Home > Articles > Home & Office Computing > Home Networking

📄 Contents

  1. Introduction
  2. Spectral Analysis of Nonperiodic Functions
  3. Statistical Methods
  4. References
  • Print
  • + Share This
This chapter is from the book

3.2 SPECTRAL ANALYSIS OF NONPERIODIC FUNCTIONS

A nonperiodic function of time is a function that is nonrepetitive over time. A stream of binary data as typically transmitted by digital communication systems is a stream of nonperiodic functions, each pulse having equal probability of being one or zero, independent of the value of other pulses in the stream. The analysis of the spectral properties of nonperiodic functions is thus an important component of the study of digital transmission.

3.2.1 The Fourier Transform

A nonperiodic waveform, v(t) say, may be represented in terms of its frequency characteristics by the following relationship:

Equation 3.1

03equ01.gif


The factor V(f) is the amplitude spectral density or the Fourier transform[1] of v(t). It is given by

Equation 3.2

03equ02.gif


Because V(f) extends from –∞ to +∞ (i.e., it exists on both sides of the zero frequency axis) it is referred to as a two-sided spectrum.

An example of the application of the Fourier transform that is useful in the study of digital communications is its use in determining the spectrum of a nonperiodic pulse. Consider a pulse v(t) shown in Fig. 3.1(a), of amplitude V, and that extends from t = –τ/2 to t = τ/2. Its Fourier transform, V(f), is given by

Equation 3.3

03equ03.gif


03fig01.gifFigure 3.1 Rectangular pulse and its spectrum.

The form (sin x)/x is well known and referred to as the sampling function, Sa(x).[1] The plot of V(f) is shown in Fig. 3.1(b). It will be observed that it is a continuous function. This is a common feature of the spectrum of all nonperiodic waveforms. We note also that it has zero crossings at ±1/ τ, ±2/ τ, . . .

The Fourier transform V(f) of an impulse of unit strength is also a useful result. By definition an impulse δ(t) has zero value except at time t = 0, and an impulse of unit strength has the property

Equation 3.4

03equ04.gif


Thus

Equation 3.5

03equ05.gif


Equation (3.5) indicates that the spectrum of an impulse δ(t) has a constant amplitude and phase and extends from –∞ to +∞.

A final example of the use of the Fourier transform is the analysis of what results in the frequency domain when a signal m(t), with Fourier transform M(f), is multiplied by a sinusoidal signal of frequency fc. In the time domain the resulting signal is given by

Equation 3.6

03equ06.gif


and its Fourier transform is thus

Equation 3.7

03equ07.gif


Recognizing that

Equation 3.8

03equ08.gif


then

Equation 3.9

03equ09.gif


An amplitude spectrum |M(f)|, band limited to the range –fm to +fm, is shown in Fig. 3.2(a). In Fig.3.2(b), the corresponding amplitude spectrum of |V(f)| is shown.

03fig02.gifFigure 3.2 (a) The amplitude spectrum of a waveform with no special component beyond fm. (b) The amplitude spectrum of the waveform in (a) multiplied by cos 2πfct. (From Taub, H., and Schilling, D., Principles of Communication Systems, McGraw-Hill, 1971, and reproduced with the permission of the McGraw-Hill Companies.)

3.2.2 Linear System Response

A linear system is one in which, in the frequency domain, the output amplitude at a given frequency bears a fixed ratio to the input amplitude at that frequency and the output phase at that frequency bears a fixed difference to the input phase at that frequency, irrespective of the absolute value of the input signal. Such a system can be characterized by the complex transfer function, H(f) say, given by

Equation 3.10

03equ10.gif


where |H(f)| represents the absolute amplitude characteristic, and θ(2πf) the phase characteristic of H(f).

Consider a linear system with complex transfer function H(f), as shown in Fig.3.3, with an input signal vi(t), an output signal vo(t), and with corresponding spectral amplitude densities of Vi(f), and Vo(f). After transfer through the system, the spectral amplitude density of Vi(f) will be changed to Vi(f) H(f). Thus

Equation 3.11

03equ11.gif


03fig03.gifFigure 3.3 Signal transfer through a linear system.

and

Equation 3.12

03equ12.gif


An informative situation is the one where the input to a linear system is an impulse function of unit strength. For this case, as per Eq. (3.5), Vi(f) = 1, and

Equation 3.13

03equ13.gif


Thus, the output response of a linear system to a unit strength impulse function is the transfer function of the system.

3.2.3 Energy and Power Analysis

In considering energy and power in communication systems, it is often convenient to assume that the energy is dissipated in a 1-ohm resistor, as with this assumption one need not keep track of the impact of the true resistance value, R say. When this assumption is made, we refer to the energy as the normalized energy and to the power as normalized power. It can be shown that the normalized energy E of a nonperiodic waveform v(t), with a Fourier transform V(f), is given by

Equation 3.14

03equ14.gif


The preceding relationship is called Parseval's theorem.[1] Should the actual energy be required, then it is simply E [as given in Eq. (3.14)] divided by R.

The energy density, De(f), of a waveform is the factor dE(f)/df. Thus, by differentiating the right-hand side of Eq. (3.14), we have

Equation 3.15

03equ15.gif


For a nonperiodic function such as a single pulse, normalized energy is finite, but power, which is energy per unit time, approaches zero. Power is thus somewhat meaningless in this context. However, a train of binary nonperiodic adjacent pulses does have meaningful average normalized power. This power, P say, is equal to the normalized energy per pulse E, multiplied by fs, the number of pulses per second; that is,

Equation 3.16

03equ16.gif


If the duration of each pulse is τ, then fs = 1/τ. Substituting this relationship and Eq. (3.14) into Eq. (3.16), we get

Equation 3.17

03equ17.gif


The power spectral density, G(f), of a waveform is the factor dP(f)/df. Thus, by differentiating the right-hand side of Eq. (3.17), we have

Equation 3.18

03equ18.gif


To determine the effect of a linear transfer function H(f) on normalized power, we substitute Eq. (3.11) into Eq. (3.17). From this substitution we determine that the normalized power, Po, at the output of a linear network, is given by

Equation 3.19

03equ19.gif


Also, from Eq. (3.11), we have

Equation 3.20

03equ20.gif


Substituting Eq. (3.18) into Eq. (3.20), we determine that the power spectral density Go(f) at the output of a linear network is related to the power spectral density Gi(f) at the input by the relationship

Equation 3.21

03equ21.gif


  • + Share This
  • 🔖 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.

Newsletters

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.

Links


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.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020