Starting from:
$30
EE381-Lab 2 Conditional Probabilities Solved
0.1. Message transmission through a noisy communication channel
A digital message "M" is created and sent through a noisy communication channel (Figure 1). The signal "S" to be transmitted consists of a series of zeros and ones:
• symbol "0" appears in the signal with probability p0
• symbol "1" appears in the signal with probability p1 = −1 p0
The transmitted signal "S" is received at the other end of the channel as signal "R". Due to noise in the communication channel, a transmitted bit may change during transmission:
• a transmitted bit 0 may be received as 1 with probability ε0(probability of transmission error for symbol 0);
• a transmitted bit 1 may be received as 0 with probability ε1(probability of transmission error for symbol 1).
The errors for different symbol transmissions are independent.
In order to create one bit of the transmitted message “S” you have to: • Generate a random number: 𝑚𝑚 = np.random.rand( )
0 if m p≤ 0
• Generate the transmitted message “S” as: S=
1 if m p 0
• Notice that this process can be implemented by using your own function: 𝑆𝑆 = nSidedDie([𝑝𝑝0 , 1 − 𝑝𝑝0]) ; 𝑆𝑆 = 𝑆𝑆 − 1
In order to create the received signal “R” you have to:
• Generate a random number: 𝑡𝑡 = np.random.rand( ). The random number t should be different than the previous random number m .
•
1 0
Generate the received signal “R” as: R =
1
0
if S = 0 and t ≤ε0
if S = 0 and t ε0
if S =1 and t ε1 if S =1 and t ≤ε1
• This process can be implemented by using your own function:
If 𝑆𝑆 = 1 ∶ 𝑅𝑅 = nSidedDie([𝜀𝜀1 , 1 − 𝜀𝜀1]) ; 𝑅𝑅 = 𝑅𝑅 − 1
If 𝑆𝑆 = 0 ∶ 𝑅𝑅 = nSidedDie([1 − 𝜀𝜀0 , 𝜀𝜀0]) ; 𝑅𝑅 = 𝑅𝑅 − 1
MESSAGE(M) SEND (S) RECEIVE(R)
Figure 1. Probabilities for symbol transmission error
Reference: "Introduction to probability", D. Bertsekas and N. Tsitsiklis, 2nd Edition, Athena Scientific, 2008.
0. Probability of erroneous transmission
Consider the following experiment, where the required probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 have been provided to you in a separate document.
• You transmit a one-bit message S and look at the received signal R . If R S= , the experiment in considered a success., otherwise it is a failure.
• You repeat this experiment N =100,000times and count the number of failures.
• Find the probability that the transmitted bit will be received incorrectly, i.e. the probability of failure.
•
Probability of transmission error
Ans.
p =
1. Conditional probability: P(R= 1S= 1)
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as you did before. The goal is to calculate the conditional probability P R( =1 S=1) . This means that you will focus only in those transmissions where S=1.
• For all the events for which the transmitted signal is S=1, look at the received bit R . If R=1, the experiment is a success, i.e. success is defined as the conditional event: (R=1 S=1)
• You repeat this experiment N=100,000 times and count the number of successes.
• Find the conditional probability P R( =1 S=1) , i.e. the probability that if you transmit the symbol S=1, it will be received correctly.
•
Conditional probability P R( =1 S=1)
Ans.
p =
2. Conditional probability: P(S= 1R= 1)
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as you did before. The goal is to calculate the conditional probability P S( =1 R=1) . This means that you will only be interested in those messages where the received signal is R=1.
• For all the events for which the received signal is R=1, look at transmitted bit
S . If S=1, the experiment is a success, i.e. success is defined as the conditional event: (S=1 R=1)
• You repeat this experiment N=100,000 times and count the number of successes.
• Find the conditional probability P S( =1 R=1) , i.e. the probability that if you receive the symbol R=1, you can correctly conclude that it actually came from a transmitted signal of S=1.
•
Conditional probability P S( =1 R=1)
Ans.
p =
3. Enhanced transmission method
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as before. In order to improve reliability, the same bit “S” is transmitted three times (S S S) as shown in Figure 2.
• The received bits “R” are not necessarily the same as the transmitted bits “S” due to transmission errors. The three received bits, shown as (R1 R2 R3) in Figure 2 will be equal to one of the following eight triplets:
(R1 R2 R3) ={ (000), (001), (010), (100), (011), (101), (110), (111) }
When you look at the received triplet (R1 R2 R3) you must decide what was the bit “S” originally transmitted by using voting and the majority rule. Here are some examples of the majority rule.
• For example, if the three received bits are (R1 R2 R3)=(001), then the majority rule will decide that the bit must be a “0”. We denote this as the decoded bit D=0.
• As another example if the three received bits are (R1 R2 R3)=(101), then the majority rule will decode the bit as D=1.
• Another example: If you send S=0 three times, i.e. (S S S) = ( 0 0 0 ) and the received string is (R1 R2 R3) = (000), (001), (010), or (100) then the symbol will be decoded as D=0 and the experiment is a success, otherwise it is a failure.
• Another example: If you transmit S=1 three times, i.e. (S S S) = ( 1 1 1 ) and the received string is (011), (101), (110), or (111) the symbol will be decoded as D=1 and the experiment is a success, otherwise it is a failure.
• This procedure as described above is considered one experiment.
• Repeat the experiment N=100,000 times and count the number of successes.
• Find the probability that the transmitted bit "S" will be received and decoded incorrectly.
• Comment on whether the voting method used in this problem provides any improvement as compared to the method of Problem 1.
Probability of error with enhanced transmission
Ans.
p =
MESSAGE SEND 3 TIMESSEND 3 TIMES RECEIVE 3 BITS DECODE BY VOTING
(M) (S1 S2 S(S S S) 3) (R1 R2 R3) (D)
Figure 2. Enhanced message transmission method
A digital message "M" is created and sent through a noisy communication channel (Figure 1). The signal "S" to be transmitted consists of a series of zeros and ones:
• symbol "0" appears in the signal with probability p0
• symbol "1" appears in the signal with probability p1 = −1 p0
The transmitted signal "S" is received at the other end of the channel as signal "R". Due to noise in the communication channel, a transmitted bit may change during transmission:
• a transmitted bit 0 may be received as 1 with probability ε0(probability of transmission error for symbol 0);
• a transmitted bit 1 may be received as 0 with probability ε1(probability of transmission error for symbol 1).
The errors for different symbol transmissions are independent.
In order to create one bit of the transmitted message “S” you have to: • Generate a random number: 𝑚𝑚 = np.random.rand( )
0 if m p≤ 0
• Generate the transmitted message “S” as: S=
1 if m p 0
• Notice that this process can be implemented by using your own function: 𝑆𝑆 = nSidedDie([𝑝𝑝0 , 1 − 𝑝𝑝0]) ; 𝑆𝑆 = 𝑆𝑆 − 1
In order to create the received signal “R” you have to:
• Generate a random number: 𝑡𝑡 = np.random.rand( ). The random number t should be different than the previous random number m .
•
1 0
Generate the received signal “R” as: R =
1
0
if S = 0 and t ≤ε0
if S = 0 and t ε0
if S =1 and t ε1 if S =1 and t ≤ε1
• This process can be implemented by using your own function:
If 𝑆𝑆 = 1 ∶ 𝑅𝑅 = nSidedDie([𝜀𝜀1 , 1 − 𝜀𝜀1]) ; 𝑅𝑅 = 𝑅𝑅 − 1
If 𝑆𝑆 = 0 ∶ 𝑅𝑅 = nSidedDie([1 − 𝜀𝜀0 , 𝜀𝜀0]) ; 𝑅𝑅 = 𝑅𝑅 − 1
MESSAGE(M) SEND (S) RECEIVE(R)
Figure 1. Probabilities for symbol transmission error
Reference: "Introduction to probability", D. Bertsekas and N. Tsitsiklis, 2nd Edition, Athena Scientific, 2008.
0. Probability of erroneous transmission
Consider the following experiment, where the required probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 have been provided to you in a separate document.
• You transmit a one-bit message S and look at the received signal R . If R S= , the experiment in considered a success., otherwise it is a failure.
• You repeat this experiment N =100,000times and count the number of failures.
• Find the probability that the transmitted bit will be received incorrectly, i.e. the probability of failure.
•
Probability of transmission error
Ans.
p =
1. Conditional probability: P(R= 1S= 1)
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as you did before. The goal is to calculate the conditional probability P R( =1 S=1) . This means that you will focus only in those transmissions where S=1.
• For all the events for which the transmitted signal is S=1, look at the received bit R . If R=1, the experiment is a success, i.e. success is defined as the conditional event: (R=1 S=1)
• You repeat this experiment N=100,000 times and count the number of successes.
• Find the conditional probability P R( =1 S=1) , i.e. the probability that if you transmit the symbol S=1, it will be received correctly.
•
Conditional probability P R( =1 S=1)
Ans.
p =
2. Conditional probability: P(S= 1R= 1)
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as you did before. The goal is to calculate the conditional probability P S( =1 R=1) . This means that you will only be interested in those messages where the received signal is R=1.
• For all the events for which the received signal is R=1, look at transmitted bit
S . If S=1, the experiment is a success, i.e. success is defined as the conditional event: (S=1 R=1)
• You repeat this experiment N=100,000 times and count the number of successes.
• Find the conditional probability P S( =1 R=1) , i.e. the probability that if you receive the symbol R=1, you can correctly conclude that it actually came from a transmitted signal of S=1.
•
Conditional probability P S( =1 R=1)
Ans.
p =
3. Enhanced transmission method
Use the same probabilities 𝑝𝑝0 ; 𝜀𝜀0; and 𝜀𝜀1 as before and consider the following experiment:
• You create and transmit a one-bit message S as before. In order to improve reliability, the same bit “S” is transmitted three times (S S S) as shown in Figure 2.
• The received bits “R” are not necessarily the same as the transmitted bits “S” due to transmission errors. The three received bits, shown as (R1 R2 R3) in Figure 2 will be equal to one of the following eight triplets:
(R1 R2 R3) ={ (000), (001), (010), (100), (011), (101), (110), (111) }
When you look at the received triplet (R1 R2 R3) you must decide what was the bit “S” originally transmitted by using voting and the majority rule. Here are some examples of the majority rule.
• For example, if the three received bits are (R1 R2 R3)=(001), then the majority rule will decide that the bit must be a “0”. We denote this as the decoded bit D=0.
• As another example if the three received bits are (R1 R2 R3)=(101), then the majority rule will decode the bit as D=1.
• Another example: If you send S=0 three times, i.e. (S S S) = ( 0 0 0 ) and the received string is (R1 R2 R3) = (000), (001), (010), or (100) then the symbol will be decoded as D=0 and the experiment is a success, otherwise it is a failure.
• Another example: If you transmit S=1 three times, i.e. (S S S) = ( 1 1 1 ) and the received string is (011), (101), (110), or (111) the symbol will be decoded as D=1 and the experiment is a success, otherwise it is a failure.
• This procedure as described above is considered one experiment.
• Repeat the experiment N=100,000 times and count the number of successes.
• Find the probability that the transmitted bit "S" will be received and decoded incorrectly.
• Comment on whether the voting method used in this problem provides any improvement as compared to the method of Problem 1.
Probability of error with enhanced transmission
Ans.
p =
MESSAGE SEND 3 TIMESSEND 3 TIMES RECEIVE 3 BITS DECODE BY VOTING
(M) (S1 S2 S(S S S) 3) (R1 R2 R3) (D)
Figure 2. Enhanced message transmission method
1 file (1.3KB)
