x
. For 3-D arrays, tqwt
interprets the first dimension as time, the second dimension as channels, and the third
dimension as a batch.
The TQWT is defined for even-length signals. If the number of samples in
x
is odd, the last sample of x
is repeated
to obtain an even-length signal.
Data Types: single
| double
Complex Number Support: Yes
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: wt = tqwt(x,Level=3,QualityFactor=2)
Level
— Decomposition level
positive integer
Decomposition level of the TQWT, specified as a positive integer between 1 and the maximum level. The maximum level depends on the signal length and quality factor. For more information, see TQWT Decomposition Levels.
Example: wt = tqwt(x,Level=3)
specifies a decomposition level of
3.
Data Types: single
| double
QualityFactor
— Quality factor
1
(default) | positive scalar
Quality factor, specified as a real-valued scalar greater than or equal to 1. The quality factor is the ratio of the center frequency to the bandwidth of the filters. If unspecified, the quality factor defaults to 1.
Example: wt = tqwt(x,QualityFactor=1.5)
specifies a quality
factor of 1.5.
Data Types: single
| double
Output Arguments
wt
— Tunable Q-factor wavelet transform
cell array
Tunable Q-factor wavelet transform, returned as a cell array.
wt
is a cell array with length equal to the maximum level of the
TQWT plus one. The ith element of wt
contains
the TQWT coefficients for the ith subband. The subbands are ordered
by decreasing center frequency. The final element of wt
contains
the lowpass subband coefficients. The wavelet coefficients in wt
match x
in data type and complexity.
If
x
is a row vector, each element ofwt
is a column vector containing the TQWT coefficients.If
x
is a matrix or 3-D array, the column and page sizes of each element ofwt
match the column and page sizes ofx
.
Data Types: single
| double
info
— Transform information
structure array
Transform information, returned as a structure array. info
has
five fields:
CenterFrequencies
— The normalized center frequencies (cycles/sample) of the wavelet subbands in the TQWT ofx
. To convert the frequencies to hertz, multiplyCenterFrequencies
by the sample rate.Bandwidths
— The approximate bandwidths of the wavelet subbands in normalized frequency (cycles/sample). To convert the bandwidths to hertz, multiplyBandwidths
by the sample rate.Level
— Level of the TQWT. Note thatinfo.Level
may be different from your specified level if you specify a level greater than the maximum supported level for your signal length and quality factor.Beta
— Highpass scaling factor. The highpass scaling factor is computed from the quality factor as2/(
. Accordingly,QualityFactor
+1)0 < Beta ≤ 1
.Alpha
— Lowpass scaling factor. The lowpass scaling factor is computed from the highpass scaling factor as1-Beta/3
. Accordingly,2/3 ≤ Alpha < 1
.
Data Types: struct
More About
TQWT Decomposition Levels
The TQWT minimum and maximum decomposition levels depend on the signal length, N, and quality factor, Q. In the description that follows, the signal length, N, is one sample larger than the input length for odd-length signals.
The maximum decomposition level is
where the ⌊ ⌋ symbols denote the floor function.
The minimum level also depends on the signal length and quality factor. The logarithm of N, must satisfy the following inequality:
If the maximum level is less than 1 and tqwt
throws an
error.
Redundancy
The TQWT algorithm depends on scaling in the frequency domain:
lowpass scaling — frequency-domain scaling by α that preserves low-frequency content
highpass scaling — frequency-domain scaling by β that preserves high-frequency content
The redundancy is defined to be
For more information, see Tunable Q-factor Wavelet Transform.
References
[1] Selesnick, Ivan W. “Wavelet Transform With Tunable Q-Factor.” IEEE Transactions on Signal Processing 59, no. 8 (August 2011): 3560–75. https://doi.org/10.1109/TSP.2011.2143711.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The tqwt
function
supports GPU array input with these usage notes and limitations:
The signal
x
is the only supported input argument.The TQWT array
wt
is the only supported output argument.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2021b
See Also
Apps
Functions
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