Digital Computers and Information n Digital computers n Number representations n Arithmetic operations n Alphanumeric codes Read MK 3-26.

Digital Computers and Information

Digital computersNumber representationsArithmetic operationsAlphanumeric codes

Read MK 3-26

1.2 - Jon Turner - 04/18/23

What’s in a Computer?

microprocessor

monitor

keyboard

FPU

CPU

MMU

InternalCache

ExternalCache

MainMemory

PeripheralBus Interface

GraphicsCard

DiskController

NetworkInterface

Microprocessor•Central Processing Unit•Floating Point Unit•Memory Mgmt. Unit•Internal Cache Memory

Main Memory•stores running programs and associated data

•typically 64-512 Mbytes

•Dynamic RAM

1.3 - Jon Turner - 04/18/23

What’s in a Computer?

microprocessor

monitor

keyboard

FPU

CPU

MMU

InternalCache

ExternalCache

MainMemory

PeripheralBus Interface

GraphicsCard

DiskController

NetworkInterface

Memory BusExternal Cache•small, fast memory

•stores recently used data

Control movement of data between memory and I/O devices

1.4 - Jon Turner - 04/18/23

What’s in a Computer?

microprocessor

monitor

keyboard

FPU

CPU

MMU

Internalcache

ExternalCache

MainMemory

PeripheralBus Interface

GraphicsCard

DiskController

NetworkInterface

Convert text & graphics to video

Peripheral Bus

Read/write data on proper physical location on disk

Transfer data between external network & memory

1.5 - Jon Turner - 04/18/23

What’s in a Chip?Intel 80286 Microprocessor

(www.micro.magnet.fsu.edu/chipshots)NEC/MIPS R4400 Microprocessor(www.micro.magnet.fsu.edu/chipshots)

Photomicrographs made using colored filters.

1.6 - Jon Turner - 04/18/23

Basic Processor & Memory

Memory stores programs and data.»organized as set of numbered storage slots»each storage slot (memory word) can hold a

number»processor can read from or write to any word

Fetch & execute cycle»read word whose address is in Program Counter

(PC) and increment PC»interpret stored value as instruction (decoding)»perform instruction using Accumulator (ACC) and

Arithmetic & Logic Unit (ALU)

10 50PC ACC

InstructionDecoder

Arithmetic& Logic Unit

AddressDecode

110123...

ffff

21-365

56

Processor MemoryAddress

Data

read/write

1.7 - Jon Turner - 04/18/23

Instruction Set0000 halt - halt execution0001 negate - ACC := ACC1xxx immediate load – if sign bit of xxx is 0 then

ACC := 0xxx else ACC := fxxx2xxx direct load - ACC := M[0xxx]3xxx indirect load - ACC := M[M[0xxx]]4xxx direct store - M[0xxx] := ACC5xxx indirect store – M[M[0xxx]] := ACC6xxx branch - PC := 0xxx7xxx branch if zero - if ACC = 0 then PC := 0xxx8xxx branch if positive - if ACC > 0 then PC :=

0xxx9xxx branch if negative - if ACC < 0 then PC :=

0xxxaxxx add - ACC := ACC + M[0xxx]

1.8 - Jon Turner - 04/18/23

Simple Program Add the values in locations 20-2f and write sum in 10.Address Instruction Comment0000 (start)1000 (ACC := 0000) initialize sum0001 4010 (M[0010] := ACC)0002 1020 (ACC := 0020) initialize pointer0003 4011 (M[0011] := ACC)0004 (loop) 1030 (ACC := 0030) if pointer = 030, quit

0005 0001 (ACC := -ACC)0006 a011 (ACC :=ACC+M[0011])0007 700f (if 0 goto 000f)0008 3011 (M[M[0011]] := ACC) sum = sum + *pointer0009 a010 (ACC :=ACC+M[0010])000a 4010 (M[0010] := ACC)000b 1001 (ACC := 0001) pointer = pointer + 1000c a011 (ACC :=ACC+M[0011])000d 4011 (M[011] := ACC)000e 6004 (goto 0004) goto loop000f (end) 0000 (halt) halt0010 Store sum here0011 Pointer to next value

1.9 - Jon Turner - 04/18/23

Representing Information in Computers

Electronic computers represent information as voltage levels.

To make the computer hardware simple and reliable, computers represent information in binary form.»example: voltages greater than 3V are interpreted as

representing one value (called “1”), voltages less than 2V are interpreted as representing another value (called “0”).

In principle, could use more voltage levels.»example: 0 to .75V represents “0”, 1 to 1.75V represents

“1”, 2 to 2.75V represents “2”, and so forth. In practice, this is rarely done.

»requires more complex circuits»circuits are more susceptible to noise, hence less reliable

1.10 - Jon Turner - 04/18/23

Noise in Computer Systems Computers, like all electronic systems, are

affected by noise.»noise has various sources (nearby signal changes,

thermal vibrations of molecules in semiconductor materials, . . . )

» in computers, noise can cause binary signals to be misinterpreted

The noise margin is the amountof noise that a system cantolerate and still correctlyidentify a logic high or low.

Undefined

High

Low

Undefined

High

Low

5v

4v

3v

2v

1v

0v

noise margin1 V

noise margin3 V

1.11 - Jon Turner - 04/18/23

Number Representation Standard decimal number representation

243.83 = 2102 + 4101 + 3100 + 8101 + 3102

Generalization to base rAn. . .A1A0.A1 . . .Am

=Anrn + . . . + A1r1 +A0r0 + A1r1 + . . . + Amrm

Binary number representation110.01 = 122 + 121 + 020 + 021 + 122

Converting binary numbers to decimal (easy).write binary expansion, replace powers of 2 with decimal values, and add up the values110.01 = 122 + 121 + 020 + 021 + 122 = 4 + 2 + 14 = 6.25Note: it helps to know your powers of 2 (hint)

1.12 - Jon Turner - 04/18/23

Decimal-Binary Conversion Repeated division by 2

625/2 = 312 with remainder of 1 least significant bit312/2 = 156 with remainder of 0156/2 = 78 with remainder of 0 78/2 = 39 with remainder of 0 39/2 = 19 with remainder of 1 19/2 = 9 with remainder of 1 9/2 = 4 with remainder of 1 4/2 = 2 with remainder of 0 2/2 = 1 with remainder of 0 1/2 = 0 with remainder of 1

So, (625)10 = (10 0111 0001)2

Works similarly for other number bases.

1.13 - Jon Turner - 04/18/23

Octal and Hexadecimal Octal (base 8) and hexadecimal (base 16) provide

more convenient way for people to write binary numbers.»110101100010 = 110 101 100 010 = (6542)8

1101 0110 0010 = (d62)16

octal conversion hexadecimal conversion000 = 0 0000 = 0 1000 = 8001 = 1 0001 = 1 1001 = 9010 = 2 0010 = 2 1010 = 10 = a011 = 3 0011 = 3 1011 = 11 = b100 = 4 0100 = 4 1100 = 12 = c

101 = 5 0101 = 5 1101 = 13 = d110 = 6 0110 = 6 1110 = 14 = e

111 = 7 0111 = 7 1111 = 15 = f

1.14 - Jon Turner - 04/18/23

Finite Data Representations Computer hardware is generally designed to operate

on words with a fixed number of bits (e.g. 16 bits). Places a limit on the number of discrete values that

can be stored in a single word (e.g. 216). If we use words to represent positive integers then

with n bits, we can represent integers 0 up to 2n1 Larger integers can be represented by multiple

words.»computer hardware operates on single words»software must combine results from single word

operations to produce desired result Or, use floating point representation for large (and

small) values; typically supported by computer hardware.

1.15 - Jon Turner - 04/18/23

How Computers Add Binary long addition similar to decimal long addition.

decimal binarycarry 1100 111100augend 2565 10110addend 6754 11011sum 9319 110001

Binary addition algorithm - add an1...a0 to bn1...b0 and put result in sn...s0

c0=0 // ci are carry bitsfor i = 0 to n1

if one or three of ai, bi or ci are = 1 then si = 1 else si = 0 if at least two of ai, bi or ci are = 1, then ci+1 = 1 else ci+1

= 0sn = cn

1.16 - Jon Turner - 04/18/23

Modular and Signed Arithmetic

Computers use modular arithmetic in which values wrap around circularly.»to add A+B, start at position

for A and then count clockwise B positions

»modular arithmetic is just like “clock arithmetic”

0011+0110 =1001

1111+0011=0010

000

000

01

1100

0010

0011

1011

0100

1010

0101

1101

0110

0111

1111

100

0

1110

1001

0 12

345

14

109 7

6

15

131211

8

1111+0011=0010

000

000

01

1100

0010

0011

1011

0100

1010

0101

1101

0110

0111

1111

100

0

1110

1001

0 12

345

-2

-6-7 7

6

-1

-3-4-5

-8

Associating certain bit patterns with negative values yields signed arithmetic.

Negate a given value by flipping all bits and adding 1.

Must pay attention to overflow.

1.17 - Jon Turner - 04/18/23

Representing Text Computers use numbers to represent

alphabetic characters, numerals and punctuation.

Most common encoding is ASCII (American Standard Code for Communication Interchange)»characters represented by 7 bit values»numerals start at (30)16

»upper case letters start at (41)16

» lower case letters start at (61)16 »see Table 1-4 in Mano for details

Unicode uses 16 bits per character, allowing it to represent far more distinct characters.

1.18 - Jon Turner - 04/18/23

Convert Numeric String to Internal Value ASCII character codes for decimal integer stored

in locations 20-24 with M.S.D. at location 20. Write internal value in location 1f.

Address Instruction Comment0000 (start)1000 (ACC := 0000) result = 00001 401f (M[001f] := ACC)0002 1020 (ACC := 0020) pointer = 0200003 401d (M[001d] := ACC)0004 (loop)1025 (ACC := 0025) if pointer = 025, quit

0005 0001 (ACC := -ACC)0006 a01d (ACC := ACC + M[001d])0007 7015 (if 0 goto 0015)0008 100a (ACC := 000a) result = 10 * result0009 b01f (ACC := ACC*M[001f]) New multiply inst.000a 401f (M[001f] := ACC)

1.19 - Jon Turner - 04/18/23

Convert Numeric String (continued)Address Instruction Comment 000b 1fd0 (ACC := -30) result = result 000c 401e (M[001e] := ACC) + (*pointer-‘0’)

000d 301d (ACC := M[M[001d]]) 000e a01e (ACC += ACC+M[001e]) 000f a01f (ACC += ACC+M[001f]) 0010 401f (M[001f] := ACC) 0011 1001 (ACC := 0001) pointer = pointer + 1

0012 a01d (ACC := ACC+M[001d]) 0013 401d (M[001d] := ACC) 0014 6004 (goto 0004) goto loop 0015 (end) 0000 (halt) halt .... 001d Pointer to next char 001e Temporary value 001f Store result here 0020 ASCII digits

1.20 - Jon Turner - 04/18/23

Convert Internal Value to Numeric String

Write ASCII character codes for value in location 1f into words 20..24 with L.S.D. in word 20.

pointer = 20loop if pointer = 25 then quit

*pointer = (value modulo 10) + ‘0’value = value / 10pointer = pointer + 1goto loop

Exercise: write corresponding machine program; assume two new instructionscxxx divide value in accumulator by value in

location xxxand leave quotient in accumulator

dxxx divide by value in xxx & put remainder in ACC

1.21 - Jon Turner - 04/18/23

Another Look at Basic Computer

Program Counter (PC) stores address of next instruction.

Instruction Register & Decoder (IREG) stores current instruction.

Indirect Address Register (IAR) stores indirect addresses. Accumulator (ACC) stores arithmetic operands and results. Arithmetic & Logic Unit (ALU) implements arithmetic

functions. Controller coordinates actions of other components. Data & Address Buses carry data between components.

Controller

PC Data Bus

Address Bus

Memory

0000000100020003000400050006000700080009000a

ffff

. . .

IREG IAR ACC

read/writeALU

1.22 - Jon Turner - 04/18/23

Execution of a Computer Program

monitoredsignals

waveforms

& buses

timeaxis

system

clock

reset(initialization

)

program counter,instruction register, accumulator, . . .

1.23 - Jon Turner - 04/18/23

Execution of a Computer Program

resetperiod

execute first

instruction - mload

fetch firstinstructio

nfetch secondinstruction

execute second

instruction – dload