Winter 2007 Issue


Basic Frequency Modulation Theory

By Bob Witte, KØNR

The concept of frequency modulation in radio communications was first described by Armstrong back in 1936.1 FM continues to be a very popular mode with radio amateurs, especially on the 144-, 222-, and 440-MHz bands. Despite the widespread use of FM, some of its characteristics are not well understood. This article is a review of the basics of FM theory and is based heavily on an article I wrote that was originally published in QST.2

Modulation Techniques

A review of the common analog modulation techniques will help put FM into perspective. Mathematically, we can represent a radio signal as a sinusoid:

v(t) = Acos(2šft + q)
A is the peak amplitude of the signal
q is the relative phase of the signal
f is the carrier frequency
t is time

If we keep the amplitude A and the phase q constant, we have an unmodulated carrier or continuous wave (CW). If we keep the phase constant but modulate the amplitude, amplitude modulation is produced. Similarly, we can adjust the phase of the carrier with the modulating to produce phase or frequency modulation.

Phase modulation (PM) and frequency modulation (FM) are closely related and together are called angle modulation. Generally, PM and FM can be derived from each other by filtering the modulating signal, often referred to as adding pre-emphasis or de-emphasis to the audio signal. Phase modulation is, in fact, the industry standard.3 However, since the conventional ham radio terminology is “frequency modulation,” I will use that terminology here. I am not going to cover pre-emphasis and de-emphasis in this article. Maintaining a flat audio response in FM repeater systems does require some understanding and care. Refer to the article by Jeff Stouffer, K6JSI, in the Winter 2005 issue of CQ VHF for more insight on how to keep your repeater audio sounding great.4

The mathematical expression for CW, AM, and FM/PM and the waveform of each signal is shown in figure 1. Single-sideband modulation is the most common form used on the HF bands. It is a version of AM that increases the efficiency of the system by eliminating the power in the carrier and one of the sidebands.5 Since the RF spectrum of an SSB signal is just a frequency-shifted version of the modulating signal, the bandwidths of the two signals are the same. Our focus in this article is on FM, but SSB provides an important standard of comparison. Figure 2 shows the spectrum of the modulating signal and that of the resulting SSB signal (in this case, upper sideband).

Frequency Modulation

When we frequency modulate a signal, we instantaneously adjust the carrier’s frequency according to the modulating signal. An FM signal is described by its frequency deviation, which is the instantaneous change away from the carrier frequency, and by the modulation index, which is defined as:

Modulation index (b) =
freq. deviation/modulation freq.

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