1. Chapter 1 Introduction 1 - 1

2. 1 - 2

3. Digital Systems Digital systems: computation, data processing, control, communication, measurement - Reliable, Integration Analog – Continuous - Natural Phenomena (Pressure, Temperature, Speed … ) - Difficulty in realizing, processing using electronics Digital – Discrete - Binary Digit  Signal Processing as Bit unit - Easy in realizing, processing using electronics - High performance due to Integrated Circuit Technology 1 - 3

4. Chapter 1 Introduction 1.1 Logic Design  Digital System Inputs & Outputs Digital System A B n inputs W X m outputs Clock  Simple Example of Digital System A system with three inputs, A,B, and C, and one output Z, such that Z = 1 iff two of the inputs are 1. ABCZ 0000 0010 0100 0111 1000 1011 1101 111 1 P. 2 1 - 27 Table 1.1

5. Binary Digit? Binary:- Two values(0, 1) - Each digit is called as a “ bit ” - Number representation with only two values (0,1) - Can be implemented with simple electronics devices (ex: Voltage High(1), Low(0) Switch On (1) Off(0) … ) Good things in Binary Number 1 - 4

6. 1 - 33 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Decimal to Binary 746 10  2 = 373 remainder 0 373  2 = 186 remainder 1 186  2 = 93 remainder 0 93  2 = 46 remainder 1 46  2 = 23 remainder 0 23  2 = 11 remainder 1 11  2 = 5 remainder 1 706 10 = 1011101010 2 5  2 = 2 remainder 1 2  2 = 1 remainder 0 1  2 = 0 remainder 1 0 10 010 1010 01010 101010 1101010 11101010 011101010 1011101010

7. 1 - 6 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Fractions, Mixed Numbers Mixed numbers: example 24.375 24 = 1 1 0 0 0.375 =. 0 1 1 24.375 = 11000.011 Mixed numbers are converted separately

8. Chapter 1 Introduction  number system: example n : number of digits r : radix or base a i : coefficients Decimal number Binary number 1 - 8

9.  Binary digits: example B =  b i 2 i m - 1 i = - n 100010 2 = 132 + 016 + 08 + 04 + 12 + 01 = 34 10 leftmost digit: most significant bit(MSB) rightmost digit: least significant bit(LSB) b m-1 b m-2 …b 1 b 0. b -1 b -2 …b -n binary point 1 - 9

10. 1 - 13 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Hexadecimal 16 digits(0 ~ 9, A ~ F) 1011101010 2 = 0010 1110 1010 = 2EA 16 4-bit string ~ 1 hexadecimal digit 2EA 16

11. Chapter 2 Number Systems and Codes 2.2 Octal and Hexadecimal Numbers  Octal number system  Hexadecimal number system 8 digits(0 ~ 7) 16 digits(0 ~ 9, A ~ F) 100011001110 2 = 100 011 001 110 = 4316 8 3-bit string ~ 1 octal digit 100011001110 2 = 1000 1100 1110 = 8CE 16 4-bit string ~ 1 hexadecimal digit 1 - 11

12. Number-System Conversions  binary  octal  binary  hexadecimal  binary  decimal  octal  binary  octal  hexadecimal  octal  decimal  hexadecimal  binary  hexadecimal  octal  hexadecimal  decimal 10111011001 2 = 10 111 011 001 = 2731 8 10111011001 2 = 101 1101 1001 = 5D9 16 10111011001 2 = 11024 + 0512 + 1256 + 1128 + 164 + 032 + 116 + 18 + 04 + 02 + 11 = 1497 10 1234 8 = 001 010 011 100 2 1234 8 = 001 010 011 100 2 = 0010 1001 1100 2 = 29C 16 1234 8 = 1512 + 264 + 38 + 41 = 668 10 C0DE 16 = 1100 0000 1101 1110 2 = 1 100 000 011 011 110 2 = 140336 8 C0DE 16 = 124096 + 0256 + 1316 + 141 = 49374 10 1 - 12

13. Representation of Negative Numbers b n1– b 1 b 0 Magnitude MSB (a) Unsigned number b n1– b 1 b 0 Magnitude Sign (b) Signed number b n2– 0 denotes 1 denotes + – MSB 1 - 14

14. Representation of Negative Numbers  Sign-Magnitude Representation A number consists of tow parts: magnitude and sign The sign is represented by a single additional bit in binary numbers sign bit 0 : positive sign bit 1 : negative 01010101 2 = + 85 10 11010101 2 = – 85 10 Signed-magnitude system has equal number of positive and negative integers Range of n-bits signed-magnitude integer : – (2 n – 1 –1) ~ + (2 n – 1 – 1) Need to compare sign and magnitude when addition and subtraction  slower than complement number system 1 - 15

15. 1 - 17 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Signed Numbers Converting steps for negative numbers in two’s complement 1.Find the binary equivalent of the magnitude 2.Complement each bit (that is, change 0’s to 1’s and 1’s to 0’s) 3.Add 1 0101 1010 1 1011 1. 2. 3. 0001 1110 1 1111 0000 1111 1 0000 -5-0

16. 1111 1110 1101 1100 1011 1010 1001 1000 - 1111 1110 1101 1100 1011 1010 1001 1000 1001 1010 1011 1100 1101 1110 1111 - -0 -2 -3 -4 -5 -6 -7 -8 0000 0001 0010 0100 0101 0110 0111 +0 +1 +2 +3 +4 +5 +6 +7 1’s complement 2’s complement N* Sign and magnitude Negative integers -N Positive integers (all systems)+N N 2 ’ s complement representation for Negative Numbers 1 - 18 Representation of Negative Numbers

17.  Subtraction Rules Two’s - Complement Addition and Subtraction 4-bit two’s complement subtractions: 0100 0011 – – 4 3 1 1101+ 1 0001 complementing the subtrahend and adding the minuend Two’s complement numbers are added and subtraction using same procedure(same logic circuit) 1 - 21

18. Two’s - Complement Addition and Subtraction  Addition Rules 0011 0111 0100 + + 3 4 7 4-bit two’s complement additions 1110 1000 1010 + + -2 -6 -8ignored carry = 1 Overflow Operation that produces a result that exceeds the range of the number number system 0100 1001 0101 + (+4) (+5) (-7) Addition overflow occurs whenever the sign of the sum is different from the sign of both addends 1 - 20

19. 1 - 23 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Binary Coded Decimal(BCD) Decimal digit8421 code5421 code2421 codeExcess 3 code2 of 5 code 00000 001111000 10001 010010100 20010 010110010 30011 011010001 40100 011101100 5010110001011100001010 6011010011100100101001 7011110101101101000110 8100010111110101100101 9100111001111110000011 unused10100101 0000 any of 10110110 0001 the 22 11000111 0010 patterns 1101 10001101 with 0, 1, 1110 10011110 3, 4, or 5 1111 10101111 1’s P. 18 Table 1.7

20. 1 - 24 Chapter 1 Introduction 1.2 A Brief Review Of Number Systems  Other Codes ASCII Code a3a2a1a0a3a2a1a0 a6a5a4a6a5a4 010011100101110111 0000space0@P`p 0001!1AQaq 0010“2BRbr 0011#3CScs 0100$4DTdt 0101%5EUeu 0110&6FVfv 0111‘7GWgw 1000(8HXhx 1001)9IYiy 1010*:JZjz 1011+;K[k{ 1100,<L\l| 1101=M]m} 1110.>N^n~ 1111/?O_odelete P. 19 Table 1.8