Table of Contents

**Modulation Index of FM and Sidebands of FM**

Modulation Index of FM:- Any modulation process produces sidebands. When a constant-frequency sine wave modulates a carrier, two side frequencies are produced. The side frequencies are the sum and difference of the carrier and the modulating frequency. In FM and PM, as in AM, sum and difference sideband frequencies are produced. In addition, a large number of pairs of upper and lower sidebands are generated. As a result, the spectrum of an FM or a PM signal is usually wider than that of an equivalent AM signal. It is also possible to generate a special narrowband FM signal whose bandwidth is only slightly wider than that of an AM signal. Fig. 5-7 shows the frequency spectrum of a typical FM signal produced by modulating a carrier with a single-frequency sine wave. Note that the sidebands are spaced from the carrier fc and from one another by a frequency equal to the modulating frequency fm. If the modulating frequency is 1 kHz, the first pair of sidebands is above and below the carrier by 1000 Hz. The second pair of sidebands is above and below the carrier by 2×1000 Hz = 2000 Hz, or 2 kHz, and so on.

Figure 5-7 Frequency spectrum of an FM signal. Note that the carrier and sideband amplitudes shown are just examples. The amplitudes depend upon the modulation index mf.

Note also that the amplitudes of the sidebands vary. If each sideband is assumed to be a sine wave, with a frequency and an amplitude as indicated in Fig. 5-7, and all the sine waves are added, then the FM signal producing them will be created. As the amplitude of the modulating signal varies, the frequency deviation changes. The number of sidebands produced, and their amplitude and spacing, depends on the frequency deviation and modulating frequency. Keep in mind that an FM signal has a constant amplitude. Since an FM signal is a summation of the sideband frequencies, the sideband amplitudes must vary with frequency deviation and modulating frequency if their sum is to produce a constant-amplitude but variable-frequency FM signal. Theoretically, the FM process produces an infinite number of upper and lower sidebands and, therefore, a theoretically infinitely large bandwidth. However, in practice, only those sidebands with the largest amplitudes are significant in carrying the information. Typically any sideband whose amplitude is less than 1 percent of the unmodulated carrier is considered insignificant. Thus FM is readily passed by circuits or communication media with finite bandwidth. Despite this, the bandwidth of an FM signal is usually much wider than that of an AM signal with the same modulating signal.

**Modulation Index** **of FM**

The ratio of the frequency deviation to the modulating frequency is known as the modulation index mf:

**mf = fd/fm **

where fd is the frequency deviation and fm is the modulating frequency. Sometimes the lowercase Greek letter delta (δ) is used instead of fd to represent deviation; then mf 5 δ/fm. For example, if the maximum frequency deviation of the carrier is +-12 kHz and the maximum modulating frequency is 2.5 kHz, the modulating index is mf = 12/2.5 = 4.8. In most communication systems using FM, maximum limits are put on both the frequency deviation and the modulating frequency. For example, in standard FM broadcasting, the maximum permitted frequency deviation is 75 kHz and the maximum permitted modulating frequency is 15 kHz. This produces a modulation index of mf = 75/15 = 5. When the maximum allowable frequency deviation and the maximum modulating frequency are used in computing the modulation index, mf is known as the deviation ratio.

Example 5-2 What is the deviation ratio of TV sound if the maximum deviation is 25 kHz and the maximum modulating frequency is 15 kHz?

**Bessel Functions**

Given the modulation index, the number and amplitudes of the significant sidebands can be determined by solving the basic equation of an FM signal. The FM equation, whose derivation is beyond the scope of this book, is υFM = Vc sin [2πfct + mf sin (2πfmt)], where υFM is the instantaneous value of the FM signal and mf is the modulation index. The term whose coefficient is mf is the phase angle of the carrier. Note that this equation expresses the phase angle in terms of the sine wave modulating signal. This equation is solved with a complex mathematical process known as Bessel functions. It is not necessary to show this solution, but the result is as follows:

The FM wave is expressed as a composite of sine waves of different frequencies and amplitudes that, when added, give an FM time-domain signal. The first term is the carrier with an amplitude given by a Jn coefficient, in this case J0. The next term represents a pair of upper and lower side frequencies equal to the sum and difference of the carrier and modulating signal frequency. The amplitude of these side frequencies is J1. The next term is another pair of side frequencies equal to the carrier +-2 times the modulating signal frequency. The other terms represent additional side frequencies spaced from one another by an amount equal to the modulating signal frequency. The amplitudes of the sidebands are determined by the Jn coefficients, which are, in turn, determined by the value of the modulation index. These amplitude coefficients are computed by using the expression

In practice, you do not have to know or calculate these coefficients, since tables giving them are widely available. The Bessel coefficients for a range of modulation indexes are given in Fig. 5-8. The leftmost column gives the modulation index mf. The remaining columns indicate the relative amplitudes of the carrier and the various pairs of sidebands. Any sideband with a relative carrier amplitude of less than 1 percent (0.01) has been eliminated. Note that some of the carrier and sideband amplitudes have negative signs. This means that the signal represented by that amplitude is simply shifted in phase 180° (phase inversion). Fig. 5-9 shows the curves that are generated by plotting the data in Fig. 5-8. The carrier and sideband amplitudes and polarities are plotted on the vertical axis; the modulation index is plotted on the horizontal axis. As the figures illustrate, the carrier amplitude J0 varies with the modulation index. In FM, the carrier amplitude and the amplitudes of the sidebands change as the modulating signal frequency and deviation change. In AM, the carrier amplitude remains constant. Note that at several points in Figs. 5-8 and 5-9, at modulation indexes of about 2.4, 5.5, and 8.7, the carrier amplitude J0 actually drops to zero. At those points, all the signal power is completely distributed throughout the sidebands. And as can be seen in Fig. 5-9, the sidebands also go to zero at certain values of the modulation index.

Example 5-3 What is the maximum modulating frequency that can be used to achieve a modulation index of 2.2 with a deviation of 7.48 kHz?

Fig. 5-10 shows several examples of an FM signal spectrum with different modulation indexes. Compare the examples to the entries in Fig. 5-8. The unmodulated carrier in Fig. 5-10(a) has a relative amplitude of 1.0. With no modulation, all the power is in the carrier. With modulation, the carrier amplitude decreases while the amplitudes of the various sidebands increase. In Fig. 5-10(d), the modulation index is 0.25. This is a special case of FM in which the modulation process produces only a single pair of significant sidebands like those produced by AM. With a modulation index of 0.25, the FM signal occupies no more spectrum space than an AM signal. This type of FM is called narrowband FM, or NBFM. The formal definition of NBFM is any FM system in which the modulation index is less than π/2 = 1.57, or mf , π/2. However, for true NBFM with only a single pair of sidebands, mf must be much less than π/2. Values of mf in the 0.2 to 0.25 range will give true NBFM. Common FM mobile radios use a maximum deviation of 5 kHz, with a maximum voice frequency of 3 kHz, giving a modulation index of mf = 5 kHz/3 kHz = 1.667. Although these systems do not fall within the formal definition of NBFM, they are nonetheless regarded as narrowband transmissions.

Example 5-4 State the amplitudes of the carrier and the first four sidebands of an FM signal with a modulation index of 4. (Use Figs. 5-8 and 5-9.)

The primary purpose of NBFM is to conserve spectrum space, and NBFM is widely used in radio communication. Note, however, that NBFM conserves spectrum space at the expense of the signal-to-noise ratio.

**FM Signal Bandwidth**

As stated previously, the higher the modulation index in FM, the greater the number of significant sidebands and the wider the bandwidth of the signal. When spectrum conservation is necessary, the bandwidth of an FM signal can be deliberately restricted by putting an upper limit on the modulation index. The total bandwidth of an FM signal can be determined by knowing the modulation index and using Fig. 5-8. For example, assume that the highest modulating frequency of a signal is 3 kHz and the maximum deviation is 6 kHz. This gives a modulation index of mf = 6 kHz/3 kHz = 2. Referring to Fig. 5-8, you can see that this produces four significant pairs of sidebands. The bandwidth can then be determined with the simple formula

**BW = 2fm N**

where N is the number of significant sidebands in the signal. According to this formula, the bandwidth of our FM signal is

**BW = 2(3 kHz)(4) = 24 kHz **

In general terms, an FM signal with a modulation index of 2 and a highest modulating frequency of 3 kHz will occupy a 24-kHz bandwidth. Another way to determine the bandwidth of an FM signal is to use Carson’s rule. This rule recognizes only the power in the most significant sidebands with amplitudes greater than 2 percent of the carrier (0.02 or higher in Fig. 5-8). This rule is

**BW = 2[fd(max) + fm (max)]**

According to Carson’s rule, the bandwidth of the FM signal in the previous example would be

**BW = 2(6 kHz + 3 kHz) = 2(9 kHz) = 18 kHz**

Carson’s rule will always give a bandwidth lower than that calculated with the formula BW = 2fm N. However, it has been proved that if a circuit or system has the bandwidth calculated by Carson’s rule, the sidebands will indeed be passed well enough to ensure full intelligibility of the signal. So far, all the examples of FM have assumed a single-frequency sine wave modulating signal. However, as you know, most modulating signals are not pure sine waves, but complex waves made up of many different frequencies. When the modulating signal is a pulse or binary wave train, the carrier is modulated by the equivalent signal, which is a mix of a fundamental sine wave and all the relevant harmonics, as determined by Fourier theory. For example, if the modulating signal is a square wave, the fundamental sine wave and all the odd harmonics modulate the carrier. Each harmonic produces multiple pairs of sidebands depending on the modulation index. As you can imagine, FM by a square or rectangular wave generates many sidebands and produces a signal with an enormous bandwidth. The circuits or systems that will carry, process, or pass such a signal must have the appropriate bandwidth so as not to distort the signal. In most equipment that transmits digital or binary data by FSK, the binary signal is filtered to remove higher-level harmonics prior to modulation. This reduces the bandwidth required for transmission.

Example 5-5: What is the maximum bandwidth of an FM signal with a deviation of 30 kHz and a maximum modulating signal of 5 kHz as determined by (a) Fig. 5-8 and (b) Carson’s rule?

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