Modulation

Acoustic information is not transmitted by radio directly, but by means of modulation. The term (not to be confused with the similar one used in a musical context) indicates the change in the characteristics of a signal (typically a pure sinusoid), called the carrier, as a function of the characteristics of the signal to be transmitted (called the modulating signal).

Analogue modulations are those traditionally used in radio broadcasting, while mobile telephony and digital transmissions use other kinds of modulation.

Without going into the details of how these modulations are produced by the transmitter and decoded by the receiver, we think it useful to give some examples. In the case of radio transmissions the transmitted waves are electromagnetic. The carrier is, of course, in the radio-frequency range, while the modulating signal is at acoustic frequencies. To make the examples audible (though not pleasant) we have instead brought both waves into the audio range. We fixed the sinusoidal carrier at 1000 Hz, and modulated its amplitude and frequency, first with a sinusoid, then with a triangular wave at various frequencies.

Amplitude modulation

The amplitude of the carrier is controlled by the amplitude of the modulating signal.

Using, for example, a triangular modulating signal at 1 Hz, we expect the envelope of the resulting wave to be a triangular profile with a period of 1 s. In the following three examples we have gradually increased the frequency of the modulating signal. When the frequency of the modulating signal is very low, the Fourier transform of the sonogram is computed over a time window that is short compared with the period of the modulating signal, and so it cannot resolve the different spectral components. The sonogram appears as a single line modulated in intensity. The waveform, instead, correctly shows what we expect: a signal at the carrier frequency, whose envelope is a signal with the shape and frequency of the modulating signal.

waveform	spectrogram	Audio
		AM_1000_tr_1.mp3 Amplitude modulation: 1000 Hz sine carrier, modulating triangle at 1 Hz

As the frequency of the modulating signal increases, the sonogram can distinguish the spectrum of the signal, which is characterized by sidebands symmetric with respect to the carrier. Each band reflects the waveform of the modulating signal (in this case that of a triangular wave), and has components spaced by multiples of the frequency of the modulating signal (100 Hz in the first case, 500 in the second).

waveform	spectrogram	Audio
		AM_1000_tr_100.mp3 Amplitude modulation: 1000 Hz sine carrier, modulating triangle at 100 Hz
		AM_1000_tr_500.mp3 Amplitude modulation: 1000 Hz sine carrier, modulating triangle at 500 Hz

To observe a simpler case, we use a pure sinusoidal modulating signal. In this case its spectrum consists of a single line and, consequently, the AM spectrum consists of three lines: the central one corresponds to the carrier, while the two side lines are the effect of the modulation.

waveform	spectrogram	Audio
		AM_1000_sin_10.mp3 Amplitude modulation: 1000 Hz sine carrier, modulating sine at 10 Hz
		AM_1000_sin_500.mp3 Amplitude modulation: 1000 Hz sine carrier, modulating sine at 500 Hz

All this can be obtained in an elementary way with a little trigonometry. Indeed, in the simplest case, calling ${\displaystyle \omega _{P}}$ and ${\displaystyle \omega _{M}}$ respectively the angular frequencies of the pure carrier

${\displaystyle p(t)=\sin(\omega _{P}t)\;}$,

and of the modulating signal, also pure,

${\displaystyle m(t)=\cos(\omega _{M}t)\;}$,

we find that the modulated signal ${\displaystyle x(t)}$ can be decomposed into the sum of three pure sinusoidal components:

${\displaystyle x(t)=[C+m(t)]p(t)=C\sin(\omega _{P}t)+\cos(\omega _{M}t)\sin(\omega _{P}t)\;}$

${\displaystyle =C\sin(\omega _{P}t)+{\frac {1}{2}}\sin[(\omega _{P}+\omega _{M})t]+{\frac {1}{2}}\sin[(\omega _{P}-\omega _{M})t]}$,

where C is an arbitrary constant.

Frequency modulation

The frequency of the carrier is controlled by the amplitude of the modulating signal.

Again, if the frequency of the modulating signal is very low, the spectrum of the signal is resolved as a single line that moves in time like the modulating signal. Note that now the signal has constant amplitude, while the triangular shape of the modulating signal can be recognized in the sonogram.

waveform	spectrogram	Audio
		FM_1000_tr_1.mp3 Frequency modulation: 1000 Hz sine carrier, modulating triangle at 1 Hz

As the frequency of the modulating signal increases, however, we see that the nature of the spectrum of the modulated signal is more complex. Sidebands appear again, but this time their profile no longer reflects the triangular waveform of the modulating signal. The analysis in the time domain, on the other hand, becomes more difficult. Now, to recognize parts of the signal corresponding to different modulation intensities, one has to measure and compare the periods (the figure highlights the time of 1 ms, corresponding to the period of the wave in the right-hand part of the image, but not to that of the wave in the left-hand part).

waveform	spectrogram	Audio
		FM_1000_tr_100.mp3 Frequency modulation: 1000 Hz sine carrier, modulating triangle at 100 Hz
		FM_1000_tr_500.mp3 Frequency modulation: 1000 Hz sine carrier, modulating triangle at 500 Hz

Even with the pure sinusoidal modulating signal several sidebands appear. Moreover, while in amplitude modulation a signal of finite bandwidth kept a finite bandwidth after modulation (see the case of the sine), in this case it can be shown that this is not so: frequency modulation requires a theoretically infinite bandwidth. Fortunately not in practice, because it can be shown (though not in an elementary way) that the amplitude of the side lines decreases with frequency.

waveform	spectrogram	Audio
		FM_1000_sin_10.mp3 Frequency modulation: 1000 Hz sine carrier, modulating sine at 10 Hz
		FM_1000_sin_500.mp3 Frequency modulation: 1000 Hz sine carrier, modulating sine at 500 Hz

In this case there is no elementary mathematical manipulation to express as a sum of sines alone the modulated signal

${\displaystyle x(t)=\cos[\omega _{P}t+M\sin(\omega _{M}t)]\;}$,

and one has to resort to higher mathematics. The amplitude of the side lines follows the behaviour of the so-called Bessel functions. Curiously, among these are the same functions that describe the modes of vibration of a circular membrane.