Determination of the Boundary Quantization Interval in Recording Spatially Distributed Parameters
Abstract Generated abstract
This paper addresses the choice of sensor spacing for recording one-dimensional spatially distributed parameters when a continuous field is reconstructed from uniformly spaced discrete readings. It develops a criterion for the boundary spatial quantization interval by equating the interpolation error of a trigonometric polynomial representation to the root-mean-square measurement error, then derives a practical equation using estimates of Fourier coefficients and the correlation function of a homogeneous random field. For an exponential correlation function, the resulting transcendental equation can be solved graphically, and an example for temperature monitoring in a cement kiln gives a boundary interval of about 28.5 m, corresponding to seven sensors over a 150 m field.
Full Text
DETERMINATION OF THE BOUNDARY QUANTIZATION INTERVAL IN RECORDING SPATIALLY DISTRIBUTED PARAMETERS
I. E. NAATS
(Presented by the joint scientific seminar of the departments of the Faculty of Physics and Engineering)
When designing devices for recording spatially distributed parameters, intended to represent a continuous distribution curve by means of a set of discrete readings (sensor indications), one of the main questions is the determination of the necessary distance between sensors.
It is quite obvious that the distance between sensors, i.e., the spatial quantization interval, must be determined by the specified magnitude of the representation (interpolation) error and by the complexity of the relief of the field being monitored. In the present work we consider the determination of the boundary quantization interval \(h_o\), when the sensors are distributed uniformly in the field and the representation of the distribution curve \(f(l)\) from the discrete readings \(f(l_k)\) is carried out with the aid of trigonometric interpolation polynomials. Here, by the boundary interval of spatial quantization \(h_o\) is meant a certain distance between neighboring sensors in the field such that recording with an interval \(h<h_o\) does not eliminate the uncertainty in the representation of the curve caused by the error of the recording means. In other words, when recording with an interval between sensors \(h_o\), the error in representing the continuous distribution curve \(f(l)\) from the discrete readings \(f(l_k)\) does not exceed the error of measuring the parameter values at the reading points.
In deriving the computational relations for determining the quantity \(h_o\), we assume that the correlation function of the parametric field (the field of the parameter is considered random) and the root-mean-square error of measuring the parameter values by the recording device are known.
As was noted above, the representation of the continuous curve \(f(l)\) of the parameter distribution (the one-dimensional case is considered) by means of a set of discrete readings \(\{f(l_k)\}\), taken at the field points
\[ l_k=\frac{Lk}{2n+1} \tag{1} \]
is carried out with the aid of a trigonometric polynomial.
\[ T_n(l)=A+\sum_{m=1}^{n}\left(a_m\cos\frac{2\pi}{L}ml+b_m\sin\frac{2\pi}{L}ml\right), \tag{2} \]
where
\[ A=\frac{1}{2n+1}\sum_{k=0}^{2n} f(l_k), \]
\[ a_m=\frac{1}{2n+1}\sum_{k=0}^{2n} f(l_k)\cos \frac{2\pi}{L}ml_k, \tag{3} \]
\[ b_m=\frac{1}{2n+1}\sum_{k=0}^{2n} f(l_k)\sin \frac{2\pi}{L}ml_k. \]
In relations (2) and (3), \(L\) is the length of the parametric field.
In practical problems, in particular when monitoring the spatial distribution of a technological parameter by means of point sensors, the values \(f(l_k)\) are known only with a certain degree of accuracy; therefore the coefficients determined by expression (3) are also approximate quantities.
The error of representing the function \(f(l)\) by the polynomial \(T_n(l)\), i.e., the interpolation error, is determined by the following expression [1]:
\[ \hat{\delta}_n=\sqrt{\frac{1}{2}\sum_{k=n+1}^{\infty}(a_k^2+b_k^2)}, \tag{4} \]
where \(a_k\) and \(b_k\) are the coefficients of the omitted terms of the series. With the addition of a new term to the series, the interpolation error decreases by the amount
\[ \hat{\delta}_k=\sqrt{\frac{1}{2}(a_k^2+b_k^2)}. \tag{5} \]
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; in doing so, the truncation criterion may be the consideration that the accuracy of interpolation should not exceed the accuracy of the initial data. Taking this into account, the condition for truncating series (2) may be written in the form
\[ \hat{\delta}_n=\hat{\delta}_0=\sqrt{\frac{1}{2}(a_n^2+b_n^2)}, \tag{6} \]
where \(\hat{\delta}_0\) is the root-mean-square error of measuring the parameter values.
Having determined from equation (6) the value of the number \(n\), which determines the degree of the interpolation polynomial, one can calculate the value of the limiting interval of spatial quantization
\[ h_0=\frac{L}{2n+1}. \tag{7} \]
The number \(n\) can be determined from expression (6) by the method of successive approximation, i.e., by computing the sequence of values \(\hat{\delta}_n\) so that \(\hat{\delta}_m>\hat{\delta}_{m+1}>\hat{\delta}_{m+2}\ldots>\hat{\delta}_0\geq \hat{\delta}_n\). The number \(n\) is determined from the condition \(\hat{\delta}_0\geq \hat{\delta}_n\).
This method of calculation is rather complicated and not always feasible; however, using relation (6), one can obtain a simpler method for determining the value of \(h_0\). For this purpose, one may use estimates of the coefficients \(a_n\) and \(b_n\), which can be obtained simply if certain analytic properties of the function \(f(l)\) are assumed to be known in advance.
In monitoring a spatial parametric field, it is natural to make the following assumptions: the distribution function \(f(l)\) is continuous on the segment \([O,L]\), together with the 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 existence of bounded variation \(V_1\) means that the field gradient has a finite number of minima and maxima on the segment \([O,L]\). Further, assuming that the functions \(f(l)\) and \(f'(l)\) satisfy the boundary conditions
\[ f(0)=f(L), \]
\[ f'(0)=f'(L), \]
the following estimates can be obtained for the Fourier coefficients [2]:
\[ \begin{aligned} |a_n| &< \frac{LV_1}{2\pi^2 n^2},\\[4pt] |b_n| &< \frac{LV_1}{2\pi^2 n^2}, \end{aligned} \tag{8} \]
and equation (6) is written in the form
\[ \delta_0^2=\frac{L^2V_1^2}{4\pi^4 n^4}. \tag{9} \]
For a random parametric field, respectively, we obtain
\[ \delta_0^2=\frac{L^2\overline{V_1^2}}{4\pi^4 n^4}. \tag{10} \]
Let us introduce the relative root-mean-square recording error
\[ \varepsilon= \sqrt{ \frac{M\left[(\bar f(l)-f(l))^2\right]} {M\left[f(l)^2\right]} } = \frac{\delta_0}{\sqrt{M\left[f(l)^2\right]}}, \tag{11} \]
where \(\bar f(l)\) is the true value of the distribution function, \(f(l)\) is the recorded value, and \(M\) is the mathematical-expectation operator.
For a homogeneous random field [3]
\[ M\left[\bar f(l)^2\right]=K(0), \tag{12} \]
where \(K(0)\) 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 segment \([O,L]\) [3] and the properties of the correlation function of a homogeneous random field [4], for the mean square of the variation of the field gradient \(f'(l)\) one can obtain the expression
\[ \overline{V_1^2}=8n^2\left[K^{(2)}(0)-K^{(2)}(h_0)\right]. \tag{13} \]
Taking into account relations (11), (12), and (13), equation (10) is rewritten in the form
\[ h_0^2\left[ \frac{K^{(2)}(0)-K^{(2)}(h_0)}{K(0)} \right] = \frac{\pi^4\varepsilon^2}{8}. \tag{14} \]
In the particular case where the correlation function of the field can be represented in the form
\[ K(h)=K(0)e^{-\alpha |h|}, \tag{15} \]
equation (15) takes the form
\[ h_0^2\left(1-e^{-\alpha h_0}\right)=\frac{\pi^4\varepsilon^2}{8\alpha^2}. \tag{16} \]
Equation (17), like equation (15) with respect to \(h_0\), is transcendental and can be solved simply by graphical means. Indeed, let us denote, for example, in equation (17)
Fig. 1.
\[ g_1(h)=1-e^{-\alpha |h|} \tag{17} \]
and
\[ g_2(h)=\frac{g_0}{h^2}, \tag{18} \]
where
\[ g_0=\frac{\pi^4\varepsilon^2}{8\alpha^2}. \tag{19} \]
Then the solution of equation (17) is determined from the equality of these two 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 plot, of the functions \(g_1(h)\) and \(g_2(h)\).
Example. To illustrate this method, let us consider the example given in [5]. In this example, the distance between sensors is calculated for monitoring the temperature field of a cement kiln. The calculation is carried out on the basis of representing the distribution function by means of Lagrange polynomials. In the present case the correlation function of the field is represented in the form
\[ K(h)=K(0)e^{-0.019|h|}. \]
For an error of the recording instruments \(\varepsilon=0.1\), \(g_0=337\), and, accordingly, the functions are
\[ g_1(h)=1-e^{-0.019|h|}, \]
\[ g_2(h)=337/h^2. \]
The graphs of these functions are shown in Fig. 1, from which the quantization interval \(h_0=28.5\ \text{m}\) is determined as the point of their intersection. For a field length \(L=150\ \text{m}\), the required number of sensors is \(N=7\).
Thus, for the given case, using the readings of seven sensors at each instant of time, with the aid of the trigonometric interpolation polynomial (2) of degree \(n=3\), it is possible to represent completely—that is, with the accuracy of the recording instruments—the continuous temperature distribution along the field.
Conclusions
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The quantization interval of the function of the spatial distribution of the parameter \(h^\circ\) is determined by its structural properties and by the mean-square measurement error.
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By using the method of truncating the series (2) for the function \(f(l)\) and the corresponding estimates of the expansion coefficients, an equation has been obtained for determining the limiting quantization interval for a specified measurement error \(\varepsilon\) and the statistical characteristics of the parametric field.
References
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A. M. Yaglom. Introduction to the theory of stationary functions. Uspekhi Matematicheskikh Nauk, vol. VII, issue 5, 1952.
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E. L. Itskovich. Determination of the distance between sensors in monitoring spatially distributed fields. Avtomatika i Telemekhanika, vol. XXIV, No. 2, 1963.