# Understanding Digital Signal Processing: Discrete Sequences and Systems

In this introduction to his book, Richard G. Lyons defines the terminology used in digital signal processing, illustrates the various ways of graphically representing discrete signals, establishes the notation used to describe sequences of data values, presents the symbols used to depict signal processing operations, and briefly introduces the concept of a linear discrete system.
This chapter is from the book

### This chapter is from the book 

Digital signal processing has never been more prevalent or easier to perform. It wasn't that long ago when the fast Fourier transform (FFT), a topic we'll discuss in Chapter 4, was a mysterious mathematical process used only in industrial research centers and universities. Now, amazingly, the FFT is readily available to us all. It's even a built-in function provided by inexpensive spreadsheet software for home computers. The availability of more sophisticated commercial signal processing software now allows us to analyze and develop complicated signal processing applications rapidly and reliably. We can perform spectral analysis, design digital filters, develop voice recognition, data communication, and image compression processes using software that's interactive both in the way algorithms are defined and how the resulting data are graphically displayed. Since the mid-1980s the same integrated circuit technology that led to affordable home computers has produced powerful and inexpensive hardware development systems on which to implement our digital signal processing designs. Regardless, though, of the ease with which these new digital signal processing development systems and software can be applied, we still need a solid foundation in understanding the basics of digital signal processing. The purpose of this book is to build that foundation.

In this chapter we'll set the stage for the topics we'll study throughout the remainder of this book by defining the terminology used in digital signal processing, illustrating the various ways of graphically representing discrete signals, establishing the notation used to describe sequences of data values, presenting the symbols used to depict signal processing operations, and briefly introducing the concept of a linear discrete system.

## 1.1 Discrete Sequences and Their Notation

In general, the term signal processing refers to the science of analyzing time-varying physical processes. As such, signal processing is divided into two categories, analog signal processing and digital signal processing. The term analog is used to describe a waveform that's continuous in time and can take on a continuous range of amplitude values. An example of an analog signal is some voltage that can be applied to an oscilloscope, resulting in a continuous display as a function of time. Analog signals can also be applied to a conventional spectrum analyzer to determine their frequency content. The term analog appears to have stemmed from the analog computers used prior to 1980. These computers solved linear differential equations by means of connecting physical (electronic) differentiators and integrators using old-style telephone operator patch cords. That way, a continuous voltage or current in the actual circuit was analogous to some variable in a differential equation, such as speed, temperature, air pressure, etc. (Although the flexibility and speed of modern-day digital computers have since made analog computers obsolete, a good description of the short-lived utility of analog computers can be found in reference [1].) Because present-day signal processing of continuous radio-type signals using resistors, capacitors, operational amplifiers, etc., has nothing to do with analogies, the term analog is actually a misnomer. The more correct term is continuous signal processing for what is today so commonly called analog signal processing. As such, in this book we'll minimize the use of the term analog signals and substitute the phrase continuous signals whenever appropriate.

The term discrete-time signal is used to describe a signal whose independent time variable is quantized so that we know only the value of the signal at discrete instants in time. Thus a discrete-time signal is not represented by a continuous waveform but, instead, a sequence of values. In addition to quantizing time, a discrete-time signal quantizes the signal amplitude. We can illustrate this concept with an example. Think of a continuous sinewave with a peak amplitude of 1 at a frequency fo described by the equation

#### Equation 1–1

The frequency fo is measured in hertz (Hz). (In physical systems, we usually measure frequency in units of hertz. One Hz is a single oscillation, or cycle, per second. One kilohertz (kHz) is a thousand Hz, and a megahertz (MHz) is one million Hz.) With t in Eq. 1–1 representing time in seconds, the fo t factor has dimensions of cycles, and the complete 2pfo t term is an angle measured in radians.

Plotting Eq. (1–1), we get the venerable continuous sinewave curve shown in Figure 1–1(a). If our continuous sinewave represents a physical voltage, we could sample it once every ts seconds using an analog-to-digital converter and represent the sinewave as a sequence of discrete values. Plotting those individual values as dots would give us the discrete waveform in Figure 1–1(b). We say that Figure Figure 1–1(b) is the "discrete-time" version of the continuous signal in Figure 1–1(a). The independent variable t in Eq. (1–1) and Figure 1–1(a) is continuous. The independent index variable n in Figure 1–1(b) is discrete and can have only integer values. That is, index n is used to identify the individual elements of the discrete sequence in Figure 1–1(b).

Do not be tempted to draw lines between the dots in Figure 1–1(b). For some reason, people (particularly those engineers experienced in working with continuous signals) want to connect the dots with straight lines, or the stair-step lines shown in Figure 1–1(c). Don't fall into this innocent-looking trap. Connecting the dots can mislead the beginner into forgetting that the x(n) sequence is nothing more than a list of numbers. Remember, x(n) is a discrete-time sequence of individual values, and each value in that sequence plots as a single dot. It's not that we're ignorant of what lies between the dots of x(n); there is nothing between those dots.

We can reinforce this discrete-time sequence concept by listing those Figure 1–1(b) sampled values as follows:

#### Equation 1–2

where n represents the time index integer sequence 0, 1, 2, 3, etc., and ts is some constant time period between samples. Those sample values can be represented collectively, and concisely, by the discrete-time expression

#### Equation 1–3

(Here again, the 2pfo nts term is an angle measured in radians.) Notice that the index n in Eq. (1–2) started with a value of 0, instead of 1. There's nothing sacred about this; the first value of n could just as well have been 1, but we start the index n at zero out of habit because doing so allows us to describe the sinewave starting at time zero. The variable x(n) in Eq. (1–3) is read as "the sequence x of n." Equations (1–1) and (1–3) describe what are also referred to as time-domain signals because the independent variables, the continuous time t in Eq. (1–1), and the discrete-time nts values used in Eq. (1–3) are measures of time.

With this notion of a discrete-time signal in mind, let's say that a discrete system is a collection of hardware components, or software routines, that operate on a discrete-time signal sequence. For example, a discrete system could be a process that gives us a discrete output sequence y(0), y(1), y(2), etc., when a discrete input sequence of x(0), x(1), x(2), etc., is applied to the system input as shown in Figure 1–2(a). Again, to keep the notation concise and still keep track of individual elements of the input and output sequences, an abbreviated notation is used as shown in Figure 1–2(b) where n represents the integer sequence 0, 1, 2, 3, etc. Thus, x(n) and y(n) are general variables that represent two separate sequences of numbers. Figure 1–2(b) allows us to describe a system's output with a simple expression such as

#### Equation 1–4

Illustrating Eq. (1–4), if x(n) is the five-element sequence x(0) = 1, x(1) = 3, x(2) = 5, x(3) = 7, and x(4) = 9, then y(n) is the five-element sequence y(0) = 1, y(1) = 5, y(2) = 9, y(3) = 13, and y(4) = 17.

Equation (1–1) is formally called a difference equation. (In this book we won't be working with differential equations or partial differential equations. However, we will, now and then, work with partially difficult equations.)

The fundamental difference between the way time is represented in continuous and discrete systems leads to a very important difference in how we characterize frequency in continuous and discrete systems. To illustrate, let's reconsider the continuous sinewave in Figure 1–1(a). If it represented a voltage at the end of a cable, we could measure its frequency by applying it to an oscilloscope, a spectrum analyzer, or a frequency counter. We'd have a problem, however, if we were merely given the list of values from Eq. (1–2) and asked to determine the frequency of the waveform they represent. We'd graph those discrete values, and, sure enough, we'd recognize a single sinewave as in Figure 1–1(b). We can say that the sinewave repeats every 20 samples, but there's no way to determine the exact sinewave frequency from the discrete sequence values alone. You can probably see the point we're leading to here. If we knew the time between samples—the sample period ts —we'd be able to determine the absolute frequency of the discrete sinewave. Given that the ts sample period is, say, 0.05 milliseconds/sample, the period of the sinewave is

#### Equation 1–5

Because the frequency of a sinewave is the reciprocal of its period, we now know that the sinewave's absolute frequency is 1/(1 ms), or 1 kHz. On the other hand, if we found that the sample period was, in fact, 2 milliseconds, the discrete samples in Figure 1–1(b) would represent a sinewave whose period is 40 milliseconds and whose frequency is 25 Hz. The point here is that when dealing with discrete systems, absolute frequency determination in Hz is dependent on the sampling frequency

#### Equation 1–5'

We'll be reminded of this dependence throughout the remainder of this book.

In digital signal processing, we often find it necessary to characterize the frequency content of discrete time-domain signals. When we do so, this frequency representation takes place in what's called the frequency domain. By way of example, let's say we have a discrete sinewave sequence x1(n) with an arbitrary frequency fo Hz as shown on the left side of Figure 1–3(a). We can also characterize x1(n) by showing its spectral content, the X1(m) sequence on the right side of Figure 1–3(a), indicating that it has a single spectral component, and no other frequency content. Although we won't dwell on it just now, notice that the frequency-domain representations in Figure 1–3 are themselves discrete.

To illustrate our time- and frequency-domain representations further, Figure 1–3(b) shows another discrete sinewave x2(n), whose peak amplitude is 0.4, with a frequency of 2fo. The discrete sample values of x2(n) are expressed by the equation

#### Equation 1–6

When the two sinewaves, x1(n) and x2(n), are added to produce a new waveform xsum(n), its time-domain equation is

#### Equation 1–7

and its time- and frequency-domain representations are those given in Figure 1–3(c). We interpret the Xsum(m) frequency-domain depiction, the spectrum, in Figure 1–3(c) to indicate that xsum(n) has a frequency component of fo Hz and a reduced-amplitude frequency component of 2fo Hz.

Notice three things in Figure 1–3. First, time sequences use lowercase variable names like the "x" in x1(n), and uppercase symbols for frequency-domain variables such as the "X" in X1(m). The term X1(m) is read as "the spectral sequence X sub one of m." Second, because the X1(m) frequency-domain representation of the x1(n) time sequence is itself a sequence (a list of numbers), we use the index "m" to keep track of individual elements in X1(m). We can list frequency-domain sequences just as we did with the time sequence in Eq. (1–2). For example, Xsum(m) is listed as

 X sum(0) = 0 (1st X sum (m) value, index m = 0) X sum(1) = 1.0 (2nd X sum (m) value, index m = 1) X sum(2) = 0.4 (3rd X sum (m) value, index m = 2) X sum(3) = 0 (4th X sum (m) value, index m = 3) . . . . . . and so on,

where the frequency index m is the integer sequence 0, 1, 2, 3, etc. Third, because the x1(n) + x2(n) sinewaves have a phase shift of zero degrees relative to each other, we didn't really need to bother depicting this phase relationship in Xsum(m) in Figure 1–3(c). In general, however, phase relationships in frequency-domain sequences are important, and we'll cover that subject in Chapters 3 and 5.

A key point to keep in mind here is that we now know three equivalent ways to describe a discrete-time waveform. Mathematically, we can use a time-domain equation like Eq. (1–6). We can also represent a time-domain waveform graphically as we did on the left side of Figure 1–3, and we can depict its corresponding, discrete, frequency-domain equivalent as that on the right side of Figure 1–3.

As it turns out, the discrete time-domain signals we're concerned with are not only quantized in time; their amplitude values are also quantized. Because we represent all digital quantities with binary numbers, there's a limit to the resolution, or granularity, that we have in representing the values of discrete numbers. Although signal amplitude quantization can be an important consideration—we cover that particular topic in Chapter 12—we won't worry about it just now.

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.