Am, FM, PM modulation technology

AM modulation — Amplitude modulation

1. Concept

   Make the amplitude of the carrier wave change according to the change law of the required transmission signal , But the modulation method with constant frequency .

2. Advantages and disadvantages

   Long propagation distance , But the anti-interference ability is poor .

3. classification

   Common amplitude modulation ：AM

   Bilateral band amplitude modulation ：DSB-AM

   Single sideband amplitude modulation ：SSB_AM

   Residual sideband banner ：VSB_AM

4. Modulation signal expression $transfersystemLetterNumber：U_{\Omega }(t)=U_{\Omega m}\cos \Omega t$

$$loadwaveLetterNumber{：U}_c(t)=U_{\text{cm}}cos(w_{c}t)$$

because AM The frequency of the modulation does not change , Adopt the frequency of carrier signal , The amplitude varies with the transmitted signal , therefore AM The expression of the modulated signal is ：

$$hastransferLetterNumber：U_{\text{AM}}(t)=U_m(t)cos(w_{c}t)$$

$$=(U_{\text{cm}}{+K_{a}U}_{\Omega m}\cos \Omega t)cos(w_{c}t)$$

$$=U_{\text{cm}}(1+K_a\frac{U_{\Omega m}}{U_{\text{cm}}}\cos \Omega t)cos(w_{c}t)$$

among $m_a$ Is the amplitude modulation coefficient ：$m_a$=$K_a\frac{U_{\Omega m}}{U_{\text{cm}}}$

Maximum amplitude of AM signal ：$U_m$(max)=($U_{\text{cm}}(1+m_a$)

Minimum amplitude of AM signal ：$U_m$(min)=($U_{\text{cm}}(1-m_a$)

So when $m_a$>1 when , Over modulation will occur , That is, the minimum value of AM signal is negative .

take $U_{\text{AM}}(t)=$$U_{\text{cm}}$(1 + $m_{\text{a}}$cos$\Omega$ t)cos($w_{\text{c}}t)$ Continue to expand to get :

$$U_{\text{AM}}(t)=U_{\text{cm}}cos(w_{c}t)+\frac{1}{2}\text{ma}{U}_{\text{cm}}cos(w_c+\Omega )t+\frac{1}{2}\text{ma}{U}_{\text{cm}}cos(w_c-\Omega )t$$

Therefore, it is known that the modulated wave contains three frequency components $w_c、w_c+\Omega (Onedgefrequency)$、$w_c-\Omega$( Lower sideband )

FM modulation ---- Frequency modulation

1. Concept

   The amplitude of the carrier wave does not change , The instantaneous angular frequency changes linearly with the modulation signal .

2. Advantages and disadvantages

   Strong anti-interference , But the transmission distance is short .

3. Modulation signal expression

   $transfersystemLetterNumber：U_{\Omega }(t)=U_{\Omega m}\cos (\Omega t)$

$$loadwaveLetterNumber{：U}_c(t)=U_{\text{cm}}cos(w_{c}t)$$

FM The instantaneous angular frequency of the modulation is ：

$$w_f(t)=w_c+k_f{U}_{\Omega }(t)=w_c+k_f{U}_{\Omega m}\cos \Omega t=w_c+\mathrm{\Delta }{w}_{\text{fm}}\cos \Omega t$$

among ,$w_c$ Is the carrier angular frequency ;

$k_f$ Is the frequency modulation sensitivity , Indicates the frequency change caused by unit modulation signal amplitude , Unit is rad/s.V perhaps hz/V;

$\mathrm{\Delta }{w}_{\text{fm}}$ Is the maximum angular frequency offset of FM wave , Express FM Amplitude of wave frequency oscillation ;$\mathrm{\Delta }{w}_{\text{fm}}$=$k_f{U}_{\Omega m}$

$transferfrequencysystemCount{m}_f=\frac{\mathrm{\Delta }{w}_{\text{fm}}}{\Omega }=\frac{k_f{U}_{\Omega m}}{\Omega }=\frac{\mathrm{\Delta }{f}_m}{F}=\mathrm{\Delta }{\phi }_{\text{fm}}$, Add the additional maximum phase offset to the phase of the carrier signal during time-frequency modulation , And $U_{\Omega m}$ In direct proportion to , And $\Omega$ In inverse proportion .

Therefore, the signal has been adjusted

$$U_{\text{fm}}(t)=U_{\text{cm}}\cos (w_f(t)\ast t)=U_{\text{cm}}\cos (w_{c}t+m_{f}sin(\Omega t))$$

Converted to $U_{\text{fm}}(t)=U_{\text{cm}}\cos (w_f(t)\ast t)=U_{\text{cm}}\cos (w_{c}t+k_f{\int }_0^t{U}_{\Omega }(t)dt)$

Come to the conclusion , FM time , The instantaneous angular frequency changes linearly with the modulated signal , The change of instantaneous phase is linear with the integral of modulated signal . FM time , Frequency offset reflects the change law of modulated signal , The phase offset is proportional to the integral of the modulated signal .

From the frequency modulation waveform , The waveform of FM wave is equal amplitude density wave , The density of the waveform reflects the magnitude of the instantaneous angular frequency of the FM wave , That is, the size of the modulated signal .

PM modulation — Phase modulation

1. Concept

   The phase of the carrier varies linearly with the modulated signal .

2. expression

$$transfersystemLetterNumber：U_{\Omega }(t)=U_{\Omega m}\cos (\Omega t)$$

$$loadwaveLetterNumber{：U}_c(t)=U_{\text{cm}}cos(w_{c}t)$$

Instantaneous phase of phase modulated signal ：

$$\phi (t)=w_{c}t+k_p{U}_{\Omega }(t)=w_{c}t+k_p{U}_{\Omega m}\cos \Omega t$$

The instantaneous angular frequency is ：

$$w(t)={\frac{d\phi (t)}{\text{dt}}=w_c+k_p\frac{{dU}_{\Omega }(t)}{\text{dt}}=w}_c+k_p{U}_{\Omega }(t)$$

among ,$k_p$ Is the modulation coefficient .

The general expression of phase modulated wave can be calculated ：

$U_{pm}(t)=U_{\text{cm}}\cos (\phi (t))=U_{\text{cm}}\cos (w_{c}t+k_p{U}_{\Omega }(t))$

3. The difference between frequency modulation and phase modulation

   Frequency modulation and phase modulation will cause the carrier to change in frequency and phase , But the law of their changes is different , Frequency modulation is that the angular frequency of the carrier varies with the modulated signal , Phase modulation means that the phase of the carrier wave changes with the modulation signal .