Wireless Mesh Networks Anatolij Zubow zubowinformatik huberlin de

Wireless Mesh Networks Anatolij Zubow (zubow@informatik. hu-berlin. de) Antennas and Propagation

Introduction • An antenna is an electrical conductor or system of conductors • Transmission - radiates electromagnetic energy into space • Reception - collects electromagnetic energy from space • In two-way communication, the same antenna can be used for transmission and reception (same characteristics) 2

Antenna • Radiation pattern • Performance of an antenna • Graphical representation of radiation properties of an antenna as a function of space coordinates • Depicted as two-dimensional cross section • Beam width (or half-power beam width) • Measure of directivity of antenna • Reception pattern • Receiving antenna’s equivalent to radiation pattern 3

Idealized Radiation Patterns • Distance from the antenna to each point on the radiation pattern is proportional to the power radiated from the antenna in that direction • Actual size of radiation pattern is arbitrary – important is relative distance (= relative power) from antenna in each direction 4

Types of Antennas • Isotropic antenna (idealized) • Radiates power equally in all directions • Dipole antennas • Half-wave dipole antenna (or Hertz antenna) • Quarter-wave vertical antenna (or Marconi antenna) • Parabolic Reflective Antenna 5

Radiation Patterns in Three Dimensions 6 Acoustics: http: //www. kettering. edu/~drussell/Demos/rad 2/mdq. html

Antenna Gain • Antenna gain • Measure of the directionality of an antenna • Power output, in a particular direction, compared to that produced in any direction by a perfect omnidirectional antenna (isotropic antenna) • Note: increased power is radiated in one direction by reducing the power radiated in other directions • E. g. antenna gain of 3 d. B • Effective area • Related to physical size and shape of antenna 7

Propagation Modes • A signal radiated from an antenna travels along one of three routes: • Ground-wave propagation (GW) • Follows contour of the earth • Frequencies up to 2 MHz • Example: AM radio • Sky-wave propagation (SW) • Signal reflected from ionized layer of atmosphere back down to earth • Signal can travel a number of hops • Examples: Amateur radio, CB radio • Line-of-sight propagation (LOS) • Transmitting and receiving antennas must be within line of sight • Ground communication – antennas within effective line of site due to refraction 8

Fundamentals of Propagation Modeling 9

Fundamentals of Propagation Modeling (cont. ) • … and would like to understand why the received power is like this: 10

Presupposed Basics • To really understand these phenomena, one needs a profound knowledge in Physics and Mathematics. • From the world of Physics: • Formulation of electromagnetic propagation • Reflection, scattering and diffraction • Many subsequent processes are random: • Notions of statistics (PDF, CDF) • Moments, mean, variance, etc. • Dependence, correlation, etc. • Many processes are in addition stochastic: • Notions of coherence, etc. 11

Presupposed Basics • Some revisions on statistics: • A random process (left) leads to a histogram (middle) and a mathematical abstraction in form of the probability density function, PDF (right) • The most important factors about the PDF are mean, std/variance, and shape • In nature, unbounded PDFs are Gaussian and bounded PDFs are uniform • Typical half-bounded PDFs: Rayleigh, Rice, Nakagami, lognormal, Gamma, etc. 12

Lognormal Gaussian Gamma Rayleigh 13

Presupposed Basics • Electromagnetic (EM) waves: • E & H are in-phase and occur together; hence, only E-field is considered normally • E-wave oscillates: in time with angular frequency ω = 2πf = 2π/T in space with spatial frequency k = 2π/λ • f is the frequency in [Hz], T the period in [s], and λ = c/f the wavelength in [m] • E = E 0 cos(ωt – kr); for convenience, we write E = Re{ E 0 ej‧(ωt – kr) } (Euler's formula ) 14

Sources of Signal Distortions • A useful signal can get distorted by: • Noise (thermal, shot): additive • Interference (self, other): additive • Wireless channel: multiplicative • Simplified, we can hence write for the received signal: • received = channel * transmitted + noise + interference • Note! • Noise and interference is always bad news, the channel not always (cf. MIMO) • Modern communication systems are dominated by interference and channel • For the additive components, important is the ratio between signal power and noise + interference powers (SNR, SINR) 15

Wireless Channel Taxonomies • Propagation Mechanisms: • • Free space propagation Reflection and refraction Diffraction Scattering (distance dependent) (from surfaces, into buildings) (from roof edges) (from surrounding trees) • Propagation Conditions: • Line-of-sight (LOS) (great visibility between Tx & Rx) • Non LOS (n. LOS) (no direct visibility between Tx & Rx) • Obstructed LOS (o. LOS) (small obstacle in-between Tx & Rx) • Distortions: • Doppler effect (caused by mobility in the channel) • Multipath propagation (signals arriving via different paths) 16

Propagation Mechanisms – Overview • Note! • All 5 effects result from the same set of equations: Maxwell's Equations • The equations are very complicated and not useful for every problem • For different ratios between object size and wavelength, different effects occur • Occurrence (object size d and wavelength λ): • Free-space propagation: always occurs for any d and λ • Reflection/refraction: d >> λ • Diffraction: λ in the order of the curvature of the edge • Scattering: d ≈ or < λ 17

Reflection of Waves from Boundaries Reflection from a HARD boundary Reflection from a SOFT boundary When a wave encounters a boundary which is neither rigid (hard) nor free (soft), part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary. From high to low speed (low high density) From low speed to high speed 18 Example: http: //www. kettering. edu/~drussell/Demos/reflect. html

Propagation Mechanisms – Free-Space • Friis' Transmission Equation: assuming • • PRx to be received and PTx transmitted powers GRx to be receive and GTx transmit antenna gains d the distance between Tx and Rx, and λ = c/f the wavelength perfect matching of Tx and Rx antennas, no multipath and aligned polarisation • In d. B, we hence get: PRx=PTx+GTx+ 20 log(c/2π) – 20 log(f) – 20 log(d) • PRx decreases with -20 d. B/dec: 19

Antenna Polarization • An antenna is a transducer that converts electric current to electromagnetic waves that are then radiated into space. • The electric field plane determines the polarization or orientation of the radio wave. • Most systems use either vertical, horizontal or circular polarization. • Vertical polarization is most commonly used when it is desired to radiate a radio signal in all directions over a short to medium range. 20

Propagation Conditions – Overview • LOS (opposite for n. LOS) has the following properties: • Advantage: strong signal • Disadvantage LOS: strong interference • o. LOS is something in-between LOS and n. LOS 21

Distortions – Doppler Effect • Doppler Formula: , where • c = 3‧ 108 m/s is the speed of light, and • v is the summed speed of the Tx and/or Rx and/or (!) reflecting objects • E. g. , little movement in the channel (left), more movement in the channel (right): 22

Doppler effect • C. Doppler, was first to describe how the observed frequency of sound waves is affected by the relative motion of the source and the detector (Doppler effect). • Doppler effect occurs for all type of waves! • Experiment: • Doppler hired a train and the trumpet section of an orchestra. • Half of the players got on the train and played a specific note, the other half did the same sitting at the station. • As the train passed by the train station, people listened to the difference between the notes played by the moving musicians and those sitting. It was possible to tell the difference between the notes! • Also possible to show that a moving observer will detect a different frequency from that of a stationary observer 23

Doppler effect (cont. ) • Moving source, stationary observer • We always measure everything relative to the air (medium transporting wave) • Source of sound moves with vs, observer stationary • Speed of sound cw • Sound waves are generated T=1/f 0 times apart • When source moves (and observer is stationary), the distance between the wavefronts will be (cw +/- vs)T • while for stationary source it is cw. T. 24

Doppler effect (cont. ) • Moving source, stationary observer • The arrival time at the listener is tin front = [(cw - vs)T]/cw • and tbehind = [(cw + vs)T]/cw • In terms of frequency: • where (-) refers to the observer being ahead and (+) is behind the source, and f 0 = 1/T and f±=1/t± • Doppler shift is defined as: Δf = f - fo 25 Example: http: //www. gmi. edu/~drussell/Demos/doppler. html

Distortions – Multipath Propagation • Assume we send two symbols of duration Ts; then, objects along the ellipses with Tx & Rx in the foci, yield same propagation delays: • Intra-symbol interference • Overlap of symbol replicas within symbol duration (same color) • This leads to mutual cancellation • Inter-symbol interference • Overlap of symbol replicas belonging to different symbols (grey shading) 26

Summed Contributions • Defining characteristic of the mobile wireless channel is the variations of the channel strength over time and over frequency. • The variations can be roughly divided into three types: • Large-scale Fading (e. g. free space) • Shadowing • Multi-path Fading (e. g. Rayleigh) Example of a frequency selective, fast changing (fast fading) channel. 27

Summed Contributions • The sum in d. B (i. e. product in linear scale!) of pathloss (blue), shadowing (red), fading (green) is our total channel (black). 28

Pathloss – Overview • Pathloss has the following characteristics: • Function of distance (as well as frequency, environment, antenna heights) • It is a 'deterministic' effect • Is obtained by averaging over 1000 λ 29

Pathloss – Important Models (1) Free space (2) Two-Ray Ground (3) Diffraction Model (4) Hybrid models or too complex ; -) 30

Pathloss – Degrees of Modeling • Free-space pathloss model: • Loss of -20 d. B/decade distance • Very simple model, but not very realistic • Application in satellite channels and over short LOS distances • Single-slope pathloss model: • n = 1. 5 (waveguides), n = 2… 4 (LOS + clutter), n = 4… 6 (n. LOS + clutter) • Application in WLANs, interference power in cellular systems, etc. • Other models • Two-Ray Ground • Considers antenna height 31

Pathloss – Degrees of Modeling • Other models • Dual-slope pathloss model (hybrid model) • dg turnover distance • Simple and more accurate model, but correct reference point dg has to be found • Application in long-range WLANs and cellular systems 32

Pathloss – Other Propagation Models • Hata-Okumura Model (empirical model, 3 GPP) • Walfish-Ikegami Model (radio propagation above roof tops) • Berg Model (pathloss calculation along streets) 33

Pathloss – Degrees of Modeling • Deterministically simulated pathloss behavior: • Ray-tracing type tools determine field behavior for given scenario • Very complex modeling approach, and not necessarily a better model • Empirically-fitted pathloss model: • Real measurements taken with P(d 0) and n fitted to give best match • Difficult to obtain, very simple model and fairly realistic • Application in simulators, planning and optimization tools, etc • Really measured pathloss behavior: • Real measurements taken and used for planning and optimization tools • Complex and memory-consuming model, but very accurate • Used by all operators and within available commercial tools 34

Shadowing – Overview • Shadowing has the following characteristics: • Function of the environment (freq. , distance, antenna heights) • Random effect due to randomly appearing and disappearing waves • Is obtained by averaging over 40 λ and subtracting the pathloss 35

Shadowing – Modeling Approach • The reasoning behind the distribution of shadowing is as follow: • Each arriving multipath component is the result of a random amount of multiple random reflections • Shadowing behavior: the (dis)appearance of waves • Shadowing has a lognormal distribution • Pathloss + Shadowing: • Local-mean power • Lognormal distribution has mean [d. B] = and STD [d. B] = σd. B = σ * 10 / ln 10 • Typical values are σd. B = 4 -10 d. B (microcell), 6 -18 d. B (macrocell) 36

Fading – Overview • Fading has the following characteristics: • Function of the environment and frequency • Random effect due to randomly wave additions/cancellations • Is obtained by subtracting the pathloss and shadowing (no averaging!) Requires multi-path propagation 37

Fading – Overview • Interference of Waves • Superposition of waves y 1 and y 2 Two signals of different but nearly equal frequencies (f 1 and f 2) 38 Example: http: //www. gmi. edu/~drussell/Demos/superposition. html

Fading – Overview • Interference of Waves • Superposition of signals with different phasing 39

Travelling wave over “Singlepath” • • Signal at source: s(t) = cos 2πft Signal at destination: s(t) = cos 2πf(t-Δt) Time to reach destination: Δt=Δd/c Over single path: cos 2πft Δt=Δd/c • Oversimplification of EM Wave: “Modeling” 40 cos 2πf(t-Δt)

Travelling wave over “Multipath” • • Path 1: Δt 1=Δd 1/c Path 2: Δt 2=Δd 2/c Path 3: Δt 3=Δd 3/c Over multiple path: cos 2πf(t-Δt 1) +cos 2πf(t-Δt 2) +cos 2πf(t-Δt 3) cos 2πft 41

Phasors to visualize “Multipath” • How to visualize the sum of “shifted cosines” cos 2πf(t-Δt 1) + cos 2πf(t-Δt 2) + cos 2πf(t-Δt 3) • We need some “convenient” representation of cosines and its shifted versions • We will use the “complex numbers” and the “phasors” 42

Phase Vector (Phasor) • Is a representation of a sine wave whose amplitude (A), phase (θ), and frequency (ω) are time-invariant. • Simplifies certain kinds of calculations • Definition: • Euler's formula indicates that sine waves can be represented mathematically as the real part of a complex-valued function: A cos(ωt+ θ) = Re{A ∙ ei(ωt+ θ)} = Re{Aeiθ ∙ eiωt} • A phasor can refer to either Aeiθ ∙ eiωt or just the complex constant, Aeiθ (encoding the amplitude + phase of the sinusoid) • The sine wave can be understood as the projection on the real axis of a rotating vector on the complex plane. 43

Phase Vector (Phasor) Animated phasor showing shadow (or projection) oscillating back and forth, simulating an oscillation. The angle θ = ωt + ϕ is shown by the angle numbers around the edge. The position of the tip of the shadow on the ruler is equal to x = r cos (ωt + ϕ). 44

Travelling wave –moving source • Recall Δt=Δd/c cos 2πft = cos θ Z = cos 2πf(t-Δt) = cos(2πft - 2πfΔt) = cos(θ-φ), 45 If the source moves, received cosine Z is a “randomly” rotating phasor and rotates at a rate at which the received phase changes

Constructive-Destructive multipath sum cos 2πf(t-Δt 1) +cos 2πf(t-Δt 2) +cos 2πf(t-Δt 3) • Recall: cos 2πft • Each path changes independently over time: 3 random vectors sum up at the dst. 46

Constructive-Destructive multipath sum (cont. ) An animated phasor diagram showing phasor addition of two oscillations, that of the woman and that of the man suspended on bungy cords. Like before, the animation shows the relation between the phasor diagram, the actual oscillation, and a graph of the oscillations. 47

Constructive-Destructive multipath sum (cont. ) An animated phasor diagram showing phasor addition of two oscillations when the frequencies of the two oscillations are different. This allows one of the rotating phasor vectors to rotate faster than the other, continuously changing the phase angle difference between the two. At times, the two phasors will add up in the maximum way, demonstrating constructive interference. At other times, they will subtract, demonstrating destructive interference. The result is that the amplitude of the sum varies with time in a repetitive way. 48

Multiplication of an oscillation by a complex constant An animated diagram showing the multiplication of a complex oscillation by the complex constant B. The value (amplitude and phase) of B, the multiplying complex constant is given in black at the top left of the animation. The yellow vector represent the initial oscillation, and the orange vector represents the product of the yellow vector (or oscillation) multiplied by B. 49

Fading – Modeling Approach • Typical distributions (usually referred to envelope): • Rayleigh (fits well under n. LOS) • Nakagami (fits well under weak LOS) • Rice (fits well under strong LOS) • Fading Models • Consider Pathloss, Shadowing and Fading • Rayleigh Fading: • Rician Fading: 50

Fading – Modeling Approach • The fading patterns for these cases is shown below: 51

Modeling Approach - Summary • A = pathloss alone (area-mean power, ) • B = Shadowing and pathloss (local-mean power, ) • C = Fading, Shadowing, and pathloss (instantaneous power, ) Received Power [d. B] Slow (A) Medium (B) Fast (C) log (d) Fig. Illustration from Goldsmith; not realistic!!! 52

Noise - Overview • Thermal Noise (seen before) • Intermodulation noise – occurs if signals with different frequencies share the same medium • Interference caused by a signal produced at a frequency that is the sum or difference of original frequencies • Crosstalk – unwanted coupling between signal paths • Impulse noise – irregular pulses or noise spikes • Short duration and of relatively high amplitude • Caused by external electromagnetic disturbances, or faults and flaws in the communications system 53 Impulse noise example: upload. wikimedia. org/wikipedia/commons/2/2 e/802_11 bg_interference. ogg

Attenuation of Radio Propagation • Frequency-dependent attenuation of radio waves with horizontal free-space propagation • Attenuation values for rain of different intensity (A) and fog (B) are to be added to gaseous attenuation (C) • Resonant local attenuation maxima caused by water vapour (at 23, 150, etc. , GHz) or oxygen (at 60 and 110 GHz) 54

Advantages & Disadvantages • Pathloss: • adv. : limits interference powers • disadv. : limits desired signal power • Shadowing: • adv. : limits interference, facilitates capture effect in ad hoc networks • disadv. : limits signal power, is difficult to predict • Fading: • adv. : (facilitates increase of capacity in MIMO channels) • disadv. : causes errors, requires strong channel code • Noise: • adv. : none • disadv. : causes errors, cannot be predicted 55

Error Compensation Mechanisms • Forward error correction • Adaptive equalization • Diversity techniques 56

Forward Error Correction • Transmitter adds error-correcting code (redundancy) to data block • Code is a function of the data bits • Receiver calculates error-correcting code from incoming data bits • If calculated code matches incoming code, no error occurred • If error-correcting codes don’t match, receiver attempts to determine bits in error and correct 57

Adaptive Equalization • Used to combat intersymbol interference (ISI) • Involves gathering dispersed symbol energy back into its original time interval • Techniques • Lumped analog circuits • Sophisticated digital signal processing algorithms • Common approach is to use linear equalizer circuit 58

Diversity Techniques • Diversity is based on the fact that individual channels experience independent fading events • Space diversity – techniques involving physical transmission path • Frequency diversity – techniques where the signal is spread out over a larger frequency bandwidth or carried on multiple frequency carriers • Time diversity – techniques aimed at spreading the data out over time 59

Simulation Platforms • Type of simulator: • Link-level (point-to-point): fading ("channel model") • System-level (entire system): pathloss + shadowing ("pathloss model") • E. g. Ad-Hoc Network • Link Level Simulator (e. g. Mat. Lab): • 1. Rayleigh fading to determine BER/PER versus SNR without shadowing/pathloss for given channel code, modulation and packet length 60

Simulation Platforms • System Level Simulator (e. g. NS-2): • 2. Randomly place nodes which determines distance between them. • 3. Obtain for given distance the deterministic pathloss and random shadowing loss. • 4. For given transmit power, obtain with these losses the received power, and hence SNR. • 5. Obtain PER from step 1 and re-run from step 2 with new locations/packets/etc. 61

s le p m a x E 62

Example – Link Budget Calculation • A link budget is the accounting of all of the gains and losses from the transmitter, through the medium (free space, cable, fiber, etc. ) to the receiver • It takes into account the pathloss, as well as the loss, or gain, due to the antenna • Random attenuations such as Shadowing and fading are not taken into account (see diversity techniques) • A simple link budget equation looks like this: • Received Power (d. Bm) = Transmitted Power (d. Bm) + Gains (d. B) - Losses (d. B) *d. Bm = power ratio in decibel (d. B) of the measured power referenced to one milliwatt (m. W) (e. g. 0 d. Bm = 1. 0 m. W, 15 d. Bm = 32 m. W, 20 d. Bm = 100 m. W). 63

Example – Link Budget Calculation (cont. ) • E. g. , we want to set up a 5 mile (8047 m) link between 2 points, using IEEE 802. 11 on channel 6 (2. 437 GHz) • Free space path loss: • At 5 miles, with no obstacles in between, you will lose 118 d. B of signal between the 2 points • Now, add up all of your gains (radios + antennas + amplifiers) and subtract your losses (cable length, connectors, …) 64

Example – Link Budget Calculation (cont. ) • Let’s assume: • We are using 802. 11 hardware like Orinoco Silver cards (15 d. Bm) and no amplifiers, • with a 12 d. Bi* sector on one side, and a 15 d. Bi yagi** on the other side, • 1 m of LMR 400 and a lightning arrestor on each side, • 0. 25 d. B loss for each connector, 1 d. B for each pigtail Site A: Radio – Pigtail – Arrestor – Connector – Cable – Connector + Antenna 15 – 1. 25 – 0. 22 – 0. 25 + 12 = 24. 03 Site B: Radio – Pigtail – Arrestor – Connector – Cable – Connector + Antenna 15 – 1. 25 – 0. 22 – 0. 25 + 15 • Subtract the pathloss from that total: 51. 06 – 118 = -66. 94 d. B *d. Bi = d. B(isotropic) - forward gain of an antenna compared to an idealized isotropic antenna. **Yagi antenna is a directional antenna (beam antenna) 65 = 27. 03

Example – Link Budget Calculation (cont. ) • The perceived signal level at the end of the link is -66. 94 d. Bm. • Q. : Is it enough for communications? • A. : Look up the receiver sensitivity specs for the Wi. Fi card (here Orinoco Silver Card): • Rx sensitivity (d. Bm): -82/-87/-91/-94 (11/5. 5/2/1 Mbps) • At 11 Mbps we have a safety factor of 15. 06 d. B (= 82 – 66. 94). • Theoretically, we should have no problems… 66

Simplified Pathloss Model • Simplified model for path loss as a function of distance is commonly used for system design (see single-slope pathloss model) • PR = PT K (d 0/d)γ • PR [d. Bm] = PT [d. Bm] + K [d. B] − 10γ log 10 (d/d 0) • To approximate empirical measurements, the value of K<1 is sometimes set to the free space path gain at distance d 0 assuming omni-directional antennas: • K [d. B] = 20 log 10(λ / 4πd 0) 67

Simplified Pathloss Model - Example • Consider the set of empirical measurements of PR/PT for an indoor system at 900 MHz • Find the pathloss exponent γ that minimizes the mean squared error (MSE) between the model and the empirical d. B power measurements • We assume that d 0 = 1 m and K is determined from the free space path gain formula at this d 0. • Q. : What is the received power at 150 m for the simplified pathloss model with this pathloss exponent and a transmit power of 1 m. W (0 d. Bm). 68

Simplified Pathloss Model – Example (cont. ) • Solution: minimized MSE error equation • where Mmeasured(di) is the measured pathloss at distance di and Mmodel(di) = K− 10γ log 10(d) is the pathloss based on the simplified model at di • Using the free space pathloss formula, K = 20 log 10(. 3333/(4π)) = − 31. 54 d. B • F(γ) = 21676. 3 − 11654. 9γ + 1571. 47γ 2 • Differentiating F(γ) relative to γ and setting it to zero yields 69 Distance (m) M = Pr/Pt 10 -70 d. B 20 -75 d. B 50 -90 d. B 100 -110 d. B 300 -125 d. B

Simplified Pathloss Model – Example (cont. ) • To find the received power at 150 m under the simplified path loss model with K = − 31. 54, γ = 3. 71, and Pt = 0 d. Bm, we have • Pr = Pt + K − 10γ log 10(d/d 0) = 0 − 31. 54 − 10 ∗ 3. 71 log 10(150/1) = − 112. 27 d. Bm. • Note: The measurements deviate from the simplified pathloss model: this variation can be attributed to shadow fading. 70

Shadow Fading • A signal transmitted through a wireless channel will typically experience random variation due to blockage from objects in the signal path (random variations of the received power at a given distance). • Such variations are also caused by changes in reflecting surfaces and scattering objects. • Thus, a model for the random attenuation due to these effects is also needed. Since the location, size, and dielectric properties of the blocking objects as well as the changes in reflecting surfaces and scattering objects are generally unknown, statistical models are used. • Log-normal shadowing is the most common model. 71

Shadow Fading (cont. ) • Performance in log-normal shadowing is typically parameterized by the log mean μψd. B , which is referred to as the average d. B path loss (in d. B). • With a change of variables we see that the distribution of the d. B value of ψ is Gaussian with mean μψd. B and standard deviation σψd. B : 72

Shadow Fading - Example • Previous example: • Exponent for the simplified pathloss model that best fits the measurements was γ = 3. 71. • Task: • Assuming the simplified pathloss model with this exponent and the same K = − 31. 54 d. B, find σ2ψd. B, the variance of log-normal shadowing about the mean pathloss based on these empirical measurements. • Solution: • The sample variance relative to the simplified pathloss model with γ = 3. 71 is • Thus, the standard deviation of shadow fading on this path is σψd. B = 3. 65 d. B. 73

Outage Probability under Path Loss and Shadowing • In wireless systems there is typically a target minimum received power level Pmin below which performance becomes unacceptable. • However, with shadowing the received power at any given distance from the transmitter is log-normally distributed with some probability of falling below Pmin. • Outage probability pout(Pmin, d) under pathloss and shadowing is the probability that the received power at a given distance d, Pr(d), falls below Pmin: pout(Pmin, d) = p(Pr(d) < Pmin). 74

Outage Probability under Path Loss and Shadowing (cont. ) • where the Q(. ) function is defined as the probability that a Gaussian random variable x with mean zero and variance one is bigger than z: • The conversion between the Q function and complementary error function is 75

Outage Probability - Example • Find the outage probability at 150 m for a channel based on the combined path loss and shadowing models of the previous example, assuming a transmit power of Pt = 10 m. W and minimum power requirement Pmin = − 110. 5 d. Bm. • Solution: We have Pt = 10 m. W = 10 d. Bm. 76

Cell Coverage Area • Defined as the expected percentage of area within a cell that has received power above a given minimum. • Consider a base station inside a circular cell of a given radius R. • All mobiles within the cell require some minimum received SNR. • Assuming a reasonable noise/interference model, the SNR requirement translates to a minimum received power Pmin • The transmit power at BS is designed for an average received power at the cell boundary (averaged over the shadowing variations). • However, shadowing will cause some locations within the cell to have received power below that value. 77

Cell Coverage Area (cont. ) • Random shadowing variations about the average form an amoeba-like shape. • It is not possible for all users at the cell boundary to receive the same power level. • Two solutions: • BS must transmit extra power to insure users affected by shadowing receive their minimum required power Pmin (causes excessive interference to neighboring cells) • Some users within the cell will not meet their minimum received power requirement 78 Contours of Constant Received Power.

Cell Coverage Area (cont. ) • The outage probability of the cell is defined as the percentage of area within the cell that does not meet its minimum power requirement Pmin, i. e. 79

Cell Coverage Area - Example • Q. : What is the coverage area for a cell • with the combined pathloss and shadowing models of the previous example, • a cell radius of 600 m, • a base station transmit power of Pt = 100 m. W = 20 d. Bm, • and Pmin = − 110 (-120) d. Bm • Solution: • For Pmin = − 110 d. Bm • a = (− 110 + 114. 6)/3. 65 = 1. 26 • b = 37. 1 ∗. 434/3. 65 = 4. 41 Unhappy Customers ; -) • For Pmin = − 120 d. Bm we get C = 0. 988 (nice) 80

Resources • Books: • • • William Stallings, Wireless Communications and Networks, Prentice-Hall, 2005. Simon Saunders “Antennas & Propagation” William Jakes “Microwave Mobile Communications” Kaveh Pahlavan “Wireless Information Networks” Articles: • • • A. Neskovic, N. Neskovic, and G. Paunovic, "Modern Approaches in Modeling of Mobile Radio Systems Propagation Environment, " IEEE Comm. Surveys, 2000. H. L. Bertoni, et al. , "UHF Propagation Prediction for Wireless Personal Communications, " Proc. IEEE, Sept. 1994, pp. 1333 -1359. V. Erceg et al. , "Urban/Suburban Out-of-Sight Propagation Modeling, IEEE Commun. Magazine, June 1992, pp. 56 -61. M. Dohler, “Fundamentals of Propagation Modelling”, presentation, 2006 http: //www. deas. harvard. edu/~jones/es 151/prop_models/propagation. html http: //www. ictp. trieste. it/~radionet/2000_school/lectures/carlo/linkloss/INDEX. HT M 81