Section 5
PROBLEMS
SECTIONS 21 AND 22
21 Convert these binary numbers to decimal.
 10110
 10001101
 100100001001
 1111010111
 10111111
22 Convert the following decimal values to binary.
 37
 14
 189
 205
 2313
 511
23 what is the largest decimal value that can be represented by an 8bit binary number? A 16bit number?
SECTION 23
24 Convert each octal number to its decimal equivalent.
 743
 36
 3777
 257
 1204
25 Convert each of the following decimal numbers to octal.
 59
 372
 919
 65,536
 255
26 Convert each of the octal values from Problem 24 to binary.
27 Convert the binary numbers in Problem 21 to octal.
28 List the binary numbers in Problem 21 to octal.
29 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 22(e).
SECTION 24
210 Convert these hex values to decimal.
 92
 1A6
 37FD
 2CO
 7FF
211 Convert these decimal values to hex.
 75
 314
 2048
 25,619
 4095
212 Convert the binary numbers in Problem 21 to hexadecimal.
213 Convert the hex values in problem 210 to binary.
214 In most microcomputers the addresses of memory locations are specified in hexadecimal. These addresses are sequential numbers that identify each memory circuit.
 A particular microcomputer store an 8bit number in each memory location. If the memory addresses range from 0000 to FFFF, how many memory locations are there?
 Another microcomputer is specified to have 4096 memory locations. What range of hex addresses does this computer use?
215 List the hex numbers in sequence from 280 to 2A0.
SECTION 25
216 Encode these decimal numbers in BCD
 47
 962
 187
 42,689,627
 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?
218 The following numbers are in BCD. Convert them to decimal.
 1001011101010010
 000110000100
 0111011101110101
 010010010010
SECTION 28
219 Represent the statement “X=25/Y” in ASCII code (excluding quotes). Attach an evenparity bit.
220 Attach an evenparity bit to each of the ASCII codes for problem 219 and give the results in hex.
221 The following code groups are being transmitted. Attach an evenparity bit to each group.
 10110110
 00101000
 11110111
SECTION 29
222 Convert the following decimal numbers to BCD code and then attach an oddparity bit.
 74
 38
 165
 9201
223. In a certain digital system, the decimal numbers from 000 through 999 are represented in BCD code. An oddparity 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.)
 1001010110000
 0100011101100
 0111110000011
 1000011000101
224 Suppose that the receiver received the following data from the transmitter of Example 210.
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
225 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 binarytodecimal conversion may be done directly, or it may be done as a binarytooctal conversion followed by an octaltodecimal conversion.
DRILL QUESTIONS
225. 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 binarytodecimal conversion may be done directly, or it may be done as a binarytooctal conversion followed by an octaltodecimal conversion.
 1417=____________
 255 =____________
 11010001=________
 11101010000100111=______________
 111010110000100111=_____________
 2497=____________
 511=_____________
 235=_____________
 4316=____________
 7A9=_____________
 3E1c=____________
 1600=____________
 38,187=___________
 865=_____________(BCD)
 10010100011 (BCD)=__________
 465=_____________
 B34=_____________
 01110100(BCD)=_____________
 111010=__________(BCD)
226. Represent the decimal value 37 in each of the following ways.
 straight binary
 BCD
 Hex
 ASCII
 Octal
227 Fill in the blanks with the correct word or words.
 Conversion from decimal to _______requires repeated division by 8.
 Conversion from decimal to hex requires repeated division by_______.
 In the BCD code, each___________ is converted to its 4bit binary equivalent.
 The_________code has the characteristic that only one bit changes in going from one step to the next.
 A transmitter attaches a________to a code group to allow the receiver to detect_______.
 The__________code is the most common alphanumeric code used in computer systems.
 __________and_________are often used as a convenient way to represent large binary numbers.
228. Write to binary number that results when each of the following numbers is incremented by one: (a) 0111 (b) 010000 (c) 1110
229. Repeat Problem 228 for the decrement operation.
230. Write the number that results when each of the following is incremented:
 7777
 7777
 2000
 2000
 9FF
 1000
231. Repeat problem 230 for the decrement operation.
CHALLENGING EXERCISES
232. Perform the following conversions between base5 and decimal
 3421=___________
 726 =___________
233 Convert the following binary number directly into its base4 equivalent: 01001110
234 Construct a table showing the binary, octal, hex and BCD representations of all decimal numbers from 0 to 15. Compare your table with Table 23.
Implications of DeMorgan’s Theorems

WHICH GATE REPRESENTATION TO USE
Some logiccircuit designers and many textbooks use only the standard logicgate 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 336.
The circuit in the Figure 336(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 336(b) and (c), however, can be analyzed more easily to determine the circuit operation.
The representation of Figure 336(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 activeHIGH 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 activeLOW 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 truthtable 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 336(d).
The representation of Figure 336(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 activeLOW output. Thus, we can say that Z will go LOW only when X=Y=1. Since X and Y are activeHIGH 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 truthtable 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 336(b).
Logic Symbol Interpretation
Each of the logigate symbols of figure 333 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 activeHIGH, when an input or output line does have a bubble on it, that line is said to be activeLOW. the presence or absence of a bubble, then, determines the activeHIGH/actIveLOW states of a circuit’s input and output, and is used to interpret the circuit operation.
To illustrate, Figure 334(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 activeLOW output and activeHIGH 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 334(b) has an activeHIGH output and activeLOW 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 334 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 logicgate 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 logicgate 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 lowcost, 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 highfidelity 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.
12 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 systemthe 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.
17 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 110 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 flipflops)
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:
 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.
 Information storage is easy. this is accomplished by special switching circuits that can latch onto information and hold it for as long as necessary.
 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.
 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.
 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.
 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 (highvalue capacitors, precision resistors, inductors, transformer) have prevented analog systems from achieving the same high degree of integration.
11 NUMERICAL REPRESENTATIONS
11 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
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