Q function and Error functions
Gaussian process is the underlying model for an AWGN channel.The probability density function of a Gaussian Distribution is given by
Generally, in BER derivations, the probability that a Gaussian Random Variable exceeds x0 is evaluated as the area of the shaded region as shown in Figure 1.
Mathematically, the area of the shaded region is evaluated as,
The above probability density function given inside the above integral cannot be integrated in closed form. So by change of variables method, we substitute
Then equation (3) can be re-written as,
Here the function inside the integral is a normalized gaussian probability density function , normalized to mean=0 and standard deviation=1.
The integral on the right side can be termed as Q-function, which is given by,
Here the Q function is related as,
Thus Q function gives the area of the shaded curve with the transformation 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).
The error function represents the probability that the parameter of interest is within a range between and and the complementary error function gives the probability that the parameter lies outside that range.
The error function is given by
and the complementary error function is given by
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.
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 ,the probability that is
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
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
Application of Q function in computing the Bit Error Rate (BER) or probability of bit error will be the focus of our next article.
Reference: Normal Distribution Function – Wolform MathWorld