Some Results for Cross-Layer Optimization in UWB

Wireless Sensor Networks

Holger Boche and Ullrich J.Monich

Heinrich-Hertz-Chair for Mobile Communications

Technische Universitat Berlin

Verbundprojekt

UWB for Wireless Sensor Networks:Cross-Layer Optimization (UWB4WSN)

Berichtskolloquium zum DFG Schwerpunktprogramm

\Ultrabreitband-Funktechniken 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 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 eects)

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.

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 (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

2i

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

2i

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

2i

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

2i

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 dicult.

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 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 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

[;] 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 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 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

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 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 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 fullled.

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 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

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

fullled.

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.

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 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

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