Signals and Systems/Engineering Functions

Basic Functions

Often times, complex signals can be simplified as linear combinations of certain basic functions (a key concept in Fourier analysis). These basic functions, which are useful to the field of engineering, receive little or no coverage in traditional mathematics classes. 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.

Unit Step Function

Shifted Unit Step function

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

$u(t)=\{\begin{array}{cc}0,& \text{if }t<0\\ 1,& \text{if }t\ge 0\end{array}$

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:

$\underset{-\mathrm{\infty }}{\overset{t}{\int }}u(s)ds=\left\{\begin{array}{cc}0,& \text{if }t<0\\ \underset{0}{\overset{t}{\int }}ds=t,& \text{if }t\ge 0\end{array}\right\}=tu(t)$

In other words, the integral of a unit step is a "romp" 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:

$u(-t)=1-u(t)$

Other Properties

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

• $u(\mathrm{\infty })=1$
• $u(-\mathrm{\infty })=0$
• $u(t)+u(-t)=1$

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:

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

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

1. $\delta (t)=0,\text{ if }t\ne 0$.

    

2. $\delta (t)\text{ is undefined for }t=0$.

    

3. $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (t)dt=1$.

Although, this definition is less rigorous then 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.

Sampling Theorem

Furthermore, for an integrable function f:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (t-A)f(t)dt=f(A)$

This is known as the sampling theorem, because 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.

The sampling theorem is written as follows:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(t)\delta (t-\alpha )dt=f(\alpha )$

Using this theory, 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.

• w:Dirac delta function
• w: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:

$\mathrm{sinc}(x)=\frac{\sin (\pi x)}{\pi x}\text{ if }x\ne 0$

and

$\mathrm{sinc}(0)=1$

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

$\underset{x\to 0}{\lim }\mathrm{sinc}(x)=1$

This fact can be proven using L'Hopital's rule:

$\underset{x\to 0}{\lim }\frac{\sin (x)}{x}\begin{array}{c}ℒ\\ \Rightarrow \\ \end{array}\underset{x\to 0}{\lim }\frac{\cos (x)}{1}=1$

Here we are using the stylized "L" over the thick arrow to denote L'Hospital decomposition. Since cos(0) = 1, we can show that at x = 0, the sinc function value is equal to 1. Also, the Sinc function approaches zero as x goes towards infinity. The envelope of sinc(x) tapers 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:

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

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), and setting the pulse width to A, which approaches zero:

$\delta (t)=\underset{A\to \mathrm{\infty }}{\lim }\mathrm{rect}\left(\frac{t-0}{A}\right)$

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

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

Here, both unit step functions are set a 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:

These two square waves have the same amplitude, but the second has a lower frequency. We can see that the period of the second is approximately twice as large as the first, and therefore that the frequency of the second is about half the frequency of the first.

These two square waves have the same frequency and the same peak-to-peak amplitude, but the second wave has no DC offset. Notice how the second wave is centered on the x axis, while the first wave is completely above the x axis.