Abstract:
A method and a system for measuring the energy of a signal is based on the use of a time-frequency distribution using as kernel a bi-dimensional filter having compact support properties. The two-dimentional filter is applied to the two-dimensional correlation function of the signal to be analyzed. The system includes an acquisition unit for providing samples of the signal, a Hilbert transformer for producing an alytical signals from those samples, and a local correlator for computing the convolutions of a kernel derived from a Gaussian kernel and having compact support properties and instantaneous autocorrelation functions so as to yield generalized instantaneous autocorrelation functions, and a Fourier transformer for determining information related to the energy of the signal from the derived generalized instantaneous autocorrelation functions. The method and system allows a better reduction of false energy observances (interference) while providing good resolution in time and in frequency.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates to non-stationary signal analysis. More specifically, the present invention is concerned with time-frequency analysis of the energy of non-stationary signals.  
         BACKGROUND OF THE INVENTION  
         [0002]    Time-frequency distributions are widely used more and more for non-stationary signal analysis. They perform a mapping of one-dimensional signal x(t) into a two dimensional function of time and frequency TFD x (t,f) that yields a signature of the variation of the spectral content of the signal with time.  
           [0003]    Many approaches are known in the art to perform the above-mentioned mapping. The most intuitive approach consists of analyzing the signal for small periods of time during which it can be assumed that the signal does not contain rapid changes. In the context of a slowly varying signal, this window concept will provide a useful indication of the variations over time.  
           [0004]    The well-known spectrogram and the short-time Fourier transform are techniques that utilize the above window concept, and have become standard techniques in the art. These known systems, however, are not useful in situations where the energy, or spectral content of the signal, varies with such rapidity that the signal cannot reasonably be considered to be stationary for almost any window duration. In this regard, it is to be noted that, as the duration of the window is decreased, the frequency resolution of the system is also decreased.  
           [0005]    As indicated, the spectrogram applies the Fourier transform for a short-time analysis window, within which it is assumed that the signal behaves reasonably within the requirements of stationarity. Moving the analysis window in time along the signal, one hopes to track the variations of the signal spectrum as a function of time. If the analysis window is made short enough to capture rapid changes in the signal, it becomes impossible to resolve frequency components that are close in frequency during the analysis window duration.  
           [0006]    The well-known Wigner-Ville distribution provides a high-resolution representation in time and in frequency for a non-stationary signal, such as a chirp. However, it suffers from significant disadvantages. For example, its energy distribution is not non-negative and it is often characterized with severe cross terms, or interference terms, between components in different time-frequency regions. These cross terms lead to false manifestation of energy In the time frequency plan.  
           [0007]    The Choi-Williams distribution allows reduction of such interferences compared to the Wigner-Ville distribution.  
           [0008]    Since the spectrogram, Short-Time Fourier transform, Wigner-Ville and Choi-Williams distributions are believed to be well known in the art, they will not be described herein in further detail.  
           [0009]    A general class of time-frequency distributions (TFD) is the Cohen&#39;s class distributions. A member of this class has the following expression:.  
                 TFD   x          (     t   ,   f     )       =     ∫     ∫     ∫       φ        (     η   ,   τ     )       ×     (       t   ′     +     τ   2       )            x   H          (       t   ′     -     τ   2       )                   -   j2π                   η                 t                   -   j2π                   τ                 f                   -   j2π                   η                   t   ′                   t   ′               τ             η                     (   1   )                               
 
           [0010]    where t and f represent time and frequency, respectively, and  H  the transposed conjugate operator.  
           [0011]    The kernel φ(η,t) characterizes the resulting TFD. It is known in the art that the use of a Cohen&#39;s class of distributions allows the definition of kernels whose main property is to reduce the interference patterns induced by the distribution itself.  
           [0012]    An example of such a kernel is the Gaussian kernel that has been described in “KCS—New Kernel Family with Compact Support in Scale Space: Formulation and Impact”, from IEEE T-PAMI, 9(6), pp. 970-982, Jun. 2000 by I. Remaki and M. Cheriet.  
           [0013]    A problem with the Gaussian kernel is that it does not have the compact support analytical property, i.e. it does not vanish itself outside a given compact set. Hence, it does not recover the information loss that occurs due to truncating. Moreover, the prohibitive processing time, due to the mask&#39;s width, is increased to minimize the loss of accuracy.  
         SUMMARY OF THE INVENTION  
         [0014]    More specifically, in accordance with the present invention, there is provided a method for measuring the energy of a signal comprising:  
           [0015]    providing a number N of samples n of the signal;  
           [0016]    processing each of the N samples of the signal through a Hilibert transform so as to yield N corresponding analytical signals;  
           [0017]    for values of n ranging from 1 to N,  
           [0018]    providing a window of analysis of length M;  
           [0019]    for values of m ranging from 1 to M,  
           [0020]    computing an instantaneous autocorrelation function corresponding to each of the values of m; and  
           [0021]    computing the convolution of a CB kernel and the corresponding instantaneous autocorrelation function, yielding a generalized instantaneous autocorrelation function for each combination of the values of m and n; the CB kernel being defined by  
         K        (     n   ,   m     )       =     {           exp                   (     C        (       1         (       n   2     +     m   2       )            (     B   M     )     2       -   1       +   1     )       )               if                   (       n   2     +     m   2       )            (     B   M     )     2       &lt;   1             0       Otherwise                                 
 
           [0022]    where B and C are predetermined parameters; and  
           [0023]    applying a Fast Fourier Transform to the generalized instantaneous autocorrelation functions, yielding information about the energy of the signal for each of the N samples.  
           [0024]    According to a second aspect of the present invention there is provided a system for measuring the energy of a signal, comprising:  
           [0025]    an acquisition unit for providing samples of the signal;  
           [0026]    a Hilbert transformer for producing analytical signals from the samples of the signal;  
           [0027]    a local correlator for computing the convolutions of a CB kernel and instantaneous autocorrelation functions in a window of analysis of length M so as to yield generalized instantaneous autocorrelation functions; the CB kernel being defined by  
         K        (     n   ,   m     )       =     {           exp                   (     C        (       1         (       n   2     +     m   2       )            (     B   M     )     2       -   1       +   1     )       )               if                   (       n   2     +     m   2       )            (     B   M     )     2       &lt;   1             0       Otherwise                                 
 
           [0028]    where B and C are predetermined parameters; and  
           [0029]    a Fourier transformer for determining information related to the energy of the signal from the generalized instantaneous autocorrelation functions.  
           [0030]    Other objects, advantages and features of the present invention will become more apparent upon reading the following non-restrictive description of preferred embodiments thereof, given by way of example only with reference to the accompanying drawings.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0031]    In the appended drawings:  
         [0032]    [0032]FIG. 1 is a graph illustrating a kernel used with a method for measuring the energy of a signal according to an embodiment of the present invention;  
         [0033]    [0033]FIG. 2 is a bloc diagram of an electroencephalogram time frequency (EEG TF) analyser, incorporating a system for determining the energy of a signal according to an embodiment of the present invention;  
         [0034]    [0034]FIG. 3 is a bloc diagram of the TFD processor. from FIG. 2;  
         [0035]    [0035]FIG. 4 is a flowchart of a method for determining the energy of a signal according to an embodiment of the present invention;  
         [0036]    [0036]FIG. 5 is a graph illustrating the variation with time of the energy spectrum of two wavelets;  
         [0037]    [0037]FIG. 6 is a graph illustrating the time variation and the TFD of the two wavelets of FIG. 5, as obtained using a method for determining the energy of a signal according to an embodiment of the present invention;  
         [0038]    [0038]FIG. 7 is a graph illustrating the time variation and the spectrogram of the two wavelets of FIG. 5, as obtained using the spectral window of Blackman-Harris;  
         [0039]    [0039]FIG. 8 is a graph illustrating the time variation and the TFD of the two wavelets of FIG. 5, as obtained using the Wigner-Ville transformation;  
         [0040]    [0040]FIG. 9 is a graph illustrating the time variation of a signal centred about a frequency of about 5 GHz with additional peaks from different frequencies blended in a Gaussian white noise;  
         [0041]    [0041]FIG. 10 is a graph illustrating a mesh and a map of the signal of FIG. 9, obtained using a method for determining the energy of a signal according to an embodiment of the present invention;  
         [0042]    [0042]FIG. 11 is a graph illustrating a mesh and a map of the signal of FIG. 9, obtained using the spectrogram with the spectral window of Blackman-Harris;  
         [0043]    [0043]FIG. 12 is a graph illustrating a mesh and a map of the signal of FIG. 9, obtained using the Wigner-Ville transformation;  
         [0044]    [0044]FIG. 13 is a graph illustrating a mesh and a map of the signal of FIG. 9, obtained using the Choi-Williams transformation;  
         [0045]    [0045]FIG. 14 is a graph illustrating two crossing chirps;  
         [0046]    [0046]FIG. 15 is a graph illustrating a mesh and a map of the signal of FIG. 14, obtained using a method for determining the energy of a signal according to an embodiment of the present invention;  
         [0047]    [0047]FIG. 16 is a graph illustrating a mesh and a map of the signal of FIG. 14, obtained using the spectrogram with the spectral window of Blackman-Harris;  
         [0048]    [0048]FIG. 17 is a graph illustrating a mesh and a map of the signal of FIG. 14, obtained using the Wigner-Ville transformation;  
         [0049]    [0049]FIG. 18 is a graph illustrating a mesh and a map of the signal of FIG. 14, obtained using the Choi-Williams transformation;  
         [0050]    [0050]FIG. 19 is a graph illustrating a Matlab™ simulation of a high-frequency spectrum having a bandwidth of 10 MHz centred about 500 MHz;  
         [0051]    [0051]FIG. 20 is a graph illustrating a mesh and a map of the signal of FIG. 19, obtained using a method for determining the energy of a signal according to the present invention;  
         [0052]    [0052]FIG. 21 is a graph illustrating a mesh and a map of the signal of FIG. 19, obtained using the spectrogram with the spectral window of Blackman-Harris;  
         [0053]    [0053]FIG. 22 is a graph illustrating a mesh and a map of the signal of FIG. 19, obtained using the Wigner-Ville transformation;  
         [0054]    [0054]FIG. 23 is a graph illustrating a mesh and a map of the signal of FIG. 19, obtained using the Choi-Williams transformation;  
         [0055]    [0055]FIG. 24 is a graph illustrating the TFD of a non-stationary signal blended in a Gaussian white noise;  
         [0056]    [0056]FIG. 25 is a graph illustrating a mesh and a map of the signal of FIG. 24, obtained using a method for determining the energy of a signal according to an embodiment of the present invention;  
         [0057]    [0057]FIG. 26 is a graph illustrating a mesh and a map of the signal of FIG. 24, obtained using the spectrogram with the spectral window of Blackman-Harris;  
         [0058]    [0058]FIG. 27 is a graph illustrating a mesh and a map of the signal of FIG. 25, obtained using the Wigner-Ville transformation;  
         [0059]    [0059]FIG. 28 is a graph illustrating a mesh and a map of the signal of FIG. 25, obtained using the Choi-Williams transformation;  
         [0060]    [0060]FIG. 29 is a graph illustrating a non-noisy signal sampled at 500 MHz between a 11.5 and 14,5 μs window;  
         [0061]    [0061]FIG. 30 is a mapped plot of the TFD resulting from an analysis of the signal of FIG. 29 performed with the method of FIG. 4;  
         [0062]    [0062]FIG. 31 is a meshed plot of the TFD resulting from an analysis of the signal of FIG. 29 performed with the method of FIG. 4;  
         [0063]    [0063]FIG. 32 is a graph illustrating a noisy signal sampled at 500 MHz between a 11.5 and 14,5 μs window;  
         [0064]    [0064]FIG. 33 is a mapped plot of the TFD resulting from an analysis of the signal of FIG. 32 performed with the method of FIG. 4;  
         [0065]    [0065]FIG. 34 is a meshed plot of the TFD resulting from an analysis of the signal of FIG. 32 performed with the method of FIG. 4;  
         [0066]    [0066]FIG. 35 is the mapped plot of FIG. 33 zoomed on the frequency band from 0 to 72 MHz; and  
         [0067]    [0067]FIG. 36 is the meshed plot of FIG. 34 zoomed on the frequency band from 0 to 72 MHz. 
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0068]    According to the present invention, a method and system are provided for measuring the energy of a signal using a time-frequency distribution based on a new kernel derived from the Gaussian kernel. This new Kernel will be referred to herein as the Cheriet-Belouchrani (CB) kemel. Unlike the Gaussian kernel, the CB kernel has the compact support analytical property. Hence, it recovers the information loss that occurs for the Gaussian kernel due to truncation and improves the processing time.  
         [0069]    The CB kernel is derived from the Gaussian kernel by transforming the IR 2  space into a unit ball through a change of variables. This transformation packs all the information into the unit ball. With the new variables, the Gaussian is defined on the unit ball and vanishes on the unit sphere. Then, it is extended over all the IR 2  space by taking zero values outside the unit ball. The obtained kernel still belongs to the space of functions with derivatives of any order. The CB kernel, also referred to as KCS (Kernel of Compact Support) is described in “KCS—New Kemel Family with Compact Support in Scale Space: Formulation and Impact”, from IEEE T-PAMI, 9(6), pp. 970-982, June 2000 by I. Remaki and M. Cheriet and has the following expression:  
         φ        (     η   ,   τ     )       =     {                  1   2          (       γ       η   2     +     τ   2     -   1       +   γ     )                   η   2     +     τ   2       &lt;   1               0                    elsewhere                                 
 
         [0070]    where γ is a parameter that controls the kernel width. FIG. 1 shows the KCS with γ=5.5.  
         [0071]    One of the advantages of a method and system for measuring the energy of a signal, using a time-frequency distribution based on the CB kernel, is that it recovers the above information loss and improves processing time and thus retaining the most important properties of the Gaussian kernel. These features are achieved due to the compact support analytical property of the CB kernel. This compact support property means that the kernel vanishes outside a given compact set.  
         [0072]    Turning now to FIG. 2, an electroencephalogram time frequency (EEG TF) analyser, incorporating a system for determining the energy of a signal according to an embodiment of the present invention is illustrated.  
         [0073]    The EEG TF analyser  10  comprises a signal acquisition unit  12 , a system for determining the energy of a signal according to an embodiment of the present invention in the form of a time frequency distribution (TFD) processor  14 , a parameter controller  16 , an output unit  18 , and an output device  20 .  
         [0074]    The signal acquisition unit 12 includes conventional input ports to receive signals from an EEG (not shown). The signal acquisition unit  12  is configured to receive signals having a frequency ranging from 0 Hz to about 40 Hz. The unit  12  includes an analog front end, and an analog/digital converter (sampler) to convert an analog signal s(t) at input  22  into a series of digital samples s(n) at its output  24 . Since signal acquisition units and analog/digital converter are believed to be well known in the art, they will not be described herein in more detail.  
         [0075]    The parameter controller  16  includes a user input interface allowing to specify different operating parameter of the TFD processor  14  as will be explained hereinbelow in more detail. The parameter controller  16  may take many forms, from a console display panel equipped with input knobs, to a user interface programmed into a computer (not shown).  
         [0076]    The output unit  18  and output device  20  are advantageously in form of a visualization unit and of a display monitor respectively. The output unit is configured to receive signals from the TFD processor  14 , and to process the received signals so as to be displayed onto the output device  20 . The output unit may be provided with a user interface or alternatively be controlled by the controller  16  or by another controller (not shown).  
         [0077]    The EEG TF analyser  10  may include a storing means (not shown) for storing input signals and/or outputs from the output unit  18 . The storing means may take many forms, including an EEPROM (Electrically Erasable Programmable Read-Only Memory), ROM (Read-Only Memory), a Hard-disk, a disk, DVD or CD-ROM drive, etc. Optionally, the EEG TF analyser  10  may be connected to a computer network, such as Internet.  
         [0078]    The TFD processor  14  and the output unit  18  may also be embodied in many ways, including hardware or software. For example, they can be in the form of Field-Programmable Gate Arrays (FPGA) advantageously programmed using a very-high level description language (VHDL).  
         [0079]    The EEG TF analyser  10 , when in the form of a computer system, includes appropriate software that enabled a method for determining the energy of a signal according to an embodiment of the present invention.  
         [0080]    Turning now to FIG. 3, the TFD processor  14  is illustrated. The TFD processor  14  includes a Hilbert transformer  26  to produce analytic signal, a local correlator  28  for producing a plurality of signals corresponding to the samples delayed and multiplicatively combined with their complex conjugates, and a Fourier transformer  30  for producing a distribution as a function of time and frequency. The local correlator  28  advantageously includes a bank of delay and adder for correlating the outputs of the multipliers in accordance with the CB-kernel as will be explained hereinbelow.  
         [0081]    The function of each of the components of the TFD processor  14  will become more apparent upon reading the following description of a method  100  for determining the energy of a signal according to an embodiment of the present invention, with reference to FIGS.  2  to  4 .  
         [0082]    It is to be noted that the method  100  allows implementing the following equations from the CB-distribution:  
         D        (     n   ,   k     )       =       FFT     m   -&gt;   k            (         K        (     n   ,   m     )       n   *          A        (     n   ,   m     )         )                             
 
         [0083]    where A(n,m)=z(n+m)z*(n−m) is the instantaneous autocorrelation,  
         FFT        (   .   )         m   -&gt;   k                           
 
         [0084]    is the Fast Fourier Transform performed over the time-lag m,  n  * is the discrete time convolution operator over n and  K(n,m)  is the CB kernel defined as:  
         K        (     n   ,   m     )       =     {           exp   (     C        (       1         (       n   2     +     m   2       )            (     B   M     )     2       -   1       +   1     )       )             if                   (       n   2     +     m   2       )            (     B   M     )     2       &lt;   1             0       Otherwise                                 
 
         [0085]    where B and C two parameters that controls the resolution and the cross term rejection, M is the length of the analysis window.  
         [0086]    The generalized instantaneous autocorrelation is the convolution of the kernel and the instantaneous autocorrelation, which is defined by: 
           G ( n,m )= K ( n,m ) n   *A ( n,m ) 
         [0087]    The CB-distribution can be also expressed as:  
         D        (     n   ,   k     )       =       ∑     m   =     -   M       M                       ∑     p   =     -   M       M            K        (       p   -   n     ,   m     )            z        (     p   -   m     )              z   *          (     p   -   m     )            exp        (       -   j4π          mK   N       )                                   
 
         [0088]    At  102 , the method start by the TFD processor  14  receiving a digital sample s(n). At  104 , the real signal s(n) is transformed to an analytical signal by an Hilbert transform. The Hilbert transformer  26  of the TFD processor  14  performs step  104 . The signal s(n) may be kept real if desired.  
         [0089]    At step  106 , a loop begins that proceeds from step  108  to step  130  for all N samples created by the acquisition unit  12 .  
         [0090]    At step  108 , a nested loop, including steps  110 - 122 , begins over time-lags that calculates half of the generalized instantaneous autocorrelation function G(n,m). Values of m range from 1 to M, M being the length of the chosen window of analysis). It is to be noted that the other half is obtained by symmetry and is processed at step  126 .  
         [0091]    At step  110 , the initial value of the generalized instantaneous autocorrelation function is set to zero at each lag beginning.  
         [0092]    Steps  112  to  120  represent a nested loop that correspond to the computation of the convolution of the CB kernel and the instantaneous autocorrelation function to produce at step  122  the generalized instantaneous autocorrelation function.  
         [0093]    More specifically, the argument x of the CB kernel is first computed in step  114 . This argument includes the parameter B, which controls the cross term rejection and the resolution of time-frequency representation. The parameter B is inputted and adjusted via the parameter controller  16  on FIG. 2.  
         [0094]    The next step ( 116 ) allows testing the argument x: If x≧1, then the convolution sum R is not updated (step  118 ), otherwise, R is updated at step  118 . The instantaneous autocorrelation is computed in step  118 , by the local correlator  28 , by producing a plurality of autocorrelated signals corresponding to the samples delayed and multiplicatively combined with their complex conjugates and weighted in accordance with the CB-kernel. It is to be noted that the CB-kemel (step  118 ) contains a second parameter C that also has an influence on the cross term rejection and resolution and is also inputted and adjusted via the parameter controller  16  (see FIG. 2). Hence, a method for determining the energy of a signal according to the present invention provides, through parameters B and C, two degrees of freedom for the controls of the quality of the time frequency representation in terms of the resolution and cross term rejection.  
         [0095]    The loop started at step  112  ends at step  120 . The loop started at step  108  in turn ends at step  124 .  
         [0096]    As explained hereinabove, the second half of the generalized instantaneous autocorrelation function is performed at step  126 .  
         [0097]    A Fast Fourier Transform (FFT) is then applied to the generalized instantaneous autocorrelation function (step  128 ). The result of the process for the current sample may be displayed or plotted at function block  130  via the output unit  18  and output device  20 .  
         [0098]    The method  100  then loops to the next sample (step  132 ).  
         [0099]    The method  100  ends when all samples have been processed (step  134 ).  
         [0100]    Returning to FIG. 10, the acquisition unit  12  may be adapted for other frequency ranges than 0-40 Hz. For example, a speech time frequency analyser would have a general configuration similar to the configuration described in FIG. 2 for the EEG TF analyser, with the exception of the acquisition unit  12 , which will be configured to operate in the range 0-8 kHz. The acquisition unit of a radio frequency (RF) analyser incorporating a TFD processor according to an embodiment of the present invention would have to be configured for frequencies ranging from 300 MHz to some GHz.  
         [0101]    A method and system for determining the energy of a signal according to embodiments of the present invention may be used in any application that requires information about the energy of a signal in relation with time and frequency. Such applications include spectral analyser, biomedical sensors (ultrasound devices, scanners, nuclear magnetic resonance, etc.) mechanical vibration analysis, X-ray photography, air-flow tubes, electromyography, spectrographs, mingographs, larygographs, seismic spectrograms, telecommunication, etc.  
         [0102]    The present invention is particularly advantageous to reduce noises and interference in a signal. Indeed, random noises tend to spread equally in a time-frequency continuum, while the wanted signal is concentrated in relatively narrow region. Consequently, the signal to noise ratio is increased substantially in the time-frequency domain with methods and systems according to the present invention. The present invention allows building time varying filters.  
       Comparison Results  
       [0103]    Turning now to FIGS.  5  to  28  of the appended drawings, time-frequency representations of the energy of different signals are shown. These representations are obtained using the method  100  as well as methods from the prior art. To provide comparable results between the method  100  and methods from the prior art, results from FIGS.  5  to  28  were obtained using Matlab™. The results are shown on a linear scale.  
         [0104]    FIGS.  6  to  8  illustrate the time-frequency representation of two wavelets.  
         [0105]    [0105]FIG. 5 illustrates the variation with time of the energy spectrum of the wavelets S 1  and S 2 . As can be seen in FIG. 5, there is a ratio of 4 between the peaks of the two wavelets S 1  and S 2 .  
         [0106]    [0106]FIG. 6 illustrates the time variation (above) and the TFD (below) of the two wavelets of FIG. 5, obtained using a method according to the present invention with a resolution parameter B of 10.  
         [0107]    As can be seen in FIG. 6, compared to methods of the prior art (FIGS. 7 and 8), a method for determining the energy of a signal according to the present invention allows an increase of the emergence of spectral peaks to smooth interference components and shows good time and frequency resolutions.  
         [0108]    [0108]FIG. 7 illustrates the time variation (above) and the spectrogram (below) of the two wavelets of FIG. 5, as obtained using the spectral window of Blackman-Harris.  
         [0109]    [0109]FIG. 8 illustrates the time variation (above) and the TFD (below) of the two wavelets of FIG. 5, as obtained using the Wigner-Ville transformation.  
         [0110]    The above comparison shows that a TFD obtained from a method according to the present invention belongs to a TFD provided with a wealth of details. This can be achieved since these distributions are defined by an integral operator that acts on a quadratic form of the signal. Those classes are parametrically defined via arbitrary kernels. Properties can be advantageously imposed on the distributions by structural constraints on the corresponding kernels.  
         [0111]    FIGS.  10  to  13  illustrate the TFD of a signal centred about a frequency of about 5 GHz with additional peaks from different frequencies blended in a Gaussian white noise (see FIG. 9).  
         [0112]    [0112]FIG. 10 illustrates a mesh (left) and a map (right) of the signal of FIG. 9, obtained using a method for determining the energy of a signal according to the present invention.  
         [0113]    [0113]FIG. 11 illustrates a mesh (left) and a map (right) of the signal of FIG. 9, obtained using the spectrogram with the spectral window of Blackman-Harris.  
         [0114]    [0114]FIG. 12 illustrates a mesh (left) and a map (right) of the signal of FIG. 9, obtained using the Wigner-Ville transformation.  
         [0115]    [0115]FIG. 13 illustrates a mesh (left) and a map (right) of the signal of FIG. 9, obtained using the Choi-Williams transformation.  
         [0116]    The above comparison again illustrates that a method for measuring the energy of a signal according to the present invention allows increasing of the emergence of spectral peaks, to smooth interference components and shows good time and frequency resolutions.  
         [0117]    FIGS.  15  to  18  illustrate the TFD of two crossing chirps (see FIG. 14).  
         [0118]    [0118]FIG. 15 illustrates a mesh (left) and a map (right) of the signal of FIG. 14, obtained using a method for determining the energy of a signal according to the present invention.  
         [0119]    [0119]FIG. 16 illustrates a mesh (left) and a map (right) of the signal of FIG. 14, obtained using the spectrogram with the spectral window of Blackman-Harris.  
         [0120]    [0120]FIG. 17 illustrates a mesh (left) and a map (right) of the signal of FIG. 14, obtained using the Wigner-Ville transformation.  
         [0121]    [0121]FIG. 18 illustrates a mesh (left) and a map (right) of the signal of FIG. 14, obtained using the Choi-Williams transformation.  
         [0122]    The comparisons between the results illustrated in FIGS.  15  to  18  shows that the use of the CB TFD allows removal of the cross terms and presents cute curves in contrast to the three other representations that cannot.  
         [0123]    FIGS.  20  to  23  illustrate a Matlab™ simulation of a high-frequency spectrum having a bandwidth of 10 MHz centred about 500 MHz (see FIG. 19).  
         [0124]    [0124]FIG. 20 illustrates a mesh (left) and a map (right) of the signal of FIG. 16, obtained using a method for determining the energy of a signal according to the present invention.  
         [0125]    [0125]FIG. 21 illustrates a mesh (left) and a map (right) of the signal of FIG. 19, obtained using the spectrogram with the spectral window of Blackman-Harris.  
         [0126]    [0126]FIG. 22 illustrates a mesh (left) and a map (right) of the signal of FIG. 19, obtained using the Wigner-Ville transformation.  
         [0127]    [0127]FIG. 23 illustrates a mesh (left) and a map (right) of the signal of FIG. 19, obtained using the Choi-Williams transformation.  
         [0128]    FIGS.  25  to  28  illustrate the TFD of a stationary signal blended in a Gaussian white noise (see FIG. 24).  
         [0129]    [0129]FIG. 25 illustrates a mesh (left) and a map (right) of the signal of FIG. 24, obtained using a method for determining the energy of a signal according to the present invention.  
         [0130]    [0130]FIG. 26 illustrates a mesh (left) and a map (right) of the signal of FIG. 24, obtained using the spectrogram with the spectral window of Blackman-Harris.  
         [0131]    [0131]FIG. 27 illustrates a mesh (left) and a map (right) of the signal of FIG. 24, obtained using the Wigner-Ville transformation.  
         [0132]    [0132]FIG. 28 illustrates a mesh (left) and a map (right) of the signal of FIG. 24, obtained using the Choi-Williams transformation.  
         [0133]    The spectrogram using the Blackman-Harris window, and the Choi-Williams and Wigner-Ville distributions are believed to be well known in the art and thus will not be described herein in more detail.  
         [0134]    It is to be noted that the energy representation provided by a method according to the present invention does not satisfy the marginal property just like the spectrogram. It is advantageously consistent with the energy conservation (φ(0,0)=1) and verifies both the reality and the time and frequency shift properties.  
         [0135]    FIGS.  29  to  36  illustrate two examples of signal analysis performed with the method  100  embodied as software on a personal computer.  
         [0136]    [0136]FIGS. 30 and 31 are respectively mapped and meshed plots of the time-frequency distribution resulting from an analysis of the signal of FIG. 29 according to the method  100 .  
         [0137]    [0137]FIG. 29 is a time representation between 11.5 and 14.5 μs of a non-noisy signal sampled with a 500 MHz frequency and a factor  3  subsampling.  
         [0138]    [0138]FIG. 30 shows that the method  100  allows to find the wanted signal around 70 MHz as its sign-on. Interference at around 73 MHz can also be seen from FIG. 30. We can also see that the method  100  yields a good time-frequency resolution and prevents cross-terms.  
         [0139]    Both the wanted signal and the interference are clearly distinguished on FIG. 31 with a good resolution.  
         [0140]    [0140]FIGS. 33 and 34 are respectively mapped and meshed plots of the time frequency distribution resulting from an analysis of the signal of FIG. 32 according to the method  100 .  
         [0141]    [0141]FIG. 32 is a time representation between 11.5 and 14.5 μs of a noisy signal sampled with a 500 MHz frequency and a factor  3  subsampling.  
         [0142]    [0142]FIG. 33 shows that the method  100  allows to find the interference at around 73 MHz. However, since the signal-interference ration is small, the wanted signal can not be clearly distinguished from the noise on FIGS.  33  or  34 .  
         [0143]    However, when a zoom is performed on the frequency band from 0 to 72 MHz (see FIGS. 35 and 36), the wanted signal and the interference are clearly distinguished. FIG. 36 also allows seeing a d.c. component at 0 MHZ.  
         [0144]    Although the present invention has been described hereinabove by way of preferred embodiments thereof, it can be modified without departing from the spirit and nature of the subject invention, as defined in the appended claims.