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howto:hambasics:sections:wavemodulationmath [2020/11/08 15:40] – va7fihowto:hambasics:sections:wavemodulationmath [2021/01/03 08:05] (current) – va7fi
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 ====== More Details: AM / FM ====== ====== More Details: AM / FM ======
- 
 Here are a few more details about the AM, SSB, and FM modulation schemes introduced on the [[wavemodulation |Wave Modulation]] page. Here are a few more details about the AM, SSB, and FM modulation schemes introduced on the [[wavemodulation |Wave Modulation]] page.
  
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 ====== AM ====== ====== AM ======
- 
 The resulting //Amplitude Modulated// radio wave is the **product** of the vertically shifted baseband signal and the radio carrier, which is also equal to the **sum** of the carrier and the two side bands: The resulting //Amplitude Modulated// radio wave is the **product** of the vertically shifted baseband signal and the radio carrier, which is also equal to the **sum** of the carrier and the two side bands:
  
-<WRAP centeralign> +\begin{align*}
- +
-\begin{align}+
 \Big(s(t)+1\Big) \times c(t) &= \Big(\cos(2 \pi f_s t) + 1\Big) \times \cos(2 \pi f_c t) \\ \Big(s(t)+1\Big) \times c(t) &= \Big(\cos(2 \pi f_s t) + 1\Big) \times \cos(2 \pi f_c t) \\
                  &=  \cos(2 \pi f_s t)\cos(2 \pi f_c t) + \cos(2 \pi f_c t)\\                  &=  \cos(2 \pi f_s t)\cos(2 \pi f_c t) + \cos(2 \pi f_c t)\\
                  &= \underbrace{\frac{1}{2} \cos\Big(2 \pi (f_c + f_s) t\Big)}_\text{USB} + \underbrace{\frac{1}{2} \cos\Big(2 \pi (f_c - f_s) t\Big)}_\text{LSB} + \underbrace{\cos(2 \pi f_c t)}_\text{Carrier}                  &= \underbrace{\frac{1}{2} \cos\Big(2 \pi (f_c + f_s) t\Big)}_\text{USB} + \underbrace{\frac{1}{2} \cos\Big(2 \pi (f_c - f_s) t\Big)}_\text{LSB} + \underbrace{\cos(2 \pi f_c t)}_\text{Carrier}
-\end{align} +\end{align*}
- +
-</WRAP>+
  
 In line 1, I distributed the bracket, which, in line 2, gave us the carrier (last term) and a product (first term).  To expend this product into the sum of the two side bands (line 3),  I added these two trig identities together: In line 1, I distributed the bracket, which, in line 2, gave us the carrier (last term) and a product (first term).  To expend this product into the sum of the two side bands (line 3),  I added these two trig identities together:
  
-<WRAP centeralign> +\begin{align*}
- +
-\begin{align}+
 \cos(A+B)  =& \cos(A)\cos(B) - \sin(A)\sin(B) \\ \cos(A+B)  =& \cos(A)\cos(B) - \sin(A)\sin(B) \\
-\cos(A-B)  =& \cos(A)\cos(B) + \sin(A)\sin(B) \\ +\cos(A-B)  =& \cos(A)\cos(B) + \sin(A)\sin(B) 
-\end{align} +\end{align*}
- +
-</WRAP>+
  
 Which gives: Which gives:
  
-<WRAP centeralign> +\begin{align*}
- +
-\begin{align}+
 &\cos(A+B) + \cos(A-B) = 2 \cos(A)\cos(B) \\ &\cos(A+B) + \cos(A-B) = 2 \cos(A)\cos(B) \\
 \Rightarrow &\cos(A)\cos(B) = \frac{1}{2}\cos(A+B) + \frac{1}{2}\cos(A-B) \Rightarrow &\cos(A)\cos(B) = \frac{1}{2}\cos(A+B) + \frac{1}{2}\cos(A-B)
-\end{align} +\end{align*}
- +
-</WRAP>+
  
 Use this animation to see what happens when you vary the individual frequencies.  You can use the check boxes to show or hide different waves. Use this animation to see what happens when you vary the individual frequencies.  You can use the check boxes to show or hide different waves.
  
-{{ggb>howto/hambasics/sections/am.ggb 800,405}}+FIXME: animation in wrong place 
 +{{ggb>/howto/hambasics/sections/am.ggb 800,405}}
  
 Some things to try: Some things to try:
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   * DC components (vertical shift) \$c_1\$ and \$c_2\$    * DC components (vertical shift) \$c_1\$ and \$c_2\$ 
  
-<WRAP centeralign> +\begin{align*}
- +
-\begin{align}+
 \Big( A_1 \cos(2 \pi f_1 t + \phi_1) + c_1 \Big) \times & \Big( A_2 \cos(2 \pi f_2 t + \phi_2) + c_2 \Big) \\ \\ \Big( A_1 \cos(2 \pi f_1 t + \phi_1) + c_1 \Big) \times & \Big( A_2 \cos(2 \pi f_2 t + \phi_2) + c_2 \Big) \\ \\
 = & A_1 A_2 \cos(2 \pi f_1 t + \phi_1) \cos(2 \pi f_2 t + \phi_2 ) \\ = & A_1 A_2 \cos(2 \pi f_1 t + \phi_1) \cos(2 \pi f_2 t + \phi_2 ) \\
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 = & \frac{A_1 A_2}{2} \cos(2 \pi (f_1+f_2) t + (\phi_1+\phi_2)) + \frac{A_1 A_2}{2} \cos(2 \pi (f_1-f_2) t + (\phi_1-\phi_2))  \\ = & \frac{A_1 A_2}{2} \cos(2 \pi (f_1+f_2) t + (\phi_1+\phi_2)) + \frac{A_1 A_2}{2} \cos(2 \pi (f_1-f_2) t + (\phi_1-\phi_2))  \\
 & + c_2 A_1 \cos(2 \pi f_1 t + \phi_1 ) + c_1 A_2 \cos(2 \pi f_2 t + \phi_2) +  c_1 c_2   & + c_2 A_1 \cos(2 \pi f_1 t + \phi_1 ) + c_1 A_2 \cos(2 \pi f_2 t + \phi_2) +  c_1 c_2  
-\end{align} +\end{align*}
- +
-</WRAP>+
  
 The last line looks like a real mess, but all it says is that the result is: The last line looks like a real mess, but all it says is that the result is:
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 Mathematically, FM is less intuitive and more complicated than AM to understand.  The first step is to modulate the frequency by adding a scaled baseband function to it: Mathematically, FM is less intuitive and more complicated than AM to understand.  The first step is to modulate the frequency by adding a scaled baseband function to it:
  
-<WRAP centeralign> +\$$2\pi f_c \quad \rightarrow \quad 2\pi f_c + 2\pi k s(t)\$$
-\$$2\pi f_c \quad \rightarrow \quad 2\pi f_c + 2\pi k s(t)\$$. +
-</WRAP>+
  
   * Here, \$f_c\$ is the frequency of the carrier, which is a constant (this is important),   * Here, \$f_c\$ is the frequency of the carrier, which is a constant (this is important),
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 Now, it might be tempting to simply substitute this sum in the wave like so: Now, it might be tempting to simply substitute this sum in the wave like so:
  
-<WRAP centeralign> 
 \$$ \cos(2\pi f_c t) \quad \rightarrow \quad \cos\Big(\big(2\pi f_c + 2\pi k s(t)\big) t\Big) \$$ \$$ \cos(2\pi f_c t) \quad \rightarrow \quad \cos\Big(\big(2\pi f_c + 2\pi k s(t)\big) t\Big) \$$
-</WRAP>+
 but that's not quite right because the frequency is derived from the change in angle.   but that's not quite right because the frequency is derived from the change in angle.  
 </WRAP> </WRAP>
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 To solve this properly, we need some calculus and deduce the angle from our new frequency: To solve this properly, we need some calculus and deduce the angle from our new frequency:
  
-<WRAP centeralign> 
 \$$ \frac{d}{dt}\theta(t) = 2\pi f_c + 2\pi k s(t) \qquad  \Rightarrow \qquad \theta(t) = 2\pi f_c t + 2\pi k \int_0^{t}s(\tau) d\tau \$$ \$$ \frac{d}{dt}\theta(t) = 2\pi f_c + 2\pi k s(t) \qquad  \Rightarrow \qquad \theta(t) = 2\pi f_c t + 2\pi k \int_0^{t}s(\tau) d\tau \$$
-</WRAP>+
  
 The frequency modulated transmission is actually given by: The frequency modulated transmission is actually given by:
  
-<WRAP centeralign> 
 \$$ \cos\Big(2\pi f_c t + 2\pi k \int_0^{t}s(\tau) d\tau\Big) \$$ \$$ \cos\Big(2\pi f_c t + 2\pi k \int_0^{t}s(\tau) d\tau\Big) \$$
-</WRAP>+
  
 In our particular example, with \$s(t) = \cos(2 \pi f_s t)\$, the modulated radio signal becomes: In our particular example, with \$s(t) = \cos(2 \pi f_s t)\$, the modulated radio signal becomes:
  
-<WRAP centeralign> +\begin{align*}
- +
-\begin{align}+
 \cos\Big(2 \pi f_c t + 2\pi k \int_0^{t}s(\tau)d\tau\Big) &= \cos\Big(2 \pi f_c t + 2\pi k \int_0^{t}\cos(2 \pi f_s \tau)d\tau\Big) \\ \cos\Big(2 \pi f_c t + 2\pi k \int_0^{t}s(\tau)d\tau\Big) &= \cos\Big(2 \pi f_c t + 2\pi k \int_0^{t}\cos(2 \pi f_s \tau)d\tau\Big) \\
                  &=  \cos\Big(2 \pi f_c t + k \sin(2 \pi f_s t)\Big)                  &=  \cos\Big(2 \pi f_c t + k \sin(2 \pi f_s t)\Big)
-\end{align}+\end{align*}
  
-</WRAP> 
  
 For more details about FM, see: [[http://www.ece.umd.edu/~tretter/commlab/c6713slides/ch8.pdf]] For more details about FM, see: [[http://www.ece.umd.edu/~tretter/commlab/c6713slides/ch8.pdf]]
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 Use this animation to see what happens when you vary the individual frequencies. You can use the check boxes to show or hide different waves. Use this animation to see what happens when you vary the individual frequencies. You can use the check boxes to show or hide different waves.
  
-{{ggb>howto/hambasics/sections/fm.ggb 800,350}}+FIXME: animation in wrong place 
 +{{ggb>/howto/hambasics/sections/fm.ggb 800,350}} 
  
 Some things to try: Some things to try:
howto/hambasics/sections/wavemodulationmath.1604878837.txt.gz · Last modified: 2020/11/08 15:40 by va7fi