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Contents


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 LTFAT - Non-stationary Gabor systems

  Florent Jaillet and Peter L. Soendergaard, 2011 - 2023.

  Transforms
    NSDGT                - Non-stationary DGT
    UNSDGT               - Uniform non-stationary DGT
    INSDGT               - Inverse NSDGT and UNSDGT
    NSDGTREAL            - Non-stationary DGT for real-valued signals
    UNSDGTREAL           - Uniform non-stationary DGT for real-valued signals
    INSDGTREAL           - Inverse NSDGTREAL and UNSDGTREAL

  Window construction and bounds
    NSGABDUAL            - Non-stationary dual windows
    NSGABTIGHT           - Non-stationary tight windows
    NSGABFRAMEBOUNDS     - Frame bounds of an NSDGT system
    NSGABFRAMEDIAG       - Diagonal of non-stationary Gabor frame operator

  Plots
    PLOTNSDGT            - Plot output coefficients from NSDGT
    PLOTNSDGTREAL        - Plot output coefficients from NSDGTREAL

  For help, bug reports, suggestions etc. please visit 
  http://github.com/ltfat/ltfat/issues

   Url: http://ltfat.github.io/doc/nonstatgab/Contents.html



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 LTFAT - Non-stationary Gabor systems



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insdgt


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 -- Function: insdgt
     INSDGT  Inverse non-stationary discrete Gabor transform
        Usage:  f=insdgt(c,g,a,Ls);
     
        Input parameters:
              c     : Cell array of coefficients.
              g     : Cell array of window functions.
              a     : Vector of time positions of windows.
              Ls    : Length of input signal.
        Output parameters:
              f     : Signal.
     
        INSDGT(c,g,a,Ls) computes the inverse non-stationary Gabor transform
        of the input coefficients c.
     
        INSDGT is used to invert the functions NSDGT and UNSDGT. Please
        read the help of these functions for details of variables format and
        usage.
     
        For perfect reconstruction, the windows used must be dual windows of the
        ones used to generate the coefficients. The windows can be generated
        using NSGABDUAL or NSGABTIGHT.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/insdgt.html>

     See also: nsdgt, nsgabdual, nsgabtight, demo_nsdgt.


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INSDGT  Inverse non-stationary discrete Gabor transform
   Usage:  f=insdgt(c...



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insdgtreal


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 -- Function: insdgtreal
     INSDGTREAL  Inverse NSDGT for real-valued signals
        Usage:  f=insdgt(c,g,a,M,Ls);
     
        Input parameters:
              c     : Cell array of coefficients.
              g     : Cell array of window functions.
              a     : Vector of time positions of windows.
              M     : Vector of numbers of frequency channels.
              Ls    : Length of input signal.
        Output parameters:
              f     : Signal.
     
        insdgt(c,g,a,Ls) computes the inverse non-stationary Gabor transform
        of the input coefficients c.
     
        insdgt is used to invert the functions NSDGT and UNSDGT. Please
        read the help of these functions for details of variables format and
        usage.
     
        For perfect reconstruction, the windows used must be dual windows of the
        ones used to generate the coefficients. The windows can be generated
        using NSGABDUAL or NSGABTIGHT.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/insdgtreal.html>

     See also: nsdgt, nsgabdual, nsgabtight, demo_nsdgt.


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INSDGTREAL  Inverse NSDGT for real-valued signals
   Usage:  f=insdgt(c,g,a,M...



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nsdgt


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 -- Function: nsdgt
     NSDGT  Non-stationary Discrete Gabor transform
        Usage:  c=nsdgt(f,g,a,M);
                [c,Ls]=nsdgt(f,g,a,M);
     
        Input parameters:
              f     : Input signal.
              g     : Cell array of window functions.
              a     : Vector of time shifts.
              M     : Vector of numbers of frequency channels.
        Output parameters:
              c     : Cell array of coefficients.
              Ls    : Length of input signal.
     
        NSDGT(f,g,a,M) computes the non-stationary Gabor coefficients of the
        input signal f. The signal f can be a multichannel signal, given in
        the form of a 2D matrix of size Ls xW, with Ls the signal
        length and W the number of signal channels.
     
        The non-stationary Gabor theory extends standard Gabor theory by
        enabling the evolution of the window over time. It is therefor necessary
        to specify a set of windows instead of a single window.  This is done by
        using a cell array for g. In this cell array, the n'th element g{n}
        is a row vector specifying the n'th window.
     
        The resulting coefficients also require a storage in a cell array, as
        the number of frequency channels is not constant over time. More
        precisely, the n'th cell of c, c{n}, is a 2D matrix of size 
        M(n) xW and containing the complex local spectra of the signal channels
        windowed by the n'th window g{n} shifted in time at position a(n).
        c{n}(m,w) is thus the value of the coefficient for time index n,
        frequency index m and signal channel w.
     
        The variable a contains the distance in samples between two
        consequtive blocks of coefficients. The variable M contains the
        number of channels for each block of coefficients. Both a and M are
        vectors of integers.
     
        The variables g, a and M must have the same length, and the result c*
        will also have the same length.
        
        The time positions of the coefficients blocks can be obtained by the
        following code. A value of 0 correspond to the first sample of the
        signal:
     
          timepos = cumsum(a)-a(1);
     
        [c,Ls]=NSDGT(f,g,a,M) additionally returns the length Ls of the input 
        signal f. This is handy for reconstruction:
     
          [c,Ls]=nsdgt(f,g,a,M);
          fr=insdgt(c,gd,a,Ls);
     
        will reconstruct the signal f no matter what the length of f is, 
        provided that gd are dual windows of g.
     
        Notes:
        ------
     
        NSDGT uses circular border conditions, that is to say that the signal is
        considered as periodic for windows overlapping the beginning or the 
        end of the signal.
     
        The phaselocking convention used in NSDGT is different from the
        convention used in the DGT function. NSDGT results are phaselocked (a
        phase reference moving with the window is used), whereas DGT results are
        not phaselocked (a fixed phase reference corresponding to time 0 of the
        signal is used). See the help on PHASELOCK for more details on
        phaselocking conventions.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsdgt.html>

     See also: insdgt, nsgabdual, nsgabtight, phaselock, demo_nsdgt.


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NSDGT  Non-stationary Discrete Gabor transform
   Usage:  c=nsdgt(f,g,a,M);
 ...



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nsdgtlength


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 -- Function: nsdgtlength
     NSDGTLENGTH  NSDGT length from signal
        Usage: L=nsdgtlength(Ls,a);
     
        NSDGTLENGTH(Ls,a) returns the length of an NSDGT with time shifts
        a, such that it is long enough to expand a
        signal of length Ls.
     
        If the returned length is longer than the signal length, the signal
        will be zero-padded by NSDGT or UNSDGT.
     
        If instead a set of coefficients are given, call NSDGTLENGTHCOEF.
     
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsdgtlength.html>

     See also: nsdgt, nsdgtlengthcoef.


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NSDGTLENGTH  NSDGT length from signal
   Usage: L=nsdgtlength(Ls,a);



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nsdgtreal


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 -- Function: nsdgtreal
     NSDGTREAL  Non-stationary Discrete Gabor transform for real valued signals
        Usage:  c=nsdgtreal(f,g,a,M);
                [c,Ls]=nsdgtreal(f,g,a,M);
     
        Input parameters:
              f     : Input signal.
              g     : Cell array of window functions.
              a     : Vector of time positions of windows.
              M     : Vector of numbers of frequency channels.
        Output parameters:
              c     : Cell array of coefficients.
              Ls    : Length of input signal.
     
        NSDGTREAL(f,g,a,M) computes the non-stationary Gabor coefficients of the
        input signal f. The signal f can be a multichannel signal, given in
        the form of a 2D matrix of size Ls xW, with Ls the signal
        length and W the number of signal channels.
     
        As opposed to NSDGT only the coefficients of the positive frequencies
        of the output are returned. NSDGTREAL will refuse to work for complex
        valued input signals.
     
        The non-stationary Gabor theory extends standard Gabor theory by
        enabling the evolution of the window over time. It is therefor necessary
        to specify a set of windows instead of a single window.  This is done by
        using a cell array for g. In this cell array, the n'th element g{n}
        is a row vector specifying the n'th window.
     
        The resulting coefficients also require a storage in a cell array, as
        the number of frequency channels is not constant over time. More
        precisely, the n'th cell of c, c{n}, is a 2D matrix of size
        M(n)/2+1 xW and containing the complex local spectra of the
        signal channels windowed by the n'th window g{n} shifted in time at
        position a(n).  c{n}(m,l) is thus the value of the coefficient for
        time index n, frequency index m and signal channel l.
     
        The variable a contains the distance in samples between two
        consequtive blocks of coefficients. The variable M contains the
        number of channels for each block of coefficients. Both a and M are
        vectors of integers.
     
        The variables g, a and M must have the same length, and the result c*
        will also have the same length.
        
        The time positions of the coefficients blocks can be obtained by the
        following code. A value of 0 correspond to the first sample of the
        signal:
     
          timepos = cumsum(a)-a(1);
     
        [c,Ls]=NSDGTREAL(f,g,a,M) additionally returns the length Ls of the input 
        signal f. This is handy for reconstruction:
     
          [c,Ls]=nsdgtreal(f,g,a,M);
          fr=insdgtreal(c,gd,a,Ls);
     
        will reconstruct the signal f no matter what the length of f is, 
        provided that gd are dual windows of g.
     
        Notes:
        ------
     
        NSDGTREAL uses circular border conditions, that is to say that the signal is
        considered as periodic for windows overlapping the beginning or the 
        end of the signal.
     
        The phaselocking convention used in NSDGTREAL is different from the
        convention used in the DGT function. NSDGTREAL results are phaselocked (a
        phase reference moving with the window is used), whereas DGT results are
        not phaselocked (a fixed phase reference corresponding to time 0 of the
        signal is used). See the help on PHASELOCK for more details on
        phaselocking conventions.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsdgtreal.html>

     See also: nsdgt, insdgtreal, nsgabdual, nsgabtight, phaselock,
     demo_nsdgt.


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NSDGTREAL  Non-stationary Discrete Gabor transform for real valued signals
  ...



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nsgabdual


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 -- Function: nsgabdual
     NSGABDUAL  Canonical dual window for non-stationary Gabor frames
        Usage:  gd=nsgabdual(g,a,M);
                gd=nsgabdual(g,a,M,L);
     
        Input parameters:
              g     : Cell array of windows.
              a     : Vector of time shift.
              M     : Vector of numbers of channels.
              L     : Transform length.
        Output parameters:
              gd : Cell array of canonical dual windows
     
        NSGABDUAL(g,a,M,L) computes the canonical dual windows of the 
        non-stationary discrete Gabor frame defined by windows given in g an
        time-shifts given by a.
        
        NSGABDUAL is designed to be used with the functions NSDGT and
        INSDGT.  See the help on NSDGT for more details about the variables
        structure.
     
        The computed dual windows are only valid for the 'painless case', that
        is to say that they ensure perfect reconstruction only if for each 
        window the number of frequency channels used for computation of NSDGT is
        greater than or equal to the window length. This correspond to cases
        for which the frame operator is diagonal.
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsgabdual.html>

     See also: nsgabtight, nsdgt, insdgt.


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NSGABDUAL  Canonical dual window for non-stationary Gabor frames
   Usage:  g...



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nsgabframebounds


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 -- Function: nsgabframebounds
     NSGABFRAMEBOUNDS  Frame bounds of non-stationary Gabor frame
        Usage:  fcond=nsgabframebounds(g,a,M);
                [A,B]=nsgabframebounds(g,a,M);
     
        Input parameters:
              g     : Cell array of windows
              a     : Vector of time positions of windows.
              M     : Vector of numbers of frequency channels.
        Output parameters:
              fcond : Frame condition number (B/A)
              A,B   : Frame bounds.
     
        NSGABFRAMEBOUNDS(g,a,Ls) calculates the ratio B/A of the frame
        bounds of the non-stationary discrete Gabor frame defined by windows
        given in g at positions given by a. Please see the help on NSDGT
        for a more thourough description of g and a.
     
        [A,B]=NSGABFRAMEBOUNDS(g,a,Ls) returns the actual frame bounds A*
        and B instead of just the their ratio.
     
        The computed frame bounds are only valid for the 'painless case' when
        the number of frequency channels used for computation of NSDGT is greater
        than or equal to the window length. This correspond to cases for which
        the frame operator is diagonal.
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*:
     <http://ltfat.github.io/doc/nonstatgab/nsgabframebounds.html>

     See also: nsgabtight, nsdgt, insdgt.


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NSGABFRAMEBOUNDS  Frame bounds of non-stationary Gabor frame
   Usage:  fcond...



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nsgabframediag


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 -- Function: nsgabframediag
     NSGABFRAMEDIAG  Diagonal of Gabor frame operator
        Usage:  d=nsgabframediag(g,a,M);
     
        Input parameters:
              g     : Window function.
              a     : Length of time shift.
              M     : Number of channels.
        Output parameters:
              d     : Diagonal stored as a column vector
     
        NSGABFRAMEDIAG(g,a,M) computes the diagonal of the non-stationary
        Gabor frame operator with respect to the window g and parameters a*
        and M. The diagonal is stored as a column vector of length L=sum(a).
     
        The diagonal of the frame operator can for instance be used as a
        preconditioner.
     
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsgabframediag.html>

     See also: nsdgt.


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NSGABFRAMEDIAG  Diagonal of Gabor frame operator
   Usage:  d=nsgabframediag(...



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nsgabtight


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 -- Function: nsgabtight
     NSGABTIGHT  Canonical tight window for non-stationary Gabor frames
        Usage:  gt=nsgabtight(g,a,M);
                gt=nsgabtight(g,a,M,L);
     
        Input parameters:
              g     : Cell array of windows
              a     : Vector of time shifts of windows.
              M     : Vector of numbers of channels.
              L     : Transform length.
        Output parameters:
              gt : Cell array of canonical tight windows
     
        NSGABTIGHT(g,a,M) computes the canonical tight windows of the 
        non-stationary discrete Gabor frame defined by windows given in g and  
        time-shifts given by a.
        
        NSGABTIGHT is designed to be used with functions NSDGT and
        INSDGT.  Read the help on NSDGT for more details about the variables
        structure.
     
        The computed tight windows are only valid for the 'painless case', that
        is to say that they ensure perfect reconstruction only if for each 
        window the number of frequency channels used for computation of NSDGT is
        greater than or equal to the window length. This correspond to cases
        for which the frame operator is diagonal.
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsgabtight.html>

     See also: nsgabtight, nsdgt, insdgt.


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NSGABTIGHT  Canonical tight window for non-stationary Gabor frames
   Usage: ...



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nsgabwin


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 -- Function: nsgabwin
     NSGABWIN  Compute a set of non-stationary Gabor windows from text or cell array
        Usage: [g,info] = nsgabwin(g,a,M);
     
        [g,info]=NSGABWIN(g,a,M,L) computes a window that fits well with
        time-shift a, number of channels M and and transform length L.
        The window itself is as a cell array containing additional parameters.
     
        The window can be specified directly as a cell array of vectors of
        numerical values. In this case, NSGABWIN only checks assumptions
        about transform sizes etc.
     
        [g,info]=NSGABWIN(g,a,M) does the same, but the windows must be FIR
        windows, as the transform length is unspecified.
     
        The window can also be specified as cell array. The possibilities are:
     
          {'dual',...}
              Canonical dual window of whatever follows. See the examples below.
     
          {'tight',...}
              Canonical tight window of whatever follows. See the examples below.
     
        The structure info provides some information about the computed
        window:
     
          info.gauss    True if the windows are Gaussian.
     
          info.tfr      Time/frequency support ratios of the window. 
                        Set whenever it makes sense.
     
          info.isfir    Input is an FIR window
     
          info.isdual   Output is the dual window of the auxiliary window.
     
          info.istight  Output is known to be a tight window.
     
          info.auxinfo  Info about auxiliary window.
        
          info.gl       Length of windows.
     
          info.isfac    True if the frame generated by the window has a fast
                        factorization.
     
     *Url*: <http://ltfat.github.io/doc/nonstatgab/nsgabwin.html>

     See also: filterbank, filterbankdual, filterbankrealdual.


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NSGABWIN  Compute a set of non-stationary Gabor windows from text or cell arr...



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plotnsdgt


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 -- Function: plotnsdgt
     PLOTNSDGT Plot non-stationary Gabor coefficients
        Usage:  plotnsdgt(c,a,fs,dynrange);
     
        Input parameters:
              coef     : Cell array of coefficients.
              a        : Vector of time positions of windows.
              fs       : signal sample rate in Hz (optional)
              dynrange : Color scale dynamic range in dB (optional).
     
        PLOTNSDGT(coef,a) plots coefficients computed using NSDGT or
        UNSDGT. For more details on the format of the variables coef and a,
        please read the function help for these functions.
     
        PLOTNSDGT(coef,a,fs) does the same assuming a sampling rate of
        fs Hz of the original signal.
     
        PLOTNSDGT(coef,a,fs,dynrange) additionally limits the dynamic range.
     
        C=PLOTNSDGT(...) returns the processed image data used in the
        plotting. Inputting this data directly to imagesc or similar
        functions will create the plot. This is useful for custom
        post-processing of the image data.
     
        PLOTNSDGT supports all the optional parameters of TFPLOT. Please
        see the help of TFPLOT for an exhaustive list. In addition, the
        following parameters may be specified:
     
          'xres',xres  Approximate number of pixels along x-axis / time.
                       The default value is 800
     
          'yres',yres  Approximate number of pixels along y-axis / frequency
                       The default value is 600
     
     *Url*: <http://ltfat.github.io/doc/nonstatgab/plotnsdgt.html>

     See also: tfplot, nsdgt, unsdgt, nsdgtreal.


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PLOTNSDGT Plot non-stationary Gabor coefficients
   Usage:  plotnsdgt(c,a,fs,...



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plotnsdgtreal


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 -- Function: plotnsdgtreal
     PLOTNSDGTREAL Plot NSDGTREAL coefficients
        Usage:  plotnsdgtreal(c,a,fs,dynrange);
     
        Input parameters:
              coef     : Cell array of coefficients.
              a        : Vector of time positions of windows.
              fs       : signal sample rate in Hz (optional).
              dynrange : Colorscale dynamic range in dB (optional).
     
        PLOTNSDGTREAL(coef,a) plots coefficients computed using NSDGTREAL or
        UNSDGTREAL. For more details on the format of the variables coef and a,
        please read the function help for these functions.
     
        PLOTNSDGTREAL(coef,a,fs) does the same assuming a sampling rate of
        fs Hz of the original signal.
     
        PLOTNSDGTREAL(coef,a,fs,dynrange) additionally limits the dynamic range.
     
        C=PLOTNSDGTREAL(...) returns the processed image data used in the
        plotting. Inputting this data directly to imagesc or similar
        functions will create the plot. This is useful for custom
        post-processing of the image data.
     
        PLOTNSDGTREAL supports all the optional parameters of TFPLOT. Please
        see the help of TFPLOT for an exhaustive list. In addition, the
        following parameters may be specified:
     
          'xres',xres  Approximate number of pixels along x-axis /time.
                       Default value is 800
     
          'yres',yres  Approximate number of pixels along y-axis / frequency
                       Default value is 600
     
     *Url*: <http://ltfat.github.io/doc/nonstatgab/plotnsdgtreal.html>

     See also: tfplot, nsdgt, nsdgtreal.


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PLOTNSDGTREAL Plot NSDGTREAL coefficients
   Usage:  plotnsdgtreal(c,a,fs,dyn...



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unsdgt


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 -- Function: unsdgt
     UNSDGT  Uniform Non-stationary Discrete Gabor transform
        Usage:  c=unsdgt(f,g,a,M);
                [c,Ls]=unsdgt(f,g,a,M);
     
        Input parameters:
              f     : Input signal.
              g     : Cell array of window functions.
              a     : Vector of time positions of windows.
              M     : Numbers of frequency channels.
        Output parameters:
              c     : Cell array of coefficients.
              Ls    : Length of input signal.
     
        UNSDGT(f,g,a,M) computes the uniform non-stationary Gabor coefficients
        of the input signal f. The signal f can be a multichannel signal,
        given in the form of a 2D matrix of size Ls xW, with Ls being
        the signal length and W the number of signal channels.
     
        The non-stationary Gabor theory extends standard Gabor theory by
        enabling the evolution of the window over time. It is therefore necessary
        to specify a set of windows instead of a single window.  This is done by
        using a cell array for g. In this cell array, the n'th element g{n}
        is a row vector specifying the n'th window. However, the uniformity
        means that the number of channels is fixed.
     
        The resulting coefficients is stored as a M xN xW
        array. c(m,n,w) is thus the value of the coefficient for time index n,
        frequency index m and signal channel w.
     
        The variable a contains the distance in samples between two consecutive
        blocks of coefficients. a is a vectors of integers. The variables g and
        a must have the same length.
        
        The time positions of the coefficients blocks can be obtained by the
        following code. A value of 0 correspond to the first sample of the
        signal:
     
          timepos = cumsum(a)-a(1);
     
        [c,Ls]=nsdgt(f,g,a,M) additionally returns the length Ls of the input 
        signal f. This is handy for reconstruction:
     
          [c,Ls]=unsdgt(f,g,a,M);
          fr=iunsdgt(c,gd,a,Ls);
     
        will reconstruct the signal f no matter what the length of f is, 
        provided that gd are dual windows of g.
     
        Notes:
        ------
     
        UNSDGT uses circular border conditions, that is to say that the signal is
        considered as periodic for windows overlapping the beginning or the 
        end of the signal.
     
        The phaselocking convention used in UNSDGT is different from the
        convention used in the DGT function. UNSDGT results are phaselocked
        (a phase reference moving with the window is used), whereas DGT results
        are not phaselocked (a fixed phase reference corresponding to time 0 of
        the signal is used). See the help on PHASELOCK for more details on
        phaselocking conventions.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/unsdgt.html>

     See also: insdgt, nsgabdual, nsgabtight, phaselock, demo_nsdgt.


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UNSDGT  Uniform Non-stationary Discrete Gabor transform
   Usage:  c=unsdgt(f...



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unsdgtreal


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 -- Function: unsdgtreal
     UNSDGTREAL  Uniform non-stationary Discrete Gabor transform
        Usage:  c=unsdgtreal(f,g,a,M);
                [c,Ls]=unsdgtreal(f,g,a,M);
     
        Input parameters:
              f     : Input signal.
              g     : Cell array of window functions.
              a     : Vector of time positions of windows.
              M     : Vector of numbers of frequency channels.
        Output parameters:
              c     : Cell array of coefficients.
              Ls    : Length of input signal.
     
        UNSDGTREAL(f,g,a,M) computes the non-stationary Gabor coefficients of the
        input signal f. The signal f can be a multichannel signal, given in
        the form of a 2D matrix of size Ls xW, with Ls the signal
        length and W the number of signal channels.
     
        As opposed to NSDGT only the coefficients of the positive frequencies
        of the output are returned. UNSDGTREAL will refuse to work for complex
        valued input signals.
     
        The non-stationary Gabor theory extends standard Gabor theory by
        enabling the evolution of the window over time. It is therefore
        necessary to specify a set of windows instead of a single window.  This
        is done by using a cell array for g. In this cell array, the n'th
        element g{n} is a row vector specifying the n'th window. The
        uniformity means that the number of channels is not allowed to vary over
        time.
     
        The resulting coefficients is stored as a M/2+1 xN xW
        array. c(m,n,l) is thus the value of the coefficient for time index n,
        frequency index m and signal channel l.
     
        The variable a contains the distance in samples between two
        consecutive blocks of coefficients. The variable M contains the
        number of channels for each block of coefficients. Both a and M are
        vectors of integers.
     
        The variables g, a and M must have the same length, and the result c*
        will also have the same length.
        
        The time positions of the coefficients blocks can be obtained by the
        following code. A value of 0 correspond to the first sample of the
        signal:
     
          timepos = cumsum(a)-a(1);
     
        [c,Ls]=UNSDGTREAL(f,g,a,M) additionally returns the length Ls of the input 
        signal f. This is handy for reconstruction:
     
          [c,Ls]=unsdgtreal(f,g,a,M);
          fr=insdgtreal(c,gd,a,Ls);
     
        will reconstruct the signal f no matter what the length of f is, 
        provided that gd are dual windows of g.
     
        Notes:
        ------
     
        UNSDGTREAL uses circular border conditions, that is to say that the signal is
        considered as periodic for windows overlapping the beginning or the 
        end of the signal.
     
        The phaselocking convention used in UNSDGTREAL is different from the
        convention used in the DGT function. UNSDGTREAL results are phaselocked (a
        phase reference moving with the window is used), whereas DGT results are
        not phaselocked (a fixed phase reference corresponding to time 0 of the
        signal is used). See the help on PHASELOCK for more details on
        phaselocking conventions.
     
     
     
        References:
          P. Balazs, M. Doerfler, F. Jaillet, N. Holighaus, and G. A. Velasco.
          Theory, implementation and applications of nonstationary Gabor frames.
          J. Comput. Appl. Math., 236(6):1481--1496, 2011.
          
     *Url*: <http://ltfat.github.io/doc/nonstatgab/unsdgtreal.html>

     See also: nsdgt, insdgtreal, nsgabdual, nsgabtight, phaselock,
     demo_nsdgt.


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UNSDGTREAL  Uniform non-stationary Discrete Gabor transform
   Usage:  c=unsd...





