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| hambasics:sections:wavemodulationmath [2021/01/03 08:05] – created - external edit 127.0.0.1 | hambasics:sections:wavemodulationmath [2026/07/10 10:31] (current) – [FM] va7fi |
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| ~~NOTOC~~ | ====== More Optional Details ====== |
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| ====== 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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| For both AM and FM examples, we'll Let: | For both AM and FM examples, we'll let: |
| * \$c(t) = \cos(2 \pi f_c t)\$ be the <fc #4682b4>radio carrier</fc> with frequency \$f_c\$ | * \$c(t) = \cos(2 \pi f_c t)\$ be the <fc #4682b4>radio carrier</fc> with frequency \$f_c\$ |
| * \$s(t) = \cos(2 \pi f_s t)\$ be the <fc #ff0000>baseband audio signal</fc> with frequency \$f_s\$ | * \$s(t) = \cos(2 \pi f_s t)\$ be the <fc #ff0000>baseband audio signal</fc> with frequency \$f_s\$ |
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| FIXME: animation in wrong place | FIXME: animation in wrong place |
| {{ggb>/howto/hambasics/sections/am.ggb 800,405}} | {{ggb>/hambasics/sections/am.ggb 800,405}} |
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| Some things to try: | Some things to try: |
| * Set <fc #ff0000>\$f_s\$ at 10</fc> and <fc #4682b4>\$f_c\$ at 200</fc> and check only the transmitted signal. You can easily imagine what the envelope (the baseband signal) should be that produced that signal. But... | * Set <fc #ff0000>\$f_s\$ at 10</fc> and <fc #4682b4>\$f_c\$ at 200</fc> and check only the transmitted signal. You can easily imagine what the envelope should be, which is what the baseband signal is. But... |
| * Decrease <fc #4682b4>\$f_c\$</fc> slowly. At some point (around 20 or 30) the baseband signal becomes unrecoverable. This illustrates the point that to transmit a high frequency baseband, a higher frequency carrier is needed (at least 3 to 4 times the frequency of the baseband signal. This is why with digital signals, the higher the transfer speed, the higher the carrier frequency needs to be. | * Decrease <fc #4682b4>\$f_c\$</fc> slowly. At some point (around 20 or 30) the baseband signal becomes unrecoverable. This illustrates the point that to transmit a high frequency baseband, a higher frequency carrier is needed (at least 3 to 4 times the frequency of the baseband signal. This is why with digital signals, the higher the transfer speed, the higher the carrier frequency needs to be. |
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| <WRAP round alert box center 80%> | <WRAP round alert box center 80%> |
| Now, it might be tempting to simply substitute this sum in the wave like so: | It might be tempting to simply substitute this sum in the wave like so: |
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| \$$ \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> | </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 to deduce the angle from our new frequency: |
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| \$$ \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 \$$ |
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| FIXME: animation in wrong place | FIXME: animation in wrong place |
| {{ggb>/howto/hambasics/sections/fm.ggb 800,350}} | {{ggb>/hambasics/sections/fm.ggb 800,350}} |
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| Some things to try: | Some things to try: |
| * Set <fc #ff0000>\$f_s\$ at 10</fc> and <fc #4682b4>\$f_c\$ at 200</fc> and check only the transmitted signal. Notice how when the <fc #ff0000>baseband</fc> is high, the **transmitted wave** is "tight" (ie, its frequency is high), and vise-versa. But... | * Set <fc #ff0000>\$f_s\$ at 10</fc> and <fc #4682b4>\$f_c\$ at 200</fc> and check only the transmitted signal. Notice how when the <fc #ff0000>baseband</fc> is high, the **transmitted wave** is "tight" (ie, its frequency is high), and vise-versa. But... |
| * Decrease <fc #4682b4>\$f_c\$</fc> slowly. At some point (around 20 or 30) that pattern becomes unnoticeable. Again, this illustrates the point that to transmit a high frequency baseband, a higher frequency carrier is needed (at least 3 to 4 times the frequency of the baseband signal. This is why with digital signals, the higher the transfer speed, the higher the carrier frequency needs to be. | * Decrease <fc #4682b4>\$f_c\$</fc> slowly. At some point (around 20 or 30) that pattern becomes unnoticeable. Again, this illustrates the point that to transmit a high frequency baseband, a higher frequency carrier is needed (at least 3 to 4 times the frequency of the baseband signal). This is why with digital signals, the higher the transfer speed, the higher the carrier frequency needs to be. |
| * 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 "Narrow Band FM" and "Wide Band FM". | * 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 "Narrow Band FM" and "Wide Band FM". |
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