Lecture 3 - EE 359: Wireless Communications - Winter 2020
Shadowing, Combined Path Loss/Shadowing,
Data Model Parameters, Statistical Multipath Model
Lecture Outline
• Log Normal Shadowing
• Combined Path Loss and Shadowing
• Outage Probability
• Model Parameters from Empirical Data
• Statistical Multipath Model
1. Log-normal Shadowing:
• Statistical model for variations in the received signal amplitude due to blockage.
• The received signal power with the combined effect of path loss (power falloff model)
and shadowing is, in dB, given by
Pr (dB) = Pt (dB) + 10 log10 K − 10γ log10 (d/dr ) − ψ(dB).
• Empirical measurements support the log-normal distribution for ψ:
" #
ξ (10 log10 ψ − µψdB )2
p(ψ) = √ exp − , ψ > 0,
2πσψdB ψ 2σψ2 dB
where ξ = 10/ ln 10, µψdB is the mean of ψdB = 10 log10 ψ in dB and σψdB is the
standard deviation of ψdB , also in dB.
• With a change of variables, setting ψdB = 10 log10 ψ, we get
" #
1 (ψdB − µψdB )2
p(ψdB ) = √ exp − , −∞ < ψdB < ∞.
2πσψdB 2σψ2 dB
• This empirical distribution can be justified by a CLT argument.
• The autocorrelation based on measurements follows an autoregressive model:
Aψ (δ) = σψ2 dB e−δ/Xc = σψ2 dB e−vτ /Xc ,
where Xc is the decorrelation distance, which depends on the environment.
2. Combined Path Loss and Shadowing
• Linear Model: γ
Pr d
=K ψ.
Pt dr
• dB Model:
Pr
(dB) = 10 log10 K − 10γ log10 (d/dr ) − ψdB .
Pt
• Average shadowing attenuation: when KdB = 10 log10 K captures average dB shadow-
ing, µψdB = 0, otherwise µψdB > 0 since shadowing causes positive attenuation.
3. Outage Probability under Path Loss and Shadowing
• With path loss and shadowing, the received power at any given distance between
transmitter and receiver is random.
• Leads to a non-circular coverage area around the transmitter, i.e. non-circular contours
of constant power above which performance (e.g. in WiFi or cellular) is acceptable.
• Outage probability Pout (Pmin , d) is defined as the probability that the received power
at a given distance d, Pr (d), is below a target Pmin : Pout (Pmin , d) = p(Pr (d) < Pmin ).
• For the simplified path loss model and log normal shadowing this becomes
!
Pmin − (Pt + KdB − 10γ log10 (d/dr ))
p(Pr (d) ≤ Pmin ) = 1 − Q .
σψdB
4. Model Parameters from Empirical Data:
• Constant KdB typically obtained from measurement at distance d0 .
• Power falloff exponent γ obtained by minimizing the MSE of the predicted model
versus the data (assume N samples):
N
X
F (γ) = [Mmeasured (di ) − Mmodel (di )]2 ,
i=1
where Mmeasured (di ) is the ith path loss measurement at distance di and Mmodel (di ) =
KdB − 10γ log10 (di ). The minimizing γ is obtained by differentiating with respect to
γ, setting this derivative to zero, and solving for γ.
• The resulting path loss model will include average attenuation, so µψdB = 0.
• Can also solve simultaneously for (KdB , γ) via a least squares fit of both parameters
to the data. Using the line equation for each data point yi that yi = mxi + KdB for
m = −10γ and xi = log10 (di ), the error of the straight line fit is
N
X
F (K, γ) = [Mmeasured (di ) − (mxi + KdB )]2 ,
i=1
• The shadowing variance σψ2 dB is obtained by determining the MSE of the data versus
the empirical path loss model with the minimizing γ = γ0 :
N
1 X
σψ2 dB = [M (di ) − Mmodel (di )]2 ; Mmodel (di ) = KdB − 10γ0 log10 (di ).
N i=1 measured
5. Statistical Multipath Model:
• At each time instant there are a random number N (t) of multipath signal components.
• At time t the ith component has a random amplitude αi (t), angle of arrival θi (t),
Doppler shift fDi = λv cos θi (t) and associated phase shift φDi = t fDi (t)dt, and path
R
delay relative to the LOS component τi (t) = (xi (t))/c.
• Thus, the received signal is given by the following expression, which implies the channel
has a time-varying impulse response.
NX(t)
r(t) = < αi (t)u(t − τi (t))ej(2πfc (t−τi (t))+φDi )
i=0
Main Points
• Shadowing decorrelates over its decorrelation distance, which is on the order of the size of
shadowing objects.
• Combined path loss and shadowing leads to outage and non-circular coverage area (cells).
• Path loss and shadowing parameters are obtained from empirical measurements through a
least-squares fit.
• Can find path loss exponent γ by a 1-dimensional least-squares-error line fit assuming a fixed
value of KdB from one far-field measurement (most common), or find path loss exponent γ
and KdB parameters simultaneously through a 2-dimensional least-squares-error line fit.
• Statistical multipath model leads to a time varying channel impulse response
