Signal detection: Consider an additive noise channel ( Y=X+W ). The signal [ X=left{egin{array}{ll} +1, & ext { with probability } frac{1}{4} \ -1, & ext { with probability } frac{3}{4} end{array} ight. ] and the noise ( W ) is a Gaussian with mean 0 and variance 4 . The observation ( Y ), the signal ( X ) and the noise ( W ) are independent. a. Write the MAP detection rule for ( X ). b. Find an optimum threshold for a simpler detection rule. c. Find the probability of error in terms of ( Q ) functions.
Added by Sherlock M.
Close
Step 1
The Maximum A Posteriori (MAP) detection rule for X is given by comparing the posterior probabilities of X given Y. We have two hypotheses, H1: X = +1 and H2: X = -1. The MAP rule is to decide for H1 if P(H1|Y) > P(H2|Y) and decide for H2 otherwise. Using Bayes' Show more…
Show all steps
Your feedback will help us improve your experience
Rachel Gore and 56 other Probability educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Recommended Videos
A pseudo-random signal (X) of 0 or 5 is transmitted over a noisy channel. X=0 is transmitted with probability of 3/4. The received signal, Y, is the sum of the sent signal and noise, where the noise is modeled as a normal Gaussian with μ = 0, σ = 2 . The received signal is decoded as 0 for Y ≤ 3 and 1 for Y > 3 . a. What is the probability of error in this system? b. Assume that answer to part a was 0.1 for mathematical simplicity. To improve the system, each signal is transmitted three times and output is determine by majority rule (2 or more). What is the new probability of error?
Sri K.
Problem 4.67: A binary transmission system transmits signal X (-1 to send a "0" bit; +1 to send a "1" bit). The received signal is Y = X + N, where noise N has a zero-mean Gaussian distribution with variance σ^2. Assume that "0" bits are three times as likely as "1" bits. (a) Find the conditional pdf of Y given the input value: fy (y|X = +1) and fy (y|X = -1). (b) The receiver decides a "0" was transmitted if the observed value of y satisfies fy(y|X = -1)P(X = -1) > fy(y|X = +1)P(X = +1), and it decides a "1" was transmitted otherwise. Use the results from part a to show that this decision rule is equivalent to: If Y < T, decide "0"; if Y > T, decide "1". (c) What is the probability that the receiver makes an error given that X = +1 was transmitted? X = -1 was transmitted? Assume σ^2 = 1/16. (d) What is the overall probability of error?
A Gaussian random process X(t) has zero mean and a power spectral density given below. Find the probability that X(t) takes a value outside the interval (-1.5 ... + 1.5). (b) The random process in (a) is affected by additive Gaussian white noise with zero mean and a variance of σ^2_NN = 0.25, resulting in a noisy process Z(t). The noise N(t) is independent from X(t). Find the probability that Z(t) takes a value outside the interval (-1.5 ... + 1.5).
Madhur L.
Recommended Textbooks
Probability with Applications in Engineering, Science, and Technology
Probability and Statistics for Engineers and Scientists
Applied Statistics and Probability for Engineers
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD