Laws of Brightness Distribution on the Screen of an Indicator with a Brightness Mark
Abstract Generated abstract
This paper examines statistical laws governing brightness variations along the sweep line of a cathode-ray indicator with beam brightness modulation, motivated by the evaluation of radar noise immunity and signal observability. It derives general transformations from voltage distribution laws to brightness distribution laws for nonlinear and linear cathode-ray tube characteristics, including the effects of cutoff and voltage limitation. Under stated assumptions about video-filter bandwidth, the analysis treats detector and tube nonlinearities as interchangeable within linear channel sections, and presents differential brightness distributions and principal parameters for several receiver characteristics in the presence of noise alone and signal plus noise.
Full Text
Laws of Brightness Distribution on the Screen of an Indicator with a Brightness Marker
G. S. Zubarev
(Presented by the scientific seminar of the Radio Engineering Faculty)
Introduction
One of the basic problems of radar is the problem of increasing the noise immunity of a radar station, i.e., its ability to withstand the interfering action of various kinds of noise.
Noise immunity determines the reliability of the information of the reflected signal and the range of action of a radar station for a given level of noise and transmitter power. With constant transmitter power, noise immunity depends mainly on a number of parameters of the receiving-indicator channel and on the type of indication employed.
To assess the influence of the parameters of one or another channel of the station on its noise immunity, and also to compare various types of indication, it is necessary to know the statistical regularities of the change in brightness on the indicator screen.
Below is given a determination of the laws of brightness distribution along the sweep line of a cathode-ray tube with brightness modulation of the beam and their principal parameters. The consideration is carried out for the case of “smooth” noise. The influence of voltage limitation and of the operating mode of the cathode-ray tube is taken into account.
General Remarks on the Laws of Brightness Distribution
The characteristic of screen luminescence when a voltage is applied from the output of the amplifier channel to the control grid of the cathode-ray tube may be characterized by the function
\[ B = \varphi(U_y), \]
where \(B\) is the brightness of screen luminescence at an arbitrary point on the sweep line;
\(U_y\) is the voltage applied to the control grid of the tube.
In the absence of a signal, the voltage at the output of the amplifier channel represents the noise voltage, whose amplitude and phase are
are random in character. Fluctuations of the voltage applied to the cathode-ray tube cause fluctuations of brightness along the sweep line.
To find the law of brightness distribution it is necessary to know the distribution law of the applied voltage and the function \(\varphi\). The latter, usually in the region of small amplitudes of the applied voltage, is a nonlinear function and is determined by the type and operating mode of the cathode-ray tube.
Given the differential distribution law of the applied voltage \(p_u(U_y)\) and a specified function \(\varphi\), the differential law of brightness distribution is determined as
\[ p_B(B)=p_u[\psi(B)]\psi'(B), \tag{1} \]
where \(\psi(B)\) is the function inverse to \(\varphi(U_y)\), and \(\psi'(B)\) is its derivative.
At small amplitudes of the applied voltage, the characteristic of the tube may be approximated by a parabola of the \(n\)-th degree, i.e.,
\[ \begin{cases} B=\alpha \cdot U_y^n & \text{for } U_y>0,\\ B=0 & \text{for } U_y\leq 0, \end{cases} \]
where \(\alpha\) is a coefficient depending on the type and operating mode of the cathode-ray tube.
The function inverse to \(\varphi(U_y)\) will be equal to
\[ \psi(B)=\sqrt[n]{\frac{B}{\alpha}}, \quad \text{whence} \]
\[ \psi'(B)=\frac{1}{n}\cdot \sqrt[n]{\frac{1}{\alpha}}\cdot \frac{1}{\sqrt[n]{B^{\,n-1}}}. \]
Consequently, on the basis of (1),
\[ p_B(B)=\frac{1}{n}\cdot \sqrt[n]{\frac{1}{\alpha}}\cdot \frac{1}{\sqrt[n]{B^{\,n-1}}}\cdot p_u\!\left(\sqrt[n]{\frac{B}{\alpha}}\right) \quad \text{or, passing to } U_y \]
\[ p_B(B)=\frac{1}{n\cdot \alpha \cdot U_y^{\,n-1}}\cdot p_u(U_y). \tag{2} \]
At large amplitudes of the applied voltage, the characteristic of the tube may be approximated by a straight line.
For the case of a linear approximation of the tube characteristic
\[ \begin{cases} B=\alpha U_y & \text{for } U_y>0,\\ B=0 & \text{for } U_y\leq 0 . \end{cases} \]
(here \(\alpha\) is the slope of the rectilinear portion of the tube characteristic) the differential law of brightness distribution, according to (1), will be
\[ p_B(B)=\frac{1}{\alpha}p_u\!\left(\frac{B}{\alpha}\right)=\frac{1}{\alpha}p_u(U_y). \tag{3} \]
Expression (3) is obtained from (2) for \(n=1\) (see also [2]).
Influence of cutoff and voltage limitation on the laws of brightness distribution
Expressions (1), (2), (3) will be valid if the entire voltage supplied to the cathode-ray tube affects the change in brightness, i.e., if there is no limitation or cutoff of it.
Cutoff and limitation of the supplied voltage lead to a change in the law of brightness distribution.
Thus, for \(U=U_y\), when
\[ U_{otc}<U<U_{ogr} \]
(where \(U_{otc}\) and \(U_{ogr}\) are the cutoff and limitation levels of the supplied voltage),
\[ \text{and}\quad \begin{cases} B=\varphi(U_y) & \text{for } U_{otc}<U<U_{ogr},\\ B=0 & \text{for } U\leqslant U_{otc},\\ B=\varphi(U_{ogr}) & \text{for } U\geqslant U_{ogr}, \end{cases} \]
we shall have
\[ p_B(B)=p_{ou}(U_y)\cdot \psi'[\varphi(U_y)] =\xi\cdot p_u(U)\psi'[\varphi(U)], \]
where, in turn, \(p_{ou}(U_y)\) is the modified, and \(p_u(U)\) the initial, differential laws of distribution of the supplied voltage.
The factor \(\xi\) characterizes the increase in the probability density of the amplitudes of the voltage \(U\) (Fig. 1), bounded by the limits of its possible values from \(U_{otc}\) to \(U_{ogr}\), so that
\[ p_u(U)=0 \quad \text{for } U<U_{otc} \text{ and } U>U_{ogr}, \]
and
\[ \xi\cdot \int_{U_{otc}}^{U_{ogr}} p_u(U)\cdot dU=1. \]
From the last expression,
\[ \xi=\frac{1}{\displaystyle \int_{U_{otc}}^{U_{ogr}} p_u(U)\cdot dU}. \]
Finally, the differential law of brightness distribution in the presence of cutoff and limitation will be:
\[ p_B(B)= \frac{1}{\displaystyle \int_{U_{otc}}^{U_{ogr}} p_u(U)\cdot dU} \cdot p_u(U)\cdot \psi'[\varphi(U)]. \tag{4} \]
On the Generality of the Analysis in the Presence of a Video Filter
In the case where a video filter is present, finding the laws of brightness distribution is made difficult by the mathematical complexity of establishing the influence of the video filter on the distribution law of the voltage applied to it from the detector output.
For a passband of the video filter
\[ \Delta F_{n.ch.} > \Delta F_{v.ch.} \]
(where \(\Delta F_{v.ch.}=\dfrac{a}{\tau}\) is the optimal passband up to the detector; \(\tau\) is the pulse duration; \(a\) is a constant depending on the shape of the envelope of the high-frequency pulse and the resonance curve), it is assumed that, to a first approximation, the video filter does not change the voltage distribution law or the observability of the signal.
Indeed, the highest frequency of the spectrum at the detector output is equal to the difference between the extreme side frequencies within the passband up to the detector, i.e., is equal to this band. Consequently, in the presence of a linear video filter with a comparatively uniform amplitude-frequency characteristic within a passband equal to the passband up to the detector, all voltage components from the detector output will pass without distortion to the video-filter output. Further widening of the video-filter band will have no effect on the observability of a weak signal.
Fig. 1. Effect of cutoff and limiting on the voltage distribution law
Therefore, when
\[ \Delta F_{n.ch.} > \Delta F_{v.ch.} \]
one may in practice assume that the voltage distribution law at the output of the video filter will be the same as at the output of the detector.
Moreover, the highest frequencies of the voltage spectrum from the detector output form the so-called “fine” structure of the pulses, characterized by small amplitude values. The observability of the signal, however, is determined mainly by the maximum voltage peaks, owing to the physiological characteristics of the observer’s eye.
Therefore, from the point of view of signal observability, the assumptions made about the identity of the voltage distribution laws at the outputs of the detector and the video filter may also be extended to the case
\[ \Delta F_{n.ch.} < \Delta F_{v.ch.} \]
Table 1
Laws of brightness distribution and their principal parameters on the screen of an indicator with a brightness marker
| No. | Receiver characteristic | Differential laws of brightness distribution: presence of signal | Differential laws of brightness distribution: absence of signal | Principal parameters of the brightness-distribution laws, with signal and noise: mean value $$\dfrac{\bar B_{c+w}}{A}$$ | Principal parameters of the brightness-distribution laws, with signal and noise: mean-square value $$\dfrac{D(B_{c+w})}{A}$$ | Principal parameters of the brightness-distribution laws, with noise only: mean value $$\dfrac{\bar B_w}{A}$$ | Principal parameters of the brightness-distribution laws, with noise only: mean-square value $$\dfrac{D(B_w)}{A}$$ | Constant multiplier \(A\) |
|---|---|---|---|---|---|---|---|---|
| 1 | Undetected output\(^1\) | $$\frac{2}{A\sqrt{2\pi^3}}\int_0^\pi e^{-\frac{(x-a\cos\theta)^2}{2}}\,d\theta$$ $$x=\frac{u}{D(U_w)}$$ |
$$\frac{2}{A\sqrt{2\pi}}\,e^{-\frac{x^2}{2}}$$ | $$\sqrt{\frac{2}{\pi}}\left(1+\frac{a^2}{4}\right)$$ | $$0.603\sqrt{1+\frac{a^2}{2}}$$ | $$\sqrt{\frac{2}{\pi}}$$ | $$0.603$$ | $$aD(U_w)$$ |
| 2 | Linear detection | $$\frac{x}{A}\,e^{-\frac{x^2+a^2}{2}}\cdot J_0(ax)$$ $$x=\frac{U_{w+c}}{K_0\cdot D(U_w)}$$ |
$$\frac{1}{A}\,x\,e^{-\frac{x^2}{2}}$$ | $$\sqrt{\frac{\pi}{2}}\left(1+\frac{a^2}{4}\right)$$ | $$0.66\sqrt{1+\frac{a^2}{2}}$$ | $$\sqrt{\frac{\pi}{2}}$$ | $$0.66$$ | $$a\cdot K\cdot D(U_w)$$ |
| 3 | Square-law detection | $$\frac{1}{2A}\,e^{-\frac{x+a^2}{2}}\cdot J_0(a\sqrt{x})$$ $$x=\frac{U_{w+c}}{\eta\cdot D^2(U_w)};\qquad \eta=\frac{\frac14\,\frac{S'}{S}}{1+\frac{1}{S R_w}}$$ |
$$\frac{1}{2A}\,e^{-\frac{x}{2}}$$ | $$2\left(1+\frac{a^2}{2}\right)$$ | $$2\sqrt{1+a^2}$$ | $$2$$ | $$2$$ | $$a\cdot\eta D^2(U_w)\cdot K_{w+c}$$ |
| 4 | Detection of the \(n\)-th order | $$\frac{1}{A\cdot n}\,x^{\frac{2-n}{n}}\,e^{-\frac{x^{2/n}+a^2}{2}}\cdot J_0\!\left(a\sqrt[n]{x}\right)$$ $$x=\frac{U_{w+c}}{\varepsilon D^n(U_w)};\qquad \varepsilon=\beta R_w\frac{C_n^{\,n/2}}{2^n}$$ |
$$\frac{1}{An}\,x^{\frac{2-n}{n}}\,e^{-\frac{x^{2/n}}{2}}$$ | $$\left(\frac{n}{2}\right)!\,2^{n/2}\left(1+\frac{n}{4}a^2\right)$$ | $$\frac{2^{n/2}\sqrt{\left(1+\frac{n}{2}a^2\right)}}{\sqrt{\left[n!-\left(\frac{n}{2}!\right)^2\right]}}$$ | $$\left(\frac{n}{2}\right)!\,2^{n/2}$$ | $$2^{n/2}\sqrt{\,n!-\left(\frac{n}{2}!\right)^2\,}$$ | $$aK_{w+c}\varepsilon D^n(U_w)$$ |
\(^1\) In the presence of clipping of the negative half-periods of the voltage.
This is confirmed by studies carried out by A. P. Belousov [1] and by a whole series of available experimental works, according to which observability does not deteriorate when
\[ \Delta F_{\text{l.f.}} \geq (0.4 \div 0.5)\Delta F_{\text{h.f.}} . \]
Thus, in finding the laws of brightness distribution in the case where a video filter is present, one may assume
\[ p_u(U_{\phi}) \approx p_u(U_{det}), \]
where \(U_{det}\) and \(U_{\phi}\) are, respectively, the voltages at the output of the detector and of the video filter.
The differential law of distribution of the voltage at the output of the detector is determined by its operating mode and by the form of the detector characteristic.
Consequently, the differential law of brightness distribution will depend on the order of nonlinearity of the characteristics of the detector and of the cathode-ray tube.
However, under the assumptions made above, when a video filter is present there is no need to consider separately the influence of the nonlinearity of the detector and of the tube characteristic on the law of brightness distribution.
It can be shown that a receiving-indicator-channel system having an \(n\)-th order detector and a linear low-frequency amplifier and cathode-ray tube characteristic is equivalent to a system having a linear detector and a low-frequency amplifier or cathode-ray tube characteristic of the same order of nonlinearity.
That is, the character of the output voltage will not change when the nonlinear element is transposed among the linear links that pass the spectrum of the amplified frequencies.
Therefore, the influence of the nonlinearity of the cathode-ray tube may be considered using the example of an \(n\)-th order detector followed by a linear channel.
Laws of Brightness Distribution
Using the known expressions for the laws of distribution of the voltage at the output of the amplification channel [2, 3, etc.] and equations (2), (3), one can obtain the differential laws of brightness distribution for the cases of a signal together with noise and of noise alone, as well as the principal parameters of these laws.
The results of the calculations are given in Table 1, where the following notation is used:
\[
U = U_c + U_{\sh}
\]
— the total voltage of the harmonic signal and noise;
\[
U_c = U_o \cdot \cos(\omega_o t + \varphi)
\]
— the signal voltage at the output of the linear system;
\[
a = \frac{U_o}{D(U_{\sh})}
\]
— the signal amplitude referred to \(D(U_{\sh})\) at the output of the linear system (or at the detector input), while \(D(U_{\sh})\) is the root-mean-square value of the fluctuations at the detector input;
\[
K_d
\]
— the detector voltage transmission coefficient;
\[ K = K_d \cdot K_{\text{l.f.}}; \]
$K_{\text{l.f.}}$ — voltage gain of the low-frequency amplifier;
$\alpha$ — slope of the characteristic of the cathode-ray tube;
$S$ — static slope of the diode current–voltage characteristic at the operating point;
$R$ — load resistance of the detector;
$\beta$ — slope of the static characteristic of the detector;
$J_{0}(x)$ — modified Bessel function of the first kind, of zero order.
REFERENCES
-
Belousov A. P. On the highest real sensitivity of a receiver, Radiotekhnika, No. 5, 1946.
-
Fialko E. I. On the question of the real sensitivity of a receiving device. Izvestiya TPI, vol. 73, 1952.
-
Rice S. O. Theory of fluctuation noises. Theory of the transmission of electrical signals in the presence of interference (collection of articles), IL, 1953.