Some Ways to Increase the Sensitivity of the Scintillation Method of Betatron Flaw Detection
Abstract Generated abstract
This paper examines ways to improve the sensitivity and reliability of scintillation-based betatron flaw detection for large metal thicknesses, focusing on noise in the detector and recording channel. Using probability theory and mathematical statistics, it treats defect-signal reception as a statistical detection problem and relates optimal reception to likelihood estimation and cross-correlation between transmitted and received signals. The analysis argues that frequency filtering, especially comb filtering matched to periodic pulse spectra from betatron systems, can suppress white noise while coherently accumulating useful signal pulses. It concludes that such filtering can improve the signal-to-noise ratio by a factor proportional to the accumulation time, making comb filters a promising method for recording circuits.
Full Text
SOME WAYS TO INCREASE THE SENSITIVITY OF THE SCINTILLATION METHOD OF BETATRON FLAW DETECTION
V. I. GORBUNOV, A. V. POKROVSKII, A. K. TEMNIK
(Presented by the scientific seminar of the Research Institute of Electron Introscopy)
The scintillation method of betatron flaw detection is finding ever wider application in the automatic inspection of large metal thicknesses [1, 2].
The questions of increasing the sensitivity of the method and the reliability of obtaining trustworthy data on the presence of a defect are presently decisive.
However, sensitivity and reliability depend in many respects on the noise arising in the receiving-recording channel.
On the basis of probability theory and mathematical statistics, we shall analyze fluctuation phenomena in the electrical circuits of the recording scheme.
Let us consider the possibilities of lowering the noise level and methods of extracting the signal at the noise level. Noise in an electrical circuit is formed as a result of current fluctuations in electron tubes (shot effect) and as a result of the chaotic motion of electrons in resistances and other circuit elements; in addition, the detector itself is a source of noise [1, 2]. The signal from the detector also fluctuates owing to the instability of the betatron radiation. Noise is an oscillatory process varying with time according to a random law. In various electrical circuits there occur fluctuation oscillations that differ in their structure. Mathematically they can be described with the aid of the apparatus of probability theory.
At each moment \(t\), the noise function has some value characterizing the probability of its occurrence at that given instant of time. For a more complete description of the process of interaction between signal and noise, 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 probability distribution law, the mutual correlation function can be determined by the formula [4]:
\[ R(t_1 t_2)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} x_1(t_1)\cdot x_2(t_2)\cdot P(x_1 x_2)\,dx_1 dx_2 = x_1(t_1)\cdot x_2(t_2). \tag{1} \]
where \(x_1(t_1)\) is the law of variation of the signal with time;
\(x_2(t_2)\) is the law of variation of the noise with time;
\(P(x_1 x_2)\) is the two-dimensional probability density.
The correlation function represents the average degree of probabilistic connection between the processes \(x_1\) and \(x_2\) at two instants of time \(t_1\) and \(t_2\) [4].
A recording scheme that ensures the best reception of a signal in the presence of noise is called optimal. Such a scheme provides the maximum probability of undistorted reception of the signal. However, under the action of noise the signal is substantially distorted, and therefore uncertainty arises in receiving the signal.
Proceeding from this, determination of the signal corresponding to a defect is a statistical problem: from the known signal \(Y\) at the output, to learn everything about the signal \(X\) at the input of the scheme.
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 scheme; therefore, in what follows we shall call this scheme ideal.
Another quantity determining reliable reception of the signal is \(P_x(Y)\)—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 indicates 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 when a disturbance in the form of “white” noise is superposed on the signal \(X(t)\), which exists during the time \(T\) in which the defect passes 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 disturbance with time.
To find the likelihood function, which is equivalent to calculating the probability \(P_x(Y)\), we shall use Kotelnikov’s 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 \(Y(t)\) on the time interval \(T\) is
\[ m=\frac{T}{\Delta t}=2F_0T. \tag{5} \]
Applying the multiplication theorem for \(m\) independent events to the discrete values, we obtain [6]
\[ L(X)=P_x(Y)=\prod_{k=1}^{m} P_x(Y)=(2\pi\sigma_{\mathrm{ш}})^{-F_0T}\cdot \exp \times \]
\[ \times \left[-\frac{1}{G}\sum_{k=1}^{2F_0T}\left|Y(t)-X(t)\right|^2\right], \tag{6} \]
where \(\sigma_{\mathrm{ш}}\) is the noise variance;
\[ G=\frac{\sigma_{\mathrm{ш}}}{F_0} \]
is the power spectral density of the noise.
The function \(L(X)\) is maximal when the transmitted and received signa-
coincide in the interval under consideration. Replacing the limiting sum in expression (6) by an integral, let us consider the constituent parts:
\[ \int_0^T [Y(t)-X(t)]^2\,dt = \int_0^T [Y(t)]^2\,dt + \int_0^T [X(t)]^2\,dt - 2\int_0^T Y(t)\cdot X(t)\,dt . \tag{7} \]
The first integral characterizes the entire received signal, the second—the entire transmitted signal, and the third expresses the interrelation between the transmitted and received signals. Therefore, in order to characterize completely the input signal \(X(t)\) from the received \(Y(t)\), it is necessary to know that the received signal \(Y(t)\) corresponds to the given original signal \(X(t)\). It follows that, to determine the likelihood function \(L(X)\), it is sufficient to compute the integral
\[ g(t)=\int_0^T Y(t)\cdot X(t)\cdot dt . \tag{8} \]
The expression obtained is an analog of the mutual-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 transmitted frequencies for a unit signal is described by the formula [4]
\[ S(\omega)=\int_0^{T_u} e^{-j\omega t}\,dt . \tag{9} \]
However, when a signal from a defect is reproduced, 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]:
\[
S(\omega)=S_0(\omega)+S_0(\omega)e^{-j\omega t}+S_0(\omega)e^{+j\omega t}+\ldots+
\]
\[
+S_0(\omega)e^{-j\omega \frac{N-1}{2}T}+S_0(\omega)e^{+j\omega \frac{N-1}{2}t},
\tag{10}
\]
where
\[ S_0(\omega)=T_u\,\frac{\sin \frac{\omega T_u}{2}}{\frac{\omega T_u}{2}} . \tag{11} \]
Using the formula for the sum of a geometric progression, we transform expression (10) [6]:
\[ S(\omega)=S_0(\omega)\frac{\sin N\frac{\omega T_u}{2}}{\sin \frac{\omega T_u}{2}} . \tag{12} \]
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 single pulses. The mechanism of optimal filtering is especially clearly seen in a comb 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 separately the parts of the signal and noise that have passed through the filter, since the laws of summation for signal and noise are different. In this case the useful-signal pulses are summed arithmetically, and the resulting signal has a voltage amplitude \(N\) times larger; accordingly, its power increases by \(N^2\) times. At the same time, the resulting noise amplitude in voltage will add according to the root-mean-square law and will be equal to \(\sqrt{N}\); consequently, the power will increase by \(N\) times. As a result of the synchronous accumulation operation performed, the signal-to-noise ratio in power will increase by
\[ \frac{N^2}{N} = N \text{ times}. \tag{13} \]
What has been said applies only to periodic pulses, which is precisely the case 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:
-
An increase in the signal-to-noise ratio in betatron flaw detection with scintillation counters can be obtained by the methods indicated above.
-
The most promising method is the use, in recording circuits, of comb filters, which provide a gain in the signal-to-noise ratio by a factor proportional to the accumulation time.
REFERENCES
- V. I. Gorbunov et al. “Welding Production,” No. 7, 1965.
- V. I. Gorbunov, A. V. Pokrovskii. “Defectoscopy,” No. 5, 1965.
- A. A. Kharkevich. Interference Suppression. Moscow, “Nauka,” 1965.
- Z. M. Konevskii, M. I. Finkelshtein. Fluctuation Noise and the Detection of Pulsed Radio Signals. Moscow, GEI, 1963.
- B. R. Levin. Theoretical Foundations of Statistical Radio Engineering. Moscow, “Sovetskoe Radio,” 1968.
- N. V. Smirnov et al. A Course in Probability Theory and Mathematical Statistics. Moscow, “Nauka,” 1965.