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** X_{c}).

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

- 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)

**Models the spatial correlation of shadowing**

First-order AR model, with autocovariance:

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

Setting ρ_{D}=1/e and D=X_{c} the following simplified model is obtained: (see the figure on the right).

*X _{c}*is 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)

Fit model to data

Path loss parameters (*K*,γ) (d_{0}known):

- “Best fit” line through dB data
- K obtained from measurements at d
_{0} - 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

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 d_{0} = 1m and K is determined from the free-space path-gain formula at this d_{0}.

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, σ ^{2}_{dB} = 13.29 dB**

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

**For the considered path loss+ lognormal shadowing model:**

where

Assume that P_{r,min} is the minimum received power (RX sensitivity), and let P_{t,min}be 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:

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).

**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

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.

**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 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.

*3*. Current and emerging wireless systems

*5*. Shadowing

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

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion