Application of Trigonometric Interpolation in Discrete Measurement Problems
Abstract Generated abstract
This paper applies trigonometric interpolation to discrete measurement of stationary processes and homogeneous random fields when samples are affected by measurement error. Using the known correlation function, it derives an upper bound for the mean square of a structural variation quantity and relates this bound to sensor spacing and the number of interpolation nodes. The argument yields formulas for choosing a reasonable number of samples for a prescribed approximation error and for estimating the root mean square error obtainable from a fixed number of samples. It concludes that, under the stated limiting assumptions, unreliable higher trigonometric terms should be discarded and the attainable total approximation error from approximate samples is bounded by a value proportional to the measurement error.
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APPLICATION OF TRIGONOMETRIC INTERPOLATION IN PROBLEMS OF DISCRETE MEASUREMENT. II
I. E. Naats
(Presented to the scientific seminar of the computing laboratory of TPI)
3. Estimation of the mean square of the quantity \(V\) for a random function
We shall assume that the realizations of some random homogeneous field (respectively, a stationary process), for which the correlation function \(K(h)\) is known, are subjected to interpolation. It is precisely such problems that one has to deal with in discrete measurement and control of technological parameters [1].
For a random function \(f(x)\), with fixed nodes \(x_k\), the sum
\[ \sum_{k=0}^{2n-1} \left| f(x_k)-f(x_{k+1}) \right| \]
takes random values from one realization to another and, consequently, is a random variable.
Let us construct an estimate of the mean square of this quantity for a homogeneous field \(f(x)\). According to (2.8), we have
\[ M[V^2] = M\left[ \left( \sum_{k=0}^{2n-1} \bigl(f(x_k)-f(x_{k+1})\bigr) \right)^2 \right]. \tag{3.1} \]
Using the inequality
\[ \left(\sum_{i=1}^{n} a_i\right)^2 \leq n \sum_{i=1}^{n} a_i^2 \]
(a special case of Cauchy’s formula), we transform the right-hand side of (3.1):
\[ M\left[ \left( \sum_{k=0}^{2n-1} \bigl(f(x_k)-f(x_{k+1})\bigr) \right)^2 \right] \leq 2nM\left[ \sum_{k=0}^{2n-1} \bigl(f(x_k)-f(x_{k+1})\bigr)^2 \right] = \]
\[ = 2n \sum_{k=0}^{2n-1} \left( M[f(x_k)^2] + M[f(x_{k+1})^2] - 2M[f(x_k)\cdot f(x_{k+1})] \right). \tag{3.2} \]
The terms in the last expression are nothing other than the values of the correlation function at the corresponding points, i.e.
\[ M[f(x_k)\cdot f(x_{k+1})]=K(x_k,x_{k+1}), \]
\[ M[f(x_k)^2]=K(x_k,x_k), \]
\[ M[f(x_{k+1})^2]=K(x_{k+1},x_{k+1}). \]
In the case of a homogeneous random field, the correlation function does not depend on the choice of the points \(x_k\) and \(x_{k+1}\); it is completely determined by the distance between these points [2].
Consequently, (3.2) finally takes the form
\[ M\left[\left(\sum_{k=0}^{2n-1}(f(x_k)-f(x_{k+1}))\right)^2\right] \leq 8n^2[K(0)-K(x_{k+1}-x_k)] . \tag{3.3} \]
For equidistant nodes,
\[ M[V^2]=\overline{V}^{\,2}\leq 8n^2[K(0)-K(h)], \tag{3.4} \]
where
\[ h=x_{k+1}-x_k=\frac{2\pi}{2n+1}. \]
4. Determination of the number of nodes for a prescribed approximation error \(\varepsilon_3\) and measurement error \(\varepsilon_0\).
If it is assumed that the correlation function \(K(h)\) is determined experimentally from readings at points of the field \(x_k\), with error of the measuring instruments \(\varepsilon_0\), then it is expedient to choose the distance between nodes not less than \(h_0\), determined by the relation [3]:
\[ K(0)-K(h_0)=\frac{\varepsilon_0^2}{2}. \]
Indeed, a further reduction of the distance between readings does not increase the accuracy of determining the correlation function, nor of any other characteristics.
On this basis, the following conclusion can be drawn: the structural characteristic \(\overline{V}^{\,2}\) of the random field \(f(x)\), determined in the present case through the correlation function according to (3.4), cannot be estimated more accurately than \(4n^2\varepsilon_0^2\), i.e.
\[ \overline{V}^{\,2} = M\left[\left(\sum_{k=0}^{2n-1}(f(x_k)-f(x_{k+1}))\right)^2\right] \leq 4n^2\varepsilon_0^2, \tag{4.2} \]
where \(n\) is determined from the equation
\[ K(0)-K\left(\frac{L}{2n+1}\right)=\frac{\varepsilon_0^2}{2}. \tag{4.3} \]
Substituting (4.2) into (2.12, part 1), we obtain
\[ \sigma_{n,n+1}=\varepsilon_0, \]
i.e., in the limiting case, when interpolating from approximate readings, the series (1, part 1) must be truncated at the term for which
\[ \sqrt{\frac{1}{2}(a_n^2+b_n^2)}\leq \varepsilon_0 . \tag{4.5} \]
All the remaining terms of the series, beginning with the \(n\)-th, are smaller in absolute value and therefore must be discarded as unreliable.
Thus, the results obtained give a solution to the following problem of discrete measurement: what number of nodes is it necessary to restrict oneself to when interpolating the relief of the field \(f(l)\) by a trigonometric polynomial, if the samples are known with root-mean-square error \(\sigma_0\).
In technical applications this problem may be formulated as follows: what number of sensors is it reasonable to restrict oneself to when monitoring the relief of a spatial parametric field, if the error of the sensors and measuring instruments does not exceed \(\sigma_0\).
We now consider the following problem of discrete measurement: with what root-mean-square error \(\sigma_m\) can the relief of the field \(f(l)\) be represented from \((2m+1)\) discrete samples, if the error in measuring the field values is \(\sigma_0\). In solving this problem we take into account that the limiting value of the interpolation error \(\sigma_m\) is equal to \(\sigma_0\) for \(m=n\).
From (2.11) we have
\[ \sigma_m^2 = \frac{\overline{V}^{\,2}}{4} \sum_{k=m+1}^{n}\frac{1}{k^2} = n^2\sigma_0^2 \sum_{k=m+1}^{n}\frac{1}{k^2}. \tag{4.6} \]
The value of the sum in (4.6) can be estimated approximately by means of Euler’s formula:
\[ \sum_{k=m+1}^{n}\frac{1}{k^2} = \left(\frac{1}{m+1}-\frac{1}{n}\right) + \frac{1}{2} \left(\frac{1}{(m+1)^2}+\frac{1}{n^2}\right) - \frac{1}{60} \left(\frac{1}{(m+1)^3}-\frac{1}{n^3}\right) +\cdots \]
Restricting ourselves to two terms of this expansion, which is quite sufficient for practical calculations, we obtain
\[ \sigma_m = \sigma_0\cdot n \sqrt{ \frac{1}{m+1} - \frac{1}{n} + \frac{1}{2} \left( \frac{1}{(m+1)^2} + \frac{1}{n^2} \right) }, \tag{4.7} \]
With the aid of this relation the inverse problem can also be solved, i.e., when it is required to determine the necessary number of interpolation nodes for a given value of the error \(\sigma_{\text{з}}\), with it known in advance that \(\sigma_{\text{з}}>\sigma_0\). The necessary number of samples \((2m+1)\) is determined by the equation
\[ \frac{1}{(m+1)^2} + \frac{2}{m+1} - \frac{2\sigma_{\text{з}}^{2}-\sigma_0}{n^2\sigma_0^2} - \frac{2}{n} = 0. \tag{4.8} \]
In conclusion, let us consider the structure of the total error of approximation from approximate samples. The total error is composed of an irreducible and a reducible error, i.e.,
\[ \sigma^2=\sigma_{\text{з}}^2+\sigma_{\text{н}}^2. \tag{4.9} \]
It was shown above that \(\sigma_{\text{н}}=\sigma_0\), and in the limiting case \(\sigma_{\text{у}}=\sigma_0\). As a result, the attainable approximation error is
\[ \sigma=\sigma_0\cdot\sqrt{2} \tag{4.10} \]
when interpolating by a trigonometric polynomial from approximate samples.
The relations for calculating the number of nodes were obtained under the assumption that realizations of the random field form a family of functions of bounded variation and, generally speaking, give an upper bound for the possible values. Indeed, usually the curves describing the distribution of physical quantities are, as a rule, smooth and continuous together with the per—
derivative (the field gradient), except, perhaps, at the ends of the interval. But in this case, as is known, \(a_n = 0\left(\dfrac{1}{n^2}\right)\), i.e., the interpolation series converges considerably faster and, consequently, a smaller number of terms is required for the same approximation error. These questions are considered in more detail in [4].
The results obtained in this work were used by the author in the development of automatic interpolators intended for recording spatial fields.
Conclusions
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An estimate is determined for the quantity \(V\), characterizing the structural properties of distribution curves, through the statistical characteristics of a random homogeneous parametric field.
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The problems of the discrete method for measuring distributed parameters are formulated, and the basic relations for calculating the required quantities are given.
LITERATURE
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E. L. Itskovich. Determination of the distance between sensors in monitoring spatially distributed fields. Avtomatika i telemekhanika, No. 2, 1963.
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A. M. Yaglom. Introduction to the Theory of Stationary Functions. Uspekhi Mat. Nauk, vol. VII, issue 5, 1952.
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E. L. Itskovich. Determination of the necessary measurement frequency in discrete monitoring. Avtomatika i telemekhanika, vol. XXII, No. 2, 1961.
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I. E. Naats. Determination of the interpolation and quantization error boundary in recording spatially distributed parameters. Izvestiya TPI, vol. 138, 1965.