第四章 模拟调制系统

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第四章 模拟调制系统 Powered By Docstoc
					          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?

                                                      48
•   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。
                                                   49
• 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
                                                                 50
•   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。




                                         51
• 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



                                                               52
•  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

                                                             53
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
                                              54
•   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:。。。。。。




                                55
总结:
• 模拟调制:带宽,抗噪声性能,比较




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