Question

Consider a binary communication channel with the following characteristics. The transmitter sends "0" and "1" bits with equal probability (P[IN=0] = P[IN=1] = 0.5). Some bit errors are introduced because of the additive noise. Without noise, the responses to inputs "0" and "1" are 0.75 and 2.5, respectively (i.e., min = 0.75 and Imax = 2.5). The noise v is random with a Gaussian distribution with zero mean and a standard deviation of 1.5. Let y = r + v be the output of the channel. Detecting the channel output is done by comparing the channel output to a threshold T. So if y < T, out = 0; if y ≥ T, out = 1, where T is the threshold.

a) Sketch the pdf of y given that the input bit is 1. Note: Clearly show in your sketch the location of the x and y coordinates of the peak in the pdf, and the pdf should drop to about 60% of its peak value at the standard deviation.

Answer 1

To sketch the PDF of 'y' when the input bit is '1' in a binary communication channel with additive noise, consider that 'y' follows a Gaussian distribution with a mean of 2.5 and a standard deviation of 1.5.

Step-by-step explanation:

For this problem, we are given a binary communication channel with certain characteristics. The transmitter sends '0' and '1' bits with equal probability. The noise introduced in the channel is random and follows a Gaussian distribution. We are asked to sketch the probability density function (PDF) of the output 'y' when the input bit is '1'.

To sketch the PDF of 'y' when the input bit is '1', we need to consider that 'y' is the sum of the channel output 'r' and the noise 'v'. Since 'v' follows a Gaussian distribution with zero mean and a standard deviation of 1.5, 'y' will also follow a Gaussian distribution with a mean equal to 2.5 (the channel response to input '1') and a standard deviation equal to 1.5. The PDF will be centered at 2.5 and drop to about 60% of its peak value at 1.5 standard deviations from the mean.