Some Results for Cross-Layer Optimization in UWB Wireless Sensor Networks

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Nov 21, 2013 (3 years and 8 months ago)

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Some Results for Cross-Layer Optimization in UWB
Wireless Sensor Networks
Holger Boche and Ullrich J.Monich
Heinrich-Hertz-Chair for Mobile Communications
Technische Universitat Berlin
Verbundprojekt
UWB for Wireless Sensor Networks:Cross-Layer Optimization (UWB4WSN)
Berichtskolloquium zum DFG Schwerpunktprogramm
\Ultrabreitband-Funktechniken fur Kommunikation,Lokalisierung und Sensorik"(UKoLoS)
Dienstag,23.Marz 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 Cross-Layer 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 Cross-Layer 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 eects)
Some Results for Cross-Layer 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.
Denition (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.
Denition (Paley-Wiener Space)
For 1  p  1we denote by PW
p

the Paley-Wiener 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 Cross-Layer 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 Cross-Layer 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 Cross-Layer 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 Cross-Layer 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

2i
x
5
3
1
1
3
5
q

x
5
3
1
3
5
Some Results for Cross-Layer 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

2i


3
3
t
f(t)
Some Results for Cross-Layer 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

2i


3
3
k
f(k)
Some Results for Cross-Layer 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

2i


3
3
k
(

f)(k)
Some Results for Cross-Layer 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 (Riemann-Lebesgue 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 Cross-Layer Optimization in UWB Wireless Sensor Networks Holger Boche 8
Properties of the Reconstruction Process A

1
For every  > 0,A

is a non-linear 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 non-linearity of the threshold operator makes the analysis dicult.
Some Results for Cross-Layer 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 Cross-Layer 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 low-pass lter are stable LTI systems.
Example (Hilbert transform)
The Hilbert transform
~
f of a signal f 2 PW
1

is dened 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 Cross-Layer 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
[;] denes 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 Cross-Layer 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 Cross-Layer Optimization in UWB Wireless Sensor Networks Holger Boche 13
Pointwise Stability
 The following theorem gives a necessary and sucient 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
discrete-time systems.
Some Results for Cross-Layer 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 fullled,then we have a good pointwise approximation behavior
because the approximation error converges to zero as the threshold  goes
to zero.
Some Results for Cross-Layer Optimization in UWB Wireless Sensor Networks Holger Boche 15
Example:Ideal Low-Pass Filter
Even for common stable LTI systems like the ideal low-pass lter there are
problems because (*) is not fullled.
Example
T
L
:ideal low-pass 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 Cross-Layer Optimization in UWB Wireless Sensor Networks Holger Boche 16
Global Stability
We can also give a necessary and sucient 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
0t1
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
continuous-time systems.
Some Results for Cross-Layer 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
fullled.
Some Results for Cross-Layer 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.
Denition
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 discrete-time impulse response fh
T
(k)g
k2Z
has only nitely many
non-zero elements.

P
1
k=1
jh
T
(k)j < 1for every FIR system
Some Results for Cross-Layer 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 Cross-Layer 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 Cross-Layer 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 Cross-Layer Optimization in UWB Wireless Sensor Networks Holger Boche 22