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## 1. Boxcar Function

The **boxcar function** is a special function that has a value of zero everywhere except one single interval where it equals a constant. Its graph has the shape of a boxcar sitting on a “railway track” (the x axis).

We can write the boxcar function as:

where H is the Heaviside Step Function.

Since you can choose any constant value and any size interval, there are an infinite number of possible versions of this function.

## Applications of Boxcar Functions

The simple versatility of this function makes it important for a wide variety of applications. It is especially useful in computer science, signal analysis and other engineering applications.

For example, in signal analysis, multiplying the boxcar function by the signal gives a representation of what happens when the signal is turned on for a brief period of time and then turned off.

## Rectangular Function

The **rectangular function** (also called the *rectangle function*, *gate function*, or *window function*) is a special case of the boxcar function. Its graph that looks like a rectangular box centered over the y axis.

The rectangular function is important in electric engineering, simulation, data processing and many other fields. It can be used to check how a system responds to a sudden input. It is also often used to model probability, as a probability density function (PDF). When a PDF is equal to the probability function, it is a special case of the continuous uniform distribution. This PDF would be useful in situations where an event is likely to happen only over a single well defined interval; with a lesser chance of happening right at the borderline.

The characteristic function of this PDF is

This is an alternative way of describing the PDF, and it describes it completely.

## Unit Rectangle (Pulse Function)

The **unit rectangle function**, also called the *pulse function*, is a special case of the boxcar function. It is defined as:

It has the shape of a 1 x 1 boxcar, centered at the origin.

## References

De Oliveira, Mauricio. Continuous-time Signals (aka Analog Signals). MAE 143A Lecture Notes. Retrieved from http://control.ucsd.edu/mauricio/courses/mae143a/lectures/1analogsignals.pdf on Feb 2, 2019.

Hrynkiewicz E. (2011) 2 Rectangular Function Π(x) and Its Application for Description of Some Logical Devices Operation. In: Adamski M., Barkalov A., Węgrzyn M. (eds) Design of Digital Systems and Devices. Lecture Notes in Electrical Engineering, vol 79. Springer, Berlin, Heidelberg. Retrieved from https://link.springer.com/chapter/10.1007/978-3-642-17545-9_2 on August 26, 2019

de Oliveira, Mauricio. Continuous-time Signals (AKA analog signals).From MAE 143 A – Signals & Systems – Lecture Notes. Published online Winter 2011. Retrieved from http://control.ucsd.edu/mauricio/courses/mae143a/lectures/1analogsignals.pdf on August 26, 2019

Yen-Ping Shih and Chyi Hwang. Application of block pulse functions in dynamic simulation. Computers & Chemical Engineering Volume 6, Issue 1, 1982, Pages 7-13. Retrieved from https://www.sciencedirect.com/science/article/pii/0098135482800021 on Feb 2, 2019.

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