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CSCE 211: Digital Logic Design

This chapter provides an introduction to digital systems, discussing their binary nature and the use of timing signals. It also covers number systems, including conversion between binary and decimal, hexadecimal notation, and binary addition. Additionally, it touches on binary-coded decimal (BCD) and other coding schemes like ASCII and Gray code.

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CSCE 211: Digital Logic Design

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  1. CSCE 211:Digital Logic Design Chin-Tser Huang huangct@cse.sc.edu University of South Carolina

  2. Chapter 1: Introduction

  3. Digital Systems • They are everywhere! • They are usually binary: operating on two-valued signals • Take an arbitrary number of inputs and produce an arbitrary number of outputs • Some systems require a timing signal called clock

  4. Examples • A system with three inputs, A, B, and C, and one output Z, such that Z = 1 if and only if two of the inputs are 1 • A system with eight inputs, representing two 4-bit binary numbers, and one 5-bit output, representing the sum

  5. Examples • A system with one input, A, plus a clock, and one output, Z, which is 1 iff the input was one at the last three consecutive clock times • A traffic controller on two streets: the light is green on each street for a fixed period of time, then goes to yellow for another fixed period and finally to red. The only input to this system is the clock

  6. Truth Table • Describe the behavior of a digital system in tabular form

  7. A Brief Review of Number Systems • Integers are usually written using a positional number system N = an-1rn-1+ an-2rn-2+ … + a2r2+ a1r + a0 where 0  ai < r

  8. Conversion between Number Systems • How to convert from binary to decimal? Evaluate the power series • Example: 1010112 = ?

  9. Conversion between Number Systems • How to convert from decimal to binary? Two algorithms • Repeatedly subtract from the number the largest power of 2 less than that number and put a 1 in corresponding position • Repeatedly divide the number by 2 and put the remainder from right to left

  10. Hexadecimal • Radix r = 16 • Why use hexadecimal? Shorthand notation for binary • Grouping 4 bits in binary to get 1 digit in hexadecimal

  11. Binary Addition

  12. One-bit Adder

  13. 4-bit Adder

  14. Overflow • Overflow occurs when the result of an arithmetic operation is out of range and indicates an error • For example, in a computer with n-bit words, if the addition of two n-bit integers produces an (n+1)-bit result, we call it overflow

  15. Binary Coded Decimal (BCD) • Most computers operate on binary numbers • However, for computers to interface with humans, the mode of communication is generally decimal • Convert from decimal to binary on input • Convert from binary to decimal on output • But the decimal output still needs to be coded into binary, digit by digit

  16. Binary-Coded Decimal Codes

  17. Other Codes • ASCII: used to transmit alphanumeric information • Gray code: consecutive numbers differ in only one bit • Particularly useful in coding the position of a continuous device and error detection

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