Chapter 4 Digital Logic and Implementation 4.1 Boolean Algebra

Chapter 4
Digital Logic and Implementation
4.1 Boolean Algebra
4.1.1 Boolean logic
• A single bit can be either 0 or 1. You can interpret 0 as the logical value false and
1 as the logical value true. In this way, a bit stored in the memory of a computer can
represent a logical value that is either false or true.
• The digital computer is based on Boolean logic or Boolean algebra.
• There are a number of logical rules, but these can all be derived from three fundamental
operations, the operations of AND, OR, and NOT, which are called fundamental logical
operations
• If you interpret a bit as a logical value, then you can apply logical operations on this bit.
Operations AND and OR are binary operations which accept 2 bit inputs to create only
1 bit output, while the operation NOT is a unary operation.
• All logical operations or rules can be defined or described as a formula, or by a truth
table, which specifies the result for all possible combinations of inputs.
4.1.2 Truth Tables for the Fundamental Logical Operations
• The rule for performing the AND operation is as follows: The result of an AND operation
on two inputs A and B is true if and only if both inputs are true. Denote by C = A·B
where C represents the result. The truth table is Table (a)
A B C = A · B
0 0 0
0 1 0
1 0 0
1 1 1
A B C = A + B
0 0 0
0 1 1
1 0 1
1 1 1
A C = A
0 1
1 0
(a) (b) (c)
• The rule for evaluating the OR operation is as follows: If one of two inputs A and B is
true, the result is true; otherwise the result is false. Denote by C = A + B. The
truth table is shown in Table (b)
1

2 CHAPTER 4. DIGITAL LOGIC AND IMPLEMENTATION
• The rule for evaluating the NOT operation is as follows: If the input A is true, the result
is false, and if the input is false, the result is true. The truth table is shown in
Table (c).
• With three fundamental operations you can make up other operations or Boolean expressions.
For example, the Exclusive-OR operation is defined as
C = A ⊕ b = (A + B) · A · B
It is easy for you to construct the truth table for the exclusive-or operation.
4.1.3 Logic Laws
• There are a number of useful laws and identities that help to manipulate Boolean equations.
• Boolean algebra operations are associative, distributive, and communicative
A + B = B + A
A + (B + C) = (A + B) + C,
A · (B + C) = A · B + A · C
A · B = B · A
A · (B · C) = (A · B) · C
• DeMorgan’s Law
A + B = A · B and A · B = A + B
• It is easy for you to prove these laws using their truth table. The method is to show the
truth tables of both sides of the equation are same. Let us prove that A · B = A + B.
For the left side, using the truth tables of the three fundamental operations we have
A B A · B A · B
0 0 0 1
0 1 0 1
1 0 0 1
1 1 1 0
For the right side, we have
A B A B A + B
0 0 1 1 1
0 1 1 0 1
1 0 0 1 1
1 1 0 0 0
You can see the column A · B in the first truth table is exactly same as the column A + B
in the second truth table. That is, for each pair of inputs A and B, the values from A · B
and A + B are same. Hence A · B = A + B

4.2. LOGIC GATES 3
4.2 Logic Gates
4.2.1 Three Fundamental Gates
• A gate is an electronic device that operates on a collection of binary inputs to produce a
binary output. Gates are constructed from transistor switches and other electronic components,
formed into integrated circuits.
• Gates can implement a wide range of different logical rules, eps., the three fundamental
logical operations.
• The following Figure shows the diagram of circuits constructing the NOT operation. In
the diagram, the collector of the transistor is connected to the power supply and the
emitter is grounded. If the control line is set to 1, then the transistor is in the ON state,
and it passes the current through to the ground. In this case, the output of the gate is
0. However, if the control line is set to 0, the transistor is the OFF state, and it blocks
passage of the current to the ground. Instead, the current is transmitted to the output line,
producing an output of 1. Thus if we consider the control line as the input of a logical
rule, and the output line as the output of the rule, then this circuit behaves exactly same
as the NOT operation. We call this gate as the NOT gate.
Power Supply
¡
¡
Collector
Output
Line
Control
Emitter
(Grounded)
• Similarly we can construct gates for OR, NOT and Exclusive-OR operations.
• Here are the standard logic gate representations for the fundamental logic operations.
4.2.2 Combinatorial Logic Circuits
• It is not difficult to manipulate the Boolean algebra to show that combinatorial Boolean
logic can be implemented entirely with a series of fundamental gates. For more examples
see Figure S1.8 and S1.9 on page 653 of the textbook.

4 CHAPTER 4. DIGITAL LOGIC AND IMPLEMENTATION
• Figure S1.10 shows a circuit called a selector. It can be used to switch the input between
A and B depending on the signal from the select line. You can use this simple selector to
build up a selector between two bytes of data, or more.
• The half adder shown in Figure S1.11 is another interesting combinatorial logic circuits.
If you review the addition table for binary numbers, you will note that 0+0 = 0, 1+0 = 1,
0 + 1 = 1 (without carries, considered as 0) and 1 + 1 = 10 ( with a carry 1). We can
split the addition into two parts:
– The sum part (S), (0, 0) → 0, (0, 1) → 1, (1, 0) → 1 and (1, 1) → 0 which can be
described by the Exclusive-OR operation
– The carry part (C), (0, 0) → 0, (0, 1) → 0, (1, 0) → 0 and (1, 1) → 1 which can be
described by the AND operation.
Combining two parts together results in a half adder. Figure S1.11 shows one of such
examples.
• In the addition of binary numbers, you must consider the carries from the previous bits.
To implement binary addition by circuits, you must include carries in the operation. Figure
S1.12 shows the so-called full adder in which two half adders are connected together
to cope with the carry from the previous bit. In the circuit, the previous carry is added to
the sum part. This addition may create another carry. Finally two possible carries are put
into a NAND gate to generate the final carry (why?).
• A full adder calculates one bit addition with possible carry from the previous bit. You
can combine a series of full adder circuits to make a large circuit for the addition of two
binary numbers with more bits.
4.2.3 Sequential Logic Circuits
• Sequential logic circuits are circuits whose output is dependent not only on the input and
the circuit configuration, but also on the previous state of the circuit.
• The key to sequential logic circuit is the presence of memory within the circuit. The state
of the circuit is stored in these memory bits.
• The basic memory element in a sequential logic circuit is called a flip-flop. The simplest
flip-flop is shown in Figure S1.13.
• There are many other types of flip-flop circuits. They are used to construct some control
circuits in a computer. This is beyond our course.
4.3 Application and Implementation
4.3.1 A Compare-For-Equality Circuit
• The first example is the compare-for-equality circuit, or CE circuit, which tests two unsigned
binary numbers for exact equality, i.e., all the corresponding bits are equal.

4.3. APPLICATION AND IMPLEMENTATION 5
• First we consider the compare-for-equality for a single bit. The comparison operation
here is defined by
A B The Result of Comparison
0 0 1
1 0 0
0 1 0
1 1 1
This task can be fulfilled by the Exclusive-OR gate.
• Suppose the binary numbers have N bits (generally N is 8 or 16 or 32 etc). Denote A =
a N−1 a N−2 ...a 0 and B = b N−1 b N−2 ...b 0 . For each pair (a i , b i ), we use a Exclusive-OR
gate to compare it. Thus we have N outputs (one per bit).
• Group the outputs subgroups of 2 outputs, and connect them into a AND gate. There will
be N/2 AND gates. Group them successively in the same way until we have only one
output which gives the result of comparison between two binary numbers.
4.3.2 An Addition Circuit
• The full adder is the circuit for the addition of two bits. You can use it to construct a full
addition circuit for two binary numbers with N bits.
• Mathematically the addition is defined as
• The full addition circuit is
a N−1 a N−2 a N−3 · · · a 0
+ b N−1 b N−2 b N−3 · · · b 0
s N s N−1 s N−2 s N−3 · · · s 0
N−bit input value b
b N−1
b N−2
. . . b 1
b 0
N−bit input value a
a
N−1
a a
1
a
0
N−2 . . .
C C C C C
Initial
N 1−ADD N−1 1−ADD N−2 . . . . 2 1−ADD 2 1−ADD Carry 0
. . . . . .
S N SN−1 SN−2 S 1 S 0
(N+1)−bit output value S
where c i represents the carry output from a full adder and the initial carry is set to 0 (no
carry at all)