ee40 lec 14ee40 lec 14 digital signal and boolean algebradigital signal and boolean...

EE40 Lec 14EE40 Lec 14

Digital Signal and Boolean AlgebraDigital Signal and Boolean Algebra

Prof Nathan CheungProf. Nathan Cheung

10/14/2009

Reading: Hambley Chapters 7.1-7.4

Slide 1EE40 Fall 2009 Prof. Cheung

Reading: Hambley Chapters 7.1 7.4

Analog Signals

Analog: signal amplitude is continuous with time.

Am plitude M odulated S ignal

1

0.2

0 .4

0 .6

0 .8

mic

rovo

lts

0 8

-0 .6

-0 .4

-0 .2

00 5 10 15 20 25 30 35 40 45 50

Sign

al in

m

-1

-0 .8

T im e in m icroseconds


Digitalized signal

Digital: signal amplitude is represented by a restricted set of discrete numbers.


Why Digital?

Di i l i l b i d i d

(For example, why CDROM audio vs. vinyl recordings?)

• Digital signals can be transmitted, received, amplified, and re-transmitted with far less degradationdegradation.

• Digital information is easily and inexpensively stored (in RAM, ROM, etc.), with arbitrary accuracy.

• Complex logical functions are easily expressed as binary functions (e g in control applications)as binary functions (e.g. in control applications).

• Digital signals are easy to manipulate (as we shall see)


shall see).

Digital Signal Representations

Binary numbers can be used to represent any quantity.y q y

We generally have to agree on some sort of “code”, and the dynamic range of the signal incode , and the dynamic range of the signal in order to know the form and the number of binary digits (“bits”) required.

Example : To encode the signal to an accuracy f 1 t i 64 (1 5% i i ) 6 bi di itof 1 part in 64 (1.5% precision), 6 binary digits

(“bits”) are needed


Digital Signals• For a digital signal, the voltage must be within

one of two ranges in order to be defined:V

“1” VOH

VDD

1VIH

undefined region increasingvoltage

• Positive Logic: “0”VIL

VOL

voltage

– “low” voltage ≡ logic state 0– “high” voltage ≡ logic state 1

VOL

0 Volts


Number Base• Number Base B ⇒ B symbols per digit:

–Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9–Base 2 (Binary): 0, 1

• Number base representation: –d31d30 ... d1d0 is a 32 digit number–value = d31 × B31 + d30 × B30 + ... + d1 × B1 + d0 × B0

E l Bi (B 2) 0 1 (I bi di it• Example : Binary (B=2): 0,1 (In binary digits called “bits”)

11010 = 1 24 + 1 23 + 0 22 + 1 21 + 0 2011010 = 1×24 + 1×23 + 0×22 + 1×21 + 0×20

= 16 + 8 + 2= 26


Here 5 digit binary number turns into a 2 digit decimal number

Decimal-Binary Conversion

• Decimal Integer to Binary– Repeated Division By 2

Example: Decimal integer 343


Decimal-Binary Conversion

• Decimal Fraction to Binary Fraction– Repeated Multiplication By 2

Example: Decimal Fraction 0.392


Other examples: 0.7510=0.112 0.3 ≈.0100112

Binary to Decimal conversion

110001.00122

= 1x25 +1x24 +0x23 +0x22 + 0x21 + 1x20= 1x2 +1x2 +0x2 +0x2 + 0x2 + 1x2+0x2-1+ 0x2-2 + 1x2-3

= 3210 + 1610 + 110 +0.1251010 10 10 10

= 49.12510


10

Adding Binary Numbers

Addition Rules

Example


Two’complement of Binary Numbers


Subtracting Binary Numbers

Find signed two’sFind signed two s complement of substrahend

Subtraction = add the signed binary number toSubtraction add the signed binary number to the signed two’s complement of substrahend

Note: If two numbers to be added have the same sign bit but the result have the


gopposite sign bit, overflow or underflow has occurred

Hexadecimal Numbers: Base 16

• Hexadecimal: 0 1 2 3 4 5 6 7 8 9 A B C D E F0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F–Normal digits + 6 more from the alphabet

Conversion: Binary⇔Hex• Conversion: Binary⇔Hex–1 hex digit represents 16 decimal values

4 bi di i 16 d i l l–4 binary digits represent 16 decimal values⇒1 hex digit replaces 4 binary digitsExample:11110110.12=11110110.10002 =F6.816


Digital Representations of Logical Functions

• Digital signals offer an easy way to perform logical functions, using Boolean algebra.– Variables have two possible values: “true” or “false”

• usually represented by 1 and 0, respectively.

• All modern control systems use this approach.

• Example: Hot tub controller with the following algorithmg g– Turn on the heater if the temperature is less than

desired (T < Tset) and the motor is on and the key it h t ti t th h t t b i l d


switch to activate the hot tub is closed.

Combinatorial Logic gates

– Combine several logic variable inputs to produce a logic variable output

– Memoryless: output at a given instant depends the input values of that instant.


Logic Functions, Symbols, & NotationTRUTH

NAME SYMBOL NOTATION TABLE

“NOT” F = AFAA F0 11 0

“OR” F A+BFAA B F0 0 00 1 1“OR” F = A+BFB0 1 11 0 11 1 1

A B F0 0 00 1 0“AND” F = ABFA

B


1 0 01 1 1

B

3-Input Gates


Boolean algebra

Th t f B l l b b• The operators of Boolean algebra may be represented in various ways. Often they are simply written as AND, OR and NOT. p y

• In describing circuits, NAND (NOT AND), NOR (NOT OR) and XOR (eXclusive OR) may also be usedused.

An excellent web site to visithttp://en.wikipedia.org/wiki/Boolean_algebra

• Mathematicians often use + for OR and • for AND (since in some ways those operations are analogous to addition and multiplication in otheranalogous to addition and multiplication in other algebraic structures) and represent NOT by a line drawn above the expression being negated.


Note: In Hambley text, there is no dot for AND opeartion

Boolean Algebra Relations

A•A = AA•A = 0

A+A = AA+A = 1A A 0

A•1 = AA 0 0

A A 1A+1 = 1A 0 AA•0 = 0

A•B = B•AA+0 = AA+B = B+A

A•(B•C) = (A•B)•C A+(B+C) = (A+B)+CA•(B+C) = A•B + A•CA•(B+C) = A•B + A•C

A•B = A + BD M ’ l


A•B = A + BDe Morgan’s laws

Boolean Expression Example

)ED)(DC(ABCCBAF ++++=

AC)BB(AC =+DECEDC

)ED(D)ED(C++=

+++

DE)EDA(CF +++=


Logic Functions, Symbols, & Notation 2

“NOR” F = A+BFAB

A B F0 0 10 1 01 0 0B 1 0 01 1 0

“NAND” F = ABFAB

A B F0 0 10 1 1

B 1 0 11 1 0

A B F0 0 00 1 11 0 1

“XOR”(exclusive OR)

F = A + BFAB


1 0 11 1 0

(exclusive OR) B

NAND Gate Implementation

• De Morgan’s law tells us that• De Morgan s law tells us that

is the same asis the same as

• By definition,

is the same as


Graphic Representation

0AAA

A1AA

0AA=+

=

A

Venn Diagram

Full square = complete set =1Yellow part = NOT(A) =AWhi i l A


White circle = A

Graphic Representation of XOR

ABAB

A B+

BAB

BAAB)BA)(BA(BABABA ++=++=+=⊕

Exclusive OR=yellow and blue part – intersection part= exactly when only one of the input is true


exactly when only one of the input is true

Circuit Realization of XOR Gate

BAAB)BA)(BA(BABABA ++=++=+=⊕

A B⊕

AA

ABB A B⊕

BB

AB


Logical Sufficiency of NAND Gates• If the inputs to a NAND gate are tied together, an

inverter results

• From De Morgan’s laws, the OR operation can be realized by inverting the input variables and combiningrealized by inverting the input variables and combining the results in a NAND gate.

• Since the basic logic functions (AND OR and NOT) can• Since the basic logic functions (AND, OR, and NOT) can be realized by using only NAND gates, NAND gates are sufficient to realize any combinational logic function.


Logical Sufficiency of NOR Gates• Show how to realize the AND, OR, and NOT functions

using only NOR gates

Since the basic logic f nctions (AND OR andSince the basic logic functions (AND, OR, and NOT) can be realized by using only NOR gates, NOR gates are sufficient to realize any


g , g ycombinational logic function.

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