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 / 24
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ECE5884 Wireless Communications Week 9: 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 / 24
Introduction
● Independent signal paths have a low probability of experiencing deep fades simultaneously.
● Diversity: is to send the same data over independent fading paths/links. These independent paths/links are combined in such a way that the fading of the resultant signal is reduced.
Figure 1: Information symbols are passed through multiple links, each of which fades independently.
● Reliable communication is possible as long as one of the links is strong.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 3 / 24
Received signal (with single antenna)
Example:h1 =−1.22+j0.67;h2 =0.5+j2.3;h3 =1.2−j0.7;h4 =0.45−j2.2
● If only Link 1 is available:
r =h1s+n⇒γ1 = ∣h1∣2Ps = ∣1.4∣2Ps
r =(h1 +h2 +h3 +h4)s+n⇒γall = ∣h1+h2+h3+h4∣ Ps = ∣0.9∣ Ps
● If all Links are available:
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 / 24
Received signal (with multiple antennas)
Figure 2: Single antenna Tx with multiple antennas Rx.
● If the antennas are spaced sufficiently far apart, it is unlikely that they
all experience deep fades at the same time.
● h1,h2,h3,h4 are random values which change every coherence time.
● Example: By selecting the antenna with the strongest signal, a technique known as selection combining, we obtain a much better signal than if we had just one (fixed) antenna.
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Multiple antennas techniques
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 6 / 24
● A diversity scheme refers to 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 / 24
Receiver Diversity – System Model
Figure 3: Linear combiner at the receiver with M-branch diversity.
● Linear combiner : the output of the combiner is just a weighted sum
of the different fading paths or branches.
● CSI at Rx: The complex fading of the ith branch is hi = ri ejθi
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 8 / 24
Branch coherent detection
Figure 4: Branch coherent detection.
● The receiver knows hi s, i.e., amplitude: ri = ∣hi ∣ and/or phase: θi .
● Combining more than one branch signal requires co-phasing, where the phase θi of the ith branch is removed through multiplication by
αi = ai e−j θi for some real-valued ai .
Without co-phasing, the branch signals would not add up coherently
in the combiner.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 9 / 24
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 .
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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 ith channel:
γi = ∣hi∣2Ps = ∣hi∣2γ ̄ = giγ ̄; i = 1,⋯,M (2)
3 Channel distribution: circularly symmetric complex Gaussian random variables with zero mean and unit variance hi ∼ CN (0, 1)
● ∣hi∣∼Rayleighdistribution,i.e.,
f∣hi ∣(x) = 2x e−x2
● gi = ∣hi ∣2 ∼ Exponential distribution, i.e.,
fgi (x) = e−x
● SNR γi = gi γ ̄ ∼ Exponential distribution, i.e.,
fγi(x)= γ ̄e γ ̄ andFγi(x)=1−e γ ̄ ECE5884 Wireless Communications @ Monash Uni.
September 19, 2022
Selection Combining (SC)
1 Selection combiner outputs the signal on the branch with the highest SNR.
2 As only one branch is used at a time, SC requires just one receiver that is switched into the active antenna branch.
3 End-to-end SNR of SC: the path output from the combiner has an SNR equal to the maximum SNR of all the branches.
γSC = max (γ1,⋯,γM) i ∈{1,⋯,M }
4 Selected antenna index
i∗ =arg 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)
M M ∣hi∣2Ps2
=∏Pr(γi <γth)=∏Fγi(γth)whereγi = N0 =∣hi∣ γ ̄=giγ ̄ (8)
● ∣hi ∣ is the multipath fading channel, e.g., Rayleigh, Rician, Nakagami-m.
● For Nakagami-m fading channels:
m m x2m−1 mx2 1
f∣hi∣(x)=2(Ω) Γ(m)e− Ω ;m≥2 m m xm−1 mx 1
(9) (10) (11)
fgi(x)=(Ω) Γ(m)e−Ω ;m≥2 mmxm−1 mx 1
fγi(x)=(Ωγ ̄) Γ(m)e−Ωγ ̄;m≥2 Γ(m, mx )
Fγ (x)=1− Ωγ ̄ i Γ(m)
ECE5884 Wireless Communications @ Monash Uni.
September 19, 2022
SC: Outage probability
● The SNR outage probability over Nakagami-m fading channels
Po,SC =∏Fγi(γth) (13)
= [Fγi (γth)]M for i.i.d. channels (14)
⎡⎢ Γ(m, mγth )⎤⎥M
= ⎢1 − Ωγ ̄ ⎥ (15)
⎢⎣ Γ(m) ⎥⎦
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022
SC: Asymptotic Analysis
● The SNR outage probability behavior at high SNR regime, i.e., γ ̄ → ∞. Let’s use Ω = 1.
⎡⎢ lim P = lim ⎢1−
Γ(m, mγth )⎤⎥M γ ̄ ⎥
γ ̄→∞ o,SC γ ̄→∞ ⎢⎣ Γ(m) ⎥⎦ ● Byusinglimx→0Γ[n,x]≈Γ[n]−xn,
⎥ =( m−1 m) γ ̄ γ ̄ → ∞ ⎢ ⎢ ⎢ Γ ( m ) ⎥ ⎥ ⎥ Γ ( m )
mγth m ⎤M ⎢(Γ[m]−(γ ̄))⎥ mγM
limP ≈⎢1− m
o,SC⎢ ⎥th−mM
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 (18) where Gc is termed the coding gain or array gain, and Gd is referred
to as the diversity gain, diversity order, or, simply diversity.
2 The diversity order Gd determines the slope of the outage versus average SNR curve, at high SNR, in a log-log scale.
3 The array gain Gc (in dB) determines the shift of the curve in SNR relative to a benchmark outage curve of γ ̄−Gd .
4 When the diversity order equals the number of independent fading paths that are combined via diversity, the system is said to achieve full diversity order.
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 16 / 24
SC: 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 (19) where Gc is termed the coding gain or array gain, and Gd is referred
to as the diversity gain, diversity order, or, simply diversity. 2 Previous example:
lim Po,SC ≈ ( th ) γ ̄−mM
γ ̄ → ∞ Γ ( m )
● Diversity order: Gd = m M which is full diversity order.
● Arraygain:Gc =(
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.
γMRC =∑γi (21)
ECE5884 Wireless Communications @ Monash Uni. September 19, 2022 18 / 24
MRC: Outage probability
● The SNR outage is
Po,MRC = Pr(γMRC < γth) = Pr(∑γi < γth) (22)
● 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
● a chi-square distribution with 2M degrees of freedom, expected value γ ̄MRC = Mγ ̄ and variance 2Mγ ̄.
● a gamma distribution with shape parameter M and scale parameter γ ̄.
fγMRC(x)= γ ̄M(M−1)!
xM−1e− x γ ̄
(x) = 1 − γ ̄ = 1 − e− x Γ(M)
γ ̄ (24) k!
ECE5884 Wireless Communications @ Monash Uni.
September 19, 2022
MRC: Outage probability
● The SNR outage probability over i.i.d. Rayleigh fading channels
Po,MRC =Pr(∑γi <γth)
γ M−1 (γth )k
γ ̄ =1−e−th ∑ γ ̄
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) (27) i=1 i=1
4 The distribution PDF and CDF of γEGC do not exist in closed form for M > 2.
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Threshold Combining
1 Select the the first signal whose SNR is above a given threshold γt .
2 Once a branch is chosen, the combiner outputs that signal as long as
the SNR on that branch remains above the desired threshold.
3 If the SNR on the selected branch falls below the threshold, the combiner switches to another branch (e.g., switch randomly to another branch).
4 There are several criteria the combiner can use for determining which branch to switch. E.g.,
● Switch-and-stay combining (SSC): Switching when the SNR falls below a threshold does not always select the branch with the highest SNR.
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References
A. Goldsmith, Wireless Communications, Cambridge University Press, USA, 2005.
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Thank You! See you again
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