Sun Coast Amateur
Radio Club Society

User Tools

Site Tools

You are here: home » howto » hambasics » sections » wavemodulationmath
howto:hambasics:sections:wavemodulationmath

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
howto:hambasics:sections:wavemodulationmath [2021/01/03 07:40] – [AM] va7fihowto:hambasics:sections:wavemodulationmath [2021/01/03 08:05] (current) – va7fi
Line 26: Line 26:
 \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*}
  
Line 38: Line 38:
 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}}</WRAP>+FIXME: animation in wrong place 
 +{{ggb>/howto/hambasics/sections/am.ggb 800,405}}
  
 Some things to try: Some things to try:
Line 51: Line 52:
   * 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 ) \\
Line 58: Line 58:
 = & \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:
Line 73: Line 72:
 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),
Line 83: Line 80:
 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>
Line 91: Line 87:
 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]]
Line 114: Line 107:
 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.1609688427.txt.gz · Last modified: 2021/01/03 07:40 by va7fi