馃攳
Lecture 05: Rayleigh Fading Channel - YouTube
Channel: unknown
[18]
welcome to another module in this mooc
[20]
on wireless communications let us now
[22]
continue with our discussion on the
[24]
probability the probability density
[26]
function or the modeling of the fading
[28]
channel coefficient h
[30]
we said that if the fading channel
[31]
coefficient h
[33]
is h equals a e to the power of
[37]
j phi
[39]
then
[40]
the joint distribution of a comma phi
[45]
equals
[47]
equals
[54]
the joint distribution of a comma phi
[56]
equals a over
[58]
pi
[59]
e to the power of minus a square
[61]
we had already found we have already
[63]
found the marginal density with respect
[65]
to the channel coefficient a that is f
[67]
of a of a
[69]
equals 2 a
[71]
e to the power of minus a square and
[74]
this we said is the rayleigh
[76]
distribution
[78]
or the rayleigh
[81]
density
[85]
let us now find the distribution of the
[88]
phase
[89]
that is f of phi
[92]
of phi the phase component and for this
[94]
i can integrate the joint
[97]
density
[98]
with respect to the
[100]
amplitude
[103]
a and the amplitude lies between 0 to
[105]
infinity because the amplitude is always
[107]
positive
[108]
so this is 0 to infinity
[110]
integral 0 to infinity of a by pi
[115]
of a by pi
[118]
e raised to
[121]
minus a square
[123]
d a
[125]
now if you look at this integral i have
[127]
a e raised to minus a
[129]
so i bring 1 by pi
[132]
outside so i have 0 to infinity now if i
[135]
multiply and divide by 2 i have 1 over 2
[138]
pi
[140]
2 a
[141]
e raised to minus a square
[144]
d a
[145]
and integral 2 a e raised to minus a
[147]
square is minus of e raised to minus so
[150]
this is 1 over 2 pi
[153]
times minus e raised to minus a square
[158]
between the limits
[160]
0 to infinity and this you can see
[163]
as basically 1 over 2 pi as minus e
[167]
raised to minus a square between the
[169]
limit zero to infinity is is one so
[172]
therefore the distribution of the phase
[177]
so we have f of phi
[180]
of phi
[181]
equals one over 2 pi between the limits
[185]
for the phase
[186]
we have minus pi
[188]
less than phi
[190]
less than or equal to pi which means
[193]
this is a constant and this is the
[195]
uniform density so if you look at this
[199]
this is the
[202]
uniform
[206]
distribution
[208]
and therefore
[210]
if you have the limits
[213]
minus pi
[215]
if you have the between the limits
[222]
minus pi to pi
[224]
this as the amplitude this has the
[227]
height 1 over 2 pi and this as we said
[230]
is the uniform
[231]
uniform
[237]
probability so
[239]
the phase is distributed uniformly
[241]
between the limits minus pi
[243]
to pi so this is a uniform distribution
[246]
and therefore we have derived both the
[247]
marginal densities that is both the
[249]
distribution of both the amplitude
[252]
of the rayleigh fading channel
[253]
coefficient and the phase of the
[254]
rayleigh fading channel coefficient and
[256]
therefore f of a
[258]
of a
[259]
equals
[261]
2 a
[262]
e to the power of minus a square
[264]
and f of
[267]
phi
[269]
equals this equals well and this is
[272]
within the limit zero less than equal to
[274]
a
[275]
less than equal to infinity and this is
[277]
between the one over two pi
[280]
between the limits
[282]
minus pi
[284]
less than phi
[285]
less than or equal to
[287]
pi
[288]
and therefore now we have the so these
[291]
are the distributions the densities of
[292]
the amplitude and the phase
[308]
that is the amplitude a and phase phi of
[311]
the wireless channel and these can now
[314]
be used to characterize ah derive
[316]
various properties of the wireless
[318]
channel and further before we proceed
[320]
further let us look at one other
[322]
interesting point if we look at the
[324]
joint distribution of the amplitude and
[327]
the phase
[329]
remember the joint distribution is given
[331]
as
[332]
a over pi
[333]
e to the power of minus a square
[336]
i can write this as 1 over
[339]
2 pi
[340]
times
[341]
twice e to the power of minus a square
[344]
which is basically equal to f of
[347]
the marginal density of with respect to
[349]
the phase
[351]
times the marginal density with respect
[353]
to the amplitude so what we have is that
[356]
the joint density
[358]
with respect to the phase and amplitude
[361]
is equal to the
[362]
product of the marginal densities with
[364]
respect to of the amplitude and the
[366]
phase so the joint density
[374]
is equal to the product
[387]
the joint density is equal to the
[390]
product of the marginal densities
[392]
therefore this implies that the
[394]
amplitude and phase are independent
[396]
random variables so the amplitude and
[398]
phase of the rayleigh fading channel are
[400]
independent random variables so this
[402]
means that these are
[404]
independent that is a
[406]
comma
[408]
phi the amplitude and phase are
[412]
independent
[423]
the amplitude and phase are independent
[426]
random variables and these densities can
[429]
be used to derive very valuable
[431]
properties of
[432]
the fading channel for instance let us
[434]
look at the following example to
[436]
understand this better let us look at
[443]
the following
[444]
example
[446]
what is the probability that the
[448]
attenuation of the channel is worse than
[450]
20 db so let us ask the following
[452]
question
[454]
what
[455]
is
[457]
the
[459]
probability
[465]
that
[466]
the
[471]
attenuation
[475]
of the
[478]
channel is
[482]
worse than
[487]
so let us look at the following example
[489]
what is the probability that the
[490]
attenuation of the channel is worse than
[493]
20 db so what we are asking is what is
[496]
the probability so let let us look at
[498]
the gain attenuation of the wireless
[500]
channel or the gain of the wireless
[502]
channel
[504]
gain of
[507]
channel equals the
[510]
amplitude square which is less than
[512]
which is equal to a square so what we
[514]
are asking if the attenuation is worse
[516]
than 20 db it means
[518]
10 log
[521]
10
[522]
of a square
[524]
is less than
[526]
minus
[527]
20 db so that animation is 20 db implies
[531]
the gain of the received signal
[534]
the power gain of the received signal is
[535]
minus 20 db or lower so that attenuation
[538]
is worse than 20 db which means log 10
[543]
of a square is less than
[545]
minus
[547]
2 which means
[550]
a square
[552]
is less than
[555]
10 to the power of minus 2
[557]
equals 0.01
[559]
which implies
[561]
a is less than
[563]
0.1 so the gain of the channel is or the
[567]
attenuation is worse than minus 20 db if
[570]
the amplitude of the channel is less
[572]
than 0.1
[574]
and now we know that the distribution of
[576]
the amplitude f of a
[578]
of a equals
[581]
2a
[582]
e to the power of minus a square
[584]
therefore the probability
[587]
the probability that the amplitude is
[589]
less than 0.1
[591]
is basically equal to the density
[594]
integrated between the limits
[598]
0
[598]
to 0.1 so this is 2a e power minus a
[602]
square integrated between the limits 0
[604]
to 0.1 integral 2 a e power minus a
[607]
square is minus
[608]
e power minus a square
[611]
integrated between between the limits 0
[614]
to 0.1 so that is basically equal to 1
[617]
minus e power minus a square minus point
[621]
one square minus point zero one
[623]
which is approximately equal to ah which
[626]
is approximately equal to
[628]
point
[629]
zero one so the probability that the
[632]
attenuation or the probability that the
[634]
attenuation of the channel is worse than
[637]
20 db is
[638]
0.01 so this is the probability what is
[642]
this this quantity that we calculated
[644]
here this is the probability
[654]
that
[656]
attenuation
[660]
is
[663]
is worse
[674]
so this is the probability that the
[676]
attenuation of the channel the wireless
[678]
channel is worse than 20 db that is 0.01
[681]
so as we have seen these probability
[684]
densities or this model that we have
[686]
developed for the fading channel
[687]
coefficient that is the joint
[689]
distribution of its amplitude and phase
[691]
that is tree and the individual
[692]
distributions of the amplitude and phase
[694]
components are very important tools they
[697]
help us characterize the channel
[698]
statistically and derive valuable
[701]
inferences derived variable statistical
[704]
properties of the channel from these
[708]
various these distributions of the
[710]
amplitude and phase components and hence
[712]
these are also going to be important
[714]
when we look at characterizing the
[716]
performance of the wireless channel in
[718]
various scenarios
[720]
so this concludes this module
[722]
and we will
[724]
continue in the subsequent module thank
[726]
you very much
Most Recent Videos:
You can go back to the homepage right here: Homepage