howto:hambasics:sections:wavemodulationmath
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howto:hambasics:sections:wavemodulationmath [2020/11/08 00:37] – [FM] va7fi | howto:hambasics:sections:wavemodulationmath [2021/01/03 08:01] – [AM] va7fi | ||
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~~NOTOC~~ | ~~NOTOC~~ | ||
- | ====== 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) \\ | ||
& | & | ||
& | & | ||
- | \end{align} | + | \end{align*} |
- | + | ||
- | </ | + | |
In line 1, I distributed the bracket, which, in line 2, gave us the carrier (last term) and a product (first term). | In line 1, I distributed the bracket, which, in line 2, gave us the carrier (last term) and a product (first term). | ||
- | <WRAP centeralign> | + | \begin{align*} |
- | + | ||
- | \begin{align} | + | |
\cos(A+B) | \cos(A+B) | ||
\cos(A-B) | \cos(A-B) | ||
- | \end{align} | + | \end{align*} |
- | + | ||
- | </ | + | |
Which gives: | Which gives: | ||
- | <WRAP centeralign> | + | \begin{align*} |
- | + | ||
- | \begin{align} | + | |
& | & | ||
\Rightarrow & | \Rightarrow & | ||
- | \end{align} | + | \end{align*} |
- | + | ||
- | </ | + | |
Use this animation to see what happens when you vary the individual frequencies. | Use this animation to see what happens when you vary the individual frequencies. | ||
- | {{ggb> | + | {{ggb>/howto/ |
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*} |
- | + | ||
- | </ | + | |
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, | Mathematically, | ||
- | <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)\$$. | + | |
- | </ | + | |
* 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) \$$ | ||
- | </ | + | |
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. | ||
</ | </ | ||
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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 | \$$ \frac{d}{dt}\theta(t) = 2\pi f_c + 2\pi k s(t) \qquad | ||
- | </ | + | |
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) \$$ | ||
- | </ | + | |
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) \\ | ||
& | & | ||
- | \end{align} | + | \end{align*} |
- | </ | ||
For more details about FM, see: [[http:// | For more details about FM, see: [[http:// | ||
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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> | + | {{ggb>/howto/ |
Some things to try: | Some things to try: | ||
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* Decrease <fc # | * Decrease <fc # | ||
* Increase and decrease **k** to see the effect it has on the transmitted wave. The greater **k**, the more bandwidth the resulting signal uses. This dictates the difference between " | * Increase and decrease **k** to see the effect it has on the transmitted wave. The greater **k**, the more bandwidth the resulting signal uses. This dictates the difference between " | ||
- | |||
See [[https:// | See [[https:// | ||
+ | |||
====== PM ====== | ====== PM ====== | ||
//Phase Modulation// | //Phase Modulation// |
howto/hambasics/sections/wavemodulationmath.txt · Last modified: 2021/01/03 08:05 by va7fi