Download - 第四章 模拟调制系统
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• 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
第四章 模拟调制系统
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• 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).
1. 调制的定义
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• modulation
Diagram of a typical modulation system
m(t)
Baseband signal
Modulating signal
Modulators(t)
Bandpass signal
Modulated signal
Local oscillator
cosωctCarrier
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• Bandpass communication system
m(t)
Baseband signal
Modulating signal
Modulators(t)
Bandpass signal
Local oscillator
cosωctCarrier
Modulated signal
channel
Demodulator m’(t)
Corrupted baseband signal
Corrupted bandpass signal
noise
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• 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
2. 线性调制系统
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• 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)
Complex envelope representation
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• 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[.].
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• 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
t
f dttmjD
ceA)(
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• 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}
Spectrum of bandpass signals
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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.
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• 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+1/2Ac
2<m2(t)>
Linear and non-linear modulation systems
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• AM system diagram----modulation:
• demodulation:
Local oscillator
m(t)
Accosωct
Ac[1+m(t)]cosωct
BPFEnvelope detectorNoisy s(t) m’(t)
BPFNoisy s(t) m’(t)
cosωct
LPF
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• Spectrum of AM waveform:
f
M(f)
B-B
1
fc
f
S(f)
1/21/2δ(f-f0)
1/2M(f-fc)
Where Ac=1
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• 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.
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• In AM signaling,we have:
E=<m2(t)>/[1+ <m2(t)>]*100%m(t)
t
t
s(t) Amax
Amin
Ac
s(t)
Ac[1+m(t)]
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• 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 1/2Ac
2/50=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 .
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• 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:
s’(t)+n(t)Accosωct
channelm(t) s(t)
BPF
cosωct
LPF
m’(t)
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• 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)
±
])()([)]([
tmtmAtmg cSSB
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h(t)=1/πt
and
H(f)=-f for f>0 and H(f)=f for f<0
M(f)
B f
1
│G(f)│
B
2Ac
f
USSB
│S(f)│
ffc
Ac
fc+B-fc-fc-B
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• 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:
-90o phase shift across band of m(t)
L.O.
-90o
m(t)
m(t)
m^(t)
s(t)
SSB signal
Phasing method
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• Diagram of SSB system(con.):
• Demodulation:
Accosωct
Sideband filterm(t)
s(t)
SSB signal
Filter method
s’(t)+n(t)
channels(t)
BPF
cosωct
LPFSSB signal
m’(t)
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• 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.
DSB modulator
VSB filter(Bandpass filter)
m(t) s(t)
DSB signal
Baseband signal
sVSB(t)
VSB signal
Hv(f)
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• Spectrum of VSB signal:
fc
f
S(f)
1/21/2M(f-fc)
Where Ac=1,DSB signal
VSB Filter(USSB)
SVSB(f)
Hv(f-fc)+Hv(f+fc) fΔ
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• 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:
Low pass filter
sVSB(t)
cosωct
m’(t)
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• 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.
3.Non-linear modulation----angular modulation
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• 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
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• Generation of FM from a phase modulator and vice versa
Integratorgain=Df /Dp
Phasemodulator
mf(t)mp(t) s(t)
FM signal out
FM from a phase modulator
Differentiatorgain=Dp/Df
Frequencymodulator
mp(t)mf(t) s(t)
PM signal out
PM from a frequency modulator
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• 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]}
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• 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.
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• 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.
Spectra of angular modulated signals
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• 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.
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• 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.
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• 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
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• 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
Integratorgain=Df
Σ
Local oscillator
-90o
+
-m(t) s(t)
NBFM
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• 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).
)]}(2
[)](2
[{2
)(2
cf
mcf
mf
c ffD
fffD
fD
fP A
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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.
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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
4. 调制系统的比较(抗噪声性能分析和比较)
sm(t)+n(t)
BPF LPFmo(t)+no(t)
解调器
sm(t)+ni(t)
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• 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) 〉 = 〈 ns
2(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= 〈 ni
2(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)>/ <no
2(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。
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• 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:• 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
si(t)+ni(t)
BPF
cosωct
LPFmo(t)
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• 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。
vin R1 C1 R1
C1
v vout
t
s(t)
Ac[1+m(t)]
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• 包络检测(续)s(t)
v
m(t)
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• 信噪比计算: Si =Ac2/2+Ac
2/2<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[ω
ct +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)>/[Ac
2+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。
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• 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
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• DSB-SC 系统的抗噪声性能si(t)=Acm(t)cosωct , Si= Ac
2/2<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?
±
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• 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
sFM(t) m(t)
信号项 噪声项 ni(t)
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• 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
a1
a2
Φ1Φ
Φ2
Φ2-Φ1
任意参考相位
a
a2
a1
Φ2Φ
Φ1
Φ1-Φ2
任意参考相位
图 a 图 b
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• 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。
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• 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< mf
2(t)>
输出噪声: no(t)= 1/(2πA)dns(t)/dt
求 ns(t) 的功率?
信号 噪声 Ans(t)
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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:〈 ns
2(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。
理想微分器d ns(t)/dtns(t)
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• 解调器的输出噪声功率为:No= <no
2(t) >= <ns’2(t) >/ 4π2A2 =1/ 4π2A2 0∫
fm P0(f) df
So/ No= 4A2 Df2 <m2(t) >/[8π2nofm
3]
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
f
P0(f)
B’
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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/ No
Si/Ni
DSB 同步检测
门限效应
a
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 n
oise(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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• 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
5. 频率分割复用( FDM )
复用 解复用
信号 1
信号 2
信号 n
信号 1
信号 2
信号 n
一路宽带信号
S(f)
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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• 模拟调制:带宽,抗噪声性能,比较总结: