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Mobile Radio Propagation: Small-scale Fading and Multipath

• Fading is rapid fluctuations of the amplitude of a radio signal over a short period of time or travel distance.

• Fading is caused by interference between two or more versions of transmitted signal, which arrives at the receiver at slightly different times.

• These multipath waves combine at the receiver antenna to give a resultant signal, which can vary in delay, in amplitude and phase.

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Multipath Effects

• Rapid changes in signal strength over a small distance or time interval.

• Random frequency modulation due to varying Doppler shift on different multipath signal.

• Time Dispersion (echoes) caused by multipath propagation delay.

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Causes of Fading • In urban areas, fading occurs because the height of mobile is <<

height of surrounding structures such as buildings, trees • Existence of several propagation paths between transmitter and

receiver.

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Factors influencing small signal fading

• Multipath propagations

• Speed of mobile (Doppler Shift)

received frequency = f +/- fd + If mobile is moving towards base station.

- If mobile is moving away from base station.

• Speed of Surrounding objects

This is considered only if the speed of the surrounding objects is greater than the mobile

• Transmission bandwidth of signal and bandwidth of

channel.

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Analysis methods of Multipath Channel

Transmitter Receiver

Spatial Position d

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Convolution model for multipath propagation

- - - - -

Received Signal: y (t) = x(t) + A1 x(t -τ1) + A2 x(t -τ2) + ...........

T R

A2 x(t-τ2)

A1 x(t-τ1)

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Time varying system model for channel

• For a fixed position d, the channel between transmitter and receiver can be modulated as a linear time varying system(LTV system)

• Impulse response of the LTI system can be given as h(d,t)

• If x(t) is the transmitted signal, the received signal can be represented as:

y(d,t) = x(t) * h(d,t) * denotes convolution h(d,t) is impulse response of the system

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Time varying system model for channel • y(d,t) = x(t) * h(d,t) t y(d,t) = ∫ x(τ) h(d,t -τ) dτ -∞ • distance d= v.t where v is constant velocity of the receiver. t y(vt,t) = ∫ x(τ) h(vt,t -τ) dτ -∞ t y(t) = ∫ x(τ) h(t,τ) dτ -∞ • This is a time varying system with impulse response of h(t,τ)

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System Definition

Linear time varying (LTV) system

x(t) y(t)

h(t,τ)

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Signal definitions

• Transmitted signal C(t) j2πfct x(t)= Re { c(t) e } • Received signal j2π fct y(t) = Re {r(t) e } • Impulse response jπfct

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h(t,τ)= Re {hb(t,τ) e }

Base band equivalent channel Impulse Response Model

hb(t,τ) c(t) r(t)

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Modeling of the Base band Impulse Response Model

• Mathematical model

r(t) = c(t) * hb(t,τ) hb(t,τ) t3 t2 t1 t0 τo τ1 τ2 τN-2 τN-1

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Excess Delay concept

• The delay axis τ, τo<= τ <= τ n-1 is divided into equal time delay

segments called excess delay bins • τi+1 – τi = ∆τ

τ0 = 0 τ1 = ∆τ τ2 = 2 ∆τ

. τN-1 = (N-1)∆ τ • All multipath signals received within the bins are represented by a

single resolvable multipath component having delay τi.

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• This modal can analyze transmitted signals having bandwidths less than 1/ (2∆τ).

Mathematical Model for Base band Impulse Response

N-1 j2πfcτi(t) jΦi (t,τ) hb(t,τ) = ∑ ai(t,τ) e e * δ[τ –τi(t)] i=0 ai(t,τ) = Real Amplitude τi(t) = Excess delays 2πfcτi(t) = Phase shift due to free space propagation Φi (t,τ) = phase shift encountered in channel

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Simplified Mathematical Model for Baseband Impulse Response

• If the channel is assumed to be time invariant, over a small period of time or over small distance interval

N-1 jθi

hb(τ) = ∑ ai e δ[τ –τi] i =0 Since r(t)= c(t) * hb(t) N-1 jθi

r(t) = ∑ ai e c[t –τi] i =0

• For measuring or predicting the impulse response a probing pulse c(t)= δ[T-t] is used.

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Relationship between bandwidth and received power -Wideband signals

j2πfct Transmitted signal x(t)=Re{p(t) e }

N-1 jθi

Received signal r(t) = ½ ∑ ai e p[t –τi] i =0 p(t) is a pulse train.

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Instantaneous Multipath received power amplitude τmax

|r(t)|2 =(1/τmax) ?r(t) r *(t) dt 0 N-1 = ∑ |ak|2 k = 0 =>Total received power = sum of the power of individual multipath components.

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Average small-scale received power

N-1 Ea,θ [PwB] = Ea,θ [ ∑ |ai exp jθI|

2]

i = 0 N-1 --- = ∑ ai 2 i = 0 Ea,θ = average overall possible values of ai and θI in a local area. __ ai 2= sample average area local measurement area, generally measured using multipath measurement equipment.

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Relationship between bandwidth and received power -Narrowband signals

Transmitted signal: c(t)= K (constant) N-1 jθi (t,τ) Received signal r(t) = K Σ ai e i = 0

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Instantaneous Multipath received power amplitude

N-1 jθi (t,τ) |r(t)|2 = | Σ ai e |2

i = 0

Average Power over a local area

N-1 jθi (t,τ) Ea,θ [PwB ]= Ea,θ [ | Σ ai e |2 ]

i = 0

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Conclusions • When the transmitted signal has a wide bandwidth >>

bandwidth of the channel multipath structure is completely resolved by the receiver at any time and the received power varies very little.

• When the transmitted signal has a very narrow bandwidth

(example the base band signal has a duration greater than the excess delay of the channel) then multipath is not resolved by the received signal and large signal fluctuations occur (fading).

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Example Assume a discrete channel impulse response is used to model urban radio channels with excess delays as large as 100 µ s and microcellular channels with excess delays not larger than 4 µ s . If the number of multipath bins is fixed at 64 find: (a) ∆τ (b) Maximum bandwidth, which the two models can accurately represent. Solution: Delays in channel ∆τ, 2 ∆ τ ….N∆ τ Maximum excess delay of channel τN = N ∆τ = 100 µs. N = 64

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∆τ = τN /N = 100 µs /64 = 1.5625 µs Maximum bandwidth represented accurately by model = 1/ (2 ∆τ) = 0.32 MHz For microcellular channel Maximum excess delay of channel τN = N ∆τ = 4 µs. N = 64 ∆τ = τN /N = 4 µs /64 = 62.5 ns Maximum bandwidth represented accurately by model = 1/ (2 ∆τ) = 8 MHz

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Example: Assume a mobile traveling at a velocity of 10m/s receives two multipath components at a carrier frequency of 1000MHz. The first component is assumed to arrive at τ = 0 with an initial phase of 0° and a power of –70dBm. The second component is 3dB weaker than the first one and arrives at τ = 1µs, also with the initial phase of 0°. If the mobile moves directly to the direction of arrival of the first component and directly away from the direction of arrival of the second component, compute the following

(a) The narrow band and wide band received power over the interval 0-0.5s (b) The average narrow band received power.

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Narrow band instantaneous power.

N-1 jθi (t,τ)

|r(t)|2 = | Σ ai e |2

i = 0 _______ Now –70dBm => 100 pw so a1 = v 100 pw _______ and -73dBm => 50 pw so a2 = v 50 pw θi = 2πd/λ = 2πvt/λ λ = (3*108)/(100*106)=0.3 m θ1=2π*10*t/0.3 = 209.4 t rad.

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θ2 = -θ1 = -209.4 t rad. t=0 ___ ___

|r(t)|2 = | v100 + v50 | 2= 291pw

t=0.1 ___ ___ |r(t)|2 = v100. e j209.4 x 0.1+ v50. e -j209.4 x 0.1 = 78.2pw t=0.2 ___ ___ |r(t)|2 = v100. e j209.4 x 0.2+ v50. e -j209.4 x 0.2

= 81.5pw

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t=0.3 |r(t)|2 = 291pw t=0.4 |r(t)|2 = 78.2pw t=0.5 |r(t)|2 = 81.5pw

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Wideband Instantaneous Power

N-1 |r(t)|2 = ∑ |ak|2 = 100 + 50 = 150 pW k = 0 Average Narrow band received power Ea,θ [PCW ] = [2(291) + 2(78.2) +2(81.5)] /6 =150.233pw The average narrow band power and wideband power are almost the same over 5m. while the narrow band signal fades over the observation interval, the wideband signal remains constant.

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Small-scale Multipath measurements.

• Direct Pulse Measurements • Spread Spectrum Sliding Correlate Measurement • Swept Frequency Measurement

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Types of Small scale fading

• 2 main propagation mechanisms ----Multipath time delay spread and Doppler spread

Multipath time delay

Doppler Spread

Flat Fading Frequency Selective Fading

Flat Fading Frequency Selective Fading

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• Two types of fading are independent of each other. Multipath Terms associated with fading

Ts = Symbol period or reciprocal bandwidth

Bs = Bandwidth of transmitted signal

Bc = coherence bandwidth of channel

Bc= 1/50στ where στ is rms delay spread __ _ στ

2 = τ 2 - ( τ )2 _ τ = (Σ ak

2 τκ) / (Σ ak2) = mean Excess delay

__

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τ 2 = (Σ ak2 τκ

2) / (Σ ak2)

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Fading effects due to Doppler spread fc = frequency of pure or transmitted sinusoid Received signal spectrum = fc +/- fd, fd = Doppler shift fc S θ V

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Doppler spread and coherence time

Doppler frequency shift : fd = ( v / λ) cos θ , Where Wavelength λ = c / fc meters Doppler Spread BD = fm = Maximum Frequency deviation = v / λ coherence time = Tc = 0.423 / fm

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Types of fading Flat Fading

• Mobile channel has constant gain and linear phase response • spectral characteristics of the transmitted signal are maintained

at receiver.

• Bs < < Bc => Ts >= στ

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Frequency Selective Fading:

• Mobile channel has a constant gain and linear phase response over a bandwidth

• Bs > Bc => Ts < 10 στ

• Received signal includes multiple versions of transmitted waveform

so received signal is distorted.

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Fast Fading Channel

• The channel impulse response changes rapidly within the symbol

duration. This causes frequency dispersion due to Doppler

spreading, which leads to signal distortion.

• Ts (Symbol period) > Tc

• Bs < BD

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Slow Fading Channel

• The channel impulse response changes at a rate much slower than the transmitted signal s(t).

•

Ts << Tc

Bs >> BD

• Velocity of mobile (or velocity of objects in channel) and base band signaling determines slow fading or fast fading.

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Rayleigh and Ricean Distributions Rayleigh Fading Distribution

• In mobile radio channels, the Rayleigh distribution is commonly

used to describe the statistical time varying nature of the received

envelope of an individual multipath component.

Pdf (Probability density function)

p(r) = (r/σ2) e –(r2/2σ2) (0 = r = ∞)

= 0 r < 0

σ → rms value of received voltage before envelope detection.

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Cumulative distribution function (cdf)

R P (R) = P( r ≤ R) = ∫ p(r ) dr 0 = 1 – e –(R2/2σ2)

8 Mean Value E[R] = ∫ r p(r ) dr 0 ___ = σ√π/2 = 1.25336 σ

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σR2 = E[R2] - {E(R)}2

8 = ∫ r2 p(r ) dr - σ2∏/2

0 = 0.4292 σ2

Median value for r => ½ =? pr dr => r (median) = 1.77σ

rmedian Median value for r = ∫ p(r ) dr => rmedian = 1.77σ

0

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Ricean Fading Distribution

• When there is a dominant (non fading) signal component present

such as line-of-sight propagation path, the small scale fading

envelope distribution is Ricean.

• This can be modeled as random, multipath components arriving at

different angles superimposed on a stationary dominant signal.

p(r) = (r/σ2) e –(r2 + A2)/ (2σ2) Io(Ar/σ2)

for A ≥ 0, r ≥ 0

= 0 for r < 0

Io → Modified Bessel function of first kind and zero order

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Ricean Factor

K(dB) = 10log(A2/2σ2) dB

= 10 log(Deterministic signal power/variance of multipath)

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Level crossing and Fading statistics

• Level crossing rate (LCR) is defined as the expected rate at which

the Rayleigh fading envelope, normalized to the local rms signal

level, crosses a specified level in a positive going direction.

8 NR = ∫ r p(R, r•) dr• 0 r• = d/dt r(t) ⇒ slope P(R, r•) → Joint density function of r and r• at r = R.

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Simplified equation for level crossing rate LCR

__ NR = ∫ r• p(R, r•) dr• = √2π fm ρe-ρ2

fm = Maximum Doppler frequency

ρ = R/Rms is the value of the specified level R, normalized to the

local rms amplitude of the fading amplitude

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Example

For a Rayleigh fading signal, compute the positive going level crossing rate for ρ = 1, when the maximum Doppler frequency (fm) is 20 Hz. What is the maximum velocity of the mobile for this Doppler frequency if the carrier is 900 MHz? Solution

ρ = 1

fm = 20 Hz

The number of zero level crossings is:

NR = √2π (20) e-1

= 18.44 Crossings/Sec

Maximum velocity of mobile = fd λ = 20 (3 X 108)/(900X106)

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= 6.66 m/s

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Average Fade Duration

• Average period of time for which the received signal is below a specified level R. For a Rayleigh fading channel, this is given by:

__ • τ= (1/NR ) Pr[r ≤ R]

R

Pr[r ≤ R] = ∫ p(r)dr 0 = 1 – e-ρ2

⇒ τ = 1 – e-ρ2 / (pfm√2π) e-ρ2

= eρ2– 1

__ ρfm√2π

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Example

Find the average fade duration for threshold level ρ = 0.01, ρ = 0.1

and ρ = 1, when the Doppler frequency is 20Hz.

Solution

ρ τ = eρ2 – 1

ρfm√2π

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

0.01 19.9µs

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0.1 200µs

1.0 3.43ms

Statistical methods for multipath Fading channels

• Clark’s model for Flat Fading

• Two-Ray Rayleigh Fading Model

• Saleh and Valenzuela Indoor statistical Model

• SIRCIM (Simulation of Indoor Radio Channels Impulse Response

Models)

• SMRCIM (Simulation of Mobile Radio Channel Impulse-Response

Models)

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