Wireless Communication Channels: Small-Scale Fading

Wireless Communication Channels (Reminder) Distance Pathloss Mobile Speed 3 Km/hr PL=137.744+ 35.225log10(DKM) d Lognormal Shadowing Mobile Speed 3 Km/hr ARMA Correlated Shadow Model Rapid Changes in Signal Strength over a small traveling distances d Small-Scale Fading Mobile Speed 3 Km/hr Jakes’s Rayleigh Fading Model d

Multi-Path Propagation From “Wireless Communications” Edfors, Molisch, Tufvesson Multi-Path in the radio channel creates small-scale fading. The three most important effects are: Rapid changes in signal strength over a small travel distance or time interval Random frequency modulation due to varying Doppler shifts on different multi-path signals Time dispersion (echoes) caused by multi-path propagation delays

Multi-Path Propagation Modeling Power Multi-Path Components τ0 τ1 τ2 Time Multi-path results from reflection, diffraction, and scattering off environment surroundings Note: The figure above demonstrates the roles of reflection and scattering only on multi-path

Multi-Path Propagation Modeling Power Multi-Path Components τ0 τ1 τ2 Time As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies

Multi-Path Propagation Modeling Power Multi-Path Components τ0 τ1 τ2 Time As the mobile receiver (i.e. car) moves in the environment, the strength of each multi-path component varies

Multi-Path = Frequency-Selective! 0.5 0.5 1 1 0.5 1 μs 1 μs f=1 MHz 1 0.5 0.5 1 0.5 -0.5 -1 -1 1 μs 1 μs f=500 KHz 1 1 0.5 0.5 0.5 -0.5 -1 -1 1 μs 1 μs

Multi-Path = Frequency-Selective! h(t) |H(f)| 0.5 0.5 1 f (MHz) 0.5 1 1.5 2 1 μs A multi-path channel treats signals with different frequencies differently A signal composed of multiple frequencies would be distorted by passing through such channel

Multi-Path Propagation Modeling From “Wireless Communications: Principles & Practice” T. Rappaport

Power Delay Profile The power delay profile depicts the spatial average of received power within the multi-path channel over a radius that is comparable to the signal wavelength From “Wireless Communications: Principles & Practice” T. Rappaport Multi-Path Profile from a 900 MHz cellular system in San Francisco

Parameters of Mobile Multi-Path Channels The power delay profile is used to derive some parameters that can help characterize the effect of the wireless channel on signal communication We will discuss the following: Time dispersion parameters Mean excess delay Rms delay spread Excess delay spread (X dB) Coherence bandwidth Doppler spread and coherence time

Time Dispersion Parameters Mean Excess Delay P(τ) RMS Delay Spread τ0 τ1 τ2 τ3 τN τ Power Delay Profile Note: These delays are measured relative to the first detectable signal (multi-path component) arriving at the receiver at τ0=0 Maximum Excess Delay (XdB) or Excess Delay Spread (XdB): Time delay during which multi-path energy falls to X dB below the maximum (Note that the strongest component does not necessarily arrive at τ0)

From “Wireless Communications: Principles & Practice” T. Rappaport Example From “Wireless Communications: Principles & Practice” T. Rappaport Example of an Indoor Multi-Path Profile; rms delay spread, mean excess delay, maximum excess delay (10 dB)

Coherence Bandwidth A statistical measure of the range of frequencies over which the channel is can be considered to be “flat” (i.e., a channel which passes all spectral components with approximately equal gain and linear phase) Coherence Bandwidth over which the frequency correlation function is 0.9 Coherence Bandwidth over which the frequency correlation function is 0.5

Doppler Shift is Given by The difference in path lengths traveled by the wave from source S to the mobile at X and Y is Δl Note: Assume SX, SY >>d such that angle of arrival is nearly equal at X and Y S Δl θ θ Phase Difference due to variation in path lengths X d Y v Doppler Shift is Given by

Doppler Spread and Coherence Time Doppler spread and coherence time are parameters which describe the time varying nature of the channel Doppler spread BD is a measure of spectral broadening due to the Doppler shift associated with mobile motion Coherence time is a statistical measure of the time duration over which the channel impulse response is essentially invariant Coherence Time is inversely proportional to Doppler spread Coherence Time over which the time correlation function is 0.5 where fm is the maximum Doppler shift given by fm=v/λ A Common Rule:

Flat Fading Vs Frequency Selective Fading P(τ) Power Delay Profile A Common Rule of Thumb: TS>10σt  Flat fading τ0 τ1 τN τ Symbol Time (Digital Communication) TS 1 1 + Minimal ISI Wireless Channel + τ0 τN τa

Flat Fading Vs Frequency Selective Fading P(τ) Power Delay Profile A Common Rule of Thumb: TS<10σt  Frequency Selective Fading τ0 τ1 τ2 τ3 τN τ Symbol Time (Digital Communication) TS 1 1 + Significant ISI Wireless Channel + τ0 τa τN

Frequency Selective Channel Simulation

Slow Fading Vs Fast Fading P(τ0,t) Power Delay Profile P(τ) P(τ0,TC) P(τ0,2TC) P(τ0,3TC) P(τ0,KTC) τ0 τ t TC 2TC 3TC KTC Consider a wireless channel comprised of a single path component. The power delay profile reflects average measurements P(τ0) shall vary as the mobile moves Fast Fading Slow Fading Frequency dispersion (time selective fading)

Stationary Receiver Tx signal = cos 2t Assume v = 0, phase difference of two paths = 2.8 radians Rx signal = cos 2t + cos (2t+2.8) S2 S1 θ Ф X d Y v Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

Slow Fading Tx signal = cos 2t Assume slow fading, phase difference of two paths = 2.8 Rx signal = cos 2.01t + cos (2.02t+2.8) S1 θ Ф X d Y v Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

Fast Fading Tx signal = cos 2t Assume slow fading, phase difference of two paths = 2.8 Rx signal = cos 2.1t + cos (2.2t+2.8) S1 θ Ф X d Y v Note: S1 and S2 represent two paths that arrive at the receiver (example two ray model)

Stationary Receiver Vs Slow Vs Fast Fading

Types of Small-Scale Fading From “Wireless Communications: Principles & Practice” T. Rappaport

Types of Small-Scale Fading From “Wireless Communications: Principles & Practice” T. Rappaport