[Circuit] Modulation

電路知識：Modulation
工具：Qucs

簡介：發射端將低頻訊號處理成高頻訊號以後，再傳送出去

主要的調變方法

• 振幅調變（amplitude modulation，AM）
• 頻率調變（Frequency modulation，FM）
• 相位調變（Phase Modulation，PM）

AM (振幅調變)

$$\begin{array}{rl}c(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ m(t)& =Mcos(2\pi f_{m}t+\varphi )\\ AM(t)& =[1+m(t)]\cdot c(t)\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c)+\frac{A_{c}M}{2}[cos(2\pi (f_c-f_m)t+{\varphi }_c-\varphi )+cos(2\pi (f_c+f_m)t+{\varphi }_c+\varphi )]\\ M& <1\end{array}$$

$$\begin{array}{rl}AM(t)& =[1+m(t)]\cdot c(t)\\ & =[1+Mcos(2\pi f_{m}t+\varphi )]\cdot A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c)+Mcos(2\pi f_{m}t+\varphi )\cdot A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c)+A_{c}Mcos(2\pi f_{c}t+{\varphi }_c)cos(2\pi f_{m}t+\varphi )\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c)+\frac{A_{c}M}{2}[cos(2\pi f_{c}t+{\varphi }_c-(2\pi f_{m}t+\varphi ))+cos(2\pi f_{c}t+{\varphi }_c+(2\pi f_{m}t+\varphi ))]\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c)+\frac{A_{c}M}{2}[cos(2\pi (f_c-f_m)t+{\varphi }_c-\varphi )+cos(2\pi (f_c+f_m)t+{\varphi }_c+\varphi )]\end{array}$$ 必要條件：$1+m(t)>1$，故 $M<1$，否則會 overmodulation，訊號就不對了，因會使得輸出振幅為負的

頻寬 $BW=2f_m$

電路驗證

角調變 定義

$$\begin{array}{rl}c(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ XM(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c+{\varphi }_m(t))\\ \\ \theta (t)& =2\pi f_{c}t+{\varphi }_c+{\varphi }_m(t)\\ f(t)& =f_c+\frac{1}{2\pi }\frac{\mathrm{d}{\varphi }_m(t)}{\mathrm{d}t}\end{array}$$ 相位偏差：${\varphi }_m(t)$
頻率偏差 $\mathrm{\Delta }f$：$\frac{1}{2\pi }\frac{\mathrm{d}{\varphi }_m(t)}{\mathrm{d}t}$

$$\begin{array}{rl}\frac{\mathrm{d}\theta (t)}{\mathrm{d}t}& =w(t)\\ & =2\pi f(t)\\ & =2\pi f_c+\frac{\mathrm{d}{\varphi }_m(t)}{\mathrm{d}t}\\ f(t)& =f_c+\frac{1}{2\pi }\frac{\mathrm{d}{\varphi }_m(t)}{\mathrm{d}t}\end{array}$$

FM (頻率調變)

$$\begin{array}{rl}c(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ m(t)& =Mcos(2\pi f_{m}t+\varphi )\\ \\ {\varphi }_m(t)& =k_f{\int }_0^{t}m(\tau )d\tau \\ FM(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c+{\varphi }_m(t))\\ & =A_c\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_k(\beta )cos(2\pi (f_c+k{f}_m)t+{\varphi }_c+k\varphi )\\ \beta & =\frac{k_{f}M}{2\pi f_m}\\ \mathrm{\Delta }f& =\beta f_m\end{array}$$ $k_f$ 為自定頻率偏差函數，單位為 Hz/Volt

根據角調變
$$\begin{array}{rl}\frac{\mathrm{d}{\varphi }_m(t)}{\mathrm{d}t}& =k_{f}m(t)\\ {\varphi }_m(t)& =k_f{\int }_0^{t}m(\tau )d\tau \\ FM(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c+{\varphi }_m(t))\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+k_f{\int }_0^{t}m(\tau )d\tau )\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+k_f{\int }_0^{t}Mcos(2\pi f_m\tau +\varphi ))\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+k_f\frac{1}{2\pi f_m}(Msin(2\pi f_{m}t+\varphi )))\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+\frac{k_{f}M}{2\pi f_m}sin(2\pi f_{m}t+\varphi ))\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+\beta sin(2\pi f_{m}t+\varphi ))\\ & =A_c[cos(2\pi f_{c}t+{\varphi }_c)cos(\beta sin(2\pi f_{m}t+\varphi ))-sin(2\pi f_{c}t+{\varphi }_c)sin(\beta sin(2\pi f_{m}t+\varphi ))]\end{array}$$ 用上面倒數第二式求 $\mathrm{\Delta }f$
$$\begin{array}{rl}\frac{\mathrm{d}\theta }{\mathrm{d}t}& =\frac{\mathrm{d}(2\pi f_{c}t+{\varphi }_c+\beta sin(2\pi f_{m}t+\varphi ))}{\mathrm{d}t}\\ & =2\pi f_c+2\pi f_m\beta cos(2\pi f_{m}t+\varphi )\\ & =w(t)\\ & =2\pi f(t)\\ f(t)& =f_c+\beta f_{m}cos(2\pi f_{m}t+\varphi )\\ & =f_c+\mathrm{\Delta }f\cdot cos(2\pi f_{m}t+\varphi )\end{array}$$ 借由 bessel function
$$\begin{array}{rl}cos(zsin\theta )& =J_0(z)+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(z)cos(2k\theta )\\ sin(zsin\theta )& =2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(z)sin((2k+1)\theta )\\ J_{-n}(z)& =(-1{)}^n{J}_n(z)\end{array}$$ 可得
$$\begin{array}{rl}& cos(2\pi f_{c}t+{\varphi }_c)cos(\beta sin(2\pi f_{m}t+\varphi ))\\ =& cos(2\pi f_{c}t+{\varphi }_c)[J_0(\beta )+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2k(2\pi f_{m}t+\varphi ))]\\ =& J_0(\beta )cos(2\pi f_{c}t+{\varphi }_c)+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi f_{c}t+{\varphi }_c)cos(2k(2\pi f_{m}t+\varphi ))\\ =& J_0(\beta )cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )[cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )+cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi )]\\ =& J_0(\beta )cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi )\\ =& J_0(\beta )cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )+\sum _{k=-\mathrm{\infty }}^{-1}(-1{)}^{2k}{J}_{2k}(\beta )cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )\\ =& \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )\\ \\ & sin(2\pi f_{c}t+{\varphi }_c)sin(\beta sin(2\pi f_{m}t+\varphi ))\\ =& sin(2\pi f_{c}t+{\varphi }_c)[2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )sin((2k+1)(2\pi f_{m}t+\varphi ))]\\ =& 2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )sin(2\pi f_{c}t+{\varphi }_c)sin((2k+1)(2\pi f_{m}t+\varphi ))\\ =& \sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )[cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi )-cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )]\\ =& \sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi )-\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )(cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )]\\ =& \sum _{k=-\mathrm{\infty }}^{-1}(-1{)}^{2k+1}{J}_{2k+1}(\beta )cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )-\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(\beta )(cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )]\\ =& -\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k+1}(\beta )(cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )\end{array}$$ 故可得 $$\begin{array}{rl}FM(t)& =A_c[cos(2\pi f_{c}t+{\varphi }_c)cos(\beta sin(2\pi f_{m}t+\varphi ))-sin(2\pi f_{c}t+{\varphi }_c)sin(\beta sin(2\pi f_{m}t+\varphi ))]\\ & =A_c[\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k}(\beta )cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi )+\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k+1}(\beta )(cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi )]\\ & =A_c[\sum _{k=even}{J}_k(\beta )cos(2\pi (f_c+k{f}_m)t+{\varphi }_c+k\varphi )+\sum _{k=odd}{J}_k(\beta )(cos(2\pi (f_c+k{f}_m)t+{\varphi }_c+k\varphi )]\\ & =A_c\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_k(\beta )cos(2\pi (f_c+k{f}_m)t+{\varphi }_c+k\varphi )\end{array}$$

當 $\beta \le 0.2$ 且 $k\ge 2$，$J_k(\beta )\cong 0$，則頻寬 $BW\cong 2f_m$

電路驗證

PM (相位調變)

$$\begin{array}{rl}c(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c)\\ m(t)& =Mcos(2\pi f_{m}t+\varphi )\\ \\ PM(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c+m(t))\\ & =A_c\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_k(M)cos(2\pi (f_c-k{f}_m)t+{\varphi }_c-k\varphi +\frac{k}{2}\pi )\\ \mathrm{\Delta }f& =M{f}_m\end{array}$$

$$\begin{array}{rl}PM(t)& =A_{c}cos(2\pi f_{c}t+{\varphi }_c+m(t))\\ & =A_{c}cos(2\pi f_{c}t+{\varphi }_c+Mcos(2\pi f_{m}t+\varphi ))\\ & =A_c[cos(2\pi f_{c}t+{\varphi }_c)cos(Mcos(2\pi f_{m}t+\varphi ))-sin(2\pi f_{c}t+{\varphi }_c)sin(Mcos(2\pi f_{m}t+\varphi ))]\end{array}$$ 用上面倒數第二式求 $\mathrm{\Delta }f$
$$\begin{array}{rl}\frac{\mathrm{d}\theta }{\mathrm{d}t}& =\frac{\mathrm{d}(2\pi f_{c}t+{\varphi }_c+Mcos(2\pi f_{m}t+\varphi ))}{\mathrm{d}t}\\ & =2\pi f_c-2\pi f_{m}Msin(2\pi f_{m}t+\varphi )\\ & =w(t)\\ & =2\pi f(t)\\ f(t)& =f_c-M{f}_{m}sin(2\pi f_{m}t+\varphi )\\ & =f_c+\mathrm{\Delta }f\cdot sin(2\pi f_{m}t+\varphi )\end{array}$$ 借由 bessel function
$$\begin{array}{rl}cos(zsin\theta )& =J_0(z)+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(z)cos(2k\theta )\\ sin(zsin\theta )& =2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(z)sin((2k+1)\theta )\\ J_{-n}(z)& =(-1{)}^n{J}_n(z)\end{array}$$ 可得
$$\begin{array}{rl}& cos(2\pi f_{c}t+{\varphi }_c)cos(Mcos(2\pi f_{m}t+\varphi ))\\ =& cos(2\pi f_{c}t+{\varphi }_c)cos(Msin(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))\\ =& cos(2\pi f_{c}t+{\varphi }_c)[J_0(M)+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)cos(2k(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))]\\ =& J_0(M)cos(2\pi f_{c}t+{\varphi }_c)+2\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi f_{c}t+{\varphi }_c)cos(2k(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))\\ =& J_0(M)cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)[cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )+cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi -k\pi )]\\ =& J_0(M)cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi (f_c+2k{f}_m)t+{\varphi }_c+2k\varphi -k\pi )\\ =& J_0(M)cos(2\pi f_{c}t+{\varphi }_c)+\sum _{k=1}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )+\sum _{k=-\mathrm{\infty }}^{-1}(-1{)}^{2k}{J}_{2k}(M)cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )\\ =& \sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )\\ \\ & sin(2\pi f_{c}t+{\varphi }_c)sin(Mcos(2\pi f_{m}t+\varphi ))\\ =& sin(2\pi f_{c}t+{\varphi }_c)sin(Msin(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))\\ =& sin(2\pi f_{c}t+{\varphi }_c)[2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)sin((2k+1)(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))]\\ =& 2\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)sin(2\pi f_{c}t+{\varphi }_c)sin((2k+1)(\frac{\pi }{2}-2\pi f_{m}t-\varphi ))\\ =& \sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)[cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi -(2k+1)\frac{\pi }{2})-cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})]\\ =& \sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)cos(2\pi (f_c+(2k+1)f_m)t+{\varphi }_c+(2k+1)\varphi -(2k+1)\frac{\pi }{2})-\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)(cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})\\ =& \sum _{k=-\mathrm{\infty }}^{-1}(-1{)}^{2k+1}{J}_{2k+1}(M)cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})-\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+1}(M)(cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})\\ =& -\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k+1}(M)(cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})\end{array}$$ 故可得 $$\begin{array}{rl}PM(t)& =A_c[cos(2\pi f_{c}t+{\varphi }_c)cos(Mcos(2\pi f_{m}t+\varphi ))-sin(2\pi f_{c}t+{\varphi }_c)sin(Mcos(2\pi f_{m}t+\varphi ))]\\ & =A_c[\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k}(M)cos(2\pi (f_c-2k{f}_m)t+{\varphi }_c-2k\varphi +k\pi )+\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{2k+1}(M)(cos(2\pi (f_c-(2k+1)f_m)t+{\varphi }_c-(2k+1)\varphi +(2k+1)\frac{\pi }{2})]\\ & =A_c[\sum _{k=even}{J}_k(M)cos(2\pi (f_c-k{f}_m)t+{\varphi }_c-k\varphi +\frac{k}{2}\pi )+\sum _{k=odd}{J}_k(M)(cos(2\pi (f_c-k{f}_m)t+{\varphi }_c-k\varphi +\frac{k}{2}\pi )]\\ & =A_c\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_k(M)cos(2\pi (f_c-k{f}_m)t+{\varphi }_c-k\varphi +\frac{k}{2}\pi )\end{array}$$

當 $M\le 0.2$ 且 $k\ge 2$，$J_k(\beta )\cong 0$，則頻寬 $BW\cong 2f_m$

電路驗證

參考

類比與數位調變
何謂 I/Q 資料 (I/Q Data)？
瞭解無線的基礎科技
電視RF訊號, 為何影像要用AM調變 / 聲音要用FM調變
頻率調製
振幅調變
Why is NTSC color carrier frequency 3.57954545 MHz and not some other number that can be remembered easily?
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