# Amplitude Modulation

**Topics:**Frequency modulation, Amplitude modulation, Modulation

**Pages:**9 (1171 words)

**Published:**April 21, 2015

A carrier is used to make the wavelength smaller for practical transmission and to permit multiplexing. The spectrum is used to measure bandwidth (the range of frequencies) and the efficiency (the power in the side-bands compared to the total power) Bandwidth can be predicted using BW = 2 fm where fm = the maximum modulating frequency Efficiency depends only on the modulating index, m (the fraction of the carrier you modulate by) AM is limited to 33% efficiency because the modulation index cannot be increased to > 1.0 without introducing distortion in the receiver. 1. Carrier signal equations

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

C (t) = C sin (ωc + 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 (ωm + 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 (ωm t + φ ] . sin(ωc t)

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

y (t) = [ A + M cos (ωm t + φ ] . sin(ωc t)

This can be expanded out using the standard trigonometric rules:

y (t) = A . sin (ωc t) + M/2 [ sin ((ωc + ωm) t + φ) + M/2 [ sin ((ωc - ωm) t - φ)

In this theory, three terms can be seen which represent the carrier, and upper and lower sidebands: Carrier :A sin (ω c t)

Upper sideband: M/2 [ sin ((ωc + ωm) t + φ)

Lower sideband: 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.

Sidebands on an amplitude modulated carrier

when modulated with a single 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...

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