# Understanding Digital Signal Processing: Discrete Sequences and Systems

• Print
This chapter is from the book

## 1.5 Discrete Linear Systems

The term linear defines a special class of systems where the output is the superposition, or sum, of the individual outputs had the individual inputs been applied separately to the system. For example, we can say that the application of an input x1(n) to a system results in an output y1(n). We symbolize this situation with the following expression:

#### Equation 1–11

Given a different input x2(n), the system has a y2(n) output as

#### Equation 1–12

For the system to be linear, when its input is the sum x1(n) + x2(n), its output must be the sum of the individual outputs so that

#### Equation 1–13

One way to paraphrase expression (1–13) is to state that a linear system's output is the sum of the outputs of its parts. Also, part of this description of linearity is a proportionality characteristic. This means that if the inputs are scaled by constant factors c1 and c2, then the output sequence parts are also scaled by those factors as

#### Equation 1–14

In the literature, this proportionality attribute of linear systems in expression (1–14) is sometimes called the homogeneity property. With these thoughts in mind, then, let's demonstrate the concept of system linearity.

### 1.5.1 Example of a Linear System

To illustrate system linearity, let's say we have the discrete system shown in Figure 1–7(a) whose output is defined as

#### Equation 1–15

that is, the output sequence is equal to the negative of the input sequence with the amplitude reduced by a factor of two. If we apply an x1(n) input sequence representing a 1 Hz sinewave sampled at a rate of 32 samples per cycle, we'll have a y1(n) output as shown in the center of Figure 1–7(b). The frequency-domain spectral amplitude of the y1(n) output is the plot on the right side of Figure 1–7(b), indicating that the output comprises a single tone of peak amplitude equal to –0.5 whose frequency is 1 Hz. Next, applying an x2(n) input sequence representing a 3 Hz sinewave, the system provides a y2(n) output sequence, as shown in the center of Figure 1–7(c). The spectrum of the y2(n) output, Y2(m), confirming a single 3 Hz sinewave output is shown on the right side of Figure 1–7(c). Finally—here's where the linearity comes in—if we apply an x3(n) input sequence that's the sum of a 1 Hz sinewave and a 3 Hz sinewave, the y3(n) output is as shown in the center of Figure 1–7(d). Notice how y3(n) is the sample-for-sample sum of y1(n) and y2(n). Figure 1–7(d) also shows that the output spectrum Y3(m) is the sum of Y1(m) and Y2(m). That's linearity.

### 1.5.2 Example of a Nonlinear System

It's easy to demonstrate how a nonlinear system yields an output that is not equal to the sum of y1(n) and y2(n) when its input is x1(n) + x2(n). A simple example of a nonlinear discrete system is that in Figure 1–8(a) where the output is the square of the input described by

#### Equation 1–16

We'll use a well-known trigonometric identity and a little algebra to predict the output of this nonlinear system when the input comprises simple sinewaves. Following the form of Eq. (1–3), let's describe a sinusoidal sequence, whose frequency fo = 1 Hz, by

#### Equation 1–17

Equation (1–17) describes the x1(n) sequence on the left side of Figure 1–8(b). Given this x1(n) input sequence, the y1(n) output of the nonlinear system is the square of a 1 Hz sinewave, or

#### Equation 1–18

We can simplify our expression for y1(n) in Eq. (1–18) by using the following trigonometric identity:

#### Equation 1–19

Using Eq. (1–19), we can express y1(n) as

#### Equation 1–20

which is shown as the all-positive sequence in the center of Figure 1–8(b). Because Eq. (1–19) results in a frequency sum (a + b) and frequency difference (ab) effect when multiplying two sinusoids, the y1(n) output sequence will be a cosine wave of 2 Hz and a peak amplitude of –0.5, added to a constant value of 1/2. The constant value of 1/2 in Eq. (1–20) is interpreted as a zero Hz frequency component, as shown in the Y1(m) spectrum in Figure 1–8(b). We could go through the same algebraic exercise to determine that when a 3 Hz sinewave x2(n) sequence is applied to this nonlinear system, the output y2(n) would contain a zero Hz component and a 6 Hz component, as shown in Figure 1–8(c)

System nonlinearity is evident if we apply an x3(n) sequence comprising the sum of a 1 Hz and a 3 Hz sinewave as shown in Figure 1–8(d). We can predict the frequency content of the y3(n) output sequence by using the algebraic relationship

#### Equation 1–21

where a and b represent the 1 Hz and 3 Hz sinewaves, respectively. From Eq. (1–19), the a 2 term in Eq. (1–21) generates the zero Hz and 2 Hz output sinusoids in Figure 1–8(b). Likewise, the b 2 term produces in y3(n) another zero Hz and the 6 Hz sinusoid in Figure 1–8(c). However, the 2ab term yields additional 2 Hz and 4 Hz sinusoids in y3(n). We can show this algebraically by using Eq. (1–19) and expressing the 2ab term in Eq. (1–21) as

#### Equation 1–22

Equation (1–22) tells us that two additional sinusoidal components will be present in y3(n) because of the system's nonlinearity, a 2 Hz cosine wave whose amplitude is +1 and a 4 Hz cosine wave having an amplitude of –1. These spectral components are illustrated in Y3(m) on the right side of Figure 1–8(d).

Notice that when the sum of the two sinewaves is applied to the nonlinear system, the output contained sinusoids, Eq. (1–22), that were not present in either of the outputs when the individual sinewaves alone were applied. Those extra sinusoids were generated by an interaction of the two input sinusoids due to the squaring operation. That's nonlinearity; expression (1–13) was not satisfied. (Electrical engineers recognize this effect of internally generated sinusoids as intermodulation distortion.) Although nonlinear systems are usually difficult to analyze, they are occasionally used in practice. References [2], [3], and [4], for example, describe their application in nonlinear digital filters. Again, expressions (1–13) and (1–14) state that a linear system's output resulting from a sum of individual inputs is the superposition (sum) of the individual outputs. They also stipulate that the output sequence y1(n) depends only on x1(n) combined with the system characteristics, and not on the other input x2(n); i.e., there's no interaction between inputs x1(n) and x2(n) at the output of a linear system.

• 🔖 Save To Your Account

### 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.

### Cookies and Related Technologies

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.