# Digital System Design with SystemVerilog: Combinational Logic Design

• Print
This chapter is from the book

## 2.5 Number Codes

Digital signals are either control signals of some kind or information. In general, information takes the form of numbers or characters. These numbers and characters have to be coded in a form suitable for storage and manipulation by digital hardware. Thus, one integer or one character may be represented by a set of bits. From the point of view of a computer or other digital system, no one system of coding is better than another. There do, however, need to be standards, so that different systems can communicate. The standards that have emerged are generally also designed such that a human being can interpret the data if necessary.

### 2.5.1 Integers

The simplest form of coding is that of positive integers. For example, a set of three bits would allow us to represent the decimal integers 0 to 7. In base 2 arithmetic, 0002 represents 010, 0112 represents 310, and 1112 represents 710. As with decimal notation, the most significant bit is on the left.

For the benefit of human beings, strings of bits may be grouped into sets of three or four and written using octal (base 8) or hexadecimal (base 16) notation. For example, 668 is equal to 110 1102 or 5410. For hexadecimal notation, the letters A to F represent the decimal numbers 10 to 15. For example, EDA 16 is 1110 1101 10102 or 73328 or 380210.

The simple translation of a decimal number into bits is sufficient for zero and positive integers. Negative integers require additional information. The simplest approach is to set aside one bit as a sign bit. Therefore, 0 1102 might represent +610, while 1 1102 would represent –610. While this makes translation between binary and decimal numbers simple, the arithmetic operations of addition and subtraction require that the sign bits be checked before an operation can be performed on two numbers. It is common, therefore, to use a different notation for signed integers: two's complement. The principle of two's complement notation is that the code for –b, where b is a binary number represented using n bits, is the code given by 2 n b. For example, –610 is represented by 100002 – 01102, which is 10102. The same result is obtained by inverting all the bits and adding 1: –610 is 10012 +1 = 10102.

The advantage of two's complement notation is that addition and subtraction may be performed using exactly the same hardware as for unsigned arithmetic; no sign checking is needed. The major disadvantage is that multiplication and division become much more complicated. Booth's algorithm, described in Section 5.7, is a technique for multiplying two's complement numbers.

### 2.5.2 Fixed Point Numbers

For many applications, non-integer data needs to be stored and manipulated. The binary representation of a fixed-point number is exactly the same as for an integer number, except that there is an implicit "decimal" point. For example, 6.25 is equal to 22 + 21 + 2 2 or 110.012. Instead of representing the point, the number 110012 (2510) is stored with the implicit knowledge that it and the results of any operations involving it have to be divided by 22 to obtain the true value. Notice that all operations, including two's complement representations, are the same as for integer numbers.

### 2.5.3 Floating-Point Numbers

The number of bits that have been allocated to represent fractions limits the range of fixed point numbers. Floating-point numbers allow a much wider range of accuracy. In general, floating-point operations are only performed using specialized hardware because they are very computationally expensive. A typical single precision floating-point number has 32 bits, of which 1 is the sign bit (s), 8 are the exponent (e), biased by an offset (2 e – 1 = 127), and the remaining 23 are the mantissa (m), such that a decimal number is represented as

 (–1) s x 1 · m x 2 e

IEEE standard 754-1985 defines formats for 32-, 64-, and 128-bit floating-point numbers, with special patterns for ± and the results of invalid operations, such as .

### 2.5.4 Alphanumeric Characters

Characters are commonly represented by 7 or 8 bits. The ASCII code is widely used. Seven bits allow the basic Latin alphabet in upper and lower cases, together with various punctuation symbols and control codes to be represented. For example, the letter A is represented by 1000001. For accented characters, 8-bit codes are commonly used. Manipulation of text is normally performed using general purpose computers rather than specialized digital hardware. Non-European languages may use 16 or 32 bits to represent individual characters.

### 2.5.5 Gray Codes

In the normal binary counting sequence, the transition from 0111 (710) to 1000 (810) causes 3 bits to change. In some circumstances, it may be undesirable that several bits should change at once because the bit changes may not occur at exactly the same time. The intermediate values might generate spurious warnings. A Gray code is one in which only 1 bit changes at a time. For example a 3-bit Gray code would count through the following sequence (other Gray codes can also be derived):

• 000
• 001
• 011
• 010
• 110
• 111
• 101
• 100

Note that the sequence of bits on a K-map is a Gray code. Another application of Gray codes is as a position encoder on a rotating shaft, as shown in Figure 2.18. Only 1-bit changes at each transition, so missed transitions are easily spotted.

### 2.5.6 Parity Bits

When data are transmitted either by wire or by using radio communications, there is always the possibility that noise may cause a bit to be misinterpreted. At the very least, it is desirable to know that an error has occurred, and it may be desirable to transmit sufficient information to allow any error to be corrected.

The simplest form of error detection is to use a parity bit with each word of data. For each 8 bits of data, a ninth bit is sent that is 0 if there is an even number of 1s in the data word (even parity) or 1 otherwise. Alternatively, odd parity can be used; in which case, the parity bit is inverted. This is sufficient if the chances of an error occurring are low. We cannot tell which bit is in error, but knowing that an error has occurred means that the data can be transmitted again. Unfortunately, if two errors occur, the parity bit might appear to be correct. A single error can be corrected by using a two-dimensional parity scheme, in which every ninth word is itself a set of parity bits, as shown in Table 2.10. If a single error occurs, both the row parity and column parity will be incorrect, allowing the erroneous bit to be identified and corrected. Certain multiple errors are also detectable and correctable.

#### Table 2.10. Two-Dimensional Parity

 Bit 7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0 Parity Word 0 0 1 0 1 0 1 1 0 0 Word 1 0 1 0 0 1 0 0 0 0 Word 2 0 1 0 1 0 0 1 1 0 Word 3 0 1 0 0 1 0 0 1 1 Word 4 0 1 0 0 0 0 1 1 1 Word 5 0 1 0 0 1 0 0 0 0 Word 6 0 1 0 0 0 1 0 0 0 Word 7 0 1 0 0 1 1 0 0 1 Parity 0 0 0 0 0 1 1 1 1

By using a greater number of parity bits, each derived from part of the word, multiple errors can be detected and corrected. The simplest forms of such codes were derived by Hamming in 1948. Better codes were derived by Reed and Solomon in 1960.

• 🔖 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.

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.