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tqwt

Tunable Q-factor wavelet transform

    Description

    example

    wt = tqwt(x) returns the tunable Q-factor wavelet transform (TQWT) of 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.

    wt = tqwt(x,Name=Value) specifies one or more additional name-value arguments. For example, wt = tqwt(x,QualityFactor=2) specifies a quality factor of 2.

    example

    [wt,info] = tqwt(___) returns the structure array, info, with information about the tunable Q-factor wavelet transform.

    Examples

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    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 ≤ inumel(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
    

    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

    Figure contains 2 axes objects. Axes object 1 with title Original Signal contains an object of type line. Axes object 2 with title Lowpass Subband Coefficients contains an object of type line.

    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 r=3. Confirm the highpass and lowpass scaling factors, info.Beta and info.Alpha respectively, satisfy the relation r=β1-α.

    info.Beta/(1-info.Alpha)
    ans = 3
    

    Input Arguments

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    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)

    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

    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

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    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 of wt 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 of wt match the column and page sizes of x.

    Data Types: single | double

    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 of x. To convert the frequencies to hertz, multiply CenterFrequencies by the sample rate.

    • Bandwidths — The approximate bandwidths of the wavelet subbands in normalized frequency (cycles/sample). To convert the bandwidths to hertz, multiply Bandwidths by the sample rate.

    • Level — Level of the TQWT. Note that info.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 as 2/(QualityFactor+1). Accordingly, 0 < Beta ≤ 1.

    • Alpha — Lowpass scaling factor. The lowpass scaling factor is computed from the highpass scaling factor as 1-Beta/3. Accordingly, 2/3 ≤ Alpha < 1.

    Data Types: struct

    More About

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    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

    log(N4Q+4)/log(3Q+33Q+1),

    where the ⌊ ⌋ symbols denote the floor function.

    The minimum level also depends on the signal length and quality factor. The logarithm of N, log(N), must satisfy the following inequality:

    log(N)log(4Q+4)log(3Q+1)+log(3Q+3).

    If log(N)<log(4Q+4)log(3Q+1)+log(3Q+3), 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

    r=β1α.

    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™.

    Introduced in R2021b