Q function and Error functions

July 16, 2012 in Probability, Random Process

Q functions are often encountered in the theoretical equations for Bit Error Rate (BER) involving AWGN channel. A brief discussion on Q function and its relation to erfc function is given here.

Gaussian process is the underlying model for an AWGN channel.The probability density function of a Gaussian Distribution is given by

$p(x) = \frac{1}{ \sigma \sqrt{2 \pi}} e^{ - \frac{(x-\mu)^2}{2 \sigma^2}} \;\;\;\;\;\;\; (1)$

Generally, in BER derivations, the probability that a Gaussian Random Variable $X \sim N( \mu, \sigma^2)$ exceeds x₀ is evaluated as the area of the shaded region as shown in Figure 1.

Gaussian PDF and illustration of Q function

Mathematically, the area of the shaded region is evaluated as,

$Pr(X \geq x_0) = \int_{x_0}^{\infty} p(x) dx = \int_{x_0}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} e^{ - \frac{(x-\mu)^2}{2 \sigma^2}} dx \;\;\;\;\;\;\; (2)$

The above probability density function given inside the above integral cannot be integrated in closed form. So by change of variables method, we substitute

$y = \frac{x-\mu}{\sigma} \;\;\;\;\;\;\; (3)$

Then equation (3) can be re-written as,

$Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = \int_{ \left( \frac{x_{0} -\mu}{\sigma}\right)}^{\infty} \frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy \;\;\;\;\;\;\; (3)$

Here the function inside the integral is a normalized gaussian probability density function $Y \sim N( 0, 1)$ , normalized to mean=0 and standard deviation=1.

The integral on the right side can be termed as Q-function, which is given by,

$Q(z) = \int_{z}^{\infty}\frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy \;\;\;\;\;\;\; (4)$

Here the Q function is related as,

$Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = Q\left(\frac{x_0-\mu}{\sigma} \right ) = Q(z)\;\;\;\;\;\;\; (5)$

Thus Q function gives the area of the shaded curve with the transformation $y = \frac{x-\mu}{\sigma}$ applied to the Gaussian probability density function. Essentially, Q function evaluates the tail probability of normal distribution (area of shaded area in the above figure).

Error functions:

The error function represents the probability that the parameter of interest is within a range between $-x/ \sigma \sqrt{2}$ and $x/ \sigma \sqrt{2}$ and the complementary error function gives the probability that the parameter lies outside that range.

The error function is given by

$erf(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-x^2} dx\;\;\;\;\;\;\; (6)$

and the complementary error function is given by

$erfc(z) = 1 - erf(z) \;\;\;\;\;\;\; (7)$

or equivalently,

$erfc(z) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-x^2} dx\;\;\;\;\;\;\; (8)$

Q function and Complementary Error Function (erfc) :

From the limits of the integrals in equation (4) and (8) one can conclude that Q function is directly related to complementary error function (erfc).
It follows from equation (4) and (8), Q function is related to complementary error function by the following relation.

$Q(z) = \frac{1}{2} erfc \left( \frac{z}{\sqrt{2}}\right) \;\;\;\;\;\;\; (9)$

Some important results:

Keep a note of the following equations that can come handy when deriving probability of bit errors for various scenarios. These equations are compiled here for easy reference.

If we have a normal variable $X \sim N(\mu, \sigma^2)$ ,the probability that $X > x$ is
$Pr \left( X > x \right) = Q \left( \frac{x-\mu}{\sigma} \right ) \;\;\;\;\;\;\; (10)$

If we want to know the probaility that X is away from the mean by an amount a (on the left or right side of the mean), then
$Pr \left( X > \mu+a \right) = Pr \left( X < \mu-a \right) = Q\left(\frac{a}{\sigma} \right ) \;\;\;\;\;\;\; (11)$

If we want to know the probability that X is away from the mean by an amount a (on both sides of the mean), then
$Pr \left( \mu-a > X > \mu+a \right) = 2 Q\left(\frac{a}{\sigma} \right ) \;\;\;\;\;\;\; (12)$