1. CPEN 315 - Digital System Design
Spring 2008
Combinational Functions
and Circuits
Chapter 3
© Logic and Computer Design Fundamentals, 4rd Ed., Mano
© 2008 Pearson Prentice Hall

2. Overview Functions and Functional Blocks Decoding Encoding Selecting
Implementing Combinational Functions Using:
Decoders and OR gates
Multiplexers (and inverter)
ROMs
PLAs
PALs

3. Functions and Functional Blocks
The functions considered are those found to be very useful in design
Corresponding to each of the functions is a combinational circuit implementation called a functional block.
In the past, many functional blocks were implemented as SSI, MSI, and LSI circuits.
Today, they are often simply parts within VLSI circuits.

4. Simple Logic Functions
Functions of a single variable
Can be used on the inputs to functional blocks.

5. Enabling Functions Enabling Permits an input signal to pass
Disabling
Permits an input signal to pass
through to an output
Blocks an input signal from
Passing through to an output
Example:
The ignition switch = EN
Lights = X

6. Decoding
Decoding - the conversion of an n-bit input code to an m-bit output code such that each valid code word produces a unique output code
Circuits that perform decoding are called decoders
Here, functional blocks for decoding are called n-to-m line decoders, and they generate 2n (or fewer) minterms for the n input variables

7. Decoder Examples
1-to-2-Line Decoder
2-to-4-Line Decoder

8. Decoder with Enable Example
In general, attach m-enabling circuits to the outputs
See truth table below for function
Note use of X’s to denote both 0 and 1
Combination containing two X’s represent four binary combinations
2-to-4-Line Decoder
with Enable

9. Encoding
Encoding - the opposite of decoding - the conversion of an m-bit input code to a n-bit output code such that each valid code word produces a unique output code
Circuits that perform encoding are called encoders
An encoder has 2n (or fewer) input lines and n output lines which generate the binary code corresponding to the input values
Typically, an encoder converts a code containing exactly one bit that is 1 to a binary code corres-ponding to the position in which the 1 appears.

10. Octal-to-Binary Encoder
INPUTS
OUTPUTS D7 D6 D5 D4 D3 D2 D1 D0 A2 A1 A0 1
Equations:
A2 = D4 + D5 + D6 + D7
A1 = D2 + D3 + D6 + D7
A0 = D1 + D3 + D5 + D7
What type of gates do
we need for this encoder?
4-input OR gates

11. Selecting
Selecting of data or information is a critical function in digital systems and computers
Circuits that perform selecting have:
A set of information inputs from which the selection is made
A single output
A set of control lines for making the selection
Logic circuits that perform selecting are called ___________?
Selecting can also be done by three-state logic or transmission gates
multiplexers

12. Multiplexers
A multiplexer selects information from an input line and directs the information to an output line
For an n selection inputs (Sn - 1, … S0) called selection inputs, 2n information inputs (I2n - 1, … I0), and one output Y
are needed.

13. 2-to-1-Line Multiplexer
Since 2 information inputs = 21  n = 1
The single selection variable S has two values:
S = 0 selects input I0
S = 1 selects input I1
The Output:
Y = I0 + SI1
The circuit: S S I 1
Decoder
Enabling
gates Y

14. 4-to-1-Line Multiplexer
In general, for an 2n-to-1-line multiplexer:
n-to-2n-line decoder
2n x 2 AND-OR S 1
Decoder Y I 2 3

15. Alternative Function Implementation
Alternative implementation techniques:
Decoders and OR gates
Multiplexers (and inverter)
ROMs
PLAs
PALs
Can be referred to as structured implementation methods since a specific underlying structure is assumed in each case

16. Decoders and OR gates - Example

17. Decoders and OR gates (continued)

18. Multiplexer Approach
Implement any m functions of n + 1 variables by using:
An m-wide 2n-to-1-line multiplexer
A single inverter
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 basic function of the final variable (0, 1, X,...)
Using the first n variables as the index, value-fix the information inputs to the multiplexer with the corresponding rudimentary functions
Use the inverter to generate the basic function X

19. Example: Gray to Binary Code
Design a circuit to convert a 3-bit Gray code to a binary code
The formulation gives the truth table on the right
It is obvious from this table that X = C and the Y and Z are more complex
Gray
A B C
Binary
x y z
1 0
0 1 1
0 1 0
1 0 1

20. Gray to Binary (continued)
Rearrange the table so that the input combinations are in counting order, pair rows, and find the simple functions.
Gray
A B C
Binary
x y z
Functions of C for y
Functions of C for z
0 0 0
0 0 1
1 1 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
F = C
F = C
F = C
F = C
F = C
F = C
F = C
F = C

21. Gray to Binary (continued)
Assign the variables and functions to the multiplexer inputs: S1 S0 A B
D03
D02
D01
D00
Out Y
8-to-1
MUX C Z

22. Implementing a 4-input Function with a Multiplexer

23. Read Only Memory
Read Only Memories (ROM) or Programmable Read Only Memories (PROM) have:
N input lines,
M output lines, and
2N decoded minterms.
Fixed AND array
Programmable OR Array
A program for a ROM or PROM is simply a multiple-output truth table
If a 1 entry, a connection is made to the corresponding minterm for the corresponding output
If a 0, no connection is made

24. Read Only Memory Example
Example: A 8 X 4 ROM (N = 3 input lines, M= 4 output lines)
The fixed "AND" array is a “decoder” with 3 inputs and 8 outputs implementing minterms.
The programmable "OR“ array uses a single line to represent all inputs to an OR gate. An “X” in the array corresponds to attaching the minterm to the OR
Read Example: For input (A2,A1,A0) = 111, output is (F3,F2,F1,F0 ) =_______. D7 D6 D5 D4 D3 D2 D1 D0 A2 A1 A0 A B C F0 F1 F2 F3 X

25. Programmable Logic - Advantages
Many programmable logic devices are field- programmable, i. e., can be programmed outside of the manufacturing environment
Most programmable logic devices are erasable and reprogrammable.
Allows “updating” a device or correction of errors
Allows reuse the device for a different design - the ultimate in re-usability!
Programmable logic devices can be used to prototype design that will be implemented for sale in regular ICs.
Complete Intel Pentium designs were actually prototype with specialized systems based on large numbers of VLSI programmable devices!

26. Programmable Configurations
Fixed
Programmable
Inputs
Programmable
AND array
Outputs
Connections
OR array
(decoder)
(a) Programmable read-only memory (PROM)
Inputs
Programmable
Programmable
Fixed
Outputs
Connections
AND array
OR array
(b) Programmable array logic (PAL) device
Inputs
Programmable
Programmable
Programmable
Programmable
Outputs
Connections
AND array
Connections
OR array
(c) Programmable logic array (PLA) device

27. Programmable Array Logic Example
AND gates inputs
Product A A B B C C D D W W
term 1 X 2 X W 3 X A 4 X 5 X F1 6 X B 7 X 8 X F2 9 X C 10 11 12 D A A B B C C D D W W

28. Programmable Logic Array Example
Fuse intact
Fuse blown 1 F 2 X A B C 3 4
A B
A C
B C
What are the equations for F1 and F2?
3-input, 3-output PLA with 4 product terms