Oscar uses his high-speed modem to connect to the internet. The modem transmits

Question

Oscar uses his high-speed modem to connect to the internet. The modem transmits zeros and ones by sending signals -1 and +1, respectively. We assume that any given bit has probability p of being a zero. The network cable introduces additive zero-mean Gaussian noise with variance sigma^2 (so, the receiver at the other end receives a signal which is the sum of the transmitted signal and the channel noise). The value of the noise is assumed to be independent of the encoded signal value. Let a be a constant between -1 and 1. The receiver at the other end decides that the signal -1 (respectively, +1) was transmitted if the value it receives is less (respectively, more) than a. What is P[error when a +1 is transmitted]? What is P[error when a -1 is transmitted]? Considering both cases in (a) and (b), what is the overall probability of making an error? Find a numerical answer for overall error probability assuming that p = 2/5, a = 1/2, and sigma^2=1/4.

Explanation / Answer

An error occurs whenever 1 was transmitted and the received signal is more than a so 1 + N > a, or whenever a +1 was sent and the received signal is less than a or 1 + N < a where N is the noise. The probability of error thus given by: P(error) = P(error|send 1)P(send 1) + P(error|send 1)P(send 1) = P(1 + N > a)p + P(1 + N < a)(1 p) = P(N > a + 1)p + P(N < a 1)(1 p) = P(N > a + 1)p + (1 P(N a 1))(1 p) = P µ N > a + 1 ¶ p + (1 P µ N a 1 ¶ )(1 p) = Q µ a + 1 ¶ p + (1 Q µ a 1 ¶ )(1 p) = Q µ a + 1 ¶ p + (1 Q µ a 1 ¶ )(1 p) = Q µ a + 1 ¶ p + Q µ 1 a ¶ (1 p) where Q(x) = Z x 1 2 exp µ z 2 2 ¶ is the complementary cumulative distribution function (CCDF) of a standard normal random variable (that you will study in the next homework!) There is no closed form for Q(·), but it is a standard function that you will be using quite often

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