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Given a positive integer, fraction, or mixed number in any base 2 through 16 ; convert to any other base. Justify the procedure used by using a power series expansion for the number. Add, subtract, multiply, and divide positive binary numbers.

Explain the addition and subtraction process in terms of carries and borrows. Justify the methods used. State when an overflow occurs. Represent a decimal number in binary-coded-decimal BCD , code, excess-3 code, etc. Given a set of weights, construct a weighted code. Study Section 1. Answer the following study questions as you go along: a Is the first remainder obtained in the division method for base conversion the most or least significant digit?

Why should you start forming the groups of four bits at the binary point instead of the left end of the number? Number Systems and Conversion 3 e Complete the following conversion table. Binary base 2 0 1 10 11 Octal base 8 Decimal base 10 Hexadecimal base 16 0 0 0 20 16 10 f Work Problems 1.

Work Problems 1. Does a carry out of the last bit position indicate that an overflow has occurred? If this number is n bits long, what number does it represent and why? It is not negative zero. Note that for some decimal digits, two different code combinations could have been used.

For example, either or could represent 4. In each case the combination with the smaller binary value has been used. If you are taking this course on a self-paced basis, you will need to pass a readiness test on this unit before going on to the next unit.

The purpose of the readiness test is to determine if you have mastered the material in this unit and are ready to go on to the next unit. Before you take the readiness test: a Check your answers to the problems against those provided at the end of this book. If you missed any of the problems, make sure that you understand why your answer is wrong and correct your solution.

Introduction Number Systems and Conversion 1. Because digital systems are capable of greater accuracy and reliability than analog systems, many tasks formerly done by analog systems are now being performed digitally.

In a digital system, the physical quantities or signals can assume only discrete values, while in analog systems the physical quantities or signals may vary continuously over a specified range. Because digital systems work with discrete quantities, in many cases they can be designed so that for a given input, the output is exactly correct. For example, if we multiply two 5-digit numbers using a digital multiplier, the digit product will be correct in all 10 digits. On the other hand, the output of an analog multiplier might have an error ranging from a fraction of one percent to a few percent depending on the accuracy of the components used in construction of the multiplier.

Furthermore, if we need a product which is correct to 20 digits rather than 10, we can redesign the digital multiplier to process more digits and add more digits to its input.

A similar improvement in the accuracy of an analog multiplier would not be possible because of limitations on the accuracy of the components. The design of digital systems may be divided roughly into three parts—system design, logic design, and circuit design. System design involves breaking the overall system into subsystems and specifying the characteristics of each subsystem. For example, the system design of a digital computer could involve specifying the number and type of memory units, arithmetic units, and input-output devices as well as the interconnection and control of these subsystems.

Logic design involves determining how to interconnect basic logic building blocks to perform a specific function. An example of logic design is determining the interconnection of logic gates and flip-flops required to perform binary addition. Inputs X1 X2 Switching Circuit Most contemporary circuit design is done in integrated circuit form using appropriate computeraided design tools to lay out and interconnect the components on a chip of silicon.

This book is largely devoted to a study of logic design and the theory necessary for understanding the logic design process. Some aspects of system design are treated in Units 18 and Circuit design of logic gates is discussed briefly in Appendix A. A switching circuit has one or more inputs and one or more outputs which take on discrete values.

In this text, we will study two types of switching circuits—combinational and sequential. In a combinational circuit, the output values depend only on the present value of the inputs and not on past values. In a sequential circuit, the outputs depend on both the present and past input values. In other words, in order to determine the output of a sequential circuit, a sequence of input values must be specified.

In general, a sequential circuit is composed of a combinational circuit with added memory elements. Combinational circuits are easier to design than sequential circuits and will be studied first. Z1 Z2 Outputs Zn The basic building blocks used to construct combinational circuits are logic gates. The logic designer must determine how to interconnect these gates in order to convert the circuit input signals into the desired output signals. The relationship between these input and output signals can be described mathematically using Boolean algebra.

Units 2 and 3 of this text introduce the basic laws and theorems of Boolean algebra and show how they can be used to describe the behavior of circuits of logic gates. Starting from a given problem statement, the first step in designing a combinational logic circuit is to derive a table or the algebraic logic equations which describe the circuit outputs as a function of the circuit inputs Unit 4. In order to design an economical circuit to realize these output functions, the logic equations which describe the circuit outputs generally must be simplified.

Algebraic methods for this simplification are described in Unit 3, and other simplification methods Karnaugh map and Quine-McCluskey procedure are introduced in Units 5 and 6.

Implementation of the simplified logic equations using several types of gates is described in Unit 7, and alternative design procedures using programmable logic devices are developed in Unit 9. The basic memory elements used in the design of sequential circuits are called flip-flops Unit These flip-flops can be interconnected with gates to form counters and registers Unit Analysis of more general sequential circuits using 8 Unit 1 timing diagrams, state tables, and graphs is presented in Unit The first step in designing a sequential switching circuit is to construct a state table or graph which describes the relationship between the input and output sequences Unit Methods for going from a state table or graph to a circuit of gates and flip-flops are developed in Unit Methods of implementing sequential circuits using programmable logic are discussed in Unit In Unit 18, combinational and sequential design techniques are applied to the realization of systems for performing binary addition, multiplication, and division.

The sequential circuits designed in this text are called synchronous sequential circuits because they use a common timing signal, called a clock, to synchronize the operation of the memory elements.

Use of a hardware description language, VHDL, in the design of combinational logic, sequential logic, and digital systems is introduced in Units 10, 17, and VHDL is used to describe, simulate, and synthesize digital hardware.

After writing VHDL code, the designer can use computer-aided design software to compile the hardware description and complete the design of the digital logic. This allows the completion of complex designs without having to manually work out detailed circuit descriptions in terms of gates and flip-flops.

The switching devices used in digital systems are generally two-state devices, that is, the output can assume only two different discrete values. Examples of switching devices are relays, diodes, and transistors. A relay can assume two states—closed or open—depending on whether power is applied to the coil or not.

A diode can be in a conducting state or a nonconducting state. A transistor can be in a cut-off or saturated state with a corresponding high or low output voltage. Of course, transistors can also be operated as linear amplifiers with a continuous range of output voltages, but in digital applications greater reliability is obtained by operating them as two-state devices.

Because the outputs of most switching devices assume only two different values, it is natural to use binary numbers internally in digital systems. For this reason binary numbers and number systems will be discussed first before proceeding to the design of switching circuits. For example, If the base is R, then R digits 0, 1,. A number written in positional notation can be expanded in a power series in R. If the arithmetic indicated in the power series expansion is done in base 10, then the result is the decimal equivalent of N.

Note: In base 3, 10 is , 7 is 21, etc. To complete the conversion, base 3 arithmetic would be used. Of course, this is not very convenient if the arithmetic is being done by hand. For hand calculation, use the power series expansion when converting from some base into base For bases greater than 10, more than 10 symbols are needed to represent the digits. In this case, letters are usually used to represent digits greater than 9.

For example, in hexadecimal base 16 , A represents , B represents , C represents , D represents , E represents , and F represents Note that the remainder obtained at each division step is one of the desired digits and the least significant digit is obtained first.

Example Convert to binary. Note that the integer part obtained at each step is one of the desired digits and the most significant digit is obtained first. Example Convert 0. It is generally easier to convert to decimal first and then convert the decimal number to the new base. Starting at the binary point, the bits are divided into groups of four, and each group is replaced by a hexadecimal digit: Binary arithmetic is carried out in much the same manner as decimal, except the addition and multiplication tables are much simpler.

Number Systems and Conversion Example 13 Add and in binary. In order to borrow 1 from the second column, we must in turn borrow 1 from the third column, etc. Binary subtraction sometimes causes confusion, perhaps because we are so used to doing decimal subtraction that we forget the significance of the borrowing process.

Before doing a detailed analysis of binary subtraction, we will review the borrowing process for decimal subtraction. A detailed analysis of binary subtraction example c follows. To subtract in the second column, we must borrow from the third column. Rather than borrow immediately, we place a 1 over the third column to indicate that a borrow is necessary, and we will actually do the borrowing when we get to the third column. This is similar to the way borrow signals might propagate in a computer.

In order to borrow 1 from the third column, we must borrow 1 from the fourth column indicated by placing a 1 over column 4. Now in column 4, we subtract off the borrow leaving 0. When adding up long columns of binary numbers, the sum of the bits in a single column can exceed , and therefore the carry to the next column can be greater than 1.

When doing binary multiplication, a common way to avoid carries greater than 1 is to add in the partial products one at a time as illustrated by the following example: multiplicand multiplier first partial product second partial product sum of first two partial products third partial product sum after adding third partial product fourth partial product final product sum after adding fourth partial product The following example illustrates division of by in binary: 10 The quotient is with a remainder of Binary division is similar to decimal division, except it is much easier because the only two possible quotient digits are 0 and 1.

In the above example, if we start by comparing the divisor with the upper four bits of the dividend , we find that we cannot subtract without a negative result, so we move the divisor one place to the right and try again. This time we can subtract from to give as a result, so we put the first quotient bit of 1 above We then bring down the next dividend bit 0 to get and shift the divisor right.

We then subtract from to get 11, so the second quotient bit is 1. When we bring down the next dividend bit, the result is , and we cannot subtract the shifted divisor, so the third quotient bit is 0. We then bring down the last dividend bit and subtract from to get a final remainder of 10, and the last quotient bit is 1. In most computers, in order to represent both positive and negative numbers the first bit in a word is used as a sign bit, with 0 used for plus and 1 used for minus.

Several representations of negative binary numbers are possible. The sign and magnitude system is similar to that which people commonly use.

The design of logic circuits to do arithmetic with sign and magnitude binary numbers is awkward; therefore, other representations are often used.

No borrows occur in this subtraction. The addition is carried out just as if all the numbers were positive, and any carry from the sign position is ignored.

This will always yield the correct result except when an overflow occurs. When the word length is n bits, we say that an 18 Unit 1 overflow has occurred if the correct representation of the sum including sign requires more than n bits. This is referred to as an end-around carry. Because most logic circuits only accept two-valued signals, the decimal numbers must be coded in terms of binary signals.

In the simplest form of binary code, each decimal digit is replaced by its binary equivalent. Note that the result is quite different than that obtained by converting the number as a whole into binary. Because there are only ten decimal digits, through are not valid BCD codes. Table shows several possible sets of binary codes for the ten decimal digits. Many other possibilities exist because the only requirement for a TABLE Binary Codes for Decimal Digits Decimal Digit Code BCD Code Excess-3 Code 2-out-of-5 Code Gray Code 0 1 2 3 4 5 6 7 8 9 22 Unit 1 valid code is that each decimal digit be represented by a distinct combination of binary digits.

To translate a decimal number to coded form, each decimal digit is replaced by its corresponding code. Thus expressed in excess-3 code is The BCD code and the code are examples of weighted codes.

The 2-out-of-5 code has the property that exactly 2 out of the 5 bits are 1 for every valid code combination. This code has useful error-checking properties because if any one of the bits in a code combination is changed due to a malfunction of the logic circuitry, the number of 1 bits is no longer exactly two.

The table shows one example of a Gray code. A Gray code has the property that the codes for successive decimal digits differ in exactly one bit. For example, the codes for 6 and 7 differ only in the fourth bit, and the codes for 9 and 0 differ only in the first bit.

A Gray code is often used when translating an analog quantity, such as a shaft position, into digital form. In this case, a small change in the analog quantity will change only one bit in the code, which gives more reliable operation than if two or more bits changed at a time.

The Gray and 2-out-of-5 codes are not weighted codes. In general, the decimal value of a coded digit cannot be computed by a simple formula when a non-weighted code is used. Many applications of computers require the processing of data which contains numbers, letters, and other symbols such as punctuation marks.

In order to transmit such alphanumeric data to or from a computer or store it internally in a computer, each symbol must be represented by a binary code. This is a 7-bit code, so 27 different code combinations are available to represent letters, numbers, and other symbols. Convert to hexadecimal. Then convert both of your answers to decimal, and verify that they are the same. Round to two digits past the hexadecimal point. Place a 1 over each column from which it was necessary to borrow.

Use a word length of 6 bits including sign and indicate if an overflow occurs. Express your answers in decimal. A do all of the arithmetic in decimal. The exact value requires an infinite repeating part in the fractional part of the number.

Show the steps of your derivation. Use your method to convert the following number to base 9: Determine the base of the numbers. Did any of the additions overflow? Hint: Represent the base b number using the power series expansion.

A weighted code? Justify your answers. A weighte code? What number does represent in this code? What numbers does represent in this code? Write B4A9 using this code. Construct a table for this weighted code. Subtract each of the following pairs of 5-bit binary numbers by adding the complement of the subtrahend to the minuend. Indicate when an overflow occurs. Perform the following operations and indicate whether overflow occurs. When you complete this unit, you should be familiar with and be able to use any of the first 12 of these.

Specifically, you should be able to: 1. Understand the basic operations and laws of Boolean algebra. Also relate these operations and laws to circuits composed of switches.

Prove any of these laws using a truth table. Apply these laws to the manipulation of algebraic expressions including: a. Factoring an expression to obtain a product of sums POS. Simplifying an expression by applying one of the laws. Finding the complement of an expression. In this unit you will study Boolean algebra, the basic mathematics needed for the logic design of digital systems. Just as when you first learned ordinary algebra, you will need a fair amount of practice before you can use Boolean algebra effectively.

However, by the end of the course, you should be just as comfortable with Boolean algebra as with ordinary algebra. Fortunately, many of the rules of Boolean algebra are the same as for ordinary algebra, but watch out for some surprises!

Study Sections 2. The latter has the advantage that it is much easier for typists, printers, and computers. Have you ever tried to get a computer to print a bar over a letter? You may use either notation in your work, but please do not mix notations in the same equation.

Initially we will use These symbols point in the direction of signal flow. This makes it easier to read the circuit diagrams in comparison with the square or round symbols used in some books. How many literals? For example, for a three-variable truth table, the first row should be , the next row , then , , , , , and Study Section 2. Your answer should require only two gates. Identify X and Y in each case. Probably the most difficult part of the unit is using the second distributive law for factoring or multiplying out an expression.

If you have difficulty with Problems 2. Now factor your answer to a to see that you can get back the original expression. Find the complement of each of the following expressions as indicated. In your answer, the complement operation should be applied only to single variables. Verify that this is true. A fully simplified expression should have nothing complemented except the individual variables.

Find the minimum product of sums for F. Work Problems 2. Review the first 12 laws and theorems on page Make sure that you can recognize when to apply them even if an expression has been substituted for a variable. Reread the objectives of this unit. If you are satisfied that you can meet these objectives, take the readiness test.

However, by the end of Unit 3, you should know all the theorems by memory. Boolean algebra has many other applications including set theory and mathematical logic, but we will restrict ourselves to its application to switching circuits in this text.

Because all of the switching devices which we will use are essentially two-state devices such as a transistor with high or low output voltage , we will study the special case of Boolean algebra in which all of the variables assume only one of two values. This two-valued Boolean algebra is often referred to as switching algebra. George Boole developed Boolean algebra in and used Boolean Algebra 35 it to solve problems in mathematical logic.

Claude Shannon first applied Boolean algebra to the design of switching circuits in We will use a Boolean variable, such as X or Y, to represent the input or output of a switching circuit. We will assume that each of these variables can take on only two different values. In a logic gate circuit, 0 usually represents a range of low voltages, and 1 represents a range of high voltages. In a switch circuit, 0 usually represents an open switch, and 1 represents a closed circuit.

In general, 0 and 1 can be used to represent the two states in any binary-valued system. The complement of 0 is 1, and the complement of 1 is 0. If a logic 0 corresponds to a low voltage and a logic 1 corresponds to a high voltage, a low voltage at the inverter input produces a high voltage at the output and vice versa. Although this looks like binary multiplication, it is not, because 0 and 1 here are Boolean constants rather than binary numbers.

The AND operation is also referred to as logical or Boolean multiplication. This type of OR operation is sometimes referred to as inclusive-OR. Next, we will apply switching algebra to describe circuits containing switches. We will label each switch with a variable. If switch X is open, then we will define the value of X to be 0; if switch X is closed, then we will define the value of X to be 1.

A 1 2 B In this case, we have a closed circuit between terminals 1 and 2 iff switch A is closed or switch B is closed. More complicated expressions are formed by combining two or more other expressions using AND or OR, or by complementing another expression.

Each expression corresponds directly to a circuit of logic gates. Figure gives the circuits for Expressions and A truth table also called a table of combinations specifies the values of a Boolean expression for every possible combination of values of the variables in the expression. The name truth table comes from a similar table which is used in symbolic logic to list the truth or falsity of a statement under all possible conditions.

We can use a truth table to specify the output values for a circuit of logic gates in terms of the values of the input variables. Figure b shows a truth table which specifies the output of the circuit for all possible combinations of values of the inputs A and B.

On the left side of Table , we list the values of the variables A, B, and C. These combinations are easily obtained by listing the binary numbers , ,. Two expressions are equal if they have the same value for every possible combination of the variables.

As before, 0 will represent an open circuit or open switch, and 1 will represent a closed circuit or closed switch. We will prove the associative law for AND by using a truth table Table On the left side of the table, we list all combinations of values of the variables X, Y, and Z. If any of the variables have the value 0, the result of the AND operation will be 0.

The result of the OR operation will be 0 iff all of the variables have the value 0. This second law is very useful in manipulating Boolean expressions. Because each expression corresponds to a circuit of logic gates, simplifying an expression leads to simplifying the corresponding logic circuit.

Boolean Algebra 43 Each of the preceding theorems can be proved by using a truth table, or they can be proved algebraically starting with the basic theorems. We will illustrate Theorem D , using switches. Y X The following example illustrates simplification of a logic gate circuit using one of the theorems.

Therefore, circuit a can be replaced with the equivalent circuit b. An expression is said to be in sum-of-products form when all products are the products of single variables. This form is the end result when an expression is fully multiplied out. When multiplying out an expression, apply the second distributive law first when possible. Both distributive laws can be used to factor an expression to obtain a productof-sums form. The truth table for the AND gate is shown in Figure 7 a.

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