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

Chapter 1: Introduction

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 1/14/2011

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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 1/14/2011

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 1/14/2011

Truth Table Describe the behavior of a digital system in tabular form 1/14/2011

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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 1/14/2011

Conversion between Number Systems How to convert from binary to decimal? Evaluate the power series Example: 1010112 = ? 1/14/2011

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 1/14/2011

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Hexadecimal Radix r = 16 Why use hexadecimal? Shorthand notation for binary Grouping 4 bits in binary to get 1 digit in hexadecimal 1/14/2011

Binary Addition 1/14/2011

One-bit Adder 1/14/2011

4-bit Adder 1/14/2011

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 1/14/2011

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 1/14/2011

Binary-Coded Decimal Codes 1/14/2011

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 1/14/2011

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