Unit 1: The wireless channelWireless communications course

Ronal D. Montoya M.http://tableroalparque.weebly.com/radiocomunicaciones.html

ronalmontoya5310@correo.itm.edu.co

September 1, 2017

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Outline I1. Shadow fading

Overview

2. The log-normal distribution

3. Path loss independent from shadowing

4. µ and σ measurements

5. Gaussian model for path loss

6. Combined path loss and shadowing

7. Outage probability under path loss and shadowing

8. Cell coverage areaOverviewModel

2/31

Shadow fading I

◦ A signal transmitted through a wireless channel will typicallyexperience random variation due to blockage from objects inthe signal path, giving rise to random variations of the receivedpower at a given distance.

◦ The location, size, and dielectric properties of the blockingobjects, random reflection surfaces and scattering objects thatcause the random attenuation.

◦ Statistical models must be used to characterize thisattenuation.

1. Shadow fading 3/31

Shadow fading II

◦ The most common model for this additional attenuation islog-normal shadowing.

◦ This model has been confirmed empirically to accurately modelthe variation in received power in both outdoor and indoorradio propagation environments.

1. Shadow fading 4/31

Path loss, shadowing and multipath vs. distance.

Figure: Path loss, shadowing and multipath vs. distance.

1. Shadow fading 5/31

The log-normal distribution I

In the log-normal shadowing model the ratio of transmit-to-receivepower ψ = Pt/Pr is assumed random with a log-normal distributiongiven by:

p (ψ) =ξ√

2πσψdBψ

exp

[−(10 log10 ψ − µψdB

)2

2σ2ψdB

], ψ > 0 (1)

Where:

◦ ξ = 10/ ln 10.

◦ σψdBis the standard deviation of ψdB in dB.

2. The log-normal distribution 6/31

The log-normal distribution II

◦ µψdBis the mean ψdB = 10 log10 ψ. It must be based on an

analytical model or empirical measurements.

◦ For empirical measurements µψdBis equals to the empirical

path loss, since average attenuation from shadowing is alreadyincorporated into the measurements.

◦ For analytical models, µψdBmust incorporate both the path loss

(e.g. from free-space or a ray tracing model) as well as averageattenuation from blockage.

2. The log-normal distribution 7/31

Path loss independent from shadowing I

◦ Path loss can be treated separately from shadowing.

◦ Note that if the ψ is log-normal, then the Pr and receiver SNRwill also be log-normal since these are just constant multiplesof ψ.

◦ For received SNR the mean and standard deviation of thislog-normal random variable are also in dB (unitless).

◦ For log-normal received power, since the random variable hasunits of power, its mean and standard deviation will be in dBmor dBW instead of dB.

3. Path loss independent from shadowing 8/31

Path loss independent from shadowing II

The mean of ψ (the linear average path gain) is given by:

µψ = E [ψ] = exp

[µψdB

ξ+σ2ψdB

2ξ2

](2)

Converting 2 from linear mean to log-linear mean:

10 log10 µψ = E [ψ] = µψdB+σ2ψdB

2ξ(3)

3. Path loss independent from shadowing 9/31

Path loss independent from shadowing III

◦ Performance in log-normal shadowing is typicallyparameterized by the log-mean µψdB

, which is refered to as theaverage dB path loss and is in units of dB.

◦ The distribution of the dB value of ψ is Gaussian with meanµψdB

and standard deviation σψdB:

p (µψdB) =

1√2πσψdB

exp

[−(ψdB − µψdB

)2

2σ2ψdB

](4)

3. Path loss independent from shadowing 10/31

Path loss independent from shadowing IV

Anotations:

◦ ψ = Pt

Pr≥ 1, then µψdB

≥ 0.

◦ 0 ≤ ψ < 1 are not considered (impossible?).

3. Path loss independent from shadowing 11/31

µ and σ for empirical measurements I

Given an empirical path loss measurements {pi}Ni=1, should themean path loss:

µψ =1

N

N∑i=1

pi (5)

Or:

µψdB=

1

N

N∑i=1

10 log10 pi (6)

4. µ and σ measurements 12/31

µ and σ for empirical measurements II

◦ In practice it’s more common to determine mean path loss andvariance based on averaging the dB values of the empiricalmeasurements for several reasons.

◦ The mathematical justification for the log-normal model isbased on dB measurements.

◦ To obtain empirical averages based on dB path lossmeasurements leads to a smaller estimation error.

◦ Power falloff with distance models are often obtained by apiece-wise linear approximation to empirical measurements ofdB power versus the log of distance.

4. µ and σ measurements 13/31

µ and σ for empirical measurements III

◦ Most empirical studies for outdoor channels support a standarddeviation σψdB

ranging from 4 to 13 dB.

◦ µψdBdepends on the path loss and building properties in the

area under consideration. It varies with distance due to pathloss and the fact that average attenuation from objectsincreases with distance due to the potential for a larger numberof attenuating objects.

4. µ and σ measurements 14/31

Gaussian model for path loss I

◦ It can be justified when shadowing is dominated by theattenuation from blocking objects.

◦ The attenuation of a signal as it travels through an object ofdepth d is approximately equal to:

s (d) = exp (−αd) (7)

◦ α : attenuation constant that depends on the object’s materialsand dielectric properties.

5. Gaussian model for path loss 15/31

Gaussian model for path loss II

◦ If α is approximately equal for all blocking objects, and thatthe ith blocking object has a random depth di , then the s of asignal as it propagates through this region is:

s (dt) = exp

(−α∑i

di

)= exp (−αdt) (8)

5. Gaussian model for path loss 16/31

Combined path loss and shadowing I

◦ Models for path loss and shadowing can be superimposed tocapture power falloff versus distance along with the randomattenuation about this path loss from shadowing.

◦ In this combined model, average dB path loss (µψdB) is

characterized by the path loss model and shadow fading, with amean of 0 dB, creates variations about this path loss.

◦ For this combined model, Pr/Pt in dB is given by:

PrPt

[dB] = 10 log10K − 10γ log10

d

d0− ψdB (9)

6. Combined path loss and shadowing 17/31

Combined path loss and shadowing II

◦ ψdB : Gauss-distributed random variable with mean zero andvariance σ2dB.

6. Combined path loss and shadowing 18/31

Outage probability under path loss and shadowing I

◦ There is typically a target minimum received power level Pminbelow which performance of the wireless systems becomesunacceptable.

◦ With shadowing, the received power at any given distance fromthe transmitter is log-normally distributed with someprobability of falling below Pmin.

◦ It’s defined the outage probability pout (Pmin, d) under path lossand shadowing to be the probability that the received power ata given distance d falls below Pmin

pout (Pmin, d) = p (Pr (d) < Pmin) (10)

7. Outage probability under path loss and shadowing 19/31

Outage probability under path loss and shadowing II

◦ For the combined path loss and shadowing model, this becomes:

p (Pr (d) ≤ Pmin) = 1−

Q

Pmin −(Pt + 10 log10K − 10γ log10

dd0

)σψdB

(11)

7. Outage probability under path loss and shadowing 20/31

Outage probability under path loss and shadowing III

◦ Where the Q-function can be expressed in terms of the errorfunction complementary:

Q (z) =1

2erfc

(z√2

)(12)

7. Outage probability under path loss and shadowing 21/31

Contours of Constant Received Power

Figure: Contours of Constant Received Power.

8. Cell coverage area 22/31

Cell coverage area I

◦ The cell coverage area in a cellular system is defined as theexpected percentage of area within a cell that has receivedpower above a given minimum.

◦ For path loss and random shadowing the contours form anamoeba-like shape due to the random shadowing variationsabout the average.

◦ The constant power contours for combined path loss andrandom shadowing indicate the challenge shadowing poses incellular system design (Pr is different for all users in the cell).

8. Cell coverage area 23/31

Cell coverage area II

◦ The BS must either transmit extra power to insure usersaffected by shadowing receive their minimum required powerPmin, which causes excessive interference to neighboring cells,or some users within the cell will not meet their minimumreceived power requirement.

◦ The Gaussian distribution has infinite tails, there is a nonzeroprobability that any mobile within the cell will have a Pr thatfalls below the Pmin, even if the mobile is close to the basestation.

8. Cell coverage area 24/31

Cell coverage area III

◦ This makes sense intuitively since a mobile may be in a tunnelor blocked by a large building, regardless of its proximity to thebase station.

◦ The following model for cell coverage area is obtained from thepath loss and shadowing model.

8. Cell coverage area 25/31

Cell coverage area I

◦ The %A within a cell where the received power exceeds theminimum required power Pmin is obtained by taking anincremental area dA at radius r from the BS in the cell.

◦ Pr (r) be the received power in dA.

◦ The total area within the cell where the Pmin requirement isexceeded is obtained by integrating overall incremental areaswhere this minimum is exceeded:

C =

∫cell area

E [1 [Pr (r) > Pmin in dA]] dA (13)

8. Cell coverage area 26/31

Cell coverage area II

◦ 1 [.] : indicator function.

◦ PA = p (Pr (r) > Pmin in dA) = E [1 [Pr (r) > Pmin in dA]]

◦ Making some sustitutions and evaluating in polar coordinates:

C =1

πR2

∫cell area

PAdA =1

πR2

∫ 2π

0

∫ R

0

PArdrdθ (14)

◦ The outage probability of the cell is defined as the %A withinthe cell that does not meet its Pmin:

pcellout = 1− C (15)

8. Cell coverage area 27/31

Cell coverage area III

◦ Using the log-normal distribution for the combined path lossand shadowing:

p (Pr (d) ≥ Pmin) = 1− pout (Pmin, r) =

Q

Pmin −(Pt + 10 log10K − 10γ log10

rd0

)σψdB

(16)

8. Cell coverage area 28/31

Cell coverage area IV

◦ Combining 14 and 16:

C =2

R2

∫ R

0

rQ(a+ b ln

r

R

)dr (17)

◦ Where:

a =Pmin − P r (R)

σψdB

(18)

b =10γ log10 e

σψdB

(19)

8. Cell coverage area 29/31

Cell coverage area V

P r (R) = Pt + 10 log10K − 10γ log10

R

d0(20)

◦ The integral yields a closed-form solution for C in terms of aand b:

C = Q (a) + exp

(2− 2ab

b2

)Q

(2− abb

)(21)

8. Cell coverage area 30/31

Cell coverage area VI

◦ If Pmin = P r (R), then a = 0 and:

C =1

2+ exp

(2

b2

)Q

(2

b

)(22)

◦ Note that with this simplification C depends only on the ratioγ/σψdB

.

◦ Due to the symmetry of the Gaussian distribution,pout (Pr (R) , R) = 0.5.

8. Cell coverage area 31/31