# Патент USA US3025010

код для вставкиMarch 13, 1962 l.. TABAcK 3,025,000 FUNCTION GENERATOR FOR GENERATTNO A FUNCTION OF Two INDEPENDENT VARIABLES Filed oct. 4, 1957 5 Sheets-Sheet l 6A T5 ou FP1/7 To _,m ß/G. 2B 2m /\ m 75@ 0266/61) VON/165 60A/meuf@ 47m/mm@ E' y. 2,4 x /NPUT V205!! for) „s/@A/AL FPO/w Pawn/0N 65N. /aa @few-í 2054 _300 60W/P4194 T0@ 200 Ew +500 e ‘500 INVENTOR 20o “ ¿e0/mf 7am/A Ú/îx’g/ígíggM H63) t’ ’500 Von/165 60A/maag@ M75/M470? Íß BTY m( W Mi ATTORNEY: March 13, 1962 |_. TABACK 3,025,000 FUNCTION GENERATOR FOR GENERATING A FUNCTION OE TWO INDEPENDENT VARIABLES Filed Oct. 4, 1957 s sheets-sheet 2 _’1 YI“ w ’ - _“ AïïEA/L/Am/a ,aPPßox/MATELY I" _*l vk Z6 FL/LL 0/v | l TI m ~- F@ Zß ATTENUATo/e APPROX/wmv Il JVi H/JLf 0N I i Zi __ AITLNL/A 70@ APPROX/MATEN LPLHB/ÉL d2 BY leo/70rd 70,6610? ¿mi IW ATTORNEYS March 13, 1962 |_. TABACK 3,025,000 FUNCTION GENERATOR FOR GENERATING A FUNCTION OF Two INDEPENDENT VARIABLES Filed Oct. 4, 1957 5 Sheets-Sheet 5 Aff“ 0 400 -300 c;? VW AMR ___/\/_>__O T0 200e F/È. Zó’ wwe/FTE@ /04 (H6. /) 404 OUTPUT „SYSTEM Fig?? ï INVENTOR ' ¿E0/mrd ï?/back BY m ATTORNEYS nite States Patent il* tiìce 1 FUNCTKON SENER-1191i FÜR GENEäATltNG A FUNCTÍÜN @if TWO HNDEPENDENT VARIABLES Leonard Tabacir, Mount Rainier, Md., assignor to the United States of America as represented by the Secre tary of Commerce Filed Get. 4, 1957, Ser. No. dtiäßäli 6 ‘Ciaima (Ci. 23S-197) 3,025,000 Patented Mar. 13, 1962 2 FIG. 2B is a schematic diagram detailing the circuit construction of the voltage-controlled pulsed attenuator of FIG. 2A; ` FfGS. 2C, 2D, and 2E are waveforms showing the operation of the voltage-controlled attenuator for various input settings; FlGS. 2F, 2G are diagrams showing the principle of operation of the vonage-controlled attenuator; FiG, 2H is an explanatory diagram; The present invention reiates to the generation of func 10 FIG. 3 is a circuit diagram showing lthe construction of tion representing signals for use in connection with elec a triangular wave generator employed with the present trical analogue computers. Specifically, the invention con invention; and templates an improved device for generating an arbitrary FiG. 4 isa diagram of the output system employed. function of, for example, two independent variables. The present invention contemplates a device comprising The need of a function generating means in order to 15 a number of individual mechanisms for generating func represent nonlinear phenomena for the simulation of phys tions of a first variable at successive fixed values of a ical problems in analogue computers is app arent. In gen second variable in combination with means for inter eral, function generators provide an output manifestation polating linearly among the function generators under which varies in some arbitrary but controllable way in re control of one of said variables. An over-all block dia gram of a preferred embodiment of the function generator in accordance with this invention is illustrated in FIG. 1. The apparatus comprises a plurality of individual function For example, in copending application, Serial No. generators indicated as 1000, 1001, 1002, etc. each of which 651,121, now Patent No. 2,998,193, for an Electronic will produce or generate a different discrete function of Analogue Computer for Radioactive Fallout Prediction, 25 an applied input variable x. The x variable is in the form tiled on April 15, 1957, by H. K. Skramstad et al., which of a signal applied in parallel to the function generators' is assigned to the assignee of the present case, it is neces 1000 through 1002 etc. While an embodiment employ sary to generate a particle radioactivity factor which is ex ing three function generators is illustrated, as indicated in pressed as a function of two independent variables, height the broken `line representation in FIG. 1, as many function and time of particle fall. No familiarity with the mathe 30 generators as is desired can be incorporated in the appara matical functions underlying the radioactivity factor is tus of the present invention. necessary in accordance with the instrument of the present Each of the function generators 1000 etc. shown in invention, since the function is incorporated in the instru FIG. 1 are of conventional type and are therefore only sponse to an 4applied input manifestation. In many in stances the function generator must provide an output which is a function of two independent input variables. ment as a family of curves and it is necessary only to symbolically shown. A typically commercially available manipulate the controls in order to obtain any desired 35 unit satisfactory for such purpose is the Goodyear Aircraft function within «the range defined by the curves. Company GN-215-N3 function generator. The output The usefulness of a function generator is measured by of each function generator 1000 through 10402 corresponds such characteristics as flexibility, that is, the diiiiculty of to an arbitrary function of x such as f0(x), f1(x), f2(x)_, setting up the function generator «and the ease with which etc. The output of each function generator, except the it can be changed from one setup to provide different 40 first one, 1000, is applied to a respective voltage-controlled types of functions; frequency response, for example, the pulsed attenuator 1011, 1012, as indicated in FIG. 1. range of input and output frequencies which can be Each function generator 1000 etc. will accordingly pro handled without introducing intolerable errors in either vide an output signal representing a different discrete func phase or magnitude; and finally, accuracy; or specifically, tion of the input signal representing the variable x. the manner in which the output conforms to the desired 45 Each of the voltage-controlled attenuators 1011, 1012 is function. The present invention is intended to improve a form of time division multiplier, the general principles upon the deficiencies of prior art devices in connection of which are described on pages 223-226 of Electronic with the above-enumerated criteria. Analog Computers, by Korn and Korn. Each attenuiator It is accordingly an immediate object of the present in-> functions to combine the particular f(x) signal generated vention to provide a function generator for generating an 50 `by the function generators 1001, 1002 with a signal rep arbitrary function of two independent variables which will resenting the second independent variable y to provide an work at electronic speeds and which is sufficiently iiexible output which is a function of both x and y. to be adaptable for the generation of a wide variety of The specific construction and mode of operation of optional functions. such pulsed attenuators is fully discussed in connection it is a further object of this invention to provide a func with the description of FIG. 2. At this point in the de~ tion generator which will accurately provide an output scription, however, it can be stated that each attenuator signal that conforms to any desired arbitrary function of provides an output signal pulse having a duration propor two independent variables. tional to the applied input variable (y), and an amplitude Another object of this invention is to provide a function corresponding to f(x), already referred to. ` generator which is completely electronic in operation. 60 The outputs of the attenuators `1011, y1012, together with . Still another object of this invention is to provide a the output from the »first function generator 1000, are function generator which is adjustable in accordance with applied to a filter circuit `102 which filters out any carrier» any desired function of two variables. signal introduced by the triangular' wave employed in the Other uses and advantages of the invention will become operation of each of the attenuators as will be described. apparent upon reference to the specification and drawings 65 In this manner, a signal which represents a function of in which: two independent variables f(x,y) is obtained for applica FIG. 1 is a block diagram illustrating the functional tion to a summer 103. arrangement among the elements comprising the present The signals representing function f1(x) and f2(x) . generated by the function generators 1001 and 1002 are FIG. 2A is a functional block diagram illustrating the 70 also applied to an inverter 104 and through a second filter invention; construction of a voltage-controlled pulsed attenuator em ployed in the present invention; 105 to the summer 103. The gain of the inverter ampli tier 10d is less than one-half and provides anoutput which i 3,025,000 3 is, subtracted from the signal applied to summer 103. The purpose of such subtractive effect is to compensate for residual signals which inherently arise from the func tion generators‘even when the outputs derived should'be zero. 4 The Equations Sa-Sc are general, permitting the ad justment for a function f(x, y) using any desired y in crement. However, it is simpler to set the function gen erators for equal increments k. Combining the above Ul Equations 5a-5c and making theA increments equal to k: Each of the attenuators `1011, 1012 are energized se quentially as> will be described in connection with the de tailed description of FIGS; 2A and 2B. afgifte@ #framprîïpmkj _111,111 Before considering the specific construction of the various elements of the invention, the principles under 10 (5a) which may be expressed in terms of the outputs of the lying the apparatus diagrammatically shown in FIG. 1 will ñrst be discussed. A representative plot of a function of two independent variables, x and y, is indicated in FIG. 2H, That is, each function generators thus (5b) curve 20, Z1, and Z2 shows an arbitrary relation between 15 For any selected value of x indicated by the broken line variables x and y for different assumed values of y. The in FIG. 2H, the value of f(x, y) represented on each of assumed value of y in connection with curve 20 is 0 and the curves 20, 21, and 22 is f0(x), f1(x) and f2(:c), re~ spectively. It will be obvious that if the function gener the curve- is accordingly designated f(x, 0). Similarly, curves 21 and 22 are labeled f(x,a) and Hach), respec tively, since they indicate the arbitrary relationship be 20 ators 1000, 1001, and 1002 shown in FIG. 1 were set to obtain such values respectively, in accordance with Equa tions 5tz-5c the apparatus would be able to provide a f(x, y) for three different ñxed values of y. It will also be obvious that by the addition of sufficient function ship between x and y can be plotted in this manner. Considering for the moment the block diagram of FlG. 25 generators as symbolized in broken lines in FIG. 1 to represent additional curves (FIG. 2H), the apparatus can 1'J it will be apparent that the output signal derived from lbe employed to implement almost any arbitrary function the apparatus of this invention is composed of the sum of two independent variables such as Í3(x), fn(x), etc. mation of the outputs of the various voltage-controlled The usev of function generators alone is, however, attenuators 1011, 1012 etc. and the output from inverter amplifier 104 together with its associated filter 105. The 30 neither sufficient nor economical in the attainment of al high degree of selectivity of a desired function. rl`he ap output, f(x, y) can be expressed as tween x and y for assumed values in which y equals a and b, respectively. It will be understood that any desired number ofy curves showing Various degrees of relation paratus of the present invention therefore also provides means for interpolating for Vvalues lying between those represented by'the'curves in FIG. 2H. The feature for accomplishing interpolation in accord» ance with the present invention is based on the follow ing considerations. As is well known, interpolation for a point y, the value of which is f(y), between two values where A is the gain of the inverting amplifier and filter f(a) and f(b) for values of the variable equal to a and b combination 104, `105 and has an approximate value of .2. 40 respectively is obtained as follows: The values of G1, G2, etc.7 the respective gains of the voltage-controlled attenuators 1011, 1012, lie between .2 b-a for fixed increments k and a=0 and> .8, the gains being controlled by the independent in put variable y in the following manner. The values of the gain G1 are (7a) G1=.2 for y<0 G1=.8 for y>0 For a given value of X Equation 6a reduces to Equa tion 7a thereby showing that the device linearly inter polates along the y axis. (3a) The various functions of x for example, f0(x), f1(x), f2(x), designatedv in FIG. 2H may readily be generated by the respective function generator 11000, 1001, 1002, etc. The values of the gain G2 are GET-@+2 for a<y<(a-i-b) Ö2=~2 for y<a shown in FIG. 1. The respective functions are indicated in FIG. 1 -a‘s output signals from each function generaÍ tor. The respective attenuators 101, 1012 connected to (3b) 02:.8 -for y> (a -Jf-b) the outputs of the f1(x) function generator «.1001 and Substituting the values of G1, G2, etc'. from Equations 3a, 3b in Equation 2: f2(x)‘ function generator 1002, provide respectively, inter polation by modifying such input signals in accordance 60 Returning to a consideration of FIG. 2H in which each of the' curves represents a particular value of y held at an arbitrary constant ‘0, a, b, etc., in order to obtain a with the a m la’ k factors Adefined in Equation 6. As will be described in value corresponding to f(x, y) the following equations 65 detail, each such attenuator receives signals correspond ing to f1(x), f2(x) and modifies such signal with a second must be satisfied applied signalcorresponding to a, .v_-ZC. k Íc 70 etc., to provide an output corresponding to -k It will be apparent that Equations 5a-5c specify the settings of the corresponding function generators 1000, 1001, 'a?'d 110011. ä, fle) and yk fax) respectively. The unmodified foQc) signal from Afunction- 75 generator 1000 together with the respective outputs from spettano E' ¿à to? attenuators Zitti and M12 are then accumulated in sum gate circuit 205 symbolically indicated in FIG. 2A is representative in FIG. 2B by the series gate tube V205A and parallel gate tube VZtiSB. The “y” input variable is mer 103 to provide an output signal corresponding to Equation 6b; namely, applied at input terminal Ztltib shown at the left-hand por~ tion of tFIG. 2B, while the triangular carrier signal ob tained from the mechanism of FIG. 3 to be described is Voltage-Controlled Pulsed Attenuazors The construction and operation of the voltage-con trolled pulled attenuators will, M312 designated in FIG. l is detailed in connection with the block diagram of FIG. 2A, the detailed circuit diagram of FIG. 2B, and the waveforms illustrated in FIGS. ZC-ZF. Referring to FiG. 2A each attenuator comprises a comparator 200 to which there is applied both a repetitions triangular input signal to input terminal 2Min and another input signal to terminal êtitib representing the y variable which is also applied at terminal Z'ätia. The voltage corresponding to the input variable "y” applied at terminal 26% is limited to a potential between 0 and ‘+10 volts above ground by the lO-volt bias on the cathode of the diode of VZtiflb and the inversely connected diode Vâtttia. The diode V2M prevents the grid of the left~hand section of the D.C. amplifier V262 from going negative. A poten tiometer RZtit? is connected to a -SOO-volt source as in dicated and therefore provides an adjustable means for determining the point at which the anode terminal of diode VZtiftb will be positive. In this manner adjustment shown in FIG. l. The output of the comparator is ap of the corresponding potentiometer R290 in each at plied to a gate circuit 205 coincidentally with the Kx) tenuator determines the amplitude value of the applied signal obtained from an appropriate one of the function 20 “y” signal that will result in energization of the attenua generators 1661, dtìíig. The output of the gate circuit 2% tor. The attenuator Mill, MM2 may thereby be energized is applied to the -íilter MBZ shown in FIG. l. The wave in sequence depending on the amplitude of the respective form of the signals are indicated in FIG. 2A adjacent y, y-Jc, etc. signals applied. The resistors RZdtiA and each of the circuit components. Rìtiûlì comprise summing resistors. The construction and operation of the voltage-con~ When the y input variable voltage applied to terminal trolled attenuator can be explained by considering the 2691; reaches a value large enough to malte the left~hand waveform shown in tFlG. 2F of the drawings. The tri~ grid of D.-C. amplifier V 2%2 positive, the resulting signal angular shaped wave shown in FIG. 2F corresponds to is amplified and applied to a gate driver tube V263. The the triangular input carrier signal indicated in the block gate driver tube Vïtiâ comprises a bistable circuit which diagram of FIG. 2A. The triangular wave illustrated in FIG. 2F is shown relative to a zero voltage reference level indicated by the broken line in FIG. 2F. As will be` described, the comparator in each voltage-controlled at tenuator is a device which is controlled by the triangular wave so that it will conduct at voltage conditions defined by the portions of the triangular wave which are above the zero reference level indicated in FIG. 2F. Speciñcally, conduction will be obtained in the regions defined by the rising and falling portions of the triangular is adapted to be driven to either of two states of con ductivity by the signal applied to the left-hand grid of the amplifier VZiiZ. That is, when a positive signal is applied to the left-hand grid of VZGZ, a positive signal will also be applied to the left-hand grid of the gate driver V263. Accordingly, the plate of the right~hand section of gate driver V263 will be positive and such positive signal will be transmitted through the upper chain of neon tubes VZddA to `the series gate tube VZÜSA. The (x) input signal applied to terminal ZtlSa Wave included between points a and b and between points 40 of the gate will therefore be transmitted through the c and d, respectively. It will therefore be obvious that series gate tube VZÈBSA to the filter 162- shown in the the resulting output signal will have a duration or pulse block diagram of FIG. l. In other words, under the width corresponding to the distances a-b, and c-d indi above-described conditions, the f1(x) output of the par cated in lFIG. 2F. It will also be apparent that each such ticular function generator ltitil applied to terminal 265e: pulse will necessarily occur at a frequency deñned by 45 of the voltage-controlled attenuator appears at the output the frequency of the triangular wave. The comparator of the gate. therefore functions as an on-off device which generates Similarly, if the resulting signal applied to the left-hand a pulse having a width or duration corresponding to the grid of the D.-C. amplifier V202 is negative then the distances a-b, c-d, etc. plate of the right-hand section of the D.-C. amplifier By shifting the triangular wave relative to the reference 50 will be negative and the resulting negative signal applied level, it will be apparent that the duration or width of the output pulse can be selectively varied. Such action is illustrated in FIG. 2G. The upper reference level line corresponds to the 0-voltage reference level line indicated in FIG. 2F, while the lower reference level line indicates the result obtained when the level of the triangular wave is increased relative to the reference level. It will be obvious from FIG. 2G that the desired pulse width will obtained at the plate of the right-hand section of the V2ti3 and applied through the chain of neon tubes VZtMA to the grid of the VZÜSA gate will cut that tube off. As a consequence the positive signal derived at the increase from an amount corresponding to r11-b1 to an gize the shunt gate VZtlSB. to the left-hand section of the gate driver V2M will cause the bistable V293 to flip. The resulting negative signal left-hand plate of the gate driver V203 `«vill be applied through the lower chain of neon tubes VîtìétB to ener The applied itx) signal amount corresponding to :z2-b2 consequent to a shift in 60 from the function generator will now be cut off by the the reference level. The “y” input signal designated in series gate and the shunt gate will maintain a low imped FIG. 2A provides the described effect of shifting the ance at the input of the filter. amplitude `of the triangular wave in the manner indicated From the above description, it will be clear that the in FIGS. 2F and 2G. The variable "y” will therefore determine the duration of the derived pulse signal. Stated 65 polarity of the y input variable signal determines both the duration of conduction of the series gate VZtiSA and in another way, the pulse width will correspond to the energization of the shunt or cutoff gate VìdSB. The factor "y.” output pulse obtained from the voltage-controlled atten FIG. 2B shows a circuit diagram implementing the uator therefore will have a duration determined by the mechanism diagrammaticaliy shown in FIG. 2A. The portions of the circuitry shown in FdG. 2B corresponding 70 y input variable. Since the f(x) signal from the func tion generator is conducted through the series gate to the blocks indicated in :FIG 2A are labeled with cor VZdSA, the amplitude of such output pulse signal will responding reference numerals. The comparator 26d be determined by the x variable. Figs. 2C, 2D, and 2E. comprises an attenuator circuit including diodes VZt'iGb, are typical oscilloscope traces showing the output ob V2M, a D_C. amplifier comprising the twin triode V2tl2, V293 and banks of neon tubes VZMA, VZMB. The 75 tained from each voltage-controlled attenuator. Since 8,025,000 7 It will be clear from the above description that theI function generator of the present invention is completelyl electronic in operation. It operates satisfactorily at sig nal frequencies up to 100 c.p.s. and a high degree of function selectivity is obtainable because of the flexibility the output signal is modulated by the triangular wave-in put as described, the output produced comprises a series» of pulses' having an envelope corresponding to the out put of the function generator. The width of each pulse comprising the envelope as shown in FÍGS. 2C, 2D, and 2E corresponds to the y input variable. FIGS. 2C, 2D, and 2E also'illustrate the eñect of adjusting the attenu ator from a full-on condition to a full-off condition. Triangular Wave Generator The triangular wave generator employed for generat in the number of individual function generators that can be used. While linear interpolation is employed it will be apparent from the description that nonlinear inter polation is obtainable by suitable shaping of the sav/tooth, signal. The device operates at electronic speeds, limited only by the sawtooth frequency as this frequency deter mines the design of the filters. 10 ing the referred-to triangular wave input signal applied to the voltage-controlled attenuator of FlG. 2B is illus t'ratedy in FIG. 3. yIt will be apparent that the embodiments shown are only exemplary and that various modifications can be made in construction and arrangement within the scope of invention as defined in the appended claims. What is claimed is: l. A device for generating an output signal represent The triangular wave generator com prises a conventional operational amplifier 400 connected as an integrator by means of the capacitor C-itifì. A bi stable» device comprising the» twin triode Vdt'ifi controls the input signal ofthe operational amplifier. Assuming that the left-hand section of the tube V400 is conducting, ing a function of two independent variables comprising: then the plate signal output of the left-hand section will 20 first means responsive to a first input signal representing be negative and the output of the integrator circuit com a first> of said variables for generating a signal which is> prising the operational ampliñer 400 will increase linearly an arbitrary function of said first variable for a fixed at a rate determined by a summing resistor R460 and value of said second variable, second means_responsive to said first input signal for generating signals representing the capacitor C400. The output of the operational am plifier is connected to the grid of the right-hand section additional functions of said first variable for correspond of the twin triode V400 and therefore as the output of ing fixed values of said second variable, third means re sponsive to said additional function generating means the integrator increases positively, the right-hand sec and to second input signals representing discrete values tion of the tube V400 will consequently be turned on of said second variable for providing output signals pro thereby extinguishing the left-hand section in a conven portional to a function of said signals generatedvby saidv tional manner. The resulting positive signal output ob second means and said second variableA signals and meansv for cumulatively'combining the outputs of said ñrst and“ third means to provide an output signal which is a func tained from the left-hand section of the two triode ap plied to the integrator comprising operational ampliñer 400 will therefore result in a linearly decreasing signal. tion of said two variables. The resulting output of the generator circuit shown in FIG. 3 is therefore a triangular wave as indicated adja 35 2. The invention of claim 1 in which said third means cent the output terminal. Such output signal comprises comprises a pulse generator for providing an output pulse the triangular wave input appiied at terminal Moa in FIG. 2B. having a duration corresponding to said second Variable and an amplitude corresponding to said first variable. 3. The invention of claim 2 including a triangular Wave generator and comparator means in said pulse gen erator responsive to said triangular wave generator and Output System The portion of the block diagram included in the broken line outline in FiG. 1 comprising the output sys tem. and. is further detailed in FIG. 4 of the drawings. The outputs from `the function generators 1061' 1002 etc. are applied through appropriate summing resistors R600 said second variable signal for determining the duration and frequency of said output pulse. 4. The invention of claim 3 including a plurality of’ said pulse generators each connected to a respective one» of said additional function generating means, and ampli tude sensitive means in each of said pulse generators re to amplifier 401 which together with resistor R401 com prises inverter 104 designated in FIG. l. As indicated in FIG. l the output of the first function generator sponsive to said second variable signal for energizing 1000 as well as the outputs from each of the voltage each of said pulse generators in sequence. controlled attenuators 1011, 1012 are applied to the filter 5. The invention of claim 4 including a filter for said 102. Such construction is detailed in FIG. 4 in which triangular wave signal connecting said pulse generators the. ñlter 102 is represented by the RC combinations to said combining means. RC402, RC403, and RC404 to which the outputs of func 6. The invention of claim 5 including means for in tion generator 1000 and attenuators 1011, 1012, respec verting the outputs of said second means and a second tively, are applied. The filter 105 designated in FIG. 1 55 trlangular wave signal filter connecting said inverting is indicated', by an RC network RC405 in FIG. 4to which means to said combining means. the output of the inverter is applied. The resulting out puts from all the referred-to i’ilters are applied through` References Cited in the file of this patent appropriate summing resistors R406, R407, and R408 to UNETED STATES PATENTS the summer 103 which comprises an operational ampli fierv 403. The output accordingly comprises a signal representing an arbitrary function of the variables x and y, as derived in accordance with the above-detailed de scription. 2,773,641 2,794,965 Baum ______________ __ Dec. ll, 1956' Yost ________________ __ June 4, 1957 2,801,351 2,878,999 Calvert et al __________ __ July 30, 1957 Lindsey et al. ________ __ Mar. 24, 1959 It will be recalled from the description of FIG. 2,v (i54 that the output waveform of the voltage-controlled pulsed attenuators 1001, 1002, etc., includes the triangular Wave carrier signal generated by the triangular wave generator. The filter 102 shown in FIGS. l and 4 functions to par t-ially filter out such carrier signal so that the signal ap plied to the. summer 103 is essentially similar to the out puts of the function generator. OTHER REFERENCES Amemiya: A New Diode Function Generator, I.R.E. Trans. on Electronic Computers, fune 1957, pages 95-97. Model F2V Function of Two Variables (Philbríck), ‘ received in Div. 23, December l1, 1957, pages l-15. A Palimpsest on the Electronic Analog Art (Paynter), 1955;` pages 266-270. . .n

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