1
00:00:14,009 --> 00:00:19,340
Hello, welcome to another module in this massive
open online course on probability and random
2
00:00:19,340 --> 00:00:21,509
variables for wireless communication.
3
00:00:21,509 --> 00:00:26,040
In the previous model but we had seen the
concept of a random variable, the random variable
4
00:00:26,040 --> 00:00:30,890
X, the concept of probability density function
for this random variable, X and we had looked
5
00:00:30,890 --> 00:00:34,550
at the exponential random variables.
6
00:00:34,550 --> 00:00:39,699
Let us now look at the relevance of this concept
of a random variable.
7
00:00:39,699 --> 00:00:43,760
Specifically, the exponential random variable
in the context of wireless communication.
8
00:00:43,760 --> 00:00:55,929
So let us now look at an example, simple example
from
9
00:00:55,929 --> 00:01:04,980
the context of
wireless communication.
10
00:01:04,980 --> 00:01:13,420
So let us look at a simple example because
remember, this is a course in probability
11
00:01:13,420 --> 00:01:18,370
and random variables, the applications of
probability and random variables, random processes
12
00:01:18,370 --> 00:01:21,690
in communications, specially wireless communication.
13
00:01:21,690 --> 00:01:26,000
So let us consider a typical wireless communication
scenario with a transmitter and receiver,
14
00:01:26,000 --> 00:01:30,170
a cellular scenario in which a base station
is transmitting to a mobile.
15
00:01:30,170 --> 00:01:37,820
So for instance, I have a typical wireless
scenario and we have already seen this to
16
00:01:37,820 --> 00:01:38,840
some extent.
17
00:01:38,840 --> 00:01:42,230
A wireless transmitter which is transmitting.
18
00:01:42,230 --> 00:01:55,070
So I have my base station
which is transmitting to the mobile.
19
00:01:55,070 --> 00:01:58,370
This is my mobile.
20
00:01:58,370 --> 00:02:02,450
And therefore, there is going to be a direct
line of sight path.
21
00:02:02,450 --> 00:02:11,790
However, there are also going to be several
non line of sight paths between the base station
22
00:02:11,790 --> 00:02:20,330
and the mobile which arise from the presence
of what are known as scatterers.
23
00:02:20,330 --> 00:02:21,390
Right?
24
00:02:21,390 --> 00:02:28,120
So we have a line of sight path and several
non line of sight paths.
25
00:02:28,120 --> 00:02:34,100
So we are looking at a wireless communication
environment.
26
00:02:34,100 --> 00:02:37,230
So in a wireless communication environment,
we have a transmitter and a receiver.
27
00:02:37,230 --> 00:02:41,780
There is no dedicated medium, there is no
guided propagation medium between the transmitter
28
00:02:41,780 --> 00:02:42,780
and the receiver.
29
00:02:42,780 --> 00:02:46,900
Therefore, there is a signal that is radiated
into the space by the transmitter.
30
00:02:46,900 --> 00:02:51,409
This propagates along a direct line of sight
path from the base station to that mobile.
31
00:02:51,409 --> 00:02:56,680
So, there is a line of sight which is also
represented by the nomenclature, LOS which
32
00:02:56,680 --> 00:02:58,579
stands for line of sight.
33
00:02:58,579 --> 00:03:09,379
So this LOS, this stands for line of sight,
and there are several obstacles such as for
34
00:03:09,379 --> 00:03:19,250
instance, trees and buildings in the propagation
path.
35
00:03:19,250 --> 00:03:32,719
These are known as, these obstacles here,
these buildings, trees, etc. these are known
36
00:03:32,719 --> 00:03:35,959
as scatterers which basically scatter the
towns voted transmitted signal.
37
00:03:35,959 --> 00:03:41,959
So, from these scattered, scatterers, there
are several scattered signal components or
38
00:03:41,959 --> 00:03:45,469
several reflected signal components.
39
00:03:45,469 --> 00:03:47,120
These deflect the signal.
40
00:03:47,120 --> 00:03:52,349
That is, the transmitted wireless signal and
the deflected single component also arrive
41
00:03:52,349 --> 00:03:53,349
at the mobile.
42
00:03:53,349 --> 00:03:58,180
So there is one line of sight and there are
several non line of sight which is denoted
43
00:03:58,180 --> 00:04:04,299
by NLOS, several non line of sight components
which are deflected or which are scattered
44
00:04:04,299 --> 00:04:09,730
by the scatterers such as trees and buildings
which also arrive at the mobile.
45
00:04:09,730 --> 00:04:19,680
So this NLOS stands for non line of sight.
46
00:04:19,680 --> 00:04:23,560
And as a result of this, what is going to
happen over here?
47
00:04:23,560 --> 00:04:29,030
These several signal components: these are
going to cause interference.
48
00:04:29,030 --> 00:04:31,780
These interfere.
49
00:04:31,780 --> 00:04:47,360
So these are going to cause interference at,
so these multiple signal components cause
50
00:04:47,360 --> 00:04:51,350
interference at the mobile receiver.
51
00:04:51,350 --> 00:04:56,750
As a result of this, the received signal quality,
the interference can be constructive or the
52
00:04:56,750 --> 00:04:58,790
interference can be destructive.
53
00:04:58,790 --> 00:05:03,890
As a result of this, the received signal quality
or the received signal level is random in
54
00:05:03,890 --> 00:05:04,890
nature.
55
00:05:04,890 --> 00:05:10,010
This is known as the fading channel because
the received signal power is fading.
56
00:05:10,010 --> 00:05:13,610
And therefore, the received signal power is
random in nature.
57
00:05:13,610 --> 00:05:19,400
So, interference, so in the wireless channel,
we have, the sequence that we said is we have
58
00:05:19,400 --> 00:05:30,420
multipath propagation.
59
00:05:30,420 --> 00:05:37,060
This multipath propagation leads to, what
does this lead to?
60
00:05:37,060 --> 00:05:39,740
Multipath purgation propagation leads to interference.
61
00:05:39,740 --> 00:05:50,150
It leads to multiple signal components which
leads to interference.
62
00:05:50,150 --> 00:05:58,610
So, multiple signal components.
63
00:05:58,610 --> 00:06:04,610
These multiple signal components in turn lead
to interference.
64
00:06:04,610 --> 00:06:09,220
This is known as multipath interference.
65
00:06:09,220 --> 00:06:12,740
This leads to interference.
66
00:06:12,740 --> 00:06:16,680
And because of interference, the received
signal is fading.
67
00:06:16,680 --> 00:06:27,020
So
this leads to a fading signal.
68
00:06:27,020 --> 00:06:34,440
And as the signal is fading, the received
power, therefore the received signal level
69
00:06:34,440 --> 00:06:54,700
or the received signal power
is random in nature.
70
00:06:54,700 --> 00:06:59,880
So there is multipath propagation arising
because of the scatterers or reflectors in
71
00:06:59,880 --> 00:07:04,040
the wireless environment that leads to several
non line of sight components.
72
00:07:04,040 --> 00:07:07,700
These non line of sight components together
with the line of sight components when they
73
00:07:07,700 --> 00:07:13,320
impinge on the mobile, they cause interference
between these multiple components, between
74
00:07:13,320 --> 00:07:15,760
these multiple signal components.
75
00:07:15,760 --> 00:07:20,970
The interference can be constructive or destructive
depending on the amplitude and phases of these
76
00:07:20,970 --> 00:07:22,870
different signal components.
77
00:07:22,870 --> 00:07:25,410
And therefore, the received signal power is
fading.
78
00:07:25,410 --> 00:07:30,480
That is, the received signal power at the
mobile is random, it is fading in nature.
79
00:07:30,480 --> 00:07:33,110
It is varying with time.
80
00:07:33,110 --> 00:07:34,110
Correct?
81
00:07:34,110 --> 00:07:38,110
And therefore the received signal power is
random in nature.
82
00:07:38,110 --> 00:07:40,980
This wireless channel is a fading Wireless
Channel.
83
00:07:40,980 --> 00:07:45,280
The received signal power is random in nature.
84
00:07:45,280 --> 00:07:50,560
And the power of the channel coefficient,
the power of the fading channel is characterised
85
00:07:50,560 --> 00:07:52,570
by the probability density function.
86
00:07:52,570 --> 00:08:20,270
Therefore, the fading power, the power of
fading channel which is denoted by
87
00:08:20,270 --> 00:08:21,270
G.
88
00:08:21,270 --> 00:08:25,800
This is characterised by the fading channel
distribution.
89
00:08:25,800 --> 00:08:28,200
This is characterised by the probability.
90
00:08:28,200 --> 00:08:29,200
This is random.
91
00:08:29,200 --> 00:08:36,090
As we said, the received signal power is random,
this is characterised by the exponential probability
92
00:08:36,090 --> 00:08:42,610
density function e to the power of minus g
where g is greater than or equal to 0 which
93
00:08:42,610 --> 00:08:46,900
is basically nothing but as we have seen in
the previous module, this is the exponential
94
00:08:46,900 --> 00:08:49,320
probability density function.
95
00:08:49,320 --> 00:08:55,750
What is this?
96
00:08:55,750 --> 00:09:10,050
This is the exponential PDF with parameter
A is equal to 1.
97
00:09:10,050 --> 00:09:14,050
So this is a very interesting example from
wireless communication in application of the
98
00:09:14,050 --> 00:09:16,910
probability theory that we have learnt so
far in wireless communication.
99
00:09:16,910 --> 00:09:20,220
We have said that the fading channel is random
in nature.
100
00:09:20,220 --> 00:09:24,270
That is the signal level and hence the power
level in the fading wireless channel because
101
00:09:24,270 --> 00:09:30,060
of the unpredictable nature of interference
arising from these random scatterers in the
102
00:09:30,060 --> 00:09:33,870
environment, the signal power is random in
nature.
103
00:09:33,870 --> 00:09:39,900
This signal power, if we denote this power
of the fading channel coefficient by G, g
104
00:09:39,900 --> 00:09:41,830
is a random variable.
105
00:09:41,830 --> 00:09:48,450
The probability density function F of g of
g is denoted by e raised to the power minus
106
00:09:48,450 --> 00:09:53,100
g for g greater than or equal to 0 because
it is a fading power.
107
00:09:53,100 --> 00:09:54,600
The power cannot be negative.
108
00:09:54,600 --> 00:10:00,960
So this probability density function is defined
only for g greater than or equal to 0.
109
00:10:00,960 --> 00:10:07,240
This is characterised by e raised to minus
g which we are saying is the exponential probability
110
00:10:07,240 --> 00:10:08,240
density function.
111
00:10:08,240 --> 00:10:13,790
Remember, in the previous module, we had seen
the exponential probability density function
112
00:10:13,790 --> 00:10:19,740
which is basically exponential probability
density function is a parametric distribution
113
00:10:19,740 --> 00:10:23,990
which is A e to the power of minus A G.
114
00:10:23,990 --> 00:10:34,940
If we set A equals 1, then we get basically
what we get is e to the power of minus g for
115
00:10:34,940 --> 00:10:38,290
g greater than or equal to 0.
116
00:10:38,290 --> 00:10:46,760
So therefore, what we are saying is, in a
fading Wireless Channel, the power is distributed
117
00:10:46,760 --> 00:10:54,990
as an exponential random variable with the
parametric A equal to 1.
118
00:10:54,990 --> 00:11:00,040
And this example is about specific fading
wireless channel which is known as the Rayleigh
119
00:11:00,040 --> 00:11:01,790
fading Wireless Channel.
120
00:11:01,790 --> 00:11:11,010
This is valid for a specific wireless channel
which is known as the Rayleigh Fading which
121
00:11:11,010 --> 00:11:24,890
is a very popular model for wireless channel.
122
00:11:24,890 --> 00:11:29,410
This is known as a Rayleigh Fading channel
in which the amplitude of the fading coefficient
123
00:11:29,410 --> 00:11:31,590
is distributed as a Rayleigh Fading channel.
124
00:11:31,590 --> 00:11:36,860
So we are looking at a fading wireless channel,
we are saying that the amplitude of the channel
125
00:11:36,860 --> 00:11:42,430
coefficient of this random channel coefficient
is distributed as a Rayleigh Fading channel.
126
00:11:42,430 --> 00:11:48,190
The power is distributed as an, power which
is the square of the amplitude is distributed
127
00:11:48,190 --> 00:11:54,552
as an exponential random variable, e to the
power of minus g with parametric which is
128
00:11:54,552 --> 00:12:00,070
an exponential probability density function
with the parameter A equals 1.
129
00:12:00,070 --> 00:12:04,500
And therefore, this probability density function
can be used to answer several interesting
130
00:12:04,500 --> 00:12:05,500
questions.
131
00:12:05,500 --> 00:12:09,100
For instance, we can use this to characterize
the behavior of the wireless channel.
132
00:12:09,100 --> 00:12:20,050
So we can now use this probability density
function to characterize the behavior and
133
00:12:20,050 --> 00:12:36,980
properties of
the wireless channel.
134
00:12:36,980 --> 00:12:50,190
We can use this to characterize the behavior
and properties of the wireless channel.
135
00:12:50,190 --> 00:12:53,270
For instance, let us look at this simple example.
136
00:12:53,270 --> 00:12:59,990
Since we are talking about the characterization
of the behavior of wireless channel, let us
137
00:12:59,990 --> 00:13:04,080
look at this simple example.
138
00:13:04,080 --> 00:13:10,080
Let us consider this Rayleigh Fading channel,
Rayleigh Fading wireless channel we are considering
139
00:13:10,080 --> 00:13:11,080
so far.
140
00:13:11,080 --> 00:13:49,320
And let us ask the question, what is the probability
that the attenuation of the wireless channel
141
00:13:49,320 --> 00:14:03,100
that is the signal attenuation of the wireless
channel is worse than that is the are asking
142
00:14:03,100 --> 00:14:07,660
the question, what is the probability that
is given this fading wireless channel in which
143
00:14:07,660 --> 00:14:13,649
the power is distributed as a exponential
random variable, what is the probability that
144
00:14:13,649 --> 00:14:17,490
the attenuation of this fading wireless channel
is worse than 20 dB.
145
00:14:17,490 --> 00:14:25,500
That is, if I transmit a signal, X of T with
a certain power P, the received signal power
146
00:14:25,500 --> 00:14:30,540
is 20 dB attenuated with respect to the transmitted
signal power., What is the probability that
147
00:14:30,540 --> 00:14:35,390
the attenuation of the signal, of the wireless
signal is worse than 20 dB?
148
00:14:35,390 --> 00:14:37,300
And we can answer this as follows.
149
00:14:37,300 --> 00:14:43,431
The answer to this is as follows, since the
attenuation, the power attenuation, the dB
150
00:14:43,431 --> 00:14:55,020
attenuation has to be, since the attenuation
is 20 dB, the dB attenuation has to be less
151
00:14:55,020 --> 00:15:02,570
than or the amplification or the power gain
has to be less than or equal to minus 20 dB
152
00:15:02,570 --> 00:15:12,899
which implies 10 log 10 of g is less than
or equal to minus 20 dB Remember, the power
153
00:15:12,899 --> 00:15:14,470
gain is G.
154
00:15:14,470 --> 00:15:18,920
The power gain in dB is 10 log the base 10
of G.
155
00:15:18,920 --> 00:15:24,220
And for the attention to be worse than 20
dB, this has to be less than or equal to minus
156
00:15:24,220 --> 00:15:25,220
20.
157
00:15:25,220 --> 00:15:36,970
So 10 log to the base 10 of g is less than
or equal to minus 20 which impliesÉ.
158
00:15:36,970 --> 00:15:43,590
Which basically implies that g is less than
or equal to 10 to the power of minus 2 is
159
00:15:43,590 --> 00:15:46,550
equal to 0.01.
160
00:15:46,550 --> 00:15:53,110
And we already know that g is greater than
or equal to 0, Because this is the power gain,
161
00:15:53,110 --> 00:15:55,740
power cannot be negative which means we are
asking,
162
00:15:55,740 --> 00:16:08,890
Éwhat is the probability that 0 is less than
or equal to g is less than or equal to 0.1.
163
00:16:08,890 --> 00:16:15,390
So the probability that the attenuation is
worse than 20 dB is the probability that the
164
00:16:15,390 --> 00:16:22,620
power gain of the wireless channel g belongs
to, lies in 0 to 0.1, it is greater than or
165
00:16:22,620 --> 00:16:26,959
equal to 0 but less than 0.1 which means it
belongs to the interval, 0 to 0.1.
166
00:16:26,959 --> 00:16:40,430
So this is the same as the probability, it
belongs to 0 to 0.01.
167
00:16:40,430 --> 00:16:46,550
What we are saying is, this is the probability
that attenuation: this is equal to the probability
168
00:16:46,550 --> 00:17:05,790
that attenuation is worse than 20 dB.
169
00:17:05,790 --> 00:17:14,020
And the probability that this belongs to 0
to 0.01, this is nothing but the probability,
170
00:17:14,020 --> 00:17:18,270
g belongs to the interval, 0 to 0.01.
171
00:17:18,270 --> 00:17:22,911
Remember, the probability that any random
variable belongs to a particular interval,
172
00:17:22,911 --> 00:17:24,940
A to B isÉ
173
00:17:24,940 --> 00:17:32,460
Integral A to B, integral 0 to 0.01, integral
of the probability density function which
174
00:17:32,460 --> 00:17:35,000
is F of g of g dG.
175
00:17:35,000 --> 00:17:42,850
We already know that this probability density
function F of g of g is distributed as the
176
00:17:42,850 --> 00:17:46,250
exponential random variable width parameter
1.
177
00:17:46,250 --> 00:17:53,080
So this is integral 0 to 0.01, e to the power
of minus g dG.
178
00:17:53,080 --> 00:17:58,280
That is the probability that it lies in the
interval 0 to 0.01.
179
00:17:58,280 --> 00:18:03,750
It is the integral of the probability density
function e to the power of minus g between
180
00:18:03,750 --> 00:18:09,220
these limits, 0 to 0.01 which is basically
nothing butÉ
181
00:18:09,220 --> 00:18:15,090
Integral of e to the power of minus g is minus
e to the power of g evaluated between the
182
00:18:15,090 --> 00:18:21,980
limits, 0 to 0.01 which is basically, you
can verify that this is one minus e power
183
00:18:21,980 --> 00:18:22,980
minus 0.01.
184
00:18:22,980 --> 00:18:34,090
Now, I am going to use an approximation for
small
185
00:18:34,090 --> 00:18:41,179
X. e to the power of minus X is approximately
equal to 1 minus e to the power of minus X.
186
00:18:41,179 --> 00:18:54,820
Therefore this probability is approximately
1 minus 1 minus 0.01 which is equal to 0.01.
187
00:18:54,820 --> 00:19:09,530
And therefore, the probability that the attenuation
is worse than 20 dB or the probability that
188
00:19:09,530 --> 00:19:16,120
the gain g is less than or equal to 10 to
the power of minus 2 that is 0.01.
189
00:19:16,120 --> 00:19:17,730
That is approximately 0.01.
190
00:19:17,730 --> 00:19:23,309
Therefore the probability that the attenuation
is worse than 20 dB of the fading wireless
191
00:19:23,309 --> 00:19:29,140
channel is 0.01 or a 1% chance that the received
wireless signal is attenuated by more than
192
00:19:29,140 --> 00:19:30,840
20 dB.
193
00:19:30,840 --> 00:19:46,660
So this probability, attenuation
194
00:19:46,660 --> 00:20:06,920
of fading wireless channel
is worse
195
00:20:06,920 --> 00:20:25,330
than 20 dB is 0.01 or basically, there is
a 1% chance of attenuation.
196
00:20:25,330 --> 00:20:35,770
There is a 1% chance of attenuation being
worse than 20 dB.
197
00:20:35,770 --> 00:20:39,080
So this is a practical example in the context
of wireless communication.
198
00:20:39,080 --> 00:20:41,850
We have said that the wireless channel his
fading in nature.
199
00:20:41,850 --> 00:20:46,740
We said that this can be characterized by
an exponential random variable with parameter
200
00:20:46,740 --> 00:20:51,669
A equal to 1 and we said, this can be used
to characterize the behavior of the wireless
201
00:20:51,669 --> 00:20:58,110
channel and we did a simple example in which
we found what is the probability that the
202
00:20:58,110 --> 00:21:02,620
attenuation of the received signal is lower
than minus 20 dB.
203
00:21:02,620 --> 00:21:03,940
So we will stop this module here.
204
00:21:03,940 --> 00:21:07,450
We will continue with the other aspects in
the subsequent modules.
205
00:21:07,450 --> 00:21:32,720
Thank you, thank you very much.