QR-Code reader for .net
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications.
In this section, we give an overview on the background and mathematical concepts in information theory in order to establish Shannon s channel coding theorem. Concepts of ergodic and outage capacities will be followed. Finally, we give several examples of channel capacity in various channels. This will form the basis for the remaining chapters in the book.
.NET qr bidimensional barcode draweron .net
using .net framework toprint qr code iso/iec18004 with web,windows application
Visual .net denso qr bar code decoderin .net
Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications.
Entropy and Mutual Information
Connect barcode with .net
using visual studio .net crystal toassign barcode for web,windows application
The rst important concept in information theory is entropy, which is a measure of uncertainty in a random variable. De nition 1.2 (Entropy of Discrete Random Variable) The entropy of a discrete random variable X with probability mass function p(X) is given by H ( X ) = - p( x) log 2 ( p( x)) = -e [log 2 ( p( X ))]
Barcode scanner with .net
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Control qr bidimensional barcode data for visual
to include qr bidimensional barcode and qrcode data, size, image with visual barcode sdk
where the expectation is taken over X. De nition 1.3 (Entropy of Continuous Random Variable) On the other hand, the entropy of a continuous random variable X with probability density function f(X) is given by H ( X ) = - f ( x) log 2 ( f ( x))dx
Qr Bidimensional Barcode barcode library in .net
generate, create qr codes none in .net projects
Control qr code image on visual basic
use visual .net qr code encoder tomake qr for visual basic
For simplicity, we shall assume discrete random variable unless otherwise speci ed. De nition 1.4 (Joint Entropy) The joint entropy of two random variables X1, X2 is de ned as H ( X 1 , X 2 ) = - p( x1 , x2 ) log 2 ( p( x1 , x2 )) = -e [log 2 ( p( X 1 , X 2 ))] (1.53)
Receive bar code on .net
using .net toget barcode for web,windows application
x1 ,x2
2D Barcode integrating in .net
using barcode creation for .net framework control to generate, create 2d matrix barcode image in .net framework applications.
where the expectation is taken over (X1, X2). De nition 1.5 (Conditional Entropy) The conditional entropy of a random variable X2 given X1 is de ned as H ( X 2 X 1 ) = - p( x1 , x2 ) log 2 ( p( x2 x1 )) = -e [log 2 ( p( X 2 X 1 ))]
Code 3 Of 9 creation on .net
using barcode integration for visual .net crystal control to generate, create barcode code39 image in visual .net crystal applications.
x1 ,x2
.net Framework Crystal usps postnet barcode integratingon .net
using barcode implementation for .net crystal control to generate, create usps postal numeric encoding technique barcode image in .net crystal applications.
Control data matrix size with visual basic
data matrix barcode size on vb
where the expectation is taken over (X1, X2). After introducing the de nitions of entropy, let s look at various properties of entropy. They are summarized as lemmas below. Please refer to the text by Cover and Thomas [30] for the proof. The rst lemma gives a lower bound on entropy. Lemma 1.2 (Lower Bound of Entropy) H (X ) 0 Equality holds if and only if there exists x0 X such that p(x0) = 1. (1.55)
Make barcode for c#
using .net toprint barcode on web,windows application
EAN-13 encoding with visual c#
generate, create european article number 13 none on visual c# projects
Proof Directly obtained from Equation (1.51). The following lemma gives an upper bound on entropy for discrete and continuous random variable X. Lemma 1.3 (Upper Bound of Entropy) If X is a discrete random variable, then H ( X ) log 2 ( X ) (1.56)
Get barcode pdf417 on visual c#
using barcode encoder for aspx crystal control to generate, create pdf417 image in aspx crystal applications.
Equality holds if and only if p(X) = 1/|X|. On the other hand, if X is a continuous random variable, then H (X ) 1 2 log 2 (2pes X ) 2 (1.57)
Assign upc a in office excel
using office excel toproduce upc a in web,windows application
2 where sX = e[|X|2] and equality holds if and only if X is Gaussian distributed 2 with an arbitrary mean m and variance sX .
.net For Windows Forms Crystal gs1-128 drawerfor vb
using .net for windows forms crystal tocompose ucc ean 128 in web,windows application
From the lemmas presented-above, entropy can be interpreted as a measure of information because H(X) = 0 if there is no uncertainty about X. On the other hand, H(X) is maximized if X is equiprobable or X is Gaussian distributed. The chain rule of entropy is given by the following lemma. Lemma 1.4 (Chain Rule of Entropy) H (X1 , X 2 ) = H (X1 ) + H (X 2 X1 ) (1.58)
Matrix Barcode implement on .net
use visual studio .net (winforms) matrix barcode creator toreceive 2d barcode for .net
On the other hand, as summarized in the lemma below, conditioning reduces entropy. Lemma 1.5 (Conditioning Reduces Entropy) H (X Y ) H (X ) (1.59)
Control qrcode data for office word
to print denso qr bar code and qr code data, size, image with office word barcode sdk
Equality holds if and only if p(XY) = p(X)p(Y) (X and Y are independent). Lemma 1.6 (Concavity of Entropy) H(X) is a concave function of p(X). Lemma 1.7 If X and Y are independent, then H(X + Y) H(X). Lemma 1.8 (Fano s Inequality) Given two random variables X and Y, let X = g(Y) be an estimate of X given Y. De ne the probability of error as