Q_func

Q_func - 1 Q-function Handout c Keith M Chugg 1 Definitions...

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1 Q -function Handout – c ± Keith M. Chugg 1 Definitions The Q -function is tail integral of a unit-Gaussian pdf, and is defined as Q ( z ) Δ = Z z 1 2 π e - x 2 2 dx. The Q -function has the following properties: lim z →∞ Q ( z )=0 lim z →-∞ Q ( z )=1 Q (0)=1 / 2 Q ( - z - Q ( z ) . There are several other common notations used to denote this integral function or a close relative. The Q -function is sometime referred to as the “Gaussian Integral Function” and denoted GIF( z ). Other functions which are closely related are the erf( · ) (error function) and erfc( · ) (complementary error function): erf( z ) Δ = Z z 0 2 π e - x 2 dx z 0 erfc( z ) Δ = Z z 2 π e - x 2 dx =1 - erf( z ) z 0 The Q -function is related to these functions by Q ( z )= 1 2 " 1 - erf ˆ z 2 !# = 1 2 erfc ˆ z 2 ! z 0 . It is clear that if X ( u ) is a mean zero, unit variance Gaussian random variable, that Q ( z - F X ( u ) ( z ) . A useful relation is that if Y ( u ) is Gaussian with mean m and variance σ 2 , then Pr { Y ( u ) >a } = Q ± a - m σ .
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Q_func - 1 Q-function Handout c Keith M Chugg 1 Definitions...

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