SPL-05501-2008.R1
1
Optimal Approximation of the Impulse
Response of Wireless Channels by Stochastic
Differential Equations
Seddik M. Djouadi, Mohammed M. Olama, and Yanan Li
Abstract—Wireless
communication
networks
are
characterized by nodes and scatters mobility which make the
propagation environment time varying and subject to fading.
These variations are captured by random time varying impulse
responses. The latter are fairly general finite energy functions of
both time and space that cannot be specified by a finite number
of parameters.
In this paper, we show that the impulse
responses can be approximated in a mean square sense as close as
desired by impulse responses that can be realized by stochastic
differential equations (SDEs). The behaviors of the SDEs are
characterized by small finite dimensional parameter sets that
characterize their behaviors.
Index Terms—Impulse
equations, Hilbert space.
T
response,
stochastic
differential
I. INTRODUCTION
ime-varying (TV) wireless channel models capture both
the space and time variations of wireless systems, which
are due to the relative mobility of the receiver, transmitter
and/or scatterers [1]-[3]. These time variations compel us to
introduce more advanced dynamical models based on
stochastic differential equations (SDEs) and represented in
stochastic state space form with a finite number of parameters.
The SDE models directly address issues such as time varying
fading and noise sources caused by the transmitter population,
and since the SDE models are parametric, they generalize to
diverse propagation environments including log-normal, flat
and frequency selective [1]-[3]. The SDE parameters and
states can be directly estimated from received (time-domain)
signal measurements, which are usually available or easy to
obtain in any wireless network. For instance, Kalman filtering
[4] together with the expectation maximization algorithm [5]
can be used in the estimation process. Moreover, the proposed
models allow the tools of system theory, identification, and
Manuscript received May 13, 2008.
S. M. Djouadi is with the Department of Electrical Engineering and
Computer Science, University of Tennessee, Knoxville, TN 37996 USA.
(phone: 865-974-5447; fax: 865-974-5483; e-mail: djouadi@eecs.utk.edu).
M. M. Olama is with the Computational Sciences and Engineering
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA. (email: olamahussemm@ornl.gov).
Y. Li Author is with the Department of Electrical Engineering and
Computer Science, University of Tennessee, Knoxville, TN 37996 USA. (email: yli24@utk.edu).
estimation [8] to be applied to this class of problems as carried
out in [2, 3, 11] using simulations and experimental data. Due
to the lack of space we refer the reader to papers [2, 3] for
numerical examples which show the effectiveness of the SDE
models in comparison with traditional models, and paper [11]
that uses experimental data to validate the SDE models. There
is a large body of literature in modeling TV processes (see for
example [12-15] to cite a few), but here the focus is on SDE
models since they capture the inphase and quadrature
components of wireless channels as well as received signal
measurements. TV models based on Autoregressive Moving
Average (ARMA) assume the channel state is completely
observable, which in reality is not the case due to additive
noise, and require long observation intervals [12, 14, 15]. Fig.
1 compares the ARMA models with the proposed SDE models
using experimental data, and show that the SDE models are a
better fit.
A necessary and sufficient condition for representing any
impulse response (IR) in stochastic state space form is that it
is factorizable into the product of two separate functions of
time and space. However, in general this is not the case for IR
of wireless channels. In this paper, we show that the IR of
wireless channels can be approximated in the mean square
sense as closely as desired by factorizable impulse responses
that can be realized by SDEs in state space form.
The remainder of this paper is organized as follows. Section
II introduces the IR for TV wireless communication channels.
In Section III, the stochastic state space channel models are
proposed and derived from the TV channel IR. Section IV
provides the conclusion.
II. IMPULSE RESPONSE FOR TIME VARYING WIRELESS
CHANNELS
The general TV model of a wireless channel is typically
represented by the following multipath band-pass impulse
response [6]
C ( t ;τ ) =
J (t )
∑(I
j =1
j
( t ,τ ) cos (ωc t ) − Q j ( t ,τ ) sin (ωc t ) ) δ (τ − τ j ( t ) ) (1)
where C ( t ;τ ) is the band-pass response of the channel at time
t, due to an impulse applied at time t −τ , J(t) is the random
number of multipath components, ωc is the carrier frequency,
{
J (t )
} j =1
and the set I j ( t ,τ ) , Q j ( t ,τ ) ,τ j ( t )
describes the random
SPL-05501-2008.R1
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TV inphase component, quadrature component, and arrival
time of the different paths, respectively. Let sl ( t ) be the low
The space L2 ([0, ∞) ) contains all finite energy signals defined
pass equivalent representation of the transmitted signal, then
the band-pass representation of the received signal is given by
[6]
The impulse response CIR ( t ;τ ) of the channel has finite
y (t ) =
J (t )
∑(
) (
I j ( t ,τ ) cos (ωc t ) − Q j ( t ,τ ) sin (ωc t ) sl t − τ j ( t )
j =1
on [0, ∞) .
energy in time and space, i.e.,
) (2)
CIR
2
2
2
∫∫
:=
CIR (t ;τ ) dτ dt < ∞,
CIR ( t ;τ )∈ L ([0, ∞) × [0, ∞) )
2
+ vI ( t ) cos(ωc t ) − vQ ( t ) sin(ωc t )
where {vI (t )}t ≥ 0 and {vQ (t )}t ≥ 0 are two independent and
identically distributed (iid) white Gaussian noise processes.
Now, we want to represent the TV IR in (1) in stochastic
state-space. The following theorem gives a necessary and
sufficient condition about the realization of the TV IR in state
space form.
Theorem 1: (7, Theorem 10.4 p. 161.) An impulse response
C ( t ;τ ) of a TV system admits a stochastic state-space
realization if and only if it is factorizable, that is, there exist
functions g (⋅) and f (⋅) such that for all t and τ we have
(3)
That is the impulse response C ( t ;τ ) can be written as the
product of a function g (⋅) of only time t and a function f (⋅)
of only τ .
It is readily seen from the expression of the IR C ( t ;τ ) of
the wireless channel that in general it is not factorizable in the
form (3) since the inphase and quadrature components
{I j (t ,τ ) , Q j (t ,τ )}
For fixed n, define the shortest distance minimization in the
i 2 -norm from the impulse response CIR ( t ;τ ) to the
subspace S, by
III. STOCHASTIC STATE SPACE MODELS FOR TIME VARYING
WIRELESS CHANNELS
C ( t ;τ ) = g (t ) f (τ )
can a priori be any functions of t and τ .
However, we will show that in general C ( t ;τ ) can be
approximated as close as desired by a factorizable IR function.
Theorem 2: Assume that the IR C ( t ;τ ) has finite energy, i.e.,
as close as desired by a factorizable IR function of the form
(3).
Proof: Let L2 ([0, ∞) × [0, ∞) ) be the Hilbert space of
Lebesgue measurable and square integrable complex valued
functions defined on [0, ∞) × [0, ∞) with the following mean
square norm
2
2
:=
2
∫∫
f (t ;τ ) dτ dt < ∞, f ( t ;τ )∈ L2 ([0, ∞ ) × [0, ∞ ))
(4)
2
Likewise define L ([0, ∞) ) as the standard Hilbert space of
Lebesgue measurable and square integrable complex valued
functions defined on [0, ∞) under the norm
2
2
(7)
2
where the subspace S is defined as
⎧⎪ n α i (t )ϕi (τ ) : α i (t ) ∈ L2 ([0, ∞) ) , ⎫⎪
S := ⎨ ∑
(8)
⎬
2
⎩⎪ i =1 ϕi (τ ) ∈ L ([0, ∞) ) ; ∀n integer ⎭⎪
Note that the distance minimization problem (7) is posed in
the infinite dimensional space L2 ([0, ∞) × [0, ∞ ) ) .
Since the transmitted and received signals are finite energy
signals, the impulse response can be viewed as an integral
operator mapping transmitted signals in L2 ([0, ∞) ) into
L2 ([0, ∞) ) , i.e., if
sl ∈ L2 ([0, ∞) ) then yl ( t ) ∈ L2 ([0, ∞))
where
yl ( t ) =
J (t )
=
∑(
∫
∞
0
CIR ( t ;τ ) sl ( t − τ ) dτ
) (
I j ( t ,τ ) cos (ωc t ) − Q j ( t ,τ ) sin (ωc t ) sl t − τ j ( t )
j =1
(9)
)
Since the impulse response is finite energy, the operator T is
a Hilbert-Schmidt or a trace class 2 operator [9]. Let us denote
the class of Hilbert-Schmidt operators acting from L2 ([0, ∞) )
into L2 ([0, ∞) ) , by C2 and the Hilbert-Schmidt norm
i
HS
is defined by
T
HS
2
∫∫
=
CIR (t ;τ ) dτ dt
(10)
[0, ∞ )[0, ∞ )
Define the adjoint of T ∗ as the operator acting from
L2 ([0, ∞) ) into L2 ([0, ∞) ) by
∞∞
< Tf , g > 2 := ∫ ∫ CIR ( t ;τ ) f (τ ) dτ g (t ) dt
0 0
=
∞
∞
0
0
(11)
∫ f (τ ) ∫ C ( t;τ ) g (t ) dt dτ =: < f , T
IR
∗
g >1
showing that
[0, ∞ )×[0, ∞ )
x
μ := inf CIR ( t ;τ ) − s (t ;τ )
s∈S
expression (6) below holds, then C ( t ;τ ) can be approximated
f
(6)
[0, ∞ )[0, ∞ )
∞
:= ∫ x(t ) dt ,
0
2
x(t ) ∈ L2 ([0, ∞) )
(5)
∞
(T ∗ g )(τ ) = ∫ CIR ( t ;τ ) g (t )dt
(12)
0
Using the polar representation of compact operators
T = U (T ∗T )1/ 2 [9], where U is a partial isometry and (T ∗T )1/ 2
is the square root of T and admits a spectral factorization of
the form [9]
SPL-05501-2008.R1
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(T ∗T )1/ 2 = ∑ λν
i i ⊗ν i
(13)
i
where ⊗ denoted the tensor product, λi > 0 and λi ↓ 0 as
i ↑ ∞ are the eigenvalues of (T ∗T )1/ 2 , and ν i form the
corresponding orthonormal sequence of eigenvectors, i.e.,
(T ∗T )1/ 2ν i = λν
. Putting Uν i =:ψ i , we can write
i i , i = 1, 2,
T = ∑ λν
i i ⊗ψ i
(14)
i
Both {ν i } and {ψ i } are orthonormal sequences in L ([0, ∞)) .
2
The sum (14) has either a finite or countably infinite number
of terms. The above representation is unique [9]. A similar
argument yields
(TT ∗ )1/ 2 = ∑ λψ
(15)
i i ⊗ψ i
i
and
∗
T =
∑ λψ
i
i
⊗ν i
(16)
i
From (14) and (16) it follows that
Tvi = U (T ∗T )1/ 2 vi = λi ψ i , T ∗ψ i = U ∗ (TT ∗ )1/ 2ψ i = λi vi
(17)
In terms of integral operators expressions, identities (17) can
be written, respectively, as
∞
ν i (t ) = ∫ CIR ( t ;τ )ψ i (τ )dτ ,
0
∞
ψ i (t ) = ∫ CIR ( t ;τ )ν i (t )dt
HS
is given by T
HS
0
=
∑λ
2
i
Pψ onto Span{ψ j , j = 1, 2,⋅⋅⋅, n} . These projections have finite
rank and since ν j ’s and ψ j ’s are orthogonal vectors in
L2 ([0, ∞)) , it can be easily verified that Pν and Pψ are given
by
n
n
j =1
j =1
( Pν f )(t ) = ∑ < f , ν j >1 ν j (t ) = ∑ ( ∫
n
n
∞
j =1
j =1
0
∞
0
f (t )ν j (t )dt )ν j (t )
(23)
( Pψ G )(τ ) = ∑ < G, ψ j >2 ψ j = ∑ ( ∫ G (τ ) ψ j (τ )d τ )ψ j (τ )
The overall orthogonal projection PS can be computed as the
tensor PS = Pν ⊗ Pψ . That is, if W ∈ C2 has spectral
∑
decomposition
i =1
ηi ui ⊗ υi ,
ηi > 0 ,
where
ui , υi ∈ L ([0, ∞)) , then
2
PSW = ∑ ηi PS (ui ⊗ υi )
i =1
(18)
In terms of the eigenvalues λi ’s of T , the Hilbert-Schmidt
norm i
Since the shortest distance minimization (20) is posed in a
Hilbert space, by the principle of orthogonality it is solved by
the orthogonal projection PS acting from C2 onto S . The
latter is computed by determining the orthogonal projection
Pν onto Span{ν j , j = 1, 2,⋅⋅⋅, n} , and the orthogonal projection
⎛⎛ n
⎛∞
⎞⎟ ⎞⎟
⎞
∞
⎜
= ∑ ηi ⎜⎜⎜⎜⎜∑ ∫ ui (t )ν j (t )dt ⎟⎟⎟ ν j ⊗ ⎜⎜⎜ ∫ υi (τ )ψ j (τ )d τ ⎟⎟ ψ j ⎟⎟⎟ (24)
⎜⎝⎜⎝⎜ j =1 0
⎠⎟
i =1
⎝⎜ 0
⎠⎟⎟ ⎠⎟
n
[9].
= ∑ θ j ν j ⊗ ψ j , ∃ scalars θ j
i
j =1
By interpreting each element of the subspace S defined in
(8) as a Hilbert-Schmidt operator as we did for CIR ( t ;τ ) , we
where the last finite sum is obtained thanks to orthogonality,
i.e., only the ui ’s and υi ’s that live in the span of ν j ’s
see that S is the subspace of Hilbert-Schmidt operators of
rank n , i.e.,
n
⎧⎪s =
⎪
ϑ f (t ) ⊗ χ j (τ ) : f j (t ) ∈ L2 ([0, ∞)) ,⎫
⎪⎪
⎪
∑
⎪
i =1 j j
S =⎨
⎬ (19)
⎪⎪χ ( x) ∈ L2 ([0, ∞)) , ϑ ∈ \
⎪
⎪
j
⎪
⎩⎪ j
⎭
In addition, the minimization (7) is then the minimal distance
from T to Hilbert-Schmidt operators of rank n . In other
terms, we have
μ = min || T − s ||HS
(20)
and ψ j ’s, respectively, are retained. For the orthogonality
s∈S
The space of Hilbert-Schmidt operators is in fact a Hilbert
space with the inner product, ( A, B) := tr ( B* A) , where tr
denotes the trace [9]. In the case where A and B are integral
operators with kernels A(t , τ ) and B (t , τ ) , respectively, the
inner product can be realized concretely by
∞ ∞
( A, B ) = ∫
0
∫
A(t , τ ) B(t , τ ) dtd τ
(21)
0
The solution to the distance minimization is given by the
orthogonal projection of T onto S . To compute the latter,
note that the eigenvectors of (TT * )1/ 2 and (T *T )1/ 2 form
orthonormal bases (by completing them if necessary) for
L2 ([0, ∞)) and L2 ([0, ∞)) , respectively. The subspace S can
be written as
S = Span{ν j ⊗ ψ j , j = 1, 2,⋅⋅⋅, n}
property we only need verify that
x ⊗ y − ( Pν ⊗ Pψ )( x ⊗ y ) ⊥ u ⊗ υ,
(26)
because Pν is the orthogonal projection of L ([0, ∞)) onto
2
Span{ν j , j = 1, 2,⋅⋅⋅, n} , and Pψ the orthogonal projection of
L2 ([0, ∞))
onto
Span{ψ j , j = 1, 2, ⋅⋅⋅, n} . The minimizer
so ∈ S in (20) is given by
n
so := PS T = ∑ λi ν i ⊗ ψi
(27)
i =1
and
n
1
μ = min || w(t , τ ) − s (t , τ ) ||2 =|| T − PS T ||HS = ( ∑ λi2 ) 2
s∈S
(28)
i = n +1
and as n ↑ ∞ , || T − PS T ||HS 2 0 . Therefore, the minimizing
function so (t , τ ) in (7) corresponds to the kernel of so , which
is given by
n
so (t , τ ) = ∑ λi ν i (t ) ψi (τ )
i =1
(22)
(25)
x ∈ L2 ([0, ∞)) , y ∈ L2 ([0, ∞)) , u ⊗ υ ∈ S
Computing the inner product, we get
< x − Pν x, u >1 < y − Pψ y, υ >2 = 0
(29)
SPL-05501-2008.R1
4
i
This implies that in the
2
-norm CIR ( t ;τ ) can be
0.4
approximated to any desired accuracy by an impulse response
∑ λ ν (t ) ψ (τ )
i
i
i
which is factorizable by putting
i =1
g (t ) := [ λ1ν 1 (t ) λ2ν 2 (t )
f (t ) := [ψ 1 (τ ) ψ 2 (τ )
0.2
λnν n (t ) ]
(30)
ψ n (τ ) ]
T
Amplitude (V)
of the form
Measurements
SDE Model
ARMA Model
0.3
n
T
where denotes the transpose .
The corresponding SDE is then given by [7]
dX ( t ) = f (t )dW (t ),
y (t ) = g (t ) X (t )
(31)
where X (t ) is the state of the channel and W (t ) is the
standard Brownian motion. Since state space realizations of
impulse responses are not unique [7], a realization of the
following form can be used
dX j ( t ) = Aj ( t ) X j ( t ) dt + B j ( t ) dW j ( t ) ,
(32)
y j (t ) = C j (t ) X j (t ) + f j (t ) + V j (t )
where y j (t ) is a vector which includes the inphase and
quadrature components, Aj (t ), B j (t ), C j ( t ) are matrices of
dimensions n × n , n × m , 1× n , respectively, X j ( t ) is a state
{
}
vector of the inphase and quadrature components, W j ( t )
t ≥0
is a vector of independent standard Brownian motions, which
correspond to the inphase and quadrature components, f j ( t )
is arbitrary vector function representing the presence or
absence of line-of-sight (LOS) of the inphase and quadrature
components, and V j ( t ) represents measurement noise for the
jth path.
The order of the state-space model determines how close
the model in (32) is to the general IR in (1). It is shown in [11]
that a 4th order state space model is usually sufficient. The
band-pass representation of the received signal in (2) is
represented as
J (t ) ⎛ C ( t ) X
( I I , j ( t ) + f jI ( t ) ) cos (ωc t ) ⎞⎟
⎜
y (t ) = ∑
sl ( t − τ j ( t ) )
(33)
j =1 ⎜ − ( C ( t ) X
t ) + f jQ ( t ) ) sin (ωc t ) ⎟
(
Q
Q
j
,
⎝
⎠
+ vI ( t ) cos(ωc t ) − vQ ( t ) sin(ωc t )
where the indices “ I ” and “ Q ” stand for the inphase and
quadrature components, respectively.
Narrow-band measured data which include samples for the
received inphase and quadrature components in a cellular
network are used for comparison with ARMA models. These
data are estimated via 2nd order SDE and ARMA models [14,
15] using least square algorithm [8] and results are presented
in Fig. 1. Note that proposed SDE model shows excellent
agreement with experimental data and a much better fit. It is
worth mentioning that the estimation process using SDE
models may require more computational cost than ARMA
models. This is because SDE models have larger number of
parameters that need to be estimated.
0.1
0
-0.1
-0.2
-0.3
-0.4
0
20
40
60
80
100
120
Samples
140
160
180
200
Fig.1: Comparison using received inphase component measurements between
proposed SDE and ARMA models.
IV. CONCLUSION
Stochastic models based on SDEs for wireless channels
have been derived. These models take into account the
statistical and time variations in wireless communication
environments. The dynamics are captured by a stochastic state
space model, which is shown to approximate the general TV
IR of the channel as closely as desired.
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