Application of Trigonometric Interpolation in Discrete Measurement Problems
Abstract Generated abstract
This paper examines the use of trigonometric interpolation for reconstructing a continuous distributed parameter from discrete measurements taken at uniformly spaced points. It formulates the interpolation polynomial through Fourier Lagrange coefficients and analyzes two components of approximation error: an unavoidable part caused by measurement uncertainty and a reducible part associated with truncating the interpolation series. The analysis shows that the root mean square unavoidable interpolation error equals the root mean square error of the individual measurements, while estimates for the reducible error are derived from bounds on the Fourier Lagrange coefficients using Abel transformation. The paper identifies the bounded variation of the sampled function as the condition governing convergence and provides formulas relevant to selecting the number of interpolation nodes.
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APPLICATION OF TRIGONOMETRIC INTERPOLATION IN DISCRETE MEASUREMENT PROBLEMS. I
I. E. NAATS
(Presented by the scientific seminar of the Computing Laboratory of TPI)
The method of discrete measurement of continuous quantities is finding the widest application in measuring technology, especially in monitoring distributed technological parameters. In this method, continuous recording of the distribution of the parameter \(P(l)\) over the length of a field \(L\) is replaced by discrete measurements at individual points of the field \(l_k\). When it is necessary to represent the continuous variation of the parameter \(P(l)\) on the interval \([0, L]\) from the discrete readings \(P(l_k)\), interpolation formulas are used.
This article considers the basic questions of applying trigonometric interpolation to solve this problem.
Trigonometric interpolation polynomials, owing to their connection with Fourier series and harmonic analysis, are very clear and convenient for applications.
The canonical form of a trigonometric polynomial is
\[ T_n(l)=A+\sum_{m=1}^{n}\left(a_m \cos \frac{2\pi}{L}ml+b_m \sin \frac{2\pi}{L}ml\right). \tag{1} \]
Here the coefficients
\[ A=\frac{1}{2n+1}\sum_{k=0}^{2n}P(l_k); \qquad a_m=\frac{2}{2n+1}\sum_{k=0}^{2n}P(l_k)\cos \frac{2\pi}{L}ml_k; \]
\[ b_m=\frac{2}{2n+1}\sum_{k=0}^{2n}P(l_k)\sin \frac{2\pi}{L}ml_k \tag{2} \]
are called the Fourier–Lagrange coefficients [1]. In these expressions \(N=2n+1\) is the number of reading points (interpolation nodes), uniformly distributed along the length of the field \(L\) with step \(h=L/2n+1\). The interpolation formula (1) can also be written in the form
\[ T_n(l)=\sum_{k=0}^{2n}P(l_k) \frac{ \sin (2n+1)\frac{2\pi}{L}\frac{l-l_k}{2} }{ (2n+1)\sin \frac{2\pi}{L}\frac{l-l_k}{2} } = \sum_{k=0}^{2n}P(l_k)t_k^{(n)}(l). \tag{3} \]
In order to simplify the notation, in what follows we shall use the normalized variable \(x=2\pi l/L\), \((0 \leq x < 2\pi)\), and a function of the actual variable \(f(x)\).
1. Definition of the unavoidable interpolation error. It is known that interpolation from approximate readings obtained as a result of measurements leads to the appearance of the so-called unavoidable interpolation error \(\sigma_{\mathrm{n}}\) [2]. In contrast to the removable error \(\sigma_{\mathrm{y}}\), the unavoidable error cannot be reduced by increasing the number of nodes: it is determined by the measurement error of the readings \(\sigma_0\).
By the unavoidable error we mean the root-mean-square deviation of the interpolation polynomial \(T_n(x)\), computed from the approximate readings \(f(x_{kj})\), from the polynomial constructed from the exact values \(f(x_k)\). In accordance with what has been said, we have
\[ \sigma_{\mathrm{n}} = \sqrt{ \frac{1}{2\pi} \int_{0}^{2\pi} \sum_{k=0}^{2n} \sum_{j=1}^{q} P_j \left| f(x_k)t_k^{(n)}(x)-f(x_{kj})t_k^{(n)}(x)\right|^2 \, dx }, \tag{1.1} \]
where
\(q\) is the number of measurements at the point \(x_k\),
\(P_j\) is the probability of occurrence of the \(j\)-th measurement result \(f(x_{kj})\).
The quantity
\[ \sum_{j=1}^{q} P_j \left| f(x_k)-f(x_{kj})\right|^2 = \sigma_{f_k}^{\,2} \tag{1.2} \]
is the root-mean-square error of measuring the parameter at the point \(x_k\). Since the measurements at all points of the field are made with one and the same instrument with error \(\sigma_0\), i.e.,
\[ \sigma_{f_k}=\sigma_{f_1}=\ldots=\sigma_{f_{2n}}=\sigma_0, \]
and
\[ \int_{0}^{2\pi} \left|t_k^{(n)}(x)\right|^2 dx = \frac{2\pi}{2n+1}, \]
then finally
\[ \sigma_{\mathrm{n}}=\sigma_0. \tag{1.3} \]
Thus, the root-mean-square unavoidable error of the trigonometric interpolation formula is equal to the root-mean-square error of measuring the readings \(\sigma_0\). The presence of an unavoidable error imposes a definite limitation on the accuracy of approximation and on the choice of the number of nodes.
2. Estimation of the Fourier--Lagrange coefficients. The root-mean-square interpolation error is determined by the sum of the discarded terms of the interpolation series (1), i.e., in the general case
\[ \sigma_n = \sqrt{ \frac{1}{2} \sum_{k=n+1}^{\infty} \left(a_k^2+b_k^2\right) }. \tag{2.1} \]
At the same time, the quantity \((\sigma_n^2-\sigma_{n+1}^2)\), by which the total error decreases when a new term is added to the series, is equal to
\[ \sigma_{n,n+1} = \sqrt{ \frac{1}{2} \left(a_n^2+b_n^2\right) }. \tag{2.2} \]
To determine the indicated quantities, it is necessary to estimate the magnitude of the coefficients \(a_n\) and \(b_n\). According to (2) we have
\[ a_m=\frac{2}{2n+1}\sum_{k=0}^{2n} f(x_k)\cos mx_k . \tag{2.3} \]
Applying Abel’s transformation to this series, we obtain
\[ a_m=\frac{2}{2n+1}\left[ \sum_{k=0}^{2n-1}\bigl[f(x_k)-f(x_{k+1})\bigr]S_k+f(x_{2n})S_{2n} \right], \tag{2.4} \]
where
\[ S_k=\sum_{\nu=0}^{k}\cos mx_\nu . \tag{2.5} \]
Since
\[ S_k=\frac{1-e^{imh(k+1)}}{1-e^{imh}}, \tag{2.6} \]
where \(h=2\pi/(2n+1)\) and \(S_{2n}=0\), for estimating the modulus of the coefficient we have the inequality
\[ |a_m|\leq \frac{2}{2n+1}\max |S_k|\cdot \sum_{k=0}^{2n-1}\left|f(x_k)-f(x_{k+1})\right|. \tag{2.7} \]
The following inequality is valid:
\[ \max |S_k|\leq \frac{2n+1}{4m}. \tag{2.8} \]
Then
\[ |a_m|\leq \frac{V}{2m}, \tag{2.9} \]
where
\[ V=\sum_{k=0}^{2n-1}\left|f(x_k)-f(x_{k+1})\right|. \tag{2.10} \]
Accordingly,
\[ \sigma_n^2=\frac{V^2}{4}\cdot \sum_{k=n+1}^{\infty}\frac{1}{k^2} \tag{2.11} \]
and
\[ \sigma_{n,n+1}^{2}=\frac{V^2}{4n^2}. \tag{2.12} \]
In the formulas obtained, the quantity \(V\) is determined only by the properties of the interpolated function \(f(x)\). The boundedness of this quantity under unbounded growth of \(n\) is a necessary and sufficient condition for convergence of the interpolation series (1).
To carry out numerical calculations of the interpolation error, it is necessary to estimate the quantity \(V\). The following article gives an estimate of this quantity for a random parametric field.
Conclusions
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The irreducible error of interpolation by trigonometric polynomials constructed from approximate readings has been determined.
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An expression is given for calculating the reducible interpolation error, determined by the number of terms of the interpolation series.
References
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A. Zygmund. Trigonometric Series. Mir Publishers, 1965.
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I. S. Berezin, N. P. Zhidkov. Computing Methods, Vol. I, Fizmatgiz, 1962.