EECE 320 Lecture 1, Spring 2015

Digital Logic Design Mohammad Fakih Numbering Systems Digital Computer Systems • • Digital systems consider discrete amounts of data. Examples –26 letters in the alphabet –10 decimal digits • Larger quantities can be built from discrete values: – – Words made of letters: Football, University Numbers made of decimal digits (e.g. 239875.32) • • Computers operate on binary values (0 and 1) Easy to represent binary values electrically – – – Voltages and currents. Can be implemented using circuits Create the building blocks of modern computers 1 Digital Logic Design M.Fakih Understanding Decimal Numbers • • Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9) Decimal number: – 8653 = 8x103 + 6x102 + 5x101 + 3x100 • What about fractions? – – 97654.35 = 9x104 + 7x103 + 6x102 + 5x101 + 4x100 + 3x10-1 + 5x10-2 In formal notation -> (97654.35)10 • Why do we use 10 digits, anyway? 2 Digital Logic Design – M.Fakih Understanding Binary Numbers • Binary numbers are made of binary digits (bits): – 0 and 1 • binary number: – (1011)2 = 1x23 + 0x22 + 1x21 + 1x20 = (11)10 • What about fractions? – (110.10)2 = 1x22 + 1x21 + 0x20 + 1x2-1 + 0x2-2 • Groups of eight bits are called a byte – (11001001) 2 bits • a nibble. Groups of four are called – (1101) 2 3 Digital Logic Design – M.Fakih Conversion Between Number Bases Decimal(base 10) Binary(base 2) Hexadecimal (base16) ° Learn to convert between bases. 4 Digital Logic Design – M.Fakih Decimal review • a weight Numbers consist of a bunch of digits, each with 1 100 6 10 2 1 . 3 1/10 7 1/100 5 1/1000 Digits Weights • These weights are this: all powers of the base, which is 10. We can rewrite 1 102 6 101 2 100 . 3 10-1 7 10-2 5 10-3 Digits Weights • To find the decimal value of a number, multiply each digit by its weight and sum the products. (1 x 102) + (6 x 101) + (2 x 100) + (3 x 10-1) + (7 x 10-2) + (5 x 10-3) = 162.375 5 Digital Logic Design – M.Fakih Converting binary to decimal • We can use the same trick to convert binary, or base 2, numbers to decimal. This time, the weights are powers of 2. – Example: 1101.01 in binary 1 23 1 22 0 21 1 20 . 0 2-1 1 2-2 Binary digits, or bits Weights (in base 10) – The decimal value is: (1 x 23) + (1 x 22) + (0 x 0 21) + (1 + x 20) + (0 x 2-1) + (1 x 2-2) = = 8 + 4 + 1 + 0 + 0.25 13.25 6 Digital Logic Design – M.Fakih Powers of 2: Useful abbreviations: 20 = 1 24 = 16 28 = 256 K = 210 = 1,024 21 = 2 25 = 32 29 = 512 M = 220 = 1,048,576 22 = 4 26 = 64 210 = 1024 G = 230 = 1,073,741,824 23 = 8 27 = 128 Converting decimal to binary • To convert a decimal integer into binary, keep dividing by 2 until the quotient is 0. Collect the remainders in reverse order. • To convert a fraction, keep multiplying the fractional part by 2 until it becomes 0. Collect the integer Example: 162.375: parts in forward order. • 162 / 2 81 / 2 40 / 2 20 / 2 10 / 2 5 / 2 2 / 2 1 / 2 = = = = = = = = 81 40 20 10 5 2 1 0 rem rem rem rem rem rem rem rem 0 1 0 0 0 1 0 1 0.375 0.750 0.500 x x x 2 2 2 = = = 0.750 1.500 1.000 • So, 162.37510 = 10100010.0112 7 Digital Logic Design – M.Fakih Base 16 is useful too • The hexadecimal system uses 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F You can convert between base 10 and base 16 using techniques like the ones we just showed for converting between decimal and binary. For our purposes, base 16 is most useful as a “shorthand” notation for binary numbers. • • – Since 16 = 24, one hexadecimal digit is equivalent to 4 binary digits. – It’s often easier to work with like B4 instead of 10110100. a number 8 Digital Logic Design – M.Fakih Decimal Binary Hex 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F Binary and hexadecimal conversions • Converting from hexadecimal to binary is easy: just replace each hex digit with its equivalent 4-bit binary sequence. 261.3516 = = 2 0010 6 0110 1 0001 . . 3 0011 516 01012 • To convert from binary to hex, make groups of 4 bits, starting from the binary point. Add 0s to the ends of the number if needed. Then, just convert each bit group to its corresponding hex digit. 10110100.0010112 = = 1011 B 0100 4 . . 0010 2 11002 C16 9 Digital Logic Design – M.Fakih Hex Binary C 1100 D 1101 E 1110 F 1111 Hex Binary 8 1000 9 1001 A 1010 B 1011 Hex Binary 4 0100 5 0101 6 0110 7 0111 Hex Binary 0 0000 1 0001 2 0010 3 0011 Hex to Decimal 8 163 x 10 Digital Logic Design – M.Fakih 7 x162 C x161 9 160 x (87C9)16 = (???)10 32,768 + 1792 + 192 + 9 (87C9)16 = (34,761)10 Convert Decimal to Hex Integer Part: Divide by the base, keep track of the remainder, and read up. 16 16 2,172 rem rem rem rem 9 12 7 8 16 = C Read up 16 0 34,76110 = 87C916 11 Digital Logic Design – M.Fakih 8 135 34,761 Questions 100101012 = ? (decimal) 12 Digital Logic Design – M.Fakih Questions 85710 = ?16 13 Digital Logic Design – M.Fakih Questions BED16 = ?2 14 Digital Logic Design – M.Fakih Addition 15 Digital Logic Design – M.Fakih Binary Addition • • Binary addition is very simple. This is best shown in an example of adding two binary numbers… 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 carries 1 1 + --------------------- 1 0 1 0 1 0 0 16 Digital Logic Design – M.Fakih Binary addition example worked out • • Some terms are given here Exercise: what are these numbers equivalent to in decimal? The initial carry in is implicitly 0 1 1 1 1 1 0 1 0 1 1 (Carries) (Augend) (Addend) 1 0 + 1 1 0 0 1 (Sum) most significant bit (MSB) least significant bit (LSB) 17 Digital Logic Design – M.Fakih Subtraction 18 Digital Logic Design – M.Fakih Binary Subtraction ° We can also perform subtraction (with borrows in place of carries). ° Let’s subtract (10111)2 from (1001101)2… 1 10 10 0 borrows 0 10 0 10 1 0 0 1 1 0 1 1 0 1 1 1 - ------------------------ 1 1 0 1 1 0 19 Digital Logic Design – M.Fakih Negative Numbers Subtract by adding 73 -35 38 73 +65 138 10’s complement Ignore carry 20 Digital Logic Design – M.Fakih Two’s Complement Shortcuts • Algorithm – Simply complement each bit and then add 1 to result. the – Finding the 2’s complement of (01100101)2 complement… and of its 2’s N = 01100101 10011010 [N] = 10011011 01100100 1 --------------- 01100101 + 1 + --------------- 10011011 21 Digital Logic Design – M.Fakih This is called one’s complement Negative Numbers Subtract by 0 1 0 0 1 0 0 1 1 1 0 1 1 1 0 1 adding 73 35 2’s 01001001 00100011 11011100 flip +1 ----------- 11011101 0 0 1 0 0 1 1 0 comp -35 22 Digital Logic Design – M.Fakih 73 -35 38 Table 2.2 Positive and Negative Binary Numbers 23 Digital Logic Design – M.Fakih Signed decimal Hex Binary Unsigned decimal -128 -127 -126 … … … -3 -2 -1 0 1 2 3 … … 125 126 127 80 81 82 … … … FD FE FF 00 01 02 03 … … 7D 7E 7F 10000000 10000001 10000010 … … … 11111101 11111110 11111111 00000000 00000001 00000010 00000011 … … … 01111101 01111110 01111111 128 129 130 … … … 253 254 255 0 1 2 3 … … … 125 126 127 Signed Numbers 4-bit: 8H = -8 to to 7H = +7 1000 0111 8-bit: 80H = -128 to 7FH = +127 16-bit: 8000H 7FFFH = = -32,768 +32,767 to 32-bit: 80000000H 7FFFFFFFH = = -2,147,483,648 +2,147,483,647 to 24 Digital Logic Design – M.Fakih Questions What is the two’s 00101100? complement of What hex number represents the decimal number -40? 25 Digital Logic Design – M.Fakih Multiplication 26 Digital Logic Design – M.Fakih Binary Multiplication • Binary multiplication is much the same as decimal multiplication, except that the multiplication operations are much simpler… 1 0 1 1 0 1 1 1 0 X ----------------------- 0 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 ----------------------- 1 1 1 0 0 1 1 0 27 Digital Logic Design – M.Fakih ASCII CODES 28 Digital Logic Design – M.Fakih ASCII • The most commonly used code for representing letters, numerals and punctuation characters (alphanumeric data) • Each character is represented with a 7-bit string; for example: ‘3’ = 00110011 ‘ ’ = 00100000 (hex 33) (hex 20) • An 8-bit extension of ASCII has also been defined 29 Digital Logic Design – M.Fakih ASCII Code • • American Standard Code for Information Interchange ASCII is a 7-bit code, frequently used with an 8th bit for error detection (more about that in a bit). 30 Digital Logic Design – M.Fakih Character ASCII (bin) ASCII (hex) Decimal A 1000001 41 65 B 1000010 42 66 C 1000011 43 67 … Z a … 1 ‘ BCD 31 Digital Logic Design – M.Fakih Binary Coded Decimal • Binary coded decimal (BCD) represents each decimal digit with four bits – Ex. 0011 0010 1001 = 329BCD 3 2 9 • • This is NOT the same as 0011001010012 Why do this? Because people think in decimal. 32 Digital Logic Design – M.Fakih Digit BCD Code Digit BCD Code 0 0000 5 0101 1 0001 6 0110 2 0010 7 0111 3 0011 8 1000 4 0100 9 1001 Putting It All Together ° ° BCD not very efficient Used in early computers (40s, 50s) Used to encode numbers for seven- segment displays. ° ° Easier to read? 33 Digital Logic Design – M.Fakih Gray code 34 Digital Logic Design – M.Fakih Gray code 35 Digital Logic Design – M.Fakih 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Decimal Binary Gray code 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000 Gray code is an un- weighted code that has a single bit change between one code word and the next in a sequence. Gray code is used to avoid problems in systems where an error can occur if more than one bit changes at a time. Floating point 36 Digital Logic Design – M.Fakih Express the speed of light, c, in single precision floating point notation. (c = 0.2998 x 109) Floating point notation is capable of representing very large or small numbers by using a form of scientific notation. A 32-bit single precision number is illustrated. S E (8 bits) F (23 bits) Sign bit Magnitude with MSB dropped Biased exponent (+127) In scientific notation, c = 1.0001 1101 1110 1001 0101 1100 0000 x 228. 0 10011011 001 1101 1110 1001 0101 1100 In binary, c = 0001 0001 1101 1110 1001 0101 1100 00002. S = 0 because the number is positive. E = 28 + 127 = 15510 = 1001 10112. F is the next 23 bits after the first 1 is dropped. In floating point notation, c = Floating point Digital Logic Design – M.Fakih 37