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EEE-241 Digital Logic Designe. Textbook Digital Design by Morris Mano , 2 nd Edition/ 3rd Edition/Digital Fundamentals. Introduction to concepts of digital logic, gates, and the digital circuits Design and analysis of combinational and sequential circuits

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EEE-241 Digital Logic Designe


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    1. EEE-241Digital Logic Designe

    2. Textbook • Digital Design by Morris Mano , 2 nd Edition/ 3rd Edition/Digital Fundamentals.

    3. Introduction to concepts of digital logic, gates, and the digital circuits Design and analysis of combinational and sequential circuits Basics of logic design of computer hardware What’s Course About?

    4. Digital Systems • Digital Computer follow a sequence of instructions, Digital System play a prominent role in this digital age • Communication, medical treatment, internet, DVD, CD, Space ,Programme.Scientific &Educational field ,ATC commercial etc. • called programs, that operate on given data • User can specify and change program or data according to needs • Like Digital Computers, most digital devices are programmable • Digital Systems have the ability to Manipulate discrete elements of information. • Any set that is restricted to a finite number of elements contains discrete information • 10 Decimal digits • 26 Alphabet letters • 52 Playing cards • 64 squares of a chessboard

    5. Course Outline • Binary Systems • Binary Algebra • Simplification of Boolean Functions • Combinational Logic • Sequential Logic • MSI Sequential Circuits

    6. Digital Systems • Digital Systems can do hundreds of millions of operations per second • Extreme reliability due to error-correcting codes • A Digital System is interconnection of digital modules • To understand Digital module, we need to know about digital circuits and their logical functions • Hardware Description Language (HDL) is a programming language that is suitable for describing digital circuit in a textual form • Simulate a digital system to verify operation before it is built

    7. Circuit Diagram

    8. Verilog /* * A 32-bit counter with only 4 bits of output. The idea is * to select which of the counter stages you want to pass on. * * Anselmo Lastra, November 2002 */ module cntr_32c(clk,res,out); input clk; input res; output [3:0] out; reg [31:0] count; always @ (posedge res or posedge clk) if(res) count <= 0; else count <= count + 1; assign out[3] = count[28]; assign out[2] = count[27]; assign out[1] = count[26]; assign out[0] = count[25]; endmodule

    9. COMPUTER Analog Computer,. It respnds to continous signals. Digital compuyer. It responds to 0 and 1. also called Binary. Main Modules. Memory Unit Processor Unit Control Unit Input Device / Output Device CPU Processor combined with Control Unit Micro Processor. CPU in a Small integrated cct Micro Computer. CPU combined with Memory and Interface control for i/p and o/p devices form a micro computer.

    10. DATA FLOW • Fetch Time. Getting data and instrs from ALU and then issue command .Fix time • Execute Time. ALU carries out execution. Time is variable • Master clock. It is in con unit and con all functions • Memory • RAM Semi conductor memory & Feritte core memory • Sequential Memory . Mag tape, mag disk, CD Foppy Mag Drum. • Address Each info has a loc and an address.

    11. DEFINATIONS MEMORY • Random Access Memory,. Access time to a loc is constant. • Sequential access memory. Access time to all locs are different • Main memory and Secondry memory. How we store • Semi coductor Magnetic Material (Hystersis loop) • Binary Req. as material can store only 1 and 0 • Three things are stored, Instructions, Data, Address.

    12. Decimal Number • 7,392= 7x103 +3x102 + 9x101 + 2x100 • Thousands, hundreds, etc…power of 10 implied by position of coefficient • Generally a decimal number is represented by a series of coefficients • a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4 • aj cofficient are any of the 10 digit (0,1,2…9) • Decimal number are base 10

    13. Binary Number • Digital Systems manipulate discrete quantities of information in binary form • Operands in calculations • Decimal Digits • Results • Strings of binary digits (“bits”) • Two possible values 0 and 1

    14. Binary Numbers • Each digit represents a power of 2 • Coefficient have two possible values 0 and 1 • Strings of binary digits (“bits”) • n bits can store numbers from 0 to 2n-1 • n bits can store 2n distinct combinations of 1’s and 0’s • Each coefficient aj is multiplied by 2j • So 101 binary is 1 x 22 + 0 x21 + 1 x 20 or 1 x 4 + 0 x 2 + 1 x 1 = 5

    15. BITs & Bytes • A bit (short for binary digit) is the smallest unit of data in a computer. • A bit can hold only one of two values: 0 or 1, corresponding to the electrical values of off or on, respectively. • Because bits are so small, you rarely work with information one bit at a time • A byte is a unit of measure for digital information. • A single byte contains eight consecutive bits • Binary Arithmic. Addition, Substraction MutiplicatioiGive example

    16. GATES

    17. Octal • Octal is base 8 • A number is represented by a series of coefficients • a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4 • aj cofficient are any of 8 digit (0,1,2…7) • Need 3 bits for representation • Example: (127.4)8= 1 X 82 +2 X 81 +7 X 80 + 4 X 8-1 64+16+7+.5= (87.5)10

    18. Hexadecimal • Hexadecimal is base 16 • A number is represented by a series of coefficients • a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4 • aj cofficient are any of 16 digit (0,1,2,3,4,5,6,7,8, 9,A,B,C,D,E,F) • Need 4 bits for representation • (B65F)16 11 X 163 +6 X 162 + 5 X 161 +15 X 160 = 11x4096 + 6x256 +5x16 +15 = 45056 + 1536 + 80 +15 = 46,687

    19. Converting Binary to Decimal • Easy, just multiply digit by power of 2 • Just like a decimal number is represented • Example follows

    20. Binary  Decimal Example What is 10011100 in decimal? 128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156

    21. Decimal to Binary • A little more work than binary to decimal • Some examples • 3 = 2 + 1 = 11 (that’s 1•21 + 1•20) • 5 = 4 + 1 = 101 (that’s 1•22 + 0•21 + 1•20)

    22. Algorithm – Decimal to Binary • Find largest power-of-two smaller than decimal number • Make the appropriate binary digit a ‘1’ • Subtract the power of 2 from decimal • Do the same thing again

    23. Decimal  Binary Example • Convert 28 decimal to binary 32 is too large, so use 16 Binary  10000 Decimal  28 – 16 = 12 Next is 8 Binary  11000 Decimal  12 – 8 = 4 Next is 4 Binary  11100 Decimal  4 – 4 = 0

    24. Decimal  Binary (Fraction) • Convert decimal 0.6875 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)10 = (0.1011)2

    25. Decimal to Octal Similar to decimal  binary. • Find largest power-of-8 smaller than decimal number • Divide by power-of-8. The integer result is Octal digit. • The remainder is new decimal number. • Do the same thing again

    26. Decimal  Octal 512 is too large, so use 64 • Convert decimal 153 to Octal Octal  200 Decimal  153 – 64X2 = 25 Next is 8 Decimal  25 – 8X3= 1 Octal  230 Decimal  1 – 1X1 = 0 Next is 1 Octal  231

    27. Decimal  Octal (Fraction) • Convert decimal 0.513 to Octal Integer Fraction Coefficient 0.513 X 8 = 4 + 0.104 a-1=4 0.104 X 8 = 0 + 0.832 a-2=0 0.832 X 8 = 6 + 0.656 a-3=6 0.656 X 8 = 5 + 0.248 a-4=5 0.248 X 8 = 1 + 0.984 a-5=1 0.984 X 8 = 7 + 0.872 a-5=7 (0.513)10= (0.406517)8

    28. Binary to Octal • Partition Binary number into group of three digits each • The corresponding octal digit is then assigned to each group • (10 110 001 101 011 . 111 100 000 100)2 • (10 110 001 101 011 . 111 100 000 100)2 = (26153.7460)8

    29. Octal to Binary • Each Octal digit is converted to its three digit binary equivalent • (26153.7460)8 = (010 110 001 101 011 . 111 100 000 100)2

    30. 0010 1010 1100 Hex to Binary • Convention – write 0x before number • Hex to Binary – just convert digits 0x2ac 0x2ac = 001010101100 No magic – remember hex digit = 4 bits

    31. 5 3 7 b Binary to Hex • Just convert groups of 4 bits 101001101111011 1011 0101  0011  0111  101001101111011 = 0x537b

    32. Hex to Decimal • Just multiply each hex digit by decimal value, and add the results. 0x2ac 2 • 256 + 10 • 16 + 12 • 1 = 684

    33. Decimal to Hex Similar to decimal  binary. • Find largest power-of-16 smaller than decimal number • Divide by power-of-16. The integer result is hex digit. • The remainder is new decimal number. • Do the same thing again

    34. Decimal to Hex 684 0x2__ 684/256 = 2 684%256 = 172 0x2a_ 172/16 = 10 = a 0x2ac 172%16 = 12 = c

    35. Arithmetic -- addition • Binary similar to decimal arithmetic No carries Carries 1+1 is 2 (or 102), which results in a carry

    36. Arithmetic -- subtraction No borrows Borrows 0 - 1 results in a borrow Borrow makes it (10)2 =(2)10

    37. Arithmetic -- multiplication Successive additions of multiplicand or zero, multiplied by 2 (102). Note that multiplication by 102 just shifts bits left.

    38. Complements • Simply Subtraction (Subtraction by addition) • R’s Complement • In Binary 2’s complement • In Decimal 10’s complement • (R-1) Complement • In Binary 1’s complement • In Decimal 9’s Complement

    39. R, complement/2,s/10,s complement For a given positive no N in base r with integer part of n digits. The r,s complement of N is defined as rn-N for N not =0 and 0 forN=0 Example 10,s Complement of(52520) is 105-52520=47480 (0.3267)=(1-0.3267=0.6733 No integer so 100=1 2,s Complement of(101100)2= 26-(101100)2=010100 0.01101 = (1-0.0110)=0.1010

    40. (r-1),s, complement/1,s/9,s complement For a given positive no N in base r with integer part of n digits and a fraction part of m digits. The( r-1),s complement of N is defined as (rn – rm- N) Example 9,s Complement of(52520) is( 105-100- 52520)=47479 (0.3267)=(1- 10-4- 0.3267=0.6732 No integer so 100=1 1,s Complement of(101100)2= (26- 1)-(101100)2=010011 0.0110 = (1-2-4)- (0.0110)=( 0.1111-0.0110) = 0.1001

    41. DUALITY METHOD • 1,S COMPLEMENT OF (10111) • REPLACE 1 BY 0 & O BY 1 • (10111)= (01000) • 2,S COMPLEMENT • REPLACE BY 1,S COMPLEMENT +1 • (10111)=01000+1=01001

    42. Subtraction with r-Complement • M-N • Add the minuend, M to r’s complement of Subtrahend, N • M+ (rn -N)= M-N+ rn • If M GTE N then sum will produce end carry . Ignore it • If M LT N (No Carry) then take r’s complement of answer (Negative)

    43. Subtraction with r’s Complement • Using 10’s complement subtract 72532-03250 • Using 10’s complement subtract 03250 -72532 • Using 2’s complement subtract 1010100 -1000011 • Using 2’s complement subtract 1000011- 1010100

    44. Subtraction with r-1 Complement • Similar to r’s complement • But since r-1 complement is 1 less than r complement, Carry is added back to get the result • If no carry, result is negative1’s complement to get the answer • 1010100-1000011 • 1000000-1010100

    45. Signed Binary Numbers • Need notation for negative values • Everything must be represented by binary digits • Signed magnitude convention • Left most bit can be used • 0  Positive • 1  Negative • 01001 is +9 and 11001 is -9 (Not 25. Convention known in advance) • Signed Complement (Store negative as comps) • Signed 1’s complement (8 bits)11110110 • Signed 2’s complement (8 bits)11110111 • Signed Magnitude (8 bits) 10001001

    46. BCD • Binary Coded Decimal • Decimal digits stored in binary • Four bits/digit (Use 10 instead of 16) • Like hex, except stops at 9 • Example 931 is coded as 1001 0011 0001 • People understand decimal system better • Written differently but decimal value is same • Decimal 15 in BCD 0001 0101 in Binary it is 1111 Since most computers store data in eight-bit , bytes • Ignore 4 extra bits • one can store two digits per byte, called "packed" BCD

    47. BCD Addition • Since each digit is max 9 Sum will always be less than 19= 9+9+1(carry) • Two BCD digits are added as binary numbers • When binary sum is more than binary (1001)2, result is invalid (unlike Hex last 6 were ignored) • Addition of 6=(0110)2 make a correct BCD and produces a carry • Binary Sum carry and Decimal Carry differ by 16-10=6 • 4+5, 4+8, 8+9 • 184+576

    48. Binary Codes for Numbers • Binary codes for decimal digits require 4 bits per digit • Many codes use 4 bits in 10 distinct possible combinations (out of 16) • 2421 and Excess 3 are self complementing (1 and 0 9’s Comp of decimal) • Contents can be interpreted differently. • What decimal value does 1100001111001001 represent in different binary codes? Dec Binary BCD Excess-3 2421 84-2-1 Octal Hexadecimal 0 0 0000 0011 0000 0000 000 0000 1 1 0001 0100 0001 0111 001 0001 2 10 0010 0101 0010 0110 010 0010 3 11 0011 0110 0011 0101 011 0011 4 100 0100 0111 0100 0100 100 0100 5 101 0101 1000 1011 1011 101 0101 • 110 0110 1001 1100 1010 110 0110 • 111 0111 1010 1101 1001 111 0111 • 1000 1000 1011 1110 1000 - 1000 • 1001 1001 1100 1111 1111 - 1001 • 1010 - - - - - 1010 A • 1011 - - - - - 1011 B …

    49. Other Codes Exist Gray Code/Reflected Code • Only one bit changes at a time • 0000,0001,0011,0010,0110,0111,0101,0100,1100,1101,1111,1110,1010,1011,1001,1000 • Why is this useful?The no changes by one digit. It is used to represent the digital data converted from Analog data.Where as in Binary all numbers changes. • 01111000 (All Four bits need to be changed)

    50. Character Codes • ASCII • Many applications require handling of not only numbers but letters and special characters • Stands for American Standard Code for Information Interchange • 7 Bits to store 128 characters • In ASCII, every letter, number, and punctuation symbol has a corresponding number, or ASCII code • This encoding system not only lets a computer store a document as a series of numbers, but also lets it share such documents with other computers that use the ASCII system.