In an earlier section we saw how *Amplitude Modulation* (AM) can be used to convey information. We also mentioned that *Frequency Modulation* (FM) and *Phase Modulation* (PM) can also be used. This section sets out the basic properties of FM/PM signals and how we can produce them.

What *is* the ‘frequency’ of a waveform? For a simple wave like a sinewave the answer appears quite obvious, we can define the wave using an expression like

and then identify *f* as the wave's frequency. An alternative way to represent such a wave would be write

where

is the wave's *Phase* at any each instant, *t*. For a simple sinewave *f* has a constant value and increases steadily with time at the rate

We can define the FM wave, produced when we modulate a *carrier frequency*, , with a *modulating signal*, , to be

where

is the *instantaneous frequency* of the wave at the instant, *t*. The term is a constant whose value depends upon the modulating system. It determines “how many Hz of frequency change we get for each volt of modulating signal.”

The FM wave can now be used to convey information about the modulating pattern in a manner similar to the AM variations we examined in an earlier section. Note that — unlike an AM wave — the FM wave doesn't have a single frequency value. This makes an FM wave obviously different to an AM one. Instead, we can define two distinct quantities; its ‘unmodulated’ (i.e. carrier) frequency, and its modulated frequency, , __which can change from instant to instant__. The *instantaneous phase* of the modulated wave at any instant can be obtained by substituting 12.6 into 12.3 and integrating to get

(for simplicity we can assume the phase is zero when .)

In a similar way, we can define a *Phase Modulated* (PM) wave as having the form

where

and determines how much “phase change per volt of modulation” the PM modulator being used produces. The instantaneous frequency of the PM wave will therefore be

Comparing 12.6 & 7 with 12.9 & 10 we find that — unless we know something about the modulation in advance — it may not be obvious whether the signal is FM or PM modulated. In both cases the wave's frequency and phase vary from moment to moment. Mathematically speaking, FM & PM are almost identical twins. The only difference is that one corresponds to a modulation pattern which is the differential of that produced by the other. The ‘good news’ is that this means we can mix FM & PM arguments and most of our conclusions about one apply to the other. The ‘bad news’ is that, in practice we often have to know in advance which type of modulation is being used if we want to recover the modulated information correctly. In general, however, both FM & PM waves are obviously different to an AM wave.

**12.2 The Spectrum of an FM signal.**

From equation 12.7 we can write an FM wave in the form

To discover the frequency spectrum of a typical FM signal we can take the example of simple sinewave modulation at a modulation frequency,

which produces an instantaneous FM signal frequency of

This expression tells us that swings up and down either side of over a range of . This range is usually described in terms of the modulated signal's *peak frequency deviation* value, defined as

since it indicates the largest swing or deviation in frequency either side of . Note that its value depends upon the magnitude of the modulation, , but __not__ upon the modulating frequency, .

By combining 12.7, 12.13, and 12.14 we can say that the instantaneous phase of the sinewave modulated FM wave is

It is conventional to define a quantity called the *modulation index*,

we can then write the FM wave in the form

This provides us with information on how the modulated signal varies with time. However we often need to know the frequency spectrum of the modulated wave – for example, in order to be able to determine the bandwidths of any filters, amplifiers, etc that we require.

By consulting a good maths textbook we can discover that this expression can (after some boring algebra!) be rewritten as

where is the *Bessel Function* (first kind, integer order, *n*) for the value, .

Clearly, expression 12.18 is much more complicated than the equivalent for an AM wave modulated with a single sinewave. The sinewave modulated AM signal has only three spectral components, at the frequencies, , , and . The FM wave has spectral components at all the frequencies, , where *n* can be any integer from to . This result is rather startling since it means that, strictly speaking, a system has to provide an infinite bandwidth to carry an accurate FM (or PM) signal! Fortunately, there is a general tendency for

and the higher order Bessel function values fall quickly with *n* when the modulation index is small. In many practical situations we can arrange that and when this is true we find that

An FM signal produced with a low modulation index (i.e. where ) is called a *narrowband FM* signal. For most purposes we can ignore the high-order Bessel function contributions and represent its spectrum with the approximation

This narrowband FM (or PM) signal is similar to AM in that it has sideband components at , hence it only requires a transmission bandwidth of . Its spectrum differs from AM in two ways. Firstly, the total amplitude of the modulated wave remains almost constant. Secondly, the two sideband components are “180 degrees out of phase” (their signs differ).

A ‘high-’ FM wave can be thought of as a carrier whose frequency is varied over a relatively wide range. It will therefore require a transmission bandwidth of at least . Combining this result with that for a narrowband FM wave leads to *Carson's Rule*, that the minimum practical bandwidth required to transmit an FM/PM signal will be

This rule is a useful guide when we have to choose a system to carry an FM signal. It should be remembered, however, that in theory FM signals require an infinite bandwidth if we want to avoid __any__ signal distortion during transmission.

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