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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：