Q-function - 2-1 2 erf ± α √ 2 ¶ ; erf( α ) = 1-2 Q (...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Last version available at www.eng.tau.ac.il/ jo/teaching Q function and error function We first note that Z -∞ e - x 2 dx = π ; Z -∞ e - ax 2 2 dx = r 2 π a For our needs in Digital Communication course, we define: Q ( α ) Δ = 1 2 π Z α e - x 2 2 dx The Q ( · ) function is monotonically decreasing. Some features: Q ( -∞ ) = 1 ; Q (0) = 1 2 ; Q ( ) = 0 ; Q ( - x ) = 1 - Q ( x ) Known bounds (valid for x > 0): 1 2 πx ± 1 - 1 x 2 e - x 2 / 2 < Q ( x ) < 1 2 πx e - x 2 / 2 Q ( x ) 1 2 e - x 2 / 2 Matlab does not have a build-in function for Q ( · ). Instead, we use its erf function: erf( α ) Δ = 2 π Z α 0 e - x 2 dx Note that erf function is defined over [0 , ) only, and erf(0) = 0 ; erf( ) = 1 The relations between the two functions are Q ( α ) = 1
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2-1 2 erf ± α √ 2 ¶ ; erf( α ) = 1-2 Q ( √ 2 α ) If we have a normal variable X ∼ N ( μ,σ 2 ), the probability that X > x is Pr { X > x } = Q ± x-μ σ ¶ Now, if we want to know the probability of X to be away from its expectation μ by at least a (either to the left or to the right) we have: Pr { X > μ + a } = Pr { X < μ-a } = Q ± a σ ¶ The probability to be away from the center where we don’t matter in which direction is 2 · Q ( a σ ). This version compiled on April 6, 2006...
View Full Document

This note was uploaded on 04/08/2010 for the course ENGINEERIN 50-22-43-2 taught by Professor Prizler during the Spring '10 term at Tel Aviv Uni..

Ask a homework question - tutors are online