ON THE INFLUENCE OF CERTAIN RADAR PARAMETERS ON THE AVERAGE HOURLY NUMBER OF DETECTED METEORS

E. I. FIALKO

(Presented by the scientific seminar of the Radio Engineering Faculty)

Introduction

The number of meteors detected by a radar station depends substantially on certain parameters of the radar. Experimental [6] and theoretical [5] works have established, in the main, the dependence of the average hourly number of meteors on the wavelength, the radiated power, the power of the receiver threshold signal, the shape of the antenna-system directional pattern, etc.

However, it remains unclear to what extent the number of detected meteors is affected by the shape of the envelope of the radiated signal and by the shape of the receiver resonance characteristic.

From the results obtained by Kaiser [5], it follows that the average hourly number of meteors \((N)\) can be determined by the approximate formula

\[ N = A \cdot \left[\frac{P_i}{\varepsilon}\right]^{\frac{3S-1}{2}}, \tag{1} \]

where \(P_i\) is the radiated power; in the case of pulsed operation \(P_i\) is the maximum power in the radiated pulse (peak power);

\(\varepsilon\) is the power of the receiver threshold signal;

\(S\) is a constant characterizing the law of distribution of meteors by mass;

\(A\) is a coefficient independent of the signal shape and of the shape of the receiver resonance characteristic.

We shall disclose the dependence of \(N\) on the shape of the envelope of the radiated pulse and on the shape of the receiver resonance characteristic, proceeding from dependence (1).

For this purpose we shall first express the power in the pulse through the average radiation power (for a given shape of the envelope of the generated pulse), and shall also determine the power of the threshold signal for the corresponding shape of the receiver resonance characteristic.

On the Pulse Power

By \(P_i\) one should understand the greatest instantaneous power in the pulse.

Usually the known quantities are the form of the voltage in the pulse as a function of time \(U(t)\), the duration of such a pulse \(\tau\), measured at a definite level from the maximum instantaneous value of the pulse voltage; the average power \(P_0\), and the transmission period \(T_i\).

The condition for comparing different systems is equality of the average powers, or equality of the energies in the pulses \((A_i)\).

Therefore let us express the peak power \(P_i\) in terms of \(P_0\) or \(A_i\).

The energy in the pulse is equal to

\[ A_i = \int_{-\frac{T}{2}}^{\frac{T}{2}} P_{i\,\mathrm{inst}}(t)\,dt, \tag{2} \]

where

\[ P_{i\,\mathrm{inst}}(t) \]

is the instantaneous power in the pulse.

Expression (2) may be represented in the form:

\[ A_i = P_i \cdot \frac{1}{P_i}\int_{-\frac{T}{2}}^{\frac{T}{2}} P_{i\,\mathrm{inst}}(t)\,dt, \tag{3} \]

where \(P_i\) is the maximum instantaneous power in the pulse.

Let us compute the expression

\[ \frac{1}{P_i}\int_{-\frac{T}{2}}^{\frac{T}{2}} P_{i\,\mathrm{inst}}(t)\,dt \]

for a rectangular pulse.

Since the power is proportional to the square of the voltage and

\[ u = \begin{cases} U_0\cos \omega_0 t, & |t| \leq \dfrac{\tau}{2},\\[6pt] 0, & |t| > \dfrac{\tau}{2}, \end{cases} \]

then

\[ \frac{1}{P_i}\int_{-\frac{T}{2}}^{\frac{T}{2}} P_{i\,\mathrm{inst}}(t)\,dt = \frac{1}{U_0^2}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} U_0^2 \cos^2 \omega_0 t\,dt. \]

Carrying out the integration, we obtain:

\[ \frac{1}{P_i}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} P_{i,\mathrm{imp}}(t)\cdot dt = \frac{\tau}{2}\left(1+\frac{T_0}{\tau}\cdot \frac{1}{2\pi}\cdot \sin 2\omega_0 \tau\right), \]

where \(T_0=\dfrac{2\pi}{\omega_0}\) is the period of the high-frequency filling of the pulse. Since \(\dfrac{T_0}{\tau}\ll 1\) and \(|\sin 2\omega_0\tau|\leq 1\), the second term on the right-hand side of the last identity may be neglected. Thus, for a pulse with a rectangular envelope,

\[ \frac{1}{P_i}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} P_{i,\mathrm{imp}}(t)\cdot dt=\frac{\tau}{2} \]

and

\[ A_i=P_i\cdot \tau\cdot \frac{1}{2}. \tag{3} \]

In the case of a pulse of arbitrary shape, by analogy with the rectangular pulse we represent the energy \(A_i\) in the form

\[ A_i=P_i\cdot \tau_{\mathrm{eff}}\cdot \frac{1}{2}, \tag{4} \]

where the effective pulse duration is, as is seen from comparison of (3) and (4),

\[ \tau_{\mathrm{eff}}=\frac{2}{P_i}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} P_{i,\mathrm{imp}}(t)\cdot dt \tag{5} \]

and, since power is proportional to the square of the voltage, we have

\[ \tau_{\mathrm{eff}}=\frac{2}{U_{\max}^{2}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} U^{2}(t)\cdot dt. \tag{6} \]

Thus, the effective pulse duration is defined as the duration of a pulse with a rectangular envelope whose energy is equal to the energy of the given pulse and whose amplitude is equal to the maximum amplitude of the given pulse.

From formula (5) or (6) one can determine the dependence between the effective pulse duration and the pulse duration at a specified level:

\[ \tau_{\mathrm{eff}}=\zeta_1(\tau). \]

In the general case this dependence will be complex, but for simple forms of envelope pulses (rectangular, triangular, probability, sinusoidal, exponential, etc.) there is a linear dependence between $\tau_{\text{eff}}$ and $\tau$:

\[ \tau_{\text{eff}} = a \cdot \tau . \tag{7} \]

Thus, the pulse shape is characterized by the coefficient $a$. As is known,

\[ A_i = P_0 \cdot T_i , \tag{8} \]

From (8) and (4) we have

\[ P_i = 2 \frac{A_i}{\tau_{\text{eff}}} = 2 \frac{P_0 \cdot T_i}{\tau_{\text{eff}}} \tag{9} \]

or

\[ P_i = 2 \frac{P_0 \cdot T_i}{a \cdot \tau} \tag{9a} \]

On the Power of the Threshold Signal

Let us now consider the power of the threshold signal $\varepsilon$. As is known, the power of the threshold signal is determined by the power of the receiver noise and external interference; by the nature of the nonstationary processes occurring in the receiver; and by the observability of weak signals on the indicator screen (or on photographic film).

Let us denote by the coefficient $\xi$ the minimum signal-to-noise ratio at the output of the linear part of the receiver (i.e., at the detector input) at which reliable¹ detection of a weak signal on the indicator screen is ensured:

\[ \left(\frac{U_c}{U_{\text{ш}}}\right)_{\substack{\text{out}\\ min}} = \xi . \tag{10} \]

The value $\xi$ depends on a number of factors: on the shape of the output pulse (and consequently on the shape of the input pulse, the form of the resonance characteristic of the linear part of the receiver, and the relation between the pulse duration and the passband of the linear part of the receiver), the form of the amplitude–frequency characteristic and the magnitude of the passband of the low-frequency amplifier, the transmission period $T_i$, the presence or absence of detection, the sweep speed, the indication mode, etc.

The amplitude of the voltage envelope of the pulse signal at the output of the linear part of the receiver is equal to:

\[ U_{c_{\text{out}}} = U_{c_{\text{in}}} \cdot K_0 \cdot \psi , \tag{11} \]

where

\[ \begin{aligned} U_{c_{\text{in}}} &\text{— amplitude of the voltage envelope of the pulse signal at the receiver input;}\\ K_0 &\text{— gain coefficient of the linear part of the receiver at the resonant frequency } \omega_0; \end{aligned} \]

\(\psi\) is a coefficient characterizing the change in the pulse amplitude due to distortions introduced by the linear part of the receiver.

Let us note that if undamped oscillations with amplitude \(U_{c\,\mathrm{in}}\) and frequency \(\omega_0\) were applied to the receiver input, then the amplitude of the output signal would be equal to \(K_0 U_{c\,\mathrm{in}}\) (and in this case \(\psi=1\)). If, however, a pulsed signal with amplitude \(U_{c\,\mathrm{in}}\) is applied to the receiver input, then, because of nonstationary processes, the output voltage, generally speaking, will not be equal to \(K_0 U_{c\,\mathrm{in}}\), which is taken into account by the coefficient \(\psi\).

The voltage of the fluctuation noise at the output of the linear part of the receiver is determined from the expression

\[ \overline{U^2}_{\mathrm{n,out}}=\overline{U^2}_{\mathrm{n,in}}\cdot K_0^2, \tag{12} \]

where \(\overline{U^2}_{\mathrm{n,in}}\) is the mean square voltage of the fluctuations at the receiver input.

Substituting (11) and (12) into (10), we have:

\[ \frac{U^2_{c\,\mathrm{in}\,\min}\cdot K_0^2\psi^2} {\overline{U^2}_{\mathrm{n,in}}\cdot K_0^2} =\xi^2, \]

or, since power is proportional to the square of the voltage,

\[ \frac{P_{c\,\mathrm{in}\,\min}\cdot \psi^2} {P_{\mathrm{n,in}}} =\xi^2. \]

Since \(P_{c\,\mathrm{in}\,\min}=\varepsilon\), it follows that

\[ \varepsilon=P_{\mathrm{n,in}}\cdot\frac{\xi^2}{\psi^2}. \]

As is known,\(^1\)

\[ P_{\mathrm{n,in}}=k\cdot T\,\Delta f_{\mathrm{n}}\,F, \]

where
\(F\) is the receiver noise factor;
\(\Delta f_{\mathrm{n}}\) is the noise bandwidth of the linear part of the receiver;
\(T\) is the absolute room temperature;
\(k\) is Boltzmann’s constant.

Thus:

\[ \varepsilon=kT\Delta f_{\mathrm{n}}F\cdot\frac{\xi^2}{\psi^2}. \tag{13} \]

The noise bandwidth is understood, as is known [3], as

\[ \Delta f_{\mathrm{n}}=\int_{0}^{\infty} y^2(f)\,df, \]

\(^1\) This formula is valid when the antenna is matched to the receiver input. Otherwise, the mismatch coefficient \(q\) [3] must be taken into account. In this case

\[ P_{\mathrm{n,in}}=q\cdot k\cdot T\cdot \Delta f_{\mathrm{n}}\cdot F \]

where \(y(f)\) is the resonance characteristic of the linear part of the receiver. Between the noise band \(\Delta f_{\mathrm{ш}}\) and the passband at a specified level\(^1\) \(\Delta F\) there exists, generally speaking, a complicated dependence

\[ \Delta f_{\mathrm{ш}}=\varphi_{2}(\Delta F). \]

But in a number of cases this is a simple linear dependence:

\[ \Delta f_{\mathrm{ш}}=b\cdot \Delta F. \tag{14} \]

Substituting (14) into (13), we have

\[ \varepsilon=\kappa\cdot T\cdot \Delta F\cdot b\cdot F\cdot \frac{\xi^{2}}{\psi^{2}} . \]

As is known [1, 2], for given forms of the resonance characteristic of the linear part of the receiver and of the envelope of the input pulse, there always exists an optimum relation between the passband of the linear part of the receiver and the pulse duration, for which the ratio \(\dfrac{\text{signal}}{\text{noise}}\) at the receiver output will be the greatest (for a given amplitude of the input signal):

\[ \Delta F_{\mathrm{opt}}=\varphi_{3}(\tau). \tag{15} \]

This dependence has the form [1, 2]:

\[ \Delta F_{\mathrm{opt}}=\frac{c}{\tau}, \tag{16} \]

where \(c\) is determined by the form of the amplitude-frequency characteristic of the linear part of the receiver and by the form of the envelope of the input pulse.

Remark. In most cases the meteor trail does not cause significant distortions of the pulse shape. However, in the case of normal polarization of the wave relative to the trail, appreciable distortions of the pulse shape may arise [7, 8]. In this case, in (15) the selective properties of the meteor trail must be taken into account.

Thus, in the case of choosing the optimum passband:

\[ \varepsilon=\frac{\kappa\cdot T\cdot F}{\tau}\cdot b\cdot c\cdot \frac{\xi^{2}}{\psi^{2}} . \tag{17} \]

Let us note that the coefficient \(b\) is determined by the form of the resonance characteristic of the receiver; the coefficients \(c\) and \(\psi\) are determined both by the form of the resonance characteristic of the receiver and by the form of the envelope of the input pulse; the coefficient \(\xi\) is determined both by the form of the signal and by the form of the resonance characteristic of the receiver, and also by the form of the amplitude-frequency characteristic and the magnitude of the passband of the low-frequency amplifier.

Studies of the passage of pulse signals through selective systems, carried out by V. I. Siforov, A. P. Belousov, S. I. Evtyanov, and others, have prepared extensive material, allow-

\(^1\) For example, at the level “0.7”.

which makes it possible to calculate the coefficients \(c\) and \(\psi\). Calculation of the coefficient \(b\) presents no difficulty.

The determination of the coefficient \(\xi\), generally speaking, is carried out experimentally.

Method for estimating the form of the resonance characteristic of the receiver and the form of the envelope of the generated pulse from the standpoint of the number of detected meteors

Substituting the expressions for the power in the pulse \(P_i\) (9 or 9a) and the power of the threshold signal \(\varepsilon\) (13 or 17) into the formula for the average hourly number of observed meteors (1), we obtain:

\[ N = A \cdot (B \cdot D \cdot P_0)^{\frac{s-1}{2}}, \]

where

\[ B = \frac{\psi^2}{\xi^2 \cdot \tau_{\mathrm{eff}} \cdot \Delta f_{\mathrm{sh}}}, \tag{18} \]

\[ D = \frac{2T_i}{\kappa \cdot T \cdot F}. \]

The ratio of the average hourly numbers of meteors detected by two radars differing only in the shapes of the resonance characteristics of the receivers and in the shapes of the envelopes of the generated pulses will be

\[ \frac{N_1}{N_2} = \left( \frac{B_1}{B_2} \right)^{\frac{s-1}{2}} . \tag{19} \]

If two radars (with different shapes of the resonance characteristics of the receivers and different pulse shapes) detect the same number of meteors, then the ratio of the average transmitted powers will be

\[ \frac{P_{01}}{P_{02}} = \frac{B_2}{B_1} \tag{20} \]

(the subscripts 1 and 2 in (19) and (20) correspond to the first and second radars).

In the case when an optimal relation is maintained between the bandwidth of the linear part of the receiver \(\Delta F\) and the duration of the input pulse \(\tau\) \((\Delta F_{\mathrm{opt}} \cdot \tau = c)\), and when there is a linear dependence between the noise bandwidth \(\Delta f_{\mathrm{sh}}\) and the bandwidth \(\Delta F\), as well as between \(\tau\) and the effective duration of the pulse \(\tau_{\mathrm{eff}}\) \((\tau_{\mathrm{eff}} = a \tau;\ \Delta f_{\mathrm{sh}} = b \cdot \Delta F)\), expression (18) for the coefficient \(B\) takes the form:

\[ B = \frac{\psi^2}{\xi^2 \cdot a \cdot b \cdot c}. \tag{21} \]

Thus, to implement the described method it is necessary to determine 5 coefficients: \(a\), \(b\), \(c\), \(\psi\), and \(\xi\).

Let us proceed to consider certain special cases. Of particular interest are cases of limiting forms of the resonance characteristics of receivers and limiting forms of the envelopes of the pulses emitted by them.

In this connection, we shall consider rectangular and Gaussian approximations of the forms of the resonance characteristics and pulse envelopes.

The case of a Gaussian approximation (a bell-shaped pulse and a bell-shaped resonance characteristic)

The pulse generated by the radar transmitter is described in this case by the equation:

\[ \mathrm{U}=\mathrm{U}_0 e^{-4\left(\frac{t}{\tau}\right)^2}\cos \omega_0 t, \tag{22} \]

where \(\tau\) is the pulse duration at the level \(\dfrac{\mathrm{U}_0}{e}\).

We assume that the pulse shape at the receiver input coincides with the shape of the generated pulse. Thus, the pulse envelope is a Gaussian, or bell-shaped, curve (a Gaussian curve); the resonance characteristic of the receiver in this case is also a Gaussian curve:

\[ y=e^{-\beta(\omega-\omega_0)^2}, \]

or, as is not difficult to show,

\[ y=e^{-2\ln 2\left(\frac{\Delta f}{\Delta F}\right)^2}, \]

where \(\Delta f=f-f_0\),
\(\Delta F\) is the passband at the level 0.7.

This case was considered by A. P. Belousov [2].

Belousov showed that the optimum passband is equal to

\[ \Delta F_{\mathrm{opt}}=\frac{0.75}{\tau}, \tag{23} \]

where the band is measured at the “0.7 level,” and the pulse duration at the level \(\dfrac{\mathrm{U}_0}{e}\). Comparing expressions (16) and (23), we conclude that \(c=0.75\).

Belousov also showed that the signal amplitude at the output of the linear part of the receiver is equal to

\[ \mathrm{U}_{c\,\mathrm{out}\,\max} = \frac{\mathrm{U}_0 K_0} {\sqrt{\,1+4\left(\dfrac{4}{2\pi\cdot \Delta F_{\frac{1}{e}}\cdot \tau}\right)^2\,}}, \]

where \(\mathrm{U}_0\) is the amplitude of the envelope of the input signal \((\mathrm{U}_0=\mathrm{U}_{c\,\mathrm{in}\,\max})\);
\(K_0\) is the gain coefficient of the linear part of the receiver at the resonant frequency;

\(\Delta F_{1/e}\) is the bandwidth of the linear part of the receiver at the “\(1/e\) level.”

For the case of the optimum bandwidth, i.e., when

\[ \Delta F=\Delta F_{opt}=\frac{0.75}{\tau} \qquad 2\pi\cdot \Delta F_{1/e,opt}=\frac{8}{\tau} \]

and

\[ U_{c\,out\,max}=\frac{U_0K_0}{\sqrt{2}}. \tag{24} \]

Comparing (11) and (24), we conclude that

\[ \varphi=\frac{1}{\sqrt{2}}. \]

Let us note that, in the case of a probabilistic resonance characteristic, the phase-frequency characteristic is linear [2].

We shall now find the relation between the noise bandwidth \(\Delta f_{sh}\) and the passband \(\Delta F_{0.7}\). In the work of R. D. Leites [4] it is stated that between \(\Delta f_{sh}\) and \(\Delta F_{0.7}\) there exists a very simple relation:

\[ \Delta f_{sh}=\frac{1}{2}\sqrt{\frac{\pi}{\ln 2}}\cdot \Delta F_{0.7} =1.065\cdot \Delta F_{0.7}. \tag{25} \]

It should be noted, however, that this relation is valid only in the case when the passband is much smaller than the resonant frequency of the amplifier \(f_0\):

\[ \frac{\Delta F_{0.7}}{f_0}\ll 1. \tag{26} \]

Assuming that condition (26) is satisfied, from comparison of (14) and (25) we conclude:

\[ b=1.065. \]

It remains to find the relation between the effective pulse duration \(\tau_{eff}\) and the pulse duration \(\tau\) at the level \(U_0/e\). Substituting expression (22) into formula (6) and taking into account identities that simplify the notation,

\[ U_{c\,in}(t)=U \quad\text{and}\quad U_{c\,in\,max}=U_0, \]

we obtain:

\[ \tau_{eff}=\frac{2}{U_0^2} \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} U_0^2\cdot e^{-2\left(\frac{2t}{\tau}\right)^2}\cdot \cos^2\omega_0 t\cdot dt. \]

Taking into account the evenness of the integrand and using the identity

\[ \cos^2 \omega_0 t=\frac{1}{2}+\frac{1}{2}\cos 2\omega_0 t, \]

we rewrite the last expression in the form:

\[ \tau_{\mathrm{eff}}= 2\int\limits_0^{T/2} e^{-2\left(\frac{2t}{\tau}\right)^2}\,dt + 2\int\limits_0^{T/2} e^{-2\left(\frac{2t}{\tau}\right)^2}\cdot \cos(2\omega_0 t)\,dt . \]

In radiolocation \(T \gg \tau\); the function \(e^{-x^2}\) decreases very rapidly. Therefore replacing the upper limit by \(\infty\) will not cause any significant error¹.

Using the tabulated integrals [9]

\[ \int\limits_0^\infty e^{-x^2}\,dx=\frac{\sqrt{\pi}}{2} \]

and

\[ \int\limits_0^\infty e^{-a^2x^2}\cos bx\cdot dx = \frac{\sqrt{\pi}}{2}\cdot \frac{e^{-\frac{b^2}{4a^2}}}{a}, \]

we obtain

\[ \tau_{\mathrm{eff}} = \tau\frac{\sqrt{2\pi}}{4} + \tau\frac{\sqrt{2\pi}}{4}\cdot e^{-\frac{1}{4}\left(\frac{\omega_0\tau}{\sqrt{2}}\right)^2}, \]

or

\[ \tau_{\mathrm{eff}} = \tau\frac{\sqrt{2\pi}}{4} \left[ 1+ e^{-\frac{1}{8}(\omega_0\tau)^2} \right]. \]

Usually

\[ \omega_0\tau=2\pi\frac{\tau}{T_0}\gg 1, \]

since the pulse contains a considerable number of periods of the intermediate and, still more, of the radio frequency.

Therefore

\[ e^{-\frac{1}{8}(\omega_0\tau)^2}\ll 1 \]

and, consequently,

\[ \tau_{\mathrm{eff}}\simeq \tau\frac{\sqrt{2\pi}}{4}. \tag{27} \]

Comparing (7) and (27), we conclude that

\[ a=\frac{\sqrt{2\pi}}{4}=0.63. \]

Assigning to the coefficients \(a\), \(b\), \(c\), and \(\psi\), in the case of a bell-shaped pulse and a bell-shaped characteristic, the index “1,” we have:

\[ \begin{aligned} a_1 &= 0.63,\\ b_1 &= 1.065,\\ c_1 &= 0.75,\\ \psi_1 &= 0.707 \end{aligned} \]

and, consequently,

\[ B_1=\frac{\psi_1^2}{a_1\cdot b_1\cdot c_1\cdot \xi_1^2} =\frac{1}{\xi_1^2}. \tag{28} \]

Case of rectangular approximation (rectangular form of the pulse envelope and rectangular form of the receiver resonance characteristic)

At the input of a linear oscillatory system having rectangular resonance and rectilinear phase-frequency characteristics, there acts a pulse of oscillatory voltage of duration \(\tau\) with a rectangular envelope curve:

\[ U= \begin{cases} U_0\cdot \cos \omega_0 t, & |t|\leq \dfrac{\tau}{2},\\[6pt] 0, & |t|>\dfrac{\tau}{2}. \end{cases} \]

This case was considered by V. I. Siforov in a number of works [1, 3, and others]. Siforov showed that the optimum frequency passband is equal to [3]

\[ \Delta F_{\text{opt}}=\frac{1.37}{\tau}. \tag{29} \]

Comparing expressions (16) and (29), we conclude that \(c=1.37\). The maximum amplitude of the signal voltage at the output of the system is equal to [3]

\[ U_{c\,\text{out max}} =\frac{2}{\pi}\cdot K_0\cdot U_0\,Si\cdot \frac{x}{2}, \]

where \(K_0\) is the gain coefficient of the system;

\[ x=\pi\cdot \tau\cdot \Delta F \]

\[ U_0=U_{c\,\text{in max}}. \]

In the case \(\Delta F=\Delta F_{\text{opt}}\), \(x=4.3\)

and

\[ U_{c\,\text{out max}} =\frac{2}{\pi}\cdot K_0\cdot U_0\cdot Si\,(2.15). \]

Since \(Si\,(2.15)=1.6682\), then

\[ U_{c\,\text{out max}}=1.062\cdot K_0 U_0. \tag{30} \]

Comparing (11) and (30), we conclude that \(\psi = 1.062\). It is quite obvious that in this case the noise band is equal to the frequency passband:

\[ \Delta f_{\mathrm{n}} = \Delta F . \tag{31} \]

Comparing (19) and (31), we have

\[ b = 1. \]

It is also quite obvious that, for a pulse with a rectangular envelope,

\[ \tau_{\mathrm{eff}} = \tau \tag{32} \]

and, thus, from (7) and (32),

\[ a = 1. \]

Assigning the index “2” to the coefficients in the case of a rectangular pulse and a rectangular characteristic, we have:

\[ \begin{aligned} a_2 &= 1,\\ b_2 &= 1,\\ c_2 &= 1.37,\\ \psi_2 &= 1.062 \end{aligned} \]

and, consequently,

\[ B_2 = \frac{\psi_2^2}{a_2 \cdot b_2 \cdot c_2 \cdot \xi_2^2} = \frac{0.825}{\xi_2^2}. \tag{33} \]

Remark on the coefficient \(\xi\)

To compare the particular cases considered, it is necessary to compare the coefficients \(\xi_1\) and \(\xi_2\), which characterize the observability of weak probabilistic and rectangular signals (passed through the linear part of the receiver) against a background of interference on the screen of a cathode-ray indicator (or on photographic film).

For an exact determination of the coefficients \(\xi_1\) and \(\xi_2\), a careful experimental investigation is required.

For an approximate solution of the problem, however, one may confine oneself to the following reasoning:

Although the rectangular signal differs sharply in form from the probabilistic one, the coefficients \(\xi_1\) and \(\xi_2\) are very close in magnitude. This is explained by the fact that, with the optimum passband, the output signal acquires a smoothed form and is very far from a rectangle, even if the input signal is rectangular.

Comparing the output rectangular and probabilistic signals, it is not difficult to show that the forms of the principal parts of the signals are very similar. Therefore, the conditions for the observability of such signals against a background of fluctuation interference cannot be substantially different and, as a consequence, the coefficients \(\xi_1\) and \(\xi_2\) cannot differ substantially from each other and, in a first approximation, may be taken as equal:

\[ \xi_1 \simeq \xi_2 . \]

Comparison of the Special Cases Considered

From (19), (21) and (28), (33) we conclude that¹

\[ \frac{N_1}{N_2} = \left[ \frac{B_1}{B_2} \right]^{\frac{s-1}{2}} = \left[ \frac{1}{\xi_1^2}\cdot \frac{\xi_2^2}{0.825} \right]^{\frac{s-1}{2}} = \left[ 1.21 \cdot \frac{\xi_2^2}{\xi_1^2} \right]^{\frac{s-1}{2}} \]

and for \(s \simeq 2\)

\[ \frac{N_1}{N_2} \simeq 1.1\,\frac{\xi_2}{\xi_1}. \]

Since, for the optimum relation between the passband and the pulse duration,

\[ \xi_1 \simeq \xi_2, \]

then

\[ \frac{N_1}{N_2} \simeq 1.1 \]

and

\[ \frac{P_{01}}{P_{02}}=\frac{B_2}{B_1}\simeq 0.83. \]

This means that a system with a bell-shaped pulse and a resonance characteristic detects a somewhat larger number of meteors than a system with a rectangular pulse and a rectangular form of the receiver resonance characteristic. This gain, however, is very small.

Since a system with a smooth pulse and a smooth form of the resonance characteristic is considerably simpler and at the same time detects no fewer meteors than a system with a rectangular resonance characteristic and a rectangular pulse envelope, the former system is the more preferable one, provided that no special requirements are imposed on the radar.

The case of a non-optimal relation between \(\tau\) and \(\Delta F\) requires additional consideration.

Conclusions

1. A simple method has been given for comparing various forms of the resonance characteristics of receiving devices and the forms of the envelopes of generated pulses from the standpoint of the number of detected meteors.

2. Two systems have been considered: a bell-shaped pulse—a bell-shaped resonance characteristic of the receiver; a rectangular pulse—a rectangular resonance characteristic of the receiver.

Consideration of the limiting cases (rectangular and probabilistic approximations) shows that the difference in the forms of pulses and resonance characteristics is of little significance from the standpoint of the number of detected meteors.

1. In the case of an optimum relation between the pulse duration and the passband, a somewhat better result is given by the system with bell-shaped forms of the pulse and of the receiver resonance characteristic.

2. Thus, if no special requirements are imposed on a radar intended for meteor observations

(for precise determination of coordinates, etc.), it is advisable to use a resonant amplifier (and not a band amplifier with a resonant characteristic close to rectangular) and a smooth (not rectangular) pulse.

This conclusion is also justified by the simplicity of such a system from the standpoint of production and operation.

REFERENCES

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

2. A. P. Belousov. On the highest real sensitivity of a pulsed receiver. Radiotekhnika, No. 5, 1946.

3. V. I. Siforov. Radio-Receiving Devices. Voenizdat, 1951.

4. R. D. Leites. Transient processes in a multistage resonant amplifier. Radiotekhnika, No. 3, 1947.

5. T. R. Kaiser. Theory of the Meteor Height Distribution Obtained from Radio-Echo Observations. Mon. Not. R. A. S., 1954, v. 114, No. 1, pp. 39–62.

6. D. W. R. McKinley. Variation of Meteor Echo Rates with Radar System Parameters. Can. Journ. Phys., 1951, v. 29, No. 5, pp. 403–426.

7. T. R. Kaiser, R. I. Closs. Theory of Radio Reflections from Meteor Trails. Phil. Mag., Ser. 7, 1952, v. 43, No. 336, pp. 1, 32.

8. E. K. Nemirova. Distortion of pulsed signals under resonant reflection from meteor trails. Izvestiya TPI, vol. 86, 1958.

9. I. N. Bronshtein and K. A. Semendyaev. Handbook of Mathematics. OGIZ, 1945.

1. This result is valid in the case of reception of unstable radio echoes. ↩↩↩