ARC Future Fellow at The University of Melbourne Sessional Lecturer at Monash University
August 8, 2022
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 1 / 17
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ECE5884 Wireless Communications Week 3 Workshop: Wireless Channel Models
Course outline
This week: Ref. Ch. 3 of [Goldsmith, 2005]
● Week 1: Overview of Wireless Communications
● Week 2: Wireless Channel (Path Loss and Shadowing)
● Week 3: Wireless Channel Models
● Week 4: Capacity of Wireless Channels
● Week 5: Digital Modulation and Detection
● Week 6: Performance Analysis
● Week 7: Equalization
● Week 8: Multicarrier Modulation (OFDM)
● Week 9: Diversity Techniques
● Week 10: Multiple-Antenna Systems (MIMO Communications) ● Week 11: Multiuser Systems
● Week 12: Guest Lecture (Emerging 5G/6G Technologies)
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 2 / 17
Doppler effect and delay
Figure 1: Illustration of the Doppler effect.
vc8 Dopplerfrequency:fD = λcosθ whereλ= f andc=3×10 m/s (1)
1 Doppler effect. 2 Scatters.
⎞ ⎤⎥ ⎢⎣⎝ i=0 ⎠ ⎥⎦
αi (t ) is fading (also a function of path loss and shadowing). φi (t ) depends on delay and Doppler. These two random processes are independent.
⎡⎢ ⎛ N ( t ) − 1
Receivedsignal r(t)=R⎢ ∑ αi(t)e−jφi(t)u(t−τi(t)) ej2πfct⎥ (2)
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 3 / 17
Fast/slow fading (w.r.t. time)
Figure 2: Received signal.
● In Coherence time (Tc ), channel is not varying.
1 Fast fading: Tc < Ts where Ts is the transmitted symbol duration.
2 Slow fading: Tc ≫ Ts, e.g. Shadowing (Log-normal model).
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Flat/frequency-selective fading (w.r.t. frequency)
Figure 3: Wireless channel as a filter.
● In coherence bandwidth (Bc ), channel response is not varying.
1 Flat fading: Bs ≪ Bc where Bs is the signal bandwidth.
2 Frequency-selective fading: Bs ≫ Bc , OFDM (Week 8).
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Intersymbol interference (ISI)
ISI is a form of distortion of a signal in which one symbol interferes with subsequent symbols.
(a) One symbol Tx (No ISI). (b) Four symbols Tx (ISI). Figure 4: Illustration of ISI effect.
● Send the next symbol after the delay spread, Tm, to avoid ISI. https://www.telecomhall.net/t/what- is- isi- inter- symbol- interference- in- lte/6370
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Narrowband/wideband communications
1 Narrowband communications use a narrow bandwidth; are used in a slower form of communication as we allow a longer time for a symbol.
2 Wideband communications use a higher bandwidth; apply Wifi, 4G LTE and beyond, HSPA.
● OFDM (Week 8)
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Multipath fading
1 Fast fading: Ts ≪ Tc
2 Flat fading: Bs ≪ BD
3 Narrowband comm.: Tm ≪ Ts
(a) Combined all. (b) Narrowband fading. Figure 5: Ref. Ch. 3 of [Goldsmith, 2005].
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System model
Thereceivedsignal: r(t)=hs(t)+n(t) (3)
● h− the multipath channel gain, usually a complex number; s(t)− the transmit signal with Ps power; and n(t)− the additive noise.
The received signal power: Multipath channel gain:
Channel envelop:
Pr = ∣h∣2Ps (4) h = hr + jhi = z ejθ (5)
∣h∣ = z = hr2 + hi2 (6)
● Additive white Gaussian noise: n(t ) = nr + jni ; noise power is constant for all frequencies; n(t) follows circularly symmetric complex Gaussian distribution with zero mean and N0 variance, i.e.,
n(t) ∼ CN(0,N0) where nr ∼ N(0,No/2) and ni ∼ N(0,No/2).
Instantaneous SNR: γ = Noise power = N
Signal power ∣h∣2Ps
We need distributions of ∣h∣ and ∣h∣2 – Multipath fading models!!! ECE5884 Wireless Communications @ Monash Uni. August 8, 2022
Rayleigh distribution
● Rayleigh fading is a model that can be used to describe the form of fading that occurs when multipath propagation exists with no Los component.
h=hr+jhi=zejθ and ∣h∣=z= hr2+hi2 (8)
● When hr and hi are two independent and identical distributed (i.i.d.) Gaussian random variables with mean zero and variance σ2, i.e., hr,hi ∼N(0,σ2),
1 The average envelope power is Ωp = 2σ2. √
hr2 + hi2 is Rayleigh distributed; 22
2 the envelop ∣h∣ = z =
fZ(z)= Ω e Ωp andFZ(z)=1−e Ωp
2z −z −z p
3 the power ∣h∣2 is Exponentially distributed;
1−t −t fZ2(t)= Ω e Ωp andFZ2(t)=1−e Ωp
ECE5884 Wireless Communications @ Monash Uni.
August 8, 2022
Rician distribution
● The channel has a LOS component with a much larger signal power than the other multipath components.
● hr ∼ N(mr,σ2) and hi ∼ N(mi,σ2);
● 2σ2 is the average power in the non-LOS multipath components and
s2 = mr2 + mi2 is the power in the LOS component.
1 Average envelope power: Ωp = s2 + 2σ2
2 the envelopis Rician/Ricean/Rice distributed;
z −(z2+s2) zs
fZ(z)=σ2e 2σ2 I0(σ2) (11)
3 The Rice factor K (fading parameter): K = s2 where K = 0 for no LoS; 2σ2
K → ∞ for no scatter; and a small K implies severe fading.
fZ(z)= wheres2=KΩp andσ2= Ωp
⎛ K(K +1)⎞
e Ωp I0 2z
2(K +1)z −K−(K+1)z
K +1 2(K +1)
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022
Nakagami-m distribution
1 The Nakagami distribution was selected to fit empirical data and is known to provide a closer match to some measurement data than either the Rayleigh, Ricean, or log-normal distributions.
2 the envelop ∣h∣ is Nakagami-m distributed; m2m−1 2
mz−mz 1 fZ(z)=2( ) e Ωp ;m≥
3 Average envelope power: Ωp ● m = 1: Rayleigh distribution.
● m = 1/2: a one-sided Gaussian distribution ● m → ∞: approaches an impulse (no fading).
● m ≈ (K +1)2 : approximation for Rician distribution. (2K +1)
4 the power ∣h∣2 is Gamma distributed;
mz−mz 1 fZ2(z)=( ) e Ωp ;m≥
Ωp Γ(m) 2 ECE5884 Wireless Communications @ Monash Uni.
August 8, 2022
SNR outage probability
● The SNR outage probability is the probability that the SNR γ falls below a certain predetermined threshold SNR γth
Pout =Pr[γ<γth] (15)
∣h∣2Ps 2 N0γth N0γth
=Pr[ N <γth]=Pr[∣h∣ < P ]=F∣h∣2 ( P ) (16)
● For Rayleigh fading (use (10))
Ps γth N0 γth 1
Pout =1−e− 2σ2 =1−e−(2σ2 Ps ) =1−e−(2σ2 γ ̄ ) (17)
γth 1 γth 1
Whenγ ̄→∞;Pout →1−(1−2σ2γ ̄)=2σ2γ ̄ Asymptoticanalysis
(18) where γ ̄ = Ps (we sometime call this as the average transmit SNR!).
● Similarly, you can evaluate the SNR outage probabilities for Rician and
Nakagami-m fading channels!
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SNR outage probability
102 101 100
10-1 10-2 10-3 10-4 10-5
Outage probability
Analytical
Asymptotic
-10 0 10 20 30 40 50
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Figure 6: Example.
SNR outage probability
102 101 100
10-1 10-2 10-3 10-4 10-5
Outage probability
Analytical
Asymptotic
-10 0 10 20 30 40 50
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Figure 7: Example.
References
A. Goldsmith, Wireless Communications, Cambridge University Press, USA, 2005.
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Thank You! See you again
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