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howto:hambasics:sections:wavemodulationmath [2021/01/03 07:42] – [Mixer] va7fihowto:hambasics:sections:wavemodulationmath [2021/01/03 08:05] (current) – va7fi
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 \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*}
  
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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/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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 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.1609688529.txt.gz · Last modified: 2021/01/03 07:42 by va7fi