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# Frequency modulation

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Frequency Modulation

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### Frequency modulation

1. 1. Angle Modulation – Frequency ModulationConsider again the general carrier vc ( t ) = Vc cos( ωc t + φc ) ( ωc t + φc ) represents the angle of the carrier. There are two ways of varying the angle of the carrier. • By varying the frequency, ωc – Frequency Modulation. • By varying the phase, φc – Phase Modulation 1
2. 2. Frequency ModulationIn FM, the message signal m(t) controls the frequency fc of the carrier. Consider thecarrier vc ( t ) = Vc cos( ωc t )then for FM we may write:FM signal v s ( t ) = Vc cos( 2π ( f c + frequency deviation ) t ) ,where the frequency deviationwill depend on m(t).Given that the carrier frequency will change we may write for an instantaneouscarrier signal Vc cos( ωi t ) = Vc cos( 2πf i t ) = Vc cos( φi )where φi is the instantaneous angle = ωi t = 2πf i t and fi is the instantaneousfrequency. 2
3. 3. Frequency Modulation dφi 1 dφiSince φi = 2πf i t then = 2πf i or fi = dt 2π dti.e. frequency is proportional to the rate of change of angle.If fc is the unmodulated carrier and fm is the modulating frequency, then we maydeduce that 1 dφi f i = f c + Δf c cos( ωm t ) = 2π dt∆fc is the peak deviation of the carrier. 1 dφiHence, we have = f c + Δf c cos( ωm t ) ,i.e. dφi = 2πf c + 2πΔf c cos( ωm t ) 2π dt dt 3
4. 4. Frequency ModulationAfter integration i.e. ∫ (ωc + 2πΔf c cos( ωm t ) ) dt 2πΔf c sin ( ωm t ) φi = ωc t + ωm Δf c φi = ωc t + sin ( ωm t ) fmHence for the FM signal, v s ( t ) = Vc cos( φi )  Δf c  v s ( t ) = Vc cos ωc t +  sin ( ωm t )    fm  4
5. 5. Frequency Modulation Δf c The ratio is called the Modulation Index denoted by β i.e. fm Peak frequency deviation β= modulating frequencyNote – FM, as implicit in the above equation for vs(t), is a non-linear process – i.e.the principle of superposition does not apply. The FM signal for a message m(t) as aband of signals is very complex. Hence, m(t) is usually considered as a single tonemodulating signal of the form m( t ) = Vm cos( ωm t ) 5
6. 6. Frequency Modulation  Δf The equation v s ( t ) = Vc cos ωc t + c sin ( ωm t )  may be expressed as Bessel    fm series (Bessel functions) ∞ v s ( t ) = Vc ∑ J ( β ) cos( ω n c + nωm ) t n= −∞where Jn(β) are Bessel functions of the first kind. Expanding the equation for a fewterms we have:v s (t ) = Vc J 0 ( β ) cos(ω c )t + Vc J 1 ( β ) cos(ω c + ω m )t + Vc J −1 ( β ) cos(ω c − ω m )t                     Amp fc Amp fc + fm Amp fc − fm + Vc J 2 ( β ) cos(ω c + 2ω m )t + Vc J − 2 ( β ) cos(ω c − 2ω m )t +                  Amp fc +2 fm Amp fc −2 f m 6
7. 7. FM Signal Spectrum.The amplitudes drawn are completely arbitrary, since we have not found any value forJn(β) – this sketch is only to illustrate the spectrum. 7
8. 8. FM Signal Waveforms. If we plot fOUT as a function of VIN:In general, m(t) will be a ‘band of signals’, i.e. it will contain amplitude and frequencyvariations. Both amplitude and frequency change in m(t) at the input are translated to(just) frequency changes in the FM output signal, i.e. the amplitude of the output FMsignal is constant.Amplitude changes at the input are translated to deviation from the carrier at theoutput. The larger the amplitude, the greater the deviation. 16
9. 9. FM Signal Waveforms.Frequency changes at the input are translated to rate of change of frequency at theoutput.An attempt to illustrate this is shown below: 17
10. 10. Significant Sidebands – Spectrum.As shown, the bandwidth of the spectrum containing significantcomponents is 6fm, for β = 1. 23
11. 11. Carson’s Rule for FM Bandwidth.An approximation for the bandwidth of an FM signal is given byBW = 2(Maximum frequency deviation + highest modulatedfrequency) Bandwidth = 2(∆f c + f m ) Carson’s Rule 25
12. 12. Narrowband and Wideband FMNarrowband FM NBFMFrom the graph/table of Bessel functions it may be seen that for small β, (β ≤ 0.3)there is only the carrier and 2 significant sidebands, i.e. BW = 2fm.FM with β ≤ 0.3 is referred to as narrowband FM (NBFM) (Note, the bandwidth isthe same as DSBAM).Wideband FM WBFMFor β > 0.3 there are more than 2 significant sidebands. As β increases the number ofsidebands increases. This is referred to as wideband FM (WBFM). 26
13. 13. Comments FM• The FM spectrum contains a carrier component and an infinite number of sidebands at frequencies fc ± nfm (n = 0, 1, 2, …) ∞ FM signal, v s (t ) = Vc ∑J n = −∞ n ( β ) cos(ω c + nω m )t• In FM we refer to sideband pairs not upper and lower sidebands. Carrier or other components may not be suppressed in FM.• The relative amplitudes of components in FM depend on the values Jn(β), where α Vm β= thus the component at the carrier frequency depends on m(t), as do all the fm other components and none may be suppressed. 28