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Giacinto Gelli » 5.Shadowing


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


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 Xc).

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 ρD correlation coefficient at distance D
Typical values:

  • suburban macrocell: D=100 m, ρD =0.82 @ fc = 900 MHz
  • urban microcell: D=10 m, ρD=0.3 @ fc = 2 GHz

Decorrelation distance

Setting ρD=1/e and D=Xc the following simplified model is obtained: (see the figure on the right).
Xcis 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

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


Consider the set of empirical measurements of Pr/Pt 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, σ2dB = 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]

Fading margin

Assume that Pr,min is the minimum received power (RX sensitivity), and let Pt,minbe the corresponding minimum TX power (neglecting shadowing) with symmetrical shadowing, there is 50% probability that the effective Pr is below Pr, min (50% outage probability).

The fading margin FdB is the excess Pt 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)


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

  • Pt = 10 mW
  • Pmin = -110.5 dBm

Answer: Pout = 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

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


Random attenuation due to shadowing modeled as log-normal (attenuation in dB is Gaussian).
Shadowing spatially decorrelates over decorrelation distance Xc (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.

I materiali di supporto della lezione

M. Gudmunson, “Correlation model for shadow fading in mobile radio systems,” Electronic Letters, pp. 2145-2146, November 1991

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (chap. 2)

Supplementary material eventually available on the website

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