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Digital Logic DesignDigital Logic Design

Engr. Kashif ShahzadEngr. Kashif Shahzad

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What’s Course About?What’s Course About? DigitalDigital logic,logic, focusing on the design of focusing on the design of

computerscomputers Stay above transistor levelStay above transistor level

Only one class on transistors and VLSIOnly one class on transistors and VLSI Each person designs a MIPS CPUEach person designs a MIPS CPU

and peripheral logic (VGA, joystick)and peripheral logic (VGA, joystick) Project like an Atari 2600 gameProject like an Atari 2600 game

High-level language High-level language Modern design practicesModern design practices

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How Can We Do This?How Can We Do This? Field Programmable Gate ArraysField Programmable Gate Arrays

Chips with a lot of circuitsChips with a lot of circuitsTens of thousands to millions of transistorsTens of thousands to millions of transistors

ProgrammableProgrammable We write “programs” describing designWe write “programs” describing design Tools translate to gates/wiresTools translate to gates/wires Download pattern to chipDownload pattern to chip

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Use This BoardUse This Board

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Schematic DiagramSchematic Diagram

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VerilogVerilog/*/* * A 32-bit counter with only 4 bits of output. The idea is* 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.* to select which of the counter stages you want to pass on. ** * Anselmo Lastra, November 2002* Anselmo Lastra, November 2002 */*/ module cntr_32c(clk,res,out);module cntr_32c(clk,res,out); input clk;input clk; input res;input res; output [3:0] out;output [3:0] out;

reg [31:0] count;reg [31:0] count;

always @ (posedge res or posedge clk)always @ (posedge res or posedge clk) if(res)if(res)

count <= 0;count <= 0;elseelse

count <= count + 1;count <= count + 1; assign out[3] = count[28];assign out[3] = count[28]; assign out[2] = count[27];assign out[2] = count[27]; assign out[1] = count[26];assign out[1] = count[26]; assign out[0] = count[25];assign out[0] = count[25];

endmoduleendmodule

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Binary SignalingBinary Signaling Zero voltsZero volts

FALSE or 0FALSE or 0 3.3 or 5 volts3.3 or 5 volts

TRUE or 1TRUE or 1Modern chips down to 1VModern chips down to 1V

Why not multilevel signaling?Why not multilevel signaling?

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Discrete DataDiscrete Data Some data inherently discreteSome data inherently discrete

Names (sets of letters)Names (sets of letters) Some quantizedSome quantized

Music recorded from microphoneMusic recorded from microphone Note that other examples like music from CD or Note that other examples like music from CD or

electronic keyboard already quantizedelectronic keyboard already quantized Mouse movement is quantizedMouse movement is quantized

Well, some miceWell, some mice

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BCDBCD Binary Coded DecimalBinary Coded Decimal Decimal digits stored in binaryDecimal digits stored in binary

Four bits/digitFour bits/digit Like hex, except stops at 9Like hex, except stops at 9 ExampleExample

931 is coded as 1001 0011 0001931 is coded as 1001 0011 0001 Remember: these are just Remember: these are just encodingsencodings. Meanings are . Meanings are

assigned by us.assigned by us.

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Other Codes ExistOther Codes Exist Non positionalNon positional Example: Gray CodeExample: Gray Code

Only one bit changes at a timeOnly one bit changes at a time 000,001,011,010,110,111,101,100000,001,011,010,110,111,101,100 Why is this useful?Why is this useful?

Actually there’s a family of Gray codesActually there’s a family of Gray codes

Ref: http://lib-www.lanl.gov/numerical/bookcpdf/c20-2.pdf

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Shaft EncoderShaft Encoder

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Character CodesCharacter Codes From numbers to lettersFrom numbers to letters ASCIIASCII

Stands for American Standard Code for Information Stands for American Standard Code for Information InterchangeInterchange

Only 7 bits definedOnly 7 bits defined UnicodeUnicode You may make up your own code for the MIPS You may make up your own code for the MIPS

VGAVGA

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ASCII tableASCII table

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Even ParityEven Parity Sometimes high-order bit of ASCII coded to Sometimes high-order bit of ASCII coded to

enable detection of errorsenable detection of errors Even parity – set bit to make number of 1’s Even parity – set bit to make number of 1’s

eveneven ExamplesExamples

A (01000001) with even parity is 01000001A (01000001) with even parity is 01000001C (01000011) with even parity is 11000011C (01000011) with even parity is 11000011

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Odd ParityOdd Parity Similar except make the number of 1’s oddSimilar except make the number of 1’s odd ExamplesExamples

A (01000001) with odd parity is 11000001A (01000001) with odd parity is 11000001C (01000011) with odd parity is 01000011C (01000011) with odd parity is 01000011

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Error DetectionError Detection Note that parity detects only simple errorsNote that parity detects only simple errors

One, three, etc. bitsOne, three, etc. bits More complex methods existMore complex methods exist Some that enable Some that enable recoveryrecovery of original info of original info

Cost is more redundant bitsCost is more redundant bits

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Today’s TopicsToday’s Topics IntroductionIntroduction Digital logicDigital logic Number systemsNumber systems ArithmeticArithmetic CodesCodes ParityParity The encoding is keyThe encoding is key

Standards are used to agree on encodingsStandards are used to agree on encodings Special purpose codes for particular usesSpecial purpose codes for particular uses

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HomeworkHomework None, but…None, but… I expect you to know number systems well and be I expect you to know number systems well and be

able to do conversions and arithmeticable to do conversions and arithmetic Decimal – BinaryDecimal – Binary Binary – DecimalBinary – Decimal Decimal – HexDecimal – Hex Hex – DecimalHex – Decimal

Can do some of the problems – 1-2, 1-4, 1-7 if you Can do some of the problems – 1-2, 1-4, 1-7 if you think you need a refresher. Answers on book website.think you need a refresher. Answers on book website.

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Binary – Powers of 2Binary – Powers of 2 Positional representation Positional representation Each digit represents a power of 2Each digit represents a power of 2

So 101 binary isSo 101 binary is 1 1 •• 2 222 + 0 + 0 • • 2211 + 1 + 1 •• 2 200

oror 1 1 • 4 + • 4 + 0 0 • 2 + • 2 + 1 1 • 1 = 5• 1 = 5

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

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Binary Binary Decimal Example Decimal Example

77 66 55 44 33 22 11 00

2277 2266 2255 2244 2233 2222 2211 2200

128128 6464 3232 1616 88 44 22 11

11 00 00 11 11 11 00 00

128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156 128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156

What is 10011100 in decimal?What is 10011100 in decimal?

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Decimal to BinaryDecimal to Binary A little more work than binary to decimalA little more work than binary to decimal Some examplesSome examples

3 = 2 + 1 = 3 = 2 + 1 = 1111 (that’s 1 (that’s 1••2211 + 1 + 1••2200))5 = 4 + 1 = 5 = 4 + 1 = 101101 (that’s 1 (that’s 1••2222 + 0 + 0••2211 + 1 + 1••2200))

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Algorithm – Decimal to BinaryAlgorithm – Decimal to Binary Find largest power-of-two smaller than decimal Find largest power-of-two smaller than decimal

numbernumber Make the appropriate binary digit a ‘1’Make the appropriate binary digit a ‘1’ Subtract the power of 2 from decimalSubtract the power of 2 from decimal Do the same thing againDo the same thing again

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Decimal Decimal Binary Example Binary Example Convert 28 decimal to binaryConvert 28 decimal to binary

77 66 55 44 33 22 11 002277 2266 2255 2244 2233 2222 2211 2200

128128 6464 3232 1616 88 44 22 11

32 is too large, so use 16Binary 10000 Decimal 28 – 16 = 12

Binary 11000 Decimal 12 – 8 = 4Next is 8

Binary 11100 Decimal 4 – 4 = 0Next is 4

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HexadecimalHexadecimal Strings of 0s and 1s too hard to writeStrings of 0s and 1s too hard to write Use base-16 or Use base-16 or hexadecimalhexadecimal – 4 bits – 4 bits

DecDec BinBin HexHex00 00000000 0011 00010001 1122 00100010 2233 00110011 3344 01000100 4455 01010101 5566 01100110 6677 01110111 77

DecDec BinBin HexHex88 10001000 8899 10011001 991010 10101010 ??1111 10111011 ??1212 11001100 ??1313 11011101 ??1414 11101110 ??1515 11111111 ??

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HexadecimalHexadecimal Letters to represent 10-15Letters to represent 10-15

DecDec BinBin HexHex00 00000000 0011 00010001 1122 00100010 2233 00110011 3344 01000100 4455 01010101 5566 01100110 6677 01110111 77

DecDec BinBin HexHex88 10001000 8899 10011001 991010 10101010 aa1111 10111011 bb1212 11001100 cc1313 11011101 dd1414 11101110 ee1515 11111111 ff

•Power of 2Power of 2•Size of byteSize of byte

Why useWhy usebase 16?base 16?

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Hex to BinaryHex to Binary

Convention – write 0x before numberConvention – write 0x before number Hex to Binary – just convert digitsHex to Binary – just convert digits

BinBin HexHex

00000000 00

00010001 11

00100010 22

00110011 33

01000100 44

01010101 55

01100110 66

01110111 77

10001000 88

10011001 99

10101010 aa

10111011 bb

11001100 cc

11011101 dd

11101110 ee

11111111 ff

0x2ac0x2ac

00100010 10101010 110011000x2ac = 0010101011000x2ac = 001010101100

No magic – remember hex digit = 4 bitsNo magic – remember hex digit = 4 bits

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Binary to HexBinary to Hex

Just convert groups of 4 bitsJust convert groups of 4 bits

BinBin HexHex

00000000 00

00010001 11

00100010 22

00110011 33

01000100 44

01010101 55

01100110 66

01110111 77

10001000 88

10011001 99

10101010 aa

10111011 bb

11001100 cc

11011101 dd

11101110 ee

11111111 ff

101001101111011101001101111011

10111011

55 33 77 bb

101001101111011 = 0x537b101001101111011 = 0x537b

0101 0101 0111 0111 0011 0011

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Hex to DecimalHex to Decimal Just multiply each hex digit by decimal Just multiply each hex digit by decimal

value, and add the results.value, and add the results.

161633 161622 161611 161600

40964096 256256 1616 11

0x2ac0x2ac

2 2 • 256• 256 + 10 + 10 • 16• 16 + 12 + 12 • 1• 1 = 684= 684

DecDec HexHex00 0011 1122 2233 3344 4455 5566 6677 7788 8899 991010 aa1111 bb1212 cc1313 dd1414 ee1515 ff

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Decimal to HexDecimal to HexAnalogous to decimal Analogous to decimal binary. binary.

1.1. Find largest power-of-16 smaller than Find largest power-of-16 smaller than decimal numberdecimal number

2.2. Divide by power-of-16. The integer result Divide by power-of-16. The integer result is hex digit.is hex digit.

3.3. The remainder is new decimal number.The remainder is new decimal number.4.4. Do the same thing againDo the same thing again

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Decimal to HexDecimal to Hex

161633 161622 161611 161600

40964096 256256 1616 11

DecDec HexHex00 0011 1122 2233 3344 4455 5566 6677 7788 8899 991010 aa1111 bb1212 cc1313 dd1414 ee1515 ff

684684684/256 = 2684/256 = 2 0x2__

684%256 = 172684%256 = 172172/16 = 10 = a172/16 = 10 = a 0x2a_

172%16 = 12 = c172%16 = 12 = c 0x2ac

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OctalOctal Octal is base 8Octal is base 8 Similar to hexadecimalSimilar to hexadecimal

ConversionsConversions Less convenient for use with 8-bit bytesLess convenient for use with 8-bit bytes

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Arithmetic -- additionArithmetic -- addition Binary similar to decimal arithmeticBinary similar to decimal arithmetic

00 11 11 00 00++ 11 00 00 00 11

11 11 11 00 11

No carries 11 00 11 11 00 0011 00 11 11 00

++ 11 00 11 11 11

11 00 11 11 00 11

Carries

1+1 is 2 (or 102), which results in a carry

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Arithmetic -- subtractionArithmetic -- subtraction

11 00 11 11 00-- 11 00 00 11 00

00 00 11 00 00

No borrows 00 00 11 11 0011 11 11 11 00

-- 11 00 00 11 11

00 11 00 11 11

Borrows

0 - 1 results in a borrow

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Arithmetic -- multiplicationArithmetic -- multiplication

11 00 11 1100 00 00 00

11 00 11 11

11 11 00 11 11 11

Successive additions of multiplicand or zero,multiplied by 2 (102).

Note that multiplication by 102 just shifts bits left.

11 00 11 11XX 11 00 11

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Hexadecimal ArithmeticHexadecimal Arithmetic SimilarSimilar If you’re doing by hand, easiest to convert If you’re doing by hand, easiest to convert

each set of digits to decimal and backeach set of digits to decimal and back Skill is not very useful…Skill is not very useful…