SOME ISSUES IN THE DESIGN OF DEVICES FOR RECORDING SPATIALLY DISTRIBUTED PARAMETERS

I. E. NAATS

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

When recording spatially distributed parameters (for example, temperature along the axis of a cement kiln, etc.) by means of point sensors, the problem arises of representing a continuous curve of the parameter distribution from discrete readings in order to determine the values of the parameter at any point of the field. Plotting the distribution curve can be performed manually, i.e., by entering the readings of individual sensors on a common graph and connecting these points by straight-line segments (with linear interpolation). Such a method of monitoring the spatial distribution of a parameter is not very efficient and is inconvenient.

Automation of the representation of the distribution curve from the values of the parameter at individual points leads to the need to use specialized computing devices—interpolators. This paper considers questions of designing devices for recording spatially distributed parameters by means of equally spaced point sensors for the case of a linearly distributed field, when interpolation of the curve is carried out by trigonometric polynomials.

1. Principle of constructing recording devices implementing trigonometric interpolation at equally spaced points

The spatial distribution of a parameter may be characterized by a function of spatial distribution \(f(l)\), which determines the value of the parameter at any point of the monitored field. When recording the spatial distribution is carried out by means of a group of sensors uniformly distributed in the field with interval \(h = l_k - l_{k-1}\), the representation of \(f(l)\) from the set of readings \(\{f(l_k)\}\) can be accomplished using the following interpolation formula [1]:

\[ f(l) = \sum_{k=0}^{2n} f(l_k)\cdot \varphi_k(l). \tag{1.1} \]

Here the basis representation functions are...

\[ \varphi_k(l)= \frac{\sin \dfrac{\pi N}{L}(l-kh)} {N\sin \dfrac{\pi}{L}(l-kh)}, \tag{1.2} \]

where \(N\) is the number of sampling points (sensors), \(n=\dfrac{N-1}{2}\) is the order of the interpolating polynomial, and \(L\) is the length of the field. Let us consider one of the possible methods for constructing recording devices that implements the representation of the parameter distribution function in the form (1.1). It consists in specially forming a time function \(f(t)\), whose profile on the time interval \([0,T]\) coincides with the profile of \(f(l)\) on the interval \([0,L]\). Formally, such a time function can be obtained by the change of variables

\[ l=\frac{L}{T}\cdot t \tag{1.3} \]

in expression (1.1), i.e.,

\[ f(t)=\sum_{k=0}^{2n} a_k \varphi_k(t), \tag{1.4} \]

where

\[ a_k=f(l_k) \tag{1.5} \]

and the basis functions are

\[ \varphi_k(t)= \frac{\sin \dfrac{\pi N}{T}(t-k\tau)} {N\sin \dfrac{\pi}{T}(t-k\tau)}. \tag{1.6} \]

The time functions \(\varphi_k(t)\) may be regarded as periodic with period \(T\); the parameter \(\tau\) is determined by the expression

\[ \tau=\frac{T}{L}\cdot h. \tag{1.7} \]

Fig. 1. a) curve of the parameter distribution along the field length \(L\); b) motion of the sensor along the field length \(L\) in a recording system with cyclic scanning; c) time signal \(f(t)\) in the channel of the recording system.

An example of implementing the transformation (1.3) is the time scanning of the field image in continuous-scanning systems, when the distribution is recorded by a sensor moving through the field at constant velocity (Fig. 1).

In the case where recording is carried out by point sensors uniformly installed in the field, time scanning of the parameter distribution can be implemented by means of a multiphase generator of periodic oscillations \(\varphi_k(t)\) (Fig. 2). The block diagram of such a recording device is shown in Fig. 3. From the generator, oscillations \(\varphi_k(t)\) \((k=0,1,\ldots,N-1)\) of unit amplitude are fed to the inputs of the corresponding devices, which, by means of amplitude modulation, form the components-

\(a_k \cdot \varphi_k(t)\) of the time signal \(f(t)\). The summation of all components of the time function \(f(t)\) is carried out at the common input of the recording device RD. A multiphase generator of oscillations \(\varphi_k(t)\)

Fig. 2. Periodic function \(\varphi_k(t)\).

Fig. 3. Structural diagram of the recording device.

can be implemented by the methods described in [2]. The total time for recording the distribution curve in such a system is equal to \(T\).

2. Determination of the Main Characteristics of the Recording Device

One of the principal questions in designing devices for recording spatially distributed parameters is the determination of the corresponding distance between sensors in the field that is necessary for recording the distribution with one or another degree of accuracy.

In solving this problem, it is natural to assume that the distance between sensors (the quantization interval) is determined by the following

data: the structural properties of the parametric field (the parameter field is regarded as random), the error of the recording means, and the method of representing a continuous distribution curve by discrete samples (the type of interpolation).

The error in representing the function \(f(l)\) by the trigonometric polynomial \(T_n(l)\), i.e., the interpolation error, is determined by the following expression:

\[ \delta_n=\sqrt{\frac{1}{2}\sum_{k=n+1}^{\infty}\left(a_k^2+b_k^2\right)}, \tag{2.1} \]

where \(a_k\) and \(b_k\) are the coefficients of the discarded terms of the series. With the addition of a new term to the series, the interpolation error decreases by the amount

\[ \delta_k=\sqrt{\frac{1}{2}\left(a_k^2+b_k^2\right)}. \tag{2.2} \]

Since the values of the samples \(f(l_k)\) are known only with a certain degree of accuracy, it is natural to truncate the series, and in this case the criterion for truncation may be the consideration that the interpolation accuracy should not exceed the accuracy of the initial data. Taking this into account, the condition for truncating series (1.1) is written in the form

\[ \delta_n=\delta_0=\sqrt{\frac{1}{2}\left(a_n^2+b_n^2\right)}, \tag{2.3} \]

where \(\delta_0\) is the root-mean-square error in measuring the parameter values.

Having determined from equation (2.3) the value of the number \(n\), which determines the degree of the interpolation polynomial, one can calculate the required distance between sensors

\[ h_0=\frac{L}{2n+1}. \tag{2.4} \]

The number \(n\) can be determined from expression (2.3) by the method of successive approximation, i.e., by calculating the series of values \(\delta_n\) so that \(\delta_m>\delta_{m+1}>\delta_{m+2}\ldots>\delta_0\geq\delta_n\). The number \(n\) is determined from the condition \(\delta_0\geq\delta_n\).

Such a calculation method is rather complicated and not always feasible; however, using relation (2.3), one can obtain a simpler method for determining the value \(h_0\). For this purpose one may use estimates of the coefficients \(a_n\) and \(b_n\), which can be obtained simply if assumptions are made in advance about the analytic properties of the function \(f(l)\). In monitoring a spatial parametric field, it is natural to make the following assumptions: the distribution function \(f(l)\) is continuous on the interval \([0,L]\), together with its first derivative \(f'(l)\), which has bounded variation \(V_1\).

The continuity condition for \(f'(l)\) means continuity of the gradient at any point of the physical field, and the presence of bounded variation \(V_1\) means that the field gradient has a finite number of minima and maxima on the interval \([0,L]\). Further, assuming that the functions \(f(l)\) and \(f'(l)\) satisfy the boundary conditions

\[ f(0)=f(L), \]

\[ f'(0)=f'(L), \]

for the Fourier coefficients the following estimates can be obtained [3]:

\[ |a_n| < \frac{L V_1}{2\pi^2 n^2}, \]

\[ |b_n| < \frac{L V_1}{2\pi^2 n^2}, \tag{2.5} \]

and equation (2.3) is written in the form

\[ \delta_0^2 = \frac{L^2 V_1^2}{4\pi^4 n^4}. \tag{2.6} \]

For a random parametric field, the total variation of the gradient \(V_1\) is also a random function; therefore expression (2.6) can be rewritten as

\[ \delta_0^2 = \frac{L^2 \overline{V_1^2}}{4\pi^4 n^4}. \tag{2.7} \]

Let us introduce the relative root-mean-square error of recording

\[ \varepsilon = \sqrt{ \frac{\overline{[\overline{f}(l)-f(l)]^2}} {\overline{[f(l)^2]}} } = \frac{\delta_0}{\sqrt{\overline{[f(l)^2]}}}, \tag{2.8} \]

where \(\overline{f}(l)\) is the true value of the distribution function, and \(f(l)\) is the recorded value. For a homogeneous random field

\[ \overline{[f(l)^2]} = K(0), \tag{2.9} \]

where \(K(l)\) is the correlation function at the point \(h=0\). It is assumed here that the correlation function is determined with an error not exceeding \(\varepsilon\).

Using the definition of the total variation of a function on the interval \([0,L]\) [4] and the properties of the correlation function of a homogeneous random field [5], for the mean square of the variation of the field gradient \(f'(l)\) one can obtain the expression

\[ \overline{V_1^2} = 8n^2 \left[ K^{(2)}(0) - K^{(2)}(h_0) \right]. \tag{2.10} \]

Taking into account relations (2.8), (2.9), and (2.10), equation (2.7) is written in the form

\[ h_0^2 \frac{K^{(2)}(0)-K^{(2)}(h_0)}{K(0)} = \frac{\varepsilon^2 \pi^4}{8}. \tag{2.11} \]

In the particular case when the correlation function of the field can be represented in the form

\[ K(h)=K(0)\cdot e^{-\alpha |h|}, \tag{2.12} \]

equation (2.11) takes the form

\[ h_0^2 \left(1-e^{-\alpha h_0}\right) = \frac{\pi^4 \varepsilon^2}{8\alpha^2}. \tag{2.13} \]

Equation (2.13), just like equation (2.11), is transcendental with respect to \(h_0\) and can be easily solved graphically. Introduce the following notation:

\[ g_1(h)=1-e^{-\alpha |h|} \tag{2.14} \]

and

\[ g_2(h)=g_0' h^2, \tag{2.15} \]

where

\[ g_0=\frac{\pi^4\varepsilon^2}{8\alpha^2}. \tag{2.16} \]

Then the solution of equation (2.13) is determined from the equality of these functions at the point \(h=h_0\), i.e.,

\[ g_1(h_0)=g_2(h_0). \]

Graphically, the point \(h_0\) is determined by the intersection on the graph of the functions \(g_1(h)\) and \(g_2(h)\).

Let us now consider the problem of determining the cycle \(T\) for scanning \(f(l)\) by the method of time transformation. In doing so, we shall proceed from the following proposition: the cycle of complete scanning \(T\) must be chosen so small that temporal changes in the field during the time \(T\) may be neglected [6]. This requirement can be written in the form of an inequality valid for any point of the field,

\[ \sqrt{[P(t)-P(t+T)]^2}\leq \delta_0, \tag{2.17} \]

where \(\delta_0\) is the root-mean-square measurement error of the recording means.

Assuming that the temporal behavior of the parameter \(P(t)\) for any point of the field can be regarded as a stationary random process, inequality (2.17) can be reduced to the form

\[ K_t(0)-K_t(T)=\frac{\delta_0^2}{2}, \tag{2.18} \]

where \(K_t(\tau)\) is the temporal correlation function of the parametric field.

The value of \(T\) can be estimated simply graphically, since the correlation function \(K_t(\tau)\), as a rule, is determined experimentally and is given in the form of a graph.

Conclusions

1. The questions of representing the curve of the spatial distribution of technological parameters from discrete readings taken by point sensors with the aid of trigonometric interpolation polynomials have been considered. A structural diagram is presented of a device that implements such a representation by forming the corresponding time signal.

2. A method is given for calculating the principal parameters of the recording device from the statistical characteristics of the parametric field: the distance between sensors in the field and the time cycle for scanning the distribution curve.

LITERATURE

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2. A. I. Stavitskii. A multifunctional converter “politron” and its application for aperiodic frequency multiplication. Proceedings of the Scientific-Technical Conference of LEIS, 1962.

3. Sh. E. Mikeladze. Numerical Methods of Mathematical Analysis. Gostekhizdat, p. 190, 1953.

4. G. E. Shilov. Mathematical Analysis. Fizmatgiz, p. 256, 1960.

5. A. M. Yaglom. Introduction to the theory of stationary functions. Uspekhi Matematicheskikh Nauk, vol. VII, issue 5, 1952.

6. E. L. Tsikovich. Determination of the necessary measurement frequency in discrete monitoring. Avtomatika i Telemekhanika, vol. XXII, No. 2, 1961.