Signals and Systems/Engineering Functions

Basic Functions
Oftentimes, complex signals can be simplified as linear combinations of certain basic functions (a key concept in Fourier analysis), which are useful to the field of engineering. These functions will be described here, and studied more in the following chapters.

Unit Step Function
The unit step function and the impulse function are considered to be fundamental functions in engineering, and it is strongly recommended that the reader becomes very familiar with both of these functions.

The unit step function, also known as the Heaviside function, is defined as such:


 * $$u(t) = \left\{

\begin{matrix} 0, & \mbox{if }t < 0 \\                 1, & \mbox{if }t \ge 0 \end{matrix} \right. $$

Sometimes, u(0) is given other values, usually either 0 or 1. For many applications, it is irrelevant what the value at zero is. u(0) is generally written as undefined.

Derivative
The unit step function is level in all places except for a discontinuity at t = 0. For this reason, the derivative of the unit step function is 0 at all points t, except where t = 0. Where t = 0, the derivative of the unit step function is infinite.

The derivative of a unit step function is called an impulse function. The impulse function will be described in more detail next.

Integral
The integral of a unit step function is computed as such:


 * $$\int_{-\infty}^t u(s)ds =

\left\{ \begin{matrix} 0, & \mbox{if }t < 0   \\ \int_0^t ds = t, & \mbox{if }t \ge 0 \end{matrix} \right\} = tu(t)$$

In other words, the integral of a unit step is a "ramp" function. This function is 0 for all values that are less than zero, and becomes a straight line at zero with a slope of +1.

Time Inversion
if we want to reverse the unit step function, we can flip it around the y axis as such: u(-t). With a little bit of manipulation, we can come to an important result:


 * $$u(-t) = 1 - u(t)$$, while $$ t \neq 0 $$

Other Properties
Here we will list some other properties of the unit step function:


 * $$u(\infty) = 1$$
 * $$u(-\infty) = 0$$
 * $$u(t) + u(-t) = 1$$, while $$ t \neq 0 $$

These are all important results, and the reader should be familiar with them.

Impulse Function
An impulse function is a special function that is often used by engineers to model certain events. An impulse function is not realizable, in that by definition the output of an impulse function is infinity at certain values. An impulse function is also known as a "delta function", although there are different types of delta functions that each have slightly different properties. Specifically, this unit-impulse function is known as the Dirac delta function. The term "Impulse Function" is unambiguous, because there is only one definition of the term "Impulse".

Let's start by drawing out a rectangle function, D(t), as such: We can define this rectangle in terms of the unit step function:


 * $$D(t) = \frac{1}{A}[u(t + A/2) - u(t - A/2)]$$

Now, we want to analyze this rectangle, as A becomes infinitesimally small. We can define this new function, the delta function, in terms of this rectangle:


 * $$\delta(t) = \lim_{A \to 0}\frac{1}{A}[u(t + A/2) - u(t - A/2)]$$

We can similarly define the delta function piecewise, as such:


 * 1) $$\delta(t) = 0\mbox{ for } t \ne 0 $$.
 * 2) $$\delta(t) = +\infty \mbox{ for } t = 0 $$.
 * 3) $$\int_{-\infty}^\infty \delta(t) dt = 1 $$.

Although, this definition is less rigorous than the previous definition.

Integration
From its definition it follows that the integral of the impulse function is just the step function:


 * $$\int \delta(t)dt = u(t)$$

Thus, defining the derivative of the unit step function as the impulse function is justified.

Shifting Property
Furthermore, for an integrable function f:


 * $$\int_{-\infty}^\infty \delta(t - A) f(t)dt = f(A)$$

This is known as the shifting property (also known as the sifting property or the sampling property) of the delta function; it effectively samples the value of the function f, at location A.

The delta function has many uses in engineering, and one of the most important uses is to sample a continuous function into discrete values.

Using this property, we can extract a single value from a continuous function by multiplying with an impulse, and then integrating.

Types of Delta
There are a number of different functions that are all called "delta functions". These functions generally all look like an impulse, but there are some differences. Generally, this book uses the term "delta function" to refer to the Dirac Delta Function.


 * Dirac delta function
 * Kronecker delta

Sinc Function
There is a particular form that appears so frequently in communications engineering, that we give it its own name. This function is called the "Sinc function" and is discussed below:

The Sinc function is defined in the following manner:


 * $$\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \mbox{ if } x \neq 0$$

and


 * $$\operatorname{sinc}(0) = 1 $$

The value of sinc(x) is defined as 1 at x = 0, since


 * $$\lim_{x \rightarrow 0}\operatorname{sinc}(x) = 1$$.

This fact can be proven by noting that for x near 0,


 * $$1 > \frac{\sin{(x)}}{x} > \cos{(x)}$$.

Then, since cos(0) = 1, we can apply the Squeeze Theorem to show that the sinc function approaches one as x goes to zero. Thus, defining sinc(0) to be 1 makes the sinc function continuous.

Also, the Sinc function approaches zero as x goes towards infinity, with the envelope of sinc(x) tapering off as 1/x.

Rect Function
The Rect Function is a function which produces a rectangular-shaped pulse with a width of 1 centered at t = 0. The Rect function pulse also has a height of 1. The Sinc function and the rectangular function form a Fourier transform pair.

A Rect function can be written in the form:


 * $$\operatorname{rect}\left(\frac{t-X}{Y} \right)$$

where the pulse is centered at X and has width Y. We can define the impulse function above in terms of the rectangle function by centering the pulse at zero (X = 0), setting its height to 1/A and setting the pulse width to A, which approaches zero:


 * $$\delta(t) = \lim_{A \to 0} \frac{1}{A}\operatorname{rect}\left(\frac{t-0}{A}\right)$$

We can also construct a Rect function out of a pair of unit step functions:


 * $$\operatorname{rect}\left(\frac{t-X}{Y} \right) = u(t - X + Y/2) - u(t - X - Y/2)$$

Here, both unit step functions are set at distance of Y/2 away from the center point of (t - X).

Square Wave
A square wave is a series of rectangular pulses. Here are some examples of square waves: