Amplitude Modulation Theory & Equations

It is possible to look at the theory of the generation of an amplitude modulated signal in four steps:

Carrier signal
Modulating signal
Overall modulated signal for a single tone
Expansion to cover a typical audio signal

These steps will be covered in greater details below:

1. Carrier signal equations

Looking at the theory, it is possible to describe the carrier in terms of a sine wave as follows:

C (t) = C \sin (ω c + φ)

Where:
carrier frequency in Hertz is equal to ωc / 2 π
C is the carrier amplitude
φ is the phase of the signal at the start of the reference time

Both C and φ can be omitted to simplify the equation by changing C to "1" and φ to "0".

2. Modulating signal equations

The modulating waveform can either be a single tone. This can be represented by a cosine waveform, or the modulating waveform could be a wide variety of frequencies - these can be represented by a series of cosine waveforms added together in a linear fashion.

For the initial look at how the signal is formed, it is easiest to look at the equation for a simple single tone waveform and then expand the concept to cover the more normal case. Take a single tone waveform:

m (t) = M \sin (ω m + φ)

Where:
modulating signal frequency in Hertz is equal to ωm / 2 π
M is the carrier amplitude
φ is the phase of the signal at the start of the reference time

Both C and φ can be omitted to simplify the equation by changing C to "1" and φ to "0".

It is worth noting that normally the modulating signal frequency is well below that of the carrier frequency.

3. Overall modulated signal for a single tone

The equation for the overall modulated signal is obtained by multiplying the carrier and the modulating signal together.

y (t) = [A + m (t)] . c (t)

The constant A is required as it represents the amplitude of the waveform.

Substituting in the individual relationships for the carrier and modulating signal, the overall signal becomes:

y (t) = [A + M \cos (ω m t + φ] . \sin (ω c t)

The trigonometry can then be expanded out to give an equation that includes the components of the signal:

y (t) = A . \sin (ω c t) + \frac{A M}{2} [\sin ((ω c + ω m) t + φ)] + \frac{A M}{2} [\sin ((ω c - ω m) t - φ)]

In this theory, three terms can be seen which represent the carrier, and upper and lower sidebands:

Carrier: A . sin (ωc t)
Upper sideband: A . M/2 [ sin ((ωc + ωm) t + φ)
Lower sideband: A . M/2 [ sin ((ωc - ωm) t - φ)

Note also that the sidebands are separated from the carrier by a frequency equal to that of the tone.

Amplitude modulation signal showing the two sidebands generated at ± 1kHz when carrier is modulated by 1 kHz tone — Spectrum (sidebands) arising from carrier modulated by 1 kHz tone

It can be seen that for a case where there is 100% modulation, i.e. M = 1, and where the carrier is not suppressed, i.e. A = 1, then the sidebands have half the value of the carrier, i.e. a quarter of the power each.

4. Expansion to cover a typical audio signal

With the basic concept of modulation and the resultant sidebands established, the same principles can be applied to the more complicated cases of modulation using speech, music or other audio sounds.

Theory can be used to break down a sound into a series of sinusoidal signals. These are linearly added to each other to form the audio spectrum of the modulating signal.

The spectrum of the modulating signal extends out either side from the carrier, one sideband is the mirror of the other, with the lowest frequencies closest to the carrier, and highest furthest away.

Amplitude modulation signal showing the two sidebands extending out either side of the carrier — Sidebands on an amplitude modulated carrier when modulated
with an audio signal containing many frequencies

It can be seen that the audio signal covers a band of frequencies either side of the main carrier. The theory and equations show that furthest extent of the sidebands from the carrier corresponds to the highest frequency of the modulating tone for the amplitude modulated signal.

Seeing a little of the theory and mathematics behind amplitude modulation gives a better understanding of how it works. This can then be applied to use this type of mode to its best, whether as amplitude modulation, single sideband, or even giving abetter understanding of how QAM operates. Understanding how the modulating waveform not only generates undulations of the envelope, but also generates sidebands, etc enables the basic concepts behind AM to be understood.