On the Actual Sensitivity of a Receiving Device

E. I. Fialko

Introduction

The problem of determining and increasing the actual sensitivity includes the following principal questions:

1. Allowance for the receiver’s own noise.
2. Passage of the signal and noise through the linear part of the receiver.
3. Passage of the signal and noise through the nonlinear part of the receiver.
4. Detectability of weak signals on the indicator.

A number of works by V. I. Siforov [1; 2], A. P. Belousov [4; 5], V. I. Bunimovich [6; 7], and others are devoted to these questions.

In particular, V. I. Siforov and V. I. Bunimovich considered the transformation of the signal and noise by the nonlinear part of a receiver (a linear and a quadratic detector) and made a comparison with the case in which there is no detector when an amplitude indication is realized.

The present work is also devoted to questions of actual sensitivity in connection with the passage of a signal and fluctuations through a nonlinear system (including a higher-order detector), but for a brightness indication.

The basic proposition in determining the minimum signal-to-noise ratio is the assumption, put forward by V. I. Siforov, that the decisive factor determining the observability of a signal is the relative change in brightness between the place where the signal is present and the noise background.

Preliminary Remarks

1. Let us find the distribution of brightness on the screen of a cathode-ray tube along the sweep line.

In the presence of fluctuations of the voltage applied to the control grid of the tube, the intensity of the glow (brightness) of the “spot” moving along the sweep line will also fluctuate (Fig. 1).

The presence of a signal will be detected, first of all, by an increase in brightness on that section of the sweep which corresponds to the duration of the signal (it is assumed that a positive signal is applied to the control grid).

As is known [1], the eye reacts to a relative change in brightness exceeding a certain minimum value \((\delta B_{\min})\).¹

¹ Generally speaking, besides the relative change in brightness, the magnitude of the background brightness, the dimensions of the luminous spot, etc., are also significant.

Therefore, the determination of sensitivity is reduced to finding such a signal as will produce a relative change in brightness exceeding this minimum value.

1. Figure 1 shows the change in brightness along the scanning line. Let us find the value of the brightness at the point \(a_1\), corresponding to the absence of a signal. We select a time interval \(\Delta t\), corresponding to the diameter of the “spot,”

Fig. 1. Change in the brightness of the “spot” with time.

\[ \Delta t = d \text{ of the spot}; \qquad B_m = \frac{\Delta t}{T}\cdot \overline{B_0}; \]

\[ \tau \text{—pulse duration} \]

so that the point \(a_1\) lies in the middle of this interval, and we find the value of the visual, or apparent, brightness [9; 10] at this point under the condition that the eye perceives a continuous image:

\[ B_{\mathrm{ш}}=\frac{1}{nT}\int_{0}^{nT} B_0(t)\cdot dt, \tag{1} \]

where \(B_{\mathrm{ш}}\) is the visual or apparent brightness (the wavy line above \(B_{\mathrm{ш}}\) denotes averaging over time),

\(T\) is the scanning period, or the repetition period if the scan is waiting,

\(nT\) is the observation time\(^1\),

\(B_0(t)\) is the instantaneous value of the brightness (the signal is absent).

Performing the integration, we obtain\(^2\)

\[ B_{\mathrm{ш}}= \frac{\Delta t}{nT}\cdot \sum_{i=1}^{n} B_{0(i)} = \frac{\Delta t}{T} \left[ \frac{1}{n}\cdot \sum_{i=1}^{n} B_{0(i)} \right]. \tag{1a} \]

Let us note that \(\Delta t=\dfrac{d}{v_p}\), where \(v_p\) is the speed of the radial scan, and \(d\) is the diameter of the spot (the base of the beam).

But

\[ \lim_{n\to\infty}\frac{1}{n}\cdot \sum_{i=1}^{n} B_{0(i)}=\overline{B}_0 \]

is the mathematical expectation (arithmetic mean) of the instantaneous brightness values (here the bar over \(B_0\) denotes averaging over the ensemble). Thus

\[ B_{\mathrm{ш}}\simeq \frac{\Delta t}{T}\cdot \overline{B}_0 . \tag{1б} \]

Let us now find the value of the brightness at the point \(a_1\), located in that part of the scan where there is a signal.

Carrying out analogous operations, we find an expression for the apparent brightness under the joint action of signal and noise, \(B_{c+\mathrm{ш}}\):

\[ B_{c+\mathrm{ш}}\simeq \frac{\Delta t}{T}\cdot \overline{B}, \tag{1в} \]

where \(\overline{B}\) is the mathematical expectation of the instantaneous brightness \(B(t)\) at the point where the signal and noise act jointly.

The relative change in visual brightness at the point where signal and noise are present, in comparison with the point where only noise is present, will be equal to:

\[ \delta B= \frac{B_{c+\mathrm{ш}}-B_{\mathrm{ш}}}{B_{\mathrm{ш}}} = \frac{\overline{B}_{c+\mathrm{ш}}}{B_{\mathrm{ш}}} -1 = \frac{\overline{B}}{\overline{B}_0}-1^{3)} . \tag{2} \]

1. Knowing the distribution law of the random variable \(B_0\), we find the value of the mean brightness \(B_0\) as the mathematical expectation of the random variable \(B_0\):

\[ \overline{B}_0= \int_{0}^{\infty} B_0\cdot P_1(B_0)\cdot dB_0, \tag{3} \]

where \(P_1(B_0)\) is the differential distribution law of the random variable \(B_0\).

\(^1\) More precisely, \(n\) is the number of scan lines “covering” the point \(a_1\) under consideration.

\(^2\) We consider the case of a “fast scan” (the time for the spot to pass over the point \(a_1\) is much less than the duration of the noise flashes: \(\Delta t \ll \dfrac{1}{\Delta F}\), where \(\Delta F\) is the receiver bandwidth). In the case of a “slow scan” the derivations become more complicated, but the result will be essentially the same. In the derivation a number of simplifying assumptions have been made: it is assumed that the brightness of the spot along the diameter of the spot is constant, etc.

\(^3\) The consideration carried out above did not take afterglow into account. It can be shown that the results obtained will also be valid in the presence of afterglow in the case of a “fast” scan; in the case of a “slow” scan these results should be regarded as very approximate.

Similarly,

\[ \overline{B}=\int_{0}^{\infty} B \cdot P_{\mathrm{II}}(B)\cdot dB . \tag{3a} \]

Here we have in fact operated with the sum of the harmonic signal and noise as with a random quantity. The justification of such an approach to the case of the joint action of fluctuation and harmonic voltages was given by V. I. Siforov and V. I. Bunimovich and is not presented here.

Let us note that, strictly speaking, the upper limit in the integrals (3) and (3a) is not \(+\infty\), but some value \(B_{max}\) (owing to the limiting properties of amplifier tubes and of the tube). In this case the expression for \(\overline{B_0}\) has the form:

\[ \overline{B_0} = \int_{0}^{B_{max}} B_0 \cdot P_{\mathrm{I}}(B_0)\,dB_0 + B_{max}\int_{B_{max}}^{\infty} P_{\mathrm{I}}(B_0)\,dB_0 . \tag{4} \]

Considering practically important cases, and not touching in the present work on the use of a limiter, we shall henceforth proceed from formulas (3a) and (3).

1. It should be noted that expressions (1b) and (1c) define truly constant (in time) quantities only in the case where changes in the instantaneous brightness values (\(B_0\) and \(B\)) at the point under consideration occur sufficiently often. Otherwise, because of insufficient inertia of the eye, the image on the screen will not be unchanged in time. The image on the screen will be “flickering,” nonuniform. In the case of a “nonuniform background,” the relative change in the mean brightnesses, given by expressions (1b) and (1c), will be a less objective criterion of signal detectability than in the case of a “uniform background.”

Fig. 2. Brightness of the screen glow as a function of the voltage on the control grid of a cathode-ray tube.

1. Thus, to find the mean values of brightness it is necessary to know the differential law of brightness distribution. Usually, however, what is known is the distribution law of the voltage at the output of the receiver (or detector), \(P_{\mathrm{III}}(u)\). If the characteristic of the cathode-ray tube is known, that is, the dependence of the screen-glow intensity (brightness) on the voltage on the control grid (Fig. 2),

\[ B=f(U_{\mathrm{ctrl}}), \tag{5} \]

then the mean value of the brightness is found from

\[ \overline{B} = \int_{u_0}^{\infty} f(U-U_{\mathrm{bias}})\cdot P_{\mathrm{III}}(U)\,dU \quad \text{(Fig. 3).} \tag{6} \]

1. Let us consider a particular case—the linear approximation of the tube characteristic

\[ B=\beta(U_{\mathrm{ctrl}}+U_{\mathrm{cut}}) \tag{7} \]

and

\[ U_0=0 \quad \text{(Fig. 4).} \]

It is quite obvious that in this case formulas (5) and (6) will take the form:

\[ B=\beta \cdot U, \tag{8} \]

\[ \overline{B}=\beta \int_{0}^{\infty} U \cdot P_{\mathrm{sh}}(U)\cdot dU, \tag{9} \]

where \(U\) is the voltage applied to the control grid of the tube.

Labels in Fig. 3: \(B=f(U_{\mathrm{ctrl}})\), \(U_{\mathrm{ctrl}}\), \(U_0\), \(U_{\mathrm{ots}}\), \(U_{\mathrm{bias}}\), \(U_i\), \(t\), \(0\).

Fig. 3. Relation between the brightness of the screen glow and the control voltage.

\[ U_{\mathrm{ctrl}} \text{ -- control voltage.} \]

\[ U_{\mathrm{bias}} \text{ -- bias voltage.} \]

\[ U_{\mathrm{ots}} \text{ -- voltage cutting off the tube.} \]

\[ U_0 = U_{\mathrm{bias}} - U_{\mathrm{ots}}. \]

Labels in Fig. 4: \(B=f(U_{\mathrm{ctrl}})\), \(U_{\mathrm{ctrl}}\), \(U_i\), \(U_{\mathrm{ots}}\), \(t\), \(0\).

Fig. 4. Linear approximation of the tube characteristic.

\(P_{\mathrm{ш}}(U)\) is the differential distribution law of this voltage. Thus, the question of the relative variation of brightness on the screen of the cathode-ray tube has been reduced to the problem of finding the voltage distribution at the output of the receiving device.

Remark. If the tube characteristic is approximated by a parabola of the \(n\)-th order, then, as is easy to show,

\[ \overline{B}=\beta\cdot \int_{0}^{\infty} U^{n}\cdot P_{\mathrm{ш}}(U)\,dU.. \tag{9a} \]

1. Strictly speaking, one must find the distribution laws at the receiver output, that is, in the case where there is a low-frequency amplifier (LFA)—at the output of the LFA. However, owing to the complexity and the insufficient study of the influence of the LFA on the voltage distribution law, we shall consider a particular case of great practical importance\(^1\): when the passband at low frequency is approximately equal to the passband at intermediate frequency (in other words, we shall consider the case in which the passband after the detector, taking into account the detector load, is not less than the passband before the detector).

In this case, as is known [6; 11], the distribution law at the output of the LFA will be the same as at the output of the detector.

Voltage distribution laws (undetected and detected outputs)

Undetected output

1. As is known [8; 11 and others], if the sum of a fluctuation voltage and a harmonic voltage is applied to the input of a linear system, then the voltage distribution law at the output of the linear system will be:

\[ p_{\mathrm{н}}(x)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\pi} e^{-\frac{(x-a\cdot \cos\Theta)^2}{2}}\,d\Theta, \tag{10} \]

where

\[ x=\frac{U}{\sigma};\qquad a=\frac{U_{0}}{\sigma};\qquad U=U_{c}+U_{\mathrm{ш}}; \]

\(\sigma\) is the rms value of the voltage \(U_{\mathrm{ш}}\);

\(U_{c}=U_{0}\cos(\omega_{0}t+\varphi)\) is the signal voltage at the output of the linear system;

\[ p_{\mathrm{ш}}(U_{\mathrm{ш}})=\frac{1}{\sqrt{2\pi}\cdot \sigma}\cdot e^{-\frac{u_{\mathrm{ш}}^{2}}{2\cdot \sigma^{2}}} \]

is the noise distribution law;

\[ p_{c}(U_{c})=\frac{1}{\pi\sqrt{U_{0}^{2}-U_{c}^{2}}} \]

is the signal distribution law.

\(^1\) A. P. Belousov showed [5] that, “with a pre-detector bandwidth \(\Delta f_{в}\sim \frac{1}{\tau}\), the LFA bandwidth may be taken equal to the pre-detector bandwidth, since this will improve the pulse shape at the output and will practically not worsen the total noise factor, compared with a low-frequency bandwidth equal to \(0.5\,\Delta f_{в}\).” It has also been established experimentally [16] that, for \(\Delta F_{\mathrm{н.ч}}=(0.4\div 30)\frac{1}{\tau}\), the observability of the signal does not change.

In the absence of a signal, the distribution law of the output of the linear system will be:

\[ p_{0н}(x)=\frac{1}{\sqrt{2\pi}}\cdot e^{-\frac{x^{2}}{2}}, \tag{10a} \]

Linear Detection

1. In the case where the sum of the voltage of a harmonic signal and fluctuations (with a normal distribution law) is applied to the input of a detector (Fig. 5) with the static characteristic

\[ I= \begin{cases} \alpha U & \text{for } U>0,\\ 0 & \text{for } U<0, \end{cases} \]

then the distribution law of the voltage across the detector load (the low-frequency output of the detector is meant) will be [11; 17]

\[ p_{1}(x)=x\cdot e^{-\frac{x^{2}+a^{2}}{2}}\cdot J_{0}(j\cdot a\cdot x), \tag{11} \]

where

\[ x=-\frac{U}{K_{det}\cdot \sigma}; \]

\(U\) — voltage across the detector load;
\(\sigma\) — rms value of the fluctuations at the detector input;
\(K_{det}\) — detector voltage transfer coefficient;
\(J_{0}(jax)\) — Bessel function of the first kind of zero order of imaginary argument.

Fig. 5. Characteristic of an ideal diode.

In the absence of a signal, formula (11) takes the form:

\[ p_{01}(x)=x e^{-\frac{x^{2}}{2}}. \tag{11b} \]

Let us note that formula (11) is exact when the condition stated above is fulfilled, namely: the passband of the detector load is not less than the passband of the selective system preceding the detector.

Quadratic Detection

1. When a signal acts on the detector together with fluctuations, if the static characteristic of the detector is \(I=\alpha U^{2}\), we have the following expression for the probability distribution law [11; 17]

\[ p_{2}(x)=\frac{1}{2}\cdot e^{-\frac{x+a^{2}}{2}}\cdot J_{0}(ja\sqrt{x}), \tag{12} \]

and in the absence of a signal

\[ p_{02}(x)=\frac{1}{2}e^{-\frac{x}{2}}. \tag{12a} \]

It should be noted that formula (12), strictly speaking, corresponds to the differential probability distribution law for the low-frequency component of the current of a square-law diode detector at zero load. In this case, by \(x\) one should understand:

\[ x=\frac{I_{\text{l.f.}}}{\dfrac{a}{2}\cdot \sigma^{2}}, \]

where \(I_{\text{l.f.}}\) is the low-frequency component of the detector current.

Under certain conditions, (12) is the distribution law of the low-frequency component of the voltage across the load of a square-law diode detector, namely—at sufficiently small voltages applied to the detector, and at small values of the detector load resistance

\[ (U_{ex}\cdot R_n \to 0). \]

Fig. 6. Static characteristic of a push-pull square-law detector.

In this case, by \(x\) one should understand:

\[ x=\frac{U_{\text{l.f.}}}{\gamma\cdot \sigma^{2}}, \]

where

\(U_{\text{l.f.}}\) is the low-frequency component of the detector output voltage;

\[ \gamma=\frac{aR_n}{2}; \]

\(R_n\) is the detector load resistance;
\(\sigma\) is the rms value of the fluctuations at the detector input.

In the case of a single-ended detector (Fig. 7)

\[ \gamma=\frac{aR_n}{4}. \]

In diode detection of weak noises and signals (diode detection of small voltages [3; 7]), \(x\) should be understood as

\[ x=\frac{U_{n.i}}{\xi\cdot\sigma^{2}}, \]

where

\[ \xi=\frac{\frac{1}{4}\cdot\frac{S'}{S}}{1+\frac{1}{S\cdot R_d}}, \]

\(S\) is the static slope of the current-voltage characteristic of the diode at the operating point.

Fig. 7. Static characteristic of a one-way quadratic detector.

\[ J= \begin{cases} \alpha U^{2}, & U>0,\\ 0, & U<0. \end{cases} \]

Detection of Higher Orders

1. Let us consider in greater detail the case of higher-order detection (above the 2nd order).

We shall restrict ourselves to considering the case of higher-order detection, but of even order, that is, when the static characteristic of the detector has the form

\[ I=\alpha U^{n}, \tag{13} \]

where \(n=2m\).

The voltage \(U\) applied to the detector, as is known [1; 6; 11], can be represented in the form of an envelope \(V\) (a random variable) and a time factor with a random phase shift \(\varphi\):

\[ U=V\cos(\omega_{0}t+\varphi), \tag{14} \]

where

\(\omega_{0}\) is the mean frequency in the passband of the linear system preceding the detector.

The differential distribution law of the envelope is [11]

\[ q(V)=\frac{V}{\sigma^{2}}\cdot e^{-\frac{V^{2}+U_{0}^{2}}{2\sigma^{2}}}\cdot J_{0}\!\left(j\frac{V\cdot U_{0}}{\sigma^{2}}\right), \tag{15} \]

where
\(\sigma\) is the rms value of the fluctuations applied to the detector input;
\(U_{0}\) is the amplitude of the harmonic voltage (signal) at the detector input.

As above, it is assumed that the fluctuations obey a normal distribution law.

The detector current, as follows from (13) and (14), in the case of zero detector load will be:

\[ I=a\cdot V^{2m}\cdot \cos^{2m}(\omega_{0}t+\varphi). \tag{16} \]

Using reference material [14a], we obtain:

\[ \cos^{2m}(\omega_{0}t+\varphi)= \frac{1}{2^{2m-1}}\sum_{k=0}^{m} C_{2m}^{k} \frac{\cos\left[(2m-2k)(\omega_{0}t+\varphi)\right]} {2^{E\left(\frac{k}{m}\right)}}, \tag{17} \]

where the symbol \(E\left(\frac{k}{m}\right)\) denotes the integer part of the fraction \(\frac{k}{m}\).

Substituting (17) into (16), we obtain

\[ I=I_{\text{l.f.}}+I_{\text{h.f.}} \tag{18} \]

where \(I_{\text{h.f.}}\) is the sum of the harmonic components from frequency \(2\omega_{0}t\) to frequency \(2m\omega_{0}t\).
\(I_{\text{l.f.}}\) is the low-frequency output of the detector, equal to

\[ I_{\text{l.f.}}=\frac{aC_{2m}^{m}}{2^{2m}}\cdot V^{2m}, \tag{19} \]

or

\[ I_{\text{l.f.}}=A\cdot V^{2m}, \tag{20} \]

where

\[ A=\frac{aC_{2m}^{m}}{2^{2m}}. \]

From (20) and (15) it is easy to find the distribution law of the low-frequency output of the detector from the known relations [13]

\[ p_{1}(I_{\text{l.f.}})=q\left[\psi(I_{\text{l.f.}})\right]\psi'(I_{\text{l.f.}}), \tag{21} \]

where

\[ \psi(I_{\text{l.f.}})=\sqrt[2m]{\frac{I_{\text{l.f.}}}{A}}=V. \tag{21a} \]

Carrying out the substitutions and transformations leads to the following differential distribution law of the low-frequency output of the detector:

\[ p_{n}(x)=\frac{1}{n}\cdot x^{\frac{2-n}{n}}\cdot e^{-\frac{x^{\frac{2}{n}}+a^{2}}{2}}\cdot J_{0}\!\left(jax^{\frac{1}{n}}\right), \tag{22} \]

where

\[ x=\frac{I_{\text{l.f.}}}{A\sigma^n}, \]

\[ a=\frac{U_0}{\sigma}, \]

\(n=2m\) is the order of detection.

From formula (22), as special cases, follow formulas (12) and (12a) for square-law detection (for \(n=2\)), as well as formulas (11) and (11a) for linear detection (for \(n=1\)), although formula (22) was derived under the assumption that \(n\) is even.

In the absence of a signal,

\[ p_{0n}(x)=\frac{1}{n}\,x^{\frac{2-n}{n}} e^{-\frac{x^{2/n}}{2}} . \tag{22a} \]

It can be shown that formula (22) expresses the differential law of distribution of the voltage across the detector load for the case of a sufficiently small voltage (signal + noise) at the detector input and a small resistance of the detector load (let us recall that complete filtering of the high-frequency components of the detector current and the absence of distortions over the entire low-frequency part of the spectrum of this current are assumed).

In this case

\[ x=\frac{U_{\text{l.f.}}}{D\sigma^n}, \tag{22b} \]

where

\(U_{\text{l.f.}}\) is the voltage across the detector load,

\(\sigma\) is the rms noise at the detector input,

\(D\) is a coefficient depending on the detector load and on the parameter of the detector static characteristic \((\alpha)\):

\[ D=A R_H=\alpha R_H\frac{C_{2m}^{m}}{2^{2m}} . \]

1. Let us note that if the output of a linear detector is fed to a low-frequency amplifier passing the entire spectrum of the voltage being amplified, then, in the case of nonlinearity of the amplifier amplitude characteristic, the law of distribution of the voltage at the amplifier output will be described by expression (22).

Knowing the distribution laws of the receiver output both in the absence and in the presence of a nonlinear system (detector), we can find the mathematical expectations of the brightness corresponding to the presence and absence of a signal, and then determine the relative change in brightness caused by the appearance of the signal.

Mathematical expectations and relative
changes in brightness (undetected and detected
outputs)

1. The mean value (mathematical expectation) of the brightness under the assumptions made above, as was shown, is:

\[ B=\beta\int_{0}^{\infty} u\,p_{\text{н}}(u)\,du . \tag{9} \]

\[ x=\frac{u}{\sigma^n D}, \]

where \(n\) and \(D\) are coefficients characterizing the type of detection (see above), we obtain:

\[ \bar{B}=D\cdot \beta \cdot \sigma^n \int\limits_0^\infty x p_{\mathrm{IV}}(x)\,dx . \tag{9a} \]

Let us proceed to consider particular cases. Note that for an undetected output \(n=1,\ D=1\); in the case of linear detection \(n=1,\ D=K_{det}\), and so on.

Undetected Output

1. From formulas (9a) and (10) we find

\[ \bar{B}_n= \frac{\beta\sigma}{\sqrt{2\pi^3}} \int\limits_0^\infty x\left[ \int\limits_0^\pi e^{-\frac{(x-a\cos\Theta)^2}{2}} \,d\Theta \right]dx . \tag{23} \]

After a series of transformations we obtain:

\[ \bar{B}_n= \frac{\beta\sigma}{\sqrt{2\pi}} \cdot e^{-\frac{a^2}{4}} \left\{ J_0\!\left(j\frac{a^2}{4}\right) + \frac{a^2}{2} \left[ J_0\!\left(j\frac{a^2}{4}\right) + J_1\!\left(j\frac{a^2}{4}\right) \right] \right\}; \tag{24} \]

where \(J_0\) and \(J_1\) are Bessel functions of the first kind of zero and first order, respectively.

It is clear that the mean brightness in the absence of a signal and in the presence of noise alone will be (put \(a=0\))

\[ \bar{B}_{on}=\frac{\beta\sigma}{\sqrt{2\pi}} . \tag{25} \]

The relative increase of brightness at the place where the signal appears above the brightness of the noise background will be, from (2), (24), and (25):

\[ \delta B_n= e^{-\frac{a^2}{4}} \left\{ J_0\!\left(j\frac{a^2}{4}\right) + \frac{a^2}{2} \left[ J_0\!\left(j\frac{a^2}{4}\right) + J_1\!\left(j\frac{a^2}{4}\right) \right] \right\} -1 . \tag{25a} \]

Representing the right-hand side of formula (25a) as a power series in \(a\) and discarding all terms containing \(a\) to powers higher than the second, we obtain:

\[ \delta B_n \simeq \frac{a^2}{4}. \tag{26} \]

Hence the minimum detectable signal, referred to \(\sigma\), at the output of the linear system will be:

\[ a_{k\,min}\simeq 2\sqrt{\delta B_{min}} . \tag{27} \]

Linear Detection

1. From formulas (9a) and (11) we have, taking into account \(n=1,\ D=K_{det}\),

\[ \bar{B}_1= \sigma K_{det}\beta e^{-\frac{a^2}{2}} \int\limits_0^\infty x^2 e^{-\frac{x^2}{2}} J_0(jax)\,dx . \]

To take the integral appearing on the right-hand side of the last expression, we shall use the known relation [14б]:

\[ \int\limits_{0}^{\infty} J_p(at)\,e^{-\beta^2 t} t^{q-1}\,df = \frac{ \Gamma\!\left(\frac{p+q}{2}\right)\cdot \left(\frac{a}{2\beta}\right)^p }{ 2\beta^q\cdot \Gamma(p+1) } \,{}_1F_1\!\left(\frac{p+q}{2};p+1;-\frac{a^2}{4\beta^2}\right), \]

where \({}_1F_1(\alpha,\gamma,z)\) is the confluent hypergeometric function.

After simple transformations we obtain:

\[ \overline{B}_1 = \sigma\cdot K_{det}\cdot \beta\cdot e^{-\frac{a^2}{2}} \sqrt{\,2\cdot \Gamma\!\left(\frac{3}{2}\right)\cdot {}_1F_1\!\left(\frac{3}{2};1;\frac{a^2}{2}\right)}. \]

As is known [14г],

\[ \Gamma\!\left(1+\frac{1}{n}\right) = \int\limits_{0}^{\infty} e^{-t^n}\,dt, \]

and for \(n=2\) we have [15]:

\[ \Gamma\!\left(\frac{3}{2}\right) = \int\limits_{0}^{\infty} e^{-t^2}\,dt = \frac{\sqrt{\pi}}{2}. \]

Thus\(^1\)

\[ \overline{B}_1 = \sigma\cdot K_{det}\cdot \beta\cdot e^{-\frac{a^2}{2}} \sqrt{\frac{\pi}{2}\, {}_1F_1\!\left(\frac{3}{2};1;\frac{a^2}{2}\right)}. \tag{28} \]

It is evident that in the absence of a signal the mean brightness will be

\[ \overline{B}_{01} = \sigma\cdot K_{det}\cdot \beta\cdot \sqrt{\frac{\pi}{2}}. \tag{29} \]

The identity \({}_1F_1\!\left(\frac{3}{2};1;0\right)=1\) follows from the representation of the hypergeometric function in the form of a series.

The relative brightness drop from (2), (28), and (29) will be:

\[ \delta B_1 = e^{-\frac{a^2}{2}}\, {}_1F_1\!\left(\frac{3}{2};1;\frac{a^2}{2}\right) -1. \tag{30} \]

Representing the right-hand side of (30) as a series [14в] in powers of \(a\) and discarding terms containing \(a\) to powers higher than the second gives:

\[ \delta B_1 \simeq \frac{a^2}{4}. \tag{31} \]

Hence the minimum signal (referred to \(\sigma\)) at the detector input that produces the required effect on the screen will be:

\[ a_{1min} \simeq 2\sqrt{\delta B_{min}}, \tag{32} \]

that is, as was to be expected, we have obtained the same result as for an undetected output.

\(^1\) (28) differs from (24) only by a constant factor, which becomes obvious if one uses the known identity expressing the hypergeometric function through Bessel functions.

Square-law detection

1. From (9) and (12), to within constant factors containing neither \(x\) nor \(a\), we have:

\[ \overline{B}_2=e^{-\frac{a^2}{2}}\int_{0}^{\infty} x\cdot e^{-\frac{x}{2}} J_0(j\sqrt{x}\,a)\,dx . \]

By the substitution \(x=y^2\) we reduce this case to the preceding one. After a number of simple transformations we obtain:

\[ \overline{B}_2=e^{-\frac{a^2}{2}}\,{}_1F_1\left(2;1;\frac{a^2}{2}\right), \tag{33} \]

\[ \overline{B}_{02}=1 \]

to within a constant factor. From (2), (33), and (34) we have:

\[ \delta B_2=e^{-\frac{a^2}{2}}\,{}_1F_1\left(2;1;\frac{a^2}{2}\right)-1 . \tag{34} \]

Using the identity [14d]

\[ {}_1F_1(\alpha,\gamma,z)=e^z\cdot{}_1F_1(\gamma-\alpha;\gamma;-z), \tag{34a} \]

we obtain:

\[ \delta B_2={}_1F_1\left(-1;1;-\frac{a^2}{2}\right)-1 . \tag{35} \]

For small \(a\), from (35) we obtain:

\[ \delta B_2\simeq \frac{a^2}{2}, \tag{36} \]

and the minimum distinguishable signal, referred to \(\sigma\), applied to the input of a square-law detector, will be:

\[ a_{2\min}\simeq \sqrt{2}\sqrt{\delta B_{\min}} . \tag{37} \]

Higher-order detection \(^{1}\)

1. From (9a) and (22), to within a constant factor containing neither \(x\) nor \(a\), we have:

\[ \overline{B}_n=\frac{1}{n}\cdot e^{-\frac{a^2}{2}}\int_{0}^{\infty} x^{\frac{2}{n}}\cdot e^{-\frac{x^{\frac{2}{n}}}{2}}\cdot J_0\left(jax^{\frac{1}{n}}\right)\,dx . \]

By the substitution \(x=y^n\) we reduce the right-hand side of the last expression to a tabulated integral (see above), and after some transformations we obtain:

\[ \overline{B}_n=2^{\frac{n}{2}}\left[\left(\frac{n}{2}\right)!\right]\cdot e^{-\frac{a^2}{2}}\,{}_1F_1\left(1+\frac{n}{2};1;\frac{a^2}{2}\right). \tag{38} \]

In the absence of a signal

\[ \overline{B}_{0n}=2^{\frac{n}{2}}\left(\frac{n}{2}\right)! . \tag{39} \]

\(^{1}\) It should be noted that detection of higher orders was considered by V. I. Bunimovich [7]. However, Bunimovich considered only the case of detecting fluctuations (without a signal), although the method proposed by him also makes it possible to solve the problem for the joint action of fluctuations and a signal.

From (2), (38), and (39), taking identity (34a) into account, we have:

\[ \delta B_n = {}_1F_1\left(-\frac{n}{2};\,1;\,-\frac{a^2}{2}\right)-1. \tag{40} \]

For small \(a\), (40) takes the form:

\[ \delta B_n \approx \frac{n a^2}{4}, \tag{41} \]

whence

\[ a_{n\,\min}\approx \frac{2}{\sqrt{n}}\cdot \sqrt{\delta B_{\min}}. \tag{42} \]

Comparison of the Cases Considered

1. Let us compare the results obtained. Assuming that the linear part of the receiving device preceding the detector remains unchanged when the mode of the output part of the receiver is changed (that is, when detection is introduced and its order is changed), we can compare the actual sensitivity of the receiving device in the cases considered, using expressions (27), (32), (37), and (42).

Let us estimate the gain in actual sensitivity for each type of detection in comparison with an undetected output, assuming that the minimum detectable brightness difference is the same for each mode, all other conditions being equal:

\[ \psi_{\min}=\frac{a_{\mathrm{nd}\,\min}}{a_{1\min}}\approx 1, \tag{43} \]

\[ \psi_{\mathrm{quad}}=\frac{a_{\mathrm{nd},\min}}{a_{2\min}}=\sqrt{2}, \tag{44} \]

\[ \psi_n=\frac{a_{\mathrm{nd},\min}}{a_{n\min}}\approx \sqrt{n}. \tag{45} \]

Thus, the higher the order of detection, the higher the sensitivity of the receiving device (with a brightness indication).

An undetected output and linear detection give the same, low result; quadratic detection improves the actual sensitivity by approximately 40% in comparison with the case of linear detection or an undetected output.

1. The improvement in actual sensitivity may be still more considerable if the nonlinear portion of the characteristic of the cathode-ray tube is used.

It should be noted that the use of very high orders of detection is not only associated with technical difficulties, but is also hardly expedient because of the very nature of dependence (45); thus, to increase the sensitivity gain, for example, from 4 to 5, it is necessary to increase \(n\) from 16 to 25.

1. Let us estimate the actual sensitivity of the receiving device in the cases considered, assuming that the signal-to-noise ratio changes little as the signal passes through the linear system.

Taking the minimum detectable brightness difference \(\delta B_{min}=0.1\), we obtain:

\[ \begin{gathered} \text{undetected output} \;—\; a_{n.min}\simeq 0.63,\\ \text{linear detection} \;—\; a_{1min}\simeq 0.63,\\ \text{quadratic detection} \;—\; a_{2min}\simeq 0.45,\\ \text{detection of the } n\text{-th order} \;—\; a_{nmin}\simeq \frac{0.63}{\sqrt{n}}, \end{gathered} \tag{46} \]

where

\[ a=\frac{U_0}{\sigma}. \]

Conclusions

1. Considering the results of the analysis carried out, we arrive at the following conclusions:

2. In receivers operating in the regime of the highest actual sensitivity (with a brightness mark), it is inadvisable to use either linear detection or an undetected output.

3. To increase the actual sensitivity, one should raise the order of detection, and also use the nonlinear part of the characteristic of the cathode-ray tube.

In conclusion, let us note that, when comparing experimental data with the results of the analysis carried out, one must take into account deviations of the experimental conditions from the assumptions stipulated above.

I consider it my pleasant duty to express gratitude to Candidate of Physico-Mathematical Sciences A. P. Belousov and Doctor of Technical Sciences L. S. Gutkin, who reviewed the manuscript and made a number of valuable comments.

REFERENCES

1. Siforov V. I. On the influence of interference on the reception of pulsed radio signals, Radiotekhnika No. 1, 1946.

2. Siforov V. I. Analysis of the passage of signal pulses and interference through receiving devices of pulse radio navigation, Proceedings of LKVVIA, issue 13, 1947.

3. Siforov V. I. Radio receiving devices. Svyazizdat, pp. 207—210, 1939.

4. Belousov A. P. On the highest actual sensitivity of a pulse receiver, Radiotekhnika No. 5, 1946.

5. Belousov A. P. On the question of the total noise factor of a receiver.

6. Bunimovich V. I. The fluctuation process as an oscillation with random amplitude and phase, ZhTF, issue 11, 1949.

7. Bunimovich V. I. Transformation of fluctuations by a nonlinear system, ZhTF, issue 6, 1946.

8. Lvovich A. A. On a method for calculating the noise immunity and sensitivity of radiotelegraph receiving devices, Radiotekhnika No. 3, 1949.

9. Kataev S. I. Fundamentals of Television, Moscow, 1940.

10. Papaleksi N. D. (ed.). Course of Physics, vol. II, Gostekhizdat, p. 602, 1947.

11. Rice S. O. Mathematical analysis of random noise, B.S.T.J., XXIII, XXIV, XXVII.

12. Bennett. Joint detection of signal and noise, B.S.T.J., XXIII.

13. Wentzel E. S. Lectures on Probability Theory, VVIA named after Zhukovsky, 1948.

14. Ryzhik I. M. Tables of integrals, sums, series, and products, OGIZ, 1948:
    a) p. 268, c; a, 3.
    b) p. 228, 25.
    c) p. 211, m.
    d) p. 206, 12.
    e) p. 318, d, 2.

15. Bronshtein N. N. and Semendyaev K. A. Handbook of Mathematics. OGIZ, 1945, p. 385, 3, 1945.

16. Vestnik informatsii, No. 17, 1948.

17. Bunimovich V. I. Fluctuation processes in radio-receiving devices, 1951.

On the Question of Space Charge in a Klystron

V. P. PANOV

Klystrons have now become widely used as low-power generators of ultrahigh frequencies. However, with the aid of klystrons, large powers can also be obtained. For this purpose so-called transmission klystrons should be used, since they possess a higher efficiency than reflection klystrons. Since obtaining large powers requires the use of large current densities as well, the question of the influence of the space charge created by the electrons on the processes in a klystron is of interest.

The principal process in a klystron is the bunching process, which consists in the formation, from a uniform electron stream, of electron condensations (bunches) as a result of the electrons acquiring different velocities in the input resonator. On the basis of the assumptions of the one-dimensional nature of the problem, of the small amplitude of the alternating voltage in the input resonator, and of the small velocity of the electrons in comparison with the velocity of light, two theories of bunching were developed. In the so-called kinematic theory the influence of space charge is not taken into account \([1; 2; 3]\). In the other theory \([2; 3; 4]\), the influence of space charge on bunching is taken into account. Both theories are constructed analogously in form, and the formulas of these theories differ only in the expression for the bunching parameter entering the formulas, which determines the degree of condensation of the electron charges. According to the kinematic theory the bunching parameter is determined as follows:

\[ r_k=\gamma_1 \zeta, \tag{1} \]

where

\(\gamma_1\) is the velocity-modulation coefficient, \(\zeta=\dfrac{\omega x}{v_0}\) is the transit angle in the drift space, \(x\) is the distance in the drift space to the point under consideration, \(v_0\) is the constant electron velocity, and \(\omega\) is the frequency.

According to the theory taking into account the influence of space charge, the bunching parameter has the following expression \([4]\)

\[ r=\frac{\gamma_1}{a}\sin a\zeta=\gamma_1 \zeta \frac{\sin a\zeta}{a\zeta}, \tag{2} \]

where

\[ a=0.0975\,\lambda \sqrt{\frac{i_0}{\sqrt{U_0}}}, \]

\(\lambda\) is the wavelength, \(i_0\) is the current density, and \(U_0\) is the constant accelerating voltage.

It is not difficult to see that when \(a\zeta \to 0\), \(r \to r_k\), i.e., the influence of space charge may be neglected. In accordance with what has been said, it is customary to distinguish two cases: 1) when the influence of space charge may be neg-