Some Ways of Increasing the Sensitivity of the Scintillation Method of Betatron Defectoscopy
Abstract Generated abstract
This paper examines how to improve the sensitivity and reliability of scintillation-based betatron flaw detection, where defect signals are limited by fluctuation noise in detector and recording circuits. Using probability theory, statistical reception theory, correlation analysis, and sampling arguments, it treats defect recognition as a problem of estimating the most likely input signal from a noisy received signal and shows that optimal reception is associated with cross-correlation and matched filtering. For periodic pulse signals typical of betatron inspection, the authors argue that comb filters and synchronous accumulation can suppress broadband noise while adding useful pulses coherently, increasing the signal-to-noise ratio in proportion to the accumulation time or number of pulses.
Full Text
SOME WAYS TO INCREASE THE SENSITIVITY OF THE SCINTILLATION METHOD IN BETATRON FLAW DETECTION
V. I. GORBUNOV, A. V. POKROVSKII, A. K. TEMNIK
(Presented by the scientific seminar of NIIEI)
The scintillation method of betatron flaw detection is finding ever wider application in the automatic inspection of large metal thicknesses \[1, 2\].
Questions of increasing the sensitivity of the method and the reliability of obtaining trustworthy data on the presence of a defect are at present decisive. However, sensitivity and reliability depend in many respects on interference arising in the receiving-and-recording channel.
On the basis of probability theory and mathematical statistics, we shall carry out an analysis of fluctuation phenomena in the electrical circuits of the recording system.
Let us consider the possibilities for lowering the noise level and methods of detecting a signal at the noise level. Noises in an electrical circuit arise as a result of current fluctuations in electron tubes (shot effect), the chaotic motion of electrons in resistances and other elements of the circuit; moreover, the detector itself is a source of noise \[1, 2\]. The signal from the detector also fluctuates because of instability of the betatron radiation. Noise is an oscillatory process that varies in time according to a random law. In various electrical circuits, fluctuation oscillations of different structure are encountered. Mathematically, they can be described by means of the apparatus of probability theory.
The interference function at each instant \(t\) has some value characterized by the probability of its occurrence at the given instant of time. For a more complete description of the process of interaction between signal and interference, a two-dimensional probability is introduced into consideration, i.e., the random variable assumes at time \(t_1\) the value \(x_1\), and at time \(t_2\) the value \(x_2\). With a known two-dimensional law of probability distribution, the mutual correlation function can be determined by the formula \[4\]:
\[ R(t_1 t_2)= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} x_1(t_1)\cdot x_2(t_2)\cdot P(x_1 x_2)\,dx_1\cdot dx_2 = \overline{x_1(t_1)\cdot x_2(t_2)}, \tag{1} \]
where
\[ \begin{aligned} x_1(t_1) &\text{— the law of variation of the signal in time,}\\ x_2(t_2) &\text{— the law of variation of the interference in time,}\\ P(x_1 x_2) &\text{— the two-dimensional probability density.} \end{aligned} \]
The correlation function is the mean degree of probability of connection between the processes \(x_1\) and \(x_2\) at two instants of time \(t_1\) and \(t_2\) [4].
A recording circuit that ensures the best reception of a signal in the presence of noise is called optimal. Such a circuit provides the maximum probability of undistorted signal reception. However, under the action of noise the signal is substantially distorted; therefore, uncertainty arises in receiving the signal.
Proceeding from this, the determination of the defect signal is a statistical problem: from the known signal \(y\) at the output, to learn everything about the signal \(X\) at the input of the circuit.
Mathematically, this problem consists in determining the a posteriori probability distribution \(P_y(x)\) [5]. This distribution is found, as a rule, by analyzing numerous experiments.
In fact, the a posteriori probability gives us everything that can be learned about the signal at the input of the recording circuit; therefore, in what follows we shall call such a circuit ideal.
Another defining quantity for reliable signal reception is the quantity \(P_x(y)\)—the conditional probability distribution that the received signal \(y\) is the result of the superposition of interference on the input signal \(X\).
\(P_x(y)\) may be regarded as a function of \(X\) for a given received signal \(y\); in this case it shows which transmitted signal \(X\) most plausibly corresponds to the received signal \(y\).
Consequently, to solve the reception problem it is necessary to perform an operation that would determine the likelihood function,
\[ L(x)=P_x(y), \tag{2} \]
where \(L(x)\) is the likelihood function.
Let us consider the case in which noise in the form of “white” noise is superposed on the signal \(x(t)\), which exists during the time \(T\) required for the defect to pass in front of the collimator window of the radiation receiver:
\[ y(t)=x(t)+n(t), \tag{3} \]
where \(n(t)\) is the law of variation of the interference with time.
To find the likelihood function, which is equivalent to calculating the probability \(P_x(y)\), we use the Kotelnikov theorem [3], which defines the function \(x(t)\) through its discrete calculated values at intervals \(\Delta t\):
\[ \Delta t=\frac{1}{2F_0}, \tag{4} \]
where \(F_0\) is the width of the frequency spectrum.
The number of discrete values representing the given function \(y(t)\) over the time interval \(T\) is equal to:
\[ m=\frac{T}{\Delta t}=2F_0T. \tag{5} \]
Applying the theorem on multiplication of \(m\) independent events to the discrete values, we obtain [6]
\[ L(x)=P_x(y)=\prod_{k=1}^{m} P_x(y) =(2\pi\sigma_{\mathrm{ш}})^{-F_0T} \exp\left[-\frac{1}{G}\sum_{k=1}^{2F_0T}|y(t)-x(t)|^2\right], \tag{6} \]
where \(\sigma_{\mathrm{ш}}\) is the noise variance,
\[ G=\frac{\sigma_{\mathrm{ш}}^{2}}{F_0} \]
is the spectral density of the noise power.
The function \(L(x)\) is maximal when the transmitted and received signals coincide over the interval under study. Replacing the limiting sum in expression (6) by an integral, let us consider its component parts:
\[ \int_{0}^{T} [y(t)-x(t)]^{2}\,dt = \int_{0}^{T} [y(t)]^{2}\,dt + \int_{0}^{T} [x(t)]^{2}\,dt - 2\int_{0}^{T} y(t)\cdot x(t)\cdot dt . \tag{7} \]
The first integral characterizes the weight of the received signal, the second the weight of the transmitted signal, and the third expresses the interrelation between the transmitted and received signals. Therefore, in order to characterize fully the input signal \(x(t)\) from the received signal \(y(t)\), it is necessary to know that the received signal \(y(t)\) corresponds to the given original signal \(x(t)\). It follows from this that, to determine the likelihood function \(l(x)\), it is sufficient to compute the integral
\[ g(t)=\int_{0}^{T} y(t)\cdot x(t)\,dt . \tag{8} \]
The expression obtained is an analogue of the cross-correlation function between the received and transmitted signals.
Thus, for the best reception of the signal, the detectors and the recording circuit must be of the cross-correlated type; however, the quality of reception depends strongly on the magnitude of the spectral density of the noise, i.e., the signal-to-noise ratio is improved when frequency filters are used in the circuits. Let us consider the frequency characteristic of a matched filter. The spectrum of the frequencies passed for a unit signal is described by formula [4]:
\[ S(\Omega)=\int_{0}^{T_u} e^{-j\omega t}\,dt . \tag{9} \]
However, when the signal from a defect is periodized, the useful information is contained in a sequence of pulses. To determine the spectrum consisting of \(N\) pulses, we shall use the linearity of the Fourier transform [6]:
\[ S(\omega)=S_{0}(\omega)+S_{0}(\omega)e^{-j\omega t}+S_{0}(\omega)e^{+j\omega t}+\cdots+ \]
\[ +S_{0}(\omega)e^{-j\omega \frac{N-1}{2}T} + S_{0}(\omega)e^{+j\omega \frac{N-1}{2}T}, \tag{10} \]
where
\[ S_{0}(\omega)=T_{u}\, \frac{\sin \dfrac{\omega T_{u}}{2}} {\dfrac{\omega T_{u}}{2}} . \tag{11} \]
Using the formula for the sum of a geometric progression, we transform expression (10) [6]
\[ S(\omega)=S_{0}(\omega)\, \frac{\sin N\dfrac{\omega T_{u}}{2}} {\sin \dfrac{\omega T_{u}}{2}} . \tag{12} \]
Formula (12) shows that, as the number of pulses \(N\) increases, the energy begins to concentrate near frequencies that are multiples of \(\omega\); the energy maxima occur at the frequencies of the spectra of individual pulses. The mechanism of optimal filtering in a comb-type ...
filter [3]. The transparency bands of the comb filter coincide with the main lobes of the spectrum of a rectangular pulse; therefore the principal part of the pulse passes without distortion. “White” noise, which has a uniform spectrum, is for the most part suppressed; therefore it is advisable to sum the parts of the signal and noise that have passed through the filter, since the laws of summation for the signal and for the noise are different. In this case the pulses of the useful signal are summed arithmetically, and the resulting signal has a voltage amplitude \(N\) times greater and, correspondingly, its power increases by \(N^2\) times. At the same time the resulting noise amplitude in voltage will be added according to the root-mean-square law and will be equal to \(\sqrt{N}\), which means that the power will increase by \(N\) times. As a result of the synchronous accumulation operation performed, the signal-to-noise ratio in power increases by
\[ \frac{N^2}{N} = N_{\text{times}}. \tag{13} \]
What has been said applies only to periodic pulses, which is precisely what occurs in flaw detection based on a betatron and other accelerators. This means that, in the spectrum passed by the filter, the contribution of frequencies falling within the frequency band of the recorded periodic pulses increases; in other words, the signal-to-noise ratio improves.
Summing up, we arrive at the following conclusions:
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An increase in the signal-to-noise ratio in betatron flaw detection with scintillation counters can be obtained by the methods indicated above.
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The most promising method is the use, in recording circuits, of comb filters that provide a gain in the signal-to-noise ratio by a number of times proportional to the accumulation time.
LITERATURE
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- A. A. Kharkevich. Combating Interference, “Nauka,” 1965.
- Z. M. Konevskii, M. I. Finkelshtein. Fluctuation Interference and the Detection of Pulsed Radio Signals, GEI, 1963.
- B. R. Levin. Theoretical Foundations of Statistical Radio Engineering, “Sovetskoye radio,” 1965.
- N. V. Smirnov et al. A Course in Probability Theory and Mathematical Statistics, “Nauka,” 1965.