Section 5

PROBLEMS

SECTIONS 2-1 AND 2-2

2-1 Convert these binary numbers to decimal.

1. 10110
2. 10001101
3. 100100001001
4. 1111010111
5. 10111111

2-2 Convert the following decimal values to binary.

1. 37
2. 14
3. 189
4. 205
5. 2313
6. 511

2-3 what is the  largest decimal value that can be represented by an 8-bit binary number? A 16-bit number?

SECTION 2-3

2-4 Convert each octal number to its decimal equivalent.

1. 743
2. 36
3. 3777
4. 257
5. 1204

2-5 Convert each of the following decimal numbers to octal.

1. 59
2. 372
3. 919
4. 65,536
5. 255

2-6 Convert each of the octal values from Problem 2-4 to binary.

2-7 Convert the binary numbers in Problem 2-1 to octal.

2-8 List the binary numbers in Problem 2-1 to octal.

2-9 When a large decimal number is to be converted to binary, it is sometimes easier to convert it first to octal, and then from octal to binary. Try this procedure for 2313 and compare it with the procedure used in problem 2-2(e).

SECTION 2-4

2-10 Convert these hex values to decimal.

1. 92
2. 1A6
3. 37FD
4. 2CO
5. 7FF

2-11 Convert these decimal values to hex.

1. 75
2. 314
3. 2048
4. 25,619
5. 4095

2-12 Convert the binary numbers in Problem 2-1 to hexadecimal.

2-13 Convert the  hex values in problem 2-10 to binary.

2-14 In most microcomputers the addresses of memory locations are specified in hexadecimal. These addresses are sequential numbers that identify each memory circuit.

1. A particular microcomputer store an 8-bit number in each memory location. If the memory addresses range from 0000 to FFFF, how many memory locations are there?
2. Another microcomputer is specified to have 4096 memory locations. What range of hex addresses does this computer use?

2-15 List the hex numbers in sequence from 280 to 2A0.

SECTION 2-5

2-16 Encode these decimal numbers in BCD

1. 47
2. 962
3. 187
4. 42,689,627
5. 1204

How many bits are required to represent the decimal numbers in the range from 0 to 999 using straight binary code? Using BCD code?

2-18 The following numbers are in BCD. Convert them to decimal.

1. 1001011101010010
2. 000110000100
3. 0111011101110101
4. 010010010010

SECTION 2-8

2-19 Represent the statement “X=25/Y” in ASCII code (excluding quotes). Attach an even-parity bit.

2-20 Attach an even-parity bit to each of the ASCII codes for problem 2-19 and give the results in hex.

2-21 The following code groups are being transmitted. Attach an even-parity bit to each group.

1. 10110110
2. 00101000
3. 11110111

SECTION 2-9

2-22 Convert the following decimal numbers to BCD code and then attach an odd-parity bit.

1. 74
2. 38
3. 165
4. 9201

2-23. In a certain digital system, the decimal numbers from 000 through 999 are represented in BCD code. An odd-parity bit is also included at the of each code group. Examine each of the code groups below and assume that each one has just been transferred from one location to another. some of the groups contain errors. Assume taht no more than two errors have occured for each group. Determine which of the code groups have a single error and which of them definitely have a double error. (Hint: Remember that this is a BCD code.)

1. 1001010110000
2. 0100011101100
3. 0111110000011
4. 1000011000101

2-24 Suppose that the  receiver received the following data from the transmitter of Example 2-10.

0 1 0 0 1 0 0 0

1 1 0 0 0 1 0 1

1 1 0 0 1 1 0 0

1 1 0 0 1 0 0 0

1 1 0 0 1 1 0 0

What errors can be the  receiver determine in these received data?

DRILL QUESTIONS

2-25 Perform each of the following conversions. For some of them, you may want to try several methods to see which one works best  for you. For example, a binary-to-decimal conversion may be  done  directly, or it may be done as a binary-to-octal conversion followed by an octal-to-decimal conversion.

DRILL QUESTIONS

2-25. Perform each of the following coversions. For some of them, you may want to try several methods to see which one works best for you. For example, a binary-to-decimal conversion may be done directly, or it may be done as a binary-to-octal conversion followed by an octal-to-decimal conversion.

1. 1417=____________
2. 255 =____________
3. 11010001=________
4. 11101010000100111=______________
5. 111010110000100111=_____________
6. 2497=____________
7. 511=_____________
8. 235=_____________
9. 4316=____________
10. 7A9=_____________
11. 3E1c=____________
12. 1600=____________
13. 38,187=___________
14. 865=_____________(BCD)
15. 10010100011 (BCD)=__________
16. 465=_____________
17. B34=_____________
18. 01110100(BCD)=_____________
19. 111010=__________(BCD)

2-26. Represent the  decimal value 37 in each of the following ways.

1. straight binary
2. BCD
3. Hex
4. ASCII
5. Octal

2-27 Fill in the blanks with the correct word or words.

1. Conversion from decimal to _______requires repeated division by 8.
2. Conversion from decimal to hex requires repeated division by_______.
3. In the BCD code, each___________ is  converted to its 4-bit binary equivalent.
4. The_________code has the characteristic that only one bit changes in going from one step to the next.
5. A transmitter attaches a________to a code group to allow the receiver to detect_______.
6. The__________code is the most common alphanumeric code used in computer systems.
7. __________and_________are often used as a convenient way to represent large binary numbers.

2-28. Write to binary number that results when each of the following numbers is incremented by one: (a) 0111 (b) 010000 (c) 1110

2-29. Repeat Problem 2-28 for the decrement operation.

2-30. Write the number that results when each of the following is incremented:

1. 7777
2. 7777
3. 2000
4. 2000
5. 9FF
6. 1000

2-31. Repeat problem 2-30 for the decrement operation.

CHALLENGING EXERCISES

2-32. Perform the following conversions between base-5 and decimal

1. 3421=___________
2. 726 =___________

2-33 Convert the following binary number directly into its base-4 equivalent: 01001110

2-34 Construct a table showing the binary, octal, hex and BCD representations of all decimal numbers from 0 to 15. Compare your table with Table 2-3.

Implications of DeMorgan’s Theorems

Let us examine these teorems (16) and (17) from the standpoint of logic circuits. First, consider theorem (16), X + y =x.y The left-hand side of the equation can be viewed as the output of a NOR gate whose inputs are x and y. The right-hand side of the equation, on the other hand, is the result of first inverting both x and y and then putting them through an AND gate. These two representations are equivalent and illustrated in Figure 3-26(a). Figure 3-26 (a) Equivalent circuits implied by theorem (16); (b)alternative symbol for the NOR function. Figure 3-27 (a) Equivalent circuits implied by theorem (17); (b) alternative symbol for the NAND function. What this means is that an AND gate with INVERTERs on each of its inputs is equivalent to a NOR gate. In fact, both representations are used to represent the NOR function. When the AND gate with inverted inputs is used to represent the NOR function, it is usually drawn as shown in Figure 3-26(b), where the small circles on the inputs represent the inversion operation. Now consider theorem (17), X.Y=X+Y The left side of the equation can be implemented by a NAND gate with inputs x and y. The right side can be implemented by first inverting inputs x and y and then putting them through an OR gate. These two equivalent representations are shown in Figure 3-27(a). The OR gate with INVERTERs on each of its inputs is equivalent to the NAND gate. In fact, both representations are used to represent the NAND function. When the OR gate with inverted inputs is used to represent the NAND function, it is usually drawn as shown in Figure 3-27(b), where the circles again represent inversion.

WHICH GATE REPRESENTATION TO USE

Some logic-circuit designers and many textbooks use only the standard logic-gate symbols in their circuit schematics. While this practice is not incorrect, it does nothing to make the circuit operation easier to follow. Proper use of the alternate gate symbols in the circuit diagram can make the circuit operation easier to follow. Proper use of the alternate gate symbols in the circuit diagram can make the circuit operation much clearer. This can be illustrated by considering the example shown in Figure 3-36.

The circuit in the Figure 3-36(a) contains three NAND gates connected to produce an output Z that depends on inputs A, B, C, D. The circuit diagram uses the standard symbol for  each of the NAND gates. While this diagram is logically correct, it does not facilitate any understanding of how the circuit functions. The improved circuit representations given in Figure 3-36(b) and (c), however, can be analyzed more easily to determine the circuit operation.

The representation of Figure 3-36(b) is obtained from from the original circuit diagram by replacing NAND gate 3 with its alternate symbol. In this diagram, output Z is taken from a Nand gate symbol that has an active-HIGH output. Thus, we can say that Z will go HIGH when either X or Y is LOW. Now, since X and Y each appear at the output of NAND symbols having active-LOW outputs, we can say that Z will go LOW only if A=B=1, and Y will go LOW only if C=D=1. Putting this all together, we can describe the circuit operation as follows:

Output Z will go HIGH whenever either A=B=1 or C=D=1 (or both).

This description can be translated to truth-table form by setting Z=1 for those cases where A=B=1, and for those cases where C=D=1. For all other cases, Z is made a 0. The resultant truth table is shown in Figure 3-36(d).

The representation of Figure 3-36(c) is obtained from the original circuit diagram by replacing NAND gates 1 and 2 by their alternate symbols. In this equivalent representation the Z ouput is taken from a NAND gate that has an active-LOW output. Thus, we can say that Z will go LOW only when X=Y=1. Since X and Y are active-HIGH outputs, we can say that X will be HIGH when either A or B is LOW, and Y will be HIGH when either C or D is LOW. Putting this all together, we can describe the circuit operation as follows:

Output Z will  go LOW only when A or  B is LOW and C or D is LOW.

This description can be translated to truth-table by making Z=0 for all cases where at least one of the A or B inputs is LOW at the same time that at least one of the C or D inputs is LOW. For all other cases, Z is made a 1. The resultant truth table is the same as that obtained for the circuit diagram for the circuit diagram of Figure 3-36(b).

Logic Symbol Interpretation

Each of the logi-gate symbols of figure 3-33 provides a unique interpretation of how the gate operates. Before we can demonstrate these interpretations, we must first establish the concept of active logic levels.

When an input or output line on a logic circuit symbol has no bubble on it, that line is said to be active-HIGH, when an input or output line does have a bubble on it, that line is said to be active-LOW. the presence or absence of a bubble, then, determines the active-HIGH/actIve-LOW states of a circuit’s input and output, and is used to interpret the circuit operation.

To illustrate, Figure 3-34(a) shows the standard symbol for a NAND gate. The standard symbol has a bubble on its output and no bubbles on its inputs. Thus, it has an active-LOW output and active-HIGH inputs. The logic operation represented by this symbol can therefore be interpreted as follows:

The output goes Low only when all the inputs are High, Note that this says that the output will go to its active state only when all the inputs are in their active states. The word ‘all’ is used because of the AND symbol.

The alternate symbol for a NAND gate shown in Figure 3-34(b) has an active-HIGH output and active-LOW inputs, and so its operation can be stated as

The output goes HIGH when any input is LOW.

Again, this says that the output will be in its active state whenever any of the inputs is in its active state. The word ‘any’ is used because of the OR symbol.

With a little thought, it can be seen that the two interpretations for the NAND symbols in figure 3-34 are different ways of saying the same thing.

Summary At this point you are probably wondering why there is a need to have two different symbols and interpretations for each logic gate. Hopefully, the reasons will become clear after reading the nest section. For now, let us summarize the important points concerning the logic-gate representations.

1- To obtain the alternate symbol for a logic gate, take the standard symbol and change its operation symbol (OR to AND, or AND to OR) and change the bubbles on both inputs and output (i.e., delete bubbles where there are none).

2. To interpret the logic-gate operation, first note which logic state, 0 or 1, is the active state for the inputs, and which is the active state for the output. Then realize that the outputs active state is produced by having all the inputs in their active state (if ANd and symbol is used), or having any of the inputs in its active state (if an OR symbol is used).

Digital

When most of us hear the term digital, we immediately think of “digital calculator” or “digital computer”. This can probably be attributed to the dramatic way that low-cost, powerful calculators and computer have become accessible to the average person. It is important to realize that calculators and computers represent only one of the many applications of digital circuits and principles. Digital circuits are used in electronic products such as video games, microwave ovens, and automobile control systems, and in test equipment such as meters, generators, and oscilloscopes. Digital techniques have also replaced a lot of the older “analog circuits” used in consumer products such as radios, TV sets, and high-fidelity sound recording and playback equipment.

In this book we are going to study the principles and techniques that are common to all digital systems from the simplest on/off switch to the most complex computer. If this book is successful, you should gain a deep understanding of how all digital systems work, and you should be able to apply this understanding to the  analysis and troubleshooting of any digital system.

We start by introducing some underlying concepts that are a vital part of digital technology; these concepts will expanded on as they are needed later in the book. We also introduce some of the terminology that is necessary when embarking on a new field of study, and add to it in every chapter. A complete glossary of terminology is presented in Appendix I.

1-2 DIGITAL AND ANALOG SYSTEMS

A digital system is a combination of devices designed to manipulate physical quantities or information that are represented in digital form; that is, they can take on only discrete values. These devices are most often electronic, but they can also be mechanical, magnetic, or pneumatic. Some of the more familiar digital systems include digital computers and calculators, digital audio and video equipment, and the telephone system-the world’s largest digital system.

An analog system contains devices that manipulate physical quantities that are represented in analog form. In an analog system, the quantities can vary over a continunous range of values. For example, the amplitude of the output signal to the speaker in a radio receiver can have any value between zero and its maximum limit. Other common analog systems are audio amplifiers, magnetic tape recording and playback eqipment, and the automobile odometer.

1-7 MEMORY

When an input signal is applied to most devices or circuits, the output somehow changes in response to the input , and when the input signal is removed, the output returns to its original state. These circuits do not  exhibit the property of memory, since their outputs revert back to normal. When an input is applied to such a circuit, the output will change its state, but it will remain in the new state even after the input is removed. This property of retaining its response to a momentary input is  called memory. Figure 1-10 illustrates nonmemory operations.

     Memory devices and circuits play an important role in digital systems because they provide means for storing binary numbers either temporarily or permanently, with the ability to change the stored information at any time. As we shall see, the various memory elements include magnetic types and those which utilize electronic latching circuits (called latches and flip-flops)   

Advantages of Digital Techniques

Advantages of Digital Techniques      An increasing majority of applications in electronics, as well as in most other technologies, use digital techniques to perform operations that were once performed using analog methods. The chief reasons for the shift to digital technology are:

1. Digital systems are generally easier to design. This is because the circuits that are used are switching circuits, where exact values of voltage or current are not important, only the range (HIGH or LOW) in which they fall.
2. Information storage is easy. this is accomplished by special switching circuits that can latch onto information and hold it for as long as necessary.
3. Accuracy and precision are greater. Digital systems can handle as many digits of precision as you need simply by adding more switching circuits. In analog systems, precision is usually limited to three or four digits because the values of voltage and cureent are directly dependent on the circuit component values.
4. Operation can be programmed. It is fairly easy to design digital systems whose operation is controlled by a set of stored instructions called a program. As technology progresses, this is becoming even easier. Analog systems can also be programmed, but the variety and complexity of the available operations is severely limited.
5. Digital circuits are less affected by noise. Spurious fluctuations in voltage (noise) are not as critical in digital systems because the exact value of a voltage is not important, as long as the noise is not large enough to prevent us from distinguishing a HIGH from a LOW.
6. More digital circuitry can be fabricated on IC chips. It is true that analog circuitry has also benefited from the tremendous development of IC technology, but its relative complexity and its use of devices that cannot be economically integrated (high-value capacitors, precision resistors, inductors, transformer) have prevented analog systems from achieving the same high degree of integration.

1-1 NUMERICAL REPRESENTATIONS

1-1 NUMERICAL REPRESENTATIONS

In science, technology, business, and, in fact, most other fields of endeavor, we are constantly dealing with quantities. Quantities are measured, monitored, recorded, manipulated arithmetically, observed, or in some other way utilized in most physical systems. It is important when dealing with various quantities that we be able to represent their values efficiently and accurately. There are basically two ways of representing the numerical value of quantities: analog and digital.

Analog Representations     In analog representation a quantity is represented by a voltage, current, or meter movement that is proportional to the value of that quantity. An example is an automobile speedometer, in which the deflection of the needle is propotional to the speed of the auto. The angular position of the needle represents the value of the auto’s speed, and the needle follows any changes that occurs as the auto speeds up or slow down.

Another example is the common room thermostat, in which the bending of the bimetallic strip is proportional to the room temperature. As the temperature changes grdually, the curvature of the strip changes proportionally.

Still another example of an analog quantity is found in the familiar audio microphone. In this device an output voltage is generated in proportion to the amplitude of the sound waves that impinge on the microphone. The variations in the output voltage follow the same variations as the input sound.

Analog quantities such as those cited above have an important characteristic: they can vary over a continuous range of values. The automobile speed can have any value between zero and, say, 100 mph. Similarly, the microphone output might be anywhere within a range of zero to 10 mV (e.g., 1mV, 2.3724 mV, 9.9999 mV).

Digital Representations     In digital representation the quantities are represented not by proportional quantities but by symbols called digits. As an example, consider the digital watch, which provides the time of day in the form of decimal digits which represent hours and minutes (and sometimes seconds). As we know, the time of day changes continuously, but the digital watch reading does not change continuously; rather, it changes in steps of one per minute (or per second). In other word, this digital representation of the time of day changes in discrete steps, as compared with the representation of time provided by an analog watch, where the dial reading changes continuously.

     The major difference between analog and digital quantities, then, can be simply stated as follows:

                               analog=continuous

                               digital=discrete (step by step)

Because of the discrete nature of digital representations, there is no ambiguity when reading the value of a digital quantity, whereas the value of an analog quantity is often open to interpretation.

Introductory Concepts

Introductory Concepts

Numerical Representations                            

Digital Circuits

Digital and Analog Systems                            

Parallel and Serial Transmission

Digital Number Systems                                  

Memory

Representing Binary Quantities                    

Digital Computers