Unit 1: The wireless channel
Wireless communications courseRonal D. Montoya M.
http://tableroalparque.weebly.com/radiocomunicaciones.html
September 1, 2017
Outline I
1. Shadow fading
Overview
2. The log-normal distribution
3. Path loss independent from shadowing 4. µ and σ measurements
5. Gaussian model for path loss
6. Combined path loss and shadowing
7. Outage probability under path loss and shadowing 8. Cell coverage area
Overview Model
Shadow fading I
◦ A signal transmitted through a wireless channel will typically
experience random variation due to blockage from objects in the signal path, giving rise to random variations of the received power at a given distance.
◦ The location, size, and dielectric properties of the blocking
objects, random reflection surfaces and scattering objects that cause the random attenuation.
◦ Statistical models must be used to characterize this
attenuation.
Shadow fading II
◦ The most common model for this additional attenuation is
log-normal shadowing.
◦ This model has been confirmed empirically to accurately model
the variation in received power in both outdoor and indoor radio propagation environments.
Path loss, shadowing and multipath vs. distance.
Figure: Path loss, shadowing and multipath vs. distance.
The log-normal distribution I
In the log-normal shadowing model the ratio of transmit-to-receive
power ψ = Pt/Pr is assumed random with a log-normal distribution
given by: p (ψ) = √ ξ 2πσψdBψ exp " −(10 log10ψ − µψdB) 2 2σψ2 dB # , ψ > 0 (1) Where: ◦ ξ = 10/ ln 10.
◦ σψdB is the standard deviation of ψdB in dB.
The log-normal distribution II
◦ µψdB is the mean ψdB = 10 log10ψ. It must be based on an
analytical model or empirical measurements.
◦ For empirical measurements µψdB is equals to the empirical
path loss, since average attenuation from shadowing is already incorporated into the measurements.
◦ For analytical models, µψdB must incorporate both the path loss
(e.g. from free-space or a ray tracing model) as well as average attenuation from blockage.
Path loss independent from shadowing I
◦ Path loss can be treated separately from shadowing.
◦ Note that if the ψ is log-normal, then the Pr and receiver SNR
will also be log-normal since these are just constant multiples of ψ.
◦ For received SNR the mean and standard deviation of this
log-normal random variable are also in dB (unitless).
◦ For log-normal received power, since the random variable has
units of power, its mean and standard deviation will be in dBm or dBW instead of dB.
Path loss independent from shadowing II
The mean of ψ (the linear average path gain) is given by:
µψ = E [ψ] = exp " µψdB ξ + σψ2 dB 2ξ2 # (2) Converting 2 from linear mean to log-linear mean:
10 log10µψ = E [ψ] = µψdB +
σ2ψ
dB
2ξ (3)
Path loss independent from shadowing III
◦ Performance in log-normal shadowing is typically
parameterized by the log-mean µψdB , which is refered to as the
average dB path loss and is in units of dB.
◦ The distribution of the dB value of ψ is Gaussian with mean
µψdB and standard deviation σψdB:
p (µψdB) = 1 √ 2πσψdB exp " −(ψdB − µψdB) 2 2σψ2 dB # (4)
Path loss independent from shadowing IV
Anotations:
◦ ψ = Pt
Pr ≥ 1, then µψdB ≥ 0.
◦ 0 ≤ ψ < 1 are not considered (impossible?).
µ and σ for empirical measurements I
Given an empirical path loss measurements {pi}
N
i=1, should the
mean path loss:
µψ = 1 N N X i=1 pi (5) Or: µψdB = 1 N N X i=1 10 log10pi (6) 4. µ and σ measurements 12/31
µ and σ for empirical measurements II
◦ In practice it’s more common to determine mean path loss and
variance based on averaging the dB values of the empirical measurements for several reasons.
◦ The mathematical justification for the log-normal model is
based on dB measurements.
◦ To obtain empirical averages based on dB path loss
measurements leads to a smaller estimation error.
◦ Power falloff with distance models are often obtained by a
piece-wise linear approximation to empirical measurements of dB power versus the log of distance.
µ and σ for empirical measurements III
◦ Most empirical studies for outdoor channels support a standard
deviation σψdB ranging from 4 to 13 dB.
◦ µψdB depends on the path loss and building properties in the
area under consideration. It varies with distance due to path loss and the fact that average attenuation from objects
increases with distance due to the potential for a larger number of attenuating objects.
Gaussian model for path loss I
◦ It can be justified when shadowing is dominated by the
attenuation from blocking objects.
◦ The attenuation of a signal as it travels through an object of
depth d is approximately equal to:
s (d) = exp (−αd) (7)
◦ α : attenuation constant that depends on the object’s materials
and dielectric properties.
Gaussian model for path loss II
◦ If α is approximately equal for all blocking objects, and that
the ith blocking object has a random depth di , then the s of a
signal as it propagates through this region is:
s (dt) = exp −α X i di ! = exp (−αdt) (8)
Combined path loss and shadowing I
◦ Models for path loss and shadowing can be superimposed to
capture power falloff versus distance along with the random attenuation about this path loss from shadowing.
◦ In this combined model, average dB path loss (µψdB) is
characterized by the path loss model and shadow fading, with a mean of 0 dB, creates variations about this path loss.
◦ For this combined model, Pr/Pt in dB is given by:
Pr
Pt
[dB] = 10 log10K − 10γ log10 d
d0
− ψdB (9)
Combined path loss and shadowing II
◦ ψdB : Gauss-distributed random variable with mean zero and
variance σdB2 .
Outage probability under path loss and shadowing I
◦ There is typically a target minimum received power level Pmin
below which performance of the wireless systems becomes unacceptable.
◦ With shadowing, the received power at any given distance from
the transmitter is log-normally distributed with some
probability of falling below Pmin.
◦ It’s defined the outage probability pout(Pmin, d) under path loss
and shadowing to be the probability that the received power at
a given distance d falls below Pmin
pout(Pmin, d) = p (Pr(d) < Pmin) (10)
Outage probability under path loss and shadowing II
◦ For the combined path loss and shadowing model, this becomes:
p (Pr(d) ≤ Pmin) = 1− Q Pmin − Pt + 10 log10K − 10γ log10 d d0 σψdB (11)
Outage probability under path loss and shadowing III
◦ Where the Q-function can be expressed in terms of the error
function complementary: Q (z) = 1 2erf c z √ 2 (12)
Contours of Constant Received Power
Figure: Contours of Constant Received Power.
Cell coverage area I
◦ The cell coverage area in a cellular system is defined as the
expected percentage of area within a cell that has received power above a given minimum.
◦ For path loss and random shadowing the contours form an
amoeba-like shape due to the random shadowing variations about the average.
◦ The constant power contours for combined path loss and
random shadowing indicate the challenge shadowing poses in
cellular system design (Pr is different for all users in the cell).
Cell coverage area II
◦ The BS must either transmit extra power to insure users
affected by shadowing receive their minimum required power
Pmin, which causes excessive interference to neighboring cells,
or some users within the cell will not meet their minimum received power requirement.
◦ The Gaussian distribution has infinite tails, there is a nonzero
probability that any mobile within the cell will have a Pr that
falls below the Pmin, even if the mobile is close to the base
station.
Cell coverage area III
◦ This makes sense intuitively since a mobile may be in a tunnel
or blocked by a large building, regardless of its proximity to the base station.
◦ The following model for cell coverage area is obtained from the
path loss and shadowing model.
Cell coverage area I
◦ The %A within a cell where the received power exceeds the
minimum required power Pmin is obtained by taking an
incremental area dA at radius r from the BS in the cell.
◦ Pr(r) be the received power in dA.
◦ The total area within the cell where the Pmin requirement is
exceeded is obtained by integrating overall incremental areas where this minimum is exceeded:
C = Z
cell area
E [1 [Pr(r) > Pmin in dA]] dA (13)
Cell coverage area II
◦ 1 [.] : indicator function.
◦ PA = p (Pr(r) > Pmin in dA) = E [1 [Pr(r) > Pmin in dA]]
◦ Making some sustitutions and evaluating in polar coordinates:
C = 1 πR2 Z cell area PAdA = 1 πR2 Z 2π 0 Z R 0 PArdrdθ (14)
◦ The outage probability of the cell is defined as the %A within
the cell that does not meet its Pmin:
pcellout = 1 − C (15)
Cell coverage area III
◦ Using the log-normal distribution for the combined path loss
and shadowing:
p (Pr(d) ≥ Pmin) = 1 − pout(Pmin, r) =
Q Pmin − Pt + 10 log10K − 10γ log10 r d0 σψdB (16)
Cell coverage area IV ◦ Combining 14 and 16: C = 2 R2 Z R 0 rQa + b ln r R dr (17) ◦ Where: a = Pmin − Pr(R) σψdB (18) b = 10γ log10e σψdB (19)
Cell coverage area V
Pr(R) = Pt + 10 log10K − 10γ log10
R d0
(20)
◦ The integral yields a closed-form solution for C in terms of a
and b:
C = Q (a) + exp 2 − 2ab
b2 Q 2 − ab b (21)
Cell coverage area VI
◦ If Pmin = Pr(R), then a = 0 and:
C = 1 2 + exp 2 b2 Q 2 b (22)
◦ Note that with this simplification C depends only on the ratio
γ/σψdB.
◦ Due to the symmetry of the Gaussian distribution,
pout(P r (R) , R) = 0.5.
