Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.10 UNIT-STEP FUNCTIONS

Definition: Unit-Step Function The unit-step function is

(B.51)

It is also known as the Heaviside step function, and the symbol H(x) is sometimes used.

The unit-step function can also be defined in terms of the signum function:

(B.52)

Since x = 0 has Lebesgue measure zero, it is generally more convenient to use the definition in (B.51), which is the form mostly used in this book, especially when u(x) defines the support of a random variable. The advantage of (B.52) is that it is consistent with the definition of the signum function, which has no ambiguity about the value at x = 0 (though, as mentioned previously sgn(0) = 1 is often used in practice). The unit-step function is obtained from the Dirac delta function as follows:

(B.53)

Definition: Discrete Unit-Step Function The discrete unit-step function is

(B.54)

It can be expressed in terms of the Kronecker delta function as follows:

(B.55)