Laplace Transform of the Received Power PDF

To facilitate the analysis of outage probabilities, the Laplace image of the probability density function (pdf) of the signal power received in a mobile channel appears a useful tool.

Laplace images of probability density functions are particularly useful in the evaluation of the joint power received from multiple interfering signals:
The pdf of the sum of powers of multiple signals is the convolution of the pdf of each component. After Laplace transformation, this convolution goes into a multiplication

The one-sided Laplace transform of a function f(x) is defined as the integral from zero to infinity over f(x) exp(-sx) , where s is the image variable.

Moreover, Laplace images are closely related to 'moment-generating functions' and 'characteristic functions' used in probability theory, although mostly the kernel 'exp(-jsx)' with j = SQRT(-1) is used for characteristic functions, rather than 'exp(-sx)'.

It can easily be shown that the probability that the power of a Rayleigh fading wanted signal sufficiently exceeds the power of an interferer equals the Laplace transform of the pdf of interference power, observed in one point. That is, the outage probability is found as a particular point on the above curve.

For Nakagami and Rician fading wanted signals the expression for outage probability is more complicated: it a series of Laplace transforms and derivatives of Laplace transforms

Click here for a discussion of how to write a PASCAL program that computes the outage probability in a channel with Rayleigh fading and shadowing.

Properties

• For s = 0, the laplace image function equals 1, because any pdf integrated over its domain yields unity.
• The n-th derivative in s = 0 equals (-1)^n time the n-th moment of the pdf
• For s goes to infinity, the image goes to zero, unless the pdf has a delta function at zero. This can be the case for an interferer that is switched off with some probability.