Mobile Radio Propagation: Small-Scale Fading and Multi-pathsaquib/EE4365/Chapter 5/Slides.pdfEE/TE 4365, UT Dallas 2 Small-scale Fading † Small-scale fading, or simply fading describes

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Mobile Radio Propagation: Small-Scale Fadingand Multi-path

1

EE/TE 4365, UT Dallas 2

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Small-scale Fading

• Small-scale fading, or simply fading describes the rapidfluctuation of the amplitude of a radio signal over a shortperiod of time or travel distance

• It is caused by interference between two or more versions of thetransmitted signal which arrive at the receiver at different times

– This interference can vary widely in amplitude and phaseover time


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Small-scale Fading Effects

• The three most important fading effects are

1. Rapid changes in signal strength over a small travel distanceor time interval

2. Random frequency modulation due to varying Dopplershifts (described later) on different multi-path signals

3. Time dispersions (echos) caused by multi-path propagationdelays


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Factors Influencing Small-scale Fading

• The following physical factors in the radio propagation channelinfluence small-scale fading

– multi-path propagation

– speed of the mobile

– speed of the surrounding objects

– the transmission bandwidth of the signal


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Multi-path Propagation

• The presence of reflecting objects and scatterers in the channelcreates a constantly changing environment

– this results in multiple versions of the transmitted signalthat arrive at the receiving antenna, displaced with respectto one another in time and spatial orientation


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Multipath Propagation (continued)

• The random phase and amplitudes of the different multipathcomponents cause fluctuations in signal strength, therebyinducing small-scale fading, signal distortion, or both

• Multipath propagation often lengthens the time required forthe baseband portion of the signal to reach the receiver whichcan cause signal smearing due to intersymbol interference


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Doppler Shift

• Definition: The shift in received signal frequency due tomotion is called the Doppler shift

– It is directly proportional to

∗ the velocity of the mobile∗ the direction of motion of the mobile with respect to the

direction of arrival of the received wave


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Doppler Shift (continued)

S

X Y

θθ∆ l

vd

• Consider a mobile moving at a constant velocity v, along apath segment having length d between points X and Y

• The mobile receives signals from a remote source S


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• Assumptions:

– d is small and S is very remote

• When the distance of S À d −→ SX is almost parallel to SY


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• The difference in path lengths traveled by the wave from sourceS to the mobile at points X and Y is

∆l = d cos θ = v∆t cos θ

where

∗ ∆t = time required for the mobile to travel from X to Y

∗ θ = angle of arrival of the wave, which is the same at pointsX and Y due to the assumptions in the previous slide


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• The transmitted signal can be expressed as

s(t) = A{exp[j2πfct]}

where

∗ A = amplitude of the signal

∗ fc = carrier frequency

• The received signal at point X is given by

rx(t) = A{exp[j2πfc (t− τx)]}

∗ τx = propagation delay


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• The received signal at point Y is given by

ry(t) = Aexp[j2πfc (t− τy)]

= Aexp[j2πfc{t− (τx −∆t)}]

= Aexp[j2πfc

{(t− τx) +

∆l

c

}]

= Aexp[j2π

{fc (t− τx) +

∆l

λ

}]

= Aexp[j2π

{fc (t− τx) +

v cos θ

λ∆t

}]


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• From the previous slide, let

Φy = 2πfct− 2πfcτx + 2πv cos θ

λ∆t

• Received frequency at point Y is

fy =12π

dΦy

dt

= fc +v cos θ

λ= fc + fd

where fd is the Doppler shift due to the motion of the mobile

• Note: fd is positive when the mobile is moving towards thesource S


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• If the mobile is moving away from the base station then

rx(t) = Aexp[j2π

{fc (t− τy)− v cos θ

λ∆t

}]

• Thus the received frequency at X is

fx = fc − fd = fc − v cos θ

λ


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Power Delay Profile

a

aa

a

a

φ

ττ τ τ τ τ

1

1

2

2

3

3

4

4

5

5

2

2

2

2

2

( )τ

Multipath Power Delay Profile

• Power delay profiles are

– used to derive many multipath channel parameters

– generally represented as plots of relative received power (a2k)

as a function of excess delay (τ) with respect to a fixed timedelay reference


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Power Delay Profile (continued)

• Power delay profiles are found by averaging instantaneouspower delay profile measurements over a local area in order todetermine an average small-scale power delay profile


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Time Dispersion Parameters

• The time dispersion parameters that can be determined from apower delay profile are

– mean excess delay

– rms delay spread

– excess delay spread

• The time dispersive properties of wide band multipath channelsare most commonly quantified by their mean excess delay (τ)and rms delay spread (στ )


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Mean Excess Delay

• The mean excess delay is the first moment of the power delayprofile and is defined as

τ =∑

k a2kτk∑

k a2k

=∑

k P (τk)τk∑k P (τk)

where

P (τk) =a2

k∑i a2

i


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RMS Delay Spread

• The rms delay spread is the square root of the second centralmoment of the power delay profile and is defined to be

στ =√

τ2 − (τ)2

where

τ2 =∑

k a2kτ2

k∑k a2

k

=∑

k P (τk)τ2k∑

k P (τk)


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Notes

• The mean excess delay and rms delay spread are measuredrelative to the first detectable signal arriving at the receiver atτ0 = 0

• τ and τ2 do not rely on the absolute power level, but only therelative amplitudes of the multipath components

• Typical values of rms delay spread are on the order of

– microseconds in outdoor mobile radio channel

– nanoseconds in indoor radio channels


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Notes (continued)

• The rms delay spread and mean excess delay are defined from asingle power delay profile which is the temporal or spatialaverage of consecutive impulse response measurementscollected and averaged over a local area

• Typically many measurements are made at many local areas inorder to determine a statistical range of multipath channelparameters for a mobile communication system over alarge-scale area


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Maximum Excess Delay

• The maximum excess delay (X dB) of the power delay profileis defined to be the time delay during which multipath energyfalls to X dB below the maximum

• If τ0 is the first arriving signal and τX is the maximum delay atwhich a multipath component is with X dB of the strongestmultipath signal (which does not necessarily arrive at τ0), thenthe maximum excess delay is defined as

τmax(XdB) = τX − τ0


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Relation between Bc and στ

• The rms delay spread and coherence bandwidth are inverselyproportional to one another, although their exact relationshipis a function of the exact multipath structure

• If the coherence bandwidth is defined as the bandwidth overwhich the frequency correlation function is above 0.9, then thecoherence bandwidth is approximately

Bc ≈ 150στ

where στ is the rms delay spread


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Relation between Bc and στ (continued)

• If the definition is relaxed so that the frequency correlationfunction is above 0.5, then the coherence bandwidth isapproximately

Bc ≈ 15στ


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Coherence Time

• Coherence time Tc is the time domain dual of Doppler spreadand is used to characterize the time varying nature of thefrequency dispersiveness of the channel in the time domain

• What is Doppler spread, Bd?

• Bd ∝ 1/Tc

• Remark: A slowly changing channel has a large coherencetime or, equivalently, a small Doppler spread


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Coherence Time (continued)

• If the coherence time is defined as the time over which the timecorrelation function is above 0.5, then it is approximated as

Tc∼= 9

16πfm=

0.179fm

where

∗ fm = maximum Doppler shift and is given by

fm = fd,max

=v

λ


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Coherence Time (continued)

• The approximation of the coherence time in the previous slideis too restrictive and a popular rule of thumb defines thecoherence time as

Tc∼=

√9

16πf2m

=0.423fm

• Note: The definition of coherence time implies that twosignals arriving with a time separation greater than Tc areaffected differently by the channel


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Types of Small-Scale Fading

• The types of small-scale fading experienced by a signalpropagating through a mobile radio channel depends on therelation between the

1. Signal parameters such as

– bandwidth– symbol period

2. Channel parameters such as

– rms delay spread– Doppler spread


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Types of Small-Scale Fading (continued)

• Based on multipath time delay spread, two types of small-scalefading are

1. Flat fading or frequency nonselective fading

2. Frequency selective fading

• Based on Doppler spread, two types of small-scale fading are

1. Fast fading

2. Slow fading


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Frequency Nonselective (flat) Fading

• Definition: If the mobile radio channel has a constant gainand linear phase response over the bandwidth Bc which isgreater than the bandwidth of the transmitted signal Bs, thenthe received signal will undergo flat fading

• In flat fading, the multipath structure of the channel is suchthat

– the spectral characteristics of the transmitted signal arepreserved at the receiver

– the strength of the received signal changes with time, due tofluctuations in the gain of the channel caused by multipath


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Flat Fading (continued)

• In a flat fading channel, all of the frequency components inSl(f) undergo the same attenuation and phase shift intransmission through the channel, which implies

– within the bandwidth occupied by Sl(f), the time varianttransfer function Hl (f, t) is a complex-valued constant inthe frequency variable


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Flat Fading (continued)

• Thus the equivalent lowpass received signal can be expressed as

rl(t) = α(t)e−jφ(t)sl(t)

where

∗ α(t) = envelope of the equivalent lowpass channel

∗ φ(t) = phase of the equivalent lowpass channel


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Transfer Function (continued)

• When α(t)e−jφ(t) is modeled as a zero-mean complex valuedGaussian random process

– the envelope α(t) is Rayleigh distributed for any fixed valueof t

– the phase φ(t) is uniformly distributed over the interval(−π, π)


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Remarks

• In a flat fading channel, the reciprocal bandwidth of thetransmitted signal is much larger than the multipath timedelay spread of the channel and thus

– hb (t, τ) can be approximated as having no excess delay −→a single delta function with τ = 0


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Remarks (continued

• Flat fading channels are known as amplitude varying channeland are sometimes referred to as narrow-band channels, sincethe bandwidth of the applied signal is narrow compared to thechannel flat fading bandwidth

• Typical flat fading channels may cause deep fades, and thusmay require 20 or 30 dB more transmitter power to achieve lowbit error rates during times of deep fades as compared tosystems operating over non-fading channel


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Summary of Flat Fading

• A signal undergoes flat fading if

Bs ¿ Bc

andTs À στ

where Bs, Bc, Ts, στ are as defined previously


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Frequency Selective Fading

• Definition: If the channel possesses a constant-gain and linearphase response over a bandwidth (coherence bandwidth) that issmaller than the bandwidth of transmitted signal, then thechannel creates frequency selective fading on the receivedsignal

– in this case, the received signal includes multiple versions ofthe transmitted waveform which are attenuated (faded) anddelayed in time, and hence the received signal is stronglydistorted by the channel


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Remarks

• Frequency selective fading is caused by multipath delays whichapproach or exceed the symbol period of the transmittedsymbol

• Frequency selective fading channels are also known as widebandchannels since the bandwidth of the signal is wider than thebandwidth of the channel impulse response

• As time varies, the channel varies in gain and phase across thespectrum of sl(t), resulting in time varying distortion in thereceived signal rl(t)


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Summary of Frequency Selective Fading

• A signal undergoes frequency selective fading if

Bs > Bc

andTs < στ

where

∗ Bs = bandwidth of the transmitted signal

∗ Ts = reciprocal bandwidth (e.g., symbol period) of thetransmitted signal

∗ στ = rms delay spread of the channel

∗ Bc = coherence bandwidth of the channel


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Remark

• A common rule of thumb is that a channel is frequencyselective if

Ts ≤ 10στ

although this is dependent on the specific type of modulationused


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Fast Fading

• In a fast fading channel, the channel impulse response changesrapidly within the symbol duration

• A signal undergoes fast fading if

Ts > Tc

andBs < Bd

where

∗ Tc = coherence time of the channel

∗ Bd = Doppler spread


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Fast Fading (continued)

• Since in a fast fading channel the coherence time of the channelis smaller than the symbol period of the transmitted signal

– this causes frequency dispersion (also called time selectivefading) due to Doppler spreading −→ leads to signaldistortion

• Viewed in the frequency domain, signal distortion due to fastfading increases with increasing Doppler spread relative to thebandwidth of the transmitted signal


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Flat and Fast Fading

• In the case of a flat fading channel, we can approximate theimpulse response to be simply a delta function (no time delay)

– a flat and fast fading channel is a channel in which theamplitude of the delta function varies faster than the rate ofchange of the transmitted baseband signal


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Frequency Selective and Fast Fading

• In the case of a frequency selective and fast fading channel, theamplitude, phases, and time delays of any one of the multipathcomponents vary faster than the rate of change of thetransmitted signal

• Remark: In practice, fast fading only occurs for very low datarates


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Slow Fading

• In a slow fading channel, the channel impulse response changesat a rate much slower than the transmitted baseband signal

• A signal undergoes slow fading if

Ts ¿ Tc

andBs À Bd


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Slow Fading (continued)

• Since in a slow fading channel, signal duration is smaller thanthe coherence time of the channel, the channel attenuation andphase shift are fixed for the duration of at least one signalinginterval

– in the frequency domain this implies that the Dopplerspread of the channel is much less than the bandwidth ofthe baseband signal

• Note: Fast and slow fading deal with the relationship betweenthe time rate of change in the channel and the transmittedsignal, and not with propagation path loss model


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Flat and Slow Fading

• When Bs ≈ 1Ts

, the conditions that the channel be frequencynon-selective and slowly fading imply that the product of στ

and Bd must satisfy the condition

στBd < 1

• The product στBd is called the spread factor of the channel

– if στBd < 1, the channel is said to be under-spread

– if στBd > 1, the channel is said to be over-spread


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Rayleigh Fading Distribution

• In mobile radio channels, the Rayleigh distribution iscommonly used to describe the statistical time varying natureof the received envelope of a flat fading signal, or the envelopeof an individual multipath component

• Remark: The envelope of the sum of two quadrature Gaussiannoise signals obeys a Rayleigh distribution


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Rayleigh Distribution

• The probability density function (pdf) of the Rayleighdistribution is given by

p(r) =

rσ2 exp

− r2

2σ2

(0 ≤ r ≤ ∞)

0 (r < 0)

where

∗ σ = rms value of the received voltage signal before envelopdetection

∗ σ2 = time-average power of the received signal beforeenvelop detection


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Rayleigh Distribution (continued)

• The probability that the envelope of the received signal doesnot exceed a specified value R is given by the correspondingcumulative distribution function (CDF)

P (R) = Prob (r ≤ R)

=∫ R

0

p(r)dr

= 1− exp− R2

2σ2


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Ricean Fading Distribution

• When there is a dominant stationary (non-fading) signalcomponent present, such as a line-of-sight propagation path,the small-scale fading envelope distribution is Ricean

• As the dominant signal becomes weaker, the Riceandistribution degenerates to a Rayleigh distribution


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Ricean Distribution

• The pdf of the Ricean distribution is given by

p(r) =

rσ2 e−

(r2+A2)2σ2 I0

−Ar

σ2

(A ≥ 0, r ≥ 0)

0 (r < 0)

where

∗ A = peak amplitude of the dominant signal

∗ I0(·) = modified Bessel function of the first kind andzero-order

∗ k = A2

2σ2 = Ricean factor


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Performance of Digital Modulation

• Goal: To evaluate the probability of error of a any digitalmodulation scheme in a slow, flat fading channel

• Recall that the flat fading channels cause a multiplicative(gain) variation in the transmitted signal s(t)


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Performance of Digital Modulation (continued)

• Since slow fading channels change much slower than theapplied modulation

– it can be assumed that the attenuation and phase shift ofthe signal is constant over at least one symbol interval

– the received signal r(t) may be expressed as

r(t) = α(t)e−jθ(t)s(t) + n(t) 0 ≤ t ≤ T

∼= α(0)e−jθ(0)s(t) + n(t)

where

∗ α(t) = gain of the channel∗ θ(t) = phase shift of the channel∗ n(t) = additive Gaussian noise


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Performance of Digital Modulation (continued)

• If θ(t) is varying slowly compared to the speed of the receiverprocessing, then

– we can estimate θ(t) and implement coherent receivers

– otherwise, we have to use non-coherent receivers


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Probability of Error

• The probability of error in slow, flat fading channels can beobtained by averaging the error in additive white Gaussiannoise (AWGN) channels over the fading probability densityfunction

• Remark: The probability of error in AWGN channels isviewed as a conditional error probability, where the condition isthat α is fixed


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Probability of Error (continued)

• For BPSK signals, the probability of error in AWGN channelsis expressed as

Pe,BPSK = Q

(√2Eb

N0

)

=12erf

(√Γb

)

where

∗ Eb = energy per bit

∗ Γb = Eb

N0= SNR


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• The probability of error in a slow, flat fading may be evaluatedas

Pe =∫ ∞

0

Pe(X)p(X)dX

where

∗ Pe(X) = probability of error for an arbitrary modulation ata specific value of signal-to-noise ratio X

∗ X = α2 Eb

N0

∗ p(X) = probability density function of X due to the fadingchannel

∗ α = amplitude values of the fading channel with respect toEb/N0


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• For Rayleigh fading channels, α has a Rayleigh distribution−→ α2 and consequently X have a chi-square distribution withtwo degrees of freedom (which is the exponential distribution)

• Thus, the pdf of X due to fading channel is expressed as

p(X) =1Γ

exp−X

Γ

X ≥ 0

where

Γ = average value of the signal-to-noise ratio

=Eb

N0α2


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• Average error probability of coherent binary PSK and coherentbinary FSK in a slow, flat Rayleigh fading channel are given by

Pe,PSK = 12

[1−

√Γ

1+Γ

](coherent binary PSK)

Pe,FSK = 12

[1−

√Γ

2+Γ

](coherent binary FSK)


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• Average error probability of differential PSK and orthogonalnon-coherent FSK in a slow, flat Rayleigh fading channel aregiven by

Pe,DPSK = 12(1+Γ) (differential binary PSK)

Pe,NCFSK = 12+Γ (non-coherent orthogonal binary FSK)


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• For large values of Eb/N0 (i.e., large values of X) the errorprobability equations may be simplified as

Pe,PSK = 14Γ (coherent binary PSK)

Pe,FSK = 12Γ (coherent FSK)

Pe,DPSK = 12Γ (differential PSK)

Pe,NCFSK = 1Γ (non-coherent orthogonal binary FSK)

• Note: At higher values of Eb/N0, Pe is a linear function of 1Γ


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Level Crossing Rate and Fade Duration

• Level Crossing Rate:

– it describes how often the envelope crosses a specified level

• Average Fade Duration:

– it describes how long the envelope remains below a specifiedlevel


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Envelope Level Crossing Rate

• Definition: The envelope level crossing rate NR at a specifiedlevel R is defined as the rate at which the envelope crosses thelevel R in the positive (or negative) going direction

• Let

∗ r(t) = received signal

∗ z(t) = envelope = |r(t)|∗ p (R, z) = joint pdf of the signal envelope z and the time

derivative of z, z at the point where z = R


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Envelope Level Crossing Rate (continued)

• The expected number of times R occurs for a given slope z andtime duration dt

NR,z = p (R, z) dzdz

• Since in small time dt, the number of times R can occur withslope z is either 1 or 0 and dz = zdt

NR,z = zp (R, z) dzdt

• The expected number of crossings NR,z(T ) of the envelopelevel R with slope z over time interval [0, T ] is given by

NR,z(T ) =∫ T

0

zp (R, z) dzdt


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• Now, since the derivative z in the positive direction will rangefrom zero to infinity, we obtain the expected number ofcrossings NR(T ) of the envelope level R over time interval[0, T ] in the positive direction (for any derivative) byintegrating NR,z(T ) over all possible derivatives

NR(T ) =∫ T

0

[∫ ∞

0

zp (R, z) dz

]dt

= T

∫ ∞

0

zp (R, z) dz


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• Thus, the expected number of crossings NR of the envelopelevel R per unit time is obtained by dividing NR(T ) by thetime interval T

NR =NR(T )

T

=∫ ∞

0

zp (R, z) dz


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• For Ricean fading, the expected number of crossings NR of theenvelope level R per unit time is expressed as

NR =√

2π (K + 1)fmρe−K−(K+1)ρ2I0

(2ρ

√K (K + 1)

)

where

∗ fm = maximum Doppler frequency

∗ K = A2

2σ2 = Ricean factor

∗ ρ = RRrms

= value of the specified level R, normalized to the

local rms amplitude of the fading envelope (i.e.,√

2σ2)


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• When the received envelope is Rayleigh distributed, K = 0

• Thus, the expected number of crossings NR of the envelopelevel R per unit time for Rayleigh fading envelope is given by

NR =√

2πfmρe−ρ2

• Note: The level crossing rate is a function of the mobile speedas is apparent from the presence of fm in the above equation


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Maximum Level Crossing Rate

• The maximum level crossing rate occurs when the derivative ofNR with respect to ρ is zero, i.e.,

dNR

dρ= e−ρ2 (

1 + 2ρ2)

= 0

⇒ ρ =1√2

• There are few crossings at both high and low levels, with themaximum rate occurring at ρ = 1/

√2

• Remark: The signal envelope experiences very deep fades onlyoccasionally, but shallow fades are frequent


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Average Fade Duration

• Definition: The average fade duration is defined as theaverage period of time for which the received signal is below aspecified level R

• For a Rayleigh fading signal, the average fade duration is givenby

τ =1

NRProb[r ≤ R]


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Average Fade Duration (continued)

• In the previous slide,

Prob[r ≤ R] =1T

∑

i

τi

= average time z(t) stays below R in one second

where

∗ τi = duration of the fade

∗ T = observation interval of the fading signal


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• The probability that the received signal r is less than thethreshold R is found from the Rayleigh distribution as

Prob[r ≤ R] =∫ R

0

p(r)dr

=∫ R

0

r

σ2e−

r2

2σ2

= 1− exp(− R2

2σ2

)

= 1− exp(−ρ2

)


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• The average fade duration as a function of ρ and fm can beexpressed as

τ =1

NRProb[r ≤ R]

=eρ2

√2πfmρ

- 1


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Remarks

• The average fade duration of a signal fade helps determine themost likely numbers of signaling bits that may be lost during afade

• Average fade duration primarily depends upon the speed of themobile, and decreases as the maximum Doppler frequency fm

becomes large assuming that ρ is fixed

v ↑→ fm ↑→ NR ↑→ τ ↓

• When the maximum Doppler fm frequency becomes small, theresults will be the other way round

Documents

Mobile Radio Propagation: Small-Scale Fading and Multi-pathsaquib/EE4365/Chapter 5/Slides.pdfEE/TE 4365, UT Dallas 2 Small-scale Fading † Small-scale fading, or simply fading describes