Higher-Order Differential Energy Operators - CiteSeerX

Higher-Order Differential Energy Operators Petros Maragos and Alexandros Potamianos Abstract


  
 
 of integer orders  are proposed to measure the cross Instantaneous signal operators  energy between a signal and its derivatives. These higher-order differential energy operators contain as a special  , the Teager-Kaiser operator. When applied to (possibly modulated) sinusoids they yield several new case, for  energy measurements useful for parameter estimation or AM–FM demodulation. Applying them to sampled signals involves replacing derivatives with differences which leads to several useful discrete energy operators defined on an extremely short window of samples.

I. H IGHER -O RDER E NERGY M EASUREMENTS Instantaneous differences in the relative rate of change between two signals

!

bracket

can be measured via their Lie

 #"%$'& )(*)& !

!

+ , ".-/0214356& -78 (9350& -/8 . Dots denote time derivatives. Note the antisymmetry:  #"%1:( 6," . If 21'& , [1], [2]  #" becomes the continuous-time Teager-Kaiser energy operator ! ; 3?(@BC A 1 + & "

because ! then

!

 

which has been used for tracking the energy of a source producing an oscillation [2], [1] and for signal and speech AM–FM demodulation [4], [5]. In the general case, if  and  represent displacements in some generalized motions, !

+ & "B1D)& E& (FG A has dimensions of energy (per unit mass), and hence we may view it as a ‘cross energy’ between  and  . This energy-like quantity H & I& (JB A was used in [2], [3] to analyze the output ; 3  V   1:( > and 1\[ . Running this recursive equation in both forward and backward order with initial conditions index L yields i ! O P   3heK 8 "61 k l [ V,u P L21 ]_^]0]abaca  3X(x^/8 >   L21 []Q`]
abaca  If the amplitude and/or frequency  of d3   3he K 8 : estimate the amplitude and frequency of a (possibly modulated) sinusoid +3.e,8=1 ! O O 3 $ O 1 3> .8 #" "1 ( O% 3 x1:( ` O 3

undamped cosine energy equations

for

LZ1 `5 

O

>

,

O , and the

, a discrete algorithm

was proposed in [6] for instantaneous frequency tracking, which is closely related to the discrete energy separation algorithm in [5]. We conclude by noting that, all the above discrete higher-order energy operators can be unified as special cases

!





P 

! ! ! ! P  3  > 3

8 3 KFL08
" P can be generated recursively from operators of lower orders L6 . In addition, each  for

III. A LTERNATIVE D ISCRETIZATIONS Instead of discretizing the Lie bracket and replacing derivatives with time shifts, an alternative approach to

O P

operators



T V

oM.N

  T%V   1   3   8 . For L21p` the asymmetric difference ; yields a 1-sample shifted version of the discrete energy operator , whereas the symmetric difference yields a ; 3-point average of , as shown in [4]. For LZ1  using the   difference yields another asymmetric discrete energy

discretizing

is to replace each

  1   3 



8



-order signal derivative involved in its expression with backward difference

or symmetric differences

velocity operator

O  3  3     8 O   3