Statistical analysis of multipath fading channels using shot-noise
Statistical Analysis of Multipath Fading Channels using Shot-Noise Analysis: An Introduction C. D. Charalambous* University of Ottawa S.I.T.E.
N. Menemenlist
Abstract - In this paper, we introduce, the use of the shot-noise analysis, brought forward by Rice [l],as a natural tool in computing the various statistical properties of the multipath fading channels, in wireless communications. By adapting and generalizing this theory we derive the various statistical properties of thc channel, including the second-order statistics, generalizations of Campbell’s theorem and central-limit theorems.
1
Introduction
A statistical temporal model which captures thc timevarying and time-spreading properties of thc channel, the so-called Multipath Fading Channel model (MFC), is represented by the following low-pass equivalent impulse response (see [2, 31):
Here Ce(t;T) denotes the response of the channel at time t , due to an impulse applied at time t - T , rx(t,T ) ,@ % ( t , ~ )denote - ~ ~ (the t ) attenuation, phase and propagation time delay, respectively, of the signal received via the ith path, and N ( t ) denotes the number of paths at time t . The phase @ % ( t , T )is typically a function of the carrier frequency, the relative velocity between the transmitter and the receiver, and the angle of arrival of the plane wave onto the receiver (see [4, 5, 61). Consider the output of the channel when the input is the low-pass *Associate Professor at the School of Information Technology and Engineering, University of Ottawa. Adjunct Professor with Department of Electrical and Computer Engineering. McGill University. Associate Member of Centre for Intelligent Machines. McGill University. This Work was supported by the Natural Science and Engineering Research Council of Canada under Grant OGP0183720. Ph.D. student Department of Electrical and Computer Eng.. McGill University. Montreal. P.Q.. Canada. #Post-doctoral Fellow at the School of Information Technology and Engineering. University of Ottawa. §AssociateProfessor at the School of Information Technology and Engineering. University of Ottawa.
0-7803-7097-1/01/$10.00 02001 IEEE
0. H. Kabranod D. Makrakiss
McGill University
University of Ottawa S.I.T.E.
signal ze(t);then N(t>
2=
1
and its band pass representation is given by
The objective of this paper is t o investigate the statistical properties of the received signal when { T i ( t ) } i ? i are the points of the counting process N ( t ) ,while for fixed sample paths of the points the distribution of the instantaneous amplitude and phase {ri(t,~ i ( t )@i(t,.ri(t))>, )! i = 1 , 2 , .. . are arbitrary. The aim is t o bring forward the statistical properties of MFC by performing an analysis which can be viewed as a generalization of the shot-noise analysis investigated by Rice [l]in the mid 1940’s. Throughout this paper we consider homogeneous and non-homogeneous Poisson processes while additional generalizations such as doubly stochastic Poisson processes which assume a random rate are found in [7]. Our analysis shows that the non-homogeneous Poisson process performs vcry well when we examine the statistical properties of the overall received signal. Furthermore, the Poisson counting process is the most natural t o start the analysis with, because of its simplicity, and because it will form the core for subsequent generalizations (see [71). Unfortunately, t o the best of the author’s knowledge, when N ( t ) is a Poisson process, no comparison between the received signal (1.2) and experimental data has been reported in the literature. Nevertheless, the statistical analysis of the received signal performed here explains many of its observed properties, by using a counting model as simple as the non-homogeneous Poisson process. Specifically, the calculation of second-order statistics, power spectrum densities, and moment generating functions reveals that when the rate of the Poisson process is sufficiently large, the received signal is normally distributed with mean and covariance functions identified. On the other hand, when the rate of the Poisson process is small, the received signal can no longer assumed t o be normally distributed. In the latter case, the probability that the individual paths overlap is negligible, while in the former case this probability is quite high.
1011
N o t a t i o n 1.1 E wall denote the expectataon operator: A /cI2 = c*c, where c E C as complex and '*" denotes complcx conjugataon: For I E C(Cm;C") a lznear operator, Itdenotes Hermztaan conjugatzon; For p E C". A n where p ~ =, Re(p,) and P I , = I m ( p z ) , l 5 z 5 n denote the real and amaganary components of p , respec-
Lemma 2.1 [a] Consader model (2.3) and Assumptzons 1.2* T h e n T S
E[he(t.~m(~))]d.r
E[ye(t)] =
ITS
a aP% b
("
-j&)/2,
aPR,
Xxe(t - T ) d r ,
1 f a f n . For f , g real or complex-valued functaons f
*g
denotes convolution operation off with g and .FT notes Fourier Transform ( F T ) . A s s u m p t i o n s 1.2 Let {X(s);O 5 s 5 t } denote the rate of the countang process { N ( s ) ; O f s 5 t } , whach as nonnegatave. For fixed r2, the random processes { h ( t , r ; m , ( t , r ) } t > o f o r a = 1 , 2 , 3 , ... are mutually zn-
dependent, adentacally dastmbuted havang the same dastrzbutzon as { h ( t ,r ;m(t,r)}t>o, and zndependent of { N ( s ) ;0 5 s 5 t } .
2
ye(t) =
r2(rz)e3d~e-3(wcfwd,)r,f3wd 2=
1
* ze(t - rz)
1
N(TS)
2
he(t,rz;mZ(rz)),0 2=
I t I T,,
(2.3)
1
X
E 1 he (t.r ;m ( r ) )
=
X I "
E
0
Ixe(t - r ) I 2 d r ,
5 t 5 T,.
(2.6)
5 T,} as a non-homogeneous Poasson process with rate X x X , ( t ) then the results of Lemma 2.1 are replaced by
5 t 5 T,.
2. Consider the periodic transmission of a pulse xq(t) =
6,
r ( t ) every T, seconds, where r ( t ) = af 0 5 t 5 T, and r ( t )= O! otherwise, where T, >> Tm, with Tm denoting the duration of the channel impulse response. Suppose the low-pass impulse response is time-invariant so the received signal is N
=
and its band-pass represcntation
[.'()I
d7
1. If { N ( t ) ; O5 t
0
N(Ta)
1'
=
(2.5)
Remark 2.2 Some observatzons concemang the results of Lemma 2.1 are now an order.
Mean, Variance and Correlation
This paper investigates the statistical properties of a simplified, non-causal, version of (1.2), namely,
0 5 t 5 Ts?
[
T S
V a r( y e ( t ) )
l
E [r(r)e3d+lwd("-T)e-/w,r
=
tavely. T h e complex demvataves wzth respect to p and p* are defined zn terms of real deravataves as follows:
riejdie-3(W=fwdi)Tifjwd.*
*
r(t - T i ) ?
i=l
(2.7)
A
1 h(t
where N,{ ~ i } y = is ~a realization of the Poisson process (i.e., known). Then the energy received over [0,Tm] at some t o E [ O , l , ] is defined by 191 - Tz;mz(Ti)), 0
5 t 5 T,,
(2.4)
z=1
where he(t.T ~m2(T2)) ; = r z ( r z ) e 3 d ~ e - 3 w ~ r ~ + ~ w dze(t ~ ( *- TTz), ~) h(t,rZ;mz(rz))= r2(r2)x(t - .rZ) and x ( t ) = R e {ze(t)e3((wc+wd.)tfdZ). Although the homogeneous Poisson process may be considered as simplistic, its analysis helps explaining many of the concepts associated with its generalizations t o the non-homogeneous and the doubly stochastic Poisson process. Moreover, when X is a random variable the results of this paper remain valid provided we include an extra integration with respect to the density of A. Mean and Vamance. The mean (expected value) and the variance of the received complex signal are computed explicitly in the next lemma.
1
0-7803-7097-1/01/$10.00 02001 IEEE
1012
which is the tame average of the second-moment of ye,N(t) based o n a single realization over the interval [O, T m ] . Further, i f the multipath components are assumed to be resolved by the probing signal r ( t )(e.g., lrz- rj I > T,, Vi # j ) then
N
i=l
,
The ensemble average power (due to a wide-band signal transmission) is N
ECVB = t=
1
Cowelation and Covariance. The correlation and covariance functions of y e ( t i ) and ye(t2) are computed next.
Lemma 2.3 [B] Consader model (2.3), under AssumpE [?-?(to)]( = N E [r2(to)] a f rt are tad). taons 1.2 and N ( T s ) a non-homogeneous Pozsson wath rate X x A&). Then
Our earlaer equataons calculate &MTB usang ensemble averages. I n partacular, the last expressaon cowesponds to expressaon (2.9) whach as obtaaned under the assumptaon that N ( T s ) = N as fixed, and assumzng E y e ( t ) = 0 , whzch yaelds
-
iv
y ~ , , ~ (=t )
1 O
N
x
5 l c Ts
t2)
Tc
TS
z=1
RYL( t 1
ITS
=
Xc(r)Ek j ( t 1 .r ;m ( r ) )
x
he(tz. 7: m ( r ) ) ] d r
+
X
E [r:(T)]%rect[t T C -2 - 7]dT
T S
I
Xc(r)E[h;(tl,r;m(r))]dr T S
x
lTs
E [r:(T)]d r .
where 0 5
x
x ~ ( ~k L)e (Et 2 , r :m ( r ) )d.r ]
, I Ts.
tl t 2
L=l
O n the other hand, under the assumptions of Lemma 2.1 and assuming Eye@) = 0 , we have from
N
3
Power Spectral Density
Throughout this section it is assumed that { r z ( ~ z ) } zare 2~ independent of T%'s,and thus we denote them by {r,},?~ and N(T,) is homogeneous Poisson. Power Spectral Densaty. The expression for the correlation function and the covariance function are [8]
=xi=, N
if~(r) r i ( t o ) 6 ( T - t o ) , t o E ( 7 , r + T C which ) is proportional to &WB .
3. Consider next the transmission into the channel (2.7) of a continuous-wave signal x e ( t ) = 1. Then the received power, given the realization of { N ( t ) 0; I t 5 T,} is
N
N
Lemma 3.1 [8] Consader model (2.3)-(2.4) and Assumptaons 1.2 with { T ~ ( T ) }independent ~~I of r . Define A a the centered processes yye,c(t) = ye(t) - jji(t), y c ( t ) = v(t)- a(t) and a
ze(t)ej(wc+wd)t+'$ . The power spect r u m density of the centered processes Ve,c(t)3y c ( t ) are
}
0-7803-7097-1/01/$10.00 02001 IEEE
1013
The characteristic function of y(t) i s
Autocorrelatzon and Power Spectral Denszty of Instantaneous Power. Next, we compute similar expressions for
A .
where s = j w and its density is
thc autocorrelation of y 2 ( t ) , under the assumption that
n ( t )= E
[I
y ( t ) = 0.
Theorem 3.2 [8] Conszder model (2.3)-(2.4) and Assumptzon 1.2 wzth {rz(r)}%21 zndependent of r. Further, assume 71( t )= X E [h(t.7; m ) ]d r = 0.
Moreover,
The power spectrum denszty of y 2 ( t ) zs where Y k ( t ) = :XJ X,(r)E[h(t,r;m(r))Ikdr < oc, is the k t h cummulant of y ( t ) and n ( t ) = E[Y(t)], 7 2 ( t ) = V a r ( ? / ( t ) ) . The charactenstic function of TJe(t) as
R e m a r k 3.3 The behavaor of the power spectral densztzes for hzgh and low rates X are obtaaned as follows. 1. Hzgh Rate Approxamatzon. If X as suficzently large, then the thard term zn (3.10) can be neglected and the power spectrum of y 2 ( t ) conszsts of only the first and second nght szde terms of (3.10). 2. Low Rate Approxamatzon. If X zs small, then the probabzlaty that t e m s h(t - rz;m,) and h(t - r3;m3) have sagnaficant overlaps for a # as very small, hence the approxzmatzon
where p
a
= p~
+ j p ~ and , its density is
Moreover, for m?n > 0 integers
N(Ts)
h2(t- r %mx). ;
y 2 ( t )=
(3.11)
z= 1
Thzs as equavalent to assumang that the paths are resolvable by the sagnal x ( t ) .
4
Distribution and Moment Generating Functions
Dastnbutaon and Moment Generatzng Functzons. The probability density function and moment generating functions of y ( t ) and are, respectively, defined by
fy(2,
t)dx
E[ ~ a
A
f ~ ( t ) a ~ } f y] L,
(.e, t ) d z e = Central Limit Theorem. The joint characteristic function
= E [esY(t)] ' @Ye (P, t ) = a . E[eJRe(p*y'(t))],s = p E C, where I{ } is the jw, indicator function.
a
of y(tl), . . . , y ( t n ) and y e ( t l ) , . . . ,ye@,), and their cummulants are obtained following the derivation of Theorem 4.1.
Theorem 4.1 [8] Consader model (2.3)-(2.4) and Assumptzons 1.2, where { N ( s ) ; O < s < Ts}zs nonhomogeneous Pozsson process wath rate X x A,.
C o r o l l a r y 4.2 [8] Consider model (2.3)-(2.4) and Assumptions 1.2, where { N ( t ) ; O t T,} i s a nonhomogeneous Poisson process with rate X x X , ( t ) .
-E [ I { y e ( t ) E d z e } ] >
@Y (s, t )
0-7803-7097-1/01/$10.00 02001 IEEE
1014
N. Menemenlist
Abstract - In this paper, we introduce, the use of the shot-noise analysis, brought forward by Rice [l],as a natural tool in computing the various statistical properties of the multipath fading channels, in wireless communications. By adapting and generalizing this theory we derive the various statistical properties of thc channel, including the second-order statistics, generalizations of Campbell’s theorem and central-limit theorems.
1
Introduction
A statistical temporal model which captures thc timevarying and time-spreading properties of thc channel, the so-called Multipath Fading Channel model (MFC), is represented by the following low-pass equivalent impulse response (see [2, 31):
Here Ce(t;T) denotes the response of the channel at time t , due to an impulse applied at time t - T , rx(t,T ) ,@ % ( t , ~ )denote - ~ ~ (the t ) attenuation, phase and propagation time delay, respectively, of the signal received via the ith path, and N ( t ) denotes the number of paths at time t . The phase @ % ( t , T )is typically a function of the carrier frequency, the relative velocity between the transmitter and the receiver, and the angle of arrival of the plane wave onto the receiver (see [4, 5, 61). Consider the output of the channel when the input is the low-pass *Associate Professor at the School of Information Technology and Engineering, University of Ottawa. Adjunct Professor with Department of Electrical and Computer Engineering. McGill University. Associate Member of Centre for Intelligent Machines. McGill University. This Work was supported by the Natural Science and Engineering Research Council of Canada under Grant OGP0183720. Ph.D. student Department of Electrical and Computer Eng.. McGill University. Montreal. P.Q.. Canada. #Post-doctoral Fellow at the School of Information Technology and Engineering. University of Ottawa. §AssociateProfessor at the School of Information Technology and Engineering. University of Ottawa.
0-7803-7097-1/01/$10.00 02001 IEEE
0. H. Kabranod D. Makrakiss
McGill University
University of Ottawa S.I.T.E.
signal ze(t);then N(t>
2=
1
and its band pass representation is given by
The objective of this paper is t o investigate the statistical properties of the received signal when { T i ( t ) } i ? i are the points of the counting process N ( t ) ,while for fixed sample paths of the points the distribution of the instantaneous amplitude and phase {ri(t,~ i ( t )@i(t,.ri(t))>, )! i = 1 , 2 , .. . are arbitrary. The aim is t o bring forward the statistical properties of MFC by performing an analysis which can be viewed as a generalization of the shot-noise analysis investigated by Rice [l]in the mid 1940’s. Throughout this paper we consider homogeneous and non-homogeneous Poisson processes while additional generalizations such as doubly stochastic Poisson processes which assume a random rate are found in [7]. Our analysis shows that the non-homogeneous Poisson process performs vcry well when we examine the statistical properties of the overall received signal. Furthermore, the Poisson counting process is the most natural t o start the analysis with, because of its simplicity, and because it will form the core for subsequent generalizations (see [71). Unfortunately, t o the best of the author’s knowledge, when N ( t ) is a Poisson process, no comparison between the received signal (1.2) and experimental data has been reported in the literature. Nevertheless, the statistical analysis of the received signal performed here explains many of its observed properties, by using a counting model as simple as the non-homogeneous Poisson process. Specifically, the calculation of second-order statistics, power spectrum densities, and moment generating functions reveals that when the rate of the Poisson process is sufficiently large, the received signal is normally distributed with mean and covariance functions identified. On the other hand, when the rate of the Poisson process is small, the received signal can no longer assumed t o be normally distributed. In the latter case, the probability that the individual paths overlap is negligible, while in the former case this probability is quite high.
1011
N o t a t i o n 1.1 E wall denote the expectataon operator: A /cI2 = c*c, where c E C as complex and '*" denotes complcx conjugataon: For I E C(Cm;C") a lznear operator, Itdenotes Hermztaan conjugatzon; For p E C". A n where p ~ =, Re(p,) and P I , = I m ( p z ) , l 5 z 5 n denote the real and amaganary components of p , respec-
Lemma 2.1 [a] Consader model (2.3) and Assumptzons 1.2* T h e n T S
E[he(t.~m(~))]d.r
E[ye(t)] =
ITS
a aP% b
("
-j&)/2,
aPR,
Xxe(t - T ) d r ,
1 f a f n . For f , g real or complex-valued functaons f
*g
denotes convolution operation off with g and .FT notes Fourier Transform ( F T ) . A s s u m p t i o n s 1.2 Let {X(s);O 5 s 5 t } denote the rate of the countang process { N ( s ) ; O f s 5 t } , whach as nonnegatave. For fixed r2, the random processes { h ( t , r ; m , ( t , r ) } t > o f o r a = 1 , 2 , 3 , ... are mutually zn-
dependent, adentacally dastmbuted havang the same dastrzbutzon as { h ( t ,r ;m(t,r)}t>o, and zndependent of { N ( s ) ;0 5 s 5 t } .
2
ye(t) =
r2(rz)e3d~e-3(wcfwd,)r,f3wd 2=
1
* ze(t - rz)
1
N(TS)
2
he(t,rz;mZ(rz)),0 2=
I t I T,,
(2.3)
1
X
E 1 he (t.r ;m ( r ) )
=
X I "
E
0
Ixe(t - r ) I 2 d r ,
5 t 5 T,.
(2.6)
5 T,} as a non-homogeneous Poasson process with rate X x X , ( t ) then the results of Lemma 2.1 are replaced by
5 t 5 T,.
2. Consider the periodic transmission of a pulse xq(t) =
6,
r ( t ) every T, seconds, where r ( t ) = af 0 5 t 5 T, and r ( t )= O! otherwise, where T, >> Tm, with Tm denoting the duration of the channel impulse response. Suppose the low-pass impulse response is time-invariant so the received signal is N
=
and its band-pass represcntation
[.'()I
d7
1. If { N ( t ) ; O5 t
0
N(Ta)
1'
=
(2.5)
Remark 2.2 Some observatzons concemang the results of Lemma 2.1 are now an order.
Mean, Variance and Correlation
This paper investigates the statistical properties of a simplified, non-causal, version of (1.2), namely,
0 5 t 5 Ts?
[
T S
V a r( y e ( t ) )
l
E [r(r)e3d+lwd("-T)e-/w,r
=
tavely. T h e complex demvataves wzth respect to p and p* are defined zn terms of real deravataves as follows:
riejdie-3(W=fwdi)Tifjwd.*
*
r(t - T i ) ?
i=l
(2.7)
A
1 h(t
where N,{ ~ i } y = is ~a realization of the Poisson process (i.e., known). Then the energy received over [0,Tm] at some t o E [ O , l , ] is defined by 191 - Tz;mz(Ti)), 0
5 t 5 T,,
(2.4)
z=1
where he(t.T ~m2(T2)) ; = r z ( r z ) e 3 d ~ e - 3 w ~ r ~ + ~ w dze(t ~ ( *- TTz), ~) h(t,rZ;mz(rz))= r2(r2)x(t - .rZ) and x ( t ) = R e {ze(t)e3((wc+wd.)tfdZ). Although the homogeneous Poisson process may be considered as simplistic, its analysis helps explaining many of the concepts associated with its generalizations t o the non-homogeneous and the doubly stochastic Poisson process. Moreover, when X is a random variable the results of this paper remain valid provided we include an extra integration with respect to the density of A. Mean and Vamance. The mean (expected value) and the variance of the received complex signal are computed explicitly in the next lemma.
1
0-7803-7097-1/01/$10.00 02001 IEEE
1012
which is the tame average of the second-moment of ye,N(t) based o n a single realization over the interval [O, T m ] . Further, i f the multipath components are assumed to be resolved by the probing signal r ( t )(e.g., lrz- rj I > T,, Vi # j ) then
N
i=l
,
The ensemble average power (due to a wide-band signal transmission) is N
ECVB = t=
1
Cowelation and Covariance. The correlation and covariance functions of y e ( t i ) and ye(t2) are computed next.
Lemma 2.3 [B] Consader model (2.3), under AssumpE [?-?(to)]( = N E [r2(to)] a f rt are tad). taons 1.2 and N ( T s ) a non-homogeneous Pozsson wath rate X x A&). Then
Our earlaer equataons calculate &MTB usang ensemble averages. I n partacular, the last expressaon cowesponds to expressaon (2.9) whach as obtaaned under the assumptaon that N ( T s ) = N as fixed, and assumzng E y e ( t ) = 0 , whzch yaelds
-
iv
y ~ , , ~ (=t )
1 O
N
x
5 l c Ts
t2)
Tc
TS
z=1
RYL( t 1
ITS
=
Xc(r)Ek j ( t 1 .r ;m ( r ) )
x
he(tz. 7: m ( r ) ) ] d r
+
X
E [r:(T)]%rect[t T C -2 - 7]dT
T S
I
Xc(r)E[h;(tl,r;m(r))]dr T S
x
lTs
E [r:(T)]d r .
where 0 5
x
x ~ ( ~k L)e (Et 2 , r :m ( r ) )d.r ]
, I Ts.
tl t 2
L=l
O n the other hand, under the assumptions of Lemma 2.1 and assuming Eye@) = 0 , we have from
N
3
Power Spectral Density
Throughout this section it is assumed that { r z ( ~ z ) } zare 2~ independent of T%'s,and thus we denote them by {r,},?~ and N(T,) is homogeneous Poisson. Power Spectral Densaty. The expression for the correlation function and the covariance function are [8]
=xi=, N
if~(r) r i ( t o ) 6 ( T - t o ) , t o E ( 7 , r + T C which ) is proportional to &WB .
3. Consider next the transmission into the channel (2.7) of a continuous-wave signal x e ( t ) = 1. Then the received power, given the realization of { N ( t ) 0; I t 5 T,} is
N
N
Lemma 3.1 [8] Consader model (2.3)-(2.4) and Assumptaons 1.2 with { T ~ ( T ) }independent ~~I of r . Define A a the centered processes yye,c(t) = ye(t) - jji(t), y c ( t ) = v(t)- a(t) and a
ze(t)ej(wc+wd)t+'$ . The power spect r u m density of the centered processes Ve,c(t)3y c ( t ) are
}
0-7803-7097-1/01/$10.00 02001 IEEE
1013
The characteristic function of y(t) i s
Autocorrelatzon and Power Spectral Denszty of Instantaneous Power. Next, we compute similar expressions for
A .
where s = j w and its density is
thc autocorrelation of y 2 ( t ) , under the assumption that
n ( t )= E
[I
y ( t ) = 0.
Theorem 3.2 [8] Conszder model (2.3)-(2.4) and Assumptzon 1.2 wzth {rz(r)}%21 zndependent of r. Further, assume 71( t )= X E [h(t.7; m ) ]d r = 0.
Moreover,
The power spectrum denszty of y 2 ( t ) zs where Y k ( t ) = :XJ X,(r)E[h(t,r;m(r))Ikdr < oc, is the k t h cummulant of y ( t ) and n ( t ) = E[Y(t)], 7 2 ( t ) = V a r ( ? / ( t ) ) . The charactenstic function of TJe(t) as
R e m a r k 3.3 The behavaor of the power spectral densztzes for hzgh and low rates X are obtaaned as follows. 1. Hzgh Rate Approxamatzon. If X as suficzently large, then the thard term zn (3.10) can be neglected and the power spectrum of y 2 ( t ) conszsts of only the first and second nght szde terms of (3.10). 2. Low Rate Approxamatzon. If X zs small, then the probabzlaty that t e m s h(t - rz;m,) and h(t - r3;m3) have sagnaficant overlaps for a # as very small, hence the approxzmatzon
where p
a
= p~
+ j p ~ and , its density is
Moreover, for m?n > 0 integers
N(Ts)
h2(t- r %mx). ;
y 2 ( t )=
(3.11)
z= 1
Thzs as equavalent to assumang that the paths are resolvable by the sagnal x ( t ) .
4
Distribution and Moment Generating Functions
Dastnbutaon and Moment Generatzng Functzons. The probability density function and moment generating functions of y ( t ) and are, respectively, defined by
fy(2,
t)dx
E[ ~ a
A
f ~ ( t ) a ~ } f y] L,
(.e, t ) d z e = Central Limit Theorem. The joint characteristic function
= E [esY(t)] ' @Ye (P, t ) = a . E[eJRe(p*y'(t))],s = p E C, where I{ } is the jw, indicator function.
a
of y(tl), . . . , y ( t n ) and y e ( t l ) , . . . ,ye@,), and their cummulants are obtained following the derivation of Theorem 4.1.
Theorem 4.1 [8] Consader model (2.3)-(2.4) and Assumptzons 1.2, where { N ( s ) ; O < s < Ts}zs nonhomogeneous Pozsson process wath rate X x A,.
C o r o l l a r y 4.2 [8] Consider model (2.3)-(2.4) and Assumptions 1.2, where { N ( t ) ; O t T,} i s a nonhomogeneous Poisson process with rate X x X , ( t ) .
-E [ I { y e ( t ) E d z e } ] >
@Y (s, t )
0-7803-7097-1/01/$10.00 02001 IEEE
1014
