The probability that symbol j is sent and symbol k is received

In summary: The probability that symbol k was sent, given that symbol k is received, is calculated using the equations 1 and 2. The probability of error incurred in using this system is also calculated.
  • #1
SamTaylor
20
0

Homework Statement


A Communication system transmits signals labeled 1, 2, and 3. The probability
that symbol j is sent and symbol k is received is listed in the table for each
pair (j,k) of sent and received symbols. For example, the probability is 0.12
that a 1 is sent and, owning to noise in the channel 3 is received.

temp.jpg


Calculate the probability that the symbol k was sent, given that symbol k is
received, for k = 1,2,3, and calculate the probability of error incurred in
using this system. An Error is defined as the reception of any symbol other
than the one transmitted.

Homework Equations


1
[tex]P(B|A)=\frac{P(A|B)*P(B)}{P(A)}[/tex]
2
[tex]P(M)=P(A)*P(M|A) + P(B)*P(M|B) + P(C)*P(M|C)[/tex]

The Attempt at a Solution



Problem 1: Symbol k was sent, given that k is received
[tex]P = P_{11} + P_{22} + P_{33}[/tex]

Problem 2: Probabilty of Error
[tex]P_e = P - 1[/tex]

Or do the numbers inside the table represent conditional probabilitys?
P(received|send) ... for example P(3|1) = 0.12
 
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  • #2
I think you need to calculate p[1|1], p[2|2], p[3|3] separately using eqs. 1 and 2.
 
  • #3
I thought about that but how do i get P(K) or P(J) ?
 
  • #4
P(k)= P(k|1)+ P(k|2)+ P(k|3)
 
  • #5
[tex]P(K_m) = P(J_n)*P(K_m|J_1) + P(J_n)*P(K_m|J_2) + P(J_n)*P(K_m|J_3)[/tex]

[tex]P(K_1) = 0.10 + 0.07 + 0.10 = 0.27[/tex]
[tex]P(K_2) = 0.06 + 0.15 + 0.15 = 0.36[/tex]
[tex]P(K_3) = 0.12 + 0.05 + 0.20 = 0.37[/tex]

a)
[tex]P(J_1|K_1) = \frac{P(J_1)*P(K_1|J_1)}{P(K_1)} = \frac{0.10}{0.27} = 0.37[/tex]
[tex]P(J_2|K_2) = \frac{P(J_2)*P(K_2|J_2)}{P(K_2)} = \frac{0.15}{0.36} = 0.4167[/tex]
[tex]P(J_3|K_3) = \frac{P(J_3)*P(K_3|J_3)}{P(K_3)} = \frac{0.20}{0.37} = 0.541[/tex]

[tex]P(J_n) = P(K_1)*P(K_1|J_n) + P(K_2)*P(K_2|J_n) + P(K_3)*P(K_3|J_n)[/tex]
[tex]P(J_1) = 0.10 + 0.06 + 0.12 = 0.28[/tex]
[tex]P(J_2) = 0.07 + 0.15 + 0.05 = 0.27[/tex]
[tex]P(J_3) = 0.10 + 0.15 + 0.20 = 0.45[/tex]

[tex]P(K_1)*P(J_1|K_1) = 0.3700 * 0.28 = 0.1036[/tex]
[tex]P(K_2)*P(J_2|K_2) = 0.4167 * 0.27 = 0.1125[/tex]
[tex]P(K_3)*P(J_3|K_3) = 0.5410 * 0.20 = 0.1082[/tex]

b)
[tex]Q = P(K_1)*P(J_1|K_1) + P(K_2)*P(J_2|K_2) + P(K_3)*P(J_3|K_3) = 0.3243[/tex]
[tex]P_e = 1-Q = 0.6757[/tex]

Is that correct? If yes how do i know that the values inside the
table are the values for P(J)*P(K|J) and not only for P(K|J)?
 
  • #6
Ok, the last post was nonsense... sorry
 

What is the meaning of "The probability that symbol j is sent and symbol k is received"?

The probability that symbol j is sent and symbol k is received refers to the likelihood that a specific symbol, denoted as j, is sent from a source and successfully received as the same symbol, denoted as k, by a receiver.

How is the probability of a symbol being sent and received calculated?

The probability of symbol j being sent and symbol k being received is calculated by dividing the number of times symbol j is sent and received as symbol k by the total number of symbols sent and received. This can be represented as P(j→k) = n(j→k)/n, where n(j→k) represents the number of times j is sent and received as k, and n represents the total number of symbols sent and received.

What factors can affect the probability of a symbol being sent and received?

The probability of a symbol being sent and received can be affected by various factors such as the quality of the transmission channel, the coding scheme used, and the presence of noise or interference during transmission.

Why is it important to calculate the probability of symbols being sent and received?

Calculating the probability of symbols being sent and received is important in communication systems as it helps in understanding the performance and reliability of the system. It also allows for the optimization of the system to ensure a higher probability of successful symbol transmission and reception.

Can the probability of symbols being sent and received be improved?

Yes, the probability of symbols being sent and received can be improved by using error-correcting codes, improving the quality of the transmission channel, and reducing interference or noise during transmission. Additionally, using efficient modulation techniques and optimizing the system design can also help improve the overall probability of successful symbol transmission and reception.

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