A digital signal receiver decodes bits of incoming signal as 0 s or 1 and makes an error in decoding a bit with probability 10−4 .
Assuming decoding success is independent for different bits, as the receiver receives more and more signals, what is the fraction of erroneously decoded bits?
10^-4
To find the fraction of erroneously decoded bits, we can use the concept of the probability of events.
Let's assume that the probability of a bit being decoded correctly is p, and the probability of a bit being decoded incorrectly (error) is q.
Since the decoding success is independent for different bits, we can say that q is the same for all the bits.
Since the receiver makes an error in decoding a bit with a probability of 10^-4, we have:
q = 10^-4
Similarly, the probability of a bit being decoded correctly is given by:
p = 1 - q
Now, let's consider a large number of bits, say N bits. The fraction of erroneously decoded bits, denoted as f, can be calculated as follows:
f = (number of erroneously decoded bits) / (total number of bits)
For a total of N bits, the number of erroneously decoded bits is N*q, because each bit has a probability of q of being decoded incorrectly.
Therefore, the fraction of erroneously decoded bits can be written as:
f = N*q / N
f = q
Substituting the value of q, we get:
f = 10^-4
Hence, as the receiver receives more and more signals (bits), the fraction of erroneously decoded bits converges to 10^-4.
To find the fraction of erroneously decoded bits, we need to consider the probability of error for each bit and then calculate the average error rate over a large number of bits.
Given that the error in decoding a bit occurs with a probability of 10^(-4), the probability of successfully decoding a bit is 1 - 10^(-4) = 0.9999.
Assuming the decoding success is independent for different bits, we can calculate the probability of error for multiple bits by multiplying the probabilities. For example, the probability of two consecutive bits being erroneously decoded is (10^(-4))^2.
Let's denote the fraction of erroneously decoded bits as p, and the probability of error for a single bit as q = 10^(-4).
For one bit, the fraction of erroneously decoded bits is p = q = 10^(-4).
For two consecutive bits, the fraction of both being erroneously decoded is p^2 = (10^(-4))^2.
Similarly, for n consecutive bits, the fraction of all bits being erroneously decoded is p^n = (10^(-4))^n.
As the receiver receives more and more signals, the fraction of erroneously decoded bits approaches 1.
In the limit, as n approaches infinity, p^n approaches 1, implying that the fraction of erroneously decoded bits approaches 100% or 1.
So, as the receiver receives more and more signals, the fraction of erroneously decoded bits tends to be 1 or 100%.