Q function and Error functions : demystified

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In simple words, The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value. The erf function gives the probability that a normally distributed variable will fall within a certain range.

Q function

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

\[p(x) = \displaystyle{ \frac{1}{ \sigma \sqrt{2 \pi}} e^{ – \frac{(x-\mu)^2}{2 \sigma^2}}}\quad\quad (1) \]

Generally, in BER derivations, the probability that a Gaussian Random Variable \(X \sim N ( \mu, \sigma^2) \) exceeds \(x_0\) is evaluated as the area of the shaded region as shown in Figure 1.

Probability density function for Gaussian distribution with tails shaded Q function error function
Figure 1: Gaussian PDF and illustration of Q function

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

\[Pr(X \geq x_0) =\displaystyle{ \int_{x_0}^{\infty} p(x) dx = \int_{x_0}^{\infty} \frac{1}{ \sigma \sqrt{2 \pi}} e^{ – \frac{(x-\mu)^2}{2 \sigma^2}} dx } \quad\quad (2) \]

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

\[\displaystyle{ y = \frac{x-\mu}{\sigma} }\]

Then equation (2) can be re-written as,

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

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

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

\[\displaystyle{Q(z) = \int_{z}^{\infty}\frac{1}{ \sqrt{2 \pi}} e^{- \frac{y^2}{2}} dy } \quad\quad (4)\]

Here the Q function is related as,

\[\displaystyle{ Pr\left( y > \frac{x_0-\mu}{\sigma} \right ) = Q\left(\frac{x_0-\mu}{\sigma} \right ) = Q(z)} \quad\quad (5)\]

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).

The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value.

Error function

The complementary error function represents the area under the two tails of zero mean Gaussian probability density function of variance \(\sigma^2 = 1/2\). The error function gives the probability that the parameter lies outside that range.

Complementary error function and error function
Figure 2: Complementary error function and error function

Therefore, the complementary error function is given by

\[\displaystyle{ erfc(z) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-x^2}} dx \quad\quad (6)\]

Hence, the error function is

\[erf(z) = 1 – erfc(z) \quad\quad (7)\]

or equivalently,

\[\displaystyle{ erf(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-x^2} dx } \quad\quad (8) \]

The erf function gives the probability that a normally distributed variable will fall within a certain range.

Q function and Complementary Error Function (erfc)

From the limits of the integrals in equation (4) and (6) one can conclude that Q function is directly related to complementary error function (erfc). It follows from equation (4) and (6), Q function is related to complementary error function by the following relation.

\[\displaystyle{ Q(z) = \frac{1}{2} erfc \left( \frac{z}{\sqrt{2}}\right)} \quad\quad (9) \]

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

\[\displaystyle{ Pr \left( X > x \right) = Q \left( \frac{x-\mu}{\sigma} \right ) } \quad\quad (10) \]

If we want to know the probability that \(X\) is away from the mean by an amount ‘a’ (on the left or right side of the mean), then

\[\displaystyle{ Pr \left( X > \mu+a \right) = Pr \left( X < \mu-a \right) = Q\left(\frac{a}{\sigma} \right ) } \quad\quad (11) \]

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

\[\displaystyle{ Pr \left( \mu-a > X > \mu+a \right) = 2 Q\left(\frac{a}{\sigma} \right ) } \quad\quad (12)\]

Application of Q function in computing the Bit Error Rate (BER) or probability of bit error will be the focus of our next article.

Applications

The Q-function and the error function (erf) are important mathematical functions that arise in many fields, including probability theory, statistics, signal processing, and communications engineering. Here are some reasons why these functions are important:

  1. Probability calculations: The Q-function and erf function are used in probability calculations involving Gaussian distributions. The Q-function gives the probability that a random variable from a normal distribution will exceed a certain threshold value. The erf function gives the probability that a normally distributed variable will fall within a certain range.
  2. Signal processing: In signal processing, the Q-function is used to calculate the probability of bit error in digital communication systems. This is important for designing communication systems that can reliably transmit data over noisy channels.
  3. Statistical analysis: The Q-function and erf function are used in statistical analysis to model data and estimate parameters. For example, in hypothesis testing, the Q-function can be used to calculate p-values.
  4. Mathematical modeling: The Q-function and erf function arise naturally in mathematical models for various phenomena. For example, the heat equation in physics and the Black-Scholes equation in finance both involve the erf function.
  5. Computational efficiency: In some cases, the Q-function and erf function provide a more efficient and accurate way of calculating certain probabilities and integrals than other methods.

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