Some Results for CrossLayer Optimization in UWB
Wireless Sensor Networks
Holger Boche and Ullrich J.Monich
HeinrichHertzChair for Mobile Communications
Technische Universitat Berlin
Verbundprojekt
UWB for Wireless Sensor Networks:CrossLayer Optimization (UWB4WSN)
Berichtskolloquium zum DFG Schwerpunktprogramm
\UltrabreitbandFunktechniken fur Kommunikation,Lokalisierung und Sensorik"(UKoLoS)
Dienstag,23.Marz 2010
Motivation I:Sensor Networks
Wireless sensor networks are one
application of UWB communication.
Their operation is usually energy limited.
A folklore result:Communication
constraints require
\importance sampling"
threshold
We analyze the impact of such constraints.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 2
Motivation I:Sensor Networks
In order to save energy,the sensors transmit only if the absolute value of
the signal f exceeds some threshold .
The receiver has to reconstruct the signal f or some transformation Tf by
using only the samples whose absolute value is larger than or equal to the
threshold .
f
Sampling
and
thresholding
Measuring procedure
Signal space PW
1
Physical process
Approximation
process TA
Reconstruction
process A
Signal processing unit
(
f)
k
a
A
f
TA
f
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 3
Motivation II
The principle of digital signal processing relies on the fact that certain
bandlimited signals can be perfectly reconstructed from their samples.
reconstruction of the signal:ff(k)g
k2Z
!f
approximation of a transformation:ff(k)g
k2Z
!Tf
perfect reconstruction only possible if the sample values are known exactly
not given in practical applications,because samples are disturbed
(quantizers with limited resolution,thresholding eects)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 4
Signal Spaces
B
is the set of all entire functions f with the property that for all > 0
there exists a constant C() with jf(z)j C() exp
( +)jzj
for all
z 2 C.
Denition (Bernstein Space)
The Bernstein space B
p
consists of all signals in B
,whose restriction to the
real line is in L
p
(R),1 p 1.
Denition (PaleyWiener Space)
For 1 p 1we denote by PW
p
the PaleyWiener space of signals f with a
representation f(z) =
1
2
R
g(!) e
iz!
d!,z 2 C,for some g 2 L
p
[;].
The norm for PW
p
is given by kfk
PW
p
=
1
2
R
j
^
f(!)j
p
d!
1=p
.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 5
The Threshold Operator
The threshold operator
sets all signal values,whose absolute value is
smaller than some threshold > 0 to zero.
For continuous functions f:R!C:
(
f)(t) =
f(t),t 2 R,where
z =
(
z jzj
0 jzj <
t
f(t)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 6
The Threshold Operator
The threshold operator
sets all signal values,whose absolute value is
smaller than some threshold > 0 to zero.
For continuous functions f:R!C:
(
f)(t) =
f(t),t 2 R,where
z =
(
z jzj
0 jzj <
k
f(k)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 6
The Threshold Operator
The threshold operator
sets all signal values,whose absolute value is
smaller than some threshold > 0 to zero.
For continuous functions f:R!C:
(
f)(t) =
f(t),t 2 R,where
z =
(
z jzj
0 jzj <
k
(
f)(k)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 6
The Quantization Operator
2 is the quantization step size
For continuous functions f:R!C:
(
f)(t) = q
f(t),t 2 R,where q
z =
Re z
2
+
1
2
2 +
Imz
2
+
1
2
2i
x
5
3
1
1
3
5
q
x
5
3
1
3
5
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 7
The Quantization Operator
2 is the quantization step size
For continuous functions f:R!C:
(
f)(t) = q
f(t),t 2 R,where q
z =
Re z
2
+
1
2
2 +
Imz
2
+
1
2
2i
3
3
t
f(t)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 7
The Quantization Operator
2 is the quantization step size
For continuous functions f:R!C:
(
f)(t) = q
f(t),t 2 R,where q
z =
Re z
2
+
1
2
2 +
Imz
2
+
1
2
2i
3
3
k
f(k)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 7
The Quantization Operator
2 is the quantization step size
For continuous functions f:R!C:
(
f)(t) = q
f(t),t 2 R,where q
z =
Re z
2
+
1
2
2 +
Imz
2
+
1
2
2i
3
3
k
(
f)(k)
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 7
The Reconstruction Process A
The threshold operator is applied on the samples ff(k)g
k2Z
of signals
f 2 PW
1
.
The resulting samples f(
f)(k)g
k2Z
are used to build an approximation
(A
f)(t):=
1
X
k=1
(
f)(k)
sin((t k))
(t k)
=
1
X
k=1
jf(k)j
f(k)
sin((t k))
(t k)
of the original signal f.
We have lim
jtj!1
f(t) = 0 (RiemannLebesgue lemma)
) the series has only nitely many summands
) A
f 2 PW
2
PW
1
.
Since the series uses all\important"samples of the signal,one could expect
A
f to have an approximation behavior similar to the Shannon sampling series.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 8
Properties of the Reconstruction Process A
1
For every > 0,A
is a nonlinear operator.
2
For every > 0,the operator A
:(PW
1
;k k
PW
1
)!(PW
2
;k k
PW
2
)
is discontinuous.
3
For some f 2 PW
1
,the operator A
is also discontinuous with respect to
.
The nonlinearity of the threshold operator makes the analysis dicult.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 9
Approximation of Stable LTI Systems
In many applications the task is to reconstruct some transformation Tf of
f 2 PW
1
and not f itself.
The goal is to approximate the desired transformation Tf of a signal f by
an approximation process,which uses only the samples of the signal that
are disturbed by the threshold operator.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 10
Stable Linear Time Invariant Systems
A linear system T:PW
1
!PW
1
is called stable linear time invariant (LTI)
system if:
T is bounded,i.e.,kTk = sup
kfk
PW
1
1
kTfk
PW
1
< 1 and
T is time invariant,i.e.,
Tf( a)
(t) = (Tf)(t a) for all f 2 PW
1
and t;a 2 R.
The Hilbert transform H and the lowpass lter are stable LTI systems.
Example (Hilbert transform)
The Hilbert transform
~
f of a signal f 2 PW
1
is dened by
~
f(t) = (Hf)(t) =
1
2
Z
1
1
i sgn(!)
^
f(!)
e
i!t
d!;
where sgn denotes the signum function.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 11
Representation of Stable LTI Systems
For every stable LTI system T:PW
1
!PW
1
there is exactly one
function
^
h
T
2 L
1
[;] such that
(Tf)(t) =
1
2
Z
^
h
T
(!)
^
f(!)
e
i!t
d!
for all f 2 PW
1
,and the integral is absolutely convergent.
Every
^
h
T
2 L
1
[;] denes a stable LTI system T:PW
1
!PW
1
.
The operator norm kTk:= sup
kfk
PW
1
1
kTfk
PW
1
is given by kTk = k
^
h
T
k
1
.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 12
System Approximation under Thresholding
If the samples ff(k)g
k2Z
are known perfectly we can use
N
X
k=N
f(k) T(sinc( k))(t) =
N
X
k=N
f(k)h
T
(t k)
to obtain an approximation of Tf.
Here:samples are disturbed.!Approximate Tf by
(T
f)(t):= (TA
f)(t) =
1
X
k=1
(
f)(k)h
T
(t k)
Goal:small approximation error
Since
j(T
f)(t) (Tf)(t)j j(T
f)(t)j +kTk kfk
PW
1
it is interesting how large sup
kfk
PW
1
1
j(T
f)(t)j can get.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 13
Pointwise Stability
The following theorem gives a necessary and sucient condition for
sup
kfk
PW
1
1
j(T
f)(t)j to be nite.
Theorem
Let T be a stable LTI system,0 < < 1=3,and t 2 R.Then we have
sup
kfk
PW
1
1
j(T
f)(t)j < 1
if and only if
1
X
k=1
jh
T
(t k)j < 1:(*)
Note that (*) is nothing else than the BIBO stability condition for
discretetime systems.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 14
Pointwise Convergence
Corollary
Let T be a stable LTI system,0 < < 1=3,and t 2 R.If
1
X
k=1
jh
T
(t k)j < 1 (*)
then we have
lim
!0
sup
f2PW
1
j(Tf)(t) (T
f)(t)j = 0:
If (*) is fullled,then we have a good pointwise approximation behavior
because the approximation error converges to zero as the threshold goes
to zero.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 15
Example:Ideal LowPass Filter
Even for common stable LTI systems like the ideal lowpass lter there are
problems because (*) is not fullled.
Example
T
L
:ideal lowpass lter,h
T
L
(t) = sin(t)=(t)
!
P
1
k=1
jh
T
L
(t k)j = 1for all t 2 Rn Z
For t 2 Rn Z and 0 < < 1=3,
sup
kfk
PW
1
1
j(T
L;
f)(t)j = sup
kfk
PW
1
1
1
X
k=1
jf(k)j
f(k)
sin((t k))
(t k)
= 1:
This shows that,for t 2 Rn Z and any with 0 < < 1=3,the
approximation error j(T
L
f)(t) (T
L;
f)(t)j can be arbitrarily large
depending on the signal f 2 PW
1
.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 16
Global Stability
We can also give a necessary and sucient condition for the uniform
boundedness on the whole real axis.
Theorem
Let T be a stable LTI system and 0 < < 1=3.We have
sup
kfk
PW
1
1
kT
fk
1
< 1
if and only if
sup
0t1
1
X
k=1
jh
T
(t k)j < 1
if and only if
Z
1
1
jh
T
()j d < 1:(**)
Note that (**) is nothing else than the BIBO stability condition for
continuoustime systems.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 17
Global Uniform Convergence
Corollary
Let T be a stable LTI system and 0 < < 1=3.If
Z
1
1
jh
T
()j d < 1:(**)
then we have
lim
!1
sup
f2PW
1
kTf T
fk
1
= 0:
This shows the good global approximation behavior of T
f if (**) is
fullled.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 18
FIR Filters
Finite impulse response (FIR) systems are an important special case of
stable LTI systems.
Denition
We call a stable LTI system T nite impulse response system if
^
h
T
is a
polynomial in
e
i!
,i.e.,if
^
h
T
has the representation
^
h
T
(!) =
M
X
k=0
c
k
e
i!k
; ! ;
for some M 2 N and c
k
2 C,k = 0;:::;M.
The discretetime impulse response fh
T
(k)g
k2Z
has only nitely many
nonzero elements.
P
1
k=1
jh
T
(k)j < 1for every FIR system
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 19
FIR Approximation
No problems on the integer lattice t = n 2 Z:
j(Tf)(n) (T
f)(n)j
1
X
k=1
jh
T
(n k)j < 1
Since
j(Tf)(n) (T
f)(n)j j(T
f)(n)j +kTk kfk
PW
1
;
it is interesting to know how large sup
kfk
PW
1
1
j(T
f)(n)j can get.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 20
FIR Approximation Error
Let M be the smallest natural number such that h
T
(k) = 0 for all k > M.
M
X
k=0
jh
T
(k)j
p
M +1 kTk ) sup
kfk
PW
1
1
j(T
f)(n)j
p
M +1 kTk
Quadratic phase function:
h
T
q
(k) =
(
1
p
M+1
exp
i
k
2
M+1
0 k M
0 otherwise
M
X
k=0
jh
T
q
(k)j =
1
p
M +1
M
X
k=0
1 =
p
M +1
The worst case approximation error on the integer lattice t = n 2 Z
increases as
p
M +1.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 21
Conclusion
For certain stable (with respect to the PW
1
norm) LTI systems the
approximation process is instable if the samples are disturbed by
thresholding/quantization,because the approximation error can be
arbitrarily large.
This holds irrespectively of how small the threshold/quantization step size
is chosen.
A complete characterization of the systems for which the approximation
process is stable under thresholding and quantization was given:
The stable LTI systems T:PW
1
!PW
1
that can be uniformly
approximated by the approximation process are the LTI systems with
h
T
2 B
1
.
Some Results for CrossLayer Optimization in UWB Wireless Sensor Networks Holger Boche 22
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