CONVERSION OF FREQUENCY MODULATION INTO AMPLITUDE MODULATION USING A WAVEGUIDE *)

The author of the paper under review indicates a new method for converting frequency modulation into amplitude modulation—by means of a waveguide. The method is based on the dependence of the phase velocity of an electromagnetic wave in a waveguide on its frequency. This dependence, for a waveguide without inhomogeneities, is expressed, as is known, by the following relation:

\[ v=\frac{c}{\sqrt{1-(f_{\mathrm{cr}}/f)^2}} =\frac{c}{\sqrt{1-(\lambda/\lambda_{\mathrm{cr}})^2}}. \tag{1} \]

Here \(v\) is the phase velocity of the wave, \(f\) its frequency, \(\lambda\) the wavelength in free space, and \(\lambda_{\mathrm{cr}}\) and \(f_{\mathrm{cr}}\) are constants depending on the geometrical dimensions of the waveguide cross section. The propagation of electromagnetic waves with a wavelength greater than \(\lambda_{\mathrm{cr}}\) (i.e., a frequency less than \(f_{\mathrm{cr}}\)) in a waveguide of the given cross section is impossible. Therefore the constant \(f_{\mathrm{cr}}\) is called the “critical frequency,” and \(\lambda_{\mathrm{cr}}\) the “critical wavelength.”

In the figure (see p. 290), at left, the vector diagram of an amplitude-modulated signal is shown:

\[ a=A(t)\sin\omega_0 t =A_0(1+M\cos\Omega t)\sin\omega_0 t = \]

\[ =A_0\sin\omega_0 t+\frac{MA_0}{2}\sin(\omega_0+\Omega)t +\frac{MA_0}{2}\sin(\omega_0-\Omega)t, \tag{2} \]

At right is shown the vector diagram for narrow-band frequency modulation. In this case

\[ a=A_0\sin\omega_0 t =A_0\sin[\omega_0 t+m\sin\Omega t] = \]

\[ =A_0\{ \cos(m\sin\Omega t)\sin\omega_0 t +\sin(m\sin\Omega t)\cos\omega_0 t\}, \tag{3} \]

*) P. S. Rogell, J. Appl. Phys. 21, 629 (1950).

where \(m \ll 1\) (narrow-band modulation). For small \(m\) one may put

\[ \sin(m\sin \Omega t) \approx m\sin \Omega t, \]

\[ \cos(m\sin \Omega t) \approx 1. \]

Then

\[ a=A_0(\sin \omega_0 t+m\sin \Omega t \cos \omega_0 t)= \]

\[ = A_0\left[\sin \omega_0 t+\frac{m}{2}\sin(\omega_0+\Omega)t-\frac{m}{2}\sin(\omega_0-\Omega)t\right]. \tag{4} \]

A frequency-modulated oscillation in this case represents the sum of three oscillations: the carrier frequency and two side frequencies. But, in contrast to the case of amplitude modulation, the phase of the oscillation of one of the side frequencies \((\omega_0-\Omega)\) is shifted by \(180^\circ\) with respect to the phase of this oscillation under amplitude modulation (the minus sign in the last term). Since the phase velocity of a wave in a waveguide depends on its frequency, the phase difference between the oscillations of the side frequencies and the carrier changes as they pass through the waveguide. The waveguide introduces some phase shift both for the wave of frequency \(\omega_0+\Omega\) and for the wave of frequency \(\omega_0-\Omega\). The shifts obtained for the different frequencies are different. This corresponds to a rotation of the vectors \(OA\), \(AC_1\), and \(AC_2\) through certain angles (not equal to one another). Suppose that after the waveguide the vector \(AC_1\) has turned through \(270^\circ\), and the vector \(AC_2\) through \(450^\circ\). The phase and frequency relations will then be the same as in amplitude modulation, i.e. the phase modulation will be transformed into amplitude modulation.

The cross section of the waveguide is chosen so that the carrier frequency of the modulated oscillation is close to the critical frequency of the waveguide: then the denominator in formula (1) is close to zero, and small frequency increments correspond to considerable increments of velocity. This makes it easier to create a considerable phase shift for the waves of the side frequencies, and the required length of waveguide becomes smaller.

The author proposes carrying out the transformation of frequency modulation into amplitude modulation at a fixed intermediate frequency \(\omega_m\), amplitude-modulated by the variable frequency of the transmitted signal \(\Omega\). Then, at the output from the waveguide, we obtain oscillations of the carrier ultra-high frequency that are amplitude-modulated by the intermediate frequency \(\omega_m\), which in turn is amplitude-modulated by the frequency of the transmitted signal \(\Omega\). The second modulation, with frequency \(\Omega\), introduces into the process of phase shift by the waveguide described above certain distortions; but

calculation shows that these distortions are sufficiently small. For $\Omega/\omega_m = 0.1$ they do not exceed, for example, 1%.

The relation between the conversion parameters is illustrated by the following numerical example. Let the carrier frequency be $f_0 = 3000$ Mc/s; the conversion of the modulation is carried out on the wave of type $H_{01}$ in a waveguide of height $5.06$ cm. This waveguide corresponds to the critical frequency $f_{\mathrm{cr}} = 2960$ Mc/s. The intermediate frequency $\omega_m$ is then chosen from the condition

\[ f_m=\frac{\omega_m}{2\pi}=\frac{240}{289}(f_0-f_{\mathrm{cr}}). \]

The calculation gives $f_m = 33.2$ Mc/s. The length of the waveguide is given by the condition

\[ l=\frac{17c}{8\,[2f_0(f_0-f_{\mathrm{cr}})]^{1/2}} \]

and is equal to $1.30$ m. The bandwidth is $73.0$ Mc/s. At signal frequencies up to $3.3$ Mc/s, the distortions at the output of the waveguide do not exceed 1%.

M. G.