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Chapter 4Henry Hexmoor-- SIUC

• Rudimentary Logic functions: • Value fixing• Transferring• Inverting

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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 a VLSI circuits.

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Rudimentary Logic Functions

• Functions of a single variable X• Can be used on the

inputs to functionalblocks to implementother than the block’sintended function

TABLE 4-1Functions of One Variable

X F = 0 F = X F = F = 1

01

00

01

10

11

X

Value fixing

invertingtransfer

0

1

F = 0

F = 1

(a)

F = 0

F = 1

VCC or VDD

(b)

X F = X

(c)

X F = X

(d)

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Multiple-bit Rudimentary Functions

• Multi-bit Examples:

• A wide line is used to representa bus which is a vector signal

• In (b) of the example, F = (F3, F2, F1, F0) is a bus.

• The bus can be split into individual bits as shown in (b)• Sets of bits can be split from the bus as shown in (c)

for bits 2 and 1 of F. • The sets of bits need not be continuous as shown in (d) for bits 3, 1, and 0 of F.• See Example 4-1

F(d)

0

F3

1 F2

F1A F0

(a)

01

A1

2 34

F0

(b)

4 2:1 F(2:1)2

F(c)

4 3,1:0 F(3), F(1:0)3

A A

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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

• See Example 4-2 Automobile…

XF

EN

(a)

ENX

F

(b)

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• Decoding - the conversion of an n-bit input code to an m-bit output code withn m 2n 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, where m 2n, and– generate 2n (or fewer) minterms for the n input

variables

Decoding4-3

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Binary n-to-2Binary n-to-2nn Decoders Decoders

• A binary decoder has n inputs and 2n outputs.• Only the output corresponding to the input value is equal

to 1.

: :

ninputs

n to 2n

decoder2n

outputs

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• 1-to-2-Line Decoder

Decoder Examples

A D0 D1

0 1 01 0 1

(a) (b)

D1= AA

D0= A

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Note that the 2-4-line made up of 2 1-to-2- line decoders and 4 AND gates.

Decoder Examples

A1

0

0

1

1

A0

0

1

0

1

D0

1

0

0

0

D1

0

1

0

0

D2

0

0

1

0

D3

0

0

0

1

(a)

D0= A1 A0

D1= A1 A0

D2= A1 A0

D3= A1 A0

(b)

A1

A0

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Decoder Expansion - Example

• 3-to-8-line decoder – Number of output ANDs = 8– Number of inputs to decoders driving output ANDs = 3– Closest possible split to equal

• 2-to-4-line decoder• 1-to-2-line decoder

– 2-to-4-line decoder• Number of output ANDs = 4• Number of inputs to decoders driving output ANDs = 2• Closest possible split to equal

– Two 1-to-2-line decoder

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Decoder Expansion – 3-to-8 line Example

• Result

3-to-8 Line decoder

1-to-2-Line decoders

4 2-input ANDs 8 2-input ANDs

2-to-4-Linedecoder

D0A 0

A 1

A 2

D1

D2

D3

D4

D5

D6

D7

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CS 315:Decoding

4-3

OR GateA

n-to-m line: An n bit binary code is converted to a unique m bit binary pattern

See Figure 4-6 for 1-to-2 line decoder

1. Let k = n.2. If k is even, divide k by 2 to obtain k/2. Use 2k AND gates driven by two decoders

of output size 2k/2. 3. If k is odd, obtain (k+1)/2 and (k-1)/2, Use 2K AND gates driven by a decoder of

output size 2(k-1)/2.4. For each decoder resulting from step 2, repeat step 2 with k equal to the values

obtained in step 2 until K = 1. For k = 1, use a 1-to-2 decoder.

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Decoder Expansion

• General procedure given in book for any decoder with n inputs and 2n outputs.

• This procedure builds a decoder backward from the outputs.

• The output AND gates are driven by two decoders with their numbers of inputs either equal or differing by 1.

• These decoders are then designed using the same procedure until 1-to-2-line decoders are reached.

• The procedure can be modified to apply to decoders with the number of outputs ≠ 2n

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Decoder Example: Seven-Segment Decoders

• A seven segment decoder

has 4-bit BCD input and

the seven segment display

code as its output:• In minimizing the circuits

for the segment outputs all

non-decimal input combinations

(1010, 1011, 1100,1101, 1110,

1111) are taken as don’t-cares

/Bl D C B A a b c d e f g 0 x x x x 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

-- d

on’t

car

e in

pu

ts -

-

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Decoder Expansion - Example

• 7-to-128-line decoder – Number of output ANDs = 128– Number of inputs to decoders driving output ANDs =

7– Closest possible split to equal

• 4-to-16-line decoder• 3-to-8-line decoder

– 4-to-16-line decoder• Number of output ANDs = 16• Number of inputs to decoders driving output ANDs = 2• Closest possible split to equal

– 2 2-to-4-line decoders

– Complete using known 3-8 and 2-to-4 line decoders

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• 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

• Alternatively, can be viewed as distributing value of signal EN to 1 of 4 outputs

• In this case, called ademultiplexer

EN

A 1

A 0

D0

D1

D2

D3

(b)

EN A1 A0 D0 D1 D2 D3

01111

X0011

X0101

01000

00100

00010

0000

Decoder with Enable

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Encoding4-4

• Encoding - the opposite of decoding - the conversion of an m-bit input code to a n-bit output code with n m 2n 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.

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Encoder

>n m

EncoderI0I1I2…I2n-1

Y0Y1…Y n-1

2n inputs

n outputs

2n – n encoder requires “n 2n-1 input” OR gatesBit j of input code is connected to OR gate j if bit j in the input is 1.Encoders are useful when the occurrence of one of several disjoint events needs to be represented by an integer identifying the event. E.g., wind direction encoder

OR0

OR1

OR2

I1,I3, I5, I7

I2, I3, I6, I7

I4, I5, I6, I7

Y0 = 0001 + 0011 + 0101 + 0111

y1

y2

0th bit

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Encoder Example 1

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Encoder Example 2

• A decimal-to-BCD encoder– Inputs: 10 bits corresponding to decimal

digits 0 through 9, (D0, …, D9)

– Outputs: 4 bits with BCD codes

– Function: If input bit Di is a 1, then the output (A3, A2, A1, A0) is the BCD code for i,

• The truth table could be formed, but alternatively, the equations for each of the four outputs can be obtained directly.

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Encoder Example (continued)

• Input Di is a term in equation Aj if bit Aj is 1 in the binary value for i.

• Equations:A3 = D8 (1000) + D9 (1001)

A2 = D4 (0100)+ D5 (0101) + D6 (0110)+ D7 (0111)

A1 = D2 (0010)+ D3 (0011) + D6 (0110) + D7 (0111)

A0 = D1 (0001)+ D3 (0011)+ D5 (0101) + D7 (0111) + D9

(1001)

• F1 = D6 + D7 can be extracted from A2 and A1.

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Priority Encoder

• If more than one input value is 1, then the encoder just designed does not work.

• One 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.

• Priority encoders are useful when inputs have a predefined priority, e.g., interrupt requests from peripheral devices

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Priority Encoder Example• Priority encoder with 5 inputs (D4, D3, D2, D1, D0) - highest priority to

most significant 1 present - Code outputs A2, A1, A0 and V where V indicates at least one 1 present.

• Xs in input part of table represent 0 or 1; thus table entries correspond to product terms instead of minterms. The column on the left shows that all 32 minterms are present in the product terms in the table

No. of Min-terms/Row

Inputs Outputs

D4 D3 D2 D1 D0 A2 A1 A0 V

1 0 0 0 0 0 X X X 0

1 0 0 0 0 1 0 0 0 1

2 0 0 0 1 X 0 0 1 1

4 0 0 1 X X 0 1 0 1

8 0 1 X X X 0 1 1 1

16 1 X X X X 1 0 0 1

A1 = D3’D2 + D3

A0 = D3’D2’D1 + D3

V = D0 + D1+ D2+ D3

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• 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 multiplexers

• Selecting can also be done by three-state logic or transmission gates

Selecting4-5

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Multiplexers

• A multiplexer selects information from an input line and directs the information to an output line

• A typical multiplexer has n control inputs (Sn

1, … S0) called selection inputs, 2n information inputs (I2

n 1, … I0), and one

output Y• A multiplexer can be designed to have m

information inputs with m 2n as well as n selection inputs

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Selecting4-5

OR GateA

Multiplexers, data selectors2n input lines + n selection inputs

See 2-to-1-line multiplexer in Figure 4-13Two input lines and one select line SY = S’I0 + S I1

S

I0

I1

DecoderEnablingCircuits

Y

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2-to-1-Line Multiplexer

• The single selection variable S has two values:– S = 0 selects input I0

– S = 1 selects input I1

• The equation: Y = S’I0 + SI1

• The circuit:

S

I0

I1

DecoderEnablingCircuits

Y

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2-to-1-Line Multiplexer (continued)

• Note the regions of the multiplexer circuit shown:– 1-to-2-line Decoder

– 2 Enabling circuits

– 2-input OR gate

• To obtain a basis for multiplexer expansion, we combine the Enabling circuits and OR gate into a 2 2 AND-OR circuit:– 1-to-2-line decoder

– 2 2 AND-OR

• In general, for an 2n-to-1-line multiplexer:– n-to-2n-line decoder

– 2n 2 AND-OR

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Example: 4-to-1-line Multiplexer: n = 2 selection bits

• 2-to-22-line decoder

• 22 2 AND-OR

S1Decoder

S0

Y

S1Decoder

S0

Y

S1Decoder

4 2 AND-ORS0

Y

I2

I3

I1

I0

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Multiplexer Width Expansion

• Select “vectors of bits” instead of “bits”

• Use multiple copies of 2n 2 AND-OR in parallel

• Example:4-to-1-linequad multi-plexer

• Figure 4-16

43 2 AND-OR

2-to-4-Line decoder

43 2 AND-OR

43 2 AND-OR

43 2 AND-OR

I0,0

I3,0

I0,1

I3,1

I0,2

I3,2I0,3

I3,3

Y 0

D0

D3

A 0

A 1

Y 1

Y 2

Y 3

.

.

.

.

.

....

.

.

.

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Using Multiplexers

• Any Boolean function can be implemented by setting the inputs corresponding to the function as inputs and the selectors as the variables.

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Shifters

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Rotator

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Parity for six bit messages

B 0

B 1

B 2

B 3

B 4

B 5

B 6

B even

B 0

B 1

B 2

B 3

B 4

B 5

B 6

B even

ERROR

Even Parity Generator Circuit

Even Parity Checker Circuit

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Example: Gray to Binary Code

• Design a circuit to convert a 3-bit Gray code to a binary code

• The formulation givesthe truth table on theright

• It is obvious from thistable that X = C and theY and Z are more complex

GrayA B C

Binaryx y z

0 0 0 0 0 01 0 0 0 0 11 1 0 0 1 00 1 0 0 1 10 1 1 1 0 01 1 1 1 0 11 0 1 1 1 00 0 1 1 1 1

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Gray to Binary (continued)• Rearrange the table so that the input combinations are in

counting order, pair rows, and find rudimentary functions

Gray

A B C

Binary

x y z

Rudimentary

Functions of C for y

Rudimentary Functions of

C for z

0 0 0 0 0 0

0 0 1 1 1 1

0 1 0 0 1 1

0 1 1 1 0 0

1 0 0 0 0 1

1 0 1 1 1 0

1 1 0 0 1 0

1 1 1 1 0 1

F = C

F = C

F = C

F = C

F = C

F = CF = C

F = C

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• Assign the variables and functions to the multiplexer inputs:

Gray to Binary (continued)

S1S0

AB

D03D02D01D00

Out Y

8-to-1MUX

C

C

C

C D13D12D11D10

Out Z

8-to-1MUX

S1S0

AB

C

C

C

CC C

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Combinational Function Implementation4-6

• Alternative implementation techniques:

– Decoders and OR gates

– Multiplexers (and inverter)

– ROMs

– PLAs

– PALs

– Lookup Tables

• Can be referred to as structured implementation methods since a specific underlying structure is assumed in each case

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Read Only Memory

• Functions are implemented by storing the truth table• Other representations such as equations more

convenient• Generation of programming information from

equations usually done by software• Text Example 4-10 Issue

– Two outputs are generated outside of the ROM– In the implementation of the system, these two functions are

“hardwired” and even if the ROM is reprogrammable or removable, cannot be corrected or updated

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Programmable Logic Array Example

XFuse intact

1Fuse blown

0

1

F1

F2

A

B

C

C B AC B A

1

2

4

3

X X

X X

X X

X XX

X

X

X X

X

X

X

X

X

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Lookup Tables

• Lookup tables are used for implementing logic in Field-Programmable Gate Arrays (FPGAs) and Complex Logic Devices (CPLDs)

• Lookup tables are typically small, often with four inputs, one output, and 16 entries

• Since lookup tables store truth tables, it is possible to implement any 4-input function

• Thus, the design problem is how to optimally decompose a set of given functions into a set of 4-input two- level functions.

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HW 4

1. Draw the detailed logic diagram of a 3-to-8-line decoder using only NOR and NOT gates. Include an enable input. Q4-9

2. Implement a binary full adder with a dual 4-to-1-line multiplexer and a single inverter. Q4-23