Log-Normal Shadowing

Cellular Networking and Mobile Radio Channel Characterization

2.3 Cellular Radio Channel Characterization

2.3.3 Large-Scale Fading: Shadowing and Foliage

2.3.3.1 Log-Normal Shadowing

The path loss models provide estimation for median signal attenuation.

However, providing an acceptable level of QoS in a wireless network cannot be achieved merely by the knowledge of estimation of median path loss. One needs to have access to the estimation of the range of SNRs as a function of TR distance. In other words, path loss model predictions must be modified in order to enable the calculation of range of actual signal loss that is encountered at any given distance from the transmitter. This implies that in addition to large-scale path loss, the effect of foliage and shadowing must also be taken into account.

Empirical data and measurements have shown that for a given value of TR distance d, the combination of path loss and shadowing at a given location is a random variable. When the overall signal attenuation (path loss plus shadowing effect) is expressed in decibel the random variable has a normal distribution, that is, the actual path loss added to shadowing attenuation has a log-normal distribution³[30]. The expected value of this log-normal random variable is reasonably estimated by large-scale path loss models or by Equation 2.21. A probabilistic model for the total path loss may, therefore, be expressed as in Equation 2.61.

…Pℓ†dBˆ …Pℓm†dB‡ …X†dB (2.61)

3 A random variable X is said to have log-normal distribution if random variable log₁₀X has a Gaussian (normal) distribution.

Here, Pℓ… †_dBis a non-zero mean Gaussian random variable representing the overall path loss, Pℓ… m†_dBis the path loss mean value in dB given by large-scale path loss models, X… †_dB is a zero-mean Gaussian random variable characterizing the shadowing effect. It is the random variable X that when expressed in straight values has a log-normal distribution. The assumption of log-normal distribution for random variable X is consis-tently supported by measured data [4,5,30].

Since X… †dB is a zero-mean random variable, its probability density function (PDF) is given by Equation 2.62:

f_XdB… † ˆx 1 ffiffiffiffiffi 2π p σX

e x²

2σ²_X (2.62)

σXis the standard deviation of the Gaussian random variable X… †dB. When the PDF of a random variable is available probability calculations become a straight forward matter. In this case, for complete characterization of the PDF, a value forσXis required. In practice, the value ofσXis determined from measured path loss data over a wide range of locations and TR separations, and with the application of linear regression that minimizes the mean square error between the measured data and estimated model.

Application of this technique on data measured and collected from four cities in Germany has resulted in a value for σX that is equal to 11.8 dB [31]. However, Ref. [5] claims that empirical studies and measured data have indicated that σXˆ 8 dB for a large number of urban and suburban areas, and the value of this standard deviation increases by 1–2 dB for dense urban areas, and is lowered by 1–2 dB in open rural environments. Having a value for σx makes it possible to compose a formula for the computation of the received signal power at distance d, as expressed in Equation 2.63.

Pr… †d

‰ Š_dBˆ P… †t _dB ‰Pℓ d… †Š_dBˆ P… †t _dB ‰Pℓm… †dŠ_dB … †X _dB (2.63) In this equation, all other losses and gains that might exist in the system, such as antenna directivity gains, have been disregarded, or can be assumed to have been included in the term…Pt†_dB. The expected value of the received power at distance d can be expressed by thefirst two terms on the right side of Equation 2.63.

E P _r… †d _dB

ˆ P… †t _dB ‰Pℓm… †dŠdB≜ μ_dB… †d (2.64) Therefore, we conclude that

P_r… †d

‰ ŠdBˆ μdB… †d … †XdB (2.65)

Hence, the received power at distance d, when expressed in dB, has a non-zero Gaussian distribution with mean valueμdB… †, and a PDF that is givend

by Equation 2.66.

f_Pr…d† ˆ 1 ffiffiffiffiffi p2π

σX

e

Pr d… † μdB d… †

‰ Š²

2σ2X (2.66)

With this PDF at hand, the probability that the received power at distance d from the BS antenna is smaller than or equal to a required value of p_dB can be calculated as follows.

Pr P r… †d_dB p_dB

ˆ Z^pdB

1

f_Pr… †dPd r… † ˆd Z^pdB

1

ffiffiffiffiffi1 p2π

σX

e

Pr d… † μdB d… †

‰ Š²

2σ2X dPr… †d

(2.67) Similarly, the probability that the received signal power at distance d is greater than a desired threshold value p_dB is given by Equation 2.68.

Pr P_r… †ddB> pdB

 

ˆ Z¹

p_dB

f_Pr… †dPd r… † ˆd Z¹

p_dB

ffiffiffiffiffi1 2π p σX

e

Pr d… † μdB d… †

‰ Š²

2σ2X dP_r… †d

(2.68) Since there is no closed-form equation for the integral of the Gaussian function, the values of these integrals are normally given either by the Q-function⁴ or the complementary error function expressions. With a simple change of variable, the Gaussian functions inside the integral signs in Equations 2.67 and 2.68 can be converted to standard Gaussian PDF, and consequently the integrals can be shown to have the values given by Equations 2.69 and 2.70.

Pr P _r… †d_dB p_dB

ˆ Q μ_dB… † pd _dB σX



(2.69)

Pr P_r… †ddB> pdB

 

ˆ 1 Q μ_dB… † pd _dB σX



ˆ Q p_dB μ_dB… †d σX



(2.70) The second part of Equation 2.70 is obtained when the Q-function property of Q… † ˆ 1 Q xx … † is applied. Often times, to make proper

4 The Q-function is used for computation of probabilities for standard Gaussian random variable, it is defined by Q x… † ≜^p1ffiffiffiffi_2πR¹

x

e^y22dy. As such the Q-function calculates the area under the upper tail of the standard Gaussian PDF. The values of Q(x) is normally looked up from a table of the Q-function. For x>3 approximate equation of Q x… † ≅ ^effiffiffiffi^x22

p2π xmight be used.

interpretation of these probabilities they are translated into percentages.

For instance, Pr P r… †d _dB> 100 dBm

ˆ 0:8 means if signal power is measured at large number of points on a circle of radius d centered on BS transmitting antenna, about 80% of data would show a power level greater than 100 dBm. If d is the approximate radius of the underlying cell, and if 100 dBm is the receiver sensitivity,⁵ then this probability value indicates that 80% of the cell boundary receives adequate signal power for the delivery of a predefined level of signal-to-noise ratio.

2.3.3.2 Estimation of Useful Coverage Area (UCA) within a Cell Footprint