第四章 模拟调制系统 1. 引言 调制的定义 调制的分类 线性调制原理 非线性调制----角度调制 调制系统的比较（抗噪声性能分析和比较） 第四章 模拟调制系统 1. 引言 调制的定义 调制的分类 线性调制原理 非线性调制----角度调制 调制系统的比较（抗噪声性能分析和比较） FDM原理 总结 重点：调制系统的抗噪声性能

1.调制的定义 Definition:A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin and negligible elsewhere. Definition:A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency f= ±fc ,where fc>>0.The spectral magnitude is negligible elsewhere. fc is called the carrier frequency.fc may be arbitrarily assigned. Definition:Modulation is the process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude and/or phase perturbation.This bandpass signal is called the modulated signal s(t),and the baseband source signal is called the modulating signal m(t).

Diagram of a typical modulation system m(t) s(t) Modulator Baseband signal Bandpass signal Carrier cosωct Local oscillator Modulating signal Modulated signal

Bandpass communication system m(t) channel Modulator Bandpass signal Baseband signal noise cosωct Carrier Local oscillator Modulated signal Modulating signal m’(t) Demodulator Corrupted baseband signal Corrupted bandpass signal

2.线性调制系统 调制系统的分类：幅度调制（线性调制），非线性调制（角度调制）和数字调制（PCM） 线性调制：AM,DSB-SC,SSB,VSB

Complex envelope representation All banpass waveforms can be represented by their complex envelope forms. Theorem:Any physical banpass waveform can be represented by: v(t)=Re{g(t)ejωct} Re{.}:real part of {.}.g(t) is called the complex envelope of v(t),and fc is the associated carrier frequency.Two other equivalent representations are: v(t)=R(t)cos[ωct+θ(t)] and v(t)=x(t)cos ωct-y(t)sin ωct where g(t)=x(t)+jy(t)=R(t) ejθ(t)

Representation of modulated signals The modulated signals a special type of bandpass waveform So we have s(t)=Re{g(t)ejωct} the complex envelope is function of the modulating signal m(t): g(t)=g[m(t)] g[.]: mapping function All type of modulations can be represented by a special mapping function g[.].

Complex envelope functions for various types of modulation Type of modulation mapping functions g(m) AM Ac[1+m(t)] linear(?) DSB-SC Acm(t) linear SSB Ac[m(t)±jm’(t)] linear PM AcejDpm(t) non-linear FM non-linear

Spectrum of bandpass signals Bandpass signal’s spectrum complex envelope’s spectrum Theorem:If a bandpass waveform is represented by: v(t)=Re{g(t)ejωct} then the spectrum of the bandpass waveform is V(f)=1/2[G(f-fc)+G*(-f-fc)] and the PSD of the waveform is Pv(f)=1/4[Pg(f-fc)+Pg(-f-fc)] where G(f)=F[g(t)], Pg(f) is the PSD of g(t). Proof: v(t)=Re{g(t)ejωct}=1/2{g(t)ejωct+g*(t)e-jωct} V(f)=1/2F{g(t)ejωct}+1/2F{g*(t)e-jωct}

We have F{g*(t)}=G*(-f) Then V(f)=1/2{G(f-fc)+G*[-(f+fc)]} The PSD for v(t) is obtained by first evaluating the autocorrelation for v(t). Rv(τ)=<v(t)V(t+τ)>=< Re{g(t)ejωct} Re{g(t+ τ)ejωc(t+ τ)} Using the identity: Re(c2)Re(c1)=1/2Re(c*2c1)+ 1/2Re(c2c*1) So we have: Rv(τ)=1/2Re<{g*(t)g(t+ τ)ejωcτ>} + 1/2Re<{g(t)g(t+ τ) ejωct ej2ωcτ>} negligible? But Rg(τ)= <{g*(t)g(t+ τ)> Rv(τ)=1/2Re<{g*(t)g(t+ τ)ejωcτ>}=1/2 Re{Rg(τ) ejωcτ} Pv(f)=F{Rv(τ)}=1/4[Pg(f-fc)+ Pg*(-f-fc)] But Pg*(f)= Pg(f),so Pv(f) is real.

Linear and non-linear modulation systems All bandpass modulated waveform can be represented by: v(t)=Re{g(t)ejωct} The desired type of modulated waveform,s(t), is defined by the mapping function g[.]. Linear modulation----Amplitude Modulation (AM) Mapping function: gAM[.]=Ac[1+.] Modulated waveform: s(t)=Re{Ac[1+m(t)] ejωct}= Ac[1+m(t)]cosωct Spectrum:S(f)=1/2Ac[δ(f+fc)+M(f+fc)+δ(f-f0)]+M(f-fc)] Normalized average power of s(t): <s2(t)>=1/2Ac2+1/2Ac2<m2(t)>

AM system diagram----modulation: demodulation: m(t) Ac[1+m(t)]cosωct Local oscillator Accosωct Envelope detector BPF Noisy s(t) m’(t) BPF LPF Noisy s(t) m’(t) cosωct

Spectrum of AM waveform: M(f) 1 -B B f S(f) 1/2δ(f-f0) 1/2 1/2M(f-fc) f fc Where Ac=1

AM modulation: Ac[1+m(t)]cosω;where │m(t)│≤1 Some definitions: AM modulation: Ac[1+m(t)]cosω;where │m(t)│≤1 The percentage of positive modulation on an AM signal is: %positive modulation=(Amax-Ac)/Ac*100=max{m(t)}*100 The percentage of negative modulation is: %negative modulation=(Ac-Amin)/Ac*100=-min{m(t)}*100 The overall modulation percentage is: %modulation= (Amax-Amin)/2Ac={max[m(t)]-min[m(t)]}/2*100 Where Amax and Amin is Ac[1+m(t)]’s maximum and minimum values, is the level of the AM envelope when m(t)=0. The modulation efficiency is the percentage of the total power of the modulated signal that convoys information.

E=<m2(t)>/[1+ <m2(t)>]*100% In AM signaling,we have: E=<m2(t)>/[1+ <m2(t)>]*100% m(t) t Ac[1+m(t)] Amax s(t) s(t) Ac Amin t

If the condition │m(t)│≤1 is not satisfied and the percentage of negative modulation is over 100%,the envelope detector can not be used. Ex. Power of an AM signal (description of the question) AM broadcast transmitter:a 5000-W transmitter is connected to a 50ohms load;then the constant Ac is given by 1/2Ac2/50=5000.So the peak voltage across the load will be Ac =707V during the times of no modulation.If the transmitter is then 100%modulated by a 100-Hz test tone, the total (carrier plus sideband) average power will be : 1.5[1/2(Ac2/50)]=7500W There we have <m2(t)>=1/2 for a sinusoidal modulation waveshape of unity (100%) amplitude. The modulation efficiency would be 33% since <m2(t)>=1/2 .

Mapping function:gDSB[.]=Ac . Modulated waveform: Linear modulation----Double-Sideband suppressed carrier modulation (DSB) Mapping function:gDSB[.]=Ac . Modulated waveform: s(t)=Re{Acm(t)] ejωct}=Acm(t)cosωct Spectrum:S(f)=1/2Ac[M(f+fc)+M(f-fc)] Normalized average power of s(t): <s2(t)>=1/2Ac2<m2(t)> Diagram of DSB system: channel BPF LPF m(t) s(t) m’(t) s’(t)+n(t) Accosωct cosωct

s(t)=Re{Acg(t)] ejωct}=Ac[m(t)cosωct m^(t) sinωct] Linear modulation----Single-Sideband modulation (SSB) Definition:An upper sideband (USSB) signal has a zero-valued spectrum for │f│<fc , where fc is the carrier frequency. A lower sideband (LSSB) signal has a zero-valued spectrum for │f│>fc , where fc is the carrier frequency. Mapping function: Modulated waveform: s(t)=Re{Acg(t)] ejωct}=Ac[m(t)cosωct m^(t) sinωct] where the upper (-) sign is used for USSB and the lower (+) is for LSSB. m^(t) denotes the Hilbert transform of m(t). m^(t)=m(t)*h(t) ±

H(f)=-f for f>0 and H(f)=f for f<0 h(t)=1/πt and H(f)=-f for f>0 and H(f)=f for f<0 M(f) │G(f)│ 2Ac 1 f B B f USSB │S(f)│ Ac f -fc-B -fc fc fc+B

S(f)=Ac{M(f-fc),for f> fc and 0 for f< fc} USSB’s Spectrum: S(f)=Ac{M(f-fc),for f> fc and 0 for f< fc} + Ac{0 for f>-fc and M(f+fc),for f<- fc } Normalized average power of SSB: <s2(t)>=1/2<│g(t)│2>= 1/2Ac2<m2(t)+ [m^(t)]2 > = Ac2<m2(t)> Diagram of SSB system: m(t) m(t) s(t) L.O. SSB signal -90o -90o phase shift across band of m(t) Phasing method m^(t)

Diagram of SSB system(con.): Demodulation: s(t) Sideband filter m(t) SSB signal Accosωct Filter method m’(t) s(t) channel BPF LPF SSB signal s’(t)+n(t) cosωct

Vestigial sideband modulation DSB spectrum resource SSB too expensive to implement A compromise between two systems:VSB The vestigial sideband modulation is obtained by partial suppression of one sideband of a DSB signal. If the bandwidth of the modulating signal m(t) is B,the VSB signal has a bandwidth between B and 2B. sVSB(t) VSB filter (Bandpass filter) m(t) DSB modulator s(t) Baseband signal DSB signal VSB signal Hv(f)

Spectrum of VSB signal: S(f) 1/2 1/2M(f-fc) f fc Where Ac=1,DSB signal VSB Filter (USSB) SVSB(f) fΔ Hv(f-fc)+Hv(f+fc)

The spectrum of VSB signal: SVSB(f)=S(f)HV(f) The VSB filter must satisfy the constraint: HV(f-fc)+HV(f+ fc)=C, │f│≤B where B is the bandwidth of modulating signal. So: SVSB(f)=Ac/2[M(f-fc)HV(f)+M(f+fc) HV(f)]] Demodulation: sVSB(t) m’(t) Low pass filter cosωct

3.Non-linear modulation----angular modulation Non-linear modulation systems----phase modulation and frequency modulation Representation of PM and FM signals Complex envelope for angular modulation: g(t)=Ace jθ(t) , the modulated signal is: s(t)= Accos[ωct+θ(t)] for PM system, the modulated signal’s phase is directly proportional to the modulating signal mp(t): θ(t)=Dpmp(t) where Dp is the phase sensibility of phase modulator and constant.

Dp is the frequency deviation constant. For FM, the phase of modulated signal is proportional to the integral of the modulating signal mf(t): θ(t)=Df -∞∫tmf(t)dt Dp is the frequency deviation constant. With the angular modulated waveform’s representation, we can not distinguish which is PM or FM.So if we have a PM signal modulated by mp(t),it is possible to represent it by a frequency modulation modulated by a different waveform mf(t).mf(t) is given by: mf(t)= Dp/Df[dmp(t)/dt] similarly,if we have an FM signal modulated by mf(t),the corresponding phase modulation on this signal is mp(t)=Dp/Df -∞∫tmf(t)dt

Generation of FM from a phase modulator and vice versa mp(t) s(t) mf(t) Integrator gain=Df /Dp Phase modulator FM signal out FM from a phase modulator mf(t) s(t) mp(t) Differentiator gain=Dp/Df Frequency modulator PM signal out PM from a frequency modulator

Definition:If a bandpass signal is represented by s(t)=R(t)cosψ(t) Some definitions: Definition:If a bandpass signal is represented by s(t)=R(t)cosψ(t) where ψ(t)=ωct+θ(t),then the instantaneous frequency of s(t) is fi(t)=1/2πωi(t)=1/2π[dψ(t)/dt] or fi(t)=fc+1/2π[dθ(t)/dt] For the case of FM,we have fi(t)=fc+1/2π[dθ(t)/dt]= fc+1/2πDfmf(t) the peak frequency deviation: ΔF=max{1/2π[dθ(t)/dt]}

ΔF=max{1/2π[dθ(t)/dt]}= 1/2πDfVp For FM signaling: ΔF=max{1/2π[dθ(t)/dt]}= 1/2πDfVp where Vp=max[mf(t)] the peak phase deviation: Δθ=max[θ(t)] so for PM signaling: Δθ=max[θ(t)]=DpVp where Vp=max[mp(t)]. Definition:The phase modulation index is given by: βp= Δθ and the frequency modulation index is: βf= ΔF/B B is the bandwidth of modulating signal.

Spectra of angular modulated signals s(t)= Re{Acg(t) ejωct}= Accos[ωct+θ(t)] the spectrum is S(f)=1/2[G(f-fc)+G*(-f-fc)] where G(f)=F[g(t)]=F[Ace jθ(t) ] In general,it is impossible to have an analytic form of angular modulation signal’s spectrum.We will use some special modulating signal (sinusoid) to estimate the spectrum.

Ex. Spectrum of a FM or PM signal with sinusoidal modulation PM case: The PM modulating waveform: mp(t)=Amsinωmt Then θ(t)=βsinωmt where β=DpAm=βp is the phase modulation index. For FM case : mf(t)=Amcosωmt and β=DfAm/ωm=βf So the complex envelope:g(t)= Ace jθ(t) = Ace jβsinωmt g(t) is a periodic function ,so it can be represented by Fourier series. We have: g(t)= Σcnejnωmt where cn is Fourier coefficients.

G(f)= Σcnδ(f-nfm)= Σ AcJn(β) δ(f-nfm) Cn=AcJn(β) The Bessel function Jn(β) can not be evaluated in analytic form,but it is well-known numerically. We have G(f): G(f)= Σcnδ(f-nfm)= Σ AcJn(β) δ(f-nfm) The G(f)’s distribution depends greatly on β. Conclusion:the bandwidth of the angle modulated signal will depends on β and fm.In general,that will be infinite. In fact , it can be shown that 98% of the total power is contained in the bandwidth: Carson’s Rule BT=2(β+1)B where β is phase or frequency modulation index. So we can estimate the bandwidth of an angle modulated signal by Carson’s Rule with sufficient precision.

s(t)=Accosωct -Acθ(t) sinωct Narrowband angle modulation when θ(t) is restricted to a small value, │θ(t)│<0.2rad, the complex envelope g(t)= Ace jθ(t) may be approximated by a Taylor’s series where only the first two terms are used. g(t)=Ac[1+j θ(t)] So we have the narrowband angle modulated signal: s(t)=Accosωct -Acθ(t) sinωct we see that the narrowband angle modulation can be considered an AM-type. Discrete carrier term Sideband power

- Σ Diagram of the narrowband Angle modulation system: + Spectrum of NBFM S(f)=Ac/2{[δ(f-fc)+δ(f+fc)]+j[Θ(f-fc)-Θ(f+fc)]} Θ(f)=F[θ(t)]=DpM(f) for PM and (Df/j2πf)M(f) for FM m(t) - Σ Integrator gain=Df s(t) NBFM + Local oscillator -90o

Wideband frequency modulation Theorem:for WBFM signaling,where s(t)= Accos [ωct+Df-∞∫tm(t)dt] βf =(Df/2πB)max[m(t)]>>1 and B is the bandwidth of m(t).The normalized PSD of the WBFM signal is approximated by: Where fm(*) is the PSD of the modulating signal m(t).

Summary: it is a non-linear function of the modulation,and consequently the bandwidth of the modulated signal increases as the modulation index increases. The real envelope of an angle-modulation signal is constant. The bandwidth can be approximated by Carson’s rule.It depends on the modulation index and the bandwidth of the modulating signal.

4.调制系统的比较（抗噪声性能分析和比较） 信噪比：通信系统抗噪声性能的体现 信噪比：信号与噪声平均功率之比 S/N 分析方法：在相同的信号传输功率和相同的Gauss白噪声功率谱密度的条件下，调制系统的解调器输出信噪比。 分析模型： 其中：ni(t)是带通（窄带）Gauss白噪声 解调器 BPF LPF mo(t)+no(t) sm(t)+n(t) sm(t)+ni(t)

ni(t)= nc(t)cosωct - ns(t)sinωct =V(t) cos[ωct+θ(t)] ni(t)， nc(t)，ns(t)有相同的平均功率，即σi= σc =σs 或〈ni2(t)〉=〈nc2(t)〉=〈ns2(t)〉 解调器前的带通滤波器的带宽为B（与调制方式及m(t)有关），故解调器的输入噪声功率为： Ni=〈ni2(t)〉=noB (no:为噪声的单边带功率谱密度） 解调器的输出信噪比为： So/No=<mo2(t)>/ <no2(t)> 系统的调制制度增益G: G=输出信噪比/输入信噪比=[So/No]/[Si/Ni] 不同的调制方式，可以获得不同的G。G越大，调制方式就抗噪声而言就越佳。

G=2 <m2(t)>/[1+ <m2(t)>] AM系统的性能： 同步检测 si(t)=Ac[1+m(t)] cosωct Si =1/2+1/2<m2(t)>, Ni=<ni2(t)>=noB so(t)=1/2m(t),So=1/4< m2(t)> no=1/2nc(t) ,N0=1/4Ni G=2 <m2(t)>/[1+ <m2(t)>] 若m(t)为方波，G=2/3 mo(t) BPF LPF si(t)+ni(t) cosωct

R1C1构成低通滤波器，当B≤1/ R1C1 ≤fc, R1C1电路只对vin的峰值的变化有响应。 包络检测（电路图） R1C1构成低通滤波器，当B≤1/ R1C1 ≤fc, R1C1电路只对vin的峰值的变化有响应。 B≤fc是为了包络清楚，此时C1在载波的两个峰值间只有微小的放电，因此v近似等于vin的包络（除高频锯齿外）， R2C2 起隔离v中的直流成分。 C1 Ac[1+m(t)] vout vin R1 C1 v R1 s(t) t

包络检测（续） v s(t) m(t)

ψ(t)=arctg{ns(t)/[Ac+ Ac m(t)+ nc(t)]} 信噪比计算： Si =Ac2/2+Ac2/2<m2(t)>, Ni=<ni2(t)>=noB 检波器输入端的信号和噪声合成为： si(t) +ni(t) =Ac[1+m(t)] cosωct+ nc(t)cosωct - ns(t)sinωct =E(t)cos[ωct +ψ(t)] 其中：E(t)={[Ac+ Ac m(t)+ nc(t)]2 + ns2(t)}1/2 ψ(t)=arctg{ns(t)/[Ac+ Ac m(t)+ nc(t)]} E(t)的信号和噪声存在非线性关系。 分析：大信噪比情况，即Ac+ Ac m(t)>> ni(t) 则：E(t)≈ Ac+ Ac m(t)+ nc(t) 故： So= Ac2 < m2(t)>， N0=noB G=2 Ac2 < m2(t)>/[Ac2+Ac2<m2(t)>] (-1≤m(t)≤1) 当max[m(t)]=1（100%调制）且为正弦波，我们有G=2/3 包络检测能达到的最大信噪比增益。

小信噪比情况，即Ac+ Ac m(t)<< ni(t) 则包络E(t)为： E(t)≈ R(t)+[Ac+ Ac m(t)]cosθ(t) 包络中的信号部分完全被噪声所淹没。门限效应。 结论：包络法在大信噪比情况下，性能与同步检测法相似，在小信噪比时系统不能解调出信号。 噪声项

si(t)=Acm(t)cosωct，Si= Ac2/2<m2(t)> 经同步检测后，输出信号和噪声为： DSB-SC系统的抗噪声性能 si(t)=Acm(t)cosωct，Si= Ac2/2<m2(t)> 经同步检测后，输出信号和噪声为： mo(t)=1/2Acm(t)， no=1/2nc(t) 因此： So= Ac2/4<m2(t)>， N0=1/4noB= 1/4Ni G=2 SSB系统的抗噪声性能 单边带解调器与双边带相同，因此有： N0=1/4noB= Ni si(t)=Ac[m(t)cosωct m^(t) sinωct]，Si=Ac2/4<m2(t)> So= Ac2/16<m2(t)> G=1 双边带（G=2）是否优于单边带？No. why? ±

sFM(t)=Acos[ωct+ φ(t)]，φ(t)=Df -∞∫tmf(t)dt 角度调制系统的抗噪声性能 FM的抗噪声性能 解调法：鉴频法 sFM(t)=Acos[ωct+ φ(t)]，φ(t)=Df -∞∫tmf(t)dt 设sFM(t)的带宽为B（不是m(t)的带宽），则鉴频法的输入信噪比为： Si/Ni=A2/2noB G=? sFM(t)+ nc(t)cosωct - ns(t)sinωct = Acos[ωct+ φ(t)]+V(t)cos[ωct+θ(t)]=V’(t)cosψ(t) 带通限幅器 鉴频器 Low-pass filter m(t) sFM(t) 信号项 噪声项ni(t)

经限幅带通滤波器后，有：Vocosψ(t) ψ(t)=？（信号和噪声的合成） 令： Acos[ωct+ φ(t)]=a1cosΦ1 V(t)cos[ωct+θ(t)]= a1cosΦ2 a1cosΦ1+a1cosΦ2= acosΦ 利用矢量表示法得： a a a2 a1 Φ2-Φ1 Φ1-Φ2 a1 a2 Φ Φ Φ1 Φ2 Φ2 Φ1 任意参考相位 任意参考相位 图b 图a

由图a得：tg(Φ- Φ1)=asin(Φ2- Φ1)/[a1+a2cos (Φ2- Φ1)] Φ= Φ1+arctg{a2sin(Φ2- Φ1)/[a1+a2cos (Φ2-Φ1)]} 由图b得： Φ= Φ2+arctg{a1sin(Φ1- Φ2)/[a2+a1cos (Φ1-Φ2)]} 根据设定的关系，有： ψ(t)=ωct+φ(t)+arctg{V(t)sin(θ(t)-φ(t))/[A+V(t)cos (θ(t)-φ(t))]} 或：ψ(t)=ωct+θ(t)+arctg{Asin(φ(t)-θ(t))/[V(t)+cos (φ(t)-θ(t))]} 鉴频器的输出正比于输入信号的瞬时频率偏移，以上表达式无法直接给出有用信号m(t)。特例分析。 两种情况：大Si/Ni和小Si/Ni情况。 分别讨论。

大Si/Ni：即A>>V(t)，因此有： V(t)sin(θ(t)-φ(t))/[A+V(t)cos (θ(t)-φ(t))]≈0 则图a成为： ψ(t)≈ωct+φ(t)+V(t)/Asin(θ(t)-φ(t)) 输出电压：vo(t)=1/2π[dψ(t)/dt]-fc = 1/2π[dφ(t)/dt]+1/(2πA)dni(t)/dt 输出的有用信号为： mo(t)= 1/2π[dφ(t)/dt]= Df/2πmf(t) So= Df2/4π2< mf2(t)> 输出噪声：no(t)= 1/(2πA)dns(t)/dt 求ns(t)的功率？ 信号 噪声 Ans(t)

Pi(f)=<ns2(t) >/B’=no f≤B’ (单边带PSD) ni(t)= V(t)cos[ωct+θ(t)]:带通噪声 ns(t)为低通型噪声，带宽由低通滤波器的截止频率确定[0，B’/2] ns(t) =V(t)sin(θ(t)-φ(t)): φ(t)信号，因此ns(t) Gauss型 有：〈 ns2(t) 〉=〈 ni2(t) 〉=noB’ 因此d ns(t)/dt的PSD Pi(f)为ns(t)的PSD乘以理想微分器的功率传递函数│H(f)│2=4π2f2 n’s(t)的PSD为Po(f): Po(f)=4π2f2 Pi(f) Pi(f)=<ns2(t) >/B’=no f≤B’ (单边带PSD) P0(f)= 4π2f2 no f≤B’ 结论： n’s(t)的PSD与频率f有关，即与f2 成正比。 d ns(t)/dt ns(t) 理想微分器

No= <no2(t) >= <ns’2(t) >/ 4π2A2 =1/ 4π2A2 0∫fm P0(f) df 解调器的输出噪声功率为： No= <no2(t) >= <ns’2(t) >/ 4π2A2 =1/ 4π2A2 0∫fm P0(f) df So/ No= 4A2 Df2 <m2(t) >/[8π2nofm3] 讨论：m(t) is a tone signal. sFM(t)=Acos[ωct+βf sinωmt], βf=Df/ ωm βf=Δf/fm So/ No=3/2 βf2(A2/2)/(nofm) Si= A2/2, nofm为(0, fm)的白噪声记作Nm So/ No=3/2 βf2(Si /Nm) fm≠B, Ni≠Nm B=2(Δf+fm) 因此： So/ No=3 βf2(βf+1) (Si /Ni) G= 3 βf2(βf+1) 例： βf=5 则G=450，此时B=2 (βf+1) fm=12 fm P0(f) f B’

与AM系统的比较： (So/ No)AM=<m2(t)>/noB,100%调制，m(t)为正弦波 (So/ No)AM=(A2/2)/(2fmno) (2fm=B) (So/ No)FM/(So/ No)AM=3βf2 结论：FM的信噪比比AM的大3βf2 代价：带宽WBFM的带宽BFM与BAM间的关系： BFM=(βf+1) BAM间 (So/ No)FM/(So/ No)AM=3(BFM/BAM)2 结论： WBFM的输出信噪比相当于调幅的改善为传输带宽的平方成正比。

小Si/Ni：即A<<V(t)，则有： ψ(t)=ωct+θ(t)+arctg{Asin(φ(t)-θ(t))/[V(t)+cos (φ(t)-θ(t))]}≈ψ(t)=ωct+θ(t)+A/V(t)sin(φ(t)-θ(t)) 无信号：门限效应 So/ No DSB同步检测 门限效应 a Si/Ni 一般情况下Si/Ni ≈10dB

各种模拟调制系统的比较 1。抗噪声性能比较 比较：有可比性 Receiver:same input power,same additive gaussian white noise(average value=0,PSD=no/2),modulating signal m(t) (<m(t)>=0,<m2(t)>=1/2,max│m(t)│=1) We have: (So/ No)DSB=(Si/noBb), (So/ No)AM= 1/3(Si/noBb) (So/ No)SSB=(Si/noBb), (So/ No)FM= 3/2βf2(Si/noBb) Bb:基带信号的带宽 见书p.82 fig.4p.82 fig.4-12 2。带宽比较 见书p.83表4-1

5.频率分割复用（FDM） FDM:Frequency-Division-Multiplexing 将若干个彼此独立的信号合并在一起在同一信道上传输 信号1 复用 解复用 信号1 一路宽带信号 信号2 信号2 S(f) 信号n 信号n f

复合调制及多级调制的概念 复合调制：SSB/SSB,SSB/FM,FM/FM 复合调制：主要用在数字通信系统 多级调制：同一基带信号经多次调制成为一高频信号 作用：。。。。。。

总结： 模拟调制：带宽，抗噪声性能，比较