SICS 154 DIGITAL COMPUTER FUNDAMENTAL

1. SICS 154 DIGITAL COMPUTER FUNDAMENTAL Prepared by Gabriel D. Kukuia

2. Reading List Digital Design by Morris Mano, published by Prentice Hall of India, 1995 Digital Systems – Principles & Applications by Tocci.R. J, published by Prentice Hall of India, 1995

3. Course Outline: Digital Computers and Digital Systems Binary Systems Binary Numbers Number Base Conversion Octal and Hexadecimal Numbers Complements Binary Codes Binary Storage and Registers Binary Logic Integrated Circuits Boolean Algebra and Logic Gates Basic Definitions Axiomatic Definition of Boolean Algebra Basic theorems and Properties of Boolean Algebra Boolean Function Logic Operations Digital Logic Gates IC Digital Logic Families

4. Outline cont... Simplification of Boolean Functions The Map Method Two- and Three- Variable Maps Product of Sums Simplification NAND and NOR Implement The Tabulation Method Combinational Logic Introduction Design Procedure Adders Subtractors Analysis Procedure Multilevel NAND Circuits Multilevel NOR Circuits

5. Outline cont... Combinational Logic with MSI and LSI Introduction Binary Parallel Adder Decimal Adder Decoders Multiplexers Read –Only Memory (ROM) Programmable Logic Array (PLA) Synchronous Sequential Logic Introduction Flip-Flops Design Procedure Design with State Equations Registers, Counters, And The Memory Unit Introduction Registers Shift Registers Counters The Memory Unit Examples of Random Access Memories

6. Outline cont.... Algorithmic State Machines (ASM) Introduction ASM Chart Design with Multiplexers PLA Control Asynchronous Sequential Logic Introduction Analysis Procedure Design Procedure Digital Integrated Circuits Introduction Transistor – Transistor Logic (TTL) Metal-Oxide Semiconductor (MOS) Complementary MOS (CMOS)

7. Digital Computers & Digital Systems Computers are used in: Scientific calculations Commercial and business data processing Space guidance The educational field and many others Digital Computers follow a sequence of instructions, called a program, that operates on given data. The user can specify and change programs and/or data according to the specific need.

8. Types of Digital Computers General-purpose digital computer e.g. PC Specific-purpose digital computer e.g. telephone switching exchanges, digital voltmeter, frequency counters, calculating machines, teletype machine.

9. Characteristics of Digital Systems It is a manipulation of discrete elements of information. Such discrete elements may be electric impulses, the decimal digits, the letters of symbols. Early digital computers were used mostly for numerical computations. An analog computer performs a direct simulation of a physical system. The variables in the analog computer are represented by continuous signals usually electric voltages that vary with time. The term analog signal is sometimes substituted for continuous signal because “analog computer” has come to mean a computer that manipulates continuous variables. To simulate a physical process a digital computer, the quantities must be quantized. When the variables of the process are presented by real time continuous signals, the latter are quantized by an analog-to-digital device.

10. Block diagram of digital computer

11. The memory unit stores programs as well as input, output, and intermediate data. The processor unit performs arithmetic. The processor unit performs arithmetic and other data processing task as specified by a program. The control unit supervises the flow of information between the various units. The control unit retrieves the instruction, one by one, from the program which is stored in memory. For each instruction, the control unit informs the processor to execute the operation specified by the instruction. Both program and data are stored in memory instructions, and the processor manipulates the data as specified by the program.

12. The program and data prepared by the user are transferred into memory unit by means of an input device such as the punch-card reader or a teletypewriter. An output device, such as printer, receives the result of the computations and the printer results are presented to the user. The input and output devices are special digital systems driven by electromechanical parts and controlled by electronic digital circuits. A processor, when combine with the control unit, forms a component referred to as a central processor unit or CPU. The CPU enclosed in small integrated circuit package is called a microprocessor The CPU combined with memory and interface control to inform a small-size computer is called a microcomputer.

13. BINARY NUMBERS A decimal number such as 7392 represents a quantity equal to 7 thousands plus 3 hundreds, plus 9 tens, plus 2 units. The thousands, hundreds, etc., are powers of 10 implied by the position of the coefficients. To be more exact, 7392 should be written as: 7X 103+3X102+9X101+2X100 In general a5 a4 a3 a2 a1 a0 a-1 a-2 a-3 aj coefficients are one of the ten digits (0,1,2,…,9) and the subscript value j gives the place value and hence the power of 10 by which the coefficient must be multiplied. 105 a5 +104a4 +103a3 +102a2 +101a1 +100a0 +10-1a-1 +10-2a-2 +10-3a-3 The decimal number system is said to be of base, or radix, 10 because it uses ten digits and the coefficients are multiplied by powers of 10. The binary system is a different number system is a different number system. The coefficients of the binary numbers system have possible values: 0 and 1. Each coefficient aj is multiply by 2j

14. Example: 11010.11 is 26.75 1X24+ 1X23+0X22+1X21+0X20+1X21+1X2-2 In general, a number expressed in base – r system has coefficients multiplied by power of r: anrn+ an-1rn-1+…+ a2r2+ a1r1+ a0+ a-1r-1 +a-2r-2+ …..+ a-mr-m (4021.2)5=4X53+0X52+2X51+1X50+2X5-1

15. For example, in the hexadecimal (base 16) number system, the first ten digits are borrowed from the decimal system. The letters A,B,C,D, E and F are used for 10,11,12,13,14,15 respectively. Example: (B65F)16 =11X163+6X162+5X161+15X160 Addition and Subtraction: 101101 and 100111 Multiplication: 1011 and 101 Number Base Conversions (1010.011)2 to base 10 =1X23+0X22+1X21+1X2-2+1X2-3 = (10.375)10 (630.4)8=6X82+3X81+4X8-1 = (408.5)10

16. Convert decimal 41 to binary Integer quotient remainder coefficient 41/2 = 20 + ½ a0=1 20/2 = 10 + 0 a1=0 10/2 = 5 + 0 a2=0 5/2 = 2 + ½ a3=1 2/2 = 1 + 0 a4=0 1/2 = 0 + ½ a5=1 (41)10 = ( a5 a4 a3 a2 a1 a0)2=(101001)2

17. Alternative Integer remainder 41 20 1 10 0 5 0 2 1 1 0 0 1

18. Convert 153 to octal: ans (231)8 Convert (0.6875)10 to binary Integer fraction coefficient 0.6875 X 2= 1 0.3750 a-1=1 0.3750 X 2= 0 0.7500 a-2=0 0.7500 X 2= 1 0.5000 a-3=1 0.5000 X 2= 1 0.0000 a-4=1 (0.6875)= (0. a-1 a-2 a-3 a-4)2=(0.1011)2 Convert (0.513)10 to octal 0.513 X 8 = 4.104 0.104 X 8 = 0.832 0.832 X 8 = 6.656 0.656 X 8 = 5.248 0.248 X 8 = 1.984 0.284 X 8 = 7.872 (0.513)10 = (0.406517…..)8

19. The conversion of decimal numbers with both integer and fraction parts is done by converting the integer and fraction separately and then combining the two answers together. (41.6875)10= (101001.1011)2 (153.513)10= (231.406517)8 EXX Convert the following (62.175)10 to base 2 (185.165)10 to base 8

20. The conversion from and to binary, octal, and hexadecimal plays an important part in digital computers. Since 23=8 and 24=16, each octal digit corresponds to three binary digits and each hexadecimal digits corresponds to four binary digits. The conversion from binary to octal is easily accomplished by partitioning the binary number into groups of three digits each, starting from the binary point and proceeding to the left and to the right. The corresponding octal digit is then assigned to each group. Example: (10 110 001 101 011 . 111 100 000 110 )2 2 6 1 5 3 7 4 0 6 (26153.7406)8 (10 1100 0110 1011 . 1111 0010)2 2 C 6 B F 2 (2C6B.F2)16 (673.124)8 = (110 111 011 . 001 010 100)2 (306.D)16 = (0011 0000 0110 . 1101)2

21. Assignment 1 1. Write the first 20 decimal digits in base 3 2. Add and multiply the following numbers in the given base without converting to decimal. (i) (1230)4 and (23)4 (ii) (135.4)6 and (43.2)6 (iii) (367)8 and (715)8 (iv) (296)12 and (57)12 3. Convert the decimal number 250.5 to base 3, base 4 base 7 and base 8, 16 4. Convert the following decimal to binary 12.0625, 104, 673.23 and 198 5.Covert the following binary numbers to decimal 10.10001, 101110.0101, 1110101.110, 1101101.111

22. Assignment 2 1. Convert the following numbers from the given base to the bases indicated: 2. Decimal 225.225 to binary, octal and hexadecimal 3. Binary 11010111.110 to decimal, octal and hexadecimal 4. Octal 62377 to decimal binary and hexadecimal 5. Hexadecimal 2AC5.D to decimal, octal and binary 6. Convert the following numbers to decimal: (i) (1001001.011)2 (ii) (12121)3 (iii) (1032.2)4 (iv) (4310)5 (v) (0.342)6 (vi) (50)7 (vii) (83)9 (viii) (198)10