Spectral Analysis of FM

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1 utumn Oct 06, 2005 Frequency Modulation nalog and Digital Communications utumn Oct 06, 2005 CS477: nalog and Digital Communications 1 Spectral nalysis of FM Exponential modulation is non-linear Bandwidth of the transmitted signals depends upon the amplitude of the information signal Hard to analyze with Fourier transforms Narrowband approximation makes analysis a little easier NBFM and NBPM For narrowband approximation to hold: þ(t) ü1rad Oct 06, 2005 CS477: nalog and Digital Communications 2 CS477: nalog and Digital Communications 1

2 utumn Oct 06, 2005 NBFM and NBPM Consider the bandpass transmitted signal: x c (t) = c cos(2ùf c t + þ(t)) x c (t) =x ci (t) cos 2ùf c t à x cq (t) sin 2ùf c t þ x ci (t) = c cos þ(t) = c [1 à 2 (t) 2! + ááá] þ x cq (t) = c sin þ(t) = c [þ(t) à 3 (t) 3! + ááá] For narrowband approximation: x ci (t) ù c ; x cq (t) ù c þ(t) Oct 06, 2005 CS477: nalog and Digital Communications 3 NBFM and NBPM Thus, for NBFM and NBPM: x c (t) ù c cos 2ùf c t à c þ(t) sin 2ùf c t X c (f) ù 1 2 c î(f à f c ) à j 2 c Φ(f à f c ); f>0 þ(t) = þ Δx(t) R PM 2ùf Δ x(t) FM Φ(f) = þ ΔX(f) PM à jf Δ X(f)/f FM Oct 06, 2005 CS477: nalog and Digital Communications 4 CS477: nalog and Digital Communications 2

3 utumn Oct 06, 2005 Tone Modulation x(t) = m sin 2ùf m t PM m cos 2ùf m t FM þ(t) = þ Δx(t) R PM 2ùf Δ x(t) FM þ(t) =ì sin 2ùf m t ì = þ Δ m PM m )f Δ FM ( fm Thus, for NBFM and NBPM: ì ü 1 Oct 06, 2005 CS477: nalog and Digital Communications 5 Tone Modulation with NB x c (t) ù c cos 2ùf c t à c þ(t) sin 2ùf c t x c (t) ù c cos 2ùf c t à c ì sin 2ùf m t sin 2ùf c t = c cos 2ùf c t à c ì 2 cos 2ù(f c à f m )t + c ì 2 cos 2ù(f c + f m )t X c (f) ù c î(f à f c ) à c ì 2 î(f à (f c à f m )) + c ì 2 î(f à (f c + f m )); f>0 Oct 06, 2005 CS477: nalog and Digital Communications 6 CS477: nalog and Digital Communications 3

4 utumn Oct 06, 2005 Tone Modulation with NB x c (t) ù c cos 2ùf c t à c ì 2 cos 2ù(f c à f m )t + c ì 2 cos 2ù(f c + f m )t X c+ (f) ù c î(f à f c ) à c ì 2 î(f à (f c à f m )) + c ì 2 î(f à (f c + f m )) c f c à f m cì 2 à cì 2 f c f c + f m f Oct 06, 2005 CS477: nalog and Digital Communications 7 Tone Modulation with WB NOTE: x c (t) = c cos(2ùf c t + þ(t)) x c (t) = c cos(2ùf c t + ì sin 2ùf m t) = c [cos(ì sin 2ùf m t) cos 2ùf c t à sin(ì sin 2ùf m t) sin 2ùf c t] cos(ì sin 2ùf m t) sin(ì sin 2ùf m t) and are periodic Try to write their Fourier series expansion! Oct 06, 2005 CS477: nalog and Digital Communications 8 CS477: nalog and Digital Communications 4

5 utumn Oct 06, 2005 Tone Modulation with WB cos(ì sin 2ùf m t)=j 0 (ì)+ P 2J n (ì) cos n2ùf m t sin(ì sin 2ùf m t)= P where n>0 n odd n even 2J n (ì) sin n2ùf m t and J n (ì) ñ 2ù 1 R ù e j(ì sin õànõ) dõ àù x c (t) = c P Jn (ì) cos 2ù(f c + nf m )t n=à Oct 06, 2005 CS477: nalog and Digital Communications 9 Tone Modulation with WB x c (t) = c P Jn (ì) cos 2ù(f c + nf m )t n=à J 0(ì) J 1(ì) J 2(ì) f c à f m J 2(ì) J 3(ì) à J 3(ì) f c f c + f m f à J 1(ì) ll values in the line spectrum are scaled by c (Do Example and Exercise in text) Oct 06, 2005 CS477: nalog and Digital Communications 10 CS477: nalog and Digital Communications 5

6 utumn Oct 06, 2005 FM Bandwidth Consider Tone Modulation Observation for ì ý 1 J n (ì) falls off quickly for n/ì > 1 Consider components only for n ôì Significant frequency components lie within f c æ ìf m Observation for ì ü 1 J n (ì) is insignificant for n > 1 Consider components only for n =1 Significant frequency components lie within f c æ f m Oct 06, 2005 CS477: nalog and Digital Communications 11 FM Bandwidth: Single Tone B =2M(ì)f m ; M(ì) õ 1 M(ì) depends upon the allowable distortion level Usually approximated by M(ì) =ì +2; ì>2 B =2(ì +2)f m =2( m f 4 +2f m ) For worst case, use: m =1,f m = W B T =2(f Δ +2W); ì>2 f What is the corresponding value of ì? (ì = Δ W ) Is this the maximum value? NO! smaller ì may result in larger B T Oct 06, 2005 CS477: nalog and Digital Communications 12 CS477: nalog and Digital Communications 6

7 utumn Oct 06, 2005 FM Bandwidth: Generic Signal Estimate BW using worst case tone modulation! f Define Deviation ratio: D = Δ W B T =2M(D)W pproximations: B T = 2DW =2f Δ; D ý 1 2W; D ü 1 Carson s rule: B T ù 2(f Δ + W) =2(D +1)W Carson s rule is applied for extreme values of deviation ratio. It underestimates the BW for practical systems. For practical systems (2 <D<10) : Use B T ù 2(f Δ +2W) =2(D +2)W; D>2 (Do Example and Exercise in text) Oct 06, 2005 CS477: nalog and Digital Communications 13 Transmission BW: Summary Condition(s) Single Tone Generic Signal D ý 1,ìý 1 2ìf m 2DW D ü 1,ìü 1 2f m 2W Carson s Rule D, ì extreme Practical designs (2 <D<10) 2(ì +1)f m 2(ì +2)f m 2(D +1)W 2(D +2)W Oct 06, 2005 CS477: nalog and Digital Communications 14 CS477: nalog and Digital Communications 7

8 utumn Oct 06, 2005 Generating NBFM Signals Direct method Use a voltage-controlled oscillator Output frequency is proportional to input voltage Use information signal as the input voltage Restricted to low power applications Most circuits unstable at carrier frequency Not suitable for commercial FM broadcast Require sophisticated feedback control Oct 06, 2005 CS477: nalog and Digital Communications 15 Generating NBFM Signals Indirect method: x c (t) ù c cos 2ùf c t à c þ(t) sin 2ùf c t (NBPM/NBFM) þ(t) =þ Δ x(t) x c (t) (NBPM) 90 0 c cos 2ùf c t Input R 1 NBPM Modulator T þ Δ =2ùTf Δ NBFM Output Oct 06, 2005 CS477: nalog and Digital Communications 16 CS477: nalog and Digital Communications 8

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