Nov 13, 2013 - All the forms have their own advantages and disadvantages. In any particular ..... P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
ELG4179: Wireless Communication Fundamentals © S.Loyka
Diversity Combining Techniques When the required signal is a combination of several waves (i.e, multipath), the total signal amplitude may experience deep fades (i.e, Rayleigh fading), over time or space. The major problem is to combat these deep fades, which result in system outage. Most popular and efficient technique for doing so is to use some form of diversity combining. Diversity: multiple copies of the required signal are available, which experience independent fading (or close to that). Effective way to combat fading. Basic principle: create multiple independent paths for the signal, and combine them in an optimum or near-optimum way.
Lecture 11
13-Nov-13
1(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Types of Diversity Space diversity: Antennas are separated in space. Frequency diversity: Multiple copies are transmitted at different frequencies. Time diversity: Multiple copies are transmitted over different time slots. Polarization diversity: Multiple copies have different field polarizations. For all types of diversity, multiple signal copies must be uncorrelated (or weakly correlated, corr. coeff. ≤ 0.5 ) for diversity combining to be most effective. Hence different types of diversity are normally efficient in different scenarios. All the forms have their own advantages and disadvantages. In any particular case, some forms are better than the other. Example: if the channel is static (or-slow-fading), time diversity is not good. If the channel is frequency-flat, frequency diversity is not good. A diversity form is efficient when the signals are independent (uncorrelated). Multiple signals have to be combined in a certain way.
Lecture 11
13-Nov-13
2(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Combining techniques • Selection combining (SC). • Maximum ratio combining (MRC). • Equal gain combining (EGC). • Hybrid combining (two different forms) Micro-diversity and macro-diversity.
Lecture 11
13-Nov-13
3(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Selection combining (SC) Key idea: at any given moment of time, pick up the branch with the largest SNR. h1 h2 hn
System model: y = hx + ξ
(11.1)
Output: select the branch with the largest SNR.
Lecture 11
13-Nov-13
4(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Selection Combining 10
dB
0
10
20
30
0
20
40
60
time
branch 1 branch 2 output
Lecture 11
13-Nov-13
.
5(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Analysis of a selection combiner Consider the SC in a frequency-flat slow-fading Rayleigh channel. Assume that some diversity form provides N independent paths (each is i.i.d. Rayleigh-fading). Instantaneous SNR in branch i is,
γi =
E 2 hi N0
(11.2)
where hi is the channel (complex) gain, E is symbol energy and N 0 is the noise spectral power density (assumed to be the same in all the branches). The PDF of γ i is
1 ρ ( γi ) = e γ0 where γ 0 = γ i =
−
γ γ0
(11.3)
E 2 hi is the average SNR. N0
The CDF (i.e., the probability that γ i < γ ) is γ
Fγ i ( γ ) = ∫ ρ ( γ i )d γ i = 1 − e −γ / γ 0 0
Lecture 11
13-Nov-13
(11.4)
6(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
This is also outage probability of i-th branch. The output SNR is
γ out = max i γ i
(11.5)
The SC outage probability is
Pout = Pr{γ out < γ} = Pr {γ1 ,.., γ N < γ}
(
N
= FN ( γ ) = ∏ Fγ i ( γ ) = 1 − e i =1
)
−γ γ 0 N
(11.6)
Asymptotically, for γ very simple!
Lecture 11
13-Nov-13
10(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Maximal Ratio Combining (MRC) Key idea: use linear coherent combining of branch signals so that the output SNR is maximized
h1
w1
h2
w2
hn
wn
The equivalent baseband model is as follows. The individual branch signals are
xn = A ⋅ hn + ξ n
(11.13)
where A - complex envelope (amplitude), hn - channel
(
)
(complex) gain , ξ n ~ CN 0, σ02 is AWGN. Note that hn acts as multiplicative noise. Lecture 11
13-Nov-13
11(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
The output of the combiner is N
xout = ∑ wn xn = A∑ wn hn + ∑ wn ξn n =1
n
(11.14)
n
signal
noise
where wn are the combining weights. The signal and noise components are given by 1st and 2nd term correspondingly. The signal and noise power at the output are: 2
Ps = A∑ wn hn n
=
1 2 A ∑ wn hn 2 n
2
(11.15)
2
Pξ =
∑ wnξn n
where σn = ξn 2
2
= ∑ wn σn2 2
n
is the branch noise power. Output SNR is 2
∑ wn hn
SNRout
Ps 1 2 n = = A Pξ 2 ∑w
2 2 n σn
(11.16)
n
We can maximize it using the Lagrange multipliers or better, Cauchy-Schwarz inequality.
Lecture 11
13-Nov-13
12(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Cuachy-Schwartz inequality: 2 * a ∑ n bn ≤ ∑ an n
2
∑ bn
n
2
(11.17)
n
With the equality achieved when an* = cbn Using (11.17), the best combining weights are
wn = hn* / σ n2
(11.18)
and the output SNR of the best combiner is
γ out = ∑ γ n
(11.19)
n
where
γn =
A2 hn 2σ2n
2
(11.20)
is n-th branch SNR. The resulting combiner is called MRC. It is the best combiner in terms of the SNR. Q.: prove (11.18) !
Lecture 11
13-Nov-13
13(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Note: wn = hn* / σ n2 is true for any channel, not necessarily Rayleigh. When all the branch noise powers are equal,
hn* σn = σ0 → wn = , γ out = N γ 0 h
(11.21)
The branch gain is proportional to the channel gain +coherent combining. Note N-fold increase in average SNR (but this is not the main gain!). Q.: explain it!
Lecture 11
13-Nov-13
14(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Outage Probability Consider a Rayleigh channel and assume there is no correlation between branches (i.i.d. channel gains). The outage probability can be expressed as
Pout = 1 − e
−
γ N γ0
( γ γ 0 )k −1 ∑ ( k − 1)! k =1
(11.22)
Asymptotically, for γ γ 0 ≪ 1 (small Pout ),
Pout ≈
( γ γ0 ) N!
N
γ = cN γ0
N N ~ Pout ,1 ~
1
( γ0 )
N
(11.23)
Diversity order = N (prove it!) Q.: Diversity gain?
Lecture 11
13-Nov-13
15(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
MRC Outage Probability
MRC 1
N=1,2,3,4,6 0.1
d =? 0.01
1 .10
3
1 .10
4
40
30
20
10
0
. 10
γ / γ0
Q.: compare to SC and EGC!
Lecture 11
13-Nov-13
16(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
MRC PDF
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
Lecture 11
13-Nov-13
17(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Implementation Issues Combiner complexity varies significantly from type to type ( as well as performance). Overall, there exists performance/complexity trade-off. Highperformance combiner (MRC) is difficult to implement; simpler combiner (SC) does not perform that well as MRC, EGC is in between. Selection diversity: a branch with the strongest signal is used; in practice - with the largest signal + noise power since it is difficult to measure SNR alone. This must be done on continuous basis—may be difficult. Can be implemented with some time constant or as scanning (switched) diversity. SC is the simplest technique. Maximum ratio combining: coherent technique, i.e., signal’s phase has to be estimated. All the signals are co-phased and weighted according to their signal voltage to noise power ratios. It requires for a lot of circuitry ( individual receivers) Provides the best performance. DSP makes it practical. The output may be acceptable even when all the inputs are not. Equal gain combining: compromise. Less complexity than MRC (all the weights are equal), but co-phasing is till required. Can still produce acceptable output from unacceptable inputs. If the individual branch signals are not independent (correlated), the performance degrades.
Lecture 11
13-Nov-13
18(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Average BER Improvement Using Diversity Combining Instantaneous SNR is a R.V. Hence, BER is a RV as well. Introduce average BER (frequency flat, slow fading), ∞
Pe = ∫ Pe ( γ ) ρ ( γ ) d γ
(11.24)
0
where ρ ( γ ) is a pdf of γ at the output of a diversity combiner. Recall that the BER can be expressed in many cases as
Pe ( γ ) = Q
(
αγ
)
(11.25)
where α = 2 for BPSK and QPSK, α = 1 for BFSK (all detected coherently) and sub-optimal DPSK. Other modulation formats have similar BER expressions (or bounds) . Hence, we evaluate average BER based on this expressions.
Lecture 11
13-Nov-13
19(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Consider N-branch MRC (the best one), N −1
i
γ −γ / γ 0 FN ( γ ) = 1 − e ∑ γ / i! i =0 0 ρ(γ ) =
(11.26)
dFN ( γ ) γ −γ / γ 0 = e dγ ( N − 1)!γ 0 N N −1
The average BER of a coherent demodulation,
1 Pe = (1 − µ ) 2
N N −1
∑
i =0
i CN −1+i
1 1 + µ ( ) 2
i
(11.27)
where µ = αγ 0 / ( 2 + αγ 0 ) , Cnk = n !/ ( k !( n − k )!) . Asymptotically, γ 0 ≫ 1, N
1 1 N Pe ≈ C ≪ 2 N −1 2 αγ 2αγ 0 0
(11.28)
Diversity order = N .
Lecture 11
13-Nov-13
20(23)
ELG4179: Wireless Communication Fundamentals © S.Loyka
The average BER for MRC and the coherent BPSK receiver. 1
Perror
0.1
0.01
1 .10
3
1 .10
4
0
10
n=1 n=2 n=3 n=4 n=5
Lecture 11
20 SNR, dB
30
40
1 1 Pe ( γ 0 ) ≈ Pout
