1. Review for Exam 1
Chapters 1 through 3

2. Chapter 1 Overview Information Representation
Number Systems [binary, octal and hexadecimal]
Base Conversion
Decimal Codes [BCD (binary coded decimal)]
Alphanumeric Codes
Parity Bit
Gray Codes

3. 1-2 Number Systems Positive radix, positional number systems Examples:
Decimal (radix r =10)
Binary (radix r =2)
Octal (radix r = )
Hexadecimal (r = )
Ex: 24.3 = 2x x100+3x10-1
Digits (0-9)
Ex: = ( . )10
Bits (0-1)
= = 13.25
Digits: 1,2,…9, A, B, C, D, E, F

4. Range of numbers Binary number: ex. a 3-bit number: n=3
000, 001 … ,111 or in decimal system: 0, 1 … 7
Total of 8 numbers (=23)
Range: from 0 to 7 (0 to 23-1)
In general a n-bit number represents:
2n different numbers
Min: 0
Max number: 2n-1
For fractions: m bits after the radix point:
Max number: (2m -1)/2m
Fractions Max number: ? 2m-1/2m
Proof: A A …. A-m2-m)
= 2-m (A-12m-1 + A-22m-2 +… A-m20)
= 2-m (2m -1)

5. Use of HEX system Short hand notation of large binary numbers:
Each HEX digits can be represented by exactly 4 bits (16=24)
Thus ( )2
Conversion from binary to HEX and HEX to binary is very easy:
( )2 = ( )16
( )2 = ( )16
B = ( )2 E
( )2 = ( 9D )16
( )2 = ( )16 =
2 B C
B39.7 =

6. Each octal digit can be represented by 3 bits
Octal system
Radix r = 8
8 digits:
0, 1, 2,…7
Ex: 2758 = 2x82 + 7x8 + 5x1 =
= 18910
Each octal digit can be represented by 3 bits

7. 1-3 Conversion Between Bases
To convert from one base to another:
1) Convert the Integer Part
2) Convert the Fraction Part
3) Join the two results with a radix point

8. Example: convert (325.65)10 to hex
Integer part: = ( )16
Fractional part: .65
325/16 = 20 and rem = 5
20/16 = 1 and rem = 4
1/ = 0 and rem = 1
Most significant
Least significant digit
Thus = 14516
0.65x16 = thus int = 10= A
0.4x16 = thus int = 6
Etc.
Most significant
Least significant
Thus = A6616
= 145.A6616

9. Conversion - Summary Hexadecimal Ai.16i Decimal Binary  Ai.8i Octal
Divisions (or x) by 16
Hexadecimal
Ai.16i
Decimal
Divisons by 2
Group in bits of 4
SAi.2i
Binary
Divisons by 8
Octal Hex: through the binary representation
Group in bits of 3
 Ai.8i
Octal

10. Example: 3-bit code can represent up to 8 different elements”
1-4 Binary Codes
A n-bit binary code is a n-bit word which can represent up to 2n different elements.
Example: 3-bit code can represent up to 8 different elements”
All quantities in a PC needs to be expressed as a binary number: e.g. the digits 0, 1, …9; characters, control characters, etc. This can be done using a “Code

11. Binary Coded Decimal (BCD)
The BCD code is the 8,4,2,1 code.
This code only encodes the first ten values from 0 to 9.
Each decimal digit is coded separately by 4 bits
Example:
(325)10 = ( )BCD
Exercise: (856)10 = ( )BCD
(325)10 = BCD
If it were a binary conversion (not BCD encoding):
(9bits)
(856)10 = BCD
856 = ( )2 3 2 5

12. Overview Chapter 2 Binary Logic and Gates Boolean Algebra
Standard Forms
Two-Level Optimization
Map Manipulation
Other Gate Types
Exclusive-OR Operator and Gates
High-Impedance Outputs
In this chapter we will introduce logic gates which are the simplest logic elements. However, they form the foundation of more complex blocks which will be discussed in later chapters.
We will also discuss the mathematical tools that are required to design, analyze and simplify logic function (Boolean Algebra): optimization techniques

13. Operator Definitions and Truth Tables
Truth table - a tabular listing of the values of a function for all possible combinations of values on its arguments
Example: Truth tables for the basic logic operations: 1
Z = X·Y Y X
AND OR X Y
Z = X+Y 1 1 X
NOT Z =

14. Existence complements
2-2 Boolean Algebra
Boolean algebra deals with binary variables and a set of three basic logic operations: AND (.), OR (+) and NOT ( ) that satisfy basic identities
Basic identities 1. X
+ 0 = 2. X
. 1 =
Existence 0 and 1 or
operations with 0 and 1 3. X + 1 = 1 4. X
. 0 =
Idempotence 5. 6.
X . X X =
X + X
Boolean algebra deals with binary variables and a set of three basic logic operations: AND, OR and NOT which satisify a set of basic identities 7. 8. =
X . X 1
X + X
Existence complements
Involution 9.
X = X
Dual
Replace “+” by “.”, “.” by +,
“0” by “1” and “1’’ by”0”

15. Boolean Algebra Dual Boolean Theorems of multiple variables 10. X + Y
Y + X =
Commutative
11. XY YX =
12.
(X + Y) Z +
X + (Y Z) =
13.
(XY) Z
X(Y
Z ) =
Associative
14.
X (Y+ Z) XY XZ + =
15. X
+ YZ
(X + Y)
(X + Z) =
Distributive
17.
X . Y
X + Y =
16.
X + Y
X . Y =
DeMorgan’s
Dual
Boolean algebra deals with binary variables and a set of three basic logic operations: AND, OR and NOT which satisify a set of basic identities

16. Other useful Theorems Dual XY + XY = Y Minimization (X + Y)(X + Y) = Y
X(X + Y) = X
X + XY = X
Absorption
X(X + Y) = XY
X + XY = X + Y
Simplification
XY + XZ + YZ = XY + XZ
Consensus
Consensus:
xy + x’z + yz = xy + x’z + yz( x+x’)
= xy(1+z) + x’z(1+y)
= xy + x’z
(X + Y)( X + Z)(Y + Z) = (X + Y)( X + Z)

17. 2-3 Standard (Canonical) Forms
It is useful to specify Boolean functions in a form that:
Allows comparison for equality.
Has a correspondence to the truth tables
Canonical Forms in common usage:
Sum of Products (SOP), also called Sum or Minterms (SOM)
Product of Sum (POS), also called Product of Maxterms (POM)

18. Maxterms and Minterms Examples: Two variable minterms and maxterms.
The index above is important for describing which variables in the terms are true and which are complemented.
Index
Minterm
Maxterm
0 (00)
x y
x + y
1 (01)
2 (10)
3 (11)

19. Index Examples – Four Variables
Index Binary Minterm Maxterm
i Pattern mi Mi
Notice: the variables are in alphabetical order in a standard form d c b a d c b a ? ? d c b a + d c b a + ?
M1 = a + b + c + d’
m3 = a’ b’ c d
m7 = a’ b c d
M 13 = a’ + b’ + c + d’ d b a c + ?
Relationship between min and MAX term? i m M =

20. Minterm Function Example
F(A, B, C, D, E) = m2 + m9 + m17 + m23
F(A, B, C, D, E) write in standard form:
Sum of Product (SOP) expression:
F = Σm(2, 9, 17, 23)
A’B’C’DE’ + A’BC’D’E + AB’C’D’E + AB’CDE m2 m9
m17
m23
F(A,B,C,D,E) = A’B’C’DE’ + A’BC’D’E + AB’C’D’E + AB’CDE

21. Expressing a function with Maxterms
Start with the SOP: F1(x,y,z) =m1 + m4 + m7
Thus its complement F1can be written as
F1 = m0 +m2 +m3 + m5 + m6 (missing term of F1)
Apply deMorgan’s theorem on F1:
(F1 = (m0 +m2 +m3 + m5 + m6)
= m0.m2.m3.m5.m6
= M0.M2.M3.M5.M6
= ΠM(0,2,3,5,6)
Thus the Product of Sum terms (POS):
also called, Big M notation ) z y
z)·(x
·(x z) (x F 1 + = x
)·( ·(

22. 2-4 Circuit Optimization
Goal: To obtain the simplest implementation for a given function
Optimization requires a cost criterion to measure the simplicity of a circuit
Distinct cost criteria we will use:
Literal cost (L)
Gate input cost (G)
Gate input cost with NOTs (GN)
Other possible criteria: Speed
Power consumption
Size and cost

23. Literal Cost Literal – a variable or its complement
Literal cost – the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram
Examples (all the same function):
F = BD + AB’C + AC’D’ L = 8
F = BD + AB’C + AB’D’ + ABC’ L =
F = (A + B)(A + D)(B + C + D’)( B’ + C’ + D) L =
Which solution is best?
2nd Literal Cost = 11
3rd Literal Cost = 10
The first solution is best NOTE: all these represent the same function: f=Sum m(5,7,8,10,11,12,13,15)

24. Karnaugh Maps (K-map) A K-map is a collection of squares
Each square represents a minterm
The collection of squares is a graphical representation of a Boolean function
Adjacent squares differ in the value of one variable
Alternative algebraic expressions for the same function are derived by recognizing patterns of squares
The K-map can be viewed as
A reorganized version of the truth table
A topologically-warped Venn diagram as used to visualize sets in algebra of sets

25. 2-5 Map Manipulation: Systematic Simplification
A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2.
A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms.
Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map.

26. Don't Cares in K-Maps
Sometimes a function table or map contains entries for which it is known:
the input values for the minterm will never occur, or
The output value for the minterm is not used
In these cases, the output value need not be defined
Instead, the output value is defined as a “don't care”
By placing “don't cares” ( an “x” entry) in the function table or map, the cost of the logic circuit may be lowered.
Example 1: A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used. The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

27. Other Gate Types: overviewB A B A B A A B
A B BUF NAND NOR XOR XNOR

28. The Tri-State Buffer Symbol Truth Table OUT= IN.EN
For the symbol and truth table, IN is the data input, and EN, the control input.
For EN = 0, regardless of the value on IN (denoted by X), the output value is Hi-Z.
For EN = 1, the output value follows the input value.
Variations:
Data input, IN, can be inverted
Control input, EN, can be inverted
by addition of “bubbles” to signals. IN EN
OUT
Truth Table EN IN
OUT X
Hi-Z 1
OUT= IN.EN

30. NAND Mapping Algorithm
Replace ANDs and ORs:
Repeat the following pair of actions until there is at most one inverter between :
A circuit input or driving NAND gate output, and
The attached NAND gate inputs.

31. NOR Mapping Algorithm Replace ANDs and ORs:
Repeat the following pair of actions until there is at most one inverter between :
A circuit input or driving NAND gate output, and
The attached NAND gate inputs.

32. Enabling Function
Enabling permits an input signal to pass through to an output
Disabling blocks an input signal from passing through to an output, replacing it with a fixed value
The value on the output when it is disable can be Hi-Z (as for three-state buffers and transmission gates), 0 , or 1
When disabled, 0 output
When disabled, 1 output
See Enabling App in text

33. 3-7 Decoding A n-bit binary code can represent up to m=2n elements:
m elements n-bit binary code
Decoding - the conversion of an n-bit input code to an m-bit output code with n ≤ m ≤ 2n such that each valid code word produces a unique output code
encoding
(ex. 256 alpha-num. chars)
(ex. 8-bit ASCII code)
decoding A0 :
An-1 D0 D1
Dm-1
n-2n
decoder
n bits
m-elements
≤ 2n

34. 2-to-4 Line Decoder circuit= A 1 2 3
Notice that the outputs of the decoder correspond to the minterms: Di=mi

35. Decoder Expansion
Larger decoders can be realized by implementing each minterm using a single AND gate:
However for large decoders this requires multiple input AND gates which is not always feasible.
Better to use a hierarchical approach: build larger ones from smaller decoders.
Approach:
Output AND gates have only 2 inputs and implement the minterms.
The output AND gates are driven by two decoders with their numbers of inputs either equal or differing by 1.

36. Rule for building large decoders
k-to-2k decoder:
One needs 2k output AND gates
If k can be divided by 2:
use two k/2-to-2k/2 decoders
If k cannot divided by 2:
use a (k+1)/2 and
use a (k-1)/2 decoder.
Previous example: 3-to-8 decoder (k=3):
Use a 2-to-4 and a 1-to-2 decoder

37. Combinational Logic Implementation - Decoder and OR Gates
Implement m functions of n variables with:
Sum-of-minterms expressions
One n-to-2n-line decoder
m OR gates, one for each output

38. Example
Design and implement a majority function F(ABC) using a 3-to-8 decoder
Truth table:
Minterms:
F=m(3,5,6,7)
Implementation using decoder:
A B C F
Indicate MSB, LSB 1 2 3 4 5 6 7 A B C 2 1 F
m(3,5,6,7)
Thus use a 3-to-8 decoder with OR gate

39. Encoding
Typically, an encoder converts a code containing exactly one bit that is 1 to a binary code corresponding to the position in which the 1 appears: ex. D1=1  output 0001
Examples: Octal-to-Binary encoder
Other examples? A0 :
An-1 D0 D1
Dm-1
encoder 1 2 3
m-1
n-1 1 1
Examples: Hex-to-Binary encoder
Decimal to BCD encoder
Decimal to binary encoder
ASCI to binary
Grades (ABCDF) to binary: A, B, C, D, F to 3 bit binary code

40. Priority Encoder
If more than one input value is 1, then the encoder just designed does not work.
An encoder that can accept all possible combinations of input values and produce a meaningful result is a priority encoder.
Among the 1s that appear, it selects the most significant input position (or the least significant input position) containing a 1 and responds with the corresponding binary code for that position. D0 D1 D2 D3 A1 A0 ? V 1 2 3
processor To

41. 3-9 Selecting (multiplexers)
Selecting of data or information is a critical function in digital systems and computers
Circuits that perform selecting have:
A set of n information inputs from which the selection is made
A set of k control (select) lines for making the selection
A single output 1 2 3 :
n-1 I0 I1 I2 I3
In-1
OUT
n ≤ 2k inputs k
Sk-1..S1 S0
k select lines

42. 4:1 MUX realization Expression for OUT Circuit implementation: SOP
4 AND gates (4 product terms)
2-to-4 line decoder (to generate the minterms)
S1 S0 OUT I0 I1 I2 I3
OUT = S1S0 I0+ S1S0 I1+ S1S0 I2+ S1S0 I3 m3 m2 m1 m0
or OUT = Σ mi Ii
i=0
2k-1

43. Exercise Build a 8:1 MUX using two 4:1 and one 2:1 muxes I0 I1 4:1 I21 2 3
1 0 1
OUT
S1 S0 I4 I5 I6 I7
4:1 1 2 3
We need 3 select signals: S2,S1 and S0 S2
1 0
Ex: S2S1S0=110 : select I6

44. Multiplexer-based combinational circuits realization- Approach 1
A mux can be easily used to implement a function defined by a truth table (lookup table)
Indeed the output F of a mux is equal to:
F = Σ mi Ii
i=0
2k-1
Example
A B OUT =F I I I I m0 m1 m2 m3
F= Σm(1,2)
Give the input Ii the
value of 0 or 1
as shown in the truth table 1
4:1 1 2 3
1 0 F
A B

45. Combinational Logic Implementation - Multiplexer Approach 2
Implement any m functions of n + 1 variables by using:
An m-wide 2n-to-1-line multiplexer
Design:
Find the truth table for the functions.
Based on the values of the first n variables, separate the truth table rows into pairs
For each pair and output, define a rudimentary function of the final variable (0, 1, X, ) X