Empirical Path Loss Models

Transcription

1 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, / 43

2 Path loss models The methods we learned in Chapter 7 are useful if we have detailed path profile knowledge In many cases this is not available; for mobile systems it is not desirable to compute results for large number of path profiles Most common in the literature to resort to empirical models in this case: commonly used for macro cell planning (i.e. distances 1 to 20 km) We will study 3 empirical models of path loss, most applicable to macro cell planning near 900 MHz As with any empirical info, avoid using outside the range of the underlying dataset! Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

3 Path loss in free space We ve previously talked about a system loss in free space as: L sys = 20 log 10 R km + 20 log 10 f MHz L gas + L rain + L pol +L imp + L coup G Tdb G Rdb (1) Eliminate atmosphere, rain, and antenna effects to define a free space path loss as: L free p = 20 log 10 R km + 20 log 10 f MHz (2) Note received power inversely proportional to R 2 and f 2 Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

4 Direct plus reflected path loss The planar-earth direct-plus-reflected model in Ch. 7 said that (for sufficiently low antennas) E tot = E 0 d 4πh 1 h 2 λd We can use this to derive the corresponding path loss as ( ) L flat 4πh1 h 2 p = 20 log 10 R km + 20 log 10 f MHz log 10 λd = log 10 R km 20 log 10 h 1 20 log 10 h 2 (4) Here h 1 and h 2 are the antenna heights in meters; note now loss increases as R 4, independent of frequency! (3) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

5 Empirical models Because the Earth environment really isn t a perfectly flat plane, we can t expect the simple direct plus reflected model to work perfectly Empirical data near 900 MHz shows that several features of the direct-plus-reflected model don t hold up in practice In particular, the exponent on R is closer to 4 than 2, but not exactly 4 Also the lack of a frequency dependence of the path loss is not observed Simple empirical models of path loss simply specify new coefficients for the dependencies on range, frequency, and antenna heights: L p = L γ log 10 R km + 10n log 10 f MHz + L 1 (h 1 ) + L 2 (h 2 ) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

6 Hata model The Hata empirical model of path loss is based on curve fits to data acquired by Okumura et al in Tokyo and surrounding areas in the 1960 s Form used here is applicable from 100 to 1500 MHz and for ranges from 1 to 20 km. Also assumes a base station transmitter at height 30 to 200 m and a mobile receiver at height 1 to 10 m. Differing values of L 0 specified for Urban, Suburban, and Open rural areas, the latter cases have an L 0 that depends on frequency The frequency scale factor n is in the Hata model The basic range scale factor γ is 4.49 However the base station antenna height term L 1 (h 1 ) also has a range dependency (see book) Mobile antenna height function L 2 depends on terrain type and frequency Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

7 900 MHz, 1 m receive height, small city 170 range 10 km, 30 m transmit height, small city Path loss (db) h 1 =30 m h 1 =200 m (+15 db) Path loss (db) h 2 =1 m h 2 =10 m (+20 db) Path loss (db) Range (km) 900 MHz, receive ht 1 m, small city km range (+30 db) 10 km range Path loss (db) Frequency (MHz) 900 MHz, transmit ht 30 m, range 10 km 170 small city large city Transmit height (m) Receiver height (m) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

8 Extension of Hata model to PCS frequencies An extended Hata model was developed by the COST-231 organization of Europe for application at frequencies 1.5 to 2 GHz, other limits similar to original Hata model Has L 0 = 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 urban environments due to the Tokyo basis of the data. Often questioned for application in suburban and rural environments Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

9 Lee model Empirical model based largely on measurements in the US Recommends that the L 0 and γ parameters be determined empirically at a given location using L 0 = P 0 10γ log 10 (1.61) 10n log 10 (900) A few known results are: P 0 = 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) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

10 Lee model (cont d) Uses n = 2 for f MHz < 850, n = 3 for f MHz > 850 Antenna height factors are L 1 (h 1 ) = 20 log 10 (h 1 /30) (13) L 2 (h 2 ) = 10χ log 10 (h 2 /3) (14) with χ = 2 for h 2 > 3 m and χ = 1 for h 2 < 3 m Most applicable near 900 MHz, for h 1 > 30 m and h 2 < 3 m Regarded as more applicable for the more open cities of North America as compared to East Asia, for example. Again only through comparison of these empirical models can we determine the level of accuracy to expect Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

11 Other models Numerous other publications are available that present values of L 0, n, and γ derived from empirical measurements; the number of models is huge! For a particular problem of interest, it is best to assess the measurement campaigns that were actually performed, and try to find the set that best matches the problem of interest One model that is somewhat widely used attempts to predict propagation in micro-cell regions (usually urban regions) with a strong influence of buildings The COST231-Walfish-Ikegami approach is a semi-analytical method for this problem that includes building properties in its predictions With complete knowledge of a site-specific area, we can try numerical EM codes or ray tracing approaches; these are very computationally expensive though Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

12 Example Consider a cellular communications system operating at 900 MHz in a Columbus, OH suburb; terrain type is suburban The transmit power is 25 Watts, all components polarization and impedance 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. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

13 Solution: P R,dbW = P T,dbW + G T,db + G R,db L p (15) = L p (16) For direct-plus-reflected model, L p = log 10 R km 20 log 10 h 1 20 log 10 h 2 (17) = db (18) For Hata model, L p = [log 10 (f MHz /28)] log 10 R km log 10 f MHz + log 10 (h 1 ) ( log 10 R km ) = db (19) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

14 Solution (cont d): For Lee model, L p = log 10 R km / log 10 f MHz / log 10 h 1 /30 10χ log 10 h 2 /3 = db (20) Predicted powers received are then 78.54, 108.3, and 99.5 db,w, respectively: a range of nearly 30 db! Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

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

16 Signal fading models By now it should be clear that there is a lot of uncertainty in modeling propagation effects The empirical models we have at best can be regarded as providing information on expectations averaged over many paths or measurements; predicting a particular measurement is very difficult without complete site information Due to these uncertainties, it is common to model propagation effects stochastically, i.e. as random quantities Signal fading models describe these behaviors using the theory of probability Today we ll review basic probability theory in order to apply it to model path loss Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

17 Types of signal fading Our statistical models will begin with the empirical models we ve learned to predict the average propagation loss, as a function of range, for example There will be a slow fading behavior on top of this, due to evolution with range in the obstacles encountered on the propagation path; changes slowly with range On top of this is possible interference among signal contributions received over many paths: multipath fading. Because this depends on the phase of received signals, it can vary rapidly. Fast fading For example, a receiver traveling 60 mph goes 27 m/sec; 27 m represents MHz wavelengths; not unusual to have 100 fast fades/sec! Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

18 (a) Average path loss estimate (b) Including slow fading Rx power (db scale) Rx power (db scale) range (log scale) range (log scale) (c) Including slow and fast fading Rx power (db scale) range (log scale) Figure: Typical signal fading behaviors Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

19 Probability theory Applied to analyze experiments with fundamentally unknown outcomes Unknown outcome of an experiment is called a random variable ; probability theory expresses information about expected outcomes averaged over many measurements Path loss due to slow fading L s (in db) is one of our random variables; adds to empirically predicted path loss value We ll also consider fast fading effects, but in terms of a random variable 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 L s requires either empirical knowledge of L s for a specific situation, or a mathematical model of the basic physical effects involved. We ll use the latter approach. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

20 Probability density function Probability density function (pdf) provides complete knowledge of a single random variable For the random variable L s, the pdf f Ls (l s ) is defined through P( L s l s < l s ) = f Ls (l s ) l s (21) The P symbol represents the probability of obtaining the experiment outcome specified in the argument of P. Here the outcome specified is a measured fading loss L s being within a small range l s of a specified value l s. The P operator always produces an output between 0 (outcome never happens) and 1 (always happens). Since P is never negative, pdf s are always positive functions; however they can exceed unity due to l s in above equation f subscript is the random variable that f is to be used with Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

21 Cumulative distribution function The cumulative distribution function (cdf) of a random variable L s (called F Ls ) is defined through: F Ls (l s ) = P(L s l s ) = ls dl f Ls (l) (22) This is the probability that the random variable takes on a value less than or equal to the value specified in the argument of F Alternatively, the probability that the random variable L s exceeds the value l s is 1 F Ls (l s ) Since F is determined by an integration over f, we can find f Ls (l s ) = d dl s F Ls (l s ) (23) CDF s are the most useful quantity in analyzing link margins, etc.; probability that loss exceeds a certain value or that power falls below a certain value Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

22 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: µ Ls = E [L s ] = dl l f Ls (l) (24) where µ Ls refers to the expected value of L s (i.e. value averaged over many independent measurements), and E[] is called the expected value operator The expected value of a function g(l s ) of a random variable can also be found through: E [g(l s )] = dl g(l) f Ls (l) (25) These definitions make sense because we are basically taking all possible values of the random variable weighted by the likelihood of those outcomes occurring: this is an average Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

23 Variance and standard deviation Even though a random variable may have a given average, results in a given measurement will fluctuate about this average One basic measure of the degree of fluctuation is the variance: [ σl 2 s = E (L s E [L s ]) 2] (26) where σl 2 s denotes the variance of random variable L s This is a measure of the average deviation of the random variable from its mean; note the square of the difference ensures a positive answer The standard deviation of a random variable σ Ls is the square root of the variance Random variables with large standard deviations relative to the mean have outcomes that fluctuate significantly in a given measurement Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

24 Stochastic processes So far we ve been talking about properties of a single random variable; 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, versus time, etc.) Measurement outcome in each case (for example, at each range) is a random variable, so we have many related random variables that we wish 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 random variables is called a stochastic process Complete description not necessary for our work, but we will discuss some basic properties One simple case: all measurements not related to each other: results are an independent set of random variables Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

25 Models of fading effects 1 Gaussian 2 Slow fading: log-normal 3 Fast fading: Rician 4 Fast fading: Rayleigh 5 Examples 6 Wideband channels Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

26 Gaussian random variables A Gaussian random variable X has: f X (x) = 1 e (x µ X ) 2 2σ X 2 (27) σ X 2π and F X (x) = 1 2 ( [ 1 + erf (x µ X )/( ]) 2σ X ) (28) where erf is the error function Table in the book provides the probability of a Gaussian random variable exceeding its mean by a specified number of standard deviations Important due to the central limit theorem of probability theory: if a new random variable is defined as the sum of N independent identically distributed random variables, the new random variable s pdf will practically always approach a Gaussian as N becomes large Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

27 Gaussian random variables (cont d) Z = X µ X σ X 100 (1 F X (Z)) Table: Percent of X outcomes exceeding a specified argument Z for a Gaussian random variable X. The value Z is specified in terms of number of standard deviations from the mean of X Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

28 Slow fading: Log-normal Recall that slow fading models the slow evolution of the obstacles encountered along a propagation path as the range or environment is varied We will describe in terms of L s, the loss due to slow fading effects in db; 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: L s = α s,db dl (29) path Here we are adding up the number of db/km due to slow fading along the total number of km If we assume that each α s,db is independent of the others, then L s is a sum of independent random variables: pdf should approach Gaussian Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

29 Slow fading: Log-normal (cont d) Thus a reasonable model for slow fading is that the loss in db (L s ) is a zero-mean Gaussian variable; typical values for σ Ls are 4 to 12 db Empirical formula for L s provided in book as function of f MHz The Gaussian table from the book can be used to determine the likelihood of the slow fading loss exceeding a threshold, if σ Ls is known It is possible to transform the pdf of L s into the pdf of S, the multiplicative slow fading factor. The result is: f S (s) = 1 sq [ln s] 2 2π e 2Q 2 (30) with Q = ln σ L s σ Ls (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 L s Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

30 Fast fading: Rician It turns out that fast fading effects are easiest to model in terms of the receiver power (in Watts, not db,w) rather than the propagation loss Do this by first finding the mean receiver power with no fast fading using the Friis formula and any model for slow fading, call this power P m for mean The Rician model of fast fading assumes that there is a dominant line of sight signal that produces P m, plus there are numerous non-line of sight paths that cause multipath interference Assume that the number of multipath fields is large: the total fast-fading field (sum of these multipaths) becomes Gaussian. For the time harmonic case, we have real and imaginary parts of the field: these are random phase and uncorrelated for a large number of paths Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

31 Fast fading: Rician (cont d) In the Rician model, the true mean received power is the sum of the mean power with no fast fading (P m ) plus the mean fast fading power (P f ): P rec,mean = P f + P m = P f (1 + K) (32) again in Watts The pdf of the received power was shown by Rice in 1944 to be f P (p) = 1 ( ) e (Pm+p) P 2 Pm p f I 0 P f P f (33) where I 0 is the modified Bessel function The standard deviation of the received power found from this pdf is σ P = P f (P f + 2P m ) = P f 1 + 2K (34) Note this increases with P m, but still vanishes for P f = 0 Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

32 Probability Density Function Rician Power Pdf Curves K= P /P rec,mean Figure: Rice power pdfs for P f = 1 W, with P m in watts indicated in the legend. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

33 Rician cdf Since we are modeling power here, the relevant question for link margins is the probability of the received power falling below a specified value; this is the CDF The Rician cdf is given by: ( ) 2K, 2p F P (p) = 1 Q 1 P f (35) where Q 1 is called the Marcum function Plot in the book illustrates the Rician cdf for varying values of P m /P f (also called the Rician K-factor ) This plot is the one to use when trying to predict link margins for Rician channels Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

34 Percent of powers less than X standard deviations below mean Gaussian X (unitless) Figure: Rice power cdfs, legend indicates P m /P f Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

35 Fast fading: Rayleigh In the Rayleigh model, we assume that only the multipath fields are received, and that there is no line-of-site link (i.e. P m = 0) Setting P m = 0 in the Rician pdf gives which is an exponential pdf The corresponding cdf is simple: f P (p) = 1 P f e p P f (36) F P (p) = 1 e p P f (37) The mean and standard deviation are both equal to P f. By linearizing the above cdf we can fin that the power level that will be exceeded q percent of the time (q > 90) is ( 1 q ) P f (38) 100 Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

36 Fade depth below mean (db) K= Percent chance that fade level is exceeded Figure: Percent of time fade depth below mean power is exceeded. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

37 Fading examples 1 Slow-fading example 2 Fast-fading examples 3 Wideband channels Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

38 Slow fading example Given a mean path loss of db, and a slow fading loss standard deviation of 4 db, find the probability that the path loss exceeds db. Slow fading loss in db is a zero-mean Gaussian random variable Here db is 8 db more than the mean; 8 db is two standard deviations Thus we are looking for the probability that a Gaussian random variable 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 would experience a path loss greater than db Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

39 Fast fading (Rician) example It is known that the received power at a location neglecting fast fading effects is dbw. The received fast fading power (in the absence of the line of sight path) is dbw. Find the probability that the total received power falls below 76 dbw. Both line of sight and fast fading powers are present: this is a Rician channel. Work in terms of received powers in Watts We find P m = microwatts and P f = microwatts; P m /P f = 10 Mean power received is microwatts and standard deviation is in microwatts µ P = P m + P f = (39) σ P = P f (P f + 2P m ) = (40) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

40 Fast fading (Rician) example (cont d) We want to find the probability that the received power is less than microwatts; this is 1.93 standard deviations below the mean value Reading the Rician cdf plots with P m /P f = 10, we can estimate that this probability is around 0.5 percent A more precise evaluation using a computer gives percent Could also read from graph using K = 10 and a fade of 7 db below mean, again around 0.5 percent Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

41 Fast fading (Rayleigh) example (cont d) The receiver is moved so that no direct line-of-site path exists. The received fast fading power remains db,w. Find the power level that 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 are asking for the power level exceeded a large percent of the time Still need to work in terms of Watts; P f = microwatts Power level exceeded 99% of the time is ( 1 q ) P f = P f /100 (41) 100 This is a 20 db fade (occurs only 1 percent of the time) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

42 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 effects are reasonably constant over the bandwidth of interest Narrowband systems fall into this category; for wideband systems, propagation effects vary appreciably within the system bandwidth Propagation environment determines the bandwidth in MHz that is narrow or wide Frequency selective fading in a wideband channel can cause signal distortion Many systems deal with this using a channel sounding algorithm so that distortion effects can be corrected Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43

43 Wideband channel parameters Though a detailed treatment of wideband channels is beyond our study, two basic parameters can be discussed The first is the channel coherence bandwidth; this parameter (in MHz) describes the separation in frequency within which propagation channel behaviors can be considered to be similar Frequencies separated much more than the coherence bandwidth are likely to experience different propagation effects Wideband channels: actual bandwidths > channel coherence bandwidth Also related to the delay spread of the channel: describes the distance in time through which a transmitted impulse is smeared upon reception Channel coherence time: the time interval within which the channel s propagation properties can be expected to stay relatively stable; important for time varying channels Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 17, / 43