Signal Propagation
Jie Gao
01/27/2010
Signal
• Signal are generated as physical representations of data
• A signal is a function of time and location
ideal
digital signal
1
0
a special type of signal, sine
waves, also called harmonics:
s(t) = A sin(2π f t + ϕ)
with frequency f, period T=1/f,
amplitude A, phase shift ϕ
t
1
0
t
2
Fourier Transform: Every Signal Can be Decomposed
as a Collection of Harmonics
∞
∞
1
g (t ) = c + ∑ an sin( 2πnft ) + ∑ bn cos(2πnft )
2
n =1
n =1
1
1
0
0
t
ideal periodical
digital signal
t
decomposition
The more harmonics used, the smaller the approximation error.
3
4
Time Domain v.s. Frequency Domain
Time domain
Frequency domain
1
1
0
t
0
f
1
1
0
0
t
f
Knowing one can recover the other.
5
Interference
Signals add up
1
1
0
0
t
t
Apply Fourier transform
1
1
0
t
0
1
f
0
t
2f
f
6
Fundamental Question: Why Not Send Digital Signal in
Wireless Communications?
1
ideal
digital signal
0
t
7
Fundamental Question: Why Not Send Digital Signal in
Wireless Communications?
• May cause interference
– suppose digital frame length T, then signal
decomposes into frequencies at 1/T, 2/T, 3/T, …
– let T = 1 ms, generates radio waves at frequencies
of 1 KHz, 2 KHz, 3 KHz, …
8
Bandwidth
9
Frequencies for Communications
twisted
pair
coax cable
1 Mm
300 Hz
10 km
30 kHz
VLF
LF
optical transmission
100 m
3 MHz
MF
HF
VLF = Very Low Frequency
LF = Low Frequency
MF = Medium Frequency
HF = High Frequency
VHF = Very High Frequency
1m
300 MHz
VHF
UHF
10 mm
30 GHz
SHF
100 µm
3 THz
EHF
infrared
1 µm
300 THz
visible light UV
UHF = Ultra High Frequency
SHF = Super High Frequency
EHF = Extra High Frequency
UV = Ultraviolet Light
Frequency and wave length:
λ = c/f
wave length λ, speed of light c ≅ 3x108m/s, frequency f
10
Spectrum
11
Frequencies and Regulations
ITU-R holds auctions for new frequencies, manages frequency bands
worldwide (WRC, World Radio Conferences)
Cellular
Phones
Cordless
Phones
Wireless
LANs
Others
Europe
USA
Japan
GSM 450 - 457, 479 486/460 - 467,489 496, 890 - 915/935 960,
1710 - 1785/1805 1880
UMTS (FDD) 1920 1980, 2110 - 2190
UMTS (TDD) 1900 1920, 2020 - 2025
CT1+ 885 - 887, 930 932
CT2
864 - 868
DECT
1880 - 1900
IEEE 802.11
2400 - 2483
HIPERLAN 2
5150 - 5350, 5470 5725
RF - Control
27, 128, 418, 433,
868
AMPS , TDMA , CDMA
824 - 849,
869 - 894
TDMA , CDMA , GSM
1850 - 1910,
1930 - 1990
PDC
810 - 826,
940 - 956,
1429 - 1465,
1477 - 1513
PACS 1850 - 1910, 1930 1990
PACS - UB 1910 - 1930
PHS
1895 - 1918
JCT
254 - 380
902 - 928
I EEE 802.11
2400 - 2483
5150 - 5350, 5725 - 5825
IEEE 802.11
2471 - 2497
5150 - 5250
RF - Control
315, 915
RF - Control
426, 868
12
Antennas and Signal Propagation
Antennas: Isotropic Radiator
Isotropic radiator: a single point
equal radiation in all directions (three dimensional)
only a theoretical reference antenna
Radiation pattern: measurement of radiation around an
antenna
y
z
z
y
x
x
ideal
isotropic
radiator
Q: how does power level decrease as a function of d, the distance from the
transmitter to the receiver?
14
Real Antennas
• Real antennas are not isotropic radiators
• Some simple antennas: quarter wave λ/4 on car roofs or half
wave dipole λ/2
size of antenna proportional to wavelength for better
transmission/receiving
λ/4
λ/2
Q: Assume frequency 1 Ghz, λ = ?
15
Dipole: Radiation Pattern of a Dipole
http://www.tpub.com/content/neets/14182/index.htm
http://en.wikipedia.org/wiki/Dipole_antenna
16
Why Not Digital Signal (revisited)
• Not good for spectrum usage/sharing
• The wavelength can be extremely large to
build portable devices
– e.g., T = 1 us -> f=1/T = 1MHz -> wavelength =
3x108/106 = 300m
17
Free-Space Isotropic Signal Propagation
Pr
 λ 
= G rG t 

Pt
 4π d 
2
Pr: received power
Pt: transmitted power
Gr, Gt: receiver and
transmitter antenna gain
λ (=c/f): wave length
• In free space, receiving
power proportional to 1/d²
(d = distance between
transmitter and receiver)
• The total radiation power
remains constant, but the
surface area of a sphere
with radius r increases like
r2.
Sometime we write path loss in log scale:
Lp = 10 log(Pt) – 10log(Pr)
18
Signal Propagation
Receiving power additionally influenced by
shadowing (e.g. through a wall or a door)
refraction depending on the density of a medium
reflection at large obstacles
scattering at small obstacles
diffraction at edges
diffraction
refraction
shadow fading
scattering
reflection
19
Signal Propagation: Scenarios
Details of signal
propagation are very
complicated
We want to understand
the key characteristics
that are important to
our objective
20
Reason I: Shadowing
• Signal strength loss after passing through
obstacles
• Some sample numbers
i.e. reduces to ¼ of signal
10 log(1/4) = -6.02
21
Distance power relationship in practice
Received power decreases proportional to 1/dr where r varies
from 2 to 6.
Long corridor, big indoor environment: r=2
Metallic building: r=6.
“Slow fading”
22
Reason II: Multipath
Signal can take many different paths between sender and
receiver due to reflection, scattering, diffraction
23
Multipath Can Reduce Signal Strength
Example: reflection from the ground: received power decreases
proportional to 1/d4 instead of 1/d² due to the destructive
interference between the direct signal and the signal reflected
from the ground
ground
For detail, see page 9:
http://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter2.pdf
24
Multipath Fading
Due to constructive and destructive interference of
multiple transmitted waves, signal strength may vary
widely as a function of receiver position
Listen to radio on a car.
25
Multipath Effect
(fixed receiver location)
Channel characteristics change over location,
frequency
example
cos(2πft )
d2
d1
(
d
α1 cos 2πf [ t − c1 ]
d1
phase
difference:
2π ( f
d1
c
−f
d2
c
)
−
(
α 2 cos 2πf [ t −
d2
c
]
)
d2
d1 − d 2
d1 − d 2
) + π = 2πf
+ π = 2π
+π
c
λ
26
Multipath
(fixed receiver location)
• Suppose at d1-d2 the two waves totally
destruct. (what does it mean?)
d1 − d 2 d1 − d 2
f
=
= integer
c
λ
• Q: can we find places where the two waves
construct?
2πf
d1 − d 2
d − d2
+ π = 2π 1
+π
c
λ
27
Option 1: Change Location
• If receiver moves to the right by λ/4:
d1’ = d1 + λ/4;
d2’ = d2 - λ/4;
-> 2π d '1 −d '2 + π
λ
= 2π
d1 − d 2
= 2π
d1 − d 2
λ
λ
λ / 4 − ( −λ / 4)
+ 2π
+π
λ
+π +π
By moving a quarter of wavelength, destructive
turns into constructive.
28
Option 2: Change Frequency
Change frequency:
1 c
f '= f ±
2 d1 − d 2
The change depends on delay spread
2πf
d1 − d 2
d − d2
+ π = 2π 1
+π
c
λ
29
Multipath Fading: A Simple Two-path
Example
d2
d1
receiver
- Wavelength is about 0.3 m for 1 GHz cellular
30
Multipath Fading with Mobility: A
Simple Two-path Example
r0
r(t) = r0 + v t, assume transmitter sends out signal cos(2π fc t)
More detail see page 16 Eqn. (2.13):
http://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter2.pdf
31
Multipath Effect
(moving receiver)
Channel characteristics change over time (location)
example
d
cos(2πft )
d1
d2
(
d
α1 cos 2πf [ t − c1 ]
d1
Suppose
d1=r0+vt
d2=2d-r0-vt
d1≈d2
)
−
(
α 2 cos 2πf [ t −
d2
c
]
)
d2
32
Derivation
cos(2πf [t −
= −2 sin(
r0 + vt
c
2 πf [ t −
]) − cos(2πf [t −
r0 + vt
]+ 2πf
c
= −2 sin(2πf [t −
[t −
2 d − r0 − vt
]
c
2
r0 + vt + 2 d − r0 − vt
2c
2 d − r0 − vt
c
) sin(
2 πf [ t −
])
r0 + vt
]− 2πf
c
]) sin(2πf [
[t−
2 d − r0 − vt
]
c
2
− r0 − vt + ( 2 d − r0 − vt )
2c
)
])
= −2 sin(2πf [t − dc ]) sin(2πf [ − r0 + vtc −d ])
= 2 sin(2πf [t − dc ]) sin(2πf [ r0 + vtc −d ])
= 2 sin(2πf [t − dc ]) sin( 2πcvf [t − dcv− r0 ])
See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)
33
Received Waveform
2 sin(2πf [t − dc ]) sin( 2πcvf [t − dcv− r0 ])
10 ms
deep fade
v = 65 miles/h, fc = 1 GHz:
fc v/c = 109 * 30 / 3x108 = 100 Hz
Why is fast multipath fading bad?
34
Small-Scale Fading
35
Multipath Can Spread Delay
signal at sender
Time dispersion: signal is
dispersed over time
LOS pulse
multipath
pulses
signal at receiver
LOS: Line Of Sight
36
Delay Spread
RMS: root-mean-square
37
Multipath Can Cause ISI
dispersed signal can cause interference
between “neighbor” symbols, Inter Symbol
Interference (ISI)
Assume 300 meters delay spread, the arrival
time difference is 300/3x108 = 1 ms
if symbol rate > 1 Ms/sec, we will have
serious ISI
signal at sender
LOS pulse
multipath
pulses
In practice, fractional ISI can already
substantially increase loss rate
signal at receiver
LOS: Line Of Sight
38
Summary: Wireless Channels
Channel characteristics change over location, time, and
frequency
Received
Signal
Power
(dB)
power
Large-scale
fading
path loss
log (distance)
small-scale fading
time
frequency
39