Abstract:
A method separates a multi-component signal by first estimating parameters of the signal. Then, using periodicity-based algebraic separation and energy-based demodulation, the signal is separated into components according to the parameters and constraints. Last, a Teager-Kaiser energy detector is applied to each component to provide a direct current signal for each component, and the constraint for each component used by the separating.

Description:
FIELD OF THE INVENTION 
     This invention relates in general to sigma processing, and in particular to separating a signal including multiple sinusoidal signals and multiple AM-FM modulated sinusoidal signals into components. 
     BACKGROUND OF THE INVENTION 
       FIG. 1  shows a conventional method for separating a multi-component signal. A mixture of amplitude-frequency modulated signals (AM-FM) and sinusoidal signals occurs frequently in acoustics applications, biological systems, and as signals received by vehicular collision avoidance radars that transmit continuous wave frequency modulation (CWFM) signals. 
     A single component AM-FM sinusoidal signal is represented as
 
 x ( t )= A ( t )cos(2π ft +φ( t )),  (1)
 
where A(t) indicates a time-varying amplitude envelope and φ is the phase angle. A K multi-component signal  110  is given by
 
 y ( t )=Σ i=1   K   A   i ( t )cos(2π f   i   t+φ   i ( t )),0&lt; t&lt;T   (2)
 
where T is a signal duration.
 
     Gianfelici et. al., in “Multicomponent AM-FM Representations: An Asymptotically Exact Approach,” IEEE Trans. Audio, Speech and Language Processing, vol. 15, no. 3, March 2007, describe a method called an Iterated Hilbert Transform (IHT). Generally, the Hilbert transform is a linear operator that takes a function, u(t), and produces a function, H(u)(t) in the same domain. The IHT can be used to estimate instantaneous frequencies of the components  150  of the signal in equation (2). The performance of IHT is suboptimal when the amplitude of a component is within a close range, e.g., A 2 /A 1 =2, in a two component case. The IHT is followed by a Teager-Kaiser energy detector (TKED) based frequency estimator  160 , which outputs  170  direct current (DC) component signals. 
     Santhanam et al., in “Multicomponent AM-FM Demodulation via Periodicity-based Algebraic Separation and Energy-based Demodulation,” IEEE Trans. Commun., vol. 48. no. 3, March 2000, describe a method called PASED  150 , which is a non-linear method that can separate mixed periodic signals with similar strengths. 
     PASED works well even when the signals have a small spectral separation. However, PASED needs to know both the period of each signal component and the number of components in the mixture. 
     Therefore, PASED is generally prefaced by a Double Differencing Function (DDF)  120  to estimate the parameters of the input signal. The parameters include the number of components and their periods. With noisy signals, DDE is also suboptimal. 
     Therefore, the conventional PASED  130  may not output an optimal separation in low signal-to-noise-ratio (SNR) cases. The PASED also uses a zero DC constraint  140  for each sinusoid expected to be in the multi-component signal. If the signal is not integrated over the correct period of a sinusoid, then the DC-constraint is violated, because the integration does not result in a zero value. 
     SUMMARY OF THE INVENTION 
     The embodiments of the invention provide a method for separating a multi-component signal into components using Periodicity-based Algebraic Separation and Energy-based Demodulation (PASED). 
     The embodiments use an enhanced PASED that iteratively determines an optimal signal separation, when the DDF is subject to errors, as described above. The enhanced PASED replaces the zero DC constraints with constraints in a form of linear prediction coefficients. 
     Instead of relying on a single estimate of the signal periods, the method iterates to improve the signal separation. The PASED is cascaded by a bank of well-known Teager-Kaiser energy detector (TKED) operators, one for each signal component. The TKED operator converts a sinusoidal signal into a direct current (DC) signal. 
     The output of the TKED is fed back to adjust the estimates of the periods used in PASED. The zero DC constraints in the conventional PASED are replaced by constraints in the sine wave in the form of linear-prediction-filter coefficients. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a conventional method for signal separation; and 
         FIG. 2  is a block diagram of a method for signal separation according to embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     As shown in  FIG. 2 , the embodiments of the invention provide a method for separating an input multi-component signal  210  into components  270  using Periodicity-based Algebraic Separation and Energy-based Demodulation (PASED). The multi-component signal can be represented as
 
 x ( t )= x   1 ( t )+ x   2 ( t ), . . . ,+x N ( t )+noise.
 
     The method can be performed in a processor  200  connected to memory and input/output interfaces by buses as known in the art. 
     PASED Method 
     Components z 1 (n) and z 2 (n) of a two component signal are respectively periodic in intervals N 1  and N 2 , such that
 
 z   1 ( n )= z   1 ( n+N   1 )
 
 z   2 ( n )= z   2 ( n+N   2 ).
 
     The multi-component signal z(n) includes z 1 (n) and z 2 (n) and noise w(n), such that
 
 z   g ( n )= z   1 ( n )+ z   2 ( n )+ w ( n )= z   1 ( n+N   1 )+ z   2 ( n+N   2 )+ w ( n ).  (3)
 
     If at least N 1 +N 2 −1 signal samples are measured, then the signals z 1 (n) and z 2 (n) can be separated. If N 1  and N 2  are estimated inaccurately, then the separation performance is degraded. 
     The composite signal samples z g (n) for n=1, 2, . . . , N can be represented in a matrix given by 
                       [             z   g     ⁡     (   1   )                   z   g     ⁡     (   2   )               …               z   g     ⁡     (   N   )             ]     =       [           I     N   1             I     N   2                 I     N   1             I     N   2               …       …         ]     ×     [             z     g   1       ⁡     (   1   )               …               z     g   1       ⁡     (     N   1     )                   z     g   2       ⁡     (   1   )               …               z     g   2       ⁢     N   2             ]         ,           (   4   )               
or alternatively as z g =Sz, where the rank of S is equal to
 
rank( S )= N   1   +N   2   −gcd ( N   1   ,N   2 ),  (5)
 
where gcd represents the greatest common divisor, and I Ni  is the identity matrix of order N i . If gcd (N 1 , N 2 )=M, then, the PASED method needs M constraints for separability. These M constraints, defined by a matrix C, are augmented to the original equation as follows.
 
     
       
         
           
             
               
                 
                   
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     The estimate of the signals produced by PASED is given by
 
 {circumflex over (z)} =( S   T   S+C   T   C ) −1   S   T   {circumflex over (z)}   g   (7)
 
such that
 
 {circumflex over (z)}   1   ={circumflex over (z)} (1,2, . . . , N   1 ) and  {circumflex over (z)}   2   ={circumflex over (z)} ( N   1 +1, . . .  N   1   +N   2 ).
 
     A sampled sinusoid may not have a zero DC when the input signal length is not an integer multiple of its fundamental period of the signal. The original PASED method ignores this fact, and therefore its performance suffers. In the present invention, constraints in a form of linear prediction coefficients  240  of a sinusoid are used to produce exact zero DC constraints. 
     A double differencing function (DDF) is used to estimate  220  parameters  221  of the input signal. The parameters include the number of components and period of the components. In the presence of additive white Gaussian noise (AWGN), the DDF based estimates  220  of the input signals are imperfect. Unfortunately, the PASED  230  is sensitive to imperfect period estimates (errors in N 1  and N 2 ). Therefore, an adaptive estimate of w is needed via a feedback loop  265  from the TKED  760 . 
     Teager-Kaiser Energy Detector (TKED) Operator 
     The TKED operator  260  functions as a frequency-to-DC converter. An output  270  of the Operator includes a DC signal for each component. The DC signals are proportional to squares of the amplitude and frequency of the input sinusoidal signal  210 . A discrete-time representation of the TKED operator is
 
φ( s ( n ))=( s ( n )) 2 −( s ( n− 1) s ( n− 2).
 
     Enhanced PASED 
     The enhanced PASED  230  takes the multi-component signal  210  as input. Let θ=[K, N 1 , . . . , N K ] denote the signal parameters  221  to be estimated, where K is the number of components in the composite signal and N i  is the period of the i th  component for i=1, 2, . . . , K. The DDF  220  is used to estimate these parameters. After {tilde over (θ)} is available, the matrix S is formed as in equation (4). 
     The enhanced PASED method uses linear prediction coefficients of sinusoids to form the zero DC constraints. For instance, for a sinusoid of v(n)=A cos(wn), where A is amplitude and is the frequency in radian, three sample linear prediction coefficient filter satisfies
 
 v ( n )−2 v ( n− 1)cos( w )+ v ( n− 2)=0.  (8)
 
     The linear prediction coefficient filter can be designed in any length, e.g., N 1 , N 2 ) Equality in equation (8) is satisfied only when w is the true frequency. The DDF  220  only provides an estimate of w. Therefore, this estimate is subject to error. The invention uses the TKED output w  265  as feedback to determine whether w is equal to the true frequency, which becomes a DC level when the values converge and are the same as a termination conditions. 
     Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the appended s to cover all such variations and modifications as come within the true spirit and scope of the invention.