Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Set Theory, Logic, Probability, Statistics
How to calculate the probability of error in an AWGN channel?
Reply to thread
Message
[QUOTE="EngWiPy, post: 5987742, member: 157016"] Eq. 4 is the distribution of the received signal ##y##, given that signal ##x_i## was transmitted, where ##x_i## can be any of the ##M## available input signals. So, what we want to find is, for a given received signal, what signal was transmitted, right? To find this, we need to find the signal that has the maximum conditional probability ##p(x_i/y)=\frac{p(y/x_i)\,p(x_i)}{p(y)}## using Bayes rule. But since all signals have the same probabilities ##p(x_i)##, and ##p(y)## is common to all possible transmitted signals, the problem is reduced to finding the signal with the maximum ##p(y/x_i)##. The received signal is ##y=x_i+n##, where ##n## is an additive white Gaussian noise with zero mean and variance ##\sigma_n^2##. So, for a given ##x_i##, ##y## becomes a Gaussian random variable with mean ##x_i## and a variance ##\sigma_n^2##. Eq. 4 describes just this. How to find the transmitted signal that maximizes ##p(y/x_i)##? Simple, find the signal point that is closest to ##y##. Now, how to quantify the error probability, you need to follow the steps I showed. So, to answer your question: Eq. 4 is the distribution of the received signal ##y##, given ##x_i## was transmitted. Which means that, if you transmit ##x_i## over an AWGN channel a large number of times, the observed received signal at the receiver will follow a Gaussian distribution centered around ##x_i##, with a variance equals to the noise variance. ##Q\left[\frac{1}{\sqrt{\sigma_n^2}}\right]## is that quantity that measures the probability of error. That is, if we transmit ##x_i## a large number of times, how many times from the total times ##y## won't be closest to ##x_i##, which results in erroneous decision. Hope this helps. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Set Theory, Logic, Probability, Statistics
How to calculate the probability of error in an AWGN channel?
Back
Top