ECE442 Communications
Lecture 2. Wireless Channels
Husheng Li
Dept. of Electrical Engineering and Computer Science
Fall, 2013
Two Features of Wireless Channels
1
Broadcast: The wireless signal can also be received by
unintended receivers, thus being interference. This is
different from wired communications. Thus, in wireless
communications, how to tackle interference is a key issue.
2
Fading: the wireless channel may experience deep fade in
time or frequency. Thus, in wireless communications, it is
important to combat fading for reliable data transmission.
Fading can also be utilized for opportunistic
communication.
Two Types of Fading
Large scale fading: characterize the signal strength over
long transmitter-receiver distance (you may need to move
100 meters to see the change of signal strength).
Small scale fading: characterize the rapid fluctuation over
a small distance (you may need to move only half a meter
to see the signal strength changing rapidly).
Transmit and Receive Signal Model
1
Wireless communication usually uses UHF (0.3 - 3GHz)
and SHF (3 - 30GHz).
2
Transmit signal: real sinusoid
h
i
s(t) = < u(t)e2jπfc t ,
where u(t) is complex baseband signal u(t) = I(t) + jQ(t),
which conveys information and fc is carrier frequency.
3
Receive signal:
h
i
r (t) = < v (t)e2jπfc t ,
where v (t) is received baseband signal v (t) = u(t) ? c(t)
and c(t) is channel response.
Doppler Shift
The Doppler frequency is given by fD =
v
λ
cos θ.
Channel Gain and Path Loss
Channel Gain and Path Loss
1
Channel power gain is given by
G = 10 log10
Pr
(dB),
Pt
where transmit power is Pt and receive power is Pr .
2
Path loss is defined by
L = 10 log10
Pt
(dB),
E [Pr ]
where the expectation is over all random environment
(shadow fading, fast fading). Path loss is fixed for all time
and all environment.
Free-space Path Loss
1
Line-of-sight (LOS): no obstructions between the
transmitter and receiver.
2
Path loss:
Pr
=
pt
!2
p
Gl λ
,
4πd
where Gl is antenna product of transmitter and receiver.
3
In free-space,
L ∝ d 2.
4
But in real-world environment, wireless signal decays
faster in distance.
Two-Ray Model
1
When h1 , h2 d, the receive power is given by
Pr (dB) = Pr (dB)+10 log10 (Gl )+20 log10 (h1 h2 )−40 log10 (d).
2
In this case, the signal decays much faster than free-space
case: L ∝ d 4 .
Two-Ray Model
1
When h1 , h2 d, the receive power is given by
Pr (dB) = Pr (dB)+10 log10 (Gl )+20 log10 (h1 h2 )−40 log10 (d).
2
In this case, the signal decays much faster than free-space
case: L ∝ d 4 .
Critical Distance
1
2
3
When d is not much larger than ht and hr , there could be
fluctuations.
The critical distance is the distance after which the signal
power falls off proportionally to d −4 .
An approximation is dc = 4ht hr /λ.
Complicated Model
1
General ray tracing
2
Empirical path-loss model
3
Olumura model
4
Hata model
5
Piecewise linear model
Hata Model
1
Hata Model (for urban environment):
L(dB) = 69.55 + 26.16 log10 (fc ) − 13.82 log10 (ht ) − a(hr )
+ (44.9 − 6.55 log10 (ht )) log10 (d),
where a(hr ) is a correction factor for the mobile height
based on the size of coverage area.
2
For suburban and rural propagation, the path loss should
be corrected (refer to Goldsmith’s book).
3
Note: Hata model is used for outdoor wireless
communications.
Model of Indoor Path Loss
1
When local area network (say, WiFi) is widely used, more
communications happen indoors. The penetration loss of
barriers (partitions, floors, et al):
L = L0 +
Nf
X
i=1
FAFi +
Np
X
PAFi ,
i=1
where FAFi and PAFi represent penetration losses of floors and
partitions, respectively.
Shadow Fading
1
Wireless signal often experience variation caused by blockage
(say buildings) in the signal path. The variation may also come
from changes in reflecting surfaces and scattering objects. We
call it shadow fading.
2
We usually model the shadow fading channel gain as a
log-normal random variable (then the shadow fading channel
gain in dB is a Gaussian random variable:
"
#
(10 log10 ψ − µpsidB )2
ξ
p(ψ) = √
exp −
2σψ2 dB
2πσψdB ψ
Path Loss and Shadow Fading
1
When the path loss and shadow fading are combined, the
ratio of received to transmitted power in dB is given by
Pr
d
dB = 10 log10 K − 10γ log10
− ψdB .
Pt
d0
2
The outage probability is the probability that the received
power at a given distance d the received power is below a
threshold:
Pout = P(Pr (d) ≤ Pmin ).
Cell Coverage: Definition
1
The cell coverage are is a cellular system is defined as the
expected percentage of locations within a cell where the
received power at these locations is above a given
minimum.
Cell Coverage: Randomness
1
All mobiles within the cell require some minimum
signal-to-noise ratio (SNR).
2
The transmit power at the base station is designed for an
average received power at the cell boundary P̄R .
3
The shadowing may make some locations within the cell to
have received power below P̄R .
Cell Coverage: Computation
1
We define the cell coverage area as
Z
1
C=E
1(P(r ) > Pmin in dA)dA .
πR 2 cellarea
2
We can prove
1
C = + exp
2
2
b2
2
Q
,
b
where the definition of b can be found in the textbook.
Homework 1
1
Problem 1. Use Matlab to implement the calculation of Eq.
(2.12) in the textbook, and plot Figure 2.5 (especially when
d is not large). Attach the Matlab codes and the figure.
2
Problem 2. Implement Matlab to calculate the cell
coverage in Eq. (2.61). Plot the curve of the cell coverage
as a function of the decay exponent γ. Plot another curve
of the cell coverage as a function of the shadow fading
variance σψdB .
3
Deadline: Sept. 9, 2013.
Small Scale Fading: Multipath Fading
1
Path loss and shadowing are both large scale, which
means that it does not change much if the location of
received is not much changed.
2
There also exists small scale fading, which changes
radically if the location is slightly changed.
Single Path Signal with Moving Receiver
1
The received signal is given by
r (t) = < [α(t)u(t) exp(j2πfc (t − τ (t)) + φD )] ,
where α(t) is amplitude, τ (t) is delay
R (time required from
transmitter to receiver) and φD = t 2πfD (t)dt is Doppler phase
shift (when receiver is moving, the carrier frequency is shifted by
fD (t) = v cos(θ)t/λ).
2
The terms α(t), τ (t) and φD (t) changes slowly when the receiver
moves. We can consider them as constants in a short period.
Resolvable Paths
1
Small scale fading results from multiple replicas of signal along
multiple propagation paths (τl means the delay along the l-th
path).
2
For two paths 1 and 2, if |τ1 − τ2 | is much smaller than a signal
symbol interval (approximately equalling to inverse bandwidth
B −1 ), these two paths cannot be distinguished (unresolvable); if
|τ1 − τ2 | is much larger than a signal symbol interval, these two
paths can be distinguished (resolvable).
Narrowband and Wideband Systems
1
In narrowband system, there are very few resolvable paths.
Each resolvable signal components may be the
superposition of several paths. The destruction or
construction effect of these paths can change rapidly in the
scale of wavelength.
2
In wideband systems, resolvable signal component may
contain only one path, whose amplitude changes slowly
when the location of transmitter changes.
Narrowband Fading Signal
1
The received signal through a fading channel in a
narrowband system is given by (suppose that there is only
one resolvable path and no LOS path)
"
!#
X
j2πfc t
−jφn (t)
r (t) = < u(t)e
αn (t)e
,
n
where n is the index of unresolvable paths, αn (t) is
amplitude and φn (t) = 2πfc τn (t) − φDn − φ0 is phase.
2
The above signal can be decomposed into in-phase and
quadrature components:
r (t) = rI (t) cos(2πfc t) − rQ (t) sin(2πfc t).
Rayleigh Fading
1
2
3
If there are sufficiently many unresolvable paths and no
LOS path, the two components rI (t) and rQ (t) are mutually
independent Gaussian variables (recall central limit
theorem). Then, it is called Rayleigh fading.
The distribution of signal amplitude r is (σ 2 is a parameter
related to the variance)
r2
r
∀r ≥ 0
f (r ) = 2 exp − 2 ,
σ
2σ
and the signal phase is a uniform random variable in
[0, 2π]. The amplitude and phase are mutually
independent. The signal power is an exponentially
distributed random variable (why?).
Rayleigh fading is bad since there may exist very deep
fade (very small receive signal power).
Ricean Fading
1
If there exists a LOS path (a constant component of the
signal), the received signal satisfies Ricean distribution:
Ar
r − r 2 +A2 2
,
p(r ) = 2 e 2σ I0
σ
σ2
where A is amplitude of the LOS path signal and I0 is
modified Bessel function of the first kind and zero-order.
2
Ricean fading has no deep fade if the LOS component is
large enough.
Rayleigh and Ricean
1
Rayleigh fading is named after Lord Rayleigh, who
discovered the Rayleigh scattering (Nobel Prize in 1904).
2
Ricean fading is named after Stephen O. Rice, who is a
pioneer of mathematical analysis of random noise and
discovered Rice distribution.
Wideband Channel Model
1
When the signal is not narrowband, we get another form of
distortion due to the multipath delay spread.
Deterministic Scattering Function
1
We denote by c(τ, t) the impulse response of the
time-varying channel.
2
We define the deterministic scattering function as
Z ∞
Sc (τ, ρ) =
c(τ, t)e−j2πρt dt.
−∞
3
The Doppler characteristics of the channel is captured by
the parameter ρ.
Random Channel Case: Autocorrelation Function
1
When c(τ, t) is random, we can define the autocorrelation
function
Ac (τ1 , τ2 ; t, δt) = E [c ∗ (τ1 , t)c(τ2 , t + δt)] .
2
Usually, we assume that the above autocorrelation is
independent of the time, thus can be written as
Ac (τ1 , τ2 ; δt).
3
We further assume that the autocorrelation is zero when
τ2 6= τ1 . Then, we write it as Ac (τ, δt).
Random Channel Case: Scattering Function
1
We define the scattering function as
Z ∞
Ac (τ, δt)e−j2πρδt dδt.
Sc (τ, ρ) =
−∞
Problem 1. Prove that the distribution of received signal
power in the Rayleigh fading is exponentially distributed.
Problem 2. If σTm = 40ns, what is the maximum symbol
rate for the environment such that a linearly modulated
signal transmitted through these environments
experiences negligible inter-symbol interference?
Problem 3. For a channel with Doppler spread Bd = 50Hz,
what time separation is required in samples of the received
signal such that the samples are approximately
independent?
Power Delay Profile
1
We call Ac (τ, 0) the power delay profile.
2
We define the average and rms delay spread as
R∞
τ Ac (τ )dτ
µ = R0 ∞
,
0 Ac (τ )dτ
and
sR
σ=
3
∞
0 (τ
R
− µ)2 Ac (τ )dτ
,
∞
0 Ac (τ )dτ
The time delay T when Ac (τ ) ≈ 0 for τ ≥ T can be used to
roughly characterize the delay spread.
Coherence Bandwidth
1
We define
Z
∞
C(f , t) =
c(τ, t)e−j2πf τ dτ.
−∞
2
Then, we obtain the autocorrelation given by
AC (f1 , f2 , ∆t) = E[C ∗ (f1 , t)C(f2 , t + ∆t)] = AC (∆f , ∆t).
3
We define
AC (∆f ) = AC (∆f , 0).
Coherence Bandwidth
1
R∞
We can define AC (∆f ) = −∞ Ac (τ )e−j2πδf τ dτ . The
frequency Bc where AC (δf ) ≈ 0, for all δf > Bc , is called
the coherence bandwidth of the frequency spectrum.
Coherence Bandwidth
1
The coherence bandwidth Bc means that the channel
responses at two frequency points having separation less
than Bc are correlated.
2
We have Bc ≈ 1/T where T is the delay spread.
Coherence Time
1
2
To characterize
R ∞the Doppler effect, we define
SC (∆f , ρ) = −∞ AC (∆f , δt)e−j2πρδt dδt
R∞
We define SC (ρ) = SC (0, ρ) = −∞ AC (δt)e−j2πρδt dδt. We
define the channel coherence time Tc to be the range of
the time over which AC (δt) is approximately nonzero.
Coherence Time
1
We call SC (ρ) the Doppler power spectrum and call its
bandwidth (BD ) as the Doppler spread of the channel.
2
Approximately, we have
BD ≈ 1/Tc ,
which measures how fast the channel changes with time.
Frequency Flat and Selective Fading Channels
1
When the coherence bandwidth is much larger than the
signal bandwidth, the signals at different frequencies
experience similar fading, called frequency flat fading.
2
When the coherence bandwidth is much smaller than the
signal bandwidth, the signals at different frequencies may
experience completely different fading, called frequency
selective fading.
3
Impact on communication systems: in frequency flat
fading, dead once deep fade; in frequency selective fading,
there are always some frequencies being at good state,
thus adding robustness. But troublesome equalization is
required for frequency selective systems.
Frequency Flat Signal Model
1
The fading channel just scales the received signal:
r (l) = a(l)s(l) + n(l),
where r (l) is complex received signal, a(l) is complex
channel gain, s(l) is transmitted signal, n(l) is noise
(assumed to be AWGN) and l is sample index.
2
Fast fading: a(l) changes with l.
3
Slow fading: a(l) is almost a constant over a period.
Frequency Selective Signal Model
1
Multipath model (M paths):
r (l) =
M
X
am (l)s(l − τm ) + n(l).
m=1
2
FIR model (delay spread N = τM ):
r (l) =
N
X
am s(l − m) + n(l)
m=1
= a ? s(l) + n(l).
3
In frequency domain, we have R(l) = A(l)S(l) + N(l),
where l is index for frequency point (flat fading on different
frequencies).
Both models are used widely in research. They are
equivalent if some coefficients in FIR model are set to 0.