Spectrum Width of Dividing Devices
Abstract Generated abstract
This paper examines the frequency limitations of analog dividing devices in comparison with multiplier devices, focusing on how the bounded dynamic range of the divisor affects passband requirements. It represents division as multiplication by the reciprocal of the divisor, models the divisor as a sinusoid superposed on a constant level, and expands the reciprocal function into a Fourier series to estimate the number of harmonics needed for a specified relative error. The analysis derives a criterion for the spectral width of the reciprocal function and hence for the total spectrum width of the quotient. It concludes that dividing devices require substantially narrower usable passbands than comparable multipliers, especially as the divisor approaches its minimum permissible level.
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SPECTRUM WIDTH OF DIVIDING DEVICES
V. P. SHERSTOBITOV
(Presented by the Department of Mathematical and Computing Instruments and Devices)
In computing and automatic systems, dividing devices are widely used to carry out the operation of division on analog time functions. There are several methods for performing this operation, in which division is obtained by means of multiplier devices [1]. For this purpose, the multiplier device is included in the feedback circuit of an operational amplifier or is supplied with a circuit that makes it possible to obtain the reciprocal time function of one variable. Alongside this, there are multiplication circuits based on the automatic variation of the transmission coefficients of two or more channels, which in principle allow the division operation without structural supplementation by auxiliary circuits.
It should be noted that physical dividing devices have greater dynamic limitations in comparison with multiplier devices, since the function modeling the divider is bounded below and above. This is connected with the fact that mathematical division by a small quantity has no physical analogue, since it is impossible to model large numbers. It will be shown below that the passband of dividing devices, for the indicated reason, proves to be narrower in comparison with the passband of multiplier devices.
The purpose of the present work is to determine a criterion for estimating the passband of dividing devices, or, equivalently, to determine the spectrum width of the quotient function for known spectrum widths of the input functions.
Academician A. A. Kharkevich showed in [2] that the total spectrum width of two time functions with spectra bounded below and above is not less than the sum and not more than twice the sum of the spectrum widths of both factors. If the lower boundaries of the spectra of the factors adjoin the zero frequency, then it is not difficult to conclude that the total spectrum width of the product of two time functions with spectra beginning in the region of zero frequencies is equal to the sum of the upper frequencies of the factors.
Obviously, for dividing devices one may use this conclusion if one finds the spectrum width of the reciprocal of the variable modeling the divider. In doing so, it is necessary to take into account the remark concerning the boundedness of the dynamic range of the divider.
Let the division operation, obtained by any of the methods mentioned, be written in the form:
\[ U_z = U_0 \frac{U_x}{U_y}. \tag{1} \]
Here \(U_x\) and \(U_y\) are time functions modeling, respectively, the dividend and the divisor; \(U_0\) is a constant coefficient; and \(U_z\) is the output function modeling the quotient of the division of two functions.
Operation (1) is analogous to multiplication if the quantity \(\frac{1}{U_y}\) is replaced by the function \(U'_y\), i.e.,
\[ U_z = U_0 \cdot U_x \cdot U'_y. \tag{2} \]
To estimate the frequency properties of the dividing device, it is expedient to represent the divisor in the form:
\[ U_y = U_{y0} + U_{ym}\cos \omega t. \tag{3} \]
\(U_{y0}\) is a certain constant level,
\(U_{ym}\) is the amplitude of the variable component varying with frequency \(\omega\) in time \(t\).
The minimum value of the divisor in this case is determined by
\[ U_{y \min} = U_{y0} - U_{ym}. \tag{4} \]
Thus, it is sufficient to consider the spectrum width of the reciprocal function \(U'_y\), taking into account its expression through \(U_y\) in the form (3).
\[ U'_y = \frac{1}{U_y} = \frac{1}{U_{y0} + U_{ym}\cdot \cos \omega t}. \tag{5} \]
Obviously, the total spectrum width of the reciprocal function is determined by the upper frequency \(\omega\) of the variable \(U_y\).
Function (5) can be expanded in a Fourier series in harmonic components that are multiples of the frequency \(\omega\). For this we use the formulas from book [3] for even functions \(f(x)\), since the function \(U'_y\) is even.
The Fourier series in this case is written as:
\[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{+\infty} a_n\cdot \cos nx, \tag{6} \]
if \(f(x)=f(-x)\), where
\(a_0\) is the constant component,
\(a_n\) is the amplitude of the harmonic component with number \(n\),
\(x\) is the independent variable.
The amplitudes of the harmonics are found by the formula:
\[ a_n=\frac{2}{\pi}\int_0^\pi f(x)\cdot \cos nx\,dx. \tag{7} \]
In accordance with (6) and (7), we write
\[ U'_y = U'_{y0}+\sum_{n=1}^{+\infty} U'_{yn}\cdot \cos n\omega t. \tag{8} \]
When determining the spectrum width, the constant component \(U'_{y0}\) is of no interest to us; therefore we find, in general form, the quantity \(U'_{yn}\),
for which we use the definite integral calculated in the book [4].
\[ \int_{0}^{\pi} \frac{\cos nx\,dx}{1+a\cos x} = \frac{\pi}{\sqrt{1-a^{2}}} \left( \frac{\sqrt{1-a^{2}}-1}{a} \right)^{n}. \tag{9} \]
for \([a^{2}<1]\).
In the present case, instead of \(x\) one should put \(\omega t\), and the coefficient \([a]\) should be determined by the expression
\[ a=\frac{U_{ym}}{U_{y0}}. \tag{10} \]
Since we are not interested in the signs of the amplitudes of the harmonics, the expression in parentheses in (9) may be multiplied by \((-1)\). Thus, the modulus \(U'_{yn}\) is calculated as
\[ U'_{yn}= \frac{2}{U_{y0}}\, \frac{1}{\sqrt{1-a^{2}}} \left( \frac{1-\sqrt{1-a^{2}}}{a} \right)^{n}. \tag{11} \]
The spectrum of the reciprocal function is infinite, and in order to limit it we allow a certain relative error \(\delta\).
\[ \delta= \frac{U_{yn}}{U'_{y\,\max}} = U'_{yn}\cdot U_{y\,\min}. \tag{12} \]
Substituting (4) into (12) and taking (10) into account, we obtain
\[ \delta= \frac{2(1-a)}{\sqrt{1-a^{2}}} \left( \frac{1-\sqrt{1-a^{2}}}{a} \right)^{n}. \tag{13} \]
Proceeding from the minimum admissible error \(\delta\) and the maximum value of \([a]\), one should find the number of the harmonic at which the spectrum of the reciprocal function may be limited.
\[ n= \lg \frac{\delta\cdot\sqrt{1-a^{2}}}{2(1-a)} \left| \lg \left( \frac{1-\sqrt{1-a^{2}}}{a} \right) \right|. \tag{14} \]
The value of \(n\) obtained must be rounded up to the nearest larger integer value \(n_{\mathrm{н}}\).
\[ n_{\mathrm{н}}\geq n. \tag{15} \]
If the width of the spectrum of the reciprocal function is denoted by \(\Omega\), then it is determined as
\[ \Omega=n_{\mathrm{н}}\cdot\omega. \tag{16} \]
The full spectrum width of division \(\Omega_{\partial}\), for equal spectrum widths of \(U_x\) and \(U_y\), is determined by the expression:
\[ \Omega_{\partial}=(1+n_{\mathrm{н}})\omega. \tag{17} \]
Thus, for a given total spectrum width of the quotient (passband) \(\Omega_{\partial}\), the spectrum width of the divider is substantially reduced (by a factor of \(n_{\mathrm{н}}\)). The indicated circumstances should be taken into account when designing dividing devices for a specified accuracy. As an illustration, numerical examples are given:
\[ \begin{aligned} &\text{For } \delta=1\% \text{ and } a=0.9 \text{, one obtains } n=8.2,\quad n_{\mathrm{н}}=9,\\ &\phantom{\text{For }} \delta=0.1\% \quad a=0.9 \text{, } n=13.1,\quad n_{\mathrm{н}}=14,\\ &\phantom{\text{For }} \delta=1\% \quad a=0.99 \text{, } n=34.7,\quad n_{\mathrm{н}}=35,\\ &\phantom{\text{For }} \delta=0.1\% \quad a=0.99 \text{, } n=51.0,\quad n_{\mathrm{н}}=51. \end{aligned} \]
Conclusions
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The passband of dividing devices, in comparison with the passband of multiplying devices, is significantly narrower, since in order to perform the division operation with high accuracy it is necessary to pass a large number of harmonics of the divider.
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When the minimum level of the divider \(U_{y\min}\) is decreased (i.e., when \(a\) is increased), the passband of the dividing device is also considerably narrowed. Thus, the passband is determined not only by the upper frequency of the divider, but also by its magnitude.
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Since a divider requires a certain minimum level below which variation of its magnitude is not permitted, a difficulty arises in taking account of the sign of the divider. This makes it necessary for the divider either to retain a constant sign or to change it at high speed (by a jump) to the minimum level of the other sign.
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A divider in the form of expression (3) may be used in testing particular circuits of dividing devices.
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The harmonic number \(n\) according to formula (14) serves as a criterion for estimating the passband of dividing devices. If the frequency \(\omega\) is the upper frequency of the divider, then \(n\omega\) will be the maximum possible spectral width of the reciprocal function for a specified error \(\delta\).
Literature
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V. B. Smolov, A. N. Lebedev et al. Continuous-Action Computing Machines. Vysshaya shkola, 1964.
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A. A. Kharkevich. Spectra and Analysis. Gostekhizdat, 1957.
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P. I. Romanovskii. Fourier Series. Field Theory. Analytic and Special Functions. Laplace Transform. Fizmatgiz, 1961.
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I. S. Gradshtein, I. M. Ryzhik. Tables of Integrals, Sums, Series, and Products. Fizmatgiz, 1963.