ARC Future Fellow at The University of Melbourne Sessional Lecturer at Monash University
September 19, 2022
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 1 / 18
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ECE5884 Wireless Communications
Week 9 Workshop: Diversity Techniques (Multiple-Antenna Systems)
Course outline
This week: Ref. Ch. 7 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: Multiple-Antenna Systems: 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. September 19, 2022 2 / 18
● Assignment #2 in this week
● Guest lecture on 10th Oct – Week 11 (please attend everyone!)
● More info about the final exam – closed book and testing fundamentals (must know procedures!)
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 3 / 18
Received signal with single antenna
h1 = −1.22 + j0.67 = 1.4ej2.6
= 2.3ej1.4 = 1.4e−j0.5
h2 = 0.5 + j2.3 h3 = 1.2 − j0.7 h4 = 0.45 − j2.2
= 2.2e−j1.4 ● OnlyLink1:r1 =h1s+n1 ⇒γ1 = ∣h1∣2Ps =1.42γ ̄
N0 AllLinks:r=(h1+h2+h3+h4)s+n⇒γall =∣h1+h2+h3+h4∣2Ps =0.92γ ̄
γ1 > γall – Do we really get benefits of having multiple paths?
We need a smarter receiving architecture!
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 4 / 18
Received signal with multiple-antenna co-phasing
● Co-phasing: The phase θi of the ith branch (hi = riejθi ) is removed through multiplication by e−jθi .
∣h1∣ = 1.4; ∣h2∣ = 2.3; ∣h3∣ = 1.4; ∣h4∣ = 2.2
● OnlyLink1:r1 =h1s+n1 ⇒γ1 = ∣h1∣2Ps =1.42γ ̄=2γ ̄
AllLinks(EGC):r =∑4 ∣h∣s+∑4 n ⇒γ = (∑4i=1 ∣hi∣)2Ps =13.3γ ̄ i=1 i i=1 i EGC 4N0
Selection combining (SC): max(∣hi ∣) ⇒ γSC = 2.32 Ps N0
MRC:γMRC==(∑4i=1∣hi∣)2Ps =53.3γ ̄ N0
September 19, 2022
ECE5884 Wireless Communications @ Monash Uni.
Multiple antennas techniques
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 6 / 18
● A diversity scheme: a method for improving the reliability of a message signal by using two or more communication channels with different characteristics.
● Diversity techniques mitigate the effect of multipath fading – microdiversity
● We need independent fading paths: use antenna array where the elements of the array are separated in distance – space diversity.
1 multiple receive antennas – receiver diversity
2 multiple transmit antennas – transmitter diversity
● Channel state information (CSI) availability:
1 CSI at Rx (will focus more on this!)
2 CSI at Tx
● We also have Time Diversity and Frequency Diversity.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 7 / 18
Diversity/combining techniques
Techniques entail various trade-offs between performance/complexity.
1 Selection Combining (SC): the combiner outputs the signal on the branch with the highest SNR.
2 Maximal-Ratio Combining (MRC): the output is a weighted sum of all branches, and the weights (αi s) are determined to maximize the SNR.
3 Equal-Gain Combining (EGC): co-phases the signals on each branch and then combines them with equal weighting.
4 Threshold Combining: outputting the first signal whose SNR is above a given threshold γT .
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 8 / 18
Selection Combining (SC)
1 Received signal over the i th channel, i ∈ {1, ⋯, M }:
yi(t)=hi s(t)+ni(t)=riejθi s(t)+ni(t);i =1,⋯,M (1)
2 Received SNR over the i th channel:
γi = ∣hi∣2Ps = ∣hi∣2γ ̄ = giγ ̄; i = 1,⋯,M (2)
3 Combiner outputs the signal on the branch with the highest SNR.
4 End-to-end SNR of SC:
5 Selected antenna index
i∗ =arg max (γ1,⋯,γM)
i ∈{1,⋯,M }
γSC = max (γ1,⋯,γM) i ∈{1,⋯,M }
ECE5884 Wireless Communications @ Monash Uni.
September 19, 2022
SC: Outage probability
● The SNR outage is
Po,SC = Pr(γSC < γth) = Pr(max(γ1,⋯,γM) < γth)
= ∏ Pr (γi < γth) = ∏ Fγi (γth) = [Fγi (γth)]M for i.i.d. channels (5)
● ∣hi ∣ is the multipath fading channel, e.g., Rayleigh, Rician, Nakagami-m.
● For Nakagami-m fading channels: m m−1
Γ (m, mx )
f (x)=(m) x e−mx andF (x)=1− γi Ωγ ̄ Γ(m) Ωγ ̄ γi
● The SNR outage probability over Nakagami-m fading channels
⎡⎢ Γ(m, mγth )⎤⎥M
Po,SC =⎢1− Ωγ ̄ ⎥ (7)
⎢⎣ Γ(m) ⎥⎦
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022
Diversity order and array gain
1 For large enough SNR (γ ̄ → ∞), the outage probability Po as a function of γ ̄ can be written as
Po ≈ (Gc γ ̄)−Gd or Po ≈ Gc γ ̄−Gd ● Gc:thecodinggainorarraygain
● Gd : the diversity gain, diversity order, or simply diversity.
2 If Gd = the number of independent fading paths that are combined
via diversity, the system is said to achieve full diversity order.
3 Asymptotic analysis for SC: By using limx →0 Γ[n, x ] ≈ Γ[n] − x n ,
⎡ mγth m ⎤M ⎢(Γ[m]−(γ ̄))⎥ mγM
⎥ =( m−1 m) γ ̄ o,SC⎢ ⎥th−mM
limP ≈⎢1− m
γ ̄ → ∞ ⎢ ⎢ ⎢ Γ ( m ) ⎥ ⎥ ⎥ Γ ( m )
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022
Maximal-Ratio Combining (MRC)
1 MRC output is a weighted sum of all branches, so the αi are all nonzero, and the weights are determined to maximize the combiner output’s SNR.
2 For a branch with hi = riejθi ,
● The signals are co-phased: e−jθi
● The optimal weight to maximize SNR is: ai = ri ● αi =rie−jθi
3 End-to-end SNR of MRC: the SNR of the combiner output is the sum of
SNRs on each branch.
M γMRC =∑γi
4 The SNR outage is
Po,MRC = Pr (γMRC < γth) = Pr (∑ γi < γth) = FγMRC (γth)
We need the CDF of γMRC .
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022
MRC: Outage probability
● For Rayleigh fading channels: i.i.d. Rayleigh fading on each branch with equal average branch SNR γ ̄, the distribution of γMRC (which is a sum of i.i.d. exponential RVs) is
∑ γ ̄ (13)
fγMRC(x)= γ ̄M(M−1)! Γ(M, x )
xM−1e− x γ ̄
γ γ ̄ F (x) = 1 − γ ̄ = 1 − e− x
The SNR outage probability of MRC over i.i.d. Rayleigh fading channels
γ M−1 (γth )k P =1−e−th∑γ ̄
k=0 k! ECE5884 Wireless Communications @ Monash Uni.
September 19, 2022
Equal-Gain Combining (EGC)
1 EGC co-phases the signals on each branch and then combines them with equal weighting.
2 For a branch with hi = riejθi ,
● The signals are co-phased: e−jθi
● The weight is: ai = 1 ● αi =e−jθi
3 End-to-end SNR of EGC: the SNR of the combiner output is Ps M 2 γ ̄ M 2
γEGC = MN0 (∑∣hi∣) = M (∑ri) (15) i=1 i=1
4 The distribution PDF and CDF of γEGC do not exist in closed form for M > 2.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 14 / 18
Numerical results (compare diversity techniques)
Outage probability vs average SNR
ngle Antenn
-10 -5 0 5 10 15
Avg. SNR (in dB)
Figure 1: Comparison of SC, MRC, EGC and no diversity.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 15 / 18
Outage probability
Numerical results (Diff. number of antennas)
Outage probability vs average SNR
-10 -5 0 5 10 15
Avg. SNR (in dB)
Figure 2: Comparison of multiple antennas receiver for MRC.
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ingle Anten
Outage probability
References
A. Goldsmith, Wireless Communications, Cambridge University Press, USA, 2005.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 17 / 18
Thank You! See you again
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