Signals & Systems Demystified

By David McMahon

The fast and easy way to learn signals and systems

Get a working knowledge of signal processing and systems--even if you don't have formal training, unlimited time, or a genius IQ. Signals and Systems Demystified offers an effective, illuminating, and entertaining way to learn this essential electrical engineering subject.

First, you'll learn methods used to calculate energy and power in signals. Next, you'll study signals in the frequency domain using Fourier analysis. Other topics covered include amplitude, frequency, and phase modulation, spectral analysis, convolution, the Laplace transform, and the z-transform. Packed with hundreds of sample equations and explained solutions, and featuring end-of-chapter quizzes and a final exam, this book will teach you the fundamentals of signals and systems in no time at all.

Simple enough for a beginner, but challenging enough for an advanced student, Signals and Systems Demystified is your shortcut to mastering this complex subject.

This hands-on, self-teaching text offers:

• An easy way to understand signal processing and systems
• Hundreds of worked examples with solutions
• A quiz at the end of each chapter to reinforce learning and pinpoint weaknesses
• A final exam at the end of the book
• No unnecessary technical jargon
• A time-saving approach to performing better on an exam or at work!

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Sep 6, 2006
• ISBN: 9780071712224

Book preview

CHAPTER 1

Introduction

A signal is a function of time that carries information. Physically, it may be a voltage across a capacitor or a current through a resistor for example, but in this book we will primarily be interested in the mathematical properties of signals. As such in general we will ignore the specific realization of a signal, beyond the understanding that it has some electrical form. It is typical to denote a signal by x(t).

Often signals are real functions of time. However, it is also possible to have a signal that is a complex function. In this book we will follow the convention used in electrical engineering and denote Therefore a complex signal can be written in the form , where x1(t) and x2(t) are real functions.

Continuous and Discrete Signals

A signal can be continuous or discrete. Basically, in this case the signal is a plain old function you are familiar with from calculus. In short, a continuous signal can assume any value in some continuous interval (a, b). In fact, a and b can range over the entire real numbers, such that the signal is defined for . A continuous signal is shown in Fig. 1-1.

Fig. 1-1. An example of a continuous signal, a sine wave.

A discrete time signal is defined at discrete times we can label with an integer n. Therefore we often define a discrete time signal by x[n]. A discrete time signal can be created from a continuous signal by sampling x(t) at regular intervals. Mathematically, we can think of a discrete signal as a sequence of numbers. If we denote the sampling interval by Ts, then ; that is, we compute the discrete values of x[n] by passing the argument t = nTs to the function x(t). In Fig. 1-2, we show a discrete time signal formed by sampling the sin function at regular intervals.

Fig. 1-2. A discrete time signal formed by sampling the sine function.

In this chapter we will examine some basic properties of continuous signals; in the next chapter we’ll do some examples with discrete signals.

As we will see throughout much of the book, it is possible to analyze signals by studying their frequency content. Later, we will see that by using a mathematical tool known as the Fourier transform we can transform a function of time into a function of frequency ω, which is denoted by X(ω). For now, we will begin by looking at some important properties that can be studied as functions of time. We often say that we are working in the time domain when looking at signals this way. When studying X(ω), we say that we are working in the frequency domain. We begin by computing the energy and power content of a signal in the time domain.

Energy and Power in Signals

Consider a voltage v(t) across a resistor R. Recalling Ohm’s law, , where i(t) is the current, the instantaneous power is given by

Equivalently, we can instead consider the current i(t) through the resistor, in which case the power is given by

Now if we calculate the power on a per-ohm basis, then we have . Then the total energy in joules is found by integrating

The average power is given by

We can generalize these notions to find the energy and power content in an arbitrary signal x(t). In general a signal can be complex, so we consider the squared modulus given by , where is the complex conjugate given by . If the signal is real then .

Therefore given a signal x(t), the normalized energy content E is given by

The normalized average power of a signal is given by

If a signal is discrete, then the normalized energy content is found by calculating

And the normalized average power becomes

Classification of Energy Signals and Power Signals

Signals can be classified as energy signals or power signals. An energy signal is one for which the energy E is finite, that is , while the average power vanishes . On the other hand, if P is nonzero but finite (i.e., and the energy is infinite, then the signal is a power signal. It is possible for a signal to be neither an energy signal nor a power signal. In the next few sections, we summarize some common signal types.

DC SIGNALS

A DC signal is simply a signal that has a constant value. In Fig. 1-3, we show a signal that maintains the constant value of unity for all times.

Fig. 1-3. A DC signal.

PERIODIC SIGNALS

In many cases of interest a signal will be periodic. This means that there exists some positive number T0 which we denote the period such that

The fundamental frequency f0 is given by the inverse of the period:

When considering periodic signals, we will examine the energy content over one period. If the energy for one period E0 is finite, then the signal is a power signal, and the power is given by

Many types of periodic functions are possible. For example, in Fig. 1-4 we have a sawtooth wave. To find the period, we look for the smallest value of t at which a feature of the function repeats (recall (1.5)). For example, in the sawtooth wave we can look at the location of each peak in the wave. By inspection we see that the peaks repeat every 2 s, and so the fundamental period is . The fundamental frequency is then found to be

Fig. 1-4. A sawtooth wave. The period is 2 s, and the frequency is 0.5 Hz.

It is also possible to have periodic discrete time signals. A discrete time signal x[n] is periodic with period N, where N is a positive integer if

A simple example of a periodic discrete time signal is found by sampling the sawtooth wave at regular intervals. This is shown in Fig. 1-5.

Fig. 1-5. A periodic discrete time signal.

SINUSOIDAL SIGNALS

The most familiar periodic functions are the trigonometric functions. In particular, we are interested in sinusoidal signals. A sinusoidal signal can be written as

Here A is a real number called the amplitude and θ is called the phase angle. The units of the argument are in radians. The fundamental period of a sinusoidal signal is given by

We call ω the angular frequency. The units of angular frequency are radians per second (rad/s). Angular frequency is related to the fundamental frequency defined in (1.6) by

When working with sinusoidal functions it is helpful to recall Euler’s formula. In particular, we can write

We can also manipulate this expression to write the cosine and sine functions in terms of exponentials. These formulas are

Computing Energy: Some Examples

Now let’s take a step back and compute the energy and power content of a few signals. First a tip.

When computing the energy of a signal, some basic mathematical facts are useful. First recall that an even function of t is one that satisfies . Such a function is symmetric about the origin. A quintessential example is the cosine function, which is shown in Fig. 1-6.

Fig. 1-6. An even function is symmetric about the origin.

We can exploit the symmetry of an even function when computing integrals. If we compute the integral of an even function over a symmetric interval a, then the following relationship holds:

This relationship also holds when integrating over the entire real line, provided that the integral in question converges

Note that dt does not converge.

An odd function is one for which . A familiar example of an odd function is sin(t). If we plot this function, it looks flipped over as we cross the origin (see Fig. 1-7).

Fig. 1-7. An example of an odd function.

Looking at the plot of the sine function it is easy to see that an integral over a symmetric interval will vanish. More generally, if f(t) is an odd function then

Or, if the integral converges when integrating over the entire real line,

Note that sin(t) dt does not converge. We will have more to say about even and odd functions later.

Before we look at some examples, let’s review some facts about energy and power signals. Consider an arbitrary signal x(t). Then

• If E as defined in (1.1) is finite, then x(t) is an energy signal.

• If x(t) is an energy signal, then the average power .

• A periodic signal is a power signal.

• If the energy is E0 over one period T0 of a periodic signal, then the power contained in the signal is .

• A signal of finite duration is an energy signal.

EXAMPLE 1-1

Find the energy content of the exponentially decreasing signal

SOLUTION 1-1

We begin by computing the square

Using (1.1) and considering that the signal is zero for , we can find the energy content of the signal. We have

To do the integral, we do the following substitution. Let . Then we have

Now, when , . At the other limit, as , we have . Putting all this together allows us to write

We can get rid of the minus sign by recalling that From calculus we also recall that , and so

The energy is finite, and so this is an energy signal.

EXAMPLE 1-2

Consider an exponentially decreasing sinusoidal signal, given by

Find the energy content of this signal.

SOLUTION 1-2

A plot of the function is shown in Fig. 1-8.

Fig. 1-8. An exponentially decreasing sinusoidal function.

Once again, we start by finding the squared modulus of the function. This signal is real so we simply square it

You will recall from calculus that to integrate the square of a sin function, it is helpful to use the trig identity

Therefore we can write

The integral to determine the energy content of the signal then becomes

We just did the first of these integrals in Example 1-1. The first term is just

The second term is a bit more complicated. We will have to resort to integration by parts. Many readers will recall the technique from calculus, but let’s go through the details. First, let’s write down the integral

When faced with an exponential times a cos or sin function, integration by parts will have to be applied twice. The integration by parts formula is

We identify and , and so

and sin 2t dt. Using (1.19) we then have

Now we apply the same procedure to sin 2t dt. We take and , so we again have

This time, . Then the integral is

The next step is to insert this result into (1.20). We obtain

Now we can move all occurrences of to the left-hand side and solve. This gives

The constants come from adding

to both sides of the relation we obtained above and then canceling the constants on the left side. Now let’s evaluate the result at the limits. At , the exponentials go to zero and so we don’t have to consider that case. At we have and , and so we obtain

This calculation was so long that it is possible to forget what we were doing! To compute the energy of the signal, we need to put this into (1.18). Recall that we had

We also found that

Putting everything together, the energy content of sin t defined for is

The energy is finite and positive, so the signal is an energy signal. For our next example, we put a twist on the exponential function.

EXAMPLE 1-3

Consider a signal . Determine the energy and power content of this signal.

SOLUTION 1-3

First we compute the squared modulus of the function

This function can be defined in the following way:

A plot of this function is shown in Fig. 1-9. Notice that this is an even function and therefore we can exploit the