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The Wireless Channel

The Wireless Channel 46
The Wireless Channel Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 1Wireless Channel is Very Different  Wireless channel “feels” very different from a wired channel.  Not a pointtopoint link  Variable capacity, errors, delays  Capacity is shared with interferers  Characteristics of the channel appear to change randomly with time, which makes it difficult to design reliable systems with guaranteed performance.  Cellular model vs reality: Cellular system designs are interferencelimited, i.e. the interference dominates the noise floor Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 2Basic Ideas: Path Loss, Shadowing, Fading  Variable decay of signal due to environment, multipaths, mobility Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book : “shiv rpi” 3Attenuation, Dispersion Effects: ISI Intersymbol interference (ISI) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: Prof. Raj Jain, WUSTL : “shiv rpi” 4Wireless Multipath Channel Channel varies at two spatial scales: Large scale fading: path loss, shadowing Small scale fading: Multipath fading (frequency selectivity, coherence b/w, 500kHz), Doppler (timeselectivity, coherence time, 2.5ms) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 5MultiPath Interference: Constructive Destructive Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 6Mobile Wireless Channel w/ Multipath Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 7Game plan  We wish to understand how physical parameters such as  carrier frequency  mobile speed  bandwidth  delay spread  angular spread impact how a wireless channel behaves from the cell planning and communication system point of view.  We start with deterministic physical model and progress towards statistical models, which are more useful for design and performance evaluation. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 8Largescale Fading: Path Loss, Shadowing Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 9Largescale fading: CellSite Planning 2  In free space, received power attenuates like 1/r .  With reflections and obstructions, can attenuate even more rapidly with distance. Detailed modelling complicated.  Time constants associated with variations are very long as the mobile moves, many seconds or minutes.  More important for cell site planning, less for communication system design. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 10Path Loss Modeling  Maxwell’s equations  Complex and impractical  Free space path loss model  Too simple  Ray tracing models  Requires sitespecific information  Empirical Models  Don’t always generalize to other environments  Simplified power falloff models  Main characteristics: good for highlevel analysis Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 11FreeSpacePropagation  If oscillating field at transmitter, it produces three components: 2 3  The electrostatic and inductive fields that decay as 1/d or 1/d 2  The EM radiation field that decays as 1/d (power decays as 1/d )  Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 12Electric (Far) Field Transfer Function  Tx: a sinusoid: cos 2ft  Electric Field: source antenna gain ( ) s  Product of antenna gains ()  Consider the function (transfer function)  The electric field is now: Linearity is a good assumption, but timeinvariance lost when Tx, Rx or environment in motion Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 13Freespace and received fields: Path Loss (power flux density P ) d Note: Electric Field (E) decays as 1/r, but 2 Power (P ) decays as 1/r d Path Loss in dB: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 14Decibels: dB, dBm, dBi  dB (Decibel) = 10 log (Pr/Pt) 10 Logratio of two signal levels. Named after Alexander Graham Bell. For example, a cable has 6 dB loss or an amplifier has 15 dB of gain. System gains and losses can be added/subtracted, especially when changes are in several orders of magnitude.  dBm (dB milliWatt) Relative to 1mW, i.e. 0 dBm is 1 mW (milliWatt). Small signals are ve (e.g. 83dBm). Typical 802.11b WLAN cards have +15 dBm (32mW) of output power. They also spec a 83 dBm RX sensitivity (minimum RX signal level required for 11Mbps reception). For example, 125 mW is 21 dBm and 250 mW is 24 dBm. (commonly used numbers)  dBi (dB isotropic) for EIRP (Effective Isotropic Radiated Power) The gain a given antenna has over a theoretical isotropic (point source) antenna. The gain of microwave antennas (above 1 GHz) is generally given in dBi.  dBd (dB dipole) The gain an antenna has over a dipole antenna at the same frequency. A dipole antenna is the smallest, least gain practical antenna that can be made. A dipole antenna has 2.14 dB gain over a 0 dBi isotropic antenna. Thus, a simple dipole antenna has a gain of 2.14 dBi or 0 dBd and is used as a standard for calibration. The term dBd (or sometimes just called dB) generally is used to describe antenna gain for antennas that operate under 1GHz (1000Mhz). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 15dB calculations: Effective Isotropic Radiated Power (EIRP)  EIRP (Effect Isotropic Radiated Power): effective power found in the main lobe of transmitter antenna.  EIRP = P G t t  In dB, EIRP is equal to sum of the antenna gain, Gt (in dBi) plus the power, Pt (in dBm) into that antenna.  For example, a 12 dBi gain antenna fed directly with 15 dBm of power has an Effective Isotropic Radiated Power (EIRP) of: 12 dBi + 15dBm = 27 dBm (500 mW). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 16Path Loss (Example 1): Carrier Frequency 10m W  Note: effect of frequency f: 900 Mhz vs 5 Ghz.  Either the receiver must have greater sensitivity or the sender must pour 44W of power, even for 10m cell radius Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book : “shiv rpi” 17Path Loss (Example 2), Interference Cell Sizing  Desired signal power:  Interference power:  SIR:  SIR is much better with higher path loss exponent ( = 5)  Higher path loss, smaller cells = lower interference, higher SIR Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: J. Andrews et al book : “shiv rpi” 18Path Loss: Range vs Bandwidth Tradeoff  Frequencies 1 GHz are often referred to as “beachfront” spectrum. Why  1. High frequency RF electronics have traditionally been harder to design and manufacture, and hence more expensive. less so nowadays 2  2. Pathloss increases O(f ) c  A signal at 3.5 GHz (one of WiMAX’s candidate frequencies) will be received with about 20 times less power than at 800 MHz (a popular cellular frequency).  Effective path loss exponent also increases at higher frequencies, due to increased absorption and attenuation of high frequency signals  Tradeoff:  Bandwidth at higher carrier frequencies is more plentiful and less expensive.  Does not support large transmission ranges.  (also increases problems for mobility/Doppler effects etc)  WIMAX Choice:  Pick any two out of three: high data rate, high range, low cost. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 19Ray Tracing  Models all signal components  Reflections  Scattering  Diffraction  Diffraction: signal “bends around” an object in its path to the receiver:  Diffraction Path loss exceeding 100 dB  Error of the ray tracing approximation is smallest when the receiver is many wavelengths from the nearest scatterer, and all the scatterers are large relative to a wavelength and fairly smooth.  Good match w/ empirical data in rural areas, along city streets (Tx/Rx close to ground), LAN with adjusted diffraction coefficients Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 20Reflection, Diffraction, Scattering s s l Reflection/Refraction: large objects () Scattering: small objects, rough surfaces (): foilage, lamposts, street signs  900Mhz:  30 cm  2.4Ghz:  13.9 cm  5.8Ghz:  5.75 cm Diffraction/Shadowing: “bending” around sharp edges, Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 21Classical 2ray Ground Bounce model Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book (derivation in book) : “shiv rpi” 222ray model observations  The electric field flips in sign canceling the LOS field, 4 2 and hence the path loss is O(d ) rather than O(d ).  The frequency effect disappears  Similar phenomenon with antenna arrays.  Nearfield, farfield detail explored in next slide:  Used for celldesign Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 232ray model: distance effect, critical distance  d h : constructive i/f t  h d d : constructive t c and destructive i/f (multipath fading upto critical distance)  d d: only destructive c interference  Piecewise linear approximation w/ slopes 0, 20 dB/decade, 40 dB/decade Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book : “shiv rpi” 242ray model example, cell design 2  Design the cell size to be critical distance to get O(d ) power 4 decay in cell and O(d ) outside  Cell radii are typically much smaller than critical distance Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book : “shiv rpi” 2510Ray Model: Urban Microcells  Ground and 13 wall reflections −2  Falloff with distance squared (d ) −2  Dominance of the multipath rays which decay as d , …  … over the combination of the LOS and groundreflected −4 rays (the tworay model), which decays as d . −γ  Empirical studies: d , where γ lies anywhere between two and six Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 26Simplified Path Loss Model  Used when path loss dominated by reflections.  Most important parameter is the path loss exponent , determined empirically.  d  0 P PK , 2 8 r t  d   Cell design impact: If the radius of a cell is reduced by half when the propagation path loss exponent is 4, the transmit power level of a base station is reduced by 12dB (=10 log 16 dB).  Costs: More base stations, frequent handoffs Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 27Typical largescale path loss Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: Rappaport and A. Goldsmith books : “shiv rpi” 28Empirical Models  Okumura model  Empirically based (site/freq specific)  Awkward (uses graphs)  Hata model  Analytical approximation to Okumura model  Cost 136 Model:  Extends Hata model to higher frequency (2 GHz)  Walfish/Bertoni:  Cost 136 extension to include diffraction from rooftops Commonly used in cellular system simulations Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 29Empirical Model: Eg: Lee Model Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 30Empirical Path Loss: Okamura, Hata, COST231  Empirical models include effects of path loss, shadowing and multipath.  Multipath effects are averaged over several wavelengths: local mean attenuation (LMA)  Empirical path loss for a given environment is the average of LMA at a distance d over all measurements  Okamura: based upon Tokyo measurements. 1100 lm, 1501500MHz, base station heights (30100m), median attenuation over freespaceloss, 10 14dB standard deviation.  Hata: closed form version of Okamura  COST 231: Extensions to 2 GHz Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: A. Goldsmith book : “shiv rpi” 31Indoor Models  900 MHz: 1020dB attenuation for 1 floor, 610dB/floor for next few floors (and frequency dependent)  Partition loss each time depending upton material (see table)  Outdoortoindoor: building penetration loss (820 dB), decreases by 1.4dB/floor for higher floors. (reduced clutter)  Windows: 6dB less loss than walls (if not lead lined) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 32Path Loss Models: Summary  Path loss models simplify Maxwell’s equations  Models vary in complexity and accuracy 2  Power falloff with distance is proportional to d in free 4 space, d in two path model  General ray tracing computationally complex  Empirical models used in 2G/3G/Wimax simulations  Main characteristics of path loss captured in simple model  P =P Kd /d r t 0 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 33Shadowing  Lognormal model for shadowing r.v. () Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 34Shadowing: Measured largescale path loss Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 35LogNormal Shadowing  Assumption: shadowing is dominated by the attenuation from blocking objects.  Attenuation of for depth d: −αd s(d) = e , (α: attenuation constant).  Many objects: −α∑ di −αdt s(d ) = e = e , t d = ∑ d is the sum of the random object depths t i  Cental Limit Theorem (CLT): αd = log s(d ) N(μ, σ). t t  log s(d ) is therefore lognormal t Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 36Area versus Distance Coverage model with Shadowing model Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 37Outage Probability w/ Shadowing dBm  Need to improve receiver sensitivity (i.e. reduce Pmin) for better coverage. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 38Shadowing: Modulation Design  Simple path loss/shadowing model:  Find Pr:  Find Noise power: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 39Shadowing: Modulation Design (Contd)  SINR:  Without shadowing ( = 0), BPSK works 100, 16QAM fails all the time.  With shadowing ( = 6dB): s BPSK: 16 QAM  75 of users can use BPSK modulation and hence get a PHY data rate of 10 MHz · 1 bit/symbol ·1/2 = 5 Mbps  Less than 1 of users can reliably use 16QAM (4 bits/symbol) for a more desirable data rate of 20 Mbps.  Interestingly for BPSK, w/o shadowing, we had 100; and 16QAM: 0 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 40SmallScale Fading: Rayleigh/Ricean Models, Multipath Doppler Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 41Smallscale Multipath fading: System Design  Wireless communication typically happens at very high carrier frequency. (eg. f = 900 MHz or 1.9 GHz for cellular) c  Multipath fading due to constructive and destructive interference of the transmitted waves.  Channel varies when mobile moves a distance of the order of the carrier wavelength. This is about 0.3 m for 900 Mhz cellular.  For vehicular speeds, this translates to channel variation of the order of 100 Hz.  Primary driver behind wireless communication system design. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 42Fading: Small Scale vs Large Scale Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 43Source 1: SingleTap Channel: Rayleigh Dist’n  Path loss, shadowing = average signal power loss  Fading around this average.  Subtract out average = fading modeled as a zeromean random process  Narrowband Fading channel: Each symbol is long in time  The channel h(t) is assumed to be uncorrelated across symbols = single “tap” in time domain.  Fading w/ many scatterers: Central Limit Theorem  Inphase (cosine) and quadrature (sine) components of the snapshot r(0), denoted as r (0) and r (0) are independent Gaussian random variables. I Q  Envelope Amplitude:  Received Power: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 44Source 2: Multipaths: PowerDelay Profile path1 path2 path3 multipath propagation path2 Path Delay path1 path3 Mobile Station (MS) Base Station (BS) Channel Impulse Response: Channel amplitude h correlated at delays . Each “tap” value kTs Rayleigh distributed (actually the sum of several subpaths) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 45 PowerEg: Power Delay Profile (WLAN/indoor) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 46Multipath: TimeDispersion = Frequency Selectivity  The impulse response of the channel is correlated in the timedomain (sum of “echoes”)  Manifests as a powerdelay profile, dispersion in channel autocorrelation function A()  Equivalent to “selectivity” or “deep fades” in the frequency domain  Delay spread: 50ns (indoor) – 1s (outdoor/cellular).  Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor)  Implications: High data rate: symbol smears onto the adjacent ones (ISI). Multipath effects O(1s) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 47Source 3: Doppler: NonStationary Impulse Response. Set of multipaths changes O(5 ms) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 48Doppler: Dispersion (Frequency) = TimeSelectivity  The doppler power spectrum shows dispersion/flatness doppler spread (100200 Hz for vehicular speeds)  Equivalent to “selectivity” or “deep fades” in the time domain correlation envelope.  Each envelope point in timedomain is drawn from Rayleigh distribution. But because of Doppler, it is not IID, but correlated for a time period Tc (correlation time).  Doppler Spread: Ds 100 Hz (vehicular speeds 1GHz)  Coherence Time: Tc = 2.55ms.  Implications: A deep fade on a tone can persist for 2.55 ms Closedloop estimation is valid only for 2.55 ms. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 49Fading Summary: TimeVarying Channel Impulse Response  1: At each tap, channel gain h is a Rayleigh distributed r.v.. The random process is not IID.  2: Response spreads out in the timedomain (), leading to intersymbol interference and deep fades in the frequency domain: “frequencyselectivity” caused by multipath fading  3: Response completely vanish (deep fade) for certain values of t: “Timeselectivity” caused by doppler effects (frequencydomain dispersion/spreading) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 50DispersionSelectivity Duality Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 51DispersionSelectivity Duality (Contd) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 52Fading: Jargon  Flat fading: no multipath ISI effects.  Eg: narrowband, indoors  Frequencyselective fading: multipath ISI effects.  Eg: broadband, outdoor.  Slow fading: no doppler effects.  Eg: indoor Wifi home networking  Fast Fading: doppler effects, timeselective channel  Eg: cellular, vehicular  Broadband cellular + vehicular = Fast + frequencyselective Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 53Fading: Details SingleTap, Narrowband Flat Fading. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 54Normal Vector R.V, Rayleigh, ChiSquared X = X , …, X is Normal random vector 1 n X is Rayleigh eg: magnitude of a complex gaussian channel X + jX 1 2 2 X is ChiSquared w/ ndegrees of freedom When n = 2, chisquared becomes exponential. eg: power in complex gaussian channel: sum of squares… Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 55Rayleigh, Ricean, Nakagamim fading Ricean used when there is a dominant LOS path. K parameter: strength of LOS to nonLOS. K = 0 = Rayleigh Nakagamim distribution can in many cases be used in tractable analysis of fading channel performance. More general than Rayleigh and Ricean. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 56Rayleigh Fading Example  Nontrivial (1) probability of very deep fades. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 57Rayleigh Fading (Fade Duration Example) L : Level Crossing Rate z Faster motion doppler better (get out of fades) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 58Effect of Rayleigh Fading Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 59Fading: Details Broadband, “FrequencySelective” Fading. Multipath Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 60Broadband Fading: Multipath Frequency Selectivity  A few major multipaths, and lots of local scatterers = each channel sample “tap” can be modeled as Rayleigh  A “tap” period generally shorter than a symbol time.  Correlation between tapped values. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 61Recall: Electric (Far) Field Transfer Function  Tx: a sinusoid: cos 2ft  Electric Field: source antenna gain ( ) s  Product of antenna gains ()  Consider the function (transfer function)  The electric field is now: Linearity is a good assumption, but timeinvariance lost when Tx, Rx or environment in motion Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 62Reflecting wall: Ray Tracing, Superposition  Superposition of phaseshifted, attenuated waves  Phase difference ( ): depends upon f r  Constructive or destructive interference  Peaktovalley: coherence distance:  Delay spread:  Coherence bandwidth: I/f pattern changes if frequency changes on the order of coherence bandwidth. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 63Power Delay Profile = InterSymbol interference Symbol Symbol Time Time  Higher bandwidth = higher symbol rate, and smaller time persymbol  Lower symbol rate, more time, energy persymbol  If the delay spread is longer than the symbolduration, symbols will “smear” onto adjacent symbols and cause symbol errors path1 path2 path3 Path Delay No Symbol Error (kbps) Delay spread Symbol Error (energy is collected 1 s If symbol rate over the full symbol period Mbps for detection) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 64 PowerEffect of Bandwidth ( taps) on MultiPath Fading Effective channel depends on both physical environment and bandwidth Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 65Multipaths Bandwidth (Contd)  Even though many paths with different delays exist (corresponding to finerscale bumps in h(t))…  Smaller bandwidth = fewer channel taps (remember Nyquist)  The receiver will simply not sample several multipaths, and interpolate what it does sample = smoother envelope h(t)  The power in these multipaths cannot be combined  In CDMA Rake (Equalization) Receiver, the power on multipath taps is received (“rake fingers”), gain adjusted and combined.  Similar to bandpass vs matched filtering (see next slide) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 66Rake Equalization Analogy: Bandpass vs Matched Filtering Simple Bandpass (low bandwidth) Filter: excludes noise, but misses some signal power in other mpath “taps” Matched Filter: includes more signal power, weighted according to size = maximal noise rejection signal power aggregation Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 67Power Delay Profile: Mean/RMS Delay Spreads Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 68Multipath Fading Example Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 69Fading: Details Doppler “Fast” Fading: Timeselectivity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 70Doppler: Approximate LTI Modeling  r → r + vt 0  vt/c phase correction (Fixed phase frequency shifts)  Doppler frequency shift of –fv/c due to relative motion  This is no longer LTI unlike wired channels  We have to make LTI approximations assuming small timescales only (t small, vt ≈ 0)  If timevarying attenuation in denominator ignored (vt ≈ 0), we can use the transfer function H(f) as earlier, but with doppler adjustment of fv/c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 71Doppler: Reflecting Wall, Moving Antenna  Doppler spread:  Note: opposite sign for doppler shift for the two waves  Effect is roughly like the product of two sinusoids Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 72Doppler Spread: Effect 5ms  Fast oscillations of the order of GHz  Slow envelope oscillations order of 50 Hz = peaktozero every 5 ms  A.k.a. Channel coherence time (Tc) = c/4fv Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 73Twopath (mobile) Example v= 60 km/hr, f = 900 MHz: c Direct path has Doppler shift of roughly 50 Hz = fv/c Reflected path has shift of +50 Hz Doppler spread = 100 Hz Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 74Doppler Spread: Effect Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 75Angular Spread: Impact on Spatial Diversity  Spacetime channel models:  Mean/RMS angular spreads (similar to multipath delay spread)  The timevarying impulse response model can be extended to incorporate AOA (angleofarrival) for the array.  A(): average received signal power as a function of AoA .  Needs appropriate linear transformation to achieve full MIMO gains. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 76Angular Spread and Coherence Distance  : RMS angular spread of a channel RMS  Refers to the statistical distribution of the angle of the arriving energy.  Large  = channel energy is coming in from many directions, RMS  Lot of local scattering, and this results in more statistical diversity in the channel based upon AoA  Small  = received channel energy is more focused. RMS  More focused energy arrival results in less statistical diversity.  The dual of angular spread is coherence distance, Dc.  As the angular spread↑, the coherence distance ↓, and vice versa.  A coherence distance of d means that any physical positions separated by d have an essentially uncorrelated received signal amplitude and phase. ↑freq =  An approximate rule of thumb is better angular diversity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 77Key Wireless Channel Parameters Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 78Fading Parameter Values Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 79SmallScale Fading Summary Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 80Fading: Design Impacts (Eg: Wimax) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 81Mathematical Models Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 82Physical Models  Wireless channels can be modeled as linear timevarying systems: where a (t) and  (t) are the gain and delay of path i. i i  The timevarying impulse response is:  Consider first the special case when the channel is time invariant: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 83TimeInvariance Assumption: Typical Channels are Underspread  Coherence time T depends on carrier frequency and c vehicular speed, of the order of milliseconds or more.  Delay spread T depends on distance to scatterers, of d the order of nanoseconds (indoor) to microseconds (outdoor).  Channel can be considered as timeinvariant over a long time scale (“underspread”).  Transfer function frequency domain methods can still be applied to this approximately LTI model Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 84Baseband Equivalence: jwt  Easier to analyze complex numbers like (e ), even though all baseband/passband are real signals involving sines and cosines. jwt  Passband signal = baseband signal (u(t)) multiplying a complex carrier (e ) signal, and extracting the real portion  u(t): complex envelope or complex lowpass equivalent signal Quadrature concept: Cosine and Sine oscillators modulated with x(t) and y(t) respectively (the Real and Quadrature parts of u(t)) Received Signal: v(t) and c(t) are baseband equivalents for received and channel Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 85Block diagram Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 86PassbandtoBaseband Conversion: Block Diagram  Communication takes place at passband  Processing takes place at baseband QAM system Note: transmitted power half of baseband power Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 87Passband vs Baseband Equivalent Spectrum  Communication at passband (allocated spectrum). Processing in baseband: modulation, coding etc. Upconvert/Downconvert.  s contains same information as s: Fourier transform hermitian around 0 b (“rotation”).  If only one of the side bands are transmitted, the passband has half the power as the baseband equivalent Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 88Perpath: Complex Baseband Equivalent Channel  The frequency response of the system is shifted from the passband to the baseband.  Each path i is associated with a delay ( ) and a i complex gain (a ). i Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 89DiscreteTime Baseband Equivalence: With Modulation and Sampling Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 90Sampling Interpretation  Due to the decay of the sinc function, the ith path contributes most significantly to the lth tap if its delay falls in the window l/W − 1/(2W), l/W +1/(2W). Discrete Time Baseband I/O relationship: where: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 91Multipath Resolution: LTI Approximation Sampled basebandequivalent channel model: where h is the l th complex channel tap. l and the sum is over all paths that fall in the delay bin System resolves the multipaths up to delays of 1/W . Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 92Baseband Equivalence: Summary  Let s(t) denote the input signal with equivalent lowpass signal u(t).  Let h(t) denote the bandpass channel impulse response with equivalent lowpass channel impulse response h (t) l  The transmitted signal s(t) and channel impulse response h(t) are both real, so the channel output r(t) = s(t) ∗ h(t) is also real, with frequency response R(f) = H(f)S(f)  R(f) will also be a bandpass signal w/ complex lowpass representation:  It can be rewritten (after manipulations as): Summary: Equivalent lowpass models for s(t), h(t) and r(t) isolates the carrier terms (f ) from the analysis. Sampled version allows discretetime processing. c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 93Multipaths in LTI Model: Flat/FrequencySelective Fading  Fading occurs when there is destructive interference of the multipaths that contribute to a tap. Delay spread Coherence bandwidth single tap, flat fading multiple taps, frequency selective Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 94Doppler: Time Variations in Model Timevarying delays Doppler shift of the i th path Doppler spread Coherence time Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 95Doppler Spread Doppler spread is proportional to:  the carrier frequency f ; c  the angular spread of arriving paths. where  is the angle the direction of motion makes i with the i th path. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 96Degrees of Freedom (Complex Dimensions) th  Discrete symbol xm is the m sample of the transmitted signal; there are W samples per second.  Continuous time signal x(t), 1 s ≡ W discrete symbols  Each discrete symbol is a complex number;  It represents one (complex) dimension or degree of freedom.  Bandlimited x(t) has W degrees of freedom per second.  Signal space of complex continuous time signals of duration T which have most of their energy within the frequency band −W/2,W/2 has dimension approximately WT.  Continuous time signal with bandwidth W can be represented by W complex dimensions per second.  Degrees of freedom of the channel to be the dimension of the received signal space of ym Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 97Statistical Models  Design and performance analysis based on statistical ensemble of channels rather than specific physical channel.  Rayleigh flat fading model: many small scattered paths Complex circular symmetric Gaussian . Squared magnitude is exponentially distributed.  Rician model: 1 lineofsight plus scattered paths Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 98Statistical Models: Correlation over Time  Specified by autocorrelation function and power spectral density of fading process.  Example: Clarke’s (or Jake’s) model. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 99Additive White Gaussian Noise (AWGN)  Complete basebandequivalent channel model:  Special case: flat fading (onetap):  Will use this throughout the course. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 100BER Effect of Fading: AWGN vs Fading Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 101Types of Channels Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 102Summary  We have understood both qualitatively and quantitatively the concepts of path loss, shadowing, fading (multipath, doppler), and some of their design impacts.  We have understood how time and frequency selectivity of wireless channels depend on key physical parameters.  We have come up with linear, LTI and statistical channel models useful for analysis and design. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 103
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