On the Choice of Local Oscillator Frequency in a VHF Receiver
Abstract Generated abstract
This paper examines how the choice of local oscillator frequency in a VHF superheterodyne receiver affects interference from cosmic noise in meteor radar observations. It models cosmic noise spectral density, receiver high-frequency gain, and image-channel transmission to derive expressions for the relative increase in noise when the image frequency lies above or below the received signal frequency. The analysis shows that choosing the local oscillator above the signal frequency reduces image-channel cosmic noise, while the practical importance of this choice depends on RF selectivity and intermediate frequency. For highly selective receivers with low intermediate frequency the effect is negligible, but in broadband or low-selectivity receivers with comparatively high intermediate frequency, choosing the local oscillator below the signal may noticeably increase the noise threshold.
Full Text
ON THE CHOICE OF LOCAL OSCILLATOR FREQUENCY IN A METER-BAND RECEIVER
E. I. FIALKO
(Presented by the scientific seminar of the Faculty of Radio Engineering)
One of the questions solved in the design of radio receiving devices is the choice of the local oscillator frequency \(f_2\).
Let us consider a particular question: how the level of interference caused by cosmic noise changes, depending on the choice of \(f_2\) (\(f_2 > f_c\) or \(f_2 < f_c\)) in receivers of radar stations intended for meteor observations.
As is known, radar observations of meteors are carried out mainly in the wavelength range \(\lambda = 4 \div 10\) m. In this range the interference level is determined chiefly by cosmic noise, whose intensity is \(\sim \lambda^m\), where \(2 < m < 3\) [1, 2].
In a superheterodyne receiver, cosmic noise passes through both the main and the image channels. In order to reduce the power of the cosmic noise passing through the image channel, it is advantageous for the frequency of the image interference to be higher than the frequency of the received signal and \(f_2 > f_c\) (Fig. 1).
Let us consider the dependence of the cosmic-noise level on the parameters of the high-frequency part of the receiver.
Let us represent the intensity of cosmic noise (referred to a unit frequency) at the receiver input in the form:
\[ P_n = \frac{a}{f^m}, \tag{1} \]
where \(a\) and \(m\) are constants.
Fig. 1. On the choice of the local oscillator frequency.
\(P_n\)—spectral density of cosmic noise; \(f\)—frequency; \(K\)—gain coefficient of the RF amplifier; \(f_c\)—signal frequency; \(f_2\)—local oscillator frequency; \(f_3\)—image-interference frequency.
The gain coefficient of the receiver’s high-frequency section is equal to
\[ K=K_o\left[1+\left(\frac{2\Delta f}{\Delta F_1}\right)^2\right]^{-\frac12(n+1)}, \tag{2} \]
where \(K_o=K_{o1}\cdot K_{o2}\ldots K_{on}\) is the resonant gain coefficient of the high-frequency section (HF section), consisting of \(n\) identical stages; \(K_{o1},\ldots,K_{on}\) are the resonant gain coefficients of each of the stages included in the high-frequency section; \(\Delta f=f-f_o\) is the detuning; \(f_o\) is the resonant frequency; \(\Delta F_1\) is the passband of one HF-section stage (we assume that all stages included in the HF section, and the input circuit, have identical passbands).
The power of the interfering signals passing through the main channel at the output of the HF section will be equal to (in the case \(f_c=f_o\)) \(P_o=P_n(f_c)\cdot G_o\cdot \Delta F_{sh}\), where \(\Delta F_{sh}\) is the effective noise passband of the receiver; \(G_o\sim K_o^2\) is the power gain coefficient.
The power of the interfering signals at the HF-section output that have passed through the image channel is equal to \(P_z=P_n(f_z)\cdot G_1\cdot \Delta F'_{sh}\), where \(G_1\sim K_1^2(f_z)\).
Strictly speaking, \(\Delta F'_{sh}\ne \Delta F_{sh}\), but in the case when the bandwidth of the intermediate-frequency amplifier is considerably smaller than the bandwidth of the HF section, \(\Delta F'_{sh}\simeq \Delta F_{sh}\).
The total power of the interfering signals that have passed through both the main and the image channels is equal to
\[ P_\Sigma=P_o+P_z=P_o\left\{1+\frac{P_n(f_z)}{P_n(f_c)}\cdot\left(\frac{K_1}{K_o}\right)^2\right\}. \]
The relative increase in the interference level due to the image channel is estimated by the ratio
\[ \frac{P_\Sigma}{P_o}=1+\frac{P_n(f_z)}{P_n(f_c)}\cdot\left(\frac{K_1}{K_o}\right)^2 \]
and, taking (1) and (2) into account,
\[ \frac{P_\Sigma}{P_o} = 1+\left(\frac{f_c}{f_z}\right)^m \cdot \left\{ 1+\left[\frac{2(f_z-f_c)}{\Delta F_1}\right]^2 \right\}^{-(n+1)}. \]
Since the difference between the frequency of the image interference and the signal frequency is equal to twice the value of the intermediate frequency, then
\[ \frac{P_\Sigma}{P_o} = 1+\left(\frac{f_c}{f_z}\right)^m \cdot \left[ 1+\left(4\frac{f_{pr}}{\Delta F_1}\right)^2 \right]^{-(n+1)}. \]
In the case \(f_z>f_c\) \((f_z=f_c+2f_{pr})\)
\[ \frac{P'_\Sigma}{P_o} = 1+ \left(1+2\frac{f_{pr}}{f_c}\right)^{-m} \cdot \left[ 1+\left(4\frac{f_{pr}}{\Delta F_1}\right)^2 \right]^{-(n+1)}. \tag{3} \]
In the case \(f_z<f_c\) \((f_z=f_c-2f_{pr})\)
\[ \frac{P''_\Sigma}{P_o} = 1+ \left(1-2\frac{f_{pr}}{f_c}\right)^{-m} \cdot \left[ 1+\left(4\frac{f_{pr}}{\Delta F_1}\right)^2 \right]^{-(n+1)}. \tag{4} \]
From consideration of (3) and (4) a number of conclusions follows. Completely
it is obvious that \(\dfrac{P_{\Sigma}''}{P_o} > \dfrac{P_{\Sigma}'}{P_o}\). It is also obvious that, with an increase in the intermediate frequency (with \(f_c\), \(\Delta F_1\), and \(n\) unchanged), \(\dfrac{P_{\Sigma}'}{P_o}\) decreases.
With high selectivity of the high-frequency path and a low intermediate frequency, cosmic noise passing through the image channel practically does not increase the overall noise level.
Indeed, for \(\dfrac{f_{\mathrm{pr}}}{f_c} \ll 1\) and \(\dfrac{f_{\mathrm{pr}}}{\Delta F_1} \gg 1\), (3) and (4) take the form
\[ \frac{P_{\Sigma}'}{P_o} \simeq 1+\alpha_1 \quad \text{and} \quad \frac{P_{\Sigma}''}{P_o} \simeq 1+\alpha_2, \]
where
\[ \alpha_1 \simeq \left(1-2 \cdot m \cdot \frac{f_{\mathrm{pr}}}{f_c}\right) \cdot \left(4 \cdot \frac{f_{\mathrm{pr}}}{\Delta F_1}\right)^{-2(n+1)} \ll 1 \]
and
\[ \alpha_2 \simeq \left(1+2 \cdot m \cdot \frac{f_{\mathrm{pr}}}{f_c}\right) \cdot \left(4 \cdot \frac{f_{\mathrm{pr}}}{\Delta F_1}\right)^{-2(n+1)} \ll 1. \]
Let us consider the case of low selectivity of the RF amplifier when a relatively high intermediate frequency is used.
If, for example, the RF amplifier consists of one stage and an input circuit whose selectivity can be neglected, formulas (3) and (4) take the form (for \(m \simeq 2.3\) [3]):
\[ \frac{P_{\Sigma}'}{P_o} \simeq 1+ \left(1+2\frac{f_{\mathrm{pr}}}{f_c}\right)^{-2.3} \left[ 1+\left(4\frac{f_{\mathrm{pr}}}{\Delta F_1}\right)^2 \right]^{-1} \tag{5} \]
and
\[ \frac{P_{\Sigma}''}{P_o} \simeq 1+ \left(1-2\frac{f_{\mathrm{pr}}}{f_c}\right)^{-2.3} \left[ 1+\left(4\frac{f_{\mathrm{pr}}}{\Delta F_1}\right)^2 \right]^{-1}. \tag{6} \]
It is easy to show that the increase in the noise level for \(f_2 > f_c\) will be quite insignificant. Indeed, even for \(f_{\mathrm{pr}} < f_c\) and
\[ \Delta F_1 \simeq f_{\mathrm{pr}} \frac{P_{\Sigma}'}{P_o} \simeq 1.06. \]
Figure 2 gives the graphical dependence of \(\dfrac{P_{\Sigma}''}{P_o}\) on \(\dfrac{f_{\mathrm{pr}}}{\Delta F_1}\) for various values of \(\dfrac{f_{\mathrm{pr}}}{f_c}\), corresponding to the choice \(f_2 < f_c\).
As follows from Fig. 2, for \(\dfrac{f_{\mathrm{pr}}}{\Delta F_1} > 5\) and \(\dfrac{f_{\mathrm{pr}}}{f_c} < 0.4\), the increase in the noise level due to the image channel is insignificant.
Thus, in this case it is immaterial whether the local-oscillator frequency is chosen higher or lower than the signal frequency.
In the case of a broadband RF amplifier and a relatively high \(f_{\mathrm{pr}}\), an incorrect choice of \(f_2\) \((f_2 < f_c)\) may lead to a significant increase in the noise level due to the image channel. For example, at
for
\[ \frac{f_{\mathrm{pr}}}{f_c} \simeq 0.45 \quad \text{and} \quad \frac{f_{\mathrm{pr}}}{\Delta F_1} \simeq 5 \]
the interference level increases approximately by a factor of 1.5.
In Fig. 3, curves are plotted that bound the regions in which
\[ \frac{P_x''}{P_o} \leq 1.1 \]
(region I, located above curve 1) and
\[ \frac{P_x''}{P_o} \geq 1.5 \]
(region II, located below curve II).
Fig. 2. Dependence of \(\dfrac{P_x''}{P_o}\) on \(f_{\mathrm{pr}}, \Delta F_1\).
1 — curve corresponding to the value \(\dfrac{f_{\mathrm{pr}}}{f_c} = 0.05\);
2 — curve corresponding to the value “—” \(= 0.2\);
3 — curve corresponding to the value “—” \(= 0.3\);
4 — curve corresponding to the value “—” \(= 0.4\);
5 — curve corresponding to the value “—” \(= 0.45\).
Thus, in receivers with low selectivity of the VHF section and a comparatively high intermediate frequency, the local-oscillator frequency should be chosen higher than the frequency of the received signal; an incorrect choice of local-oscillator frequency \((f_2 < f_c)\) may lead to a noticeable increase in the interference level. When considering individual special cases, one should, using formulas (3) and (4), compare
\[ \frac{P_x'}{P_o} \quad \text{and} \quad \frac{P_x''}{P_o}, \]
and, depending on the difference between these quantities, choose \(f_2\), taking into account the factors usually considered in the design of receiving devices.
Fig. 3. On the decrease in sensitivity with an unsuccessful choice of \(f_2\).
1 — curve corresponding to the value
\[ \frac{P_x''}{P_o} = 1.1; \]
2 — curve corresponding to the value
\[ \frac{P_x''}{P_o} = 1.5. \]
Conclusions
- An incorrect choice of the local-oscillator frequency \((f_2 < f_c)\) may lead to a substantial increase in the threshold signal owing to an increase in the relative weight of cosmic interference passing through the image channel. This may occur in the case of receivers with low selectivity of the high-frequency path at a comparatively high intermediate frequency.
- In the case of high selectivity of the high-frequency path and a comparatively low intermediate frequency, both in the case \(f_{\mathrm{g}}>f_{\mathrm{s}}\) and in the case \(f_{\mathrm{g}}<f_{\mathrm{s}}\), the level of interference due to the image-frequency channel is very small in comparison with the level of interference due to the main channel.
REFERENCES
-
Edited by B. F. Vysotskii, Maximum Range of a Radar Station, Sovetskoe radio, 1947.
-
Shklovskii I. S., On the Nature of Radio Emission of the Galaxy, Astronomical Journal, 29, no. 4, 1952.
-
H. V. Cottony, J. R. Johler. Cosmic Radio Noise Intensities in the VHF Band, Proc. I. R. E., 40, No. 9, 1952.