TABLE 69.1 Complex Envelope Functions for various Types of Modulation Corresponding Quadrature Corresponding Amplitude and Mapping Functions Modulation Phase modulation Modulation x(r) e(r Remarks AM 1+m(t) 0m()> L m(r)>-l required for L Coherent detection required. u80,m()<o cos[D,m(t)] sin[Dm(o) 1 D m(r) NL D, is the phase deviation onstant(radian/volts) D, is the frequency deviation constant(radian/wolt-sec) )2+i(n)2 [m(/m(oI L Coherent detection required SSB-PM siMr画 e"sin(D m(nI Dm(r dD, m(oyo mo)do D, m(oyo SB-EVa ellaI +m(a) i l+mMnl [1+ m(r)] cos In[I +m(oll +[l +m(o]sin(ln[l +m(oll 1+ m(r) ±ln[+m(切 m(r)>-l is required so that the In will have a real value SSB-SQca+臧∥川1+ m(t>-1 is required n1=m( ±v1+m) siny-In(l+mt ±li[+m the In will have a real value m()+m2(t) (r) tarIm (t)/m, o) L Used in NtSC color tele. L=linear, NL=nonlinear, [ is the Hilbert transform(ie, -g0 phase-shifted version)of [-]. The Hilbert transform is x(t=x(* 1_1 aUse upper signs for upper sideband signals and lower signs for lower sideband si bIn the strict sense, AM signals are not linear because the carrier term does not satisfy the linearity(superposition) condition© 2000 by CRC Press LLC TABLE 69.1 Complex Envelope Functions for Various Types of Modulation Corresponding Quadrature Corresponding Amplitude and Type of Mapping Functions Modulation Phase Modulation Modulation g[m] x(t) y(t) R(t) q(t) Linearity Remarks AM 1 + m(t) 1 + m(t) 0 *1 + m(t)* Lb m(t) > –1 required for envelope detection. DSB-SC m(t) m(t) 0 *m(t)* L Coherent detection required. PM ejDpm(t) cos[Dpm(t)] sin[Dpm(t)] 1 Dpm(t) NL Dp is the phase deviation constant (radian/volts). FM 1 NL Df is the frequency deviation constant (radian/volt-sec). SSB-AM-SCa m(t) ± jmˆ (t) m(t) ± mˆ (t) tan–1[± mˆ (t)/m(t)] L Coherent detection required. SSB-PMa ejDp[m(t)± jmˆ (t)] e7Dpmˆ(t ) cos[Dpm(t)] e7Dpmˆ(t) sin[Dpm(t)] e7Dpmˆ (t) Dpm(t) NL SSB-FMa NL SSB-EVa e{ln[1 + m(t)]± j ln[1 + ˆ m(t )]} [1 + m(t)] cos {ln[1 + ˆ m(t)]} ±[1 + m(t)]sin{ln[1 + ˆ m(t)]} 1 + m(t) ±ln[1 + ˆ m(t)] NL m(t) > –1 is required so that the ln will have a real value. SSB-SQa e(1/2){ln[1 + m(t )]± j ln[1 + ˆ m(t )]} NL m(t) > –1 is required so that the ln will have a real value. QM m1(t) + jm2(t) m1(t) m2(t) tan–1[m2(t)/m1(t)] L Used in NTSC color tele￾vision: requires coherent detection. L = linear, NL = nonlinear, [ˆ.] is the Hilbert transform (i.e., –90° phase-shifted version) of [·]. The Hilbert transform is a Use upper signs for upper sideband signals and lower signs for lower sideband signals. bIn the strict sense, AM signals are not linear because the carrier term does not satisfy the linearity (superposition) condition. 0 1 180 1 , ( ) – , ( ) – m t m t > ° < Ï Ì Ó ¸ ˝ ˛ 0 0 180 0 , ( ) , ( ) m t m t > ° < Ï Ì Ó ¸ ˝ ˛ e jDf m d t (s s ) Ú-• cos ( ) – D m d f t s s Ú • È Î Í ˘ ˚ ˙ sin ( ) – D m d f t s s Ú • È Î Í ˘ ˚ ˙ D m d f t ( ) – s s Ú • [m t( ) [m t ˆ( )] 2 2 + e jDf m jm d t [ (s)± ˆ(s)] s Ú-• e D m d D m d f t f t m ˆ( ) – – cos ( ) Ú • Ú • È Î Í ˘ ˚ ˙ s s s s e D m d D m d f t f t m ˆ( ) – – sin ( ) Ú • Ú • È Î Í ˘ ˚ ˙ s s s s e D m d f t m ˆ( ) Ú–• s s D m d f t ( ) – s s Ú • 1 1 2 + = 1 Ï Ì Ó ¸ ˝ ˛ m t( ) cos l m t n[ˆ ( )] ± + + Ï Ì Ó ¸ ˝ ˛ 1 1 2 m t( ) sin l 1 m t n[ˆ ( )] 1 + m( )t ± + 1 2 ln[ˆ 1 m t( )] m t m t 1 2 2 2 ( ) + ( ) D ˆ( ) ( ) * ( ) x t x t t x t = = d -• - • Ú 1 1 p p l l l