Introduction to Chapter 6 Digital circuits are frequently used for arithmetic operations ...

Introduction to Chapter 6 Digital circuits are frequently used for arithmetic

operations Fundamental arithmetic operations on binary

numbers and digital circuits which perform arithmetic operations will be examined.

Ronald Tocci/Neal Widmer/Gregory MossDigital Systems: Principles and Applications, 10e

6-1 Binary Addition Binary numbers are added like decimal numbers. In decimal, when numbers sum more than 9 a carry

results. In binary when numbers sum more than 1 a carry

takes place. Addition is the basic arithmetic operation used by

digital devices to perform subtraction, multiplication, and division.

6-2 Representing Signed Numbers Since it is only possible to show magnitude

with a binary number, the sign (+ or ) is shown by adding an extra “sign” bit.

A sign bit of 0 indicates a positive number. A sign bit of 1 indicates a negative number. The 2’s complement system is the most

commonly used way to represent signed numbers.

6-2 Representing Signed Numbers In order to change a binary number to 2’s complement it

must first be changed to 1’s complement. To convert to 1’s complement, simply change each bit to its

complement (opposite). To convert 1’s complement to 2’s complement add 1 to the 1’s

complement. A positive number is true binary with 0 in the sign bit. A negative number is in 2’s complement form with 1 in

the sign bit.

6-2 Representing Signed Numbers A number is negated when converted to the opposite

sign. A binary number can be negated by taking the 2’s

complement of it.

6-3 Addition in the 2’s Complement System Perform normal binary addition of magnitudes. The sign bits are added with the magnitude bits. If addition results in a carry of the sign bit, the

carry bit is ignored. If the result is positive it is in pure binary form. If the result is negative it is in 2’s complement

6-4 Subtraction in the 2’s Complement System

The number subtracted (subtrahend) is negated. The result is added to the minuend. The answer represents the difference. If the answer exceeds the number of magnitude bits

an overflow results.

6-5 Multiplication of Binary Numbers This is similar to multiplication of decimal numbers. Each bit in the multiplier is multiplied by the

multiplicand. The results are shifted as we move from LSB to

MSB in the multiplier. All of the results are added to obtain the final

product.

6-6 Binary Division This is similar to decimal long division. It is simpler because only 1 or 0 are possible. The subtraction part of the operation is done using

2’s complement subtraction. If the signs of the dividend and divisor are the same

the answer will be positive. If the signs of the dividend and divisor are different

the answer will be negative.

6-7 BCD Addition When the sum of each decimal digit is less than 9,

the operation is the same as normal binary addition. When the sum of each decimal digit is greater than

9, a binary 6 is added. This will always cause a carry.

BCD subtraction – a more complicated operation – is not discussed in the book.

6-8 Hexadecimal Arithmetic Hex addition:

Add the hex digits in decimal. If the sum is 15 or less express it directly in hex digits. If the sum is greater than 15, subtract 16 and carry 1 to the

next position. Hex subtraction – use the same method as for binary

numbers. When the MSD in a hex number is 8 or greater, the

number is negative. When the MSD is 7 or less, the number is positive.

6-9 Arithmetic Circuits An arithmetic/logic unit

(ALU) accepts data stored in memory and executes arithmetic and logic operations as instructed by the control unit.

6-9 Arithmetic Circuits Typical sequence of operations:

Control unit is instructed to add a specific number from a memory location to a number stored in the accumulator register.

The number is transferred from memory to the B register. Number in B register and accumulator register are added

in the logic circuit, with sum sent to accumulator for storage.

The new number remains in the accumulator for further operations or can be transferred to memory for storage.

6-10 Parallel Binary Adder The A and B variables represent 2 binary numbers to be

added. The C variables are the carries. The S variables are the sum bits.

6-11 Design of a Full Adder Construct a truth table of 3 inputs (2 numbers to be

added and carry in) and 2 outputs (sum and carry out).

Use algebraic methods or K-maps to simplify the resulting SOP form.

The result is the logic circuit shown in Figure 6-7. The design process for a half adder is similar, but

there is no carry in so there will be only 2 input bits.

6-12 Complete Parallel Adder With Registers Register notation – to indicate the contents of a

register we use brackets:[A]=1011 is the same as A3=1, A2=0, A1=1, A0=1

A transfer of data to or from a register is indicated with an arrow [B][A] means the contents of register B have been transferred to register A. This is a common form of notation and is also found in HDL.

6-12 Complete Parallel Adder With Registers The following describes adding binary 1001 and

0101 using the circuit of Figure 6-9: A CLR pulse is applied at t1

The first binary number 1001 is transferred from memory to the B register at t2

The sum of 1001 and 0000 is transferred to the A register at t3

0101 is transferred from memory to the B register at t4

The sum outputs are transferred to the A register at t5

The sum of the two numbers is now present in the accumulator.

6-13 Carry Propagation Parallel adder speed is limited by carry propagation

(also called carry ripple). Carry propagation results from having to wait for the

carry bits to “ripple” through the device. Additional bits will introduce more delay. Various techniques have been developed to reduce

the delay. The look-ahead carry scheme is commonly used in high speed devices.

6-14 Integrated Circuit Parallel Adder The most common parallel adder is a 4 bit device

with 4 interconnected FAs and look-ahead Carry circuits.

The The A and B lines each represent 4 bit numbers to be added. The C0 is the carry in, the C4 is the carry out, and the lines are the sum of the 2 numbers.

6-14 Integrated Circuit Parallel Adder Parallel adders may be cascaded together as shown

to add larger numbers, in this case two 8 bit numbers.

6-15 2’s Complement System Addition of negative and positive numbers using adders

is done by placing the negative number into 2’s complement form and performing normal addition.

Subtraction is done by converting the number to be subtracted (subtrahend) to 2’s complement and adding to the minuend.

An adder can be used to perform addition and subtraction by designing a way to take the 2’s complement for subtraction as described in Figure 6-13.

6-16 ALU Integrated Circuits ALUs can perform different arithmetic and

logic functions as determined by a binary code on the function select inputs.

There are many different devices. The 74LS382 (TTL) and HC382 (CMOS) is a typical device with 8 possible functions.

6-17 ALU Integrated Circuits

6-17 Troubleshooting Case Study Read the case study

in the text and determine the most likely fault in the circuit shown, given the test results described.

6-18 Using TTL Library Functions with ALTERA

Altera offers pre-defined logic circuits in macrofunctions.

ALUs may be defined graphically, as shown in Figure 6-20.

Graphic description may seem intuitive, but it is often easier to define a device using text and macrofunctions.

6-19 Logical Operations on Bit Arrays HDL techniques in two main areas:

Specifying groups of bits in arrays Using logical operations to combine arrays using Boolean

expressions To refer to three most significant bits of d as a set:

AHDL: VARIABLE d[3..0] :NODE VHDL: SIGNAL d:BIT_VECTOR (3 DOWNTO 1)

6-21 Expanding the Bit Capacity of a Circuit

Constants are fixed numbers that are represented by a symbol. The symbol is used in the code to refer to the value.

A circuit can be modified by simply changing the numbers the symbols refer to.

The examples in this section do not have look ahead carry circuits for simplicity.

AHDL – Refer to Figure 6-27. Keyword CONSTANT is followed by the symbolic name

and value assigned. VHDL – Refer to Figure 6-28.

Packages contain constants. Within the package the keyword CONSTANT is followed

by the symbolic name, type, and the value to be assigned.

The VHDL generate statement can be used to concisely replicate several components that are cascaded together.

Altera offers a library of parameterized modules (LPMs) which offer generic solutions for the various logic circuits used in digital systems.

Introduction to Chapter 6 Digital circuits are frequently used for arithmetic operations ...

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