Post on 08-Nov-2014
The Wireless Channel (2)
Wireless Communication
www.ee.ui.ac.id/wasp
Review What we discussed last lecture:
The large-scale fading because of path loss The empirical path loss formulas
Today, we will discuss about the small-scale fading and the statistical models represent it
Introduction The small-scale fading is usually called
“fading” It is caused by multipath signal, so it is also
called “multipath fading” Multipath signal causes constructive and
destructive addition of the received signal
Introduction If a single pulse is transmitted in the multipath
channel, it will yield a train of pulses with delay time
Delay spread ( ): the time delay between the arrival of the first received signal component and the last received signal component associated with a single transmitted pulse
LOS
Reflected components
dT
Introduction If the delay spread is small compared to the
(1/BW), then there is little time spreading in the received signal
If the delay spread is relatively large, there is little time spreading of the received signal, i.e. signal distortion
Multipath channel is also time-varying that means either the transmitter or the receiver is moving
It also causes the location of the reflectors will change over time
We will limit the model to be narrowband fading, i.e. the BW is small compared to (1/delay spread)
Introduction Physical factors influencing fading:
Multipath propagation Speed of the mobile Speed of surrounding objects The transmision bandwidth of the signal
Review of Doppler Shift The received signal may experience Doppler shift
If the receiver is moving towards the transmitter, the Doppler freq is positive, otherwise it is negative
v eff
Example Consider a transmitter which radiates a carrier
of 1850 MHz. For a vehicle moving 26.82 mps, compute the received carrier frequency if the mobile is moving: Directly towards the transmitter Directly away from the transmitter In a direction which is perpendicular to the
direction of arrival of the transmitted signal
Carrier freq = 1850 MHz Wavelength = Vehicle speed = 26.82 m/s Vehicle moving towards the transmitter
means positive Doppler frequency
Vehicle moving towards the transmitter means negative Doppler frequency
Vehicle is moving perpendicular means
Solution
8
6
3 100.162
1850 10c
cm
f
26.821850 cos0 1850.00016
0.162c df f f MHz
26.821850 cos 0 1849.999834
0.162c df f f MHz
90 26.82
1850 cos90 18500.162c df f f MHz
Doppler Spread Doppler spread is given by
Where, and
E.g. If the mobile is moving at 60 kph and f = 900 MHz, the the Doppler spread is
2 1:sD D D
1
fvD
c 2
fvD
c
6
8
2
900 10 16.672 100
3 10
s
fv fv fvD
c c c
Hz
Time-Varying Channel Impulse Response We have already known that the transmitted
signal is
Then, the received signal in multipath channel is
n = 0 corresponds to the LOS pathN(t) is the number of resolvable multipath components is corresponding delay is Doppler phase shift
is amplitude
2 cos 2 sin 2cj f tc cs t u t e u t f t u t f t
Time-Varying Channel Impulse Response The n-th resolvable multipath component may
correspond to the multipath associated with a single reflector or multiple reflectors clustered together
Time-Varying Channel Impulse Response If single reflector exists, the amplitude is based on the
path loss and shadowing, its phase change associated with delay and Doppler phase shift of
If reflector cluster exists, two multipath components with delay and are resolvable if
If the criteria is not satisfied, then it is nonresolvable since
The nonresolvable components are combined into a single multipath component with delay and an amplitude and phase corresponding to the sum of different components
2 c nj f tn t e
2N ND D
t
f t dt
1 21
1 2 uB
1 2u t u t
1 2
Time-Varying Channel Impulse Response The amplitude of the summed signal will
undergo fast variations due to the constructive and destructive combining of the nonresolvable multipath components
Wideband channels have resolvable multipath components the parameters change slowly
Narrowband channels tend to have nonresolvable multipath components the parameters change quickly
Time-Varying Channel Impulse Response We can simplify by letting
The received signal is then
The received signal is obtained by convolving the baseband input signal with equivalent lowpass time-varying channel impulse response of the channel, and then upconverting the carrier frequency
r t
2nn c n Dt f t
Time-Varying Channel Impulse Response The represents the equivalent lowpass
response of the channel at time t to an impulse at time
,c t
t
Parameters of Mobile Multipath Channels Time dispersion parameters Coherence bandwidth Doppler spread and coherence time
Time Dispersion Parameters The time dispersive properties of wideband
multipath channels are most commonly quantified by their mean excess delay and rms delay spread
The mean excess delay:
The rms delay spread is the square root of the second central moment of the power delay profile
2
2
k k k kk k
k kk k
a P
a P
22
2 2 2
22
k k k kk k
k kk k
a P
a P
Time Dispersion Parameters The delays are measured relative to the first
detectable signal arriving at the receiver at The maximum excess delay (X dB) of the
power delay profile is defined to be the time delay during which multipath energy falls to X dB below the maximum.
The maximum excess delay sometimes called excess delay spread, which can be expressed asWhere is the maximum delay at which a multipath component is within X dB of the strongest arriving multipath signal and is the first arriving signal
0
0X
X
0
Time Dispersion Parameters
Coherence Bandwidth Coherence bandwidth is a statistical measure
of the range of frequencies over which the channel can be considered “flat”
Flat fading is a channel which passes all spectral components with approximately equal gain and linear phase
The coherence bandwidth can be expressed as
(above 90% correlation)
(above 50% correlation)
1
5cB
1
50cB
Example Compute the mean excess delay, rms delay
spread, and the maximum excess delay for the following power delay profile
Estimate the 50% coherence bandwidth of the channel
Solution Using the definition of maximum excess delay
(10 dB), it can be seen that The mean excess delay:
The second moment
The rms delay spread:
The coherence bandwidth:
10 4dB s
1 5 0.1 1 0.1 2 0.01 04.38
0.01 0.1 0.1 1s
2 2 2 2
2 21 5 0.1 1 0.1 2 0.01 021.07
0.01 0.1 0.1 1s
221.07 4.38 1.37 s
1 1
1465 5 1.37cB kHz
s
Doppler Spread and Coherence Time Doppler spread has been discussed before The coherence time is related with Doppler
spread (Doppler shift)
0.423cT v
Types of Small-Scale Fading
Flat Fading If the mobile radio channel has a constant
gain and linear phase response over a bandwidth which is greater than the bandwidth of the transmitted signal, then the received signal will undergo flat fading
Flat Fading Flat fading channels are also known as
amplitude varying channels It is also sometimes referred to as narrowband
channels The most common amplitude distributions
are: Rayleigh, Rician, and Nakagami Summarize: a signal undergoes flat fading if
s cB B
sT
Frequency Selective Fading If the channel has a constant-gain and linear
phase response over a bandwidth that is smaller than the bandwidth of transmitted signal, then the channel creates frequency selective fading on the received signal
Frequency Selective Fading The received signal includes multiple versions
of the transmited waveform which are attenuated and delayed in time, and hence the received signal is distorted
Frequency selective fading is due to time dispersion of the transmitted symbols within the channel
Thus, the channel induces intersymbol interference (ISI)
The modeling for this kind of channel is more difficult since each multipath signal must be modeled and channel must be considered to be a linear filter
The common model: 2-ray Rayleigh fading
Frequency Selective Fading It is sometimes called wideband channels
since the bandwidth of the signal is wider than the bandwidth of the channel impulse response
Summarize: a signal undergoes frequency selective fading if s cB B
sT
Fast Fading In a fast fading channel, the channel impulse
response changes rapidly within the symbol duration
In other words, the coherence time of the channel is smaller than the symbol period of the transmitted signal
This causes frequency dispersion (time selective fading) due to Doppler spread, which lead to signal distortion
Signal distortion due to fast fading increases with increasing Doppler spread relative to the bandwidth of the transmitted signal
Summarize: a signal undergoes fast fading if
s cT T
s DB B
Slow Fading In a slow fading channel, the channel impulse
response changes at a rate much slower than the transmitted signal
The channel may be assumed to be static over one or several reciprocal bandwidth interval
The Doppler spread of the channel is much less than the bandwidth of the baseband signal
Summarize: a signal undergoes slow fading if
s cT T
s DB B
Summary
Remarks When a channel is specified as a fast or slow fading
channel, it does not specify whether the channel is flat fading or frequency selective
Fast fading only deals with the rate of change of the channel due to motion
In flat fading channel, we can approximate the impulse response to be simply delta function
A flat fading, fast fading channel is a channel in which the amplitude of the delta function varies faster that the rate of the transmitted baseband signal
A frequency selective, fast fading channel, the amplitudes, phases, and time delays of any one of the multipath components vary faster than the rate of change of the transmitted signal
Rayleigh Fading The Rayleigh distribution is commonly used to
describe the statistical time varying nature of the received envelope of a flat fading signal
Rayleigh distributed signal:
Rayleigh Fading The Rayleigh distribution has pdf
The probability that the envelope of the received signal does not exceed a specified value R is
the rms value of the received voltage signal before envelope detection
2 the time-average power of the received signal before envelope detection
Rayleigh Fading The mean value of Rayleigh distribution is
The variance of the Rayleigh distribution (represent the ac power)
The median value is
The median is often used in practice
Rayleigh Fading The corresponding Rayleigh pdf is
Level Crossing and Fading Statistics The level crossing rate (LCR) is defined as the
expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified level in a positive-going direction
The number of level crossing per second is given by
Where is time derivative of r(t) (the slope)
is the joint density function of r and at r = R
2
0
, 2R DN rp R r dr f e
,p R rr
r
rmsR R
Example For a Rayleigh fading signal, compute the
positive-going level crossing rate for when the maximum Doppler frequency is 20 Hz
What is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900 MHz?
1
Solution Use the equation for LCR
Use equation of Doppler frequency
12 20 1 18.44RN e
20 1 3 6.66 /Dv f m s
Level Crossing and Fading Statistics The average fade duration is defined as the
average period of time for which the received signal is below a specified level R.
For a Rayleigh fading signal, it is given by
So, the average fade duration can be expressed as
1Pr
R
r RN
2
0
1Pr
1 exp
ii
R
r RT
p r dr
2
1
2D
e
f
Example Find the average fade duration for threshold
levelswhen the Doppler frequency is 200 Hz
Solution Average fade duration is
0.01
20.01 119.9
0.01 200 2
es
Rician Fading Distribution When there is a dominant stationary
(nonfading) signal component present, such as line-of-sight propagation path, the small-scale fading envelope distribution is Rician
Random multipath componnets arriving at different angles are superimposed on a stationary dominant signal
The Rician distribution is given by
2 2
2202 2
for 0, 0
0 for 0
r Ar Ar
p r e I A r
r
Rician Fading Distribution The Rician distribution is described in terms of
a parameter K
As we have Rayleigh fading As we have no fading, channel has no
multipath, only LOS component
2
22
AK
2
210log
2
AK dB
0K K
Rician Fading Distribution The Rician pdf is
Conclusions Small-scale fading is variation of signal
strength over distances of the order of the carrier wavelength
It is due to constructive and destructive interference of multipath
Key parameters:Doppler spread coherence timeDelay spread coherence
bandwidth Statistical small-scale fading: Rayleigh fading
and Rician fading flat fading