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Hello, welcome to another module in this massive
open online course on probability and random
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variables for wireless communications.
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So in the previous module, we have looked
at Bayes Theorem.
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Let us now look at an application of Bayes
Theorem in the context of digital communication
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systems and wireless communications.
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As we said, this is a course on applied probability
theory, so we have looked at Bayes Theorem.
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Let us look at an important application of
this in the context of wireless communications.
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So what we are going to look at today is known
as the MAP
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receiver and this map: this acronym MAP stands
for
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Maximum
Aposteriori Probability or basically the MAP
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receiver.
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This is a very important receiver in the context
of wireless communication or even a traditional
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digital communication.
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That is digital information symbol based communication
system.
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This MAP receiver which computes the Aposteriori
probabilities that we had illustrated previously
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in the context of Bayes Theorem.
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So let us now consider a digital communication
system with binary information symbols in
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which the sample space is S consisting of
the binary information symbols, 0 and 1.
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So let us consider a digital communication
system or a digital communication link based
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wireless communication system in which the
sample space consists of the binary information
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symbols.
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So this is my sample space.
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and these are my
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binary information symbols.
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Therefore, now, so binary information symbols
are 0 and 1 and the sample space contains
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these 2 binary information symbols, 0 and
1.
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Now let us consider the events A0 equal to
0.
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This is a very simple example.
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So there are 2 events, A0 equal to 0, A1 equal
to 1.
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That is A0 corresponds to the binary symbol,
0 and A1 corresponds to the binary symbol,
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1.
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So there are 2 binary symbols, 0 and 1.
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A0 corresponds to the event of the transmitted
binary symbol being 0.
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A1 corresponds to the transmitted binary symbol
being 1.
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And therefore now, you can clearly see that
A0 and A1 are mutually exclusive and exhaustive
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because A0 union A1 equals 0 union 1 which
is basically equal to 0, 1 which is the entire
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sample space, S. So, A0 union A1 equals S.
So, A0 union A1 equals the entire sample space.
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Because A0 is 0, A1 is 1, their union is the
set, 0, 1 which is the entire sample space.
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And further, A0 and A1 are clearly mutually
exclusive because A0 intersection A1 is the
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null event, phi.
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You can see, again it is not difficult to
see that A0 intersection A1 equals Phi which
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is our null event or our empty event.
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Therefore, A0 union A1 is equal to S and A0
intersection A1 is equal to phi which implies
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A0, A1 are
mutually
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exclusive and exhaustive.
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So, A0 and A1 are therefore for this binary
information symbol based digital or wireless
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communication system, A0 and A1 are mutually
exclusive and exhaustive.
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Therefore this satisfies one of the prerequisite
conditions for our Bayes Theorem.
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Remember, for our Bayes Theorem, we said whatever
are the sets, A0 and A1, these have to be
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mutually exclusive, that is their intersection
has to be the null event and they have to
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be exhaustive that is the union of these has
to be the entire sample space, S. So this
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satisfies the prerequisite for the Bayes Theorem.
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Now let us look at the application of this
in the context of communication.
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We consider what is known as a binary symmetric
channel.
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Let us consider a standard channel in communication.
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This
is a binary symmetric channel and this can
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be described as follows.
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I am showing a schematic representation of
a binary symmetric channel where I can have
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the left side is the transmitter
and the right side is the receiver.
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I can transmit either a 0 or 1.
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That is A0 as yet seen before, A0 corresponds
to the event is 0, A1 corresponds to the event
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that the transmitted symbol is 1.
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Let us say that the probability of A0, P0
equals 0.1
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The probability of A1, P1 equals 0.9.
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So we have a binary symmetric channel in which
I have the transmitter side and I have the
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receiver side and the transmitter side, I
am saying that we can transmit a symbol that
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is A0 that is a symbol, 0, that is A0 corresponds
to the transmission of the symbol, 0.
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Event A1 corresponds with the transmission
of symbol, 1.
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And we are also saying that the probability
of A0 is 0.1.
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That is, the probability of the symbol, 0
is transmitted is 0.1.
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The probability that the symbol, 1 is transmitted
that is the probability of A1 which is equal
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to P1 is equal to 0.9.
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Therefore, probability of A0 equals, we are
denoting by P0 is equal to 0.1.
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Probability of A1 is equal to P1 which is
equal to 0.9.
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Further, at the receiver also, we can have
0 that is the event B which corresponds to
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receiving 0 or the event B tilda which corresponds
to receiving.
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Further, a transmitted 0 can either be received
as a 0, the probability of which is equal
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to 0.8 which is equal to 1 minus P or the
probability of 0 can get flipped to 1, the
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probability of which is P equal to 0.2.
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So therefore we are saying that this is an
interesting scenario where a transmitted 0
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can either be received as a 0 with a probability
0.8 or the transmitted 0 can be flipped to
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A1.
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That is, there is an error which occurs on
the channel due to which the 0 that is transmitted
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gets flipped to a 1 and the probability of
this happening is 0.2.
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Similarly, the 1 that is transmitted, can
be either received as 1, the probability corresponding
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to which is 0.8 which is a call to 1 minus
P. Or it can get flipped to a 0 the probability
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of which happening is P equal 0.2.
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So these 2 links, these 2 things that we show
here, these corresponds to the flipping.
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What do we mean by flipping?
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That is, a 1 gets flipped to a 0 or a 0 gets
flipped to a 1 because of errors on the channel
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and this channel is known as a binary symmetric
channel basically because it represents binary
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information symbols, 0 and 1 which are transmitted
as 0 and 1 and can be received as either 0
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or 1.
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And symmetric because these different probabilities
are symmetric between 0 and 1.
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The probability that a 0 can get flipped to
a 1 and the probability that a 1 can get flipped
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to a 0, both these probabilities are symmetric
and these probabilities are equal to 0.2.
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Therefore, this is known as a binary symmetric
channel.
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This is a very important channel which is
used to model the transmission of digital
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information symbols across both, a digital
communication channel and also, a wireless
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communication channel.
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A wireless communication channel which uses
digital symbols of course is also based, is
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a special case of a digital communication
system.
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So this is a binary symmetric channel which
is used to model the transmission of binary
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information symbols over a communication channel.
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And let us now look at this.
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What we have given?
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We have given that the probability of A0 that
is the probability that the transmitted symbol
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is 0 is 0.1.
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The probability of A1 which is equal to the
P1 which is equal to 0.9 is basically the
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probability that the transmitted symbol is
1.
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And as we had said before, these are the prior
probabilities.
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So these here, these are the prior probabilities.
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I hope everyone remembers from the previous
module.
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These are the prior probabilities.
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Now also observed, the probability of flipping
that is the probability of B given A1 which
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is equal to the probability, the probability
B given A1 is probability, received symbol
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is 0, given transmitted symbol is 1 which
is equal to the probability of B tilda given
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A0 that is the probability that the received
symbol is 1 given transmitted symbol is 0.
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And these are the flipping probabilities which
are 0.2 and probability of B given A0 equals
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probability of B tilda given A1.
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These are the non flipping probabilities.
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That is, the probability that a 0 is received
as a 0 and a 1 is received as a 1.
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These are non flipping probabilities.
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These conditional probabilities are known
as the likelihoods.
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As we had described in the previous module,
these probabilities are known as the likelihoods.
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So we are given the prior probabilities, we
are given the likelihoods.
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So now we would like to employ the Bayes Theorem
to compute the Aposteriori probabilities.
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What is the Aposteriori probability?
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We would like to compute, what is the probability
of A0 given B?
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Remember, this is the Aposteriori probability.
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Remember what is the Aposteriori probability?
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P of A0 given B. That corresponds to the probability,
A0 is the event remember corresponding to
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the transmission of 0 and B is the event corresponding
to the reception of 0.
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Therefore, P A0 given B corresponds to the
probability that a 0 has been transported
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given that a 0 has been received.
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So what we are asking is the question, what
is the probability or what we are asking is,
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given that, given B
equal to 0 has been received, what is the
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probability A0 equal to 0 has been transmitted.
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That is given that 0 has been received, what
is the probability that the actual transmitted
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symbol is a 0?
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That is the question that we are asking.
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What is the probability of A0 given B?
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That is the Aposteriori probability of A0
given B.
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And now, let us compute this.
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We know that the Aposteriori
probability of P of A0 given B, we know from
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the Bayes theorem that this is equal to the
probability of B given A0 times the probability
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of A0 divided by the probability of B given
A0 into the probability of A0 plus probability
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B given A1 into probability of A1.
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And now if we substitute this quantity, we
know that, let us go back and take a glimpse
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at this channel.
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Probability of B given A0, we know is a probability
0 is received corresponding to 0 transported.
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That is 0.8.
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And these various values are given, the prior
probabilities, the probabilities of A0 is
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0.1, probability of A1 is 0.9 and so on.
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So these values are given.
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Now, all we have to do is substitute these
values in the Bayes Theorem and this can be
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calculated as
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probability of B given A0 that is 0.8 into
probability of A0 that is 0.1 divided by probability
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of B given A0 that is 0.8 into probability
of A0 that is 0.1 plus probability of probability
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of B given A1, this is the flipping probability,
probability of 0 given 1 has been transmitted
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that is 0.2 times the probability of A1, the
prior probability of A1 that is 0.9.
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Therefore, this is equal to 0.08 divided by
0.26.
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That is equal to 8 divided by 26.
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So the probability, the Aposteriori probability
of A0 given B is equal to 8 divided by 26.
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So what we have seen?
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Using the Bayes Theorem, we have calculated
the Aposteriori probability of A0 given B.
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That is what is the probability that a 0 has
been transported given that a 0 has been received
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at the receiver.
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Now let us compute the other Aposteriori probability.
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What is the probability that a 1 has been
transmitted given that a 0 has been received?
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And this can be calculated as the probability
of A1 given B which is equal to probability
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of B given A1 times probability of A1 divided
by probability of B given A0 divided by probability
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of A0 probability of B given A1 times probability
of A1 which is equal to probability of B given
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A1 is again the flipping probability 0.2 times
probability of A1 which is 0.9 divided by
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probability of B given A0 which is 0.8 times
the probability of A1 which is 0.1 plus probability
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of B given A1 which is 0.2 times probability
of A1 which is a 0.9.
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And therefore, this is equal to, again you
can see 0.18 divided by 0.26 equals 18 divided
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by 26.
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And now you can observe, if you look at these
2 quantities, the probability of A1 given
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B
and the probability of A0 given B, that is
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the probability corresponding to transmission
of 0 given a 0 has been observed.
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And the probability of A1 given B that is
the probability corresponding to a transmission
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of 1 given a 0 has been observed.
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Very interestingly, you observe that the probability
of A1 given B which is 18 by 26 is higher
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than the probability of A0 given B which is
8 by 26.
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So what you observe is something that is very
fascinating.
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What you observe is that probability or the
posterior probability of A1 given B which
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is equal to 18 by 26 is greater than 8 by
26 which is the posterior probability of A0
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given B. Right?
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So we have, basically what is this?
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This is the posterior probability corresponding
to, what is this?
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This is the posterior probability or Aposteriori
probability
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corresponding to 1.
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And this is the Aposteriori
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probability corresponding to 0.
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And what we are saying is, the Aposteriori
probability even though a 0 has been received,
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the Aposteriori probability corresponding
to the transmission of 1 is higher than the
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Aposteriori probability corresponding to the
transmission of 0.
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Therefore, based on this, at the receiver
even though we are observing 0, that is the
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event B which corresponds to the reception
of 0, we conclude that the actual transmitted
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symbol must have been 1 with a higher probability
than 0.
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So based on Maximum Aposteriori, so based
on this computation, we choose or we basically
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decide or detect 1 at the receiver since it
corresponds to the Maximum Aposteriori probability.
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And this receiver is therefore known as the
Maximum Aposteriori Probability receiver.
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That is, corresponding to the observation,
we compute what are the Aposteriori probabilities
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of the various transmitted symbols and we
choose that transmitted symbol which has the
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Maximum Aposteriori Probability and this is
known as the Maximum Aposteriori Probability
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receiver.
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That is, even though we observe a symbol 0,
we compute the Aposteriori probability corresponding
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to the transmission of 0, we compute the Aposteriori
probability corresponding to transmission
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of 1 and we conclude that transmission of
1 has a greater Aposteriori probability.
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Therefore we decide that as a 1 has been transmitted
by the transmitter.
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This is known as the Maximum Aposteriori Probability
receiver or the MAP receiver.
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And this minimizes the probability of error
at the receiver.
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This is the optimal receiver which minimizes
the probability of error at the receiver.
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So this is known as the MAP receiver.
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This principle is known as the MAP principle.
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This is a very important principle in the
context of communication systems.
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This is known as the MAP or the Maximum Aposteriori
Probability probability and this leads to
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minimizes
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the probability of error, this minimizes the
probability of error at the receiver.
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The MAP receiver minimizes the probability
of error at the receiver.
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Therefore, this MAP receiver which is built
on the principle of the Bayes Theorem which
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computes Aposteriori probabilities of the
various transmitted symbols and chooses the
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transmitted symbol which has the Maximum Aposteriori
Probability is known as the MAP receiver.
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All right?
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So the MAP receiver is very important.
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On the other hand, if you look at a simple
maximum likelihood receiver, now if you look
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at the likelihoods, what are the different
likelihoods?
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If you look at the likelihoods, the likelihood
of this B given A0 is 0.8 and likelihood of
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B given A1 is 0.2.
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So now if you look at the likelihood.
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Simply if you look at the likelihoods, probability
of A0 given that is the probability of B that
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is of the received observation 0 corresponding
to the transmission of 0 equals 0.8, Probability
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of B given A1 equals 0.2, This is the flipping
probability and these are the likelihoods,
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and based simply on the likelihood, we can
see that A0 has L. Because the probability
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of B given A0 is higher than the probability
of B given A1, we can see that A0 of the transmission
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of 0 has a higher likelihood.
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Therefore, if we decide based on this that
the transmitted symbol is 0, this is known
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as the Maximum likelihood receiver.
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So, Maximum likelihood receiver, so therefore
likelihood, we are saying that P of B given
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A0 is greater than or we are saying, if I
have to write it clearly, P of B given A0
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which is equal to 0.8 is greater than 0.2
which is equal to P of B given A1.
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Therefore, likelihood of 0, not the Aposteriori
probability, likelihood of 0 is greater than
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the likelihood of the binary information symbol
1.
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Therefore, Maximum likelihood receiver or
ML receiver
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decides 0.
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So, Maximum likelihood receiver which purely
looks at the likelihood decides the 0 as against
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1 because the likelihood of 0 is greater than
the likelihood of 1.
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However, the MAP receiver looks at the Aposteriori
probability and decides a 1 rather than 0.
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And remember, the Maximum Aposteriori Probability
minimizes the probability of error, not the
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maximum likelihood receiver.
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So the maximum likelihood receiver, although
it is an easy receiver to look at, it is appealing,
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it does not minimize the probability of error.
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So the optimal receiver is the MAP receiver,
not the Maximum likelihood receiver.
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So this is, so we have looked at 2 different
kind of receivers, one is the MAP receiver
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and the Maximum likelihood receiverÉ
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Ébut the Maximum likelihood receiver, this
does not
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minimize the probability of error.
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So while it is appealing to use the Maximum
likelihood receiver, it is suboptimal as to
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the MAP received which computes the, which
chose the symbol with the Maximum Aposteriori
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Probability is the optimal receiver and that
uses the Bayes Theorem.
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So the Bayes Theorem is very important.
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In fact, I have to say, it is one of the key
principles used in digital communication or
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wireless communication system.
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Therefore, it is very important to understand
the Bayes Theorem and as well as the application
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of this Bayes Theorem in the context of modern
wireless and digital communication symbols
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because that leads to the optimal receiver
which minimizes the probability of error at
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the receiver.
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So this module, while the previous module
has explained the theoretical or abstract
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concept of Bayes Theorem, this module has
justified, has illustrated a very important
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application of this Bayes Theorem for computing
this Aposteriori probabilities and then choosing
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the symbol with the Maximum Aposteriori Probability
at the receiver in the context of a wireless
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communication system or the MAP receiver.
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So we will conclude this module here.
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We will explore other aspects of probability
and random variables in subsequent modules.
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Thank you very much.