UDC 681.172.088

ESTIMATION OF THE TIME QUANTIZATION INTERVAL IN DISCRETE MEASUREMENT OF THE RATE OF SLOWLY VARYING SIGNALS

V. K. Yasel’skii, V. S. Moskvin

(Presented by the scientific seminar of the Faculty of Physics and Technology)

Determining the time derivative of a slowly varying signal with a frequency spectrum from zero to several hertz has certain special features. In this case the derivative signal is small, while the receiving device has finite sensitivity. Therefore the differentiator must provide considerable amplification of the derivative signal. The inertia of the differentiator, however, is of less importance.

These conditions are quite fully satisfied by discrete-action differentiators, in which the differentiation operation is replaced by determining the increment of the input quantity over a finite time interval [1].

In constructing such differentiators, the problem arises of determining the time quantization interval of the signal that satisfies the required accuracy of derivative computation.

Let us estimate the error of discrete measurement of the rate for the function \(f(t)\). Suppose that at the initial time \(t\), preceding determination of the derivative, the device measures and stores the quantity \(f(t)\). After a fixed time interval, the new value of the quantity \(f(t+\Delta t)\) is measured and stored. The first-order finite difference is then determined,

\[ f'(t) \simeq \frac{f(t+\Delta t)-f(t)}{\Delta t}, \tag{1} \]

which we shall regard as the measured rate.

Taking into account that measurement of the function \(f(t)\) is performed with a certain static error \(\varphi_c\), the difference between the computed and the true rate

\[ \varphi = \frac{f(t+\Delta t)\pm \varphi_c - f(t)\pm \varphi_c}{\Delta t} - f'(t) \tag{2} \]

characterizes the error of discrete rate measurement [2].

The maximum value of this error is equal to

\[ \varphi_{\max} = \frac{f(t+\Delta t)-f(t)}{\Delta t} - f'(t) + \frac{2\varphi_c}{\Delta t}. \tag{3} \]

Formula (3) may be used to estimate the value of the interval \(\Delta t_{\mathrm{opt}}\), which provides the minimum upper bound of the measurement error—

of the velocity at a given value of \(\varphi_c\). For this, in expression (3) we expand the function \(f(t+\Delta t)\) in a Taylor series

\[ f(t+\Delta t)=f(t)+\frac{\Delta t}{1!}f'(t)+\frac{\Delta t^2}{2!}f''(t)+\cdots \tag{4} \]

Restricting ourselves to the first three terms of the series, we obtain

\[ \varphi_{\max}\approx \frac{1}{\Delta t} \left[ \frac{\Delta t^2}{2!}\,|f''(t)|+2\varphi_c \right]. \tag{5} \]

To estimate the upper limit of the error, the quantity \(|f''(t)|\) in expression (5) is replaced by \(|f''(t)|_{\max}\). Then

\[ \varphi_{\max}\approx \frac{\Delta t}{2}|f''(t)|_{\max} +\frac{2\varphi_c}{\Delta t}. \tag{6} \]

Examining relation (6) for a minimum with respect to \(\Delta t\), we find that

\[ \Delta t_{\mathrm{opt}}\approx 2\left(\frac{\varphi_c}{|f''(t)|_{\max}}\right)^{1/2}. \tag{7} \]

When estimating the value \(\Delta t_{\mathrm{opt}}\) according to (7), it is necessary to know the analytic expression of the function under study \(f(t)\), which is quite often difficult. In this case it is expedient to use an approximate representation of the signal, provided that the degree of approximation does not exceed certain admissible values.

Usually the spectral properties of the measured signal are such that the main part of the spectral energy is concentrated in some frequency range from zero to \(\omega_c\). Outside this range, the total spectral energy is sufficiently small. This assumption is all the more justified if the function \(f(t)\) is supplied to the differentiator from the output of the instrument (it may be the sensor of the measured quantity), which is a filter for the higher frequencies of the spectrum of the function.

The error of an approximate representation of a signal with an unlimited spectrum is estimated by the relative value of the root-mean-square error, determined by the inequality [3]

\[ \left(\frac{E_m}{E}\right)^{1/2} \le \left( \frac{ \displaystyle \int_{-\infty}^{+\infty} [f(t)-f_1(t)]^2\,dt }{ \displaystyle \int_{-\infty}^{+\infty} f^2(t)\,dt } \right)^{1/2} \le 1.73\left(\frac{E_m}{E}\right)^{1/2}, \tag{8} \]

where \(E_m\) is the energy contained in the high-frequency part of the spectrum;
\(E\) is the total energy of the spectrum;
\(f_1(t)\) is the approximate function.

Inequality (8) makes it possible, for a given value of the root-mean-square error of the approximate representation of the function \(f(t)\), to determine the maximum spectral frequency \(\omega_c\).

If the input signal is considered to be a function \(f_1(t)\) with a limited frequency spectrum, then, applying Bernstein’s inequality to estimate the second derivative of the signal under study, we obtain

\[ |f''(t)|\le \sup |f_1(t)|\cdot \omega_c^2. \tag{9} \]

Consequently, for the value \(\Delta t_{\mathrm{opt}}\) one may write

\[ \Delta t_{\mathrm{opt}}\ge 2\left( \frac{\varphi_c}{\sup |f_1(t)|\cdot \omega_c^2} \right)^{1/2}. \tag{10} \]

Knowing the numerical value of the permissible static error, whose magnitude is determined on the basis of the specific operating conditions of the device, and the maximum spectral frequency of the input signal, one can estimate the value of the optimal interval \(\Delta t_{\mathrm{opt}}\).

It should be noted that a refined value of the interval \(\Delta t\) can be found through a more detailed analysis of the spectrum of the signal under study.

REFERENCES

1. S. N. Diligenskii. Differentiation of slowly varying signals. Avtomatika i telemekhanika, 1960, vol. 21, No. 4.

2. V. A. Levidov, O. N. Tikhonov. On the upper limit of the error in discrete measurement of velocities and accelerations. Izv. vuzov SSSR “Priborostroenie”, 1965, vol. 18, No. 3.

3. A. V. Solodov. Information Theory and Its Application to Problems of Automatic Control and Monitoring. Moscow, Nauka, 1967.