1

Binary CodesDigital systems use 2-state devices that understand only 2 binary values (0 and 1). But we communicate using various symbols and methods. Codes are needed to allow us to communicate. Codes translate our language into the computers language and vice versa.

Lecture #2 EGR 277 – Digital Logic

Reading Assignment: Chapters 1 and 2 in Digital Design, 3rd Edition by Mano

How many bits are needed to encode a set of elements? In general,

N bits are needed to encode up to 2N elements

N = log2(number of elements)

2

Lecture #2 EGR 277 – Digital Logic

Examples: Determine the number of bits required and develop a code to encode

each of the following:A) 8 symbols

B) 10000 symbols

C) the 366 days of the year in order

D) the 366 days of the year using the month and day

3

byte – group of 8 bitsword – an n-bit code forms n-bit words. In a computer system, a certain number of bytes may form a word. For example, a 16-bit system might refer to words as consisting of 2 bytes.

Lecture #2 EGR 277 – Digital Logic

Decimal Codes – codes used to encode the digits 0 – 9Several examples are shown below (Table 1-5 from the text)

The BCD code is the most common.

Know the BCD code for tests. Other codes will be given if needed.

4

Lecture #2 EGR 277 – Digital Logic

Gray Code The Gray code is a 4-bit binary code (different from a BCD code in that it encodes all 16 4-bit combinations, not just 10 combinations).

This code is interesting in that successive code words only change by one bit.

This code is sometimes used with stepper motors. Each time the code increases, only one bit changes, and the stepper motor turns a specified amount (angle).

5

Lecture #2 EGR 277 – Digital Logic

Example: Show how a digital voltmeter would convert binary information into BCD format, including the use of binary-to-BCD converters and BCD-to-7-segment decoders.

Application: BCD codes are routinely used on equipment to display numerical values. Digital circuits work in binary, but people prefer to read numbers in base 10, so BCD codes are used to convert the data to base 10.

6

Lecture #2 EGR 277 – Digital Logic

Error Detection and Correction Codes

Extra bits are often added to codewords (using sometimes complex schemes) so that when the word is transmitted, the received can detect if errors occurred in the transmission and possibly correct some of the errors.

Parity – perhaps the simplest error detection code involves the addition of a parity bit

Discuss even parity vs odd parity

Example: Show a BCD code with a trailing odd parity bit.

Error Correction Codes - Discuss.

Example: Show a simple 2-bit code with extra bits added for error correction.

7

Lecture #2 EGR 277 – Digital Logic

Alphanumeric codes

Used to encode keyboard symbols and control characters.

ASCII and EBCDIC codes are the most common alphanumeric codes.

The ASCII code is shown below in Table 1-7 from the text.

Since the ASCII code is a 7-bit code, a parity bit is commonly added.

Example: Write the word “Digital” using an ASCII code with an even leading parity bit.

8

Lecture #2 EGR 277 – Digital Logic

Chapter 2 - Boolean Algebra - comparison to regular algebra

 

Any algebra is built upon:1) A set of elements2) A set of operators3) A set of postulates

Boolean Algebra is built upon:1) A set of elements: {0, 1}2) A set of operators: {+, • } – Define these in class3) A set of postulates: the Huntington Postulates are the most common

Huntington Postulates – The following 6 postulates, along with the set of elements and set of operators shown above, uniquely and completely define Boolean algebra.

1) Closure for the operations {+, • } - Discuss

9

Lecture #2 EGR 277 – Digital Logic

2) Two identity elements: - Illustrate by considering all possible values for x

A) 0: 0 + x = x + 0 = x

B) 1: 1 • x = x • 1 = x

3) Commutative Laws: - Illustrate by considering all possible values for x and y

A) x + y = y + x

B) xy = yx

4) Distributive Laws: - Prove by truth table

A) x • (y + z) = xy + xz

B) x + yz = (x + y) • (x + z)

10

Lecture #2 EGR 277 – Digital Logic

5) Existence of a Complement: - Illustrate by considering all possible values for x

Define

by the following truth table:

x' = x = "the complement of x" = "NOT x"

x x’

0 1

1 0

A) x + x’ = 1

B) x • x’ = 0

6) At least two non-equal elements: {0, 1} - Discuss

Common Theorems

Boolean algebra has already been completely defined. Additional theorems are also often used, not because they are required, but because they are useful. Some of the most common theorems are shown below. Note that each theorem could be formally proven using the postulates.

1) Idempotency: (“same power”)

A) x + x = x – Prove this using the postulates

B) x • x = x

Example: Show related examples using this theorem.

11

Lecture #2 EGR 277 – Digital Logic

2) (no name) – Discuss

A) x + 1 = 1

B) x • 0 = 0

3) Involution: – Discuss

x’’ = = x

4) Associative Laws: – Discuss (show logic gate application)

A) x + (y + z) = (x + y) + z

B) x(yz) = (xy)z

5) DeMorgan’s Theorems: - Prove 5A by truth table

A)

B)

Example: Show related examples using DeMorgan’s theorem.

x + y = x yx y = xy = x y

12

Lecture #2 EGR 277 – Digital Logic

6) Absorption:

A) x + xy = x

B) x (x+y) = x

Example: Show related examples using this theorem.

7) (no name)

A) x + x’y = x + y

B) x (x’ + y) = xy

Example: Show related examples using this theorem.

8) Concensus:

A) xy + x’z + yz = xy + x’z

B) (x + y)(x’ + z)( y + z) = (x + y)(x’ + z)

Example: Show related examples using this theorem.

13

Lecture #2 EGR 277 – Digital Logic

Order of operations

Example: f = ab+cd Note: spacing is often used to make it clearer: f = ab + cd

Operation Precedence

Parentheses Higher

NOT

AND

OR Lower

Boolean Functions – Simplifying Boolean functions corresponds to minimizing the amount of circuitry (logic gates) to be used.

Truth table Boolean function minimized with Boolean algebra

implement with logic circuits

Minimizing Boolean functions

No specific rules. In general we use Boolean algebra (postulates and theorems) to reduce the number of terms, literals, logic gates, or IC’s.

Literal – a primed (complemented) or unprimed variable

Example: How many literals are in the expression f = ab + a’c + bc’d ?

14

Lecture #2 EGR 277 – Digital Logic

Examples – Minimize the following Boolean functions:

 1) F = AB + A(B + C) + B(B + C)

 

2) F = AB’(C + BD) + A’B’

3) F(A,B,C,D) = A + A’BC + C’

15

Lecture #2 EGR 277 – Digital Logic

Examples – Minimize the following Boolean functions (continued):

 4) F = [(x’y)’ + z’]’

 

5) CD) ABF)(E (CD AB F

16

Lecture #2 EGR 277 – Digital Logic

Examples – Minimize the following Boolean functions (continued):

 6) f(x,y,z) = x’y(z + y’x) + y’z

 

7) ca dcb dcb c)b(aba d)c,b,f(a,