Chapter 4 Digital Logic and Implementation 4.1 Boolean Algebra

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.