SU765802A1

SU765802A1 - Device for extracting the third root

Info

Publication number: SU765802A1
Application number: SU782568047A
Authority: SU
Inventors: Роман Антонович Воробель; Валерий Богданович Дудыкевич
Original assignee: Физико-Механический Институт; Львовский политехнический институт
Priority date: 1978-01-06
Filing date: 1978-01-06
Publication date: 1980-09-23

Description

Изобретение относится к вычислительной технике и может быть использовано для вычисления кубического корня.The invention relates to computer technology and can be used to calculate the cubic root.

Известны устройства для извлече- 5 ния корня кубического, содержащие счетчик, два сумматора, три группы схем совпадения и линии задержки fl}.Known devices for extracting a cubic root 5, comprising a counter, two adders, three groups of matching schemes and delay lines fl}.

Устройство характеризуется чрезмерной сложностью и недостаточной точностью.The device is characterized by excessive complexity and lack of accuracy.

Наиболее близким по технической сущности к изобретению является устройство для извлечения кубического корня, содержащее первый счетчик, 15 разрядные выходы которого соединены с управляющими входами элементов И первой группы, второй счетчик и вторую группу элементов И [2}.Closest to the technical nature of the invention is a device for extracting a cubic root containing a first counter, 15 bit outputs of which are connected to the control inputs of the elements of the first group, the second counter and the second group of elements [2}.

,Кроме того, устройство содержит 20 дополнительный счетчик.In addition, the device contains 20 additional counter.

’Устройство работает следующим образом. На вход устройства поступает частота. f₀ .На выходе первого и второго счетчиков частоты 6, и рав- 25 ны соответственно f\=F_o/N м £_г=£,/Ы, (1) где N - число, соответствующее коду в третьем счетчике. 30'The device operates as follows. The input frequency of the device. f _0. At the output of the first and second counters, the frequencies are 6, and 25 are respectively f1 = F _o / N m £ _r = £, / Ы, (1) where N is the number corresponding to the code in the third counter. thirty

Поскольку третий счетчик суммирующий , то число N, соответствующее коду в третьем счетчике, определяется выражением ΐχ ».Since the third counter is summing, the number N corresponding to the code in the third counter is determined by the expression ΐχ ".

N‘-Kdt, (2) оN‘-Kdt, (2) about

где t_A- время, в течение’ которого частота f_o поступает на вход устройства.where t _A is the time during which the frequency f _o enters the input of the device.

С учетом выражений (1) и (2) полу-, чим мЛзн_х (3») где N_x =f_o t_x - количество импульсов частоты f_o поступивших на вход устройства в течение:времени t_x .Taking into account expressions (1) and (2), we obtain, therefore _, ML _x (3 ") where N _x = f _o t _x is the number of pulses of frequency f _o received at the input of the device during: time t _x .

Однако данное устройство чрезмерно сложно, так как содержит три счетчика, обладает ограниченными функциональными возможностями, поскольку определяет кубический корень только из количества входных импульсов и обладает недостаточной точностью при больших значениях числа входных импульсов. Например, если в третьем счетчике находится код числа N,_г то для того, чтобы код третьего счетчика соответствовал числу М_г необходимо, в соответствии с выражением (3) подать на вход устройства количество импульсов, равное (,4) из чего следует, что если количество' входных импульсов меньше д , при -5However, this device is excessively complicated, since it contains three counters, it has limited functionality, since it determines the cubic root only from the number of input pulses and has insufficient accuracy for large values of the number of input pulses. For example, if the third counter is a code number N, _r then to the third counter code corresponds to the number M _r required in accordance with the expression (3) fed to the input unit pulse number equal to (4) from which it follows that if the number of input pulses is less than d, at -5

Nj-N,* 1, то код третьего счетчика не изменится и будет соответствовать числу N, , что свидетельствует о значительной погрешности округления при вычислении результата (3) Ю £ _: _2д--¹ N. 4 1 ¹ N + 1Nj-N, * 1, then the code of the third counter will not change and will correspond to the number N, which indicates a significant rounding error when calculating the result (3) 10 £ £ _: _2d-- ¹ N. 4 1 ¹ N + 1

11

Цель изобретения - повышение точности и упрощение устройства. , 15The purpose of the invention is to improve the accuracy and simplification of the device. , fifteen

Достигается это тем, что в устройство для извлечения кубического корня, содержащее первый счетчик, разрядные входы которого соединены с управляющими входами элементов И первой группы, второй счетчик и вто- ^zрую группу элементов И введен блок сложения-вычитания частот, первый вход которого является входом устройства, второй вход соединен с выводами элементов И второй группы, а 25 выход подключен к входу первого счетставляет собой двоичный умножитель частоты, управляемый кодом числа z (· t) счетчика 1. Поэтому частота ) на выходе группы 4 элементов И определяется выражениемThis is achieved by the fact that in the device for extracting the cubic root containing the first counter, the bit inputs of which are connected to the control inputs of the elements of the first group, the second counter and the second ^z group of elements, and an addition / subtraction unit of frequencies is introduced, the first input of which is the input devices, the second input is connected to the terminals of the AND elements of the second group, and the 25 output is connected to the input of the first one, it is a binary frequency multiplier controlled by the code of the number z (· t) of counter 1. Therefore, the frequency) at the output of the group of 4 elements Comrade And is defined by the expression

N_m N _m

N_m=2ⁿ- коэффициент пересчета счетчика 2) η - количество двоичных разрядов счетчика 2.N _m = 2 ⁿ - counter conversion factor 2) η - number of binary bits of counter 2.

Подставив выражение (6) в (7),получим где (в) суммирующий и на с опреТак как счетчик его вход поступает выхода блока 5, то деляется выражениемSubstituting expression (6) into (7), we obtain where (c) is the summing and on with the definition. Since the counter of its input arrives at the output of block 5, it is divided by the expression

2.£t)--j£₅(i)dt, о где t - текущее время.2. £ t) - j £ ₅ (i) dt, о where t is the current time.

Продифференцировав выражение получим частота %( t) его код z(t) (9) чика, разрядные выходы которого соединены с первыми входами элементов И второй группы, вторые входы которых подключены к разрядным входам 30 второго счетчика, вход которого соединен с выходами элементов И первой группы.Differentiating the expression, we obtain the frequency% (t) of its code z (t) (9) of the chip, the bit outputs of which are connected to the first inputs of the elements of the second group, the second inputs of which are connected to the bit inputs 30 of the second counter, the input of which is connected to the outputs of the elements of the first groups.

На фиг. 1 представлена структурная схема устройства; на фиг. 2 - 35 возможный вариант реализации блока сложения-вычитания частот.In FIG. 1 shows a block diagram of a device; in FIG. 2 - 35 is a possible implementation of a frequency addition / subtraction unit.

Схема (см.фиг.1) содержит счетчики 1 и 2 , группы 3 и 4 элементов И, блок 5 сложения-вычитания частот, в состав которого входит блок 6 сложения частот и блок 7 вычитания частот.The circuit (see Fig. 1) contains counters 1 and 2, groups 3 and 4 of AND elements, a frequency addition / subtraction unit 5, which includes a frequency addition unit 6 and a frequency subtraction unit 7.

Устройство для извлечения кубического корня работает следующим образом.A device for extracting a cubic root works as follows.

В исходном состоянии триггеры счетчиков 1 и 2 находятся в нулевом 45 состоянии. На первый вход блока 5 слоdz£t)In the initial state, the triggers of the counters 1 and 2 are in the zero state 45. At the first input of block 5, slots dz £ t)

4^--dT (ίο)4 ^ - dT (ίο)

Подставив выражение (10) в выражение (8), получим dz(t) z\t)Substituting expression (10) into expression (8), we obtain dz (t) z \ t)

N² ¹¹ m ξ|( t) с выхода И поступает на 5 сложения-вычина первый вход котороt частота как частота 4 элементов вход блока ου жения-вычитания частот поступает тактовая частота f₄ , а с выхода блока 5 частота f₅(t) поступает на вход счетчика 1.N ² ¹¹ m ξ | (t) from the output And arrives at 5 additions-reasons; the first input is the frequency as the frequency of 4 elements; the input of the unit ου frequency subtraction-subtracts the clock frequency f ₄ , and from the output of the unit 5 the frequency f ₅ (t) arrives at the input of counter 1.

. Так группы второй тания частот, го поступает тактовая частота f₀то, в соответствии с фиг.2, fg(t) на его выходе определяется выражением f_s^)=£_o*£_s(t)-£₄(t). Since the group of the second frequency melts, the clock frequency f ₀ arrives, then, in accordance with Fig. 2, fg (t) at its output is determined by the expression f _s ^) = £ _o * £ _s (t) - £ ₄ (t)

Подставив выражение (11) жение (12), получим рSubstituting expression (11) and (12), we obtain p

N²/ di ’ (12) в выра(15)N ² / di '(12) to vyrah (15)

Счетчик 1 с группой 3 элементов И представляет собой двоичный умножитель частоты, управляемый кодом числ$ z(t) счетчика 1, поэтому частота 6^( t) на выходе группы 3 элементов И определяется выражением где Ν^= откудаCounter 1 with a group of 3 elements AND is a binary frequency multiplier controlled by the code number $ z (t) of counter 1, therefore the frequency 6 ^ (t) at the output of a group of 3 elements AND is determined by the expression where Ν ^ = where

Проинтегрировав выражение получим (15) откуда при £6) t = tx mIntegrating the expression, we obtain (15) whence for £ 6) t = tx m

коэффициент пересчета счетчика 1;counter conversion factor 1;

η - количество двоичных разрядов счетчика 1,η is the number of binary bits of the counter 1,

С выхода группы 3 частота f^( t) поступает на вход счетчика 2. Счетчик 2 с группой 4 схем совпадения пред- 65 где ты заFrom the output of group 3, the frequency f ^ (t) goes to the input of counter 2. Counter 2 with group 4 of matching circuits pre 65 where you

N_x- количество импульсов fo поступивших на вход устройства время t = t_x (N_x = f₀ t_x ) .N _x - the number of pulses fo received at the input of the device time t = t _x (N _x = f ₀ t _x ).

Таким образом, после поступления на вход устройства N_x импульсов в ' счетчике Сформируется код числа (16), часто£16)Thus, after the input of the device N _x pulses in the 'counter, a code of the number (16) is formed, often £ 16)

765802.765802.

пропорционального кубическому корню от количества входных импульсов N_x.proportional to the cubic root of the number of input pulses N _x .

В соответствии с фиг.1 на вход счетчика 2 поступает частота f_s(t) с выхода группы 3 элементов И. Поскольку счетчик 2 суммирующий, то его код N определяется выражениемIn accordance with figure 1, the frequency f _s (t) from the output of group 3 of elements I is received at the input of counter 2. Since counter 2 is summing, its code N is determined by the expression

Подставив выражение(!О) в выражение (6) , получим ³ atSubstituting the expression (! O) into expression (6), we obtain ³ at

Подставив выражение (18) в выражение (17)> получимSubstituting expression (18) into expression (17)> we obtain

С учетом выражения (16) из выражения (19) получимGiven expression (16) from expression (19) we obtain

Таким образом, после поступления на на вход устройства N* импульсов в счетчике 2 формируется код числа (20), пропорционального кубическому от квадрата количества входных импуль повысить его точность (погрешность округления уменьшена в Vn*™ раз) и расширить его функциональные возможности (появилась возможность получать кубический корень из квадрата количества входных импульсов).Thus, after N * pulses are received at the device input, a code of number (20) is generated in the counter 2, which is proportional to the cubic of the square of the number of input pulses, increase its accuracy (rounding error is reduced by a factor of Vn * ™) and expand its functionality (the opportunity get the cubic root of the square of the number of input pulses).

Claims

The invention relates to computing and can be used to calculate the cubic root. BACKGROUND OF THE INVENTION There are known devices for extracting a cubic root, containing a counter, two adders, three groups of coincidence circuits, and delay lines. Device: excessive complexity and insufficient accuracy. The closest to the technical essence of the invention is a device for extracting a cubic root, the first counter containing discharge outputs of which are connected to control inputs of elements AND of the first group, second counter and third group of elements AND. In addition, the device contains an additional counter. The device works in the following way. The device receives the frequency, f. At the output of the first and second frequency counters, respectively, f fo / N n VVN, where N is the number corresponding to the code in the third counter. Since the third counter is summing, the number N corresponding to the code in the third counter is determined by the expression tx N - j, dt, where tf is the time during which the frequency fo arrives at the input of the device. Taking into account expressions (1) and (2), we get S / D; 1b) where NX fo tx is the number of pulses of the fo frequency received at the device input during the time tx. However, this device is excessively complex, since it contains three counters, has limited functional capabilities, since it determines the cube root only from the number of input pulses and has insufficient accuracy with large values of the number of input pulses. For example, if in the third counter there is a code of the number N ,, then in order for the code of the third counter to correspond to the number H, j it is necessary MO, in accordance with the expression (3) to feed the device input the number of pulses equal to -K) - (i from which implies that if the number of input pulses is less than l, with, "1, then the code of the third counter will not change and will correspond to the number NT, which indicates a significant rounding error when calculating the result (3). The purpose of the invention is to improve the accuracy and simplify the device. this is what thu The device for extracting a cubic Korn, containing the first counter, the bit inputs of which are connected to the control inputs of the AND elements of the first group, the second counter and the second group of elements, and the addition-subtraction unit, the first input of which is the device input, the second input is connected to you; by the moves of the elements of the second group, and the output is connected to the input of the first counter, the bit outputs of which are connected to the first inputs of the elements of the second group, the second inputs of which are connected to the bit inputs of the second tchika whose input soy union of the outputs of AND gates of the first group. FIG. 1 shows a block diagram of the device; in fig. 2 A possible embodiment of the addition-deduction block, Scheme (see Fig. 1) contains counters 1 and 2, groups 3 and 4 of elements AND, addition-deduction block 5, which includes block 6 in terms of frequencies and block 7 frequency subtraction. The device for extracting the cubic root works as follows: In the initial state, the triggers of counters 1 and 2 are in the zero state. To the first input of block 5 from the flip-off frequency, the given frequency fq arrives, and from the output of block 5, the frequency fg goes to the input of counter 1. Counter 1 with a group of 3 elements I is a binary frequency multiplier controlled by a code of numbers Z ( t) counter 1, therefore the frequency f,) at the output of a group of 3 elements And is determined by the expression, tV3 ("where N is the conversion ratio of counter 1) n is the number of binary digits of counter 1, C of the output of group 3, frequency fij, (t ) is fed to the input of counter 2. Counter 2 with a group of 4 matching schemes is a binary multiply Only frequencies controlled by the code of the number r (t) of the counter 1. Therefore, the frequency f4ft) at the output of the group 4 of the elements AND is determined by the expression fb (t) -l (t) where the counter conversion factor 2 n is the number of binary bits of counter 2. Substituting expression (6) in (7), we get f5 (M) Since counter 1 is summing and the frequency fjt arrives at its input t) from the output of block 5, its code z (t) is determined by the expression 2-Ct) -Jf5 (t) dt, where t is the current time. Differentiating expression (9); get; 1 O) by expressing the expression (8), we get r, N; ... Since the frequency t (t) from the output of a group of 4 elements And goes to the second input of block 5 addition-subtraction of frequencies, the first input of which receives the clock frequency fo, then, in accordance with figure 2, the frequency fg (t) output is determined by the expression) Substituting the expression (11 into the expression (12), we get i (t) dzct). whence 22 (t) drCt). Integrating expression (14), we obtain r-.fNHl 1 1о oNjT, t.g, whence when. . ,, where NX is the number of pulses of the frequency fo received at the device input during the time t tx (). Thus, after the NX pulses arrive at the input of the device, counter 1 forms a code of number (16) proportional to the cube root of the number of input pulses MD. In accordance with FIG. 1, the input of counter 2 receives the frequency fj (t) from the output of group 3 of the elements I. As the counter 2 is summing, its code N is defined by the expression Ctydit C17 Substituting the expression (I 0) into expression (6), get #) - C18 Substituting expression (18) into expression (17), we get. , .. (t m- | 8 "„ il о. / Taking into account expression (16), we get from expression (19). Thus, after the input of the device NX pulses in counter 2, a code of the number is formed (20 proportional to cubic from Comparing expression (3) with (16), we see that in the proposed device the rounding error is reduced by a factor of X. Consequently, the inclusion of a frequency addition unit in the device and changing communications made it possible to significantly simplify the device (the number of counters decreased from three to two boost it Intensity (rounding error is reduced by VN times) and expanding its functional capabilities (it was possible to obtain a cube root from the square of the number of input pulses). Apparatus of Invention A device for extracting a cubic root containing the first counter, the discharge outputs of which are connected to the control The first inputs of the elements of the first group, the second counter and the second group of elements AND, differing from the fact that., in order to increase the accuracy, a subtraction frequency addition unit is inserted in it, the first input of which is the input of the device, the second input is connected to the outputs of elements AND of the second group, and the output is connected to the input of the first counter, the bit outputs of which are connected to the first inputs of elements AND of the second group, the second inputs of which are connected to the discharge outputs of the second counter, the INPUT of which is connected to the outputs elements And the first group. Sources of information taken into account during the examination 1. USSR author's certificate 517891, cl. G 05 F 7/38, 1S73.

2. Melnikov A.A. , Rizhevsky A.G., Trifonov E.F. Processing of frequency and time pulse signals. M., Energie, 1976, p.92, fig.77 (prototype).