definition

Ordinary function

Mathematically, if a function in the real number field can be expressed by the finite linear combination of indicator function in the semi-open interval, then this function is a step function. Step function is a combination of finite segment piecewise constant functions.

Step function is singular function, t

generalized function

According to the generalized function theory, the unit step function ε(t) is defined as:

That is, the function of step function ε(t) and the function of test function φ(t) is to give it a numerical value, which is equal to the definite integral of φ(t) in the interval of (0, ∞).

Relationship with unit pulse function

The unit impulse function is equal to the derivative of the unit step function to the time variable:? ;

Conversely, the unit step function is equal to the integral of the unit impulse function:? .

Extended data:

App application

Signal processing

Using step signal to represent complex signal can simplify the study of some characteristics of complex signal.

The linear combination of step signal and its delayed step signal is expressed or approximated, and then the spectrum of more complex signals can be discussed by using the superposition principle of the system and the spectrum and frequency domain characteristics of simple signals such as unit step signal. Thereby reducing the difficulty of calculating the spectrum of complex signals.

integral transformation

When doing integral transformation, the original function and the image function defined by segments must be processed in segments, which is often troublesome and easy to make mistakes.

Step functions can be used to represent piecewise defined functions in a unified form. Cutting the function or uniformly expressing the function defined by segments as a function defined on the integer axis often makes the transformation simple, simplifies the operation and reduces the error.

References:

Step function-Baidu Encyclopedia