Evaluation of Model:

For the case of a carrier modulated by a single sine wave, the resulting frequency spectrum can be calculated using Bessel functions of the first kind as a function of the sideband number and the modulation index. The carrier and sideband amplitudes are illustrated for different modulation indices of FM signals. For particular values of the modulation index, the carrier amplitude becomes zero and all the signal power is in the sidebands.Since the sidebands are on both sides of the carrier, their count is doubled, and then multiplied by the modulating frequency to find the bandwidth.

Knowing the modulation index, you can compute the number and amplitudes of the significant side-bands. This is done through a complex mathematical process known as the Bessel functions.

The typical Bessel function graph is shown in Table 1.1a. The left column gives the modulation index. The remaining column indicates the relative amplitudes of the carrier and the various pairs of side-bands. Any side-bands with relative carrier amplitude of less than 1% (0.01) have been eliminated. The total bandwidth of an FM signal can be determined by knowing the modulation index and using the Table.

For example, assuming the modulation index is 2. Referring to Table 1.1b (3.1b), you can see that this produces significant pairs of side-bands. The Bandwith can then be determined with the simple formula

BW = 2Fm x Number of significant side-bands.

Using the example above and assuming a highest modulating frequency of 2.5KHz, the bandwidth of the FM signal is given as

BW = 2(2.5)(4) = 20KHz.

An FM signal with a modulation index of 2 and a highest modulating frequency of 2.5KHz will then occupy a bandwidth of 20KHz.

Table 1.1a. Bessel function of the first kind

148 Table 1.1b.Bessel Function with Sideband amplitude

On using Bessel functions, it may be shown that Equ.1.16 may be expanded to yield,

149 _ = _x[ #₇()sin_x^l+#₈(){sin (_x+ ₄) − sin(_x− ₄)a}

+ # _(&){sin(_x+ 2₄)a + sin(_x− 2₄)a} + #(){sin(x+ 3₄)a − sin(4− 3₄) a} +#_"(){sin(_x+ 4₄)a + sin(_x− 4₄) a}

+………etc. (1.17)

Equ.1.17reveals that the sinusoidal carrier voltage after frequency modulation by another sinusoidal voltage consists of the following frequency terms,

(i) Carrier voltage reduced in magnitude by the factor #₇().

(ii) Infinite number of sideband terms on both lower and upper frequency sides of the carrier frequency at intervals equal to the modulation frequency. The amplitude of these sideband terms are _x multiplied by various Bessel function of the first and different orders denoted by the subscripts.

Bessel function # () is given by,

#

( ) = (

)

Î

−

+

+

+ ⋯ Ñ

In order to find the amplitude of a given pair of sideband terms and the magnitude of the carrier, it is necessary to known the value of the corresponding Bessel function. It is, however, not necessary to evaluate the Bessel function using Equ.1.18 since the magnitude of Bessel function of this type are readily available (in Table form) as in Table 1.1 or in graphical form as in Figure1.2.

Figure 1.2. Bessel functions of the first kind and different order.

Recall that the frequency modulation can be classified as narrowband if the change in the carrier frequency is about the same as the signal frequency, or as wideband if the change in the carrier frequency is much higher (modulation index >1) than the signal frequency.

For example, narrowband FM is used for two way radio systems such as Family Radio Service, in which the carrier is allowed to deviate only 2.5 kHz above and below the center frequency

150 with speech signals of no more than 3.5 kHz bandwidth. Wideband FM is used for FM broadcasting, in which music and speech are transmitted with up to 75 kHz deviation from the center frequency and carry audio with up to a 20-kHz bandwidth and subcarriers up to 92 kHz.The following conclusions are drawn from the foregoing study and from Table 1.1b and Figure1.2.

(i) In AM only three frequencies namely the carrier and the two sidebands are involved. Fm, on the other hand, has carrier and an infinite number of sideband terms recurring at frequency interval of

¾

(ii) The # coefficients, in general, decrease with increase of order ¥ but not in a simple way.

Figure 1.2. Shows that the values of # fluctuate on both sides of zero and diminish gradually.

Each # coefficient represents the amplitude of the corresponding pair of sideband terms.

Hence the amplitudes of the sideband terms also eventually decrease but not past a certain value of ¥. The modulation index (or H_R) thus determines the number of the significant sidebands, i.e. sidebands having amplitude at least 1% of the unmodulated carrier amplitude _.

(iii)The sidebands at equal frequency intervals from ¾x have equal amplitudes. Thus the sideband distribution is symmetrical about the carrier frequency.

Table 1.1c. Bessel Function of n order

(iv) From Table 1.1c, we find that as increases, value of any particular # coefficient say #₈ also increases. But is inversely proportional to the modulating frequency. Hence the relative amplitude of the distant sideband increases when the modulation frequency is reduced.

151 (v) In AM, with the increases of depth of modulation, the sideband power and hence the total power increase. In FM, on the other hand, the total transmitted power remains constant.

However, with increased value of modulation index (orH_R), the required bandwidth for relatively undistorted signal gets increased.

(vi) From Equ. 1.17, it is evident that the theoretical bandwidth required in FM is infinite. In practice. However, bandwidth used is one including all the significant sidebands under most exacting condition. This implies that using maximum deviation by the highest modulation frequency, no significant sidebands are excluded.

(vii) In AM, the amplitude of the carrier component remains constant. But in FM, the carrier component is #₇ which a function is of.

(viii) In FM, it is possible for the carrier component to disappear completely. From Figure. 1.2, we find that this happens for values of approximately equal to 24, 5.5, 8.6, 11.8 etc. these values of are called Eigen values

.