Midterm Review October 23, 2001. Administrivia – The Midterm Problem solving –What logic gates do –How to build circuits that realize truth tables Addition.

Midterm Review

October 23, 2001

Administrivia – The Midterm

• Problem solving– What logic gates do– How to build circuits that realize truth tables

• Addition circuitry• Memory

– State machines to model situations– Building the logic for state machines– Putting it together to build a computer– Programming the computer using machine language– Representing information

• Understanding key ideas– Von Neumann machine– Moore’s law

Extra Midterm help

• DPD office hours Wednesday 2—4– My office, CS 219

• Check with qlv and wtcorrea for other hours

More for Less --Moore’s Law

• Everything doubles every 18 months– Memory/dollar– Disk/dollar– Transistors per inch– Processor speed– Network bandwidth

Practical Details

• Problem sets 20%

• Lab reports 10%

• Midterm exam 25%

• Final exam 25%

• Class Participation 20%

Building a computer

Layers of Abstraction• Build more and more powerful tools out

of simpler ones.

ReallySimpleStuff

Computers

A Simple Logic Puzzle• Frank will go to the party if

Ed goes AND Dan does NOT.

• Dan will go if Bob does NOT go OR if Carole goes.

• Ed will go to the party if Alice AND Bob go.

• Alice and Bob decide to go,but Carol stays home.

• Will Frank go to the party?

Using 0’s and 1’s• What do 0’s and 1’s mean?

• For now, we’ll take “Natural meanings:”

• For example, if we have a variable Alice forwhether Alice goes to the party,

• If Alice goes, we write Alice = 1

• If Alice doesn’t, we write Alice = 0

0 = False 1 = True

Logic Gates • Computers are circuits made of Logic

Gates.

• Logic gates manipulate 0’s and 1’s (False and True) by letting electrons flow or not.

• We’ll look at three types of Logic Gates:

• AND are all inputs true?

• OR is one input true?

• NOT flip the truth value

AND Gate

ANDW

Y

ZX W X Y Z

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1

OR Gate

W X Y Z

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

ORW

Y

ZX

NOT Gate • “Yanni will go to the party if Xena does NOT

go.”X Y

Truth Table

NOT

X Y

Shorthand:

X Y

0 1

1 0

OR

AND

Logic Puzzle Circuit

AND

Ed

Dan

Frank

Dan will go if Bob does NOT go OR if Carole goes.

Carole

Alice

Bob

Given Any Truth Table

A B C F

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 0

Given Any Truth Table

ANDA

C

B

ANDA

C

ANDA

C

B

B

A B C F

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 1

1 1 1 0

Build AND gate for each output 1

OR

Given Any Truth Table

ANDA

C

B

ANDA

C

ANDA

C

B

B

Finally, combine ANDswith an OR gate.

F

Can build logic for any Truth table this way

Universality• Note same idea works no matter how

manyinput variables. So for ANY Truth Table we can write down,we can make a circuit for it using only 3 Logic Gates: AND, OR, NOT

• This gives us a very powerful tool !

• But, in practice, we are limited in size we can realize – numeracy exercise.

Our First Abstract ToolUniversal Method: Circuits for ANY Truth Table

Computers

We are here

UniversalMethod

0’s&1’s

Powers of 2 (cont.)• Some rough numbers:

210 1,000 (103 ) kilo

220 1,000,000 (106 ) mega

RAM

230 109 giga

disk

240 1012 tera BIG

disk

250 1015 peta

260 1018 exa

knowledge

Representing Information• We represent information with 0’s and 1’s

• Numbers

• Use binary or hexadecimal

• Letters

• Use ASCII

• Sounds

• Use samples

• Images

• Use pixels

Adding Binary Numbers

X Y sum

0 0 0

0 1 1

1 0 1

1 1 0

ORsum

ANDX

Y

ANDX

Y

carryAND

X

Y

X Y carry

0 0 0

0 1 0

1 0 0

1 1 1

Addition

We can use 1 building block to add numbers of any length.

Black box operation

ith bit of X

ith bit of Y

carry bit

ith bit of Z

carry bit out

Abstraction in action -- This is a piece of a carry-ripple adder

Carry-Ripple Adder

Z0

C0

UniversalCircuit

X0

Y0

Z1

C1

UniversalCircuitFor Z2 and carry

X1

Y1

Z2

C2

UniversalCircuitFor Z2 and carry

X2

Y2

X2 X1 X0 Y2 Y1 Y0============C2 Z2 Z1 Z0

0

Fixed at 0

Memory

Enter Rita• Matt doesn’t like Rita

• Matt decides to go to the party if Sue decides to go OR:

• If he (Matt) already feels like going AND Rita decides NOT to go.

OR

AND

Sue

Rita

Matt

Feedback Wire

The Flip-Flop

OR

AND

Set

Reset

M

• M becomes 1 if Set is turned on

• M becomes 0 if Reset is turned on

• Otherwise (if Set and Reset are both 0), M just remembers its value

The Flip-Flop

• M becomes 1 if Set is turned on

• M becomes 0 if Reset is turned on

• Otherwise (if Set and Reset are both 0), M just remembers its value

S

R

M

The Data Flip-Flop

• If Write = 0, M just keeps its value.(It ignores D.)

• If Write = 1, then M becomes set to D

D

Write

M

RAM• Group 8 of them together.

• 8 bits (b)is called abyte (B).

• Most RAMis arrangedin bytes.

DM

W

DM

W

DM

W

DM

W

DM

W

DM

W

DM

W

DM

W

RAM (cont.)• We want a HUGE number of such 1 byte

memory cells.

• Problem: How do we indicate which byte ofmemory we want to use at any given time?

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

1B

RAM (cont.)

220

bytes of RAM

(1 Mega-byte)Write

Address

Data input Data Output

20 bits of address

8 bits (1 byte)of data

ReviewOn our way to building a computer…

Computers

We are here

LogicCircuits:UniversalMethod

0’s&1’s

RepresentInfo with0’s & 1’s(aka bits)

MemoryCircuits

The System Clock

• “500 Mhz Pentium III computer…”

• What does “500 Mhz” mean?

• It refers to the speed of the System Clock. (actually this is only one of the clocks…)

• All digital systems have such “Clocks,” even traffic lights and elevator control systems.

Our System:A Traffic Light

A Traffic Light

• Intersection of two one-way roads

CarSensors

B

A

Light

ATraffic Light Behavior

IF A=1

AND B=0

Always

IF A=0

AND B=1

Otherwise

Light BOtherwise

Always

Note:Clock beatsevery 4 sec.So Light is Yellow for

4 sec.

Light

ATraffic Light State Machine

IF A=1

AND B=0

Always

IF A=0

AND B=1

Otherwise

Light BOtherwise

Always

Note:Clock beatsevery 4 sec.So Light is Yellow for

4 sec.

Light

ATraffic Light States

IF A=1

AND B=0

Always

IF A=0

AND B=1

Otherwise

Light BOtherwise

Always

00 01

1011

Traffic Light BehaviorBuild A Truth Table

for next state / Output

M1 M2 A B D1 D2 Light A Red Light B Red …

0 0 1 0 0 1 1 0

0 1 * * 1 0 1 0

1 0 0 1 1 1 0 1

1 1 * * 0 0 0 1

0 0 0 0 0 0 1 0

… … … … … … …

Traffic Light Design

2-bitMemoryRegisterClock

Sensor A

Sensor B

Logic For

Next State

&Output 6 Outputs:

for eachLight

Current State

Design for Any State Machine

MemoryRegisterClock

Inputs

Logic For

Next State

&Output Outputs

Current State

Manybits

State Machinesin Real Life

• Elevator control systems

• Car control systems

• VCR’s

• Alarm Clocks

• Personal Computers

• … Just about everything!

Computers

Computer Architecture

Bus

CPU

RAM

Keyboard

HardDisk

Display

CD-ROM

The Bus

Bus

• Communication on the bus is via postcards

• CPU puts “Keyboard, did the user type anything?” (represented in some way) on the Bus.

CPU Keyboard Display

“Keyboard, did the user type anything?”

The Bus

Bus

• Each device (except CPU) is a State Machine that constantly checks to see what’s on the Bus.

CPU Keyboard Display

“Keyboard, did the user type anything?”

The Bus

Bus

• Keyboard notices that its name is on the Bus,and reads info. Other devices ignore the info.

CPU Keyboard Display

“Keyboard, did the user type anything?”

The Bus

Bus

• Keyboard then writes “CPU: Yes, user typed ‘a’.” to the Bus.

CPU Keyboard Display

“CPU: Yes, user typed ‘a’.”

The Bus

Bus

• At some point, CPU reads the Bus, and getsthe Keyboard’s response.

CPU Keyboard Display

“CPU: Yes, user typed ‘a’.”

The CPU (cont.)Memory Registers

Register 0

Register 1

Register 2

Register 3

Instruction Register

Instr. Pointer (IP)

Arithmetic/ LogicUnit

Control Unit(State Machine)

Important Point• One of the important features of the von

Neumann model is the fact that:

• The instructions themselves

And

• The data the instructions manipulate

are all stored in the same RAM.

• This was one of the revolutionary features of the modern computer, totally unlike Punch-Card computers proposed in the past.

Important Point• This innovation is crucial to modern

computing.

• It even allows for the possibility of programs that change themselves as they are executed.

• With great power comes great risk…

• Computer Viruses!

Machine Language

• We now have a machine that can execute instructions.

• Basic Questions:

• What instructions?

• How do computers understand these instructions? (Representation)

• What does software look like to acomputer?

Instructions

• 16 possible Op-Codes:0: halt1: add2: subtract3: multiply4: bus output5: jump6: jump if positive7: jump & count8: bus input9: loadA: storeB: load direct/addr.C: NANDD: ANDE: Shift RightF: Shift Left

Adding Numbers• Simple Program to calculate

1 + 2 + 3 + 4 + 5 + 6 = 21 = 15 (hex)10: Load R0 0001 (always 1)11: Load R2 0000 (running total)12: Load R1 0001 (current number)

13: Add R2 R2 + R1 (R2=1)14: Add R1 R1 + R0 (R1=2)

15: Add R2 R2 + R1 (R2=3)16: Add R1 R1 + R0 (R1=3)

17: Add R2 R2 + R1 (R2=6)18: Add R1 R1 + R0 (R1=4)

19: Add R2 R2 + R1 (R2=A)1A: Add R1 R1 + R0 (R1=5)

1B: Add R2 R2 + R1 (R2=F)1C: Add R1 R1 + R0 (R1=6)

1D: Add R2 R2 + R1 (R2=15)1E: halt

Adding Numbers• Alternate Loop using Jump & Count for

1 + 2 + 3 + 4 + 5 + … + N10: Load R1 0006 (N)11: Load R2 0000 (running total)

12: Add R2 R2 + R1 (add in N)

13: Jump to 12 and (If N isn’t 0 yet, decrease R1 if (R1>0) Let N=N-1, and go back)

14: halt (N is now 0, and R2 = 1+2+…+N)

Sound is a waveform

Sampling the waveform

The approximation

Even finer sampling

Digital vs. Analog

• A sound file is just a sequence of bytes– On a CD, the sampling rate is 44,100 samples per second.

Each sample is quantized into a number in the range from 0 to 65,535, So a CD has 44,100 numbers (each 2 bytes) per channel per second of music.

• A graphic image is just a sequence of bytes– An image is made up of pixels– Each pixel might be represented by 3 bytes giving color

intensities.

• Manipulating sounds or images just involves manipulating numbers

Numeracy

• CD has 44,100 2 byte samples per second. • 44,100 x 2 x 60 x 2 (channels) = 10.09

megabytes per CD minute• A CD is 74 minutes (746 megabytes) long. • MP3 players typically use memory 32 or 64

megabytes in size. (or 3-6 minutes)

• Compression is needed

The need for compression

• A CD is 746 megabytes long

• A printed page requires 86.4 Megabytes of storage

Simple compression

• Simple compression – Use redundancy

• More complex compression– Lossy vs. lossless– Ideas for lossy

• Drop frequencies you can’t hear

• Drop small differences you won’t notice

More complex compression

• Lossy compression– Can save space by eliminating parts of the

image/sound that don’t matter very much– Big savings compared to lossless, but the signal

cannot be fully recovered.– Ideas

• Dropping frequencies you can’t hear• Dropping small differences you won’t notice

– Generally large savings compared to lossless

Pause for experiment