Hata model Lee model Other models Examplesjohnson/5010/ch8vuframe.pdf · Hata model The Hata empirical model of path loss is based on curve ts to data acquired by Okumura et al in

Empirical Path Loss Models

1 Free space and direct plus reflected path loss

2 Hata model

3 Lee model

4 Other models

5 Examples

Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, 2018 1 / 43

Path loss models

The methods we learned in Chapter 7 are useful if we have detailedpath profile knowledge

In many cases this is not available; for mobile systems it is notdesirable to compute results for large number of path profiles

Most common in the literature to resort to empirical models in thiscase: commonly used for “macro cell” planning (i.e. distances 1 to 20km)

We will study 3 empirical models of path loss, most applicable tomacro cell planning near 900 MHz

As with any empirical info, avoid using outside the range of theunderlying dataset!


Path loss in free space

We’ve previously talked about a system loss in free space as:

Lsys = 20 log10 Rkm + 20 log10 fMHz + 32.44 + Lgas + Lrain + Lpol

+Limp + Lcoup − GTdb − GRdb (1)

Eliminate atmosphere, rain, and antenna effects to define a free spacepath loss as:

Lfreep = 20 log10 Rkm + 20 log10 fMHz + 32.44 (2)

Note received power inversely proportional to R2 and f 2


Direct plus reflected path loss

The planar-Earth direct-plus-reflected model in Ch. 7 said that (forsufficiently low antennas)

|E tot | =E0

d

4πh1h2

λd(3)

We can use this to derive the corresponding path loss as

Lflatp = 20 log10 Rkm + 20 log10 fMHz + 32.44− 20 log10

(4πh1h2

λd

)= 120 + 40 log10 Rkm − 20 log10 h1 − 20 log10 h2 (4)

Here h1 and h2 are the antenna heights in meters; note now lossincreases as R4, independent of frequency!


Empirical models

Because the Earth environment really isn’t a perfectly flat plane, wecan’t expect the simple direct plus reflected model to work perfectly

Empirical data near 900 MHz shows that several features of thedirect-plus-reflected model don’t hold up in practice

In particular, the exponent on R is closer to 4 than 2, but not exactly4

Also the lack of a frequency dependence of the path loss is notobserved

Simple empirical models of path loss simply specify new coefficientsfor the dependencies on range, frequency, and antenna heights:

Lp = L0 + 10γ log10 Rkm + 10n log10 fMHz + L1(h1) + L2(h2)


Hata model

The Hata empirical model of path loss is based on curve fits to dataacquired by Okumura et al in Tokyo and surrounding areas in the1960’s

Form used here is applicable from 100 to 1500 MHz and for rangesfrom 1 to 20 km. Also assumes a “base station” transmitter at height30 to 200 m and a “mobile” receiver at height 1 to 10 m.

Differing values of L0 specified for “Urban”, “Suburban”, and “Openrural areas’, the latter cases have an L0 that depends on frequency

The frequency scale factor n is 2.616 in the Hata model

The basic range scale factor γ is 4.49

However the base station antenna height term L1(h1) also has arange dependency (see book)

Mobile antenna height function L2 depends on terrain type andfrequency


0 5 10120

130

140

150

160

170900 MHz, 1 m receive height, small city

Range (km)

Pat

h lo

ss (

dB)

h1=30 m

h1=200 m (+15 dB)

0 500 1000 1500140

150

160

170range 10 km, 30 m transmit height, small city

Frequency (MHz)

Pat

h lo

ss (

dB)

h2=1 m

h2=10 m (+20 dB)

0 50 100 150 200145

150

155

160

165900 MHz, receive ht 1 m, small city

Transmit height (m)

Pat

h lo

ss (

dB)

1 km range (+30 dB)10 km range

0 5 10130

140

150

160

170900 MHz, transmit ht 30 m, range 10 km

Receiver height (m)

Pat

h lo

ss (

dB)

small citylarge city

3.8in3.3in

Figure: Sample Hata model predictions


Extension of Hata model to PCS frequencies

An “extended” Hata model was developed by the COST-231organization of Europe for application at frequencies 1.5 to 2 GHz,other limits similar to original Hata model

Has L0 = 46.3 (dB) for medium cities, 49.3 for “large cities”

Uses n = 3.39, other parameters same as original Hata model

Hata models are generally regarded as more applicable to dense urbanenvironments due to the Tokyo basis of the data.

Often questioned for application in suburban and rural environments


Lee model

Empirical model based largely on measurements in the US

Recommends that the L0 and γ parameters be determined empiricallyat a given location using

L0 = 50.3 + P0 − 10γ log10(1.61)− 10n log10(900)

A few known results are:

P0 = 49 Open terrain (5)

= 62 Suburban (6)

= 70 Philadelphia (7)

= 64 Newark (8)

γ = 4.35 Open terrain (9)

= 3.84 Suburban (10)

= 3.68 Philadelphia (11)

= 4.31 Newark (12)


Lee model (cont’d)

Uses n = 2 for fMHz < 850, n = 3 for fMHz > 850

Antenna height factors are

L1(h1) = −20 log10(h1/30) (13)

L2(h2) = −10χ log10(h2/3) (14)

with χ = 2 for h2 > 3 m and χ = 1 for h2 < 3 m

Most applicable near 900 MHz, for h1 > 30 m and h2 < 3 m

Regarded as more applicable for the more open cities of NorthAmerica as compared to East Asia, for example.

Again only through comparison of these empirical models can wedetermine the level of accuracy to expect


Other models

Numerous other publications are available that present values of L0,n, and γ derived from empirical measurements; the number of“models” is huge!

For a particular problem of interest, it is best to assess themeasurement campaigns that were actually performed, and try to findthe set that best matches the problem of interest

One model that is somewhat widely used attempts to predictpropagation in micro-cell regions (usually urban regions) with a stronginfluence of buildings

The COST231-Walfish-Ikegami approach is a semi-analytical methodfor this problem that includes building properties in its predictions

With complete knowledge of a site-specific area, we can try numericalEM codes or ray tracing approaches; these are very computationallyexpensive though


Example

Consider a cellular communications system operating at 900 MHz in aColumbus, OH suburb; terrain type is “suburban”

The transmit power is 25 Watts, all components polarization andimpedance matched

Base station antenna: gain 9 dBi, height 30 m

Receive antenna: gain 2 dBi, height 2 m, range 3 km

Compare powers received under the direct plus ground reflected,Hata, and Lee models.


Solution:

PR,dbW = PT ,dbW + GT ,db + GR,db − Lp (15)

= 24.98− Lp (16)

For direct-plus-reflected model,

Lp = 120 + 40 log10 Rkm − 20 log10 h1 − 20 log10 h2 (17)

= 103.52 db (18)

For Hata model,

Lp = 64.15− 2 [log10 (fMHz/28)]2 + 44.9 log10 Rkm + 26.16 log10 fMHz

+ log10(h1) (−13.82− 6.55 log10 Rkm)

= 133.3 db (19)


Solution (cont’d):For Lee model,

Lp = 50.3 + 62 + 38.4 log10 Rkm/1.609 + 30 log10 fMHz/900

−20 log10 h1/30− 10χ log10 h2/3

= 124.5 db (20)

Predicted powers received are then −78.54, −108.3, and −99.5 db,W,respectively: a range of nearly 30 dB!


Brief review of random variables

1 Motivation for signal fading models

2 Random variables, pdf, and cdf

3 Expected value and standard deviation

4 Stochastic processes


Signal fading models

By now it should be clear that there is a lot of uncertainty inmodeling propagation effects

The empirical models we have at best can be regarded as providinginformation on expectations averaged over many paths ormeasurements; predicting a particular measurement is very difficultwithout complete site information

Due to these uncertainties, it is common to model propagation effectsstochastically, i.e. as random quantities

Signal “fading” models describe these behaviors using the theory ofprobability

Today we’ll review basic probability theory in order to apply it tomodel path loss


Types of signal fading

Our statistical models will begin with the empirical models we’velearned to predict the average propagation loss, as a function ofrange, for example

There will be a “slow fading” behavior on top of this, due toevolution with range in the obstacles encountered on the propagationpath; changes slowly with range

On top of this is possible interference among signal contributionsreceived over many paths: multipath fading. Because this depends onthe phase of received signals, it can vary rapidly. “Fast fading”

For example, a receiver traveling 60 mph goes 27 m/sec; 27 mrepresents 90 900 MHz wavelengths; not unusual to have 100 fastfades/sec!


(a) Average path loss estimate

range (log scale)

Rx

pow

er (

dB s

cale

)

(b) Including slow fading

range (log scale)

Rx

pow

er (

dB s

cale

)

(c) Including slow and fast fading

range (log scale)

Rx

pow

er (

dB s

cale

)

Figure: Typical signal fading behaviors


Probability theory

Applied to analyze experiments with fundamentally unknownoutcomes

Unknown outcome of an experiment is called a “random variable”;probability theory expresses information about expected outcomesaveraged over many measurements

Path loss due to slow fading Ls (in dB) is one of our randomvariables; adds to empirically predicted path loss value

We’ll also consider fast fading effects, but in terms of a randomvariable for the total received power

Both are continuous random variables (i.e. can take on a continuous,not discretized, range of values)

Trying to describe probabilistic properties of Ls requires eitherempirical knowledge of Ls for a specific situation, or a mathematicalmodel of the basic physical effects involved. We’ll use the latterapproach.


Probability density function

Probability density function (pdf) provides complete knowledge of asingle random variable

For the random variable Ls , the pdf fLs (ls) is defined through

P(|Ls − ls | < ∆ls) = fLs (ls)∆ls (21)

The P symbol represents the probability of obtaining the experimentoutcome specified in the argument of P.

Here the outcome specified is a measured fading loss Ls being withina small range ∆ls of a specified value ls .

The P operator always produces an output between 0 (outcome neverhappens) and 1 (always happens).

Since P is never negative, pdf’s are always positive functions; howeverthey can exceed unity due to ∆ls in above equation

f subscript is the random variable that f is to be used with


Cumulative distribution function

The cumulative distribution function (cdf) of a random variable Ls

(called FLs ) is defined through:

FLs (ls) = P(Ls ≤ ls) =

∫ ls

−∞dl fLs (l) (22)

This is the probability that the random variable takes on a value lessthan or equal to the value specified in the argument of F

Alternatively, the probability that the random variable Ls exceeds thevalue ls is 1− FLs (ls)

Since F is determined by an integration over f , we can find

fLs (ls) =d

dlsFLs (ls) (23)

CDF’s are the most useful quantity in analyzing link margins, etc.;probability that loss exceeds a certain value or that power falls belowa certain value


Expected value

Given the pdf of a random variable, it is possible to compute the“expected value” (also called the mean or average value) through:

µLs = E [Ls ] =

∫ ∞−∞

dl l fLs (l) (24)

where µLs refers to the expected value of Ls (i.e. value averaged overmany independent measurements), and E [] is called the expectedvalue operator

The expected value of a function g(Ls) of a random variable can alsobe found through:

E [g(Ls)] =

∫ ∞−∞

dl g(l) fLs (l) (25)

These definitions make sense because we are basically taking allpossible values of the random variable weighted by the likelihood ofthose outcomes occurring: this is an average


Variance and standard deviation

Even though a random variable may have a given average, results in agiven measurement will fluctuate about this average

One basic measure of the degree of fluctuation is the variance:

σ2Ls

= E[(Ls − E [Ls ])2

](26)

where σ2Ls

denotes the variance of random variable Ls

This is a measure of the average deviation of the random variablefrom its mean; note the square of the difference ensures a positiveanswer

The standard deviation of a random variable σLs is the square root ofthe variance

Random variables with large standard deviations relative to the meanhave outcomes that fluctuate significantly in a given measurement


Stochastic processes

So far we’ve been talking about properties of a single randomvariable; this is the outcome of a specific measurement

In many propagation studies, many non-identical, but related,measurements are performed (e.g. path loss versus range, versustime, etc.)

Measurement outcome in each case (for example, at each range) is arandom variable, so we have many related random variables that wewish to describe

As the number of random variables of interest approaches infinity (i.e.path loss as a continuous function of range), the set of randomvariables is called a “stochastic process”

Complete description not necessary for our work, but we will discusssome basic properties

One simple case: all measurements not related to each other: resultsare an “independent” set of random variables


Models of fading effects

1 Gaussian

2 Slow fading: log-normal

3 Fast fading: Rician

4 Fast fading: Rayleigh

5 Examples

6 Wideband channels


Gaussian random variables

A Gaussian random variable X has:

fX (x) =1

σX

√2π

e− (x−µX )2

2σ2X (27)

and

FX (x) =1

2

(1 + erf

[(x − µX )/(

√2σX )

])(28)

where erf is the “error function”

Table in the book provides the probability of a Gaussian randomvariable exceeding its mean by a specified number of standarddeviations

Important due to the central limit theorem of probability theory: if anew random variable is defined as the sum of N independentidentically distributed random variables, the new random variable’spdf will practically always approach a Gaussian as N becomes large


Gaussian random variables (cont’d)

Z = X−µXσX

100 (1− FX (Z ))

1.0000 15.871.2816 10.001.6449 5.002.0000 2.282.3263 1.003.0000 0.133.0902 0.104.0000 3.167× 10−3

Table: Percent of X outcomes exceeding a specified argument Z for a Gaussianrandom variable X . The value Z is specified in terms of number of standarddeviations from the mean of X


Slow fading: Log-normal

Recall that slow fading models the slow evolution of the obstaclesencountered along a propagation path as the range or environment isvaried

We will describe in terms of Ls , the loss due to slow fading effects indB; assume the mean value is 0 dB since obstacles can either increase(focusing) or decrease mean power

Develop a model following process similar to that for gas attenuation:

Ls =

∫path

αs,db dl (29)

Here we are adding up the number of dB/km due to slow fadingalong the total number of km

If we assume that each αs,db is independent of the others, then Ls is asum of independent random variables: pdf should approach Gaussian


Slow fading: Log-normal (cont’d)

Thus a reasonable model for slow fading is that the loss in dB (Ls) isa zero-mean Gaussian variable; typical values for σLs are 4 to 12 db

Empirical formula for Ls provided in book as function of fMHz

The Gaussian table from the book can be used to determine thelikelihood of the slow fading loss exceeding a threshold, if σLs is known

It is possible to transform the pdf of Ls into the pdf of S , themultiplicative slow fading factor. The result is:

fS(s) =1

sQ√

2πe− [ln s]2

2Q2 (30)

with

Q =ln 10

10σLs ≈ 0.2303σLs (31)

This pdf is called a “log-normal” pdf, since the log of S is Gaussian;we’ll just stick with the Gaussian pdf for Ls


Fast fading: Rician

It turns out that fast fading effects are easiest to model in terms ofthe receiver power (in Watts, not dB,W) rather than the propagationloss

Do this by first finding the mean receiver power with no fast fadingusing the Friis formula and any model for slow fading, call this powerPm for “mean”

The Rician model of fast fading assumes that there is a dominant lineof sight signal that produces Pm, plus there are numerous non-line ofsight paths that cause multipath interference

Assume that the number of multipath fields is large: the totalfast-fading field (sum of these multipaths) becomes Gaussian.

For the time harmonic case, we have real and imaginary parts of thefield: these are random phase and uncorrelated for a large number ofpaths


Fast fading: Rician (cont’d)

In the Rician model, the true mean received power is the sum of themean power with no fast fading (Pm) plus the mean fast fading power(Pf ):

Prec,mean = Pf + Pm = Pf (1 + K ) (32)

again in Watts

The pdf of the received power was shown by Rice in 1944 to be

fP(p) =1

Pfe− (Pm+p)

Pf I0

(2√

Pmp

Pf

)(33)

where I0 is the modified Bessel function

The standard deviation of the received power found from this pdf is

σP =√

Pf (Pf + 2Pm) = Pf

√1 + 2K (34)

Note this increases with Pm, but still vanishes for Pf = 0


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

P /Prec,mean

Pro

babi

lity

Den

sity

Fun

ctio

n

Rician Power Pdf Curves

K=01510100

Figure: Rice power pdfs for Pf = 1 W, with Pm in watts indicated in the legend.


Rician cdf

Since we are modeling power here, the relevant question for linkmargins is the probability of the received power falling below aspecified value; this is the CDF

The Rician cdf is given by:

FP(p) = 1− Q1

(√2K ,

√2p

Pf

)(35)

where Q1 is called the Marcum function

Plot in the book illustrates the Rician cdf for varying values of Pm/Pf

(also called the Rician “K-factor”)

This plot is the one to use when trying to predict link margins forRician channels


0 0.5 1 1.5 2 2.50.01

0.1

1

10

100

X (unitless)

Per

cent

of p

ower

s le

ss th

an X

sta

ndar

d de

viat

ions

bel

ow m

ean

00.51510100Gaussian

Figure: Rice power cdfs, legend indicates Pm/Pf


Fast fading: Rayleigh

In the Rayleigh model, we assume that only the multipath fields arereceived, and that there is no line-of-site link (i.e. Pm = 0)

Setting Pm = 0 in the Rician pdf gives

fP(p) =1

Pfe− p

Pf (36)

which is an “exponential” pdf

The corresponding cdf is simple:

FP(p) = 1− e− p

Pf (37)

The mean and standard deviation are both equal to Pf .

By linearizing the above cdf we can fin that the power level that willbe exceeded q percent of the time (q > 90) is(

1− q

100

)Pf (38)


0.001 0.01 0.1 1 10 10040

35

30

25

20

15

10

5

0F

ade

dept

h b

elow

mea

n (d

B)

Percent chance that fade level is exceeded

K=00.51510100

Figure: Percent of time fade depth below mean power is exceeded.


Fading examples

1 Slow-fading example

2 Fast-fading examples

3 Wideband channels


Slow fading example

Given a mean path loss of 94.35 dB, and a slow fading loss standarddeviation of 4 dB, find the probability that the path loss exceeds 102.35dB.

Slow fading loss in dB is a zero-mean Gaussian random variable

Here 102.35 dB is 8 dB more than the mean; 8 dB is two standarddeviations

Thus we are looking for the probability that a Gaussian randomvariable exceeds its mean by 2 standard deviations

Using the table in the book, this is 2.28 percent

Thus we could estimate that 2.28 percent of measurements wouldexperience a path loss greater than 102.35 dB


Fast fading (Rician) example

It is known that the received power at a location neglecting fast fadingeffects is −69.37 dbW. The received fast fading power (in the absence ofthe line of sight path) is −79.37 dbW. Find the probability that the totalreceived power falls below −76 dbW.

Both line of sight and fast fading powers are present: this is a Ricianchannel. Work in terms of received powers in Watts

We find Pm = 0.1156 microwatts and Pf = 0.01156 microwatts;Pm/Pf = 10

Mean power received is

µP = Pm + Pf = 0.1272 (39)

microwatts and standard deviation is

σP =√

Pf (Pf + 2Pm) = 0.0530 (40)

in microwatts


Fast fading (Rician) example (cont’d)

We want to find the probability that the received power is less than0.0251 microwatts; this is 1.93 standard deviations below the meanvalue

Reading the Rician cdf plots with Pm/Pf = 10, we can estimate thatthis probability is around 0.5 percent

A more precise evaluation using a computer gives 0.5365 percent

Could also read from graph using K = 10 and a fade of 7 dB belowmean, again around 0.5 percent


Fast fading (Rayleigh) example (cont’d)

The receiver is moved so that no direct line-of-site path exists. Thereceived fast fading power remains −79.37 db,W. Find the power levelthat will be exceeded 99 percent of the time.

This is a Rayleigh fading problem since there is no line-of-site path

Question can be answered using our simple equation since we areasking for the power level exceeded a large percent of the time

Still need to work in terms of Watts; Pf = 0.01156 microwatts

Power level exceeded 99% of the time is(1− q

100

)Pf = Pf /100 (41)

This is a 20 dB fade (occurs only 1 percent of the time)


Wideband channels

Our focus so far has been on path loss behaviors at a single frequency;reasonable for a real system so long as bandwidth is “small enough”

“Small enough” means that path loss, slow, and fast fading effectsare reasonably constant over the bandwidth of interest

“Narrowband” systems fall into this category; for “wideband”systems, propagation effects vary appreciably within the systembandwidth

Propagation environment determines the bandwidth in MHz that is“narrow” or “wide”

“Frequency selective fading” in a wideband channel can cause signaldistortion

Many systems deal with this using a channel “sounding” algorithm sothat distortion effects can be corrected


Wideband channel parameters

Though a detailed treatment of wideband channels is beyond ourstudy, two basic parameters can be discussed

The first is the channel coherence bandwidth; this parameter (inMHz) describes the separation in frequency within which propagationchannel behaviors can be considered to be similar

Frequencies separated much more than the coherence bandwidth arelikely to experience different propagation effects

Wideband channels: actual bandwidths > channel coherencebandwidth

Also related to the delay spread of the channel: describes thedistance in time through which a transmitted impulse is smearedupon reception

Channel coherence time: the time interval within which thechannel’s propagation properties can be expected to stay relativelystable; important for time varying channels


Documents

Hata model Lee model Other models Examplesjohnson/5010/ch8vuframe.pdf · Hata model The Hata empirical model of path loss is based on curve ts to data acquired by Okumura et al in