RESTORATION OF THE RET PHASE FUNCTION USING DECONVOLUTION

Huajian Cui

Outline

• Why Deconvolution and ill-posed problems

• Methods and results

• Methods of measurements

• Brief introduction to the RET theory

Background of the research

• It is important to accurately predict excess loss when radio waves propagating through vegetation for Radiowave channel planner such as Ofcom

• Following a 15-month project to develop a generic model of 1-60 GHz narrowband radio signal attenuation in vegetation, Oct. 2002

• A contract between the UK Radiocommunications Agency (RA) an a QinetiQ-led consortium comprised of QinetiQ, the Rutherford Appleton Laboratory (RAL), and the Universities of Portsmouth and Glamorgan

Convolution Distortion

• The received signal can be obtained by summing the arriving scattered radio waves from all directions with appropriate weighting

1 1

2 2

2 2

( ) ( ) ( )

( ) ( )

...

( ) ( )

( ) ( )i

RX RX RX RX

RX RX

RX RX

i RX i RX

P I G

I G

I G

I G

Where PRX is the received directional spectrum, GRX is the radiation pattern of the receiver

( )nI

1( )I

2( )I

RX

( )nI

1( )I

2( )I

RX

Deconvolution

• Deconvolution is the inverse operation of the convolution

• Convolution theorem y(t) = x(t) * h(t) (1) Y(w) = X(w)H(w) (2) H(w)=Y(w)/X(w) (3)

• In practical applications, the deconvolution problem is

mathematically classified as an ill-posed problem. The cause of the

“ill-posed problems” occurring is that the information represented by

the data (sequences) or the equations (continuous functions) are

incomplete

Cause of the ill-posed problems

• Inaccuracy of sampling or digitisation and data acquisition process

• Natural variability of the signal• Always present random noise

Deconvolution Methodology

• The optimum compensation iterative deconvolution

• The automated regularisation iterative deconvolution

*

1 2

( )

( )

XF

X

*

2 2 4

( )

( )

XF

X

Deconvolution Results

The RET Modelling

The Radiative Energy Transfer Theory is an theoretical model utilised to predict the

excess attenuation and directional spectra while radio waves propagating through

vegetation. It can be expressed as follow:

P

P

P k

2R RP

Max R P

ˆ2 - -R

M RP

mM

m PR M PRm 1 P

ˆ ˆ2 - -j R SR

kj

P ( ; , )exp{ ( ) }

P

{[e -e ] q ( )4

W1 e ( ) [q ( )-q ( )]}

m!

F ( ) F {-e [A e .

2 P

P

R R

N N

n R

N 1 n0k2

k

( )]}

1-n

s

Directional Spectra

• The theoretical directional spectra can be 3-D demonstrated as following based on mathematical calculations

The Phase FunctionThe Phase Function is assumed to be Gauss-like, and expressed by:

( ) ( ) (1 )P f 22 ( / )( ) 2f e

Graphical depiction its characteristics depending on various parameters

-30 -20 -10 0 10 20 30-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Angle,

Rec

eive

d A

mpl

itude

, dB

-30 -20 -10 0 10 20 30-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Angle,

Rec

eive

d A

mpl

itude

, dB

-30 -20 -10 0 10 20 30-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Angle,

Rec

eive

d A

mpl

itude

, dB

-30 -20 -10 0 10 20 30-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Angle,

Rec

eive

d A

mpl

itude

, dB

=2=3=4=5=10=0.9

=2=3=4=5=10=0.7

=2=3=4=5=10=0.5

=2=3=4=5=10=0.2

Other methods considered

• Maximum likelihood estimation

• Maximum/minimum entropy

• Monte Carlo method

• Finite element method

• Project onto convex sets

• Neural method