On a Linear Transformation of a Poisson Flow of Pulses of Random Intensity
Abstract Generated abstract
This paper analyzes the output of a linear filter driven by a Poisson pulse flow whose intensity is a stationary random function of time, a model relevant to errors in radiometric isotope instruments inspecting heterogeneous materials or rough surfaces. By approximating the continuous random intensity process with a discrete representation and passing to the limit, the authors derive expressions for the mean and variance of the filter output without earlier restrictions on the sign of the correlation function or on the smallness of intensity fluctuations. The result shows that stationary intensity fluctuations do not bias the mean output, while the variance contains the usual Poisson term plus a term determined by the intensity variance, its correlation function, and the filter response. Asymptotic cases and an extension to independent random pulse amplitudes are also given for use in instrument design.
Full Text
ON A LINEAR TRANSFORMATION OF A POISSON FLOW OF PULSES OF RANDOM INTENSITY
F. M. Zavyalkin, M. S. Kvasnitsa
(Presented by the scientific seminar of the Scientific Research Institute of Electron Introscopy)
In the inspection of materials and products with a heterogeneous structure or with a rough surface, the problem arises of estimating the contribution of fluctuations of the parameters of the object under inspection to the error in the operation of a radiometric isotope instrument. This problem is, in essence, the problem of determining the variance \(D[Y(t)]\) and the mean \(\overline{Y(t)}\) at the output of a linear filter when a Poisson flow of electrical pulses is applied to its input, the intensity of which is described by a stationary random function of time \(\lambda(t)\). It is assumed that the mean \(\overline{\lambda(t)}=\lambda_0\) and the correlation function \(B(\tau)=\sigma_\lambda^2 R(\tau)\) of the process \(\lambda(t)\) are known.
In [1] this problem was solved under the restrictions of nonnegativity of \(R(\tau)\) and smallness of the fluctuations of the process \(\lambda(t)\). These restrictions made it possible to consider the correlated flow of pulses at the filter input as a superposition of independent flows of a simpler structure and to compute the central moments of any order of the process \(Y(t)\).
However, it is of practical interest to compute \(\overline{Y(t)}\) and \(D[Y(t)]\) as the most important characteristics of the process \(Y(t)\) in determining the quality of operation of a radiometric instrument without imposing the above restrictions. The process
\[ Y(t)=\sum_{j=1}^{N} h(t-t_j). \]
where \(h(\tau)\) is the response of the filter to an individual pulse;
\(N\) is the number of pulses that have arrived during the observation interval \((0,T)\);
\(t_j\) are the moments of appearance of the pulses.
To solve this problem, let us represent the continuous process \(\lambda(t)\) by a discrete random process. In Fig. 1, using as an example an arbitrary sample function of the process \(\lambda(t)\), the data of the representation are shown and the introduced notation is defined. As \(n\to\infty\) and \(\Delta\to 0\), the error of this representation tends to zero in the limit.
In this case
\[ \overline{Y(t)} \simeq \sum_{i=1}^{n}\sum_{j=1}^{m_i} h\left[\Delta(n+1)-\Delta i-\tau_{ij}\right], \tag{1} \]
where \(m_i\) is the number of pulses that have arrived in the \(i\)-th time interval;
\(\tau_{ij}\) are the moments of appearance of the pulses in the \(i\)-th interval.
Fig. 1. Representation of the random process \(\lambda(t)\).
\[ \overline{Y(t)} \simeq \sum_{i=1}^{n} \left\langle\left\langle \sum_{j=1}^{m_i} h\,[\Delta(n+1-i)-\tau_{ij}] \right\rangle_{\tau_{ij}}\right\rangle_{m_i}, \]
where the symbol \(\langle\cdot\rangle_x\) denotes averaging with respect to the argument \(x\).
Taking into account that \(\tau_{ij}\) are ordered observations from a uniform distribution on the interval \(\Delta\), and \(\overline{m_i}=\lambda_i\Delta\),
\[ \overline{Y(t)} \simeq \left\langle \sum_{i=1}^{n} \lambda_i \int_{\Delta(n-i)}^{\Delta(n-i+1)} h(\tau)\,d\tau \right\rangle_{\lambda}. \tag{2} \]
Expression (2), as \(n\to\infty,\ \Delta\to 0\), has as its limit the following dependence:
\[ \overline{Y(t)}=\int_{0}^{t}\lambda(u)h(t-u)\,du \]
or
\[ \overline{Y(t)}=\lambda_0\int_{0}^{t}h(\tau)\,d\tau . \tag{3} \]
Consequently, stationary fluctuations of the flow intensity do not lead to a bias in the results of measuring the characteristic of the product.
\[ \begin{aligned} \overline{Y^2(t)} \simeq\;& \sum_{i=1}^{n} \left\langle\left\langle \sum_{j=1}^{m_i} h^2[\Delta(n+1-i)-\tau_{ij}] \right\rangle_{\tau_{ij}}\right\rangle_{m_i} \\ &+2\sum_{i=1}^{n} \left\langle\left\langle \sum_{\substack{k,j=1\\ k\ne j}}^{m_i} h[\Delta(n+1-i)-\tau_{ij}] h[\Delta(n+1-i)-\tau_{ik}] \right\rangle_{\tau}\right\rangle_{m_i} \\ &+2\sum_{\substack{i,f=1\\ i\ne f}}^{n} \left\{ \left\langle\left\langle \sum_{j=1}^{m_i} h[\Delta(n+1-i)-\tau_{ij}] \right\rangle_{\tau_{ij}}\right\rangle_{m_i} \right\} \times \\ &\quad\times \left\{ \left\langle\left\langle \sum_{j=1}^{m_f} h[\Delta(n+1-f)-\tau_{ij}] \right\rangle_{\tau_{fj}}\right\rangle_{m_{fj}} \right\}. \tag{4} \end{aligned} \]
Denote by \(I_1, I_2, I_3\) the groups of terms in expression (4), respectively.
\[ I_1=\sum_{i=1}^{n}\lambda_i \int_{\Delta(n-i)}^{\Delta(n-i+1)} h^2(\tau)\,d\tau, \]
\[ \lim_{\substack{n\to\infty\\ \Delta\to 0}} I_1 = \int_0^t \lambda(u) h^2(t-u)\,du = \lambda_0 \int_0^t h^2(\tau)\,d\tau . \tag{5} \]
\[ I_2 = 2\sum_{l=1}^{n} \left\langle \sum_{i=1}^{m_l}\frac{1}{\Delta^2} \left[ \int_{\Delta(n-i)}^{\Delta(n-i+1)} h(\tau)\,d\tau \right]^2 \right\rangle m_i = \sum_{l=1}^{n}\lambda_i\Delta^2 \{h[\Delta(n-l+1)]\}^2, \]
\[ \lim_{\substack{n\to\infty\\ \Delta\to 0}} I_2 = \int_0^t \lambda^2(u)\,[h(t-u)]^2\,du = \overline{Y(t)}^{\,2} + \sigma_\lambda^2 \left[ \int_0^t h(\tau)\,d\tau \right]^2 . \tag{6} \]
\[ I_3 = 2\left\langle \sum_{\substack{f,i=1\\ i\ne f}}^{n} \left[ \lambda_i \int_{\Delta(n-i)}^{\Delta(n+1-i)} h(\tau)\,d\tau \right] \left[ \lambda_f \int_{\Delta(n-f)}^{\Delta(n-f+1)} h(\tau)\,d\tau \right] \right\rangle , \tag{7} \]
\[ \lim_{\substack{n\to\infty\\ \Delta\to 0}} I_3 = -\sigma_\lambda^2 \left[ \int_0^t h(\tau)\,d\tau \right]^2 + \sigma_\lambda^2 \int_0^t\int_0^t R(\tau-\theta)h(\theta)h(\tau)\,d\tau\,d\theta . \]
Finally we obtain
\[ D[Y(t)] = \lambda_0\int_0^t h^2(\tau)\,d\tau + \sigma_\lambda^2 \int_0^t\int_0^t R(\theta-\tau)h(\tau)h(\theta)\,d\tau\,d\theta . \tag{8} \]
Let us consider some asymptotic expressions for computing \(D[Y(t)]\).
- Let the control rate be arbitrarily small. In this case
\[ B(\tau)\simeq \sigma_\lambda^2 . \]
Then, from expressions (5–7),
\[ D[Y(t)] = \lambda_0\int_0^t h^2(\tau)\,d\tau + \sigma_\lambda^2 \left[ \int_0^t h(\tau)\,d\tau \right]^2 . \tag{9} \]
- Let the control rate be an arbitrarily large quantity. In the present case
\[ R(\tau)= \begin{cases} 1, & \tau=0,\\ 0, & \tau\ne 0, \end{cases} \]
that is, \(\lambda_i\) and \(\lambda_f\) are uncorrelated random variables. Also, from expressions (5–7),
\[ D[Y(t)] = \lambda_0\int_0^t h^2(\tau)\,d\tau . \]
Expression (8) can be generalized to the case of independent fluctuations of the amplitudes of the pulses at the filter input, by averaging expression (4) over the pulse amplitudes. Finally we obtain
\[ D[Y(t)] = \lambda_0(1+\delta_0^2)\int_0^t h^2(\tau)\,d\tau + \sigma_\lambda^2 \int_0^t\int_0^t R(\tau-\theta)h(\theta)h(\tau)\,d\tau\,d\theta . \tag{10} \]
where \(\delta_0^2\) are the relative fluctuations of the pulse amplitudes \(A\)
\[ (\delta_0^2=D[A]/\overline{A}^{\,2}); \]
\(h(\tau)\) is the response of the filter to a pulse with mean amplitude.
The expressions obtained (8–10) can be used in the development of radiometric instruments (flaw detectors, density meters, etc.).
REFERENCES
- F. M. Zavyalkin, M. S. Kvasnitsa. Defektoskopiya, 1973, No. 6, 17.