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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
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).
2
Diagram of a typical modulation system
• modulation
m(t) s(t)
Modulator
Baseband signal
Bandpass signal
Carrier cosωct
Modulating Local
signal oscillator
Modulated signal
3
• Bandpass communication system
m(t) s(t)
Modulator channel
Baseband signal Bandpass
signal
noise
Carrier cosωct
Local
Modulating oscillator Modulated signal
signal
m’(t)
Demodulator
Corrupted Corrupted
baseband signal bandpass signal
4
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
5
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)
6
• 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[.].
7
• 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
t
FM jD f m ( t ) dt non-linear
Ac e
8
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}
9
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.
10
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)>
11
• AM system diagram----modulation:
m(t) Ac[1+m(t)]cosωct
Local
oscillator
Accosωct
• demodulation:
Envelope
BPF
Noisy s(t) detector m’(t)
BPF LPF
Noisy s(t) m’(t)
cosωct 12
• 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
13
• 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.
14
• In AM signaling,we have:
E=<m2(t)>/[1+ <m2(t)>]*100%
m(t)
t
Ac[1+m(t)]
s(t) Amax
s(t)
Ac
Amin
t
15
• 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 .
16
• 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)
Accosωct s’(t)+n(t)
cosωct
17
• 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: g SSB [m(t )] Ac [m(t ) m(t )]
• 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)
18
h(t)=1/πt
and
H(f)=-f for f>0 and H(f)=f for f<0
M(f) │G(f)│
2Ac
1
B f
B f
USSB
│S(f)│
Ac
f
-fc-B -fc fc fc+B
19
• 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.
-90o SSB signal
o
-90 phase
shift across
band of m(t) m^(t) Phasing method
20
• Diagram of SSB system(con.):
s(t)
Sideband
filter
m(t)
SSB signal
Accosωct
Filter method
• Demodulation:
s(t) m’(t)
channel BPF LPF
SSB signal
s’(t)+n(t)
cosωct
21
• 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)
m(t) DSB s(t) VSB filter
modulator (Bandpass filter)
Baseband DSB VSB signal
signal signal
Hv(f)
22
• Spectrum of VSB signal:
S(f)
1/2
1/2M(f-fc)
f
fc
Where Ac=1,DSB signal
SVSB(f) VSB Filter
(USSB)
Hv(f-fc)+Hv(f+fc) fΔ
23
• 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
24
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.
25
• For FM, the phase of modulated signal is proportional to
the integral of the modulating signal mf(t):
θ(t)=Df -∞∫tmf(t)dt
Df 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)=Df/Dp -∞∫tmf(t)dt
26
• Generation of FM from a phase modulator and vice versa
mp(t) s(t)
mf(t)
Integrator Phase
gain=Df /Dp modulator
FM signal
out
FM from a phase modulator
mf(t) s(t)
mp(t)
Differentiator Frequency
gain=Dp/Df modulator
PM signal
out
PM from a frequency modulator
27
• 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]}
28
• 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.
29
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.
30
• 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.
31
• 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.
32
• 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
Discrete carrier Sideband
term power
we see that the narrowband angle modulation can be
considered an AM-type.
33
• Diagram of the narrowband Angle modulation system:
m(t)
Integrator - Σ
s(t)
gain=Df
NBFM
+
Local
-90o
oscillator
• 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
34
• 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:
2
A 2 2
P( f ) c
{ fm[ ( f f c )] f m [ ( f f c )]}
2D f Df Df
Where fm(*) is the PSD of the modulating signal m(t).
35
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.
36
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：
BPF 解调器 LPF
mo(t)+no(t)
sm(t)+n(t)
sm(t)+ni(t)
其中：ni(t)是带通（窄带）Gauss白噪声
37
• 我们有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。
38
• 系统的调制制度增益G与采用的解调方式有密切的关
系
• AM系统的性能：
mo(t)
• 同步检测 BPF LPF
si(t)+ni(t) cosωct
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
39
• 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)]
vin R1 C1 v R1 vout
s(t)
t
40
• 包络检测（续） v
s(t)
m(t)
41
• 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。
42
• 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)
噪声项
• 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。
43
• 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?
44
• 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
Low-pass
带通限幅器 鉴频器
filter m(t)
sFM(t)
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)
信号项 噪声项ni(t)
45
• 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：
a2 a a1 a
Φ2-Φ1 Φ1-Φ2
a1 a2
Φ Φ1 Φ Φ2
Φ2 Φ1
任意参考相位 任意参考相位
图a 图b
46
• 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。
47
• 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))
信号 噪声 Ans(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？
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• 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’
ns(t) d ns(t)/dt
理想微分器
• 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。
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• 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] P0(f)
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) B’ f
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
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• 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。
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• 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 / N o
DSB同步检测
门限效应
a Si/Ni
一般情况下Si/Ni ≈10dB
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• 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
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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
信号1
一路宽带信号
信号2 复用 解复用 信号2
S(f)
信号n
信号n
f
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• 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：。。。。。。
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总结：
• 模拟调制：带宽，抗噪声性能，比较
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