CS302 Highlighted Notes

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Digital Logic and Design (Virtual University of Pakistan)

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Highlighted notes · Digital Logic and Design (Virtual University of Pakistan) · Digital Logic Design CS302 · Digital representing of quantities

Digital quantities unlike Analogue quantities are not continuous but represent quantities measured at discrete intervals.

Electronic Processing of Continuous and Digital Quantities

Electronic Processing of the continuous quantities or their Digital representation requires that the continuous signals or the discrete values be converted and represented in terms of voltages. There are basically two types of Electronic Processing Systems.

• Analogue Electronic Systems: · Digital Electronic Systems: · Digital Systems and Digital Values

Digital systems are designed to work with two voltage values. A +5 volts represents a logic high state or logic 1 state and 0 volts represents a logic low state or logic 0 state. Digital Systems are based on the Binary Number system which allows more than two or multiple values to be represented very conveniently.] Any quantity can be represented through digital values. The number of digits (0s and 1s) that represents a quantity is proportional to the range of values that are to be represented. Computers process information that is represented digitally in the form of Binary Numbers. Data can be easily and precisely reproduced: The picture quality and the sound quality of digital videos are far more superior to those of analogue videos. The reason being that the digital video stored as digital numbers can be exactly reproduced whereas analogue video is stored as a continuous signal cannot be reproduced with exact precision. Digital circuits occupy small space: Digital circuits are based on two logical states. Characters used in writing text are represented through yet another standard known as ASCII which allows alphabets, punctuation marks and numbers to be represented through a combination of 0s and 1s. Large and complex digital system such as a computer is built using combinations of these basic Logic Gates. These basic building blocks are available in the form of Integrated Circuit or ICs.

Combinational Logic Circuits and Functional Devices

The logic gates which form the basic building blocks of a digital system are designed to perform simple logic operations. Digital circuits that generate a new output on the basis of some previously stored information and the new input are known as Sequential circuits. PLD Programmable Logic Devices can be used to implement Combinational and Sequential Digital Circuits. RAM and ROM are an essential part of a digital system.

Caveman number system

A number system discovered by archaeologists in a prehistoric cave indicates that the caveman used a number system that has 5 distinct shapes Σ, Δ, >, Ω and ↑.

Binary to Decimal conversion

Most real world quantities are represented in Decimal Number System. Digital Systems on the other hand are based on the Binary Number System. Therefore, when converting from the Digital Domain to the real-world, Binary numbers have to be represented in terms of their Decimal equivalents. There are two methods to convert binary to decimal.

1. Sum-of-Weights Method

2. Sum-of-non-zero terms

The Sum-of-non-zero terms method is a quicker method to determine decimal equivalents of binary numbers without resorting to writing an expression.

Decimal to Binary conversion:

The Sum-of-weights and repeated division by 2 methods are used to convert a Decimal number to equivalent Binary.

Sum-of-weights · 4310 · H Weight · Binary Num · Converting Decimal fractions to Binary:

A Binary 1 is marked to represent the bit which contributed its weight in the Sum representing the decimal equivalent. The weight is subtracted from the sum decimal equivalent. The next highest weight included in the sum term is found. A binary 1 is marked to represent the bit which contributed its weight in the sum term and the weight is subtracted from the sum term. This process is repeated until the sum term becomes

Repeated Multiplication-by-2 Method:

Binary Arithmetic

Digital systems use the Binary number system to represent numbers. Therefore, these systems should be capable of performing standard arithmetic operations on binary numbers.

Binary Multiplication by shifting left

00011 (3) original binary number 00110 (6) binary number shifted left by 1 bit 01100 (12) binary number shifted left by 2 bits 11000 (24) binary number shifted left by 3 bits

Binary Division by Shifting right

10100 (20) original binary number 01010 (10) binary number shifted right by 1 bit 00101 (5) binary number shifted right by 2 bits

Signed and Unsigned Binary Numbers

Digital systems not only handle positive numbers but both positive and negative numbers. In a digital system which uses the Binary number system, the positive and negative signs cannot be represented as + and -. As mentioned in the Overview all forms of numbers, text, punctuation marks etc. are represented in the form of 1s and 0s. Thus the positive and negative signs are also presented in terms of binary 0 and 1.

MSB set to 1 indicates a negative number

MSB set to 0 indicates a positive number

Thus +13 and -13 are represented as 01101and 11101 respectively. The bits 1101 represent the number 13 and the MSBs 0 and 1 represent positive and negative signs respectively.

1’s & 2’s complement

Informing the digital system how to treat a binary number is not very efficient. A better way is to represent negative signed numbers in their 2’s complement form. 01101 The number 13 10010 1’s complement of 13 is obtained by inverting all the five bits. + 1 (The num is -13) 10011 2’s complement of 13 is obtained by adding a 1 to its 1’s complement. In a 2’s complement number system all negative numbers are represented in their 2’s complement form and all positive numbers are represented in their actual form.

Addition and Subtraction Operations with Signed Binary

An additional benefit of using 2’s complement representation for signed numbers is that both add and subtract operations can be performed by addition.

Range of Signed and Unsigned Binary numbers

Three different types of Binary representations have been discussed. The Unsigned Binary representation can only represent positive binary numbers. The Sign-Magnitude can represent both positive and negative numbers. The 2’s complement signed representation also allows positive and negative numbers to be handled.

Range of Numbers and Overflow

When arithmetic operation such as Addition, Subtraction, Multiplication and Division are performed on numbers the results generated may exceed the range of values specified by the Binary representations. The values that exceed the specified range cannot be correctly represented and are considered as Overflow values.

Determining Overflow Conditions for 2’s Complement Numbers

The Overflow condition can be easily determined when two numbers represented in 2’s Complement form are added together.

Floating-Point Numbers

Modern computers can handle large binary numbers such as 64-bit unsigned number, the maximum decimal number that can be represented using the 64-bit unsigned representation is 264-1 which is nearly equal to1.84 x 1019. The number 6918.3125 can be written as 6.9183125 x 103.

Arithmetic Operations on Floating Point Numbers

Arithmetic operations can be directly performed on floating point numbers by manipulating the mantissa and exponent parts of the floating point numbers. represented in f.p. as exponent 2 mantissa 7.23 + 134 represented in f.p. as exponent 2 mantissa 1.34 In this method the number to be converted is repeatedly multiplied by the Base Number to which the number is being converted to, in this case 2. A new number having an Integer part and a Fraction part is generated after each multiplication. The Integer part is noted down and the fraction part is again multiplied with the Base number 2. The process is repeated until the fraction term becomes