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The oretical Study of Pulse-Frequency Modulation.

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    1949 199PROCEEDINGS OF THE I.R.E. 1277 spikes with a discriminator of inadequate range impairs the tendency of positive and negative loop areas of the curve to cancel and thus augments the magnitude of low harmonics. The effect is somewhat analogous to the detection of a high-frequency modulated wave with a rectifier. It appears, then, that much of the nonlinear distortion observed in FM systems with multipath propagation is attributable to nonlinearity in the discriminator, especially in the case of wide-band systems where very large spikes may occur in the wave shape. It should be noted that no increase in bandwidth of the rf amplifier is required, since this portion of the re- ceiver may be assumed to be linear and responsive to each signal individually. It is assumed also that limiter action is unimpaired, even when the carrier amplitude fades to a minimum. It may be further noted that the foregoing analysis tends to indicate a method for reducing common-channel interference between two FM signals which are of nearly equal strength and which may be similarly or differently modulated. In case of dissimilar modulation, it is, of course, necessary to assure that the same signal always remains the stronger of several arriving simultaneously at the receiver. Theoretical Study of Pulse-Frequency Modulation* ARNOLD E. ROSSt, SENIOR MEMBER, IRE Summary-ln this paper, we study the pulse-frequency modula- tion method in pulse communication. We do not restrict ourselves to the case when the number of samples per period of our signal, as well as per unit time, is very large. Hence, the accuracy in reconstruction of the signal will depend not only on the number of sampling points per period, but also upon their distribution. We shall assume a fixed average pulse frequency (sampling rate), and will seek the maxial range of the signal frequency for which the corresponding periodic signal may be transmitted with acceptable accuracy. Thus, we are interested in a kind of threshold problem. It is our aim to consider pulse-frequency modulation independently of the circuits used to realize it, inasfar as this is consistent with lucidity of presentation. In Sections 12, 15, 16, and 17, we study in detail the behavior of sampling in case the relative frequency v (the ratio of the signal frequency to the pulse frequency) falls into the ranges (0.4855, 0.51625), (0.33599, 0.33863), (0.6450, 0.6626), and (0.402645, 0.403030), respectively. This information supplies the reason for the limitation of the useful range of the relative frequency spectrum. Because of the extreme nonlinearity of pulse-frequency modulation, we cannot deduce the behavior of the general periodic signal from the behavior of its sinusoidal components, but we have to apply methods described in this paper directly to the particular wave form studied. This is illustrated by the example in the Appendix. INTRODUCTION 1. Pulse Communication rf HE FUNDAMENTAL idea in pulse communication is the transmission of samples of a continuous signal in such a way that, from these discrete samples, the original signal can be reconstructed with accuracy sufficient for the recognition of the message contained therein. * Decimal classification: R148.6. Original manuscript received by the Institute, May 18, 1948; revised manuscript received, April 21, 1949. This paper is based on work done for the Office of Naval Research under Contract No. N60ri-187 with the Stromberg-Carlson Company. Presented, 1948 IRE National Convention, New York, N. Y., March 24, 1948. t University of Notre Dame, Notre Dame, Ind. The signal may be sampled by varying the height of the pulses in an equally spaced train of pulses, as shown in Fig. 1. This manner of sampling is spoken of as "pulse-amplitude modulation." nn n n nn %. t, t. Fig. 1-The modulated sawtooth and the corresponding pulse train. The distance between successive pulse positions t4+j and t, is a linear function of the value of the signal at ti+,. The original message may be incorporated into the pulse train also by varying, in accordance with the signal, the width of the individual pulses, or by varying the distance of the pulses from their original positions in the equally spaced pulse train. These two schemes are called "pulse-width modulation" and "pulse-position modulation" (with fixed reference). 2. Pulse-Frequency Modulation In the present paper we propose to discuss still another type of pulse modulation in which the information is transmitted through the variation of the distance between the pulses, and not in the variation of the distance of the pulses from a fixed reference (see Section 1). This type of modulation is called "pulse-frequency modulation" (pulse-position modulation without fixed reference). It may be achieved by employing a sawtooth wave in the manner shown in Fig. 1. 1278 PROCEEDINGS OF THE I.R.E. November The positions of the pulses (the sampling points) coin- nal of frequency equal to 2/3 of the pulse frequency fp cide with those of the vertical edges of the modulated was transmitted, the receiver also picked up a signal of sawtooth wave whose peaks ride the curve of the modu- frequency 1/3 fp. Far from being a defect in coding or lating signal. Thus the information (in our case, a simple decoding circuits, this proved to be a peculiarity of the harmonic wave) is contained in the fluctuation of the pulse-frequency modulation, and similar subharmonics distances between the pulses.' were predicted and observed for other frequencies. In developing the first formal version of the theory of pulse-frequency modulation we need not go into a dis- THE FORMAL MATHEMATICAL THEORY cussion of the circuits employed, since the geometrical picture in Fig. 1 furnishes us with an accurate description of the performance of such circuits. We will therefore use the geometric representation in Fig. 1 as the starting point of our theory. 5. Preliminary Remarks We will assume a fixed unmodulated pulse rate fp,. We will study the response of the circuit to a wide range of signal frequencies, and will seek the maximal range of the signal frequency for which the corresponding peri- 3. Nonlinearity of Pulse-Frequency Modulation Because of the extreme nonlinearity of pulse-frequency modulation, the mathematical theory of this type of modulation cannot be constructed in the usual manner, with the harmonic analysis of the modulated pulse train as the starting point. The essential features and the type of regularity characteristic of this apparently random method of sampling begin to appear when we realize that these sampling points are iterations of a one-to-one continuous transformation (induced by the method of sampling) of the interval 0 to 27r/w (the pe- riod of our signal). [See Section 9. ] A study of the prop- erties of such transformations2 yields information which, when applied to the particular case of the transformation of the type 4{X(t) in Section 9, yields information regarding the position of sampling points at once. It is surprising that, after the equilibrium is reached, the positions of the sampling points do not depend upon the starting point, but only upon the ratio f,/fp of the frequencyf. of the signal to the unmodulated pulse repetition rate f,, and upon the amplitude of the signal. odic signal may be transmitted with acceptable ac- curacy. The troublesome part of the spectrum, namely, the principal intervals of stability (0.4855, 0.51625), (0.33599, 0.33863), (0.6450, 0.6626), and (0.402645, 0.403030), is studied in detail in Sections 12, 15, 16, and 17. The study of response will consist of a comparison be- tween a periodic signal as it enters the coder in the transmitter, and the decoded version at the output of the receiver.' This comparison will be made upon the basis of the intrinsic properties of the given type of modulation. The degree of faithfulness in the reproduced signal, as deduced on this basis, will represent the best theoretical performance of this particular type of modulation. At the receiver, the process of extracting the informa- tion (decoding) contained in the train of pulses, gener- ated by the sawtooth modulated as above, consists of reconstruction of the sawtooth, replacement of the sawtooth by a step function approximating the signal curve, and finally in averaging this step function. The discontinuities of this step-function approxima- 4. The Principal Aim of This Study The principal aim in this paper is to study the intrin- sic properties of pulse-frequency modulation. The whole process of coding (incorporating information into the pulse train) and decoding (extracting this information at the receiver end) in this type of modulation is achieved by fairly complex configurations of nonlinear tion occur at the positions of the vertical edges of the sawtooth and the peak value of each tooth persists until the next discontinuity. Thus, the effect is that of approximating the signal curve by selecting the points of intersection with the sawtooth edges, i.e., the values of the signal at the sampling points, and constructing the step function upon the basis of this selection (see Fig. 2). circuits whose behavior cannot always be simply ana- lyzed. Many of the behavior characteristics of pulse- frequency modulation are quite unexpected. It becomes desirable, therefore, even in the study of design, to be able to distinguish those features of the behavior which belong strictly to the type of modulation employed, from those which may be due to the defects of a special circuit pattern employed to achieve this type of modu- lation. To illustrate this, we may mention the phenomenon of the occurrence of "subharmonics." Thus, when a sig- S"tRreoIpmoTbrheitrsgo-ntCyaPpruellssoeofnMpoRudelupsloearttcioobmnym,u"HnaDirecoacltedimoGbnoelrdsb2ce7h,regm1,9e4n4ow,awasnatsuutnghpgeuebsBltuiersdehaeiudn of Standards, Washington, D. C. ' A. E. Ross, Bull. Amer. Math. Soc., vol. 53, p. 287; 1947 Fig. 2-The step function approximation to the signal. I See Section 19. 1949 Ross: Pulse- Frequency Modulation 1279 It will be shown that the accuracy in the reconstruction of the signal depends not only upon the number of sampling points per period of the signal, but also upon their distribution. We will proceed with a detailed discussion of our problem. 6. The Sawtooth Wave The ordinary sawtooth wave is the graph of a function and unless- sin w(tk + P) = sin wt& wP = 2nir. 7r Xr tk - tk1 = N ,- while tk - gk'-I Co LO n, N, gk integers. (k = 1, 2, ) - - f(t)-mt qqT) qT < <(q + )T m m m q = O, 1, Here, T is the height of each tooth and m is the slope of each edge. This function f(t) is periodic with the period Tlm, and is continuous except at T in where it jumps by the amount T. Such a function may be represented by a Fourier series. That is, f(t) = aO + F°°, (ak cos kw,t + bk sin kw,t), k-1 w,, = 2,rm T 7. Modulated Sawtooth Wave If, in the definition of f(t) in paragraph six, we do not use the straight line y = T, but the curve y = T + AT sin ct, oscillating with amplitude AT and frequency w, as the locus upon which the peaks of the sawtooth wave lie, then we obtain a new function f(t) which will usually not be periodic, and whose graph (see Fig. 2) may be called a modulated sawtooth wave. The sine curve 8. The Points of Discontinuity t, off(t) The function whose graph is the modulated sawtooth wave is not necessarily periodic, i.e., it is not necessarily the resultant of a number of simple harmonic oscillations whose frequencies are integral multiples of a fundamental frequency. One may then ask if this function f(t) can be thought of as the resultant of sinusoidal oscillations of various frequencies not necessarily simply related to one an- other, i.e., if f(t) is not periodic, is it "almost periodic"? It seems that the most natural approach to the determination of the behavior of f(t) is through the study of the distribution of the points of discontinuity, tk of f(t), when plotted on a circle of radius 1/ and circumference 27r/co, the period of the modulating signal. If points tk, t. lie near one another on this circle, then f(tk) and f(t,) are nearly equal, and if tk and t. lead to the same point of this circle, then f(tk) =f(t.). 9. Plotting of tk on the c-l- Circle Plotting t, upon a circle of radius 1/w consists in measuring off along its circumference the distance tk in a counterclockwise direction from an arbitrary reference point. Thus, two values tk and t,, differing by a multiple of 2r/w, will be represented by the same point on the circumference of this circle (see Fig. 3). S = y -T = AT sin wt may be called the modulating signal. This new function f(t), determined by the modulated sawtooth, may be characterized as follows: f(t) = m(t - to); to < t < t1; T + AT sin wti tl= to+ in f(t) = m(t - t); ti < t 1- t2; (1) T + AT sin Wt2 t2 = tI + m Thus, the peaks are always considered as part of the graph. We notice that if f(l) is periodic of period P, i.e., if f(t + P) = f(t) Lp Fig. 3-The c4--circle. In view of the relations in (1), the two successive points of discontinutiy, to and t1, of the modulated signal I.e., for example: to % -." T l~~~~~-1sr.- 7 for every t, then tk+P, as well as th, is a point of discontinuity. Hence 1280 PROCEEDINGS OF THE I.R.E. Novemtber are connected by the formula T AT to = i - --- sinwt1. (2) mm The function A T AT A 1(I iC ) =t - -sin cot m- m (3) is a single-valued steadily increasing function of t as long as its slope is positive. That is, as long as ATwo - <1. (4) m This has very simple physical significance regarding the relative positions of sampling points on the oscilloscope, when the full sweep corresponds to the period of our signal. Inequality (4) may also be written in the following form: AT Ifp _~ < - _ ( (5) T 2r f. wherefp = m/T is the unmodulated sawtooth (pulse) frequency. This restricts the value of the ratio of the signal voltage AT to the voltage height of the unmodulated sawtooth. For T= 100 v, the following table gives the approxi- mate value of an upper bound B (measured in volts) of AT forflf/f.=1, 2, 3. fp 123 fe B 16 32 48 (Volts) AT 5 B. tl = Af(tO), t2 = Xf 2(to), * . (7) of to obtained by using the powers5 of the transformation 4/f(t). The problem of the distribution of the discon- tinuities of the sawtooth is therefore reduced to the study of the behavior of the chain (7) of images of the one-to-one continuous transformation {Af(t) of a circle into itself. A detailed study of the fundamental proper- ties of such transformations has been made elsewhere.2 In what follows we will apply the general results to the special transformations q,fl(t) induced by the modulated sawtooth through (3). 10. The Significance of Fixed Sampling Points Let 4'f/ be the smallest power of x&f for which sampling points repeat. Then, there exists a finite chain of points to, tl, . , tk-1 such that ti+i=ltf(t;), and to=V1f(tk_j). These points are the fixed sampling points at which the sawtooth samples the sinusoidal signal of frequency f,. Since the plot on the co-l-circle does not distinguish be- tween points which are a distance N 2ir/1w apart (N an integer), these sampling points to = O"(t0),Ii =Pf(to), * *.*., = Ok1-l(to) (8) need not all be in the same period, but may be spread over two or more, say n periods. In this case, the sawtooth is the graph of a periodic function with the period n- 2ir/co and the fundamental frequency f.,n. We shall designate such a closed chain of sampling points as an "equilibrium configuration." The case n=2, k=3 is illustrated in Fig. 4. It is clear that the relationship between the audio signal and the step-function counterpart in the decoder and, hence, between this signal and its final reproduction in the receiver, depends upon the distribution of the sampling points. Therefore, it is important to study the above-mentioned equilibrium configurations belonging to various signal frequencies f.. Since these configurations are determined by the position of the fixed points2 of the transformation Afr(t), one is led to the study of such fixed points. However, the value of AT/T (the "depth of modulation") may be restricted in a more stringent manner by other physical characteristics of the system, as for example, by the pulsed power supply of a magnetron. Thus, the condition expressed by the inequality (5) or (4) is fulfilled in practice. To create an appropriate mathematical model which would make it possible for us to study in detail the behavior of sampling points, we note that as t varies from 0 to 27r/co, 4(t) varies from - Tlm to -Tim+2ir/co, and, hence, 4(2) maps the co-'-circle into itself. Since b(j) is steadily increasing, its inverse AIf(t) is also single-valued. Thus, {f+(t) is a one-to-one continuous transformation of the co-'-circle into itself. Moreover, the set of all the discontinuities of the modu- lated sawtooth commencing at to is exactly the set of the images, INTERVALS OF STABILITY 11. Approximate Estimate: The Case n =1, k = 2 In the following discussion, a fixed pulse repetition rate f, and a fixed depth of modulation AT/T are as- sumed. The question which then arises is: What can be said about the value of the signal frequency f. if it is known that it leads to an equilibrium configuration with a fixed n and a fixed k? An estimate can first be made, as follows: Consider the case n =1, k = 2. The finite chain (8) of images consists of two points to and t1 such that T AT. ti to = - - - - sin wti (9) mm 5 Here, t2 =t'12(to) =4'fftf(tO)] = 'f(tl), where ti = I,y(to). Similarly, to = f'f°(to) =Pff[xfG(to)I 4lf (tgo1)-. 1949 Ross: Pulse- Frequency Modulation 1281 27r T AT ti = to + -- - sin wto. (10) co m m That is, til-=t(o) to =-\f(to). Here, c = 27rf8. Adding (9) and (10) transposing terms, and multiplying both members of the resulting equality by m/IAT, the result is m /27 2T\ -(w- ) = sin Coto + sincti. (11) For a fixed depth of modulation AT/T and a fixed pulse frequency fp, this equation determines the ratio U=f./fp as a single-valued function 4(coto) of the position in the period of the locking point wto. The region of frequencies f8 for which f./f, lies between the maximum and the minimum of k(coto) for 0. to < 2wr/co, is the region of frequencies for which there exist equilibrium configurations with n = 1, k = 2. Fig. 5, in which the u I I I I I I~~~~ ~~ ~~~~~~~~~~~~~~~~~~~tIII rN Since the right-hand member does not exceed two irl absolute value, m /2r 2T - 2 < - l-AT\co -_ m < 2, 0.5IC10 -- - and hence, first obtaining the inequality for (f8/f,)-'1, we have 0.50C0 - 11 f, 1 1 2 AT fp 2 AT T T because w=27rf8 and m/T=fp. This inequality silows that every signal frequency f8 which leads to an eXqui- librium configuration with n = 1 and k = 2 must lie in an interval aboutfp/2. The lower and the upper bounds for f8/fp depend upon the depth of modulation AT/T. 0.495 -- 0.490 -T 0-..4-8-50o II- 2 3 4 5 Fig. 5 Graph of U= O(wto) for n-1, k=2. 12. Precise Determination of the Interval of Stability Since the position of t1 is known when that of to is known, the whole configuration is determined by the position of its first fixed (or locking) point. Replaci:ng t1 in (11) by its value in (10), mutliplying by AT/T and transposing some terms, (J ) - 2 +- [sin 2rf8 to + sin 2rf8 * to - 2wf --f2, 7prf -T--t -p - siA nI27rT fT 8toIto (12) G) CIRCLE \ \t J°0 3 graph of (coto) for AT/ T = 0.2 is drawn to a large scale, gives the values 0.4855 and 0.51625 for the minimum and the maximum of 4(wto), respectively. In constructing the graph of this function, the solution of (12) was carried out graphically. All the fixed points of i,f2(t) are given by the values t, of the points of intersection of a horizontal straight line drawn at U=f8/fp, and the curve U=-(Coto). There are four such intersections for each value off.. The first and the third of these determine one equilibrium configuration, the second and fourth determine another. Thus, in Fig. 5, fe/fr = 0.5 and the values ti of the points of inter- section of the line f8/pf=0.5 are 0, 1.852/1, 3.14/1, 4.43/co (= -1.852/co). Of these, the chain 0, 3.14/1 1'.P8={5f2r//c2o(,O)4.4g3i/vceos=4otfn,e/2(e1q.8u5i2l/icbor)iguimvescotnhfeigoutrhaetri.on, and Fig. 4-The fixed sampling points for n =2, k -3. INTERVALS OF STABILITY-THE GENERAL CASE 13. An Estimate of the Location of the Intervals of Stability Next to be investigated are the conditions imposed upon f8 by the requirement that the kth power of 4lf(t) should have fixed poipts. Here Af(t) is the transformation of the co-l-circle induced by the signal through (2). In view of the interpretation of (2) in Fig. 3, two points, t and t, on the time axis are represented by the same point Po = p on the co-l-circle if, and only if, t -t is 1282 PROCEEDINGS OF THE I.R.E. November an integral multiple of 2ir/w. Thus, if a chain In view of (13), to = ti - T - m - A-T m sin wt, witk-X = COtk- 27r-(1 + Tsin ) T AT tl = t2- - m - - m sin wt2 (13) )tk-2 = (.otkI - 2 1 + T sin tk-I (20) T AT tk-i tk sin (otk m m of images to, L*i, *,i is considered, then the point po= p1 is a fixed point of lfk(t) if, and only if 2n7r tkAto±+ 1 (14) where n is an integer. Substituting the value (14) into (13), adding all the equalities (13) and canceling similar terms, 2n7r kT AT 0= (sin-oto + m m + sin w4_i) or Wtl = cot2- 2r f (1 + T sint2) wto = cot1-2r r(1 AsTin wt. If tk is set equal to to+2nir/w, then the first (k-1) equalities in (20) give cot1 as a continuous and differentiable function of both f./f, and coto for j= 1, * * *, k -1. It is assumed thatfp is fixed. Thus, the right-hand member of (19) is a continuous and differentiable function of both f, and wto. Write AT k + (sin wto + - + sinctk-1) T = F f-,IWto) . (21) m /2n7r kT A-(TInlcor --1im?~/) = sin coto + * + sin Wtk-_ (15) Then, The right-hand member of (15) lies between -k and k. Let -01k and 02k be its minimum and its maximum, respectively. Then 01< I and 02.1. Also, in view of (15), m In kT\ -01k <-o-- )-2k. (16) AT ml Hence, n 3mkxk I 1 ATIP 8o T n 1 =Omin (17) k AT 1 + 02 T Thus, for a given k and a given n in (14), the corresponding frequencies f. lie in the interval k(l-~T<)F(f wto) _ k(I+T) (22) On the other hand, the left-hand member (see equation 19) is m n Tf -- = n/ -- (23) T f* fp a steadily decreasing function off,/fp alone, and its minimum and maximum in the interval (18) are, respec- tively, AT( k 1 -01 and k(1+ 02T). (24) T/ Comparing these values with the bounds in the inequalities of (22), it is seen that, for each fixed value of coto in the interval (0, 2X), there exists a real value of f,/f, for which fp4-rnax 2ff,s fp -4min. (18) 14. Frequency f. in an Interval of Stability, as a RealValued Continuous Function of cwto for 0 < coto _ 2r To prove that for every cot. there exists a unique corresponding frequencyf. in (18) such that to is one of the points in a finite chain of k images, (to, t1, t2, - - , tk-1), (15) may be rewritten as follows: -*-m n = k+ AT oto T f. T + sin- tia). (19) n/ -=F -, wtoj, (25) fp i.e., for which (19) holds true. The corresponding value f. is the frequency for which cto is one of the points in an equilibrium configuration of k sampling points. As in Section 12: f,/f,,=4(cWto). Since all the functions in- volved are continuous and differentiable in all the variables under consideration, k(wto) is continuous in the neighborhood of any pair of values f,/fp, Coto satisfying (25). 1949 Ross: Pulse- Frequency Modulation EXAMPLES OF INTERVALS OF STABILITY FOR ADDITIONAL VALUES OF n AND k 15. The Case n=1, k=3, AT/T=0.2 Here the third power of ilfr(t) should have fixed points. Therefore, in view of (20) IFEFE /' 0.660--- W wto = ct, - 27r.( 1+Tsinwt,) 1283 EI E- S/X - wtil = wt2 - 27r.-( 1 +-- sin Wt2) (26) 0.650 - - - -- At2 = wto + 2n'r - 2ir.-(l + --= sin wto) and hence (see (19)), for n = 1, AT 1/U = 3 + - (sin coto + sin cot1 + sin t2), (27) where U=f/f,. In view of (26), this equation deter- mines U as a single-valued continuous function, O(wto), of wto. This function is plotted in Fig. 6 for AT/T= 0.2. Here the nminimum and maximum values of q are 0.33599 and 0.33863, respectively. u A IF 0.3380 - - - - - - - - - 0.3375 n 52 K=3 060- 1 2 --11.. 3 4 5 Fig. 7-Graph of U=c(coto) for n = 2, k = 3. U =f./fs, with the curve do not give the sampling points in their proper sequence. For example, if f8/fp = 0.6530, the points as given by Fig. 7 are: 1.47 2.83 4. 77 1 ,~) and However, the actual sampling points are distributed over two periods, as follows: 1.47 4.77 to = nI~ tl= Co 9.11 2.83+27r t2 = = Co co Here the minimum and maximum values of X are 0.6450 and 0.6626, respectively. 0.3370 _ _ 0.336~ - - 17. The Case n=2, k=5, AT/T= 0.2 Here the fifth power of of(t) should have fixed points. Fig. 8 is the graph makes which it possible for us to de- 0.336- n= K a=3 0 .-3 14 Fig. 6-Graph of U=4(coto) for n = 1, k = 3. 16. The Case n=2, k=3, AT/T=0.2 As in Section 15, here the third power of i/'j(t) should have fixed points. Hence (26) is again used and n is set equal to 2. From this is obtained AT 2/U = 3 + T (sin wto + sin cwt, + sin cWt2,) (28) where U =f,/fp. The function U=45(wto) determined by this expression is plotted in Fig. 7. In determining the sampling points of the equilibrium positions correspond- ing to a fixed value of f./Lf it should be noted that in this case the points of intersection of the straight line Fig. 8-Graph of U=4o(wto) for n=2, k=5. termine the position of the fixed points in the spectrum region 0.402645<.f,/fp<0.403030. We have made numerous oscilloscope photographs in which markers indicated the actual position of sampling 1284 PROCEEDINGS OF THE I.R.E. November points in the period of our signal. These pictures are in remarkable agreement with theoretical prediction. Such a photograph for the case n = 1, k = 3 is given in Fig. 9. COMPARISON OF THE EXPERIMENTAL DATA WITH THE THEORETICAL PREDICTIONS 18. The Apparent Phase Shift When the location of the points of equilibrium configurations was determined from the graph in Fig. 6, and when these results were compared with Fig. 9, the ob- served positions seemed to be shifted by the amount -r/co from the predicted locations. The reason for the apparent shift lies in the fact that the theoretical signal curve in Fig. 1 is in reality the negative of the actual signal translated vertically through the distance T. Fig. 10 shows the actual signal, the ficticious signal, and the sampling action of the sawtooth. . ~~~~~~(SAWTOOTH PLUS SIGNAL) Y= m(t-to)+(-l) hT sinwt Fig. 9-Signal with markers to indicate the positions of sampling points for n= 1, k =3. (SAWTOOTH PLUS SIGNAL) Fig. 10-The action of the sawtooth on the actual and fictitious signals. 19. An Experimental Study of Six Stages in the Coding and Decoding of a Sinusoidal Signal Figs. 11 to 16 represent a study of what happens to the incoming information (here, a sinusoidal signal) at Fig. 11 Fig. 14 Fig. 12 Fig. 15 Fig. 13 Fig. 16 Figs. 11-16-The photographs of the stable equilibrium conditions at various stages of coding and decoding. 1949 Ross: Pulse-Frequency Modulation 1285 various stages of transmission in pulse communication. Fig. 11 shows the incoming signal with markers which indicate the position of sampling points. Fig. 12 corresponds tp the theoretical representation of the sawtooth in Fig. 10; its lower part contains the information. Fig. 13 shows the modulated pulse train as it leaves the coder. The information is contained in the variation of the distance between successive pulses. The actual means of transmitting and receiving the coded energy are irrelevant here, so these topics are omitted. This allows the discussion to be resumed at the decoder input. Fig. 14 shows reconstruction of that part of the sawtooth in Fig. 12 which contains the signal, while Fig. 15 contains a step function approximation (see Fig. 2) to the signal. Fig. 16 shows the fundamental of the stepfunction approximation. All other components have been effectively stopped by a low-pass filter. The curve in this picture indicates how the information in Fig. 11 appears at the receiving end. sin wt become respectively 0.198 and -0.108. To find the Fourier series expansion for the resulting signal, move the origin in Fig. 17 (7r+0.2) units to the right. Then, al = + 0 0.198 cos cwtd(wt) 7r - 1r - J 0090.108 coswtd(cot) = -0.0275 7 -o.09 bi = =1 -0.01.9198 sin wotd(cot) 7r _T. -- -0.9 0. 108 sin wtd(wt) = - 0. 1944. 7 _o.0os 2 3 7 Cit FREQUENCY RESPONSE OF PULSE-FREQUENCY MODULATION SCHEME 20. General Considerations It was established in Section 5 that for every fixed value f,/f, in a wide range of such values, fixed positions are obtained for the sampling points. If fixed points belonging to a stable equilibrium position are chosen, and the values of the rignal at these points are used to construct a step function, the theoretical equivalent of the information received by the coder will be obtained. If this step function is represented in its Fourier series form, the effect of the low-pass filter in the output results in selection of the first few terms of the series. In the case of a sinusoidal signal, the information to be considered relevant must be only that term of the Fourier series which corresponds to the frequency of the original signal. It should be noted, however, that this frequency is not necessarily the fundamental frequency of the step function (i.e., the Fourier series). In fact, for fixed values of n and k, this fundamental frequency is f8/n. Thus, the proper signal frequency f8 occurs as the nth harmonic. This leads to a phenomenon which may be termed "frequency division." By the study of the frequency response of a pulse-frequency modulation scheme is meant the study of the variation in the amplitude of the term belonging to the frequency f. in the output of the decoder as a function of f8, or better, f,/f,. It is as- sumed, of course, that the amplitude of the coded signal of frequency f. is of constant magnitude. 21. Frequency Analysis The computational procedure employed may be understood from the text and Figs. 17 through 19. (a) The case of n = 1, k = 2, fl/f = 0.495. For the fixed or locking points (0.2, 3.25) the values of Fig. 17-Locking points (0.2, 3.25) and (1.69, 4.25) n=1, k =2, f8/fp 0.495. 2 3 4 5 6 7 ( Fig. 18-Locking points (0.48, 3.42) and (1.45, 4.05) nX1, k-2, fJ/f = 0.49. 0 923 _ 113c. Fig. 19-Locking points 0.08, 2.8, 4.82, 7.85, 10.09, 12.64 n=1, k =5, f./f,=0.402. The amplitude of the f, component (f,-0.495 fp), as it appears in the decoder output is Va,2 + b12 = 0.1968. Similarly, for the locking points (1.69, 4.25) a1 -0.3305, b1= -1.1068 and V/a12+b,2 =1.158. This analysis and the figure (Fig. 17) contain both of the equilibrium configurations. (b) The case of n = 1, k=2,f/fp=0.49. n k locking points ai b1 Va-'±bs flf, 1 2 0.48 3.42 -0.0464 -0.4643 0.4666 0.49 1 2 1.45 4.05 -0.2912 -1.052 1.091 0.49 Since the amplitude of the incoming signal is unity, the values given above of the amplitude of the same signal at the output of the coder give a clear idea of the degree of attenuation of the signal of this particular frequency. 1286 PROCEEDINGS OF THE I.R.E. November (c) A case where fundamental frequency of the step APPENDIX function is equal to 1/2f, is given in Fig. 19. Here the amplitude of the component corresponding tof, is given by the second coefficients of the Fourier series. It is The previous theory has been extended to include nonsinusoidal periodic signals and it is shown that here, too, there are stability intervals and chains of sampling /a22 + b22 = 0.7520. points. The sampling points and all of the limits of the stability intervals may be found by the same methods 22. Conclusion used for the sinusoidal signals. This more general theory has been developed without An analysis of pulse-frequency modulation has indi- the simplifying assumption that the edges of the saw- cated that the accuracy in the reconstructed signal de- tooth are linear. The relationship between successive pends not only upon the number of sampling points per sampling points is given by the formula period of the signal, but also upon their distribution. In general, the distribution of sampling points does not seem to be periodic, but for a given pulse repetition rate t, = ti+l + RC ln I1- T + E a(wtj+j) (unmodulated) and depth of modulation there are cer- tain frequencies for which the sampling becomes pe- where R, C, and E are circuit constants and where a(wt) riodic, and for which the sampling points occur at is a general periodic signal of period 2xr/w. definitely determined points of the signal wave. These points have been designated as fixed or locking points, and the conditions which produce them have been called 0.32040 equilibrium conditions. For a given set of conditions, the frequency range for which locking points may occur can be determined, and for a given frequency within this range the actual locking points can be found. V-O.54. The theoretical analysis has been made on the basis of mathematical considerations from topology. This analy- 0.02 sis indicates that the positions of the locking points do not depend upon the position of the first point at which 0.Q4900 the signal wave is sampled, but depend only upon the ratio of the frequency of the signal to the unmodulated pulse repetition rate and upon the ratio of the signal 0.4600 ./ 2(.t.)Xw[Ez.t.+- S.Z 2 t + _ _ + _Z I RC At.+0+f EiK +sco-I], amplitude to the unmodulated amplitude of the sampling wave. This leads to the suggestion that, if the first I o _ _ __ _ _ a~~~~~~~r(wtj) z 2Fs.- wt. s, z C'] 0.47 00 6-.-29~ 0 i1 ± 0 5 .9 .1 .9.40 a. sampling does not occur at a locking point, there must exist a transient interval during which the sampling Fig. 20-A stability interval for a nonsinusoidal signal. process reaches an equilibrium condition. This question is to be studied in a subsequent paper. It has been shown also, in every case studied, that two sets of equilibrium Fig. 20 describes the behavior of the nonsinusoidal conditions exist. That is, the signal may be sampled at signal either of two sets of locking points. The question as to whether both of these sets of locking points correspond a(wt) = 2 [sin wt + sin 2wt] to conditions of stable equilibrium or whether one set in the interval of stability (0.4776, 0.5260) of the relacorresponds to an unstable condition is also considered tive frequency spectrum. in the subsequent paper. The significance of this present study is that it lays a firm theoretical foundation upon which more extended TABLE OF SYMBOLS theoretical and experimental studies of pulse-frequency modulation may be based. ACKNOWLEDGMENTS The author wishes to acknowledge his indebtedness to H. Boehmer for his help in obtaining the photographs, v, relative frequency .................... Qf(t) ......................... f8, signal frequency ..................... fp, pulse frequency ..................... ( ) .................................. AT/T, depth of modulation ............. Summary Section 3 Section 3 Section 3 Section 9 Section 9 and to Miss M. Tinlot for the very extensive calcula- f,/fp (Cto) = U = ....................... Section 12 tions involved in the construction of the theoretical 4max =maximum of (c(wto) ............... Section 13 graphs. grsmin = minimum of 4(wto) .............. .. Section 13

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