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Published byCoral Tucker Modified over 4 years ago

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**Wireless Communication Channels: Small-Scale Fading**

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**Clarke’s Model for Flat Fading**

Assumptions: Mobile traveling in x direction Vertically polarized wave Multiple waves in the x-y plane arrive at the mobile antenna at the same time Waves arrive at different angles α z y in x-y plane α x For N waves incident at the mobile antenna Each wave arriving at an angle αn will experience a different Doppler shift fn E0 amplitude of the local average E-field Cn random variable representing the amplitude of individual waves fc carrier frequency φn random phase shift due to distance traveled by the nth wave

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**Clarke’s Model for Flat Fading**

Given that: Φn uniformly distributed over 2π N is sufficiently large (i.e., the central limit theorem is applicable) Therefore: Both Tc(t) and Ts(t) may be modeled as: Gaussian Random Processes

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**Clarke’s Model for Flat Fading**

Power received at mobile antenna Rayleigh Distribution

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**Rayleigh Fading Distribution**

Main Assumption: No LOS All waves at the mobile receiver experience approximately the same attenuation z y dα in x-y plane α x 0.6065/σ constant p(r) σ2: Time average received power σ : rms value of received voltage r σ

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**Rayleigh Fading Statistics**

Probability the received signal does not exceed a value R Mean value of the Rayleigh distribution Variance of the Rayleigh distribution Median of the Rayleigh distribution

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**Ricean Fading Distribution**

Main Assumption: LOS There is a dominant wave component at the mobile receiver in addition to experience multiple waves that experience approximately the same attenuation z y dα in x-y plane α x A : Peak amplitude of the dominant signal I(.): Modified Bessel function of the first kind and zero-order σ2: Time average received power of the non-dominant components

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**Riciean & Rayleigh Fading**

Define K called the Ricean Factor: The ratio between the deterministic signal power and the power of the non-dominant waves p(r) K=-∞ dB Rayleigh Distribution K=6 dB r

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**Level Crossing Rate and Mean fade Duration for Rayleigh Fading Signals**

Level Crossing Rate Statistic: The expected rate at which Rayleigh fading envelope normalized to local rms level crosses a specified level in a positive–going direction ρ:= R/Rrms fm: Maximum Doppler shift Mean Fade Duration Statistic: The average period of time for which the received signal is below a specified level R Mean Fade duration is a very important statistic that helps define the time correlation behavior of BER performance at the receiver

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**How Wireless Channels Components Fit Together**

Distance Pathloss Mobile Speed 3 Km/hr PL= 35.225log10(DKM) d Lognormal Shadowing Mobile Speed 3 Km/hr ARMA Correlated Shadow Model d Small-Scale Fading Mobile Speed 3 Km/hr Jakes’s Rayleigh Fading Model d

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**How Wireless Channels Components Fit Together**

PTGT GR Wireless Channel PR=PTGTGR x Distance Pathloss x Shadowing Parameters x Small-Scale Fading Power

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**System Modeling of Wireless Networks: Example**

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Diversity Techniques

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What is Diversity? Diversity techniques offer two or more inputs at the receiver such that the fading phenomena among these inputs are uncorrelated If one radio path undergoes deep fade at a particular point in time, another independent (or at least highly uncorrelated) path may have a strong signal at that input By having more than one path to select from, both the instantaneous and average SNR at the receiver may be improved

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**Diversity Techniques: Space Diversity**

Receiver Space Diversity M different antennas appropriately separated deployed at the receiver to combine uncorrelated fading signals 1 Transmitter Receiver 2 M

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**Diversity Techniques: Space Diversity**

Transmitter Space Diversity M different antennas appropriately separated deployed at the transmitter to obtain uncorrelated fading signals at the receiver The total transmitted power is split among the antennas 1 Receiver Transmitter 2 M

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**Diversity Techniques: Frequency Diversity**

Modulate the signal through M different carriers The separation between the carriers should be at least the coherent bandwidth Bc Different copies undergo independent fading Only one antenna is needed The total transmitted power is split among the carriers f Δf>Bc t

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**Diversity Techniques: Time Diversity**

Transmit the desired signal in M different periods of time i.e., each symbol is transmitted M times The interval between transmission of same symbol should be at least the coherence time Tc Different copies undergo independent fading f Δt>Tc t

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**Diversity Combining Techniques**

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**Selection Combining Select the strongest signal Transmitter Receiver**

Channel 1 Channel 2 Channel M SNR Monitor Select MAX SNR=γmax

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Selection Combining Consider M independent Rayleigh fading channels available at the receiver Average SNR at all Diversity Branches SNR = Γ Instantaneous SNR at Diversity Branch i SNR = γi Rayleigh Fading Voltage means Exponentially Distributed Power Outage Probability of a Single Branch Outage Probability of of Selection Diversity Combining

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**Maximal Ratio Combining**

Selection Combining does not benefit from power received across all diversity branches Maximal Ratio Combining conducts a weighted sum across all branches with the objective of maximizing SNR G1 r1 Channel 1 ∑ G2 r2 Channel 2 Transmitter Receiver GM rM Channel M

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**Maximal Ratio Combining**

Consider M independent Rayleigh fading channels available at the receiver Envelope applied to receiver detector Total Noise Power applied to detector SNR at the receiver detector Cauchy’s Inequality γMRC is maximized when Gi=ri (MRC requires channel measurements)

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**Maximal Ratio Combining**

γMRC is maximized when Gi=ri (MRC requires channel measurements) Rayleigh Fading Voltage means Exponentially Distributed Power SNR γMRC is Gamma distributed (sum of M exponential random variables) Outage Probability of of Maximal Ratio Diversity Combining

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Equal Ratio Combining Maximal Ratio Combining requires estimation of the channel across all diversity branches Equal Gain Combining conducts a sum across all branches (i.e. Gi=1 for all i) r1 Channel 1 ∑ r2 Channel 2 Transmitter Receiver rM Channel M

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Equal Gain Combining Consider M independent Rayleigh fading channels available at the receiver Envelope applied to receiver detector Total Noise Power applied to detector SNR at the receiver detector EGC is a special case of MRC with Gi=1 SNR and outage probability performance in EGC is inferior to that of MRC

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