On Higher Order Positive Differential Energy Operator

0.13 MB·

by  Amirhossein Javaheri

1
On Higher Order Positive Differential Energy
Operator
∗
Amirhossein Javaheri , Mohammad Bagher Shamsollahi, Senior Member, IEEE
Abstract—The higher order differential energy operator mth-orderderivativesin(1)withbackward x[n]−x[n− 1]or
(DEO), denoted via Υk(x), is an extension to the second order symmetric1/2(x[n+1]−x[n−1])differencesofsameorder,
famous Teager-Kaiser operator. The DEO helps measuring the
7 wherex[n]=x(nTs).Aswiththecontinuous-timederivative,
higher order gauge of energy of a signal which is useful for
1 the mth orderdiscrete differenceis obtainedby consecutively
AM-FMdemodulation.However, theenergycriteriondefinedby
0 the DEO is not compliant with the presumption of positivity of applying the first order difference m times. i.e.:
2
energy.Inthispaperweintroduceahigherorderoperatorcalled ∆(m)(x)≡∆ (∆(m−1)(x)) (4)
n Positive Differential Energy Operator (PDEO). This operator t t t
a which can be obtained using alternative recursive relations, where t denotes the type of difference (backward or symmet-
J resolves the energy sign problem. The simulations demonstrate
9 that the proposed operator can outperform DEOs in terms of ric). With ∆b representingthe first-order backward difference
we have:
AverageSignaltoErrorRatio(ASER)inAM/FMdemodulation.

] ∆ ≡∆(1)(x)=x[n]−x[n−1]=∆ [n]∗x[n] (5)
D b b b
IndexTerms—Teager-KaiserEnergyOperator,PositiveDiffer-
S ential Energy Operator, Discrete EnergySeparation Algorithms, In fact∆b is a linear operatorwith ∆b(z)=(1−z−1), the Z
s. Average Signal to Error Ratio. transform of its impulse response ∆b[n]. Thus for backward
c difference of higher order we have ∆(m)(z) = (1−z−1)m.
b
[ Hence, applying the inverse Z transform we get:
I. INTRODUCTION

v1 THE-continuous-timeDEO of order k denoted via Υk(x) ∆(m)(x[n])= m (−1)j m x[n−j] (6)
is defined as follows [1]: b j
4 j=0 (cid:18) (cid:19)
X
3
Υ (x)≡[x,x(k−1)]=x˙x(k−1)−xx(k),k =0,1,2,... (1) Similar approach for symmetric difference of order m yields:
8 k
03 where [,] denotes the Lie bracket and x(k) denotes the kth- ∆(m)(x[n])= 1 m (−1)j m x[n+m−2j] (7)
order derivative of x(t). The DEOs are generally applied s 2m j
1. in algorithms which help demodulate AM-FM signals also Xj=0 (cid:18) (cid:19)
0 knownasEnergySeparationAlgorithms(ESA)[2],[3].These Now replacing x(m)(t) with backward or symmetric discrete
7
algorithms are used in several applications in biomedical and difference of order m, the discrete symmetric and backward
1
: communications signal processing [4], [5], [6], [7], [8]. The DEOs are obtained [1]. For backward DEO denoted by Υkb
v continuous-time Teager-Kaiser Energy Operator (TKEO) [9], we have:
i
X [10],[11]alsodenotedbyΨ(x)isaspecialcaseofDEOwith Υ (x )=∆ (x )∆(k−1)(x )−x ∆(k)(x ) (8)
r k =2.Thereisalsoarecursiverelationbetweenacontinuous- kb n b n b n n b n
a Also for symmetric DEO (Υ ) we get:
time DEO of order k and DEOs of preceding orders [1]: ks
d Υks(xn)=∆s(xn)∆s(k−1)(xn)−xn∆(sk)(xn) (9)
Υ (x(t))= Υ (x(t))−Υ (x˙(t)) (2)
k k−1 k−2
dt Now usingthe definitionin (6), we can restate (8) in termsof
The discrete-time DEO is obtained by discretizing the ordinary DEOs:
continuous-time definition of Υ (x). There are several ap-
k k
k−1
proaches to discretization of (1) as proposed in [1]. One Υ (x )= (−1)j Υ (x ) (10)
kb n j−1 j n−j+1
approachis to first discretize the Lie bracketand then replace j=1 (cid:18) (cid:19)
X
derivatives with respective time shifts (ordinary differences),
Generalizing the definition of (3) to account for negative
which yields:
values of k, we may write:
Υk(x[n])=x[n]x[n+k−2]−x[n−1]x[n+k−1] (3) Υk(xn)=−Υ2−k(xn+k−1), k <0 (11)
The resulting operator is a discrete DEO which hereafter we Thus for symmetric DEO it can be shown that:
callitdiscreteordinaryDEO1.Anotherchoiceistoreplacethe
k−1
1 k−1
∗Amirhossein Javaheri and Mohammad Bagher Shamsollahi are both Υks(xn)= 2k (−1)j j Υk−2j(xn)
with Sharif University of Technology, Tehran, Iran. (email: java- (Xj=0 (cid:18) (cid:19)(cid:16)
heri [email protected]; email:[email protected])
1Samenotationisusedinthispaperforbothcontinuous anddiscrete-time +Υ (x ) (12)
k−2j n+1
ordinaryoperators obtained viasimplediscrete timeshift )
(cid:17)
2
The DEOs are generally used to help demodulate AM-FM with the initial conditions Υ (x ) = 0 and Υ (x ) =
1a n 2a n
signals like x(t) = a(t)cos(φ(t)). Therefore it is important Ψ(x ). Using equation (6) the output of the alternative DEO
n
forDEOstooutputasmuchsimplifiedcombinationofinstan- for a discrete cosine would also be acquired as:
taneous amplitude and frequency of the signal as possible.
D (A,Ω):=Υ (Acos(Ωn+θ))=
ThesimplestAM-FMsignalis a simplecosine.Theoutputof ka ka
continuous-time DEO to this signal is [1]: 0, k=3,5,...
Ω Ω (19)
0, k =1,3,5,... ((−1)1+k2A2sink cos2 , k=2,4,...
Υ (Acos(ωt+θ))= (13) 2 2
k (−1)1+k2A2ωk, k =2,4,6... (cid:16) (cid:17) (cid:16) (cid:17)
(cid:26)
For the discrete counterpart x[n] = Acos(Ωn+θ), discrete II. POSITIVE DEO
ordinary DEO yields [1]: We can obtain other types of DEOs by considering (2) in
otheralternativediscreteforms.Forinstance ifwe discardthe
D (A,Ω):=Υ (Acos(Ωn+θ))=A2sin(Ω)sin[(k−1)Ω]
k k
separatecasesof evenand oddk in recursivedefinitionofthe
(14)
alternative DEO (18) and consider the general case:
Using the expansions (6) and (7) the output of discrete
backwardandsymmetricDEOstox[n]wouldalsobeobtained
Υ (x )=∆ (Υ (x ))−Υ (∆ (x )) (20)
as: ka′ n b k−1a′ n k−2a′ b n
we obtain a new operator which may be formulated as:
D (A,Ω):=Υ (Acos(Ωn+θ))=
kb kb
2k(−1)k+21A2sink(Ω2)sin (k2 −1)Ω2 +sin k2Ω Υka′(xn)= ⌊k2⌋(−1)i k−i2−i ∆b(k−2−2i) Ψ ∆(bi)(xn)
2k(−1)1+k2A2sink(Ω)cos(cid:0)(k2 −1)Ω(cid:1)k =+1co,3s(cid:0),5k2,Ω.(cid:1).. The output oXif=t0his ope(cid:18)rator to x[n(cid:19)]=Acos(Ω(cid:16)n+(cid:0) θ) is: (2(cid:1)1(cid:17))
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
k =2,4,6...
 (15) Dk0a,′(A,Ω):=Υka′(Acos(Ωn+θ))= k =3,5,...
And
(−1)1+k2Ψ ∆(k2−1) Acos(Ωn+θ) , k =2,4,...
D (A,Ω):=Υ (Acos(Ωn+θ))  b !
ks ks (cid:16) (cid:17)
(22)
0, k =1,3,5,...
= (16) 
(−1)1+k2A2sink(Ω), k =2,4,6... Inspiring from this relation, we can introduce a new operator
(cid:26)
which we call it Positive Differential Energy Operator or
The discrete symmetric DEO is preferred to discrete ordinary
PDEO. The term positive emphasizes the positivity of total
andbackwardDEOsinthesensethatityieldsidenticallyzero
energy measured via this operator. The total energy E of a
output for odd k and is much more similar to the output of t
signal x(t) is defined to be the integral of energy operator
continuous-timeDEO (13). Butthere isstill anotherapproach
output of the signal O(x), over time domain, i.e.,
to discretize continuous DEO as proposed in [1] which is
referred to as discrete alternative DEO. This approach is ∞
E = O(x(t))dt (23)
based on shifting the center point time location of the sample t
windows where discrete differences used to approximate (1) Z−∞
aredefinedupon.Infacttheaimofalternativediscretizationis ForthepopularSquaredEnergyOperator(SEO),wecanstate
to equalize the center point of sampling windows for discrete the total energy in terms of energy spectral density of the
approximation of both terms x˙x(k−1) and xx(k) (For more signal:
details refer to [1]). Thus the definition of discrete alternative ∞ 1 ∞
DEO is as follows: Et = x2(t)dt= |X(jω)|2dω (24)
2π
Z−∞ Z−∞
Υ (x )=
ka n wherewehaveassumedx(t)isareal-valuedsquareintegrable
∆b(xn)∆b(k−1)(xn+k−1)−xn∆(bk)(xn+k−1) k =3,5,... signal. This criterion always yields a positive value for the
(∆b(xn+1)∆b(k−1)(xn−21+k2)−xn∆(bk)(xn+2 k2) k =2,4,... emnaeyrgoybtoafinthaenseiggnaatilv.eHvoawlueeveorf, tfootralDeEnOergdyefifonredsovmiae(k1.)Fwoer
(17)
exampletheDEO-basedtotalenergyofarealcontinuous-time
It can be easily shown that the alternative DEO satisfies the signal x(t) is:
recursive relation below for k >2:
∞ 1 ∞
Υk(x(t))=(−1)1+⌊k2⌋ |X(jω)|2ωkdω (25)
Υ (x )= π
ka n Z−∞ Z−∞
∆ (Υ (x ))−Υ (∆ (x )) k =3,5,...
b k−1a n k−2a b n+1 whichalwaysyieldsa negativevaluefork =4l.Theproblem
∆ (Υ (x ))−Υ (∆ (x )) k =2,4,...
(cid:26) b k−1a n+1 k−2a b n of energy sign also appears for discrete-time DEOs assuming
(18)
that the total energy of a discrete-time signal is computedvia
3
E = ∞ O(x[n]). For ordinary discrete-time DEO we its time derivative is identically zero and thus we can discard
n n=−∞
have: d(Υ (x)) from (2) and only write:
P dt k−1
∞ 1 π
Υk(x[n])= Dk(|X(ejΩ)|,Ω)dΩ Υkp′(x)=−Υk−2p′(x˙) (32)
π
n=−∞ Z−π
X when applied for even values of k this recursive relation with
1 π
= |X(ejΩ)|2sin[(k−1)Ω]sin(Ω)dΩ initial condition Υ (x)=Ψ(x), yields:
π 2p′
Z−π
(26) 0 k =1,3,5,...
Υ (x)= (33)
whichgives−41 forX(ejΩ)=jsin(Ω) andk =4. Ingeneral kp′ (cid:26)(−1)1+k2Ψ(x(m2)) k =2,4,6,...
for discrete DEOs it can be shown that:
which only differs in sign from what we defined in (29) for
∞ 1 π even values k.
Υ (x[n])= D (|X(ejΩ)|,Ω)dΩ (27)
kt π kt
n=−∞ Z−π
X
III. DISCRETE PDEO
where t denotes the type of DEO (backward, symmetric or
alternative) and D denotes the (desired) output to a simple Similar approach to what was explained in section I for
kt
cosine as obtained in (15), (16) and (19). For example for discretization of continuous-time DEO can be applied to
symmetric DEO we may write (x[n] is real): discretize(29).Ifwe usesimpletime-shift(orordinarydiffer-
ence)toapproximatediscrete-timederivate,thenformth-order
∞ 1 π
n=−∞Υks(x[n])=(−1)1+⌊k2⌋π Z−π|X(ejΩ)|2sink(Ω)dΩ cboenxti(nmu)o(ut)s-≈timxend−erxivna−timve.,Stuhbesdtiitsuctrinetgetahpisprionx(i2m9a)tiaonndwuosiunlgd
X (28) (11) we get:
whichisalwaysnegativefork =4l.Sincea negativevalueof
energy is not physically meaningful, it is required to define a Ψ(xn−xn−m)=Ψ(xn)+Ψ(xn−m)−Υ−m+2(xn)
newoperatorforaccuratelymeasuringthehigherorderenergy −Υ (x ) (34)
m+2 n−m
ofa signal. Thisoperatorshouldalwaysyielda positivevalue
for the total energy and the proposed operator, PDEO, does Substituting m from (30) yields the ordinary discrete PDEO:
so. The PDEO of order k for a continuous-time real-valued
Υ (x )=
signal is defined as: kp n
Ψ x(mk) , k =1,3,5,... Ψ(xn)+Ψ(xn−k−21)−Υ−k2+5(xn)−Υk+23(xn−k−21)
k =1,3,5,...
Υ (x)= (29) 
kp Υ−(cid:16)1 x(m(cid:17)k) , k =2,4,6... Ψ(xn)+Ψ(xn−k2+1)−Υ−k2+3(xn)−Υk2+1(xn−k2+1)
k =2,4,6,...
(cid:16) (cid:17)
whereΥkp(x)denotesthekth-orderPDEOandx(mi) denotes  (35)
them th-ordercontinuous-timederivative.Thevalueofm is
i k Note that Υ (x ) ≡ Υ (x ) ≡ 0. The symmetric or
chosen such that a PDEO of order k measures the kth-order 1p n 2p n
backward difference approximation of (29) would yield quite
energy or kth-order spectral moment of energy of a signal.
amorecomplexexpression.Tosymmetricallyapproximatethe
Hence:
continuous-timederivative,we take use of (7). Thuswe have:
k−1
, k =1,3,5,...
mk =k 2 (30) Υkps(x)≡Ψ(∆(smk)(xn))=
2−1, k =2,4,6... 1 mk mk mk mk (−1)i+jx x −
Now if we measure the total energy of a real-valued signal 22mk i=0j=0(cid:18) i (cid:19)(cid:18) j (cid:19) n−mk−2i n−mk−2j
XX
based on PDEO, we will get 1 mk mk m m
k k (−1)i+jx x
Z−∞∞Υkp(x(t))dt= π1 Z−∞∞ωk|X(jω)|2dω (31) 22mk Xi=0Xj=0(cid:18) i (cid:19)(cid:18) j (cid:19) n−1+mk−2i n+1+mk(−362j)
The advantage of (31) over (25) is that for even k, the inte-
where Υ (x) denotes the symmetric PDEO of order k and
grand is an even functionof ω and thus E is always positive kps
t it should be noted that m is a function of k as in (30). This
k
(sincewehaveassumedx(t)isrealandconsequently|X(jω)|
long expression can be restated in terms of discrete ordinary
is realandeven[12]).This impliesthatthe PDEO based total
DEOs in a more compact form:
energy is always of non-negative sign. We may have another
approachto (29) by consideringthe recursive relation (2) and Υ (x)=
eliminatingtheterm ddt(Υk−1(x)).Weconsequentlyobtainan kps 1 mk mk m m
operator similar to (29) for the case of even k. In fact since k k (−1)i+jΥ (x )
the output of DEO for a simple cosine is considered to be 22mk i=0j=0(cid:18) i (cid:19)(cid:18) j (cid:19) 2(i+j−1) n+mk−2j
XX
either zero (for odd k) or a constant function of A and Ω, (37)
4
The same procedure can also be applied to obtain the mathe- For comparison we consider two separate classes of AM and
matical expression for backward PDEO. i.e.: FM signals. The FM signals are produced as:
Υ (x)≡Ψ(∆(mk)(x ))
kpb b n x [n] =Acos Ω [n] , Ω [n]=Ω (1+βq[n])
= mk mk mik mjk (−1)i+jΥ(i−j+2)(xn−i) FM λ,β Xn i ! i c (44)
Xi=0Xj=0(cid:18) (cid:19)(cid:18) (cid:19) where Ωc is the carrier frequency, β is the variable denoting
(38)
the ratio of FM modulation, i.e., β = Ω /Ω , and q[n] is a
∆ c
Furthermoreitcanbeshownthatthetotalenergyofadiscrete normalizedmessage signal. We assume q[n] is a cosine in the
signal based on discrete PDEOs satisfies equation (27) where form of q[n]=cos(λΩc+θ) where λ is the ratio of message
thedesiredtermsD ,D ,D forordinary,backwardand signalbandwidthtothecarrierfrequency.TheAM signalsare
symmetric PDEOs rkepspecktpivbely,kpasre given as follows: also produced according to:
Dkp(A,Ω)=Υkp(Acos(Ωn+θ))= xAM[n]λ,β =Ai[n]cos(Ωcn+θ), Ai[n]=A(1+βq[n(4])5)
k−1
4A2sin2(Ω)sin2 Ω k =1,3,5,... We consider combinations of AM and FM signals in which
4
4A2sin2(Ω)sin2(cid:16)k−4 2Ω(cid:17) k =2,4,6,... 0b.o0t1h:λ1)a.nFdorβeavcahryv.aWlueecohfoβosweeλav∈er(a0g:e0.t0h2e:A0S.5E)Ranodveβrλ∈a(n0d:
(cid:16) (cid:17) (39) thenwe plotthe ASER in dBversusβ. Fig. 1 and2 showthe
 ASER performance of DEOs and PDEOs of different orders
for Ω = π. As depicted in Fig. 1, discrete symmetric PDEO
D (A,Ω)=Υ (Acos(Ωn+θ))= c 2
kpb kpb outperformsdiscretesymmetricDEO andmostly achievesthe
2k−1A2sin2(Ω)sink−1 Ω k =1,3,5,... highestASERforbothAMandFMdemodulationforβ ≈1.It
2
2k−2A2sin2(Ω)sink−2 Ω k =2,4,6,... isalsoobviousthatordinaryandbackwardPDEOshavebetter
(cid:26) (cid:0)2(cid:1)
(40) performance compared to their DEO counterparts, especially
(cid:0) (cid:1)
for large values of β in AM/FM demodulation.
D (A,Ω)=Υ (Acos(Ωn+θ))=
kps kps
A2sink+1(Ω) k =1,3,5,... V. CONCLUSION
(41)
A2sink(Ω) k =2,4,6,... In this paper we introduced a higher order energy operator
(cid:26)
called PDEO which resolves the energy sign problem and
It is clear that the desired terms above are even functions of
yieldspositivevalue of totalenergyforall k. We showedthat
Ω and hence the PDEO-based total energy is positive.
this operator may also be derived using both the definition of
thediscretealternativeDEOandthecontinuous-timerecursive
IV. NUMERICAL ERROR ANALYSIS relation(2)discardingthetermcorrespondingtooddorderop-
erators.We obtainedrelationsforboth the continuousandthe
In this section we compare the performance of different
discrete-time definitions of PDEO. We used two approaches
types of discrete DEOs we have already studied based on the
todiscretizePDEOusingbackwardandsymmetricdifferences
errorofapproximation.Theapproximationerrorisdefined(as
to approximate continuous-time derivatives. We also obtained
in[13])tobetheundesiredtermsavailableintheoutputofthe
formulas for the output of discrete DEOs and PDEOs to
energy operator to an AM-FM input signal while the desired
simple periodic signals. The simulation results showed that
term refers to the multiplication of instantaneous amplitude
the proposed PDEO can outperform higher order discrete
and frequency. This definition of course is valid for second-
DEOs in terms of average signal to error ratio. This helps
order operator, but we can extend it to higher order operators
developing energy separation algorithms applicable for more
as well. In fact the output approximation error of a higher
accurate amplitude and frequency extraction which can also
order operator is defined as the differencebetween the output
be more robust to noise
of the operator to the AM-FM signal x[n] = A[n]cos(φ[n])
andtheoutputtoasimplecosinex[n]=Acos(Ωn+θ)where
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ave
5
ASER of Energy Operators of Order 4 ASER of Energy Operators of Order 6 ASER of Energy Operators of Order 8
25 20 20
Ordinary DEO Ordinary DEO Ordinary DEO
Backward DEO Backward DEO Backward DEO
Symmetric DEO Symmetric DEO Symmetric DEO
20 AOlrtienranrayt iPveD EDOEO 15 AOlrtienranrayt iPveD EDOEO 15 AOlrtienranrayt iPveD EDOEO
Backward PDEO Backward PDEO Backward PDEO
15 Symmetric PDEO Symmetric PDEO Symmetric PDEO
B) B)10 B)10
R(d10 R(d R(d
E E E
S S S
A A 5 A 5
5
0 0
0
-5 -5 -5
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
β β β
(a)k=4 (b)k=6 (c)k=8
Fig.1. ASERperformance ofdiscrete DEOsandPDEOsverseβ forFMmodulation (44)withΩc= π2.
ASER of Energy Operators of Order 4 ASER of Energy Operators of Order 6 ASER of Energy Operators of Order 8
22 20 20
Ordinary DEO Ordinary DEO Ordinary DEO
20 BSaymckmweatrrdic D DEEOO 18 BSaymckmweatrrdic D DEEOO 18 BSaymckmweatrrdic D DEEOO
18 AOlrtienranrayt iPveD EDOEO 16 AOlrtienranrayt iPveD EDOEO 16 AOlrtienranrayt iPveD EDOEO
16 BSaymckmweatrrdic P PDDEEOO 14 BSaymckmweatrrdic P PDDEEOO 14 BSaymckmweatrrdic P PDDEEOO
R(dB)14 R(dB)1102 R(dB)1102
SE12 SE SE
A A 8 A 8
10
6 6
8 4 4
6 2 2
4 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
β β β
(a)k=4 (b)k=6 (c)k=8
Fig.2. ASERperformance ofdiscrete DEOsandPDEOsverseβ forAMmodulation (45)withΩc= π2.
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