On the Question of the Optimal Bandwidth of a Pulse Receiver

E. I. Fialko

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

Preliminary Remarks. Statement of the Problem

In designing radio receivers for pulse signals, it is often necessary to ensure the highest real sensitivity or, more precisely, the minimum threshold signal.

For this purpose, in particular, it is necessary to choose correctly the bandwidth of the linear part of the receiver.

The question of the optimal bandwidth has been considered in a number of works \[1, 2, and others\].

However, although in these works the optimal bandwidth is found from the condition of obtaining the maximum ratio of the output-signal amplitude to the effective value of the noise, in practice it is usually assumed that such a bandwidth also ensures the minimum threshold signal.

In reality, however, the observability of a signal is determined not only by the ratio of the signal and noise levels, but also by the ratio of the durations of the signal and noise pulses, which also depends on the receiver bandwidth.

Finding the optimal bandwidth corresponding to the minimum threshold signal, in the general case, presents serious difficulties and is associated with considerable analytical and experimental investigations.

In the present work, one of the possible methods of solving the problem and certain special cases are considered; the purpose of the consideration is to obtain approximate quantitative relations.

Let us introduce the following notation:

$H$ — observability of the signal, which is numerically estimated by the probability of correct recognition of the signal against the background of noise;

$U_c$ — level (amplitude) of the signal;

$U_{\text{ш}}$ — level (effective value) of the noise;

$U_{c\text{ш}}$ — level (effective value) of the voltage equal to the sum of the signal and noise;

$\tau_c$ — duration of the envelope of the signal pulse;

\(\tau_{\mathrm{sh}}\) — duration of the envelope of the noise pulse;

\(\tau_{\mathrm{s\,sh}}\) — duration of the envelope of the total voltage—signal and noise;

\(\Delta F\) — passband of the linear part of the receiver;

\(\Delta F_{\mathrm{opt}}\) — optimal passband, i.e., the band corresponding to the highest observability of the signal for a given signal amplitude at the receiver input;

\(F'_{\mathrm{opt}}\) — passband corresponding to the maximum of the ratio

\[ \frac{U_{\mathrm{s}}}{U_{\mathrm{sh}}} \]

at the output of the linear part of the receiver for a given signal amplitude at the receiver input.

\(U_{\mathrm{s}}, U_{\mathrm{sh}}, U_{\mathrm{s\,sh}}, \tau_{\mathrm{s}}, \tau_{\mathrm{sh}}\) and \(\tau_{\mathrm{s\,sh}}\) — parameters of the signal and noise at the output of the linear part of the receiver.

\[ H = H\left(\frac{U_{\mathrm{s\,sh}}}{U_{\mathrm{sh}}};\ \frac{\tau_{\mathrm{s\,sh}}}{\tau_{\mathrm{sh}}}\right) \tag{1} \]

and since

\[ \frac{U_{\mathrm{s\,sh}}}{U_{\mathrm{sh}}} = \frac{U_{\mathrm{s\,sh}}}{U_{\mathrm{sh}}}(\Delta F), \]

\[ \frac{\tau_{\mathrm{s\,sh}}}{\tau_{\mathrm{sh}}} = \frac{\tau_{\mathrm{s\,sh}}}{\tau_{\mathrm{sh}}}(\Delta F), \]

then

\[ H = H(\Delta F). \]

In simplified form, the basic equation (1) may be represented as

\[ H = H\left(\frac{U_{\mathrm{s}}}{U_{\mathrm{sh}}};\ \frac{\tau_{\mathrm{s}}}{\tau_{\mathrm{sh}}}\right). \tag{2} \]

Such a simplification may prove to be forced by the considerable difficulties in determining the function \(\tau_{\mathrm{s\,sh}}(\Delta F)\). It should be borne in mind that, when using formula (2), the conclusions for the case of small

\[ \frac{U_{\mathrm{s}}}{U_{\mathrm{sh}}} \]

may require corrections.

A number of questions arise: does \(\Delta F_{\mathrm{opt}}\) exist in the general case? how much does \(\Delta F_{\mathrm{opt}}\) differ from \(\Delta F'_{\mathrm{opt}}\)? etc.

Basic relations

The optimal passband may be found from the condition:

\[ \frac{dH}{d(\Delta F)} = 0. \tag{3} \]

Since

\[ \frac{dH}{d(\Delta F)} = \frac{\partial H}{\partial\left(\frac{U_{\mathrm{s\,sh}}}{U_{\mathrm{sh}}}\right)} \cdot \frac{d\left(\frac{U_{\mathrm{s\,sh}}}{U_{\mathrm{sh}}}\right)}{d(\Delta F)} + \frac{\partial H}{\partial\left(\frac{\tau_{\mathrm{s\,sh}}}{\tau_{\mathrm{sh}}}\right)} \cdot \frac{d\left(\frac{\tau_{\mathrm{s\,sh}}}{\tau_{\mathrm{sh}}}\right)}{d(\Delta F)}, \]

then condition (3) takes the form:

\[ \frac{d\left(\dfrac{U_{cш}}{U_{ш}}\right)}{d(\Delta F)} = -\frac{d\left(\dfrac{\tau_{cш}}{\tau_{ш}}\right)}{d(\Delta F)} \cdot \frac{ \dfrac{\partial H}{\partial\left(\dfrac{\tau_{cш}}{\tau_{ш}}\right)} }{ \dfrac{\partial H}{\partial\left(\dfrac{U_{cш}}{U_{ш}}\right)} }, \tag{4} \]

and, in the case of using the simplified equation (2):

\[ \frac{d\left(\dfrac{U_c}{U_{ш}}\right)}{d(\Delta F)} = -\frac{d\left(\dfrac{\tau_c}{\tau_{ш}}\right)}{d(\Delta F)} \cdot \frac{ \dfrac{\partial H}{\partial\left(\dfrac{\tau_c}{\tau_{ш}}\right)} }{ \dfrac{\partial H}{\partial\left(\dfrac{U_c}{U_{ш}}\right)} }. \tag{5} \]

Solving equation (5) or (4) with respect to \(\Delta F\), we find \(\Delta F_{opt}\) (in the case where it exists).

It should be noted, however, that finding the function \(H\left(\dfrac{U_c}{U_{ш}};\dfrac{\tau_c}{\tau_{ш}}\right)\) may turn out to be more cumbersome than directly taking the dependence \(H(\Delta F)\).

Let us consider several special cases:

1. The detectability of the signal depends on the amplitude ratios of the signal and noise to a substantially greater degree than on the ratios of the durations of the signal and noise pulses:

\[ \frac{\partial H}{\partial\left(\dfrac{U_c}{U_{ш}}\right)} \gg \frac{\partial H}{\partial\left(\dfrac{\tau_c}{\tau_{ш}}\right)}. \]

(This case corresponds, for example, to the detectability of a signal with amplitude indication.)

Condition (5) will take the form:

\[ \frac{d\left(\dfrac{U_c}{U_{ш}}\right)}{d(\Delta F)}=0. \tag{6} \]

As is known, the solution of this equation will be [1, 2, etc.]

\[ \Delta F=\Delta F_{opt}'=\frac{a}{\tau_{c0}}, \]

where \(a=0.75 \div 1.37\), depending on the approximation of the envelope of the si-

of the signal and the resonance characteristic of the linear part of the receiver, \(\tau_{c0}\) is the duration of the signal at the receiver input. Usually one assumes \(\alpha \simeq 1\).

1. The amplitude and time ratios of the signal and noise pulses are not equivalent from the point of view of signal observability, but they can be represented in the form:

\[ \frac{\partial H}{\partial\left(\dfrac{U_c}{U_{\sh}}\right)} = \frac{\partial H}{\partial\left(\dfrac{\tau_c}{\tau_{\sh}}\right)}\,\alpha, \]

where \(\alpha=\mathrm{const}\) over a certain interval of values of \(\dfrac{U_c}{U_{\sh}}\) and \(\dfrac{\tau_c}{\tau_{\sh}}\). (The case \(\alpha \simeq 1\), apparently, occurs with a brightness indication of the signal and with photographing of the screen of a cathode-ray tube.)

Condition (5) takes the form:

\[ \frac{d\left(\dfrac{U_c}{U_{\sh}}\right)}{d(\Delta F)} = - \frac{d\left(\dfrac{\tau_c}{\tau_{\sh}}\right)}{d(\Delta F)} \cdot \frac{1}{\alpha}. \tag{7} \]

To solve equation (7) with respect to \(\Delta F\), it is necessary to specify the shape of the signal envelope and the shape of the resonance characteristic of the receiver.

Let us consider one of the frequently encountered methods of approximating the pulse shape and the resonance characteristic—the case of a Gaussian approximation.

The case of a Gaussian approximation of the pulse shape and the resonance characteristic of the receiver

At the input of a receiver with resonance characteristic

\[ y=\frac{K}{K_0} = e^{-4\left(\dfrac{\omega-\omega_0}{\Delta\omega}\right)^2} \]

(where \(K\) is the gain of the linear part of the receiver at frequency \(\omega\); \(K_0\) is the resonant gain; \(\Delta\omega\) is the passband at the level \(\dfrac{K_0}{e}\)) a signal is applied

\[ U=U_0 e^{-4\left(\dfrac{t}{\tau_0}\right)^2}\cdot \cos\omega_0 t, \]

where \(U_0\) is the signal amplitude,

\(\tau_0\) is the signal duration at the level \(\dfrac{U_0}{e}\). The voltage amplitude of the signal at the output of the linear part will be [2]

\[ U_c=\frac{U_0\cdot K_0}{\sqrt{1+\left(\frac{8}{\Delta\omega\,\tau_0}\right)^2}} . \tag{8} \]

The rms noise voltage at the output of the linear system of the receiver, as is not difficult to show (in the case where the antenna is matched to the receiver input), is

\[ U_{\mathrm{ш}}=\sqrt{k\cdot T\cdot R_A\cdot \Delta F_{\mathrm{ш}}\cdot N}\,K_0, \tag{9} \]

where

\[ \begin{aligned} k &\text{— Boltzmann’s constant;}\\ T &\text{— room temperature;}\\ R_A &\text{— radiation resistance of the antenna;}\\ \Delta F_{\mathrm{ш}} &\text{— effective noise bandwidth of the receiver;}\\ N &\text{— noise figure of the receiver.} \end{aligned} \]

Denoting

\[ \Delta\omega_{\mathrm{ш}}=2\pi\Delta F_{\mathrm{ш}} \]

and

\[ Q=\frac{U_c}{U_{\mathrm{ш}}}, \tag{10} \]

from (8), (9), and (10) we obtain

\[ Q= \frac{\sqrt{2\pi}\cdot U_0} {\sqrt{1+\left(\frac{8}{\Delta\omega\cdot\tau_0}\right)^2}\, \sqrt{\Delta\omega_{\mathrm{ш}}}\,\sqrt{k\cdot T\cdot R_A\cdot N}} . \tag{11} \]

As is known [2],

\[ Q=Q_{\max}\quad \text{for}\quad \Delta\omega=\Delta\omega'_{\mathrm{opt}}=\frac{8}{\tau_0}, \]

and in this case (11) takes the form:

\[ Q_{\max}= \frac{\sqrt{2\pi}\cdot U_0} {\sqrt{2}\cdot \sqrt{\Delta\omega'_{\mathrm{ш}\,\mathrm{opt}}}\cdot \sqrt{k\cdot T\cdot R_A\cdot N}} . \tag{12} \]

Dividing (11) by (12), we obtain

\[ \frac{Q}{Q_{\max}}= \frac{\sqrt{2}} {\sqrt{1+\left(\frac{8}{\Delta\omega\cdot\tau_0}\right)^2}} \cdot \sqrt{\frac{\Delta\omega'_{\mathrm{ш}\,\mathrm{opt}}}{\Delta\omega_{\mathrm{ш}}}} . \]

Since [2]

\[ \Delta\omega_{\mathrm{ш}}=\frac{1}{2}\sqrt{\frac{\pi}{2}}\cdot\Delta\omega, \tag{13} \]

then

\[ Q = \frac{\sqrt{2}}{\sqrt{1+\left(\frac{8}{\Delta \omega \cdot \tau_0}\right)^2}} \cdot \sqrt{\frac{\Delta \omega'_{opt}}{\Delta \omega}} \cdot Q_{\max}, \tag{14} \]

or

\[ Q = \frac{\sqrt{2}\cdot \sqrt{\Delta \omega'_{opt}}} {\sqrt{\Delta \omega+\frac{1}{\Delta \omega}\left(\frac{8}{\tau_0}\right)^2}} \cdot Q_{\max}. \]

Differentiating \(Q\) with respect to \(\Delta F\) and carrying out elementary transformations, we obtain

\[ \frac{d\left(\dfrac{U_c}{U_{\sh}}\right)}{d(\Delta F)} = \sqrt{2}\cdot \pi \cdot \sqrt{\Delta \omega'_{opt}} \cdot \frac{\left(\dfrac{8}{\Delta \omega \cdot \tau_0}\right)^2-1} {\sqrt{\left\{\Delta \omega\left[1+\left(\dfrac{8}{\Delta \omega \tau_0}\right)^2\right]\right\}^3}} \cdot Q_{\max}, \]

or, since \(\Delta \omega'_{opt}=\dfrac{8}{\tau_0}\),

\[ \frac{d\left(\dfrac{U_c}{U_{\sh}}\right)}{d(\Delta F)} = 4\pi \cdot \frac{\left(\dfrac{8}{\Delta \omega \cdot \tau_0}\right)^2-1} {\sqrt{\tau_0\left\{\Delta \omega\left[1+\left(\dfrac{8}{\Delta \omega \cdot \tau_0}\right)^2\right]\right\}^3}} \cdot Q_{\max}. \tag{15} \]

Let us proceed to consider the right-hand side of equation (7). The duration of the output signal at the level \(\dfrac{U_{c\max}}{e}\) is equal to [2]

\[ \tau_c=\tau_0\sqrt{1+\left(\frac{8}{\Delta \omega \cdot \tau_0}\right)^2}. \tag{16} \]

The mean duration of a noise-envelope excursion \(\tau_{\sh}\) at the level \(U\) is equal to [3]

\[ \tau_{\sh}=\frac{1}{\Delta F_{\sh}\cdot x}, \tag{17} \]

where

\[ x=\frac{U}{U_{\sh}}, \tag{18} \]

\(U_{\sh}\) is the rms noise voltage at the output of the linear system of the receiver.

Taking (13) and (18) into account, formula (17) assumes the form

\[ \tau_{\mathrm{ш}}=\frac{\sqrt{2\pi\cdot 4}}{\Delta\omega\cdot x}. \]

Comparing the duration of the signal and noise pulses at one and the same level, for example at the level \(U=\dfrac{U_c}{e}\), we have:

\[ x=\frac{U}{U_{\mathrm{ш}}}=\frac{U_c}{U_{\mathrm{ш}}}\cdot\frac{1}{e}=Q\cdot\frac{1}{e}. \tag{20} \]

Substituting (20) into (19), we obtain

\[ \tau_{\mathrm{ш}}=\frac{4\sqrt{2\pi}\cdot e}{\Delta\omega\cdot Q}. \tag{21} \]

Remark. The question of at what levels one should compare the durations of the signal and noise pulses is subject to separate consideration.

From (16) and (21)

\[ \frac{\tau_c}{\tau_{\mathrm{ш}}} = \frac{ \Delta\omega\cdot\tau_0\cdot \sqrt{1+\left(\frac{8}{\Delta\omega\tau_0}\right)^2} }{ 4\cdot\sqrt{2\pi}\cdot e }\cdot Q \]

and, taking (14) into account,

\[ \frac{\tau_c}{\tau_{\mathrm{ш}}} = \frac{ \tau_0\cdot\sqrt{\Delta\omega'_{\mathrm{opt}}}\cdot\sqrt{\Delta\omega} }{ 4\cdot\sqrt{\pi}\cdot e }\cdot Q_{\max}. \]

Since

\[ \Delta\omega'_{\mathrm{opt}}=\frac{8}{\tau_0}, \]

then

\[ \frac{\tau_c}{\tau_{\mathrm{ш}}} = \frac{\sqrt{\tau_0\cdot\Delta\omega}}{\sqrt{2\pi}\cdot e}\cdot Q_{\max} \]

and

\[ \frac{ d\left(\dfrac{\tau_c}{\tau_{\mathrm{ш}}}\right) }{ d(\Delta F) } = \sqrt{\frac{\pi\cdot\tau_0}{2\cdot\Delta\omega}}\cdot\frac{Q_{\max}}{e}. \tag{22} \]

Substituting (15) and (22) into (7), we obtain

\[ 4\pi\cdot \frac{ \left(\dfrac{8}{\Delta\omega\cdot\tau_0}\right)^2-1 }{ \sqrt{ \tau_0 \left\{ \Delta\omega \left[ 1+\left(\dfrac{8}{\Delta\omega\cdot\tau_0}\right)^2 \right] \right\}^{3} } } \cdot Q_{\max} = \]

\[ = \sqrt{\frac{\pi \tau_0}{2 \Delta \omega}} \cdot \frac{Q_{\max}}{e} \cdot \frac{1}{\alpha}. \]

Thus, equation (7) for the case of the probability approximation has the form:

\[ \frac{ 1 - \left(\frac{8}{\Delta \omega \tau_0}\right)^2 }{ \sqrt{\left[1 + \left(\frac{8}{\Delta \omega \tau_0}\right)^2\right]^3} } = \frac{\Delta \omega \tau_0}{4\sqrt{2\pi}\, e} \cdot \frac{1}{\alpha}. \tag{23} \]

Denoting \(\Delta \omega \tau_0 = y\), we represent (23) in the form

\[ \frac{ 1 - \left(\frac{8}{y}\right)^2 }{ \sqrt{\left[1 + \left(\frac{8}{y}\right)^2\right]^3} } = \frac{y}{8e\sqrt{\frac{\pi}{2}}} \cdot \frac{1}{\alpha}. \tag{24} \]

Solving equation (24) graphically, we find that for \(\alpha < 1.07\) there is no real root; for \(\alpha \geq 1.07\)

\[ 8 < y_{opt} < 18. \]

Thus, in the case of the probability approximation,

\[ 1 < \frac{\Delta \omega_{opt}}{\Delta \omega'_{opt}} < 2.25 \]

for

\[ 1.07 < \alpha < \infty. \]

The case \(\alpha < 1.07\) requires additional consideration.

However, it is obvious that in the case of interest to us, when the observability of the signal depends almost equally on the amplitude and temporal relations of signal and noise, the optimal pass band is two to three times larger than the band that provides the maximum ratio of the signal and noise levels.

Extending the conclusion obtained to other types of approximation, we conclude that in the case of interest to us

\[ \Delta F_{opt} \simeq \frac{2 \div 3}{\tau_0}, \]

since it is usually taken that

\[ \Delta F'_{opt} \simeq \frac{1}{\tau_0} \]

(here \(\tau_0\) is the duration of the signal at the receiver input).

Conclusions

1. In the case when the observability of a signal is determined not only by the ratio of amplitudes (levels), but also by the ratio of the durations of the signal pulses and the noise, the optimum bandwidth \(\Delta F_{\text{opt}}\) is greater than the bandwidth \(\Delta F'_{\text{opt}}\) that provides the maximum

\[ \frac{\text{signal}}{\text{noise}} . \]

1. In the particular case when the magnitude of the threshold signal depends almost equally on the amplitude and time ratios of the signal and the noise, the optimum bandwidth may be chosen from the condition

\[ \Delta F_{\text{opt}} \approx \frac{2 \div 3}{\tau_0}, \]

where \(\tau_0\) is the duration of the signal at the receiver input.

Literature

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

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

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