44
The Distortion of Ultra-Wideband Signals in the Environment PhD Viva Presentation Anastasios Karousos Supervisors: Dr. Costas Tzaras Dr. Tim Brown
The Distortion of Ultra-Wideband Signals in the Environment
PhD Viva Presentation
Anastasios Karousos
Supervisors:Dr. Costas TzarasDr. Tim Brown
A. Karousos 2
Presentation Outline
UWB Communications
Radio Wave Propagation
Signal Prediction
Conclusions
Introduction
UWB Radio
Benefits / Challenges
Propagation Mechanisms
TD Representation
Multiple Interaction Phenomena
Results Ray-Trace Algorithm
Outdoor/Indoor Results
UWB Measurements
TD Ray-Trace
A. Karousos 3
UWB Communications
A. Karousos 4
Introduction
• Wired communications (telephone lines, optical fibres) are costly and complex
WHEREAS
• Wireless technology (mobile, satellite, WLAN) offers simplicity and mobility
HOWEVER
• Frequency crowding in the available spectrum
• Interference issues
prohibits proper exploitation of wireless systems
A. Karousos 5
UWB Radio
• FCC: “an intentional radiator that, at any point in time, has a fractional bandwidth equal or greater than 0.20 or has a bandwidth equal to or greater than 500 MHz, regardless of the fractional bandwidth”
100
101
102
−100
−90
−80
−70
−60
−50
−40
Frequency (GHz)
EIR
P S
pect
ral D
ensi
ty (
dBm
/MH
z)
FCC Spectral MaskEC Spectral MaskOfcom recommendations
The issued spectral masks from FCC and EC, as well as Ofcom’s recommendations for unlicensed radio transmission
• The fractional bandwidth is:
• The bandwidth is the frequency band which is bounded by the points that are 10 dB below the highest radiation emission
• Similar regulation from EC
medianLH
LHFC f
BffffB
)(2
A. Karousos 6
Benefits of UWB
• The increased bandwidth offers more capacity and higher data-rates – Shannon law (noise-like signals with small power are more preferable than high-powered NB signals)
• Multipaths are not an ‘enemy’; multipaths can be resolved, enhancing system’s performance
• Low probability-of-detection (LPD), proper for covert and secure communications (essential for the military)
• Location and tracking applications
• Ground penetration radars for geophysical prospecting, archaeology, medicine
• Low-cost and low-complexity equipment (almost true)
• Spectrum sharing
A. Karousos 7
Challenges
• A RAKE receiver with 50 or more fingers, would be necessary to exploit the multipath diversity
• Use of fast ADC, which may consume a lot of power
• Timing synchronisation is also important. A small timing mismatch would degrade the system’s performance
• The complex propagation effects of the channel would introduce distortion in the signal, preventing an optimal operation
• Channel models are treated as tap-delay lines, where signal distortion is assumed either known ‘a priori ’ or negligible
A. Karousos 8
Radio Wave
Propagation
A. Karousos 9
Radio Wave Propagation
• A traversing signal is reflected, diffracted, scattered or transmitted through the objects of the environment
• Since we use impulses, it is more natural and more efficient to treat such phenomena directly in the time-domain
• Parameters like number of multipaths, delay and power of every path are easily obtained
• TD closed-form solutions should be found through inverse Fourier or Laplace transform integrals to describe such phenomena
• The received signal is the convolution of the transmitted signal in the time-domain with TD coefficients. The numerical IFFT will be used for comparison results
A. Karousos 10
Reflection
• Fresnel reflection coefficients
• The received reflected field in the TD is written as:
• where
• and α , κs,h and Κs,h depend on the electrical parameters of the medium and the impinging angle
)(*)(*)()()( , shsir tttrtesAte
tK
hs
hshshs
hsetKtr
2/12,
,,,
,
12
)()(
3.8 3.9 4 4.1 4.2−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)N
orm
alis
ed A
mpl
itude
rIFFT
rTD
3.8 3.9 4 4.1 4.2−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
rIFFT
rTD
Reflection of a Gaussian doublet on a wall with
εr=5, σ=0.1 S/m and θ=45°.
Soft polarisation on top and hard polarisation on the bottom.
Time shiftSpreading factor
A. Karousos 11
Diffraction
• Waves ‘bend’ around objects – Huygens-Fresnel principle
• Diffraction theory is based on the solution of the Fresnel-Kirchhoff integrals – computational intensive
• Uniform Theory of Diffraction (UTD) describes accurately such phenomena for a number of obstacles, by treating the waves as rays, similarly to Geometrical Optics
• An incident ray results into infinite number of diffracted rays,placed on the surface of a cone (Keller’s cone)
• The diffracted field will be given by:
)(*)(*)()()( sid tttdtesAte
TD-Diffraction Coefficient
A. Karousos 12
Time-Domain Diffraction Coefficients
cLandwhere
tuttc
Ltd EdgeKnife
/2/cos2
)()2/cos(2
)(
2/
n
aandn
a
na
naandifor
caLnttc
nLtdwhere
tdtrtdtr
tdtdtrtrtd
iii
ii
hnshos
hnshoshs
Wedge
22
,2
,2
4,...,1
/)(sin2,)2sin(22
)(
)(*)()(*)(
)()(*)(*)()(
/
4
/
3
/
2
/
1
22
4,3,
21,,,
Transmitter Receiver
s1
knife-edge
s0
φ
φ'
Transmitter Receiver
s1
wedge
s0
φ
φ'
o-fa
ce n-face
where and
for
where
and
and
A. Karousos 13
Diffracted Pulses
18.5 18.6 18.7 18.8 18.9 19 19.1 19.2 19.3 19.4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
rifft
rTD
18.5 18.6 18.7 18.8 18.9 19 19.1 19.2 19.3 19.4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
rifft
rTD Diffracted pulse on a non-perfectly
conducting wedge with εr=4.4, σ=0.018 S/m and αint=π/3.
Diffracted pulse on an absorbing knife-edge
A. Karousos 14
Layer (Slab) Model
• Walls are not infinite in width, but finite
• The wave is partially reflected and refracted on the boundary interfaces
• These multiply reflected signals can carry significant energy
• Proper knowledge would avoid ISI and increases performance
• Easy-to-use and accurate formulations, predicting a large number of internal reflections
d
θi
Ei
Et1
Et2
Et3
Et4
Er1
Er2
Er3
Er4
Assumed path of total transmitted
wave
Assumed path of total reflected
wave
z εr, μr
A. Karousos 15
Time-Domain Coefficients
)2/(,sin/2
)(14
)()(
)(*)()()(
02
02,
,2,
2,
,,
rr
n
at
hs
hsnhs
annhsR
hsRa
Rdhs
acdand
ntuean
KnteKth
trthethtr
1 1.5 2 2.5 3 3.5
x 10−8
−4000
−2000
0
2000
4000
6000
IFFTTD Solution
0.97 0.98 0.99 1 1.01 1.02
x 10−8
−3000
−2000
−1000
0
1000
2000
3000
4000
5000
6000
1.53 1.54 1.55 1.56 1.57 1.58 1.59
x 10−8
−700
−600
−500
−400
−300
−200
−100
0
100
200
300
2.65 2.7 2.75 2.8 2.85 2.9 2.95 3
x 10−8
−0.2
−0.1
0
0.1
Time (10 s)
Am
plitu
de (
Vol
ts)
−8
1 1.5 2 2.5 3 3.5
x 10−8
−6000
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000 IFFTTD Solution
−8Time (10 s)
Am
plitu
de (
Vol
ts)
1.8 1.81 1.82 1.83 1.84 1.85
x 10−8
−60
−40
−20
0
20
40
60
80
100
120
2.21 2.22 2.23 2.24 2.25 2.26
x 10−8
−3
−2
−1
0
1
2
3
4
5
6
7
0.97 0.98 0.99 1 1.01 1.02
x 10−8
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000
0
2,,2
,
,2,
2,
,
2/,
)(14
)(
14
)(
/2/)(
n
anhshs
hs
hsnanhs
hs
hsT
Tad
hs
ntueaKKn
nteK
th
cdthet
TD Reflection Coefficient:
TD Transmission Coefficient:
Reflection (top) and transmission (bottom) of a hard polarised Gaussian doublet on a wall with
εr=4.4, θ=0° and d=20 cm.
and
A. Karousos 16
Multiple-Diffraction Phenomena
• UTD multiple-diffracted waves cannot be calculated as a concatenation of single-diffraction incidences, especially when the objects are in the transition zones of the previous objects
• Higher-order diffracted fields are needed for accurate prediction
• A new slope-UTD algorithm that only includes second order diffraction terms is implemented into the time-domain
• It incorporates the derivative and the derivative of the slope coefficient in the TD
A. Karousos 17
L-parameters
• The L-parameters enforce continuity in the field prediction
• The L-parameters for the amplitude and slope terms are given by
• They depend on the value of the field at the previous object and the value of the field if the current object was absent
• They are a function of frequency
3/2
2
)(/)(/)(
)()()(
nk
nk
jksnkrmnmn
nkmnmnnkmnk
jksnkrmnmn
nkmnmnmnk
esAnsEnssEsLs
esAsEssEL
A. Karousos 18
Time-Domain Approach
• The multiply diffracted field in the TD is generally written as:
• eder(t) is the directional derivative, which equals to:
• For the knife-edge case, the derivative and the derivative of the slope coefficient are given by:
)/(*)();(*)()(*)()( 111/
1111 cstsAtdtetdtete NNNderN
derNNNN
)/(*)(
)(*)();(*)()( 22
22,22221 cst
ssA
tdtetdtete NN
NNslopederN
derN
derNN
derN
)(
)(
)2/(sin12
)2/cos(22
)(
)()(
)2sin(22
),;(
2
2
,
2/3/
tut
acLtt
acLtd
tut
aLtd
slopederEdgeKnife
derEdgeKnife
Amplitude term Slope term
A. Karousos 19
• The TD derivative of the diffraction coefficient for a non-perfectly conducting wedge is
• When differentiating with respect to φ, the derivative of ros,h(t) is set to zero, whereas in the other case, i.e. differentiating with respect to φ/, rns,h(t) is set to zero
• di/(t) and di
der(t;φ,φ/) are given by:
Wedge Slope-Terms
),;(*)()(*)(),;(*)()(*)(),;(
),;(*)(*)()(*)(*)()(*)(*)(),;(/
4,/4,
/3,
/3,
/2
/1,,
/1,,
/1,,
/,
tdtrtdtrtdtrtdtrtd
tdtrtrtdtrtrtdtrtrtdder
hnsder
hnsder
hosder
hosder
derhnshos
derhnshosshn
derhos
derhs
)()(
/222
),;(
)()(
)cot(22
)(
2/3
2
//
2/1/
tut
cLntnca
td
tut
an
ctd
i
iideri
i
iii
A. Karousos 20
Derivative of the Slope-Term
• The derivative of the slope-term in the TD is given by:
• where
)(*)();(*)(
)(*)();(*)()()(*)(*)(
);(*)(*)();(*)(*)()(*)(*)()(
sec,4,
/4,
sec,3,3,
sec,2
sec,1,,
/1,,1,,
int1,,
,,
tdtrtdtr
tdtrtdtrtdtdtrtr
tdtrtrtdtrtrtdtrtrtd
slopehns
slopederhns
slopehos
slopederhos
slopeslopehnshos
slopederhnshos
slopehns
derhos
derhns
derhos
slopederhs
)()(
))(sin2(23tan1)2sin(
2)(
)()(costan)(sin2),;(
)(tan)2sin(2
)(
2
22
1/
sec,
21
2
//
1int
tut
caLntt
tacLnaatd
tut
tataLnatd
tutacLntd
i
i
iii
iislopei
i
i
ii
iislopei
ii
ii
A. Karousos 21
Grazing Incidence
0 1 2 3 4 5
x 10−8
−300
−200
−100
0
100
200
300
400
500
600
700
Magnification of thesignal
TD UTD
IFFT UTD
3.96 3.97 3.98 3.99 4 4.01 4.02 4.03
x 10−8
−300
−200
−100
0
100
200
300
400
500
600
700
Transmitter Receiver
2m 2m 2m 2m 2m2m
Am
plitu
de (
V/m
)
Time (10 s)−8
0 1 2 3 4 5
x 10−8
−2
−1
0
1
2
3
4
x 104
TD UTD
IFFT UTD
Magnification of thesignal
3.96 3.97 3.98 3.99 4 4.01 4.02 4.03
x 10−8
−2
−1
0
1
2
3
4
x 104
Transmitter Receiver
Time (10 s)−8
2m 2m 2m 2m 2m 2m
Am
plitu
de (
V/m
)
Diffraction for the grazing incidence of five absorbing knife-edges, which are spaced 2m apart
Diffraction for the grazing incidence of five metallic wedges with internal angle π/5 radsand are spaced 2m apart
A. Karousos 22
Transition Regions
• The approximation on the L-parameters introduces an error in the prediction, especially close to the shadow boundaries
• Also the path response in such a scenario is very sharp and a more tedious convolution is needed
0 1 2 3 4 5 6 7
x 10−8
−200
−150
−100
−50
0
50
100
150
200
250
300
TD UTD
IFFT UTD
Magnification of thesignal
The predicted signalif dt is reduced
4.97 4.98 4.99 5 5.01 5.02 5.03 5.04
x 10−8
−150
−100
−50
0
50
100
150
200
250
300
4.97 4.98 4.99 5 5.01 5.02 5.03 5.04
x 10−8
−150
−100
−50
0
50
100
150
200
250
300
Time (10 s)A
mpl
itude
(V
/m)
−8
hTx h
Rxh
1
αint
h2
αint
h3
αint
h4
αint
3m
Transmitter
3m 3m 3m 3m
o−fa
ce n−face o−fa
ce n−face o−fa
ce n−face
o−fa
ce n−face
Shadow Boundary 12Shadow Boundary 01
Shadow Boundary 23Shadow Boundary 34
Receiver
Propagation Path
A diffracted path close to the shadow boundaries
The error decreases as the time resolution is finer
A. Karousos 23
Cascade of Different Objects
• The algorithm can be applied for different objects in the path
• The source transmits a pulse every 5 ns and in each transmissiontime, the height of middle object increases by 1 m, with initial height 0 m
0 0.2 0.4 0.6 0.8 1
x 10−7
−60
−40
−20
0
20
40
60
80
TD UTD
IFFT UTD
hwαintTransmitter
2m
2 m 2 m
2m 2m 2m
Receiver
8.6 8.65 8.7
x 10−8
−15
−10
−5
0
5
10
15
20
25
4.85 4.9 4.95 5 5.05 5.1
x 10−8
−6
−4
−2
0
2
4
6
6.75 6.8 6.85 6.9
x 10−8
−40
−20
0
20
40
60
Time (10 s)−8
Am
plitu
de (
V/m
)
• The outer objects are knife-edges with height 2 m and the middle object is a non-perfectly conductive wedge with εr=10, σ=0.1 S/m and internal angle π/5 rads
A. Karousos 24
Signal Prediction
A. Karousos 25
Ray-Trace Algorithm
• A novel 3D ray-trace model was constructed based on the database preprocessing
• It operates in two stages, the preprossecing of the surrounding and the actual ray-trace
• The positions of the buildings are read from a GIS file
• The environment is then discretised into tiles and segments and the angles between them are calculated and stored into a file
A. Karousos 26
Path Search
• The Tx is inserted and its angles with the environment elements are computed
• The actual ray-trace commences. If certain conditions are fulfilled, reflection or diffraction occurs
• The path search is reduced into a search in a look-up table and the construction of the tree of the predicted paths
• If the Tx position is altered, only the top level will change, and therefore similar operations are avoided
• It combines image theory with ray-launching
A. Karousos 27
Comparison Results - Outdoor
528.4 528.6 528.8 529 529.2
181.3
181.4
181.5
181.6
181.7
181.8
181.9
182
182.1
182.2
182.3
x−coordinates (km)
y−co
ordi
nate
s (k
m)
error (dB)
−20
−15
−10
−5
0
5
10
15
20
25
30
0 100 200 300 400 500 600 700 80040
60
80
100
120
140
160
Receiver Point
Pat
h Lo
ss (
dB)
MeasurementPrediction
0 100 200 300 400 500 600 70050
60
70
80
90
100
110
120
130
140
Receiver Point
Pat
h Lo
ss (
dB)
MeasurementPrediction
530.4 530.6 530.8 531 531.2 531.4
181.6
181.7
181.8
181.9
182
182.1
182.2
182.3
182.4
182.5
182.6
x−coordinates (km)
y−co
ordi
nate
s (k
m)
error (dB)
−30
−20
−10
0
10
20
30
40
A. Karousos 28
Comparison Results - Outdoor• Measurements were conducted in various locations in London at 2.1 GHz
• The small scale effects were cancelled out by averaging
• The Tx and Rx were set at various heights (0.5 ~ 3 m)
• The ray-trace prediction tracks the changes of the received signal quiteaccurately
• Errors occur due to the simplification of the buildings shapes, the approximation of their effective electrical parameters, movement in the measuring channel (lorries, buses) and errors in the translation of the measurements on the map
9.714.720.50.5Kingsland
8.251.071.51.5Kingsland
8.38-3.031.51.5Holborn
11.46-2.370.53Holborn
9.25-4.460.53Portland
8.46-3.811.53Portland
Std (dB)Mean Error (dB)Rx Height (m)Tx Height (m)PlaceComparison results for the ray-trace predictions
A. Karousos 29
Comparison Results - Indoor
0 10 20 30 40 50 60 70 80 9050
55
60
65
70
75
80
85
90
Point
Pat
h Lo
ss (
dB)
Line 1 Line 2 Line 3 Line 4
MeasurementsPrediction
0 50 100 15040
50
60
70
80
90
100
Point
Pat
h Lo
ss (
dB)
Line 1 Line 2 Line 3
MeasurementsPrediction
Corridor NLOS
A. Karousos 30
Comparison Results - Indoor
• The channel response for various locations in the CCSR building was measured for the frequency of 4.5 GHz
• The radiation patterns of the antennas were measured in the anechoic chamber and taken into account
• The predictions are quite accurate for most of the cases
• Incorrect electrical parameters, errors in the modelling of the building, clutter inside the rooms, but also problems with the measuring apparatus that were diagnosed after the postprocessingof the data may have increased the prediction error (especially in the NLOS scenario)
8.37-0.50NLOS
5.750.04Corridor
Std (dB)Mean Error (dB)ScenarioComparison results for the ray-trace predictions
A. Karousos 31
UWB Propagation Measurements
• The channel response for the 3 GHz – 6 GHz band was measured with a VNA for the CCSR building
• The increased bandwidth offers less fractional margin
0 10 20 30 40 50 60 70 80 9045
50
55
60
65
70
75
80
85
90
PointP
ath
Loss
(dB
)
Line 1 Line 2 Line 3 Line 4
UWBNB
0 50 100 150
40
50
60
70
80
90
100
Point
Pat
h Lo
ss (
dB)
Line 1 Line 2 Line 3
UWBNB
CorridorNLOS
A. Karousos 32
TD Ray-Trace
30 35 40 45 50 55 60 65 70 750
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
MeasurementPrediction
5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
MeasurementPrediction
40 45 50 55 60 65 70 75 80 850
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
MeasurementPrediction
Rx close to the Tx Rx half way along the corridor
Rx in a deep NLOS position
A. Karousos 33
TD Prediction vs. Measurement
20 40 60 80 100 120 14045
50
55
60
65
70
75
80
85
Point
Pat
h lo
ss (
dB)
MeasurementPrediction
50 55 60 65 70 75 8045
50
55
60
65
70
75
80
85
Measured path loss (dB)
Pre
dict
ed p
ath
loss
(dB
)
10 20 30 40 50 60 70 8055
60
65
70
75
80
Point
Pat
h lo
ss (
dB)
MeasurementPrediction
55 60 65 70 7555
60
65
70
75
80
Measured path loss (dB)
Pre
dict
ed p
ath
loss
(dB
)
Corridor NLOS
A. Karousos 34
Conclusions
• TD formulations can offer correct prediction of the received signal
• They need to be described in closed-form solutions
• Import of these solutions in a deterministic tool gives fairly accurate results
• Novel ray-trace that can be used for indoor/outdoor scenarios and narrowband/ultra wideband radio
• Limitations on the knowledge of the environment characteristics (accurate dimensions, electrical properties of the walls, objects/clutter in the channel) induce an error in the prediction that is unavoidable
A. Karousos 35
Thank you for your attention.
Is there anything you may like to ask?
A. Karousos 36
Back Up
Slides
A. Karousos 37
UWB Waveforms
• They need to spread the power effectively and efficiently in the frequency-domain, avoiding interference issues.
• Fast rise and fall times, zero DC component for effective radiation.
• Such pulses are Gaussian, Rayleigh, Laplacian, cubic, orthogonal prolate spheroidalwaveforms etc.
0 2 4 6 8 10 12 14−30
−25
−20
−15
−10
−5
0
Frequency (GHz)
Nor
mal
ised
Mag
nitu
de (
dB)
Gaussian pulseGaussian monocycleGaussian doubletDamped sine wave
−1 −0.5 0 0.5 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
Gaussian pulseGaussian monocycleGaussian doubletDamped sine wave
A. Karousos 38
Modulation of UWB Radio
• Single-band modulation, where the whole band is used or multiband modulation, where the band is partitioned into smaller parts
• In single-band, modulation is different than narrowband case; information is transmitted by generating pulses at specific time instances
• Binary phase shift keying (BPSK), pulse amplitude modulation (PAM), on-off keying (OOK), pulse position modulation (PPM), pulse interval modulation (PIM), pulse shape modulation (PSM)
• Multiband modulation is a carrier based modulation, where the frequency band is divided into smaller bands with at least 500 MHz bandwidth
• It offers flexibility in conforming to local regulations, by turning bands on or off, an ability in avoiding strong NB interferers and advanced spectral efficiency
• Possible multiband modulation techniques are MB-UWB, MB-OFDM and DS-UWB
A. Karousos 39
Single-band Modulation
• Modulation is inserted either in the polarity of the pulse or in the position of the pulse inside the frame or in both of them
• Time dithering is used for smoothing the strong spectral lines, due to the frame repetition time
0 1 2 3 4 5 6 7−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
BPSKPPM
0 2 4 6 8 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (nsec)
Nor
mal
ised
Am
plitu
de
TH−PPMDS−PPMDS−TH−PPM
A. Karousos 40
Multiband UWB
0 10 20 30 40 50−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (ns)
Nor
mal
ised
Am
plitu
de
MB-UWB
MB-OFDM
• MB-UWB: signals with 500 MHz bandwidth, shifted in the appropriate band
• MB-OFDM: each band is divided into subcarriers and it is transmitted according to a time-frequency code
• DS-UWB: two bands with 1.75 GHz (3.1-4.85 GHz) and 3.5 GHz (6.2-9.7 GHz) bandwidth respectively
A. Karousos 41
Inverse Techniques
• The TD solution can be found from the FD one, using inverse Fourier transform integrals, i.e.
• We can have an one-sided integral which is
• However, since the interaction mechanisms are causal functions, the analytic function can be written as
• where H[f(t)] is the Hilbert transform of f(t)
deFtf tj)(
21)(
0
)(1)(~
deFtf tj
)()()(~ tfjHtftf
• Therefore, the real part is the wanted solution in the time-domain
A. Karousos 42
Lossy Slab
1 1.5 2 2.5 3 3.5
x 10−8
−6000
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000 IFFTTD Solution
−8Time (10 s)
Am
plitu
de (
Vol
ts)
1.8 1.81 1.82 1.83 1.84 1.85
x 10−8
−60
−40
−20
0
20
40
60
80
100
120
2.21 2.22 2.23 2.24 2.25 2.26
x 10−8
−3
−2
−1
0
1
2
3
4
5
6
7
0.97 0.98 0.99 1 1.01 1.02
x 10−8
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
1000
2000
3000
1 1.5 2 2.5 3 3.5
x 10−8
−4000
−2000
0
2000
4000
6000
8000
IFFTTD Solution
−8Time (10 s)
Am
plitu
de (
Vol
ts)
1.08 1.09 1.1 1.11 1.12 1.13
x 10−8
−4000
−2000
0
2000
4000
6000
8000
1.9 1.91 1.92 1.93 1.94 1.95 1.96
x 10−8
−15
−10
−5
0
5
10
15
20
25
30
2.31 2.32 2.33 2.34 2.35 2.36 2.37
x 10−8
−0.5
0
0.5
1
1.5
Reflection of a soft polarised Gaussian doublet on a wall with
εr=4.4, σ=0.018 S/m , θ=π/8 and d=20 cm.
Transmission of a soft polarised Gaussian doublet through a wall with
εr=4.4, σ=0.018 S/m , θ=π/8 and d=20 cm.
A. Karousos 43
Shadow Boundaries
• The incident shadow boundary (ISB) signifies the boundary between the LOS and NLOS areas
• The reflection shadow boundary (RSB) signifies the boundary between the areas where reflection can or cannot exist
• These boundaries depend on the relative position of the source with respect to the edge
Reflection Shadow Boundary Incident Shadow Boundary
φ'
φRSB
φISB
A. Karousos 44
Derivative of the reflection coefficient
• The derivative of the reflection coefficient can be easily obtained by differentiating rs,h(t)
• Therefore, it will be
• where for the o-case, ψ=π/2-φ/ and the minus corresponds to soft polarisation and the plus to hard one, whereas, for the n-case, ψ=π/2-nπ+φ and the signs are the opposite from above
• Finally
)()1(
12)(
)1(2);( ,)2/1(
4,
,2,
2,
,, tue
atattr hsatK
hs
hshs
hs
derhs
hs
2/32sin)1(sin
r
rs
2sincos
)1(sin
rr
rhand
