Algorithms for (Quadratic) Time-Frequency Distributions

This collection of M-files generate time-frequency distributions (TFDs) with as few computations and as little memory as possible. The algorithms are described in [1]; the discrete TFD definition they compute is in [2].

There are two sets of algorithms. The first set (Set1) computes the exact TFD and minimises on oversampling. The second set (Set2) computes a decimated TFD, and therefore is not exact but may be useful for some applications.

Each set computes four different types of kernels:

nonseparable kernel
separable kernel
Doppler-independent kernel
lag-independent kernel (see [1] for more details)

The code is organised as follows:

folder full_DTFD contains the first set of algorithms (computes exact TFDs)
folder dec_DTFD contains the second set of algorithms (computes decimated TFDs)
folder common and utils contain extra functions used by both algorithm sets.

These programs should run in either Matlab (v7.9.0) or Octave (v3.2.4)

Quick Start

To start, load the paths:

>> load_paths_DTFDs;

Next, generate a test signal:

>> N=128; x=gen_LFM(N,0.1,0.3);

Second, generate a TFD with a separable kernel:

>> tf=dtfd_sep1(x,{51,'hamm',0,1},{171,'hann'},256,128);

Third, plot the TFD:

>> vtfd(tf,x);

Description

functions to compute exact TFDs (in folder 'full_DTFD'):
- dtfd_DI.m computes a TFD with Doppler-independent kernel
- dtfd_LI.m computes a TFD with a lag-independent kernel
- dtfd_nonsep.m computes a nonseparable-kernel TFD
- dtfd_sep1.m computes a TFD with a separable kernel
- dtfd_sep2.m computes a TFD with a separable kernel using a different algorithm to the dtfd_sep1.m
functions to compute a decimated TFD (in folder dec_DTFD)
- dec_dtfd_DI.m computes a decimated TFD with a Doppler-independent kernel
- dec_dtfd_LI.m computes a decimated TFD with a lag-independent kernel
- dec_dtfd_nonsep.m computes a decimated TFD with a nonseparable kernel
- dec_dtfd_sep.m computes a decimated TFD with a separable kernel
- dwvd_grid1.m computes a decimated Wigner-Ville distribution
- dwvd_grid2.m computes a decimated Wigner-Ville distribution with a different decimation grid to that grid in dwvd_grid1.m
Other miscellaneous functions (in folder utils) include:
- get_analytic.m computes a discrete analytic signal, as defined in [3]
- gen_LFM.m computes a linear frequency modulated (LFM) signal
- fft_complex.m is a function for alternative FFT routines (for example to take advantage of parallel computing); likewise are fft_conj_summ.m, ifft_complex.m, and ifft_conj_symm.m

Version and Date

Version: 0.23
Last update: [13-05-2013]

Examples

In all the examples, start by adding the paths by running the command

>> load_paths_DTFDs;

Choi-Williams using the exact TFD algorithm (from Set1 [1]):

% 1. generate test signal:
N=256; 
x=gen_LFM(N,0.05,0.15)+gen_LFM(N,0.2,0.35);
% 2. generate TFD
c=dtfd_nonsep(x,'cw',{30}); 
% 3. plot
clf; vtfd(c,x);

Separable-kernel TFD using exact TFD algorithm (from Set1 [1])

% 1. generate test signal:
N=10000; 
x=gen_LFM(N,0.1,0.3)+gen_LFM(N,0.4,0.1);
% 2. generate TFD
Ntime=256; Nfreq=256;
c=dtfd_sep1(x,{51,'hamm',0,1},{271,'hann'},Ntime,Nfreq); 
% 3. plot
clf; vtfd(c,x);

Doppler-independent kernel TFD using decimated TFD algorithm (from Set2 [1]) generating selective time portions of the signal:

% 1. generate test signal:
N=171; 
x=gen_LFM(N,0.1,0.4);
% 2. generate TFD:
Nfreq=258; 
time_dec=[20:3:160,191:250,255,256]; freq_dec=3;
tf=dec_dtfd_DI(x,{51,'hamm'},Nfreq,time_dec,freq_dec); 
% 3. plot
clf; vtfd(tf);

Separable kernel TFD using decimated TFD algorithm from Set2 [2]

% 1. generate test signal:
N=10000; 
x=gen_LFM(N,0.1,0.3)+gen_LFM(N,0.4,0.1);
% 2. generate TFD:
Ntime=512; Nfreq=256;
time_dec=4; freq_dec=2;
c=dec_dtfd_sep(x,{51,'hamm',0,1},{271,'hann'}, ...
               Ntime,Nfreq,time_dec,freq_dec); 
% 3. plot
clf; vtfd(c,x);

Test computer setup

hardware: Intel Core Duo CPU, 2.13GHz; 2GB memory.
operating system: Ubuntu GNU/Linux i686 distribution (Natty, 11.04), with Linux kernel 2.6.38-8-generic
software: Octave 3.2.4 (using Gnuplot 4.4 patchlevel 2) and Matlab (R2009b, R2012a, and R2013a)

Copyright

Copyright (c) 2010 John M. O' Toole, The University of Queensland
All rights reserved.
Email: 	  j.otoole@ieee.org

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following
conditions are met:
    * Redistributions of source code must retain the above
      copyright notice, this list of conditions and the following
      disclaimer.
    * Redistributions in binary form must reproduce the above
      copyright notice, this list of conditions and the following
      disclaimer in the documentation and/or other materials
      provided with the distribution.
    * Neither the name of the The University of Queensland nor the 
      names of its contributors may be used to endorse or promote 
      products derived from this software without specific prior 
      written permission.

THIS SOFTWARE IS PROVIDED BY JOHN M. O' TOOLE ''AS IS'' AND ANY
EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL JOHN M. O' TOOLE BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT
OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
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References

J.M. O' Toole and B. Boashash, "Fast and memory-efficient algorithms for computing quadratic time–frequency distributions", Applied and Computational Harmonic Analysis, vol. 35, no. 2, pp. 350–358, 2013.
J.M. Oʼ Toole, M. Mesbah, and B. Boashash, “Improved discrete definition of quadratic time–frequency distributions,” IEEE Transactions on Signal Processing, vol. 58, Feb. 2010, pp. 906-911.
J.M. O' Toole, M. Mesbah, and B. Boashash, "A New Discrete Analytic Signal for Reducing Aliasing in the Discrete Wigner-Ville Distribution", IEEE Transactions on Signal Processing, vol. 56, no. 11, pp. 5427-5434, Nov. 2008.
J.M. Oʼ Toole, M. Mesbah, and B. Boashash, “Algorithms for discrete quadratic time–frequency distributions,” WSEAS Transactions on Signal Processing, vol. 4, May. 2008, pp. 320-329.

Contact

John M. O' Toole

Neonatal Brain Research Group,
Irish Centre for Fetal and Neonatal Translational Research (INFANT),
Department of Paediatrics and Child Health,
University College Cork,
Western Gateway Building, Room 2.17,
Cork, Ireland

email: j.otoole -AT- ieee.org
web: http://otoolej.github.io/

otoolej / fast_TFDs