DIGITAL SYSTEM FUNDAMENTALS (ECE421) DIGITAL …

Mohd Uzir Kamaluddin / July 2019 page 1

COURSE / CODE DIGITAL SYSTEM FUNDAMENTALS (ECE421)

DIGITAL ELECTRONICS FUNDAMENTAL (ECE422)

Medium Scale Integrated (MSI) devices

It’s sometimes not reasonable to do all the design work at the gate-level. For example, it is a common

practice to use full-adders as the building blocks in building a ripple-carry adder. When designing

circuits, there are certain building blocks that are used over and over again. Thus, being familiar with

a set of building blocks that are commonly used can make designing digital devices easier.

Building blocks that are more complex than gates are sometimes called medium-scale integrated

devices, or MSI devices for short. Exactly what makes a device “medium” in scale rather than “large”

or “very large” (as in VLSI) is not always agreed on, but generally it’s all right stick with devices that

are fairly simple and where small versions of them can be implemented in a couple dozen gates or so.

Special Combinational Logic Circuits

1. Adders

A half adder can add two bits. It has two inputs, generally labeled A

and B, and two outputs, the sum S and carry C. S is the two-bit XOR of

A and B, and C is the AND of A and B. Essentially the output of a half

adder is the sum of two one-bit numbers, with C being the most

significant of these two outputs.

A half adder is a logical circuit that performs an addition operation on

two binary digits. The half adder produces a sum and a carry

value which are both binary digits.

The drawback of this circuit is that in case of a multi-bit addition,

it cannot include a carry.

Sum(S) = BA and Carry(C) = AB

A full adder is capable of adding three bits: two bits and one carry bit of earlier calculation. It has

three inputs - A, B, and carry Cin, such that multiple full adders can be used to add larger numbers.

Hence, a full adder can perform an addition operation on three binary digits. The full adder produces

a sum and carry value, which are both binary digits.

Exercises:

Q1. Draw the half-adder circuit using NAND gates only.

Q2. Draw the truth table for a full adder and write down the output expressions.

Q2. Implement the operation of a full adder using two half-adders and an OR gate.

Q3. What are the advantages of full-adder over half-adder?


Parallel Binary Adders

As we discussed that a single full adder performs the addition of two one bit numbers and an input

carry. For performing the addition of binary numbers with more than one bit, more than one full

adder is required depends on the number bits. Thus, a parallel adder is used for adding all bits of the

two numbers simultaneously.

By connecting a number of full adders in parallel, n-bit parallel adder is constructed. From the below

figure, it is to be noted that there is no carry at the least significant position, hence we can use either a

half adder or made the carry input of full adder to zero at this position.

The figure below shows a parallel 4-bit binary adder which has three full adders and one half-adder.

The two binary numbers to be added are A3A2A1A0 and B3B2B1B0 which are applied to the

corresponding inputs of full adders. This parallel adder produces their sum as C4S3S2S1S0 where C4 is

the final carry.

In the 4-bit adder, first block is a half-adder that has two inputs as A0B0 and produces their sum S0 and

a carry bit C1. Next block should be full adder as there are three inputs applied to it. Hence this full

adder produces their sum S1 and a carry C2. This will be followed by other two full adders and thus

the final sum is C4S3S2S1S0.

Most commonly Full Adders designed in dual in-line package integrated circuits. A typical 74LS83 is a

4-bit full adder. Arithmetic and Logic Unit of a unit computer consist of these parallel adders to

perform the addition of binary numbers.


2. Subtractor

Subtractor is a circuit which is used to subtract two binary number (digit) and provides Difference and

Borrow as output.

Half Subtractor is used for subtracting one single bit binary digit from another single bit binary digit.

A logic circuit which is used for subtracting three single bit binary digit is known as Full Subtractor.

Parallel Binary Subtractors

To perform the subtraction of binary numbers with more than one bit is performed through the

parallel subtractors. This parallel subtractor can be designed in several ways, including combination of

half and full subtractors, all full subtractors, all full adders with subtrahend complement input, etc.

The below figure shows a 4-bit parallel binary subtractor formed by connecting one half subtractor

and three full subtractors.


In this subtractor, 4-bit minuend A3A2A1A0 is subtracted by 4-bit subtrahend B3B2B1B0 and gives the

difference output D3D2D1D0. The borrow output of each subtractor is connected as the borrow input to

the next preceding subtractor.

It is also possible to design a 4 bit parallel subtractor 4 full adders as shown in the below figure. This

circuit performs the subtraction operation by considering the principle that the addition of minuend

and the complement of the subtrahend is equivalent to the subtraction process.

We know that the subtraction of A by B is obtained by taking 2’s complement of B and adding it to A.

The 2’s complement of B is obtained by taking 1’s complement and adding 1 to the least significant

pair of bits.

Hence, in this circuit 1’s complement of B is obtained with the inverters (NOT gate) and a 1 can be

added to the sum through the input carry.

Parallel Adder / Subtractor

The operations of both addition and subtraction can be performed by a one common binary adder.

Such binary circuit can be designed by adding an Ex-OR gate with each full adder as shown in below

figure. The figure below shows the 4-bit parallel binary adder/subtractor which has two 4 bit inputs as

A3A2A1A0 and B3B2B1B0.

The mode input control line M is connected with carry input of the least significant bit of the full

adder. This control line decides the type of operation, whether addition or subtraction.


When M = 1, the circuit is a subtractor and when M = 0, the circuit becomes adder. The Ex-OR gate

consists of two inputs to which one is connected to the B and other to input M. When M = 0, B Ex-OR

of 0 produce B. Then full adders add the B with A with carry input zero and hence an addition

operation is performed.

When M = 1, B Ex-OR of 0 produce B complement and also carry input is 1. Hence the complemented

B inputs are added to A and 1 is added through the input carry, nothing but a 2’s complement

operation. Therefore, the subtraction operation is performed.

Note: BB 0 and BB 1

Exercises:

Q1. Show on the parallel adder/subtractor circuit the operation of 10012 + 01112.

Q2. Show on the parallel adder/subtractor circuit the operation of 11012 - 10012.

3. Multiplier

A binary multiplier is an electronic circuit used in digital electronics, such as a computer, to multiply

two binary numbers. It is built using binary adders. A simple 2-bit multiplier, A2A1 x B2B1 can be

designed using logic gates as shown in the truth table below.

Another method commonly used is to do multiplication by

repeated addition. The multiplicand is added by itself the

number of times as indicated by the multiplier. See the block

diagram.

To increase the speed of the multiplication process, other

method of multiplication is used such as:

Shift and add multiplier

Serial multiplier


Serial/Parallel multiplier

Array multiplier

Booth multiplier

Exercise: The multiplication of a 3-bit number B by a 4-

bit number A is illustrated below.

i) Based on the multiplication process shown,

implementation a 4-bit x 3-bit multiplier using

adders.

ii) Show the multiplication of 10112 by 1002 by

clearly indicating the values at the inputs and

outputs.

4. Comparator

A digital comparator or magnitude comparator is a hardware

electronic device that takes two numbers as input in binary

form and determines whether one number is greater than, less

than or equal to the other number. Comparators are used in

central processing units (CPUs) and microcontrollers (MCUs).

An XNOR gate is a basic comparator, because its output is "1"

only if its two input bits are equal.

The truth table shows a 1-bit comparator and its logic circuit is

given below.

Figure below shows the logic diagram and truth table for a 2-bit comparator.


A 4-bit magnitude comparator TTL 74LS85.

An 8-bit word comparator.

For comparators that compares 3 bits or more, the truth table method of designing comparators will

result in a huge truth table.

Design of a 4-bit comparator using the deduction method.


Exercises:

a) What does a comparator circuit do? What are the output of a comparator?

b) A comparator determines when two binary numbers are equal, true or false?

c) If a comparator compares two 2-bit numbers A, B and produces the output

D1 = 1 if A<B and

D2 = 1 if A=B

Derive the expressions for D1 and D2 and implement the comparator using logic gates.

d) A1 and B1 are the MSB of inputs A and B of a 2-bit comparator respectively. The comparator

operates as follows:

O1 = 1 if A < B

O2 = 1 if A = B

O3 = 1 if A > B

Derive the expressions for output O1, O2 and O3, then implement the comparator circuit for

the outputs using suitable logic gates.

5. Decoders and Encoders

In digital electronics, a decoder can take the form of a multiple-input, multiple-output logic circuit

that converts coded inputs into coded outputs, where the input and output codes are different. e.g. n-

to-2n, binary-coded decimal decoders. Enable inputs must be on for the decoder to function, otherwise

its outputs assume a single "disabled" output code word. Decoding is necessary in applications such as

data multiplexing, 7 segment display and memory address decoding.

The example decoder circuit would be an AND gate because the output of an AND gate is "High" (1)

only when all its inputs are "High." Such output is called as "active High output". If instead of AND

gate, the NAND gate is connected the output will be "Low" (0) only when all its inputs are "High".

Such output is called as "active low output".

The diagram below shows the 2 to 4 binary decoder with active HIGH output.


The IC Decoder 74LS138 is shown in the diagram. It is a 3 to 8

decoder, with active high output. Some binary decoders have an

additional input pin labelled “Enable” that controls the outputs

from the device. This extra input allows the decoders outputs to

be turned “ON” or “OFF” as required. These types of binary

decoders are commonly used as “memory address decoders” in

microprocessor memory applications.

A 4 to 16 decoder can be designed using two 3 to 8 decoders as shown.


But some binary decoders are constructed using NAND gates instead of AND gates for their decoded

output, since NAND gates are cheaper to produce than AND’s as they require fewer transistors to

implement within their design.

The use of NAND gates as the decoding element, results in an active-“LOW” output while the rest

will be “HIGH”. As a NAND gate produces the AND operation with an inverted output, the NAND

decoder looks like this with its inverted truth table.

Implementation of logic functions using a decoder:

A decoder can be used to implement logic functions. Since

all the minterms of the function are available at the output

then there is no need for simplification. All is needed is an

OR gate for each function to sum the required minterms. As

an example, implement the following functions using an

appropriate decoder and OR gates.

S(x,y,z) = ∑m(1,2,4,7) and C(x,y,z) = ∑m(3,5,6,7)

The implementation using 3 to 8 decoder active high output and OR gates is as shown.

If an active low output decoder is used,

the implementation uses a NAND gate.

Exercise 1:

a) Explain what is a decoder circuit.

b) A decoder detects the presence of a specified combination of input bits, true or false?

c) Explain why some decoder has active-high output and some has active-low output.

d) The 4-line to-10-line decoder and the 1-of-10 decoder are two different types, true or false?


e) Design a circuit that will detect the three-bit binary code at its input. The output must be logic

HIGH to indicate the corresponding binary code. (Hint: decoder circuit).

f) Design a circuit that will detect the three-bit binary code at its input. The output must be logic

LOW to indicate the corresponding binary code. (Hint: decoder circuit).

Exercise 2:

Given the function )]([ DCBABCAF

a) Implement the function F using 4 to 16 decoder with active high output.

b) Implement the function F using 3 to 8 decoder with active high output.

c) Implement the function F using 4 to 16 decoder with active low output.

Exercise 3:

a) Design a full-adder logic circuit using an active-low

2-to-4 decoder(s) and suitable logic gates.

b) A 3 to 8 decoder is connected as shown, where x, y

and z are inputs (z is the MSB) and F is an output.

Write down the expression for F.

An encoder is a digital function that performs the inverse operation of a decoder. An encoder has 2n

(or fewer) input lines and n output lines. The output lines generate the binary code corresponding to

the input value.

One of the main disadvantages of standard digital encoders is that they can generate the wrong

output code when there is more than one input present at logic level “1”.

One simple way to overcome this problem is to “Prioritize” the level of each input pin and if there was

more than one input at logic level “1” the actual output code would only correspond to the input with

the highest designated priority. Then this type of digital encoder is known commonly as a Priority

Encoder or P-encoder for short.


Exercises:

a) Explain what is an encoder circuit.

b) An encoder essentially performs a reverse decoder function, true or false?

c) What happen if more than one input of an encoder is active?

d) An incomplete truth table for an octal-to-binary priority encoder with valid bit indicator is

given below. Complete the table. D7 has the highest priority and D0 the lowest.

6. Multiplexer (MUX) and De-Multiplexer (DEMUX)

A multiplexer or MUX is a device that performs multiplexing; it selects one of many analog or digital

input signals and forwards the selected input into a single output line. A multiplexer of 2n inputs has n

select bits, which are used to select which input line to send to the

output.

A multiplexer makes it possible for several signals to share one

device or resource, for example one A/D converter or one

communication line, instead of having one device per input signal.

A simple 2-line to 1-line (2-to-1) multiplexer

from basic logic NAND gates.


Logic diagrams of 4 input multiplexer, a) using NAND gates only b) using AND, OR and NOT gates.

A de-multiplexer (or DEMUX) is a device taking a single input signal and

selecting one of many data-output-lines, which is connected to the single

input. A multiplexer is often used with a complementary de-multiplexer

on the receiving end. A DEMUX is also known as data distributor.

The 1 to 4 DEMUX can be designed using logic gates as shown.


Another way to implement a 1 to 4 DEMUX

is by using a 2 to 4 decoder with AND gates

as shown.

Implementation of logic functions using Multiplexer

A multiplexer can be used to implement logic functions. There are a few ways to design such circuit,

depending on the multiplexer used. Consider the function: CABCBACBAf ),,(

a) Using an 8 to 1 multiplexer

The function f(A,B,C) is tabulated in the form of truth table, and then the logic function is

implemented directly using the 8-1 multiplexer.

A B C f

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1

b) Using a 4 to 1 multiplexer

For this design, the input A and B is connected to the select lines C1 and C2. The expression for

mux input is obtained by comparing the output f with the input which is not used, that is C.

Thus the method is called residue method.

A B C f MUX Input

0 0 0 0

I0 = C

0 0 1 1

0 1 0 1

I1 = 1

0 1 1 1

1 0 0 0

I2 = 0

1 0 1 0

1 1 0 0

I3 = C

1 1 1 1


Exercise 1:

a) What is the function of a multiplexer circuit?

b) A multiplexer is a logic circuit that allows digital information from a single source to be routed

onto several lines, true or false?

c) Draw the block diagram of a 4 to 1 MUX and its truth table.

d) For the MUX in part b) above, determine the output for the following states:

D0=0, D1=1, D2=1, D3=0, S0=1, S1=0.

e) Design a logic circuit to produce a HIGH output if and only if the input, represented by a 4-bit

binary number, is greater than twelve or less than three. First develop the truth table, write the

Boolean function, simplify it using K-map and then draw the logic diagram.

f) Implement the logic circuit in part d) above using 8 to 1 MUX, and other logic gates if

necessary.

g) What is the function of a de-multiplexer circuit?

h) Given the function )]([ DCBABCAF

i) Implement the function F using 16 -1 MUX.

ii) Implement the function F using 8 - 1 MUX.

iii) Implement the function F using 4 - 1 MUX.

iv) Implement the function F using 2 - 1 MUX.

Exercise 2:

A Boolean function F of four variables A, B, C and D is given by the sum of minterms:

F = ∑m(1 ,2, 4, 7, 8, 10, 12)

a) Implement the function F using a 16-to-1 line multiplexer.

b) Implement the function F using an 8-to-1 line multiplexer.

c) Implement the function F using an 4-to-1 line multiplexer.

d) Implement the function F using an 2-to-1 line multiplexer.

Exercise 3:

Determine the Boolean function that the following circuits implements.

A special decoder: 7-segment decoder/driver

A seven-segment display is a form of electronic display device for

displaying decimal numerals that is an alternative to the more complex dot-

matrix displays. Seven-segment displays are widely used in digital clocks,

electronic meters, and other electronic devices for displaying numerical

information.


The common way of driving the 7-segment display is

by the use of a special IC called the 7-segment

decoder/driver. This IC receives input as BCD and

output the 7-segment codes for the 7-segment

display. The 74LS47 has active-low outputs, and can

only be used to drive common anode 7-segment

display unit.

BCD input Display

0011 3

0101 5

1001 9

Exercise Q1

Implement a full adder circuit using:

a) a 3-to-8 active high output decoder.

b) 4-to-1 multiplexers.

Exercise Q2

a) Draw the logic circuit of the following function using a 3-to-8 active high output decoder.

F = ∑m(1 ,2, 4, 6, 7)

b) Draw the logic circuit of the following function using a 3-to-8 active low output decoder.

F = ∑m(0 ,2, 3, 5, 6)

Exercise Q3

A Boolean function F of four variables A, B, C and D is given by the sum of minterms:

F = ∑m(1 ,2, 4, 7, 8, 10, 12, 14)

a) Implement the function F using a 16-to-1 line multiplexer.

b) Implement the function F using an 8-to-1 line multiplexer.

c) Implement the function F using a 4-to-1 line multiplexer.

Exercise Q4

By creating the truth table for the following decoder circuit (inputs are a,b,c,d and output is F),

determine the SOP short hand expression for F.

(Answer: F=Σm(0,1,2,3,4,5,12,13))

Date post:	20-Dec-2021
Category:	Documents
Upload:	others
View:	11 times
Download:	1 times

DIGITAL SYSTEM FUNDAMENTALS (ECE421) DIGITAL …

Documents