AN ABSTRACT OF THE THESIS OF in Jaes Willia Walker for the MSEE (Nae) (Degree) Electrical Engineering presented on yfra.,- /4/,'d) (Date) Title: A PROGRAMMED INTRODUCTION TO MODULATION TECHNIQUES Abstract approved J. Weber This prograed text covers the basic eleents of the theory of odulation. It is by no eans a coprehensive reference to odulation topics. The text is designed to introduce the student to odulation in general ters. After successful copletion of this text, the student should be able to advance to coplex odulating systes by building on the ideas presented here. The ideas presented in this text are neither new nor novel. Any textbook on odulation techniques would have essentially the sae inforation. The approach in this text is on coprehension of subject aterial which is not necessarily present when odulation, a subject unknown to a student, is presented in textbook for.
A Prograed Introduction to Modulation Techniques by Jaes Willia Walker A THESIS subitted to Oregon State University in partial fulfillent of the requireents for the degree of Master of Science June 1970
APPROVED: Pro essor of Elec ical Enginee ng in charge of ajor dead, Departent of E ectrical and Electronic Engineering Dean of Graduate School Date thesis is presented Typed by Barbara Glenn for Jaes Willia Walker
ACKNOWLEDGMENTS The author wishes to express his gratitude first to Professor L. J. Weber, whose original and continuing suggestions helped forulate this text, and second, to the any students who evaluated the several revisions.
TABLE OF CONTENTS Introduction A Prograed Introduction to Modulation Techniques 5 Suary 107 Bibliography 108 Appendix 1 09
A PROGRAMMED INTRODUCTION TO MODULATION TECHNIQUES INTRODUCTION The field of higher education, like other related fields, will always be subject to change. Educators and instructors will vary instructional ethods fro the classical fors of learning. This thesis presents one new for of learning for university seniors ajoring in counication engineering. In October, 1965, the proposal for this thesis outlined this step-by-step procedure: 1. Literature search and review in the field of prograed instruction. Although the field of prograed instruction is relatively new, there are any ideas and philosophies discussing the pros and cons of various prograing ethods. 2. Practice at progra writing. Most authorities in the field agree that progra writing is ore a atter of trial and error with student evaluation rather than skill in writing ability. This is one field where students are the ain source of progra evaluation. This thesis is no exception. If students cannot satisfactorily, read through the instructions, then the progra, not the student, needs changing.
Revisions brought about by student reaction are necessary for successful progras. 3. Review of selected technical literature for pertinent facts to be included in the prograed instruction text. 4. Preparation of the progra for trial evaluation. This requires preparation and approval by copetent evaluators before the progra is presented to students for their participation and reaction. 5. Evaluation of the prograed instruction text, priarily by student participation. 6. Inclusion of training aids in addition to the prograed instruction text, to suppleent student learning. 7. Final evaluation and report. During the last few years a new technique of instruction has appeared on all levels of Aerican education. Instructors refer to the technique as prograed learning, prograed instruction, auto instruction, or reinforced learning. Although the philosophy of prograed instruction has been known for any years, it has been applied to educational learning for only about a decade. Opinions on the proper approach of prograed instruction to educational instruction vary aong educators. As yet, sufficient experiental data is not available to substantiate any one theory of presentation. However, one opinion is fir: prograed learning is at least as effective as 2
classical ethods of classroo instruction. Although scholars disagree on the proper approach of prograed instruction, ost authorities in the field agree on these learning characteristics as essential points (Hughes, 1962): 1. A relatively sall unit of inforation is presented to the student at a tie. A stateent to be copleted or a question to be answered involving this inforation is included. This is known technically as the stiulus. 2. The student is required to coplete the stateent or answer the question about that specific bit of inforation. In technical ters, he is aking a response to the stiulus presented. The stateent or question is designed to ake it probable that the student will give the correct response. 3. The student is then iediately infored whether his response is correct or not. If his response is wrong, he ay even by told why. By this kind of feedback he is rewarded (told he is correct) if he gives the correct answer. In technical ters, his response is reinforced. Results fro the earliest of learning experients indicate that reinforceent increases the probability of aking the correct response to the sae stiulus in the future. 4. The student is next presented with the second unit of inforation, and the cycle of stiulus-response-feedback 3
is repeated. Each student works individually on the prograed instruction aterial at his own pace. It would be difficult to predict accurately whether the use of prograed instruction will transfor present educational procedures. However, in nearly every case where it has been used, prograed instruction has led to either a reduction in learning tie, or an increase in the knowledge acquired by the student, or both. There are, of course, any other principles and features pertaining to the eleents of prograed instruction. This thesis is not eant to be a anual on how to prepare aterial of this type. Rather, it illustrates one application with satisfactory results of the principles of prograed instruction applied to one phase of counication engineering. 4
5 A PROGRAMMED INTRODUCTION TO MODULATION TECHNIQUES by J. W. Walker
6 TABLE OF CONTENTS Introduction Instructions 7 9 Set 1 Definition of Modulation 12 2 Signals in a Modulation Syste 13 3 Ters and Sybols Defining Modulating, Signals 15 4 Modulation Types 18 5 Aplitude Modulation 21 6 Modulation Index of an AM Signal 25 7 Deterining the Modulation Index 28 8 Deterining the Frequency Response of Signals 32 9 Frequency Coponents in an AM Signal 35 10 The Square-Law Modulator 39 11 The Product Modulator 44 12 Modulation in a General Non-Linear Device 46 13 Balanced Modulators 54 14 Probles on Aplitude Modulation 58 15 The Deodulation Process for AM Signals 62 16 Second Haronic in the Deodulated Signal 64 17 Aplifiers Used as Aplitude Modulators 68 18 Oscillators Used as Aplitude Modulators 71 19 Sideband Redundancy 73 20 SSB and DSB-SC 75 21 Vestigial Sideband Modulation 77 22 Angle Modulation 79 23 Frequency Modulation 81 24 Phase Modulation 88 25 Frequency Coponents in an FM Signal 90 26 Probles on Angle Modulation 97 27 The Deodulation Process for FM Signals 103
7 INTRODUCTION You are about to study a prograed anual on odulation principles. Prograed anuals basically involve the presentation of ideas in gradual steps, leading fro siple definitions to ore coplex ideas. These are the essential characteristics of any prograed anual: 1. A relatively sall unit of inforation is presented to you at a tie. A stateent to be copleted, or a question to be answered about this inforation, is also included. This is known technically as the stiulus. 2. You are required to coplete the stateent or answer the question about that specific bit of inforation. In technical ters, you are aking a response to the stiulus. The stateent or question has been designed to help you give the correct responses. 3. You are then iediately infored whether your response is correct or not. By this kind of feedback you are rewarded (told you are correct) if you give the correct answer; in technical ters, your response is reinforced. 4. You are next presented with the second unit of inforation, and the cycle of presentation-answer-feedback is repeated.
Each unit of inforation is called a frae. A series of fraes presenting a ajor thought is called a set. In soe fraes one or ore words are issing. You will be required to supply the answer before going on to the next frae. This is in no way a test. The correct answers to each frae should have been supplied fro previous reading. Since each response should be evident, the text is erely guiding your progress through a series of fraes. 8
INSTRUCTIONS The aterial in this text is presented in a series of nubered stateents. Each nubered stateent is called a "frae" and each group of fraes with the sae first nuber is called a "set. " For exaple, the stateents nubered 3, 3-A, 3-B, 3-C, would be different fraes of the sae set. In each of the fraes there is a stateent to be copleted, or questions to be answered. The answer to each frae is given between the lines below the frae. For exaple, the answer to frae 3-A would be found below frae 3-A. There are three types of fraes in each set. The first frae of each set is a gating frae intended to allow the student who can answer it correctly to skip the set if he feels he knows the inforation in that set. If the gating frae cannot be answered or if the student wants ore inforation on the subject then he should continue with the teaching fraes. The last frae in each set is a criterion frae. The preceding teaching frae should have provided the necessary inforation to answer the criterion frae. If the criterion frae cannot be answered, the teaching fraes in that set should be reviewed. gating frae is only an indication of the subject aterial within the The
set. It ight be copared to a section heading in a book. You ust evaluate your understanding of the subject aterial; then deterine whether you wish to go through the teaching fraes or jup: to the criterion frae. Consider for exaple this gating frae: A circuit is at resonance when the voltage and are in phase. This gating frae iplies the set concerns resonant circuits. The answer "current" is fairly siple to guess. However, unless the student feels he understands resonance he should continue with the teaching fraes even though his response of "current" is the correct answer to the gating frae. This is not a test. You are not copeting with anyone but yourself for speed through this aterial or correct answers to all fraes. If you feel your answers and understanding of the aterial are satisfactory then ove through the text as rapidly as you can. If you answer the teaching fraes or the criterion fraes incorrectly, go back and review the previous frae or fraes until the answer to each frae can be given correctly. It is ost iportant that you answer each question and see why the answer is correct. If your answer does not atch, review the aterial before going on. The first frae in each set is a gating frae. The last frae is a criterion frae. All sets in between are teaching fraes. The answer to the gating frae should not necessarily be evident. If it can be answered, then the student ay skip the teaching fraes and 10
go to the criterion frae designated by a double asterisk (**). If both the criterion and gating fraes are answered correctly, go on to the next set. If the answers to either the gating frae or the criterion frae cannot be given, continue with the teaching fraes in the set. 11
12 SET 1--DEFINITION OF MODULATION 1. The process that occurs when any signal alters or changes soe characteristics of another signal is called odulation 1 -A. 1 -B. Modulation is defined as the alteration of one or ore of the characteristics of one signal by another signal. A signal ay be thought of as any voltage variation. The alteration is called odulation. Thus one signal odulates another signal, when it causes the characteristics of that signal to change. no answer needed When any voltage variation causes the characteristics of soe signal to vary, it is said that the voltage variation the signal and the process is known as odulates odulation 1-C. The process of, involves the altering or changing of one signal by another signal. When one signal a second signal, the of that second signal are by the first signal. odulation odulates characteristics altered or varied or odulated **1-D. Two signals are involved in the odulation process. When one of the signals another signal the process is called odulates odulation
13 SET 2--SIGNALS IN A MODULATION SYSTEM 2. In a odulation syste, the signal being odulated is referred to as the and the signal causing odulation is referred to as the. The signal generated in the odulation process is called the carrier odulating signal odulated signal odulating signal carrier Modulator odulated signal Figure 2t-1. Signals in a odulation syste 2-A. The three coponent signals of iportance in any odulation syste are, (a) the signal causing odulation or odulating signal, (b) the signal being odulated or carrier and, (c) the resultant signal generated fro the odulation process or odulated signal. When the odulating signal odulates the carrier, the process is called. These three signals are shown in Figure 2-1. Notice that two signals are applied to the odulator with one signal out of the odulator. odulation 2-B. A signal generated in the odulation process is known as the and is the result of the odulating signal odulating the odulated signal carrier
14 2-C. **2-D. A device used in the odulation process is a Two signals applied to the odulator are the odulating signal and the. The output signal is called the signal. odulator carrier odulated The three principal signals in a odulation syste are the odulating, and signals. Modulation occurs when the odulating signal the carrier signal. The resultant signal is called the carrier odulated odulate s odulated signal
SET 3--TERMS AND SYMBOLS DEFINING MODULATING SIGNALS 15 3. The signal A(t)cos 4)(0 represents a tie varying signal where A(t) is the as a function of tie, and LIAO is the as a function of tie. When the aplitude is constant and the angle changes at a constant rate, the signal is a signal. The axiu value of a sinusoidal signal is called the aplitude angle sinusoidal aplitude 3-A. A tie varying signal ay be represented by A(t) cos 44t) where A(t) represents the aplitude as a function of tie and 4)(0 represents the angle of the cosine function with respect to tie. The angle ip(t) is generally represented by A siple exaple is the sinusoidal carrier signal cot + 4 (t). A c cos(u) c t + 4)) where the aplitude has a constant value of A, and the angle varies with a constant angular velocity of w. The ter cot + 4 represents the value of the angle at any value of tie, t. The function Accos(cot + 4)) is called a sinusoidal function and is shown in Fig. 3,-1. Although the function ay be described ore accurately as a cosinusoidal function, the ter sinusoidal is used here to indicate any signal that varies as a sine wave or a cosine wave. A C B Figure 3-1,
16 A CB Figure 3-1. 3 -B. Using a cosine function, a sinusoidal function ay be represented by Acos wt where A is the and w is the angular aplitude velocity 3 -C, The aplitude of a sinusoidal signal is the axiu value of the signal. Thus, in Figure 3-1, the aplitude whose sybol is (A, B, C) is a constant. A or B 3 -ll. The ter aplitude ust be distinguished fro instantaneous value. The instantaneous value of a signal is the value at a particular tie, t and ay be found fro the equation 1 A cos w1t1 of the curve in Figure 3-1. On the other hand, the aplitude is the value of the signal and is a constant for the sinusoidal signals of Figure 3-1. axiu 3-E. The instantaneous value of the sinusoidal signal in Figure 3-1 varies whereas the aplitude is a and represents the value of the signal. The instanteneous value of Acos w t in Figure 3-1 at t = t is 1 1 sinusoidally constant axiu
17 3 -F. The ter w is correctly called the angular velocity or soeties called the angular frequency. It is equal to 2-rrf where f is the frequency in hertz. Often ties the angular frequency w is called frequency. Fro here on w and f will both be called frequency. However, it should be reebered that wand f differ by a factor of. The units of w are radians/second. The units of f are Hertz. Since iii(t) is the angle as a function of tie, each ter in the function i(t) ust have units of radians or degrees. 2 11 3-G. To convert fro Hertz to radians/second ultiply by The function of angle 4(t) ust have units of degrees. LIT radians or --;,* 3 H. The axiu value of a sinusoidal signal is called the and should be distinguished fro the instananeous value which is the value of the signal at any instant of Frequency and angular frequency are related by. The units of qi(t) are or degrees. In the signal A(t) cos 00, the sybol A(t) represents the as a function of tie and tp(t) represents the as a function of tie. aplitude tie 2 radians aplitude angle
SET 4--MODULATION TYPES 4. A sinusoidal carrier ay be odulated by two distinctly different types of odulation. These two are a odulation and an odulation. Aplitude odulation is of the for A(t) cos w t where the aplitude is a function of. Angle odulation is of the for A cos 4(t) where the angle, 4(t), varies with 4-A. aplitude angle tie tie The signal A(t) cos Lp(t) represents a tie varying signal where A(t) is the as a function of tie, and LIAO is the as a function of tie. aplitude angle 4-B. A sinusoidal carrier ay be represented by A cos (wct +4)). the aplitude is by soe eans varied withtie, aplitude odulation is obtained. The odulated signal would be of the for A(t) cos (w t + ). If instead 4) is varied with tie, the result is angqe odulation. The angle odulated signal would be of the for A cos (wct + 4) (t)). c 4-C. no answer needed Often ties the aplitude varies with tie while the angle of the cosine function changes at a constant rate of wc. This for of odulation would be odulation. The odulated signal would be of the for aplitude A(t) cos (wct + 4)) 18 4-D. The function A(t) cos (wct +4)) represents a function whose angle and aplitude vary with. The angle changes at a constant rate of tie cac
19 4-E. When the aplitude of the signal reains constant but the angle varies as a function of tie, the signal ay be written as A cos cp(t). A special case of this signal is one in which the angle changes at a constant rate. Then the signal is called a signal. sinusoidal 4-F. Figure 4-1. Sinusoidal 'signal w = constant The function A cos tp(t) represents a function with constant but whose varies with tie. If the angle changes at a constant rate it ay be represented by and is called a signal. An exaple of signal with constant aplitude with an angle as a function of tie is shown in Figure 4-2. In this case the angle varies at other than a constant rate. aplitude angle cot sinusoidal Figure 4-2. A cos (cot +.1)(t))
**4-G. Aplitude odulation is of the for. Angle odulation is of the for. In aplitude odulation, the varies with tie. When the angle varies with tie at other than a constant rate the for of odulation is called odulation. A(t) cos (,) t A cos iii(t)c aplitude angle 20
21 SET 5--AMPLITUDE MODULATION 5. In the process of aplitude odulation (AM), the odulating signal odulates the aplitude of the in proportion to the instantaneous of the odulating signal. The odulation envelope follows the instantaneous values of the carrier value odulating signal a Modulation signal I V 1 b Carrier frequency c Modulated signal d Modulation envelope Figure 5-1. Signals in an aplitude odulation syste
5-A. Review the curves in Figure 5-1. These are the signals in an AM syste. The three signals are the odulating signal, the carrier, and the. When reading these fraes, ake sure you distinguish between odulated and odulating signals. There is a difference. odulated signal 22 5-B. 5-C. 5-D. Look for the relationship between the odulating signal and the odulated signal in Figure 5-1. Notice that when the odulating signal is positive the aplitude of the odulated signal is greater than the aplitude of the carrier. When the odulating signal is zero, the aplitude of the odulated signal and the carrier are equal. Verify these stateents fro the curves in Figure 5-1. Notice that when the odulating signal is negative, the aplitude of the odulated signal is (less, ore) than the aplitude of the carrier. less When a carrier is aplitude odulated by a odulating signal, the aplitude of the odulated signal at any tie varies in proportion to the instantaneous value of the odulating signal. The aplitude of the odulated signal is greater than or less than the aplitude of the carrier, depending on whether the instantaneous value of the odulating signal is positive or negative. For exaple, it can be seen in Figure 5-1 that the aplitude of the odulated signal will be greater than the aplitude of the carrier if the instantaneous value of the odulating signal is (negative, positive). positive In AM systes the aplitude of the odulated signal varies directly with the instantaneous value of the signal. odulating 5-E. If the instantaneous value of the odulating signal is zero, the aplitude of the odulated signal will be (less than, equal to, greater than) the aplitude of the carrier. equal to
23 5-F. The aplitude of the odulated signal is dependent on the value of the odulating signal. instantaneous I I I I a Modulated signal I I t Modulation envelope Figure 5-2. Signals in an aplitude odulation syste 5-G. Refer to Figure 5-2. The odulated signal is shown in Figure 5-2a. The heavy line in Figure 5-2b, that traces the aplitude of the odulated signal is known as the odulation envelope. It can be seen, by coparing the odulating signal and the odulated signal, that the odulation envelope of Figure 5-2b follows the instantaneous value of the odulating signal of Figure 5 -la. Copare the top half of the odulation envelope with the odulating signal. The two are syetrical. no answer needed
24 5-H. 5-I. **5-j. The odulation envelope of Figure 5-2b follows the instantaneous aplitude or peaks of the odulated signal. When a odulating signal aplitude odulates a carrier, the curve that traces the instantaneous aplitude of the odulated signal is called the odulation envelope The odulation envelope follows the instantaneous aplitude of the signal and the instantaneous value of the signal. odulated odulating In an AM syste the aplitude of the carrier is varied in proportion to the instantaneous of the odulating signal. The odulation envelope follows the instantaneous value of the signal. value odulating signal
a5 SET 6--MODULATION INDEX OF AN AM SIGNAL 6. Equation A (1 + Mcos w t) cos wt is the equation for an c signal, wherce M is called the odulation, given by the ratio A/Ac w is the frequency of the signal, and wc is the frequency of the aplitude,odulated index odulating carrier 1116411116411 6-A. 6-B. Figure 6-1. An aplitude-odulated signal In AM, the aplitude of the odulated signal is proportional to the instantaneous value of the signal. odulating If the odulating signal is soe function of tie, f(t), then the aplitude of the odulated signal is A + Mf(t) where M is the proportionality constant. If f(t) has an instantaneous value of A then the aplitude of the odulated signal at that c tie is. If f(t) has a value of -A then the aplitude of the odulated signal is. The sybol M is the proportionality constant and is called the odulation index. A (1 + M) Acc(1 - M)
26 6-C. Consider the case where the odulating signal is A cos (A) t and the carrier is A c cos co t. In this special case where both the carrier and the odulating signals are sinusoidal, the odulated signal is given by where M Ac(1 + M cos (A) A Ac t) cos 0) ct, In this equation the aplitude, A(t), is. The frequency is A (1 + M cos c,) t) c c 6 -D. The ter M is referred to as the odulation index or when given in percent, as the percent odulation. M is the ratio of A to A c 6-E. The percent odulation of an AM syste with a odulation index of 0. 50 would be 50% 6-F. 6-G. The ter M in the equation A (1 + M cos w t) cos w t c c is referred to as the. When M, the odulating index, has a value of one, the aplitude of the odulated signal varies between 2Ac and. In this case, the carrier is said to be 100% odulated. odulation index zero When M has a value of 0. 5 the aplitude of the odulated signal A (1 + M cos (,) t) cos wct varies between and TY), 5 The carrier is then aid sa to be % odulated. 0. 5 Ac 50
6 -H. The equation for an AM signal when a carrier of angular frequency wcis odulated by a single frequency odulating signal, w, is A ( c ) cos w t. c 1 + M cos w t 6 -I. The equation Ac(1 + M cos wt) cos Oct is the equation for an_am signal. The sybol M is called the The frequency of the odulating signal is. The frequency of the carrier is odulation index ca (t) c 27 J. When a single frequency odulating signal aplitude odulates a carrier, the AM signal ay be represented by The ter M is known as the, If the value for M is given in percent, it is known as the odulation. When the carrier is 40% odulated, the odulation index is. The odulation index is the ratio of the aplitude of the odulating signal to the aplitude of the signal. Ac(1 + M cos wt) cos w t odulation index percent 0. 4 carrier
28 SET 7--DETERMINING THE MODULATION INDEX 7. The odulation index can be calculated fro the AM signal by the forula A - B A + B where A is the (axiu, iniu) aplitude of the AM signal and B is the (axiu, iniu) aplitude of the AM signal. Values of the odulation index range fro zero to When the odulation envelope does not follow the instantaneous value of the odulating signal, the AM signal is said to be odulated. axiu iniu one over- 4 3 1111111111611-61 -2-3 - 4-5 11,14111111, 7-A. Figure 7-1. An AM Signal The percent odulation can be calculated fro the AM signal by the forula A - B A + B where A is the axiu aplitude and 13 is the iniu aplitude of the AM signal, For exaple, the AM signal of Figure 7-1 whose axiu aplitude is 5 volts and whose iniu aplitude is 3 volts, would have a odulation index of
29 M = 5-3 =.25 5 + 3 Thus the AM signal of Figure 7-1 is % odulated. 25 7-B. If A is the axiu aplitude of an AM signal and B is the iniu aplitude, the odulation index ay be calculated fro the forula A - B A +B - 5-10, -15 20 15 10 5-0 - 5-10 - 15-20 25 20 15 10 5 0-5 - 10-15 - 20-25 c. t Figure 7-2.
30 7-C. The odulation index of the AM signal in Figure 7-2a is 0. 5 7-D. The odulation index of the AM signal in Figure 7-2b is 1. 0 7-E. Values of odulation index range fro zero to one. If the AM signal is copletely interrupted for a period of tie as shown in Figure 7-2c, the odulation envelope no longer follows the instantaneous value of the signal, and the AM signal is said to be overodulated. Overodulation causes distortion in the odulation syste and thus should norally be avoided. A value of the odulation index cannot be found fro A - B A + B since the odulation index is essentially greater than unity. In this case it is sufficient to say that the AM signal is overodulated. odulating 7-F. When the odulation index is less than unity, the odulation envelope follows the instantaneous value of the signal. In this case the odulating signal can be recovered fro the AM signal. However, during over-odulation, the envelope no longer follows the instantaneous of the odulating signal. The odulating signal cannot be copletely recovered. Thus distortion results whenever the AM signal is odulating odulation value over -odulated
31 **7-G. For the undistorted AM systes, the value of the odulation index has a range of. The odulation index ay be calculated fro the AM signal by the forula A - B A + B where A and B are the and aplitudes, re - spectively, of the AM signal. When the odulation envelope does not correspond to the odulating signal, the AM signal is said to be and distortion results. 0-1 axiu iniu over -odulated
32 SET 8--DETERMINING THE FREQUENCY RESPONSES OF SIGNALS 8. When it is required to find the frequency spectru of a odulated signal written as the product or power of sinusoidal functions, it is necessary to expand the function, using trigonoetric identities, into a Fourier series of and cosine ters. 8-A. sine Often ties a odulated signal is written as the product or power of sinusoidal functions. The aplitude odulated (AM signal) [Ac ( 1 + M cos w t) cos wet] is an exaple of non-fourier representation since it involves products of sinusoidal signals. To find the frequency spectru of this aplitude odulated signal, we ust expand the expression into a series of sine and cosine ters such as A cos w 1 t + B cos w 2 t + C sin wit +.. The expansion involves the use of appropriate trigonoetric identities. Fro the expressions below select those that require further expansion to find the frequency spectru of each signal. (Reeber each ter in the final expansion ust be in the for Al cos wt or B sin wt. If the for of the expression is in powers or products, it ust be expanded.) a) cos w,i t sin w 1 t b) cos w t + sin wit, c) cos to.), -w )t -F cos d) cos 2 w i 2 1 e) cos,-, wi 1 t + sin2 w t f) cos (w/2) t 2 g) cos (r sin wt) a, d, e, g w2t
33 8-B. As an exaple, consider the function in (a) above and its trigonoetric expansion into a series of cosine and sine ters. cos colt sin wit = 1/2 sin 2 wit This identity indicates that if a cosine ter and sine ter of identical frequency, w, are ultiplied, the frequency of the product will be 20) 8-C. 8-D. A wave analyzer is a frequency selective volteter that will easure a voltage only at the frequency for which it is tuned. By tuning the wave analyzer over a range of frequencies, it will easure the frequency coponents and respective aplitude of a given signal. If a wave analyzer or frequency selective volteter were used to easure the frequency spectru of cos w 1 t sin w t it would indicate a 1, voltage at a frequency of and nothing at w. 1 2,77 1 As another exaple, consider the product of two different frequencies such as cos w t cos w t. Expand this function 1 into a series of cosine ters using the trigonoetric identity cos a cos b = 1/2 cos (a-b) + 1/2 cos (a+b) to get cos wit cos w 2 t = 1/2 cos ( )t + 1/2 cos ( )t 2 (any order) wl*w2 8-E. If a wave analyzer were to easure the frequency spectru of cos colt cos w 2 t it would indicate a voltage only at a frequency of wi - wz and
34 **8-F. The frequency spectru of any odulated signal can be found by expanding the function using identities into a Fourier of cosine and sine ters. trigonoetric series
35 SET 9-- FREQUENCY COMPONENTS IN AN AM SIGNAL 9. The frequency coponents in increasing order present in an aplitude odulated signal are C.) 0.>" 9-A. c c w An aplitude odulated signal ay be expressed as A c (1 + M cos w t) cos w t and the sinu- where the frequency of the carrier is soidal odulating signal is c OJ c MA cos w c 9 - B. If it is desired to find the frequency coponents of any odulated signal it is necessary to expand the expression into a of cosine and sine ters using trigonoetric identities. The frequency of the carrier will be assued to be uch greater than the odulating signal frequency. 9-C. (o) >>(,) ) c series Expand the aplitude odulated expression Ac (1 + M cos w t) cos wct. Ac ( 1 + M cos w t) cos wct = A cos wct + c t MA c cos 0.)ct cos w t 9-D. Further expansion of MA cos wt cos w t requires the use of trigonoetric identity c c cos a cos b = 1/2 [cos (a+b) + cos (a-b)].
36 In the expansion, the angle a of the trigonoetric identity would be and the angle b would be wct t (any order) 9 -E, The expansion of MAc cos wct cos wt would be 1/2 MAc cos ( )t + 1/2 MA cos ( )t. c coc - (any order) Figure 9-1. Frequency Frequency coponents of an aplitude odulated signal 9 -F. Thus the expression for the aplitude odulated signal has frequency coponents of increasing order of and. These frequencies coponents are shown in Figure 9-4 with the respective aplitudes. coc c + (A) 9-G. For each frequency in the odulating signal two new frequencies are generated in the aplitude odulation process. One new frequency is equal to the carrier frequency plus the frequency and the other is the carrier frequency inus the odulating frequency. If the odulating signal frequency is w, the new frequencies generated would be
and 37 odulating 0.) (A) +(A) (any order) 9-H. If a wave analyzer were used to easure the frequency spectru of an AM signal it would indicate voltages at frequencies of increasing order of, and 9-I. a) In practice, the odulating signal ay correspond to usic or speech which is ade up of a large nuber of frequencies. In this case it is coon to refer to the group of odulating frequencies as the baseband, and the two new frequency groups as sidebands. Thus a odulating signal causes sidebands on either side of the frequency as shown in Figure 9-2. carrier Frequency spectru of odulating signal lower sideband carrier frequency upper sideband bandwidth Figure 9-2. Frequency spectru of a odulating signal and the corresponding AM signal
9 - J. The bandwidth of a signal is the difference between the highest frequency coponent and the lowest frequency coponent in the signal. The band wtith of the odulated signal will be (equal to, twice, proportional to) the highest frequency contained in the odulating signal. twice 38 9-K. If the highest frequency coponent of a odulating signal is 5000 Hz and the carrier frequency is 100 khz, the bandwidth of the odulated signal would be. In the frequency spectru, the odulated signal would occupy the frequencies fro to 10 khz 95 khz 105 khz ** 9 - L. When the odulating signal contains a group of frequency coponents it is called the signal. The new frequencies generated about the carrier frequency in the odulation process are called. When the odulating signal is a single frequency, w, the frequency coponents in the aplitude odulated sigril in increasing order will be and baseband sidebands w -w wc +w
39 SET 10--THE SQUARE-LAW MODULATOR 10. Consider the su of the odulating signal of frequency 0.) and carrier of frequency (.0 applied to a non-ideal square- 171 law odulator with characteristics E out = Ao + Al Ein + A2Ein 2 The coponent frequencies of the output signal in increasing order would be and e. =A cos 0.) t + A cos co t in c c 10-A. Figure 10-1. Non-Ideal Square-Law Modulator = Ao + Al ein + A2 ein 2 out Input-Output relationships for a square-law odulator One of the siplest and ost coon type of odulators is the square -law odulator in which the output signal and the input signal are related as shown in Figure 10-1. eout = Ao + Al ein + A2 ein 2 The output will contain a DC ter, a ultiple of the input signal and a ultiple of the of the input signal. The ter involving the square of the input signal is called the second order ter. Likewise the ter involving the input signal to the first power is called the order ter. square first
40 10-B. If the input signal is the su of a odulating signal of frequency w and carrier of frequency w, it ay be represented by c A c cos w t + A cos w t c 10-C. Fro the first order ter A e. it is easy to see that the 1 in odulating frequency, w, and tie carrier frequency, w, will be present in the ouillut signal. To find the other c frequency coponents in the output signal, it is necessary to expand a second order expression of the for (a + b)2 = a 2 + 2ab + b 2. If a and b represent the odulating signal and carrier respectively, then the second order ter in the output signal would be A 2 cos 2 w t +.2A A cos w t cos w t+ A 2 cos 2 w t C M Table 10-1. (1) cos 2 a = 1/2 + 1/2 cos 2a (2) cos a cos b = 1/2 cos (a-b) + 1/2 cos (a+b) Trigonoetric identities used in the expansion of the output signal of a square-law odulation 10-D. Study the trigonoetric identities in Table 1021. 'ping trigonoetric identity (1) the expansion of A cos w t would give frequencies of increasing order o and DC 2wc 10-E. Again using trigonoetric identity (1) fro Table 10-1, the expansion of A2 cos2 w t would give frequencies of in. creasing order 01 and DC 2w
2w w 41 10-F. 10-G. Using trigonoetric identity (2) fro Table 10-1, the expansion of 2A c A cos w ct cos w t would give frequencies of increasing order of and (A) c c + (A) The second order ter generates new frequencies at the output that were not present in the input signal. These frequencies in increasing order would be and DC - c + co 2wc 10-H. 10-I. The first order ter does not add any new frequency coponents to the output signal. If w and w are frequency coponents of the input signal then these frequencies will also appear in the signal. output The frequency coponents in the output signal would be those due to the first-order ter and second-order ter. In increasing order these would be. The frequency spectru of the output signal of a square-law odulator is shown in Figure 10-2. DC wc +w 2wc
42 10-J. Figure 10-2. Frequency coponents in the output signal of a square-law odulator The frequency coponents of the output signal of a non-ideal square-law odulator are the first and haronics of the and carrier frequency, and the first haronic of the odulating signal centered about the carrier. These last two frequencies w - w and w + w are known as and represent sus and diffgrenas of the two input frequencies. second odulating sidebands 10-K. The sideband frequencies, w - w and w + w and the carrier frequency w, are the three c frilequenccy coponents in an AM signal. Thesce frequency coponents are also in the odulated signal at the output of the square-law odulator along with soe other unnecessary frequency coponents. This points out that the non-ideal square-law odulator does produce an AM signal since it produces the carrier and frequencies. The other frequency coponents generated are usually discarded or rejected by filtering. sideband
**10-L. An AM signal is produced using a non-ideal square-law odulator when the su of the odulating and signal is applied to the input. Other unnecessary frequency coponents generated include the haronics of the odulating and frequencies. The output of the nonideal square-law odulator is related to the input signal by the equation carrier second carrier A + Al e. 2 + A e 2 in 43
44 SET 11--THE PRODUCT MODULATOR 11. In a product odulator, the coponent frequencies present at the output when a sinusoidal odulating frequency w and carrier frequency wc are applied to the input and and c c + -" (A) Modulating signal) carrier product odulator odulated signal Figure 11-1. eo = (Modulating signal)(carrier) Input and output relations of a product, odulato r 11-A. A product odulator is a device having an output that is proportional to the product of two input signals. If the two input signals are the odulating signal and the carrier, then the output will be the product of the carrier and the 11-B. odulating signal If one input signal to a product odulator is A and another is B, the output would be AB 11-C. If the odulating signal is represented by A cos w t and the carrier by A c cos w t then the output signal woufla be the product of these two c signals represented by A A cos w t cos wt c c 11 -D. A odulator that ultiplies two input signals is known as a odulator. The output of such a odulator is proportional to the of the signals. product product input
11 -E. Since the frequency coponents are of interest at the of the product odulator, the product of cosine ters ust be expanded using trigonoetric identities. Fro the identity cos a cos b = 1/2(cos (a+b) + cos (a-b)) the output frequency coponents of the product odulator in which wc and w are the input frequencies are and 45 11-F. output - coc + If two signals with frequencies of 100 Hz and 1000 Hz are applied to the input of a product odulator, the output signal will contain frequency coponents of and 900 Hz 1100 Hz 11-G. If one input signal to a product odulator has baseband fre quencies fro 300 to 3000 Hz and the other reains constant at 10kHz, the output frequency coponents will vary fro to and to 7000 Hz to 9700 Hz 10300 Hz to 13000 Hz **11 -H. New frequencies generated in a product odulator include sus and of the input signal frequencies. The carrier frequency (does, does not) appear in the output signal. The output of the product odulator can be found by ultiplying the signals. differences does not input
46 SET 12--MODULATION IN A GENERAL NON-LINEAR DEVICE 12. In general when the su of the odulating frequency and the carrier frequency is applied to any non-linear device, the frequency coponents at the output will be haronics of the odulating and frequencies and haronics of the odulating frequency centered about each haronic of the frequency. carrier carrier Ac cos co t + A cos 0,) t in c a. block diagra non-linear device e = A + A e + out o 1 in 2 2 3 A e. + A e. 2 in 3 in -1.0 -.8 -.6 -.4 -.2-1.10 e. in - 2-3 b. Characteristics of device Figure 12-1. Input and output relations for a general nonlinear device acting as a odulator
12-A. Many electronic devices have non-linear characteristics. Electron tubes or transistors for exaple have non-linear operating regions. For this set, consider any of these nonlinear devices used as a odulator. Figure 12-1 shows a non-linear device whose input signal is the su of the odulating frequency and the carrier frequency. e. Ac cos wct + A cos (,) t in In general, the output of any non-linear device ay be represented as the infinite su of ultiples of powers of the input signal. For exaple, the output eout ay be related to the input, ein, by the series eout = Ao + Al ein + A2 ein2 + A3 ein 3 + A4 ein4 +. The tenth-order ter in the expression of eout would be 10 Al0 ein 47 12-B. Using a suation sign the output ay be represented by oo n = 0 An ei: 12-C. An exaple of a non-linear device previously considered is the non-ideal square-law odulator in which the output signal was represented by 2 Ao + Al ein + A2 ein 12-D. If e in is the su of the odulating frequency and the carrier frequency it ay be represented by e = in A c cos wct + A cos co t
48 12-E. Substitute the su of the odulating frequency and carrier frequency into the series expression for e The output out. signal will then be eout = Ao + Al (A c cos w t + A cos w t) + A3(A c cos w t + A cos w t) 3 +.. A2(A c cos wct + A cos w t) 12-F. 12-G. In order to see what the frequency response will be, assue for the present the non-linearity is such that the fourth and all higher order ters are negligibly sall. Then the output signal is that represented in 12-C. For exaple, if e 0.1 e. + e. 2 + e. 3 the characteristic curve would look like Figure 12-b. Before the frequency coponents can be deterined it will be necessary to expand the expression fore out into a of cosine and sine ters of the for A cos wt or B sin wt. series Consider the frequency coponents contributed by each of the ters in e. First, the ter Al ein will contribute frequency coponents out in increasing order of and. These frequency coponents are shown in Figure 12-2. 0.) c A c A 0 Figure 12-2. Frequency--,- Frequency coponents in the output signal due to the first order ter A e 1 in
49 12-H. The second-order ter A2 e in 2 ay be represented as A2(A c cos wct + Arn cos w t) A 2(Ac2 cost wct + ZA c A cos wct cos w t + At cost w t ( 1 ) cos 2 a = 1/2 + 1/2 cos 2a (2) cos a cos b = 1/2 cos (a + b) + 1/2 cos (a - b) Table 12-1. Trigonoetric identities used in the expansion of the second-order ter 12-I. Recall fro the square-law odulator or fro trigonoetric identity (1) of Table 12-1, that the ter cost wct will contribute frequency coponents of and DC 2w 12-J. The ter cos we t cos wt will contribute frequencies of and as can be seen fro trigonoetric identity (2) of Table 12-1. 12-K. CA.) (A) c O.) The ter cos w t will contribute frequency coponents of and DC Zw 12-L. Thus it can be seen that the second-order ter will contribute frequency coponents in increasing order of and. These coponents are shown in Figure 12-3.
DC 20.) co - co wc + 20) 50 2w We -W 0.) +co c c Frequency Figure 12-3. Frequency coponents due to second-order ter 12-M. Fro Figure 12-3, the second-order ter generates the (first, second) haronic of the odulating signal and carrier, plus the first haronic of the odulating signal centered about the first haronic of the frequency. second carrier 12-N. Consider now the effect of the third-order ter A e. It involves the expansion of the for (a + b)3 = a 3 + 3a 2 b + 3ab 2 + b 3. A3(A c cos w t + A cos 0) t)3 = A3(A c 3 cos 3 co t + 3A 2 A cos 2 w t cos co t + c c c + A 3 cos w t). 3A A 2 cos 0) t cost 0) t c (1) cos 3a = 3/4 cos a + 1/4 cos 3a (2) cos 2a cos b = 1/4 [(cos (2a+b) + cos (2a-b) +1/2 cos b)] Table 12-2. Trigonoetric identities used in the expansion of the third-order ter in 3
12-0. Study the identities in Table 12-2. Fro identity (1) expan- sion of A cos 3w t would give frequencies of increasing c order of c and wc 3wc 12-P. Also fro identity (1) expansion of A 3 cos 3 w t would give frequencies of increasing order of and 51 3w 12-Q. Fro identity (2) of Table 12-2, the ter 3A 2A cos w t would give new frequencies of increasing order of, and cos 2 wct 2w -0) c +w 12-R. The ter 3A A 2 cos w t cost w t would give new frequen- cies of increasing orderof, and 12-S. C wc c + 20) Fro all of these identities the third-order ter generates new frequencies in increasing order of, and These frequency coponents are shown in Figure 12-4. 3w wc - wc + 2w c c.w -0) 20) + 2w 3coc
52 12-T. Figure 12-4. w c c + w we -wt Frequency*- c C w -2w 2w +w Frequency coponents due to third-order ter Review the frequency coponents due to the second- and third-order ters, Figures 12-3 and 12-4. The secondorder ter generated the second haronic of the odulating signal and the carrier frequency plus the first haronic of the odulating signal centered about the first haronic of the carrier. Now look at Figure 12-4. The third-order ter generated the third haronic of the odulating signal and frequency plus the second haronic of the odulating signal centered about the haronic of the carrier and the first haronic of the odulating signal centered about the haronic of the carrier. carrier first second 12 U. Other higher order ters would have a siilar effect. Figure 12-5 shows the frequency coponents generated by the fourth-order ter. As an exercise expand the fourthorder ter Aa e. 4 using the trigonoetric identities in Table 12-3. V.elay your frequency coponents with those shown in Figure 12-5. (1) cos4a = 3/8 + 1/2 cos 2a + 1/8 cos 4a (2) cos 3a cos b = 3/8 [cos (a+b)] + cos (a-b) + 1/8 [cos (3a+b) + cos (3a-b)} (3) cost a cos2b = [1/4 + 1/4 cos 2b + 1/4 cos 2a + 1/8 cos (2a-2b) - 1/8 cos (2a+2b)] Table 12-3. Trigonoetric identities.
53 a) c 4o) \\0) \\2wc w -3w 2w -2w c Figure 12-5. +3w C 3w +0) C- 2wc +2w 41wc Frequency' Frequency coponents due to fourth-order ter 12-V. Fro Figure 12-5 we see that the fourth-order ter would generate the haronic of the odulating signal and carrier, plus 3(*) + w, 240 -F 20), and as well as c other frequencies c coon to lower order ters. fourth 0) + c 12-W. The ost iportant point to reeber is that the aplitude odulation process can take place only in a (linear, non-linear) device. Most odulators are designed so that haronics of the odulating signal about the carrier do not exist. However, it is iportant to reeber that whenever a signal is passed through a non-linear device, new are generated that were not originally in the signal. non-linear frequencies **12-X. When any signal is applied to a non-linear device, the frequency coponents at the output of the device will be all in -. put frequencies as well as sus and differences of all haronics of the signal. input
54 SET 13--BALANCED MODULATORS 13. The advantage of using a balanced odulator is that certain unwanted or unnecessary frequency coponents are cancelled in the odulated signal. Aong the cancelled frequency coponents are the (odulating, carrier) frequency and (even, odd) haronics of the odulating signal centered about haronics of the carrier frequency. carrier even output signal Figure 13-1. Balanced Modulator 13-A. A balanced odulator is shown in Figure 13-1. The odulating signal is represented by e and the carrier by e. With the polarities as shown the su of two input signafs, ec + e, is applied to one non-linear device. The difference, e - e is applied to another identical device. The circuit is connected so that the balanced odulator output signal is the difference of the outputs of the two non-linear devices. Fro Figure 13-1, the output signal is e tout - e out 13-B. As was the case with the general non-linear device, consider the non-linearity such that only the third-order ters need be included in the output expression. This eans that the output of each non-linear device ay be written as,
55 elout = A (e - e ) + A 1 c 2 e2out = A (e + e) + 1 c (e - e )2 c 3 A3 (ec - e)2 A (e + e 2 c A3 (ec + e )3 13-C. Use this expansion in the next frae (a + b)3 + a 3 + 3a 2 b + 3ab 2 + b 3 The expansion of elout is then elout Alec -Ae +Ae + A2 e + A e l 2c 2 3 c 3 + 3A 3ecern2 13-D. 2A e e 2 c2 3A3ec e A e 3 The expansion of e2out fro 13-B is, e2out = Alec +A 1 e +A 2 e c 2 + 2A2 ece + + A3 e c 3 + + 3A3 e c e 2 + 2 A2e 2 e 3A 3 e c 3 A3 e
56 13-E. The difference of e2out and el out is the balanced odulator. e2out = Al ec + Al e + A2 e c 2 + 2A2 e c e + A2 e A3ec3 2e 3 + 3A3 e + 3A3 e e 2 + A c c 3 e e 1 out = Alec - Al e + A 2 e c 2-2A 2 ece + A2 e2 + A3 e c 3 e 3A3ec2e + 3A3e c e 2 - A3 e 3 ee + + 2A e = 2A e + tout - e lout 1 4A 2 c 3 2 6A3 ec e 13-F. If ec A c cos w t, and e = A cos w t, then the frequency coponents in tie balanced owulator output would coe fro these ters: 3 2Ale would have frequency coponents of w. 4A2ece would have frequency coponents of and C 6A3ec2e would have frequency coponents of, and c + + w any order 2A 3 e 3 would have frequency coponents of and and 3w (If these frequency coponents are not evident, then review the trigonoetric identities in Set 10 and Set 12. )