digital abstraction

8/14/2019 Digital Abstraction

1/206.002 Fall 2000 Lecture 14

6.002 CIRCUITS ANDELECTRONICS

The Digital Abstraction


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6.002 Fall 2000 Lecture 24

Review

Discretize matter by agreeing toobserve the lumped matter discipline

Analysis tool kit: KVL/KCL, node method,superposition, Thvenin, Norton

(remember superposition, Thvenin,Norton apply only for linear circuits)

Lumped Circuit Abstraction


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Discretize value Digital abstraction

Interestingly, we will see shortly that thetools learned in the previous threelectures are sufficient to analyze simpledigital circuits

Reading: Chapter 5 of Agarwal & Lang

Today


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Analog signal processing

But first, why digital?In the past

By superposition,

The above is an adder circuit.

2

21

11

21

20

VRR

VRR

V+

+

+

=

If ,21 RR =

2

21

0

VVV

+=

1V

1

2+

2V +

0V

and

might represent the

outputs of two

sensors, for example.

1V 2V


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Noise Problem

noise hampers our ability to distinguishbetween small differences in value e.g. between 3.1V and 3.2V.

Receiver:

huh?

add noise onthis wire

t


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Value Discretization

Why is this discretization useful?

Restrict values to be one of two

HIGH

5V

TRUE

1

LOW

0V

FALSE

0

like two digits 0 and 1

(Remember, numbers larger than 1 can berepresented using multiple binary digits andcoding, much like using multiple decimal digits torepresent numbers greater than 9. E.g., thebinary number 101 has decimal value 5.)


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Digital System

sender receiverS

VRV

noise

SV

0 01

0V

2.5V

5V HIGH

LOW

t

RV

0 01

0V

2.5V

5V

t

VVN

0=

NV

SV

0 01

2.5V t

With noiseVV

N2.0=

SV

0 01

0V

2.5V

5V

t

0.2V

t


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Digital System

Better noise immunityLots of noise margin

For 1: noise margin 5V to 2.5V = 2.5VFor 0: noise margin 0V to 2.5V = 2.5V


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Voltage Thresholdsand Logic Values

1

0

1

0

sender receiver

1

0

0V

2.5V

5V


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forbidden

region

VH

V L

3V

2V

But, but, but What about 2.5V?

Hmmm create no mans landor forbidden region

For example,

senderreceiver

0V

5V

1 1

0 0

1 V 5V

0 0V V

H

L


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sender receiver

But, but, but Wheres the noise margin?

What if the sender sent 1: ?VHHold the sender to tougher standards!

5V

0V

11

00

V

0H

V0L

VIH

VIL


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sender receiver

VH

But, but, but Wheres the noise margin?

What if the sender sent 1: ?Hold the sender to tougher standards!

5V

0V

1 noise margin:

0 noise margin:

VIH

- V0H

VIL

- V0L

11

00

V

0H

V0L

VIH

VIL

Noise margins


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Digital systems follow static discipline: ifinputs to the digital system meet valid inputthresholds, then the system guarantees itsoutputs will meet valid output thresholds.

sender

receiver

0 1 0 1

t

5VV

0H

V0L

0V

VIH

VIL

0 1 0 1

t

5VV

0H

V0L

0V

VIH

VIL


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Processing digital signals

Recall, we have only two values

Map naturally to logic: T, F

Can also represent numbers

1,0


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Processing digital signalsBoolean Logic

If X is true and Y is trueThen Z is true else Z is false.

Z = X AND YX, Y, Z

are digital signals0 , 1

Z = X YBoolean equation

Enumerate all input combinations

Truth table representation:

ZX Y

AND gateZX

Y

0 0 0

0 1 01 0 0

1 1 1


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Adheres to static discipline Outputs are a function of

inputs alone.

Combinational gateabstraction

Digital logic designers do not

have to care about what is

inside a gate.


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Demo

Noise

ZXY

Z = X Y

Z

Y

X


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Z = X Y

Examples for recitation

X

t

Y

t

Z

t


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In recitation

Another example of a gateIf (A is true) OR (B is true)

then C is true

else C is false

C = A + B Boolean equation

OR

OR gate

CA

B

ZX

Y NAND

Z = X Y

More gates

B B

Inverter


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Boolean Identities

AB + AC = A (B + C)

X 1 = XX 0 = XX + 1 = 1X + 0 = X

1 = 0

0 = 1

output

BC B C

A

Digital Circuits

Implement: output = A + B C

digital abstraction

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