uency spectra V+ AV From Equation 18.22, a sinusoidal carrier modulated by a sinusoidal baseband signal may be represented by Equation 18.34. = RVE (18.34) Frequency However Equation 18.35 holds, where J (B) is the Bessel function cfc+sf of the first kind of order n, resulting in Equation 18.36. (18.35) Frequency (18.36) Thus, the spectrum of the angle modulated wave has an upper and a lower sideband, each containing frequencies separated from the carrier by all the harmonics of the baseband signal. Equation 18.36 shows that angle modulation is non linear. If the Figure 18.11 Principles of tuned circuit discriminators: baseband signal contains two frequencies, f and f2, the sidebands (a) frequency to amplitude conversion; (b) frequency to contain many components at frequencies phase conversion 1,2,3,..., r = 1,2,3,...) in addition to f H nf, and the spectrum of the baseband signal contains many components, the spectrum of each sideband is extremely complicated (Black, 1953). It can be represented by a band of Gaussian noise (Roberts, 1977). Tables of J ,(x) have been published (Jahnke and Emde, 1945). Some values of J ( B ) are given in Table 18.1 and some typical voltage spectra of angle modulated signals are shown in Figure 18.13. It will be seen that for low index modulation only the components at fc=fm are significant. Thus, the bandwidth required for transmission is no greater than for AM. However, for high index modulation, the bandwidth is much greater than for AM. In general, the magnitudes of J (B) are small for n>B+1. As a result, the bandwidth W required is approximately given by Equa- tion 18.37 (Carson's rule). Figure 18.12 Phase locked loop circuit for demodulating W = 2 F m ( 1 + B) (18.37) controlled oscillator (VCO) has zero control voltage, its output frequency is that of the unmodulated carrier. When the frequency of Thus, for FM, Equation 18.38 may be obtained where AF is the the input signal begins to change, the resulting phase difference maximum frequency deviation. between it and the outpu