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Giacinto Gelli » 5.Shadowing

Outline

Log-normal shadowing
Combined path loss and shadowing
Model parameters from empirical measurements
Outage probability
Cell coverage area

Shadowing

Shadowing models attenuation from obstructions:

statistical model needed due to random number and type of obstructions

Typically follows a log-normal distribution.
The shadowing decorrelates over distances of the order of the dimensions of the blocking objects (decorrelation distance X_c).

Log-normal model

Due to shadowing, the path loss in dB is modeled as a normal (Gaussian) distribution:

$\psi_{dB} = 10 \, \log_{10} \frac{P_t}{P_r} \sim N( \mu_{dB}, \sigma_{dB}^2 )\quad \quadp( \psi_{dB} ) = \frac{1}{\sqrt{2 \pi}\sigma_{dB}} \, \exp \left[ - \frac{\left( \psi_{dB} - \mu_{dB} \right)^2}{2 \sigma_{dB}^2}\right]$

the path loss Ψ expressed in natural units exhibits a log-normal distribution
it is convenient to work with dB values, table/plots of the Gaussian distribution readily available
the mean path loss is evaluated using one of the previous models (free-space, two-ray, empirical models, etc.)
standard deviation typically between 4 dB and 13 dB (experimental data)

Gudmunson model

Models the spatial correlation of shadowing

First-order AR model, with autocovariance:

$A(\delta) = E\left{\left[\psi_{dB}(d) - \mu_{dB} \right] \left[\psi_{dB}(d+\delta)-\mu_{dB}\right] \right}=\sigma_{dB}^2 \,\rho^{| \delta |/D}$

where ρ_D correlation coefficient at distance D
Typical values:

suburban macrocell: D=100 m, ρ_D =0.82 @ f_c = 900 MHz
urban microcell: D=10 m, ρ_D=0.3 @ f_c = 2 GHz

Decorrelation distance

Setting ρ_D=1/e and D=X_c the following simplified model is obtained: (see the figure on the right).
X_cis the decorrelation distance:

distance at which the autocovariance equals 1/e ≈ 0.36 of its maximum value
typically of the order of the size of the blocking object or cluster of these objects (50-100m in outdoor)

Combined path loss + shadowing

Source: Stanford University

Model parameters from empirical measurements

Fit model to data
Path loss parameters (K,γ) (d₀known):

“Best fit” line through dB data
K obtained from measurements at d₀
Exponent γ is line slope (MMSE estimate)
Captures average PL due to shadowing

Shadowing variance:

variance of data relative to path loss model (red line) with MMSE estimate for γ

Source: Stanford University

Example

Consider the set of empirical measurements of P_r/P_t given in the Table for an indoor system at 900 MHz.

Find the path loss exponent δ that minimizes the MSE between the simplified model and the empirical dB power measurements, assuming that d₀ = 1m and K is determined from the free-space path-gain formula at this d₀.

Find the variance of log-normal shadowing about the mean path loss based on these empirical measurements.

Answer: γ = 3.71, K = -31.54 dB, σ²_dB = 13.29 dB

Outage probability

Probability that the received power Pr(d) is below the minimum acceptable value:

$p_{out}(P_{min}, d) = p\left( P_r(d) < P_{min} \right)$

For the considered path loss+ lognormal shadowing model:

$p\left( P_r(d) < P_{min} \right)=1-Q\left[ \frac{P_{min}-\left( P_t + 10\,\log_{10}K-10\,\gamma \,\log_{10} (d/d_0) \right)}{\sigma_{\psi_{\dB}}} \right]$
where
$Q(z)=p(X>z)=\int_{z}^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}\,dy$

Fading margin

Assume that P_r,min is the minimum received power (RX sensitivity), and let P_t,minbe the corresponding minimum TX power (neglecting shadowing) with symmetrical shadowing, there is 50% probability that the effective P_r is below P_{r, min} (50% outage probability).

The fading margin F_dB is the excess P_t needed to reduce the outage probability below a given value (typically 10%).
With the lognormal model it can be shown that:

$p_{out} = 1 - Q \left( \frac{F_{dB}}{\sigma_{dB}}\right) = G \left( \frac{F_{dB}}{\sigma_{dB}}\right)$

Example

Find the outage probability at 150 m for a channel based on the combined path loss + shadowing model of the previous examples, assuming:

P_t= 10 mW
P_min = -110.5 dBm

Answer: P_out = 0.0121
You need a chart of the Q(x) or G(x) = 1-Q(x) function to solve this example (see following slide).

The G(x) function

Cell shapes (ideal vs. real)

Hexagonal cell

Fictitious

Circular cell

Uniform path loss

Amoeba cell

Non-uniform path loss

Amoeba cell with holes in coverage

Non-uniform path loss + shadowing

Source: Stanford University

Cell coverage area

Expected percentage of locations within a cell where the received power is above a given minimum.
Increases as shadowing variance decreseas.
Large percentage is an indication of interference to other cells.
Can be evaluated analytically assuming the lognormal shadowing model.
In practice it is evaluated by using sophisticated software tools.

Shadowing countermeasures

Increase transmitted power (fading margin)

Drawback: more interference to other cells

Macroscopic diversity

Combine signals emitted by different base stations
Requires coordination between base stations

Power control

Adaptively adjust the transmit power in order to assure that the received power is above the minimum
Increased system complexity

Conclusions

Random attenuation due to shadowing modeled as log-normal (attenuation in dB is Gaussian).
Shadowing spatially decorrelates over decorrelation distance X_c (Gudmunson model).
Path loss and shadowing parameters can be obtained from empirical measurements.
Combined path loss and shadowing leads to outage, amoeba-like cells, and holes in coverage.
Cell coverage area dictates the percentage of locations within a cell that are not in outage.
Shadowing can be counteracted by increasing transmitted power, macroscopic diversity, and power control.