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jxu
Characteristics of the radio channelmodels;Envelope and spatial correlation;Level crossing rates and fade duration
08, Dec. 2001
jxu
Scattering environments in mobile radiosystem
The height and placement of the BSantennas
In the macro-cell:a fairly small angle ofarrival (AoA) spread
in the micro-cell:a larger AoA spread (thespecular component +…´) Riceandistributed envelope
Local scatterers usually surround MS.
No direct LoS component. Two-
dimensional isotropic scattering model.
Rayleigh distributed envelope
jxu
Received signal correlation and spectrum1(4) Flat fading channel
tfgtftgtr ctqcI ππ 2sin2cos)()( )(−=
∑=
Φ=N
nnnI tCtg
1
)(cos)( ∑=
Φ=N
nnnQ tCtg
1
)(sin)(
The auto-correlation of r(t) is
τπτφτπτφττ cgQgIcgIgIrr fftrtrE 2sin)(2cos)())()(()( −=+=Φ
[ ] [ ] [ ]
[ ] [ ] ∑=
=+=Ω
Ω=
Ω=+=
N
nnQIp
nDp
mp
IIgIgI
CtgEtgE
fEfEtgtgE
1
222
,,
)()(
)2cos2
)cos2cos(2
)()()( τπθτπττφ θθθτ
[ ] [ ])cos2sin(2
)()()( , θτπττφ θθτ mp
QIgIgQ fEtgtgEΩ
=+=
jxu
Received signal correlation and spectrum2(4) for 2-D isotropic scattering and an isotropic antenna
∫− =Ω
=π
πθθτπ
πτφ 0)cos2sin(
2
1
2)( dfm
pgIgQ
)2(2
)sin2cos(1
2)()()cos2cos(
2)( 00
τπθθτππ
θθθθτπτφππ
π mp
mp
mp
gIgI fJdfdGpfΩ
=Ω
=Ω
= ∫∫−
jxu
Received signal correlation and spectrum3(4) Power density spectrum
• The power density spectrum of gI(t) and gQ(t) is the Fouriertransform of gIgI(t)or gQgQ(t). For the auto-correlation, thecorresponding psd is
• The autocorrelation of the received complex envelope g(t)=gI(t) + jgQ(t) is
• And its power spectral density (Doppler power spectrum)
)()()( fjSfSfS gIgQgIgIgg +=
)()()( τφτφτφ gIgQgIgIgg j+=
[ ]
≤
−
Ω
==)(0
)()/(1
12
)()(2
otherwise
fffff
FfS
m
mm
p
gIgIgIgI
πτφ
jxu
Received signal correlation and spectrum3(4)
• With 2-D isotropic scattering and an isotropic antenna
gIgQ(t)=0 and Sgg(f)=SgIgI(f), so that
mc
m
cm
prr fff
f
ffffS ≤−
−−
Ω= ,
1
1
4)(
2π
jxu
Received envelope and phase distribution 1(3)
Rayleigh fading
Ω−
Ω=
ppa
xxxp
2
exp2
)( π21
)()( =Φ xp t
•NLoS environments
•WSS complex Gaussian random process.
•2-D isotopic scattering,gI(t) and gQ(t) are IID zero-meanGaussian random variables at any time t1, with variancebo.The magnitude of the received complex envelope a(t)=|g(t)| has a Rayleigh distribution at any time t1.
jxu
Received envelope and phase distribution 2(3)
Ricean fading•Specular component
•micro-cellular and mobile satellite applications
•gI(t) and gQ(t) are Gaussian randomprocesses with non-zero means mI(t) and mQ(t)
+−=
00
0
22
0 2exp)(
b
xsI
b
sx
b
xxpa
Where s2= mI2 (t)+ mQ 2 (t)
,)1(
2)1(
exp)1(2
)( 0
2
Ω+
Ω+−−
Ω+=
pppa
KKxI
xKK
Kxxp
Rice factor K = s2 /2bo
jxu
Received envelope and phase distribution 3(3)
Nakagami fading•characterize rapid fading in long distance HF channels
•often used for the following reasons:
A) empirical justification.B) It can model fading conditions that are either more orless severe than Rayleigh fading. When m=1, theNakagami distribution becomes the Rayleigh distribution.C) Close relation between the Rice factor K and Nakagamishape factor m.
( ))12(
1 2
2
2
++=
−−−=
K
Km
mmm
mmK
Ω−
ΩΓ=Ρ
−
ppm
mm
a
mx
m
xmx
212
exp)(
2)( (m>=0.5)
jxu
Level Crossing Rate (LCR)& Average Fade duration (AFD)
Two important second order statistics associated withenvelope fading
LCR: How often the envelope crosses a specified level AFD: How long the envelope remains below a specified level
t1 t2t3 t4
NR=4Signalstrength dB
Time,seconds
10
jxu
LCR and AFD for a Rayleigh fading signal 1(2)
Useful for designing error control codes and diversityschemes
Level crossing rate
is the time derivative of r(t)
is the joint pdf
∫ −==2
2),( ρρπ efrdrRprN mR
r)rP(R,
rmsRR _/=ρ
jxu
LCR and AFD for a Rayleigh fading signal 2(2)
Average fade duration
[ ]
[ ]
πρτ
τ
ρ
ρ
21
1)(Pr
1
2
2
0
m
R
R
R
f
e
edrrpRr
RrPN
−=
∫ −==≤
≤=
−
For the scattering environmentshown in left figure, the averageenvelope fade duration for it shown
jxu
Spatial CORRELATIONS
τλ l
fv cm ==
)/2(16
)(
)/2(2
)(
2co
paa
cop
gIgI
lJl
lJl
λππ
µ
λπφ
Ω=
Ω=
Therefore, inpractice, sufficiently un-correlated diversity branches can beobtained at the MS by spacing the antenna elements a
distance apart.
For the case of isotropic scattering become, respectively
A fundamental question that arises is theantenna separation needed to provideuncorrelated antenna diversity branches.