Figure 8.4 shows a block diagram of a composite microscopic-plus-macroscopic diversity system in which transmission from a mobile is received by N different base stations. Each station employs an L-branch microscopic diversity system, which may employ any of the diversity-combining techniques discussed previously, and produces one output per base station. Thus N base stations produce a total of N outputs. A macroscopic diversity scheme is then used to produce one output. In principle, the macroscopic diversity scheme may use any one of the previous diversity-combining schemes to produce one output from N branches.
In this section, a scheme in which a selection diversity is employed to select one of the N branches is analyzed [Tur91]. Assuming that the signals on N branches are log-normally distributed, the pdf of the N-branch selection-diversity scheme is given by
(8.8.12)
where F(.) is the cumulative normal distribution function.
f e
L
MR
L γ L
γ
γ γ
( )= (−1−− )
1
Γ
Γ !
P P f d
L k L k
e e
k L k L
Γ MR
Γ Γ
Γ
( )= ( ) ( )
= − − +
( − ) ( + )
∞
= − +
∫
∑
γ γ γ
π
γ 0
1 1 2
1 2
1
2 2
2 !1 0 5.
P P f d
L k
L k d
e e
k L
L k
d
= ( ) ( )
= − − +
( ) ( − ) ( + ) −
( − )
∞
= − + 1
∞
∫
∑ ∫
Γ Γ Γ
Γ Γ
Γ
Γ Γ Γ
Γ 0
1 0 2
2 2
1 2
5 1
2
2 10 1 2
10 2
ln ! exp log
πσ σ
f N
SD F
d d
N
Γ Γ
Γ
Γ Γ Γ Γ
( )= ( ) −( − )
−
10 −
10 2
10 2
2 10
2
1
σ ln πexp logσ logσ
The average BER to include the shadowing effect may be calculated by averaging the conditional BER at the output of microscopic diversity combiner, that is,
(8.8.13)
where Pe(Γ) denotes the average BER at the output of microscopic diversity combiner for a given mean SNR. For CFSK system operating in the Rayleigh fading environment, for SC and MRC, it is given by (8.8.7) and (8.8.10), respectively.
Let PSCMe and PMRMe denote the average BER when a composite system uses SC as mac- roscopic diversity with SC and MRC as microscopic diversity , respectively. Using (8.8.7) in (8.8.13) along with (8.8.12) yields [Tur91]
(8.8.14)
and using (8.8.10) in (8.8.13) yields [Tur91]
(8.8.15)
FIGURE 8.4
Block diagram of a macroscopic diversity combiner.
M Microscopic Diversity Combiner
Base 1 1
L M
Microscopic Diversity Combiner
Base 2 1
L M
Microscopic Diversity Combiner
Base N 1
L M
N Branch Macroscopic Diversity Combiner
Output Output
Output Base N Output Base 1
Base 2
Pe Pe f d
=∫∞ ( ) ( )Γ ΓSD Γ Γ 0
P L
k
N
k F d
e
SCM k
k L
d d
N
= ( )−
+
−( − )
−
=
∞ −
∑
∫
1 5
10 2 1
1 2
10 2
10
0
0
2 2
1
σ π
σ σ
ln
exp log log
Γ Γ
Γ Γ Γ Γ Γ
P
N L k
L k
F d
e MRM
k L
L k
d d
N
= − − +
( − )
( + ) −
( − )
−
=
− +
∞ −
∑
∫
1 2
5 1
2
2 10
1 2
10 2
10
1
1 0 2
2 2
1
Γ
Γ Γ
Γ Γ Γ Γ Γ
πσ
σ σ
ln !
exp log log
Notation and Abbreviations
AWGN additive white Gaussian noise BER bit error rate
BPSK binary phase shift keying CDC cascade diversity combiner
CFSK coherent orthogonal frequency shift keying DPSK differentially binary phase shift keying EGC equal gain combiner
GSC generalized selection combiner MGF moment generating function MRC maximum ratio combiner
NCFSK noncoherent orthogonal frequency shift keying OC optimal combiner
cdf cumulative distribution function pdf probability density function RV random variable
SC selection combiner
SDC switched diversity combiner
SIR signal power to interference power ratio CS channel gain vector for signal
CIj channel gain vector for jth interference Fγ cdf of γ
fγ pdf of γ
fˆγ pdf of γ in weight errors Ii total interference power
IOC total interference power at the output of OC IEG interference power at the output of EGC IMR interference power at the output of MRC Ij
– mean power due to jth interference, identical on all branches
–I
mean interference power due to identical interferences on all branches Iij instantaneous power on ith branch due to jth interference
K number of interferences
L number of branches
LC number of selected branches L(f) Laplace transform of f
Mx MGF of x
m Nakagami fading parameter N uncorrelated noise power
ni noise on ith channel n(t) noise vector
P( ) probability of ( ) Pe average BER Pe(γ) conditional BER
PSCe average BER at output of SC PGSe average BER at output of GSC
PSC2e average BER at output of two-branch GSC PSC3e average BER at output of three-branch GSC PCDe average BER in CDC
PMRe average BER in MRC
PˆMRe average BER in MRC with weight errors PSWe average BER in SDC
POCe average BER in OC PCDo outage probability of CDC PEGo outage probability of EGC PSCo outage probability of SC PSWo outage probability of SDC PGSo outage probability of GSC PMRo outage probability of MRC
PˆMRo outage probability of MRC in weight errors POCo outage probability of OC
pIj power of jth interference source pS power of signal source
qij amplitude of the jth interference received on ith branch R array correlation matrix
RS,RI,RN array correlation matrix of signal, interference, and noise only, respectively r signal amplitude
ri signal amplitude received on ith branch SC2 selection combiner with two branches selected SC3 selection combiner with three branches selected S–
mean signal power identical on all branches Si signal power on ith branch
SOC signal power at output of OC SEG signal power at output of EGC SMR signal power at output of MRC wi weight on the ith branch
w weight vector
woc weight vector of optimal combiner xi(t) received signal on ith branch x(t) array signal vector
y(t) combiner output.
Γ mean SNR at a branch ΓEG mean SNR of EGC
ΓCD mean SNR at output of CDC ΓSC mean SNR at output of SC ΓGS mean SNR at output of GSC ΓMR mean SNR at output of MRC Γi mean SNR at ith branch α inverse of Γ
αi,θi channel attenuation and phase on ith branch α0 an arbitrary constant
ψij phase of jth interference received on ith branch ψr characteristic function of an RV r
θi(t) signal phase on ith branch
φx cumulant generating function of x
γ SNR
γ0 threshold value of SNR γl SNR of lth branch
γ(l) ordered SNR of lth branch γ(l) mean value of γ(l)
γCD SNR of CDC
γGS SNR of GSC γSC SNR of SC
ξ0 threshold value of power
ξˆ0 optimum value of threshold power ρ correlation coefficient
à signal power to interference power ratio à0 threshold value of SIR
àSC SIR of SC àEG SIR of EGC àSW SIR of SDC
à˜ average signal power to average interference power ratio
à˜EG average signal power to average interference power ratio of EGC àMR
– mean SIR of MRC
àOC
– mean SIR of OC
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