Have you ever wondered how radio stations work? What about WiFi, or cell phones? These technologies (and countless others) all use modulation to send data at the same time, without interfering with each other! If electromagnetics wasn’t part of your college coursework, or you haven’t spent hours browsing Wikipedia, modulation of waveforms can be difficult to understand. To understand how wireless data transfer happens, we need knowledge of a handful of topics, which I plan to cover briefly in this blog post. For details, check out the full article.

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Sending a signal that is a pure sine wave is called a "tone". It carries no real information, and doesn't sound that great either.

Single tone (time domain)

How about a signal composed of many tones of varying frequencies? We can see the signal is too complex to understand in the time domain.

Multi-tone signal (time domain)

For complex signals, engineers use a different way of graphing a signal in the frequency domain by using something called a Fourier transform. Let's see what our three signals above look like in this representation (skipping to the solution). Instead of plotting a signal’s voltage in time, we are plotting the power of the signal by frequency.

Multi-tone signal (frequency domain) of 10 tones, of varying amplitude

Notice the clear spikes? That is the mathematical representation of a sine wave at that particular frequency (x-axis). The multi-tone signal that was unreadable in the time domain has been clearly chopped into small spikes, representing all the frequencies that were summed to create the signal. A final example would be to show an audio signal.

Audio Signal

This is how the spectrum of most signals appear, especially analog ones. The human voice and instruments do not play as discreet frequencies, and thus there is frequency content over an entire range. In theory, we can represent this analog signal as the sum of an infinite number of tones added together. That’s a fun one to wrap your head around.

Modulation is what takes a signal from low frequencies (we call this the message) and pulls it up to a higher frequency (the carrier). The idea is simple: Multiply your message by a high frequency carrier, such as 680kHz. Let's look at a few mathematical relationships. In this case, theta is the message and phi is the carrier.

The nifty relationships above show us that two signals multiplied can be represented as two signals added together! What does an audio spectrum look like when it's been modulated? We basically shift the signal up and around the carrier frequency.

Modulation of a sound clip to 700 kHz

Just as expected, we see two signals. One is carrier + message, one is carrier - message (even notice how it is reversed). We can change the carrier frequency (radio station) to transmit a second, different signal at the same time. This is basically how a transmitter works in a radio tower. Now let's talk about receivers. Fortunately, it’s easy to bring our audio signal back to "baseband" (near 0 Hz instead of the carrier). We simply multiply everything by the carrier again. More math!

That's a bunch of cosines, parenthesis, and f's all over the place. But it's correct, and we see that there are four signals that result from it. f_{C} is the carrier frequency, and f_{M} is the message.

- ¼ power signal, (2*carrier + message)
- ¼ power signal, (message)
- ¼ power signal, (2*carrier – message)
- ¼ power signal, (-message)

Let’s immediately disregard the last term with a negative frequency. It is a mathematical artifact which occurs quite often when talking about modulation and the math involved, but isn’t really, well… real. The two signals at double the carrier (assuming the carrier is much larger than the message, they are almost the same) can be filtered out with a low pass filter, leaving us with the original message at 25% power. Here's a picture of it, but backwards. Using this process, we can now hear the audio message that was transmitted at the 700 kHz carrier!

In summary, the purpose of this post was to give a 30,000 foot view of how radio transmission and signal modulation works. By taking multiple audio (or baseband) signals and mathematically multiplying them by different higher frequencies (the carrier), we can successfully transmit multiple signals over the same channel (our atmosphere) without interference. Multiplying it by the carrier again brings the modulated signal back to baseband, and a low pass filter and amplifier clean up and magnify the signal for our listening pleasure! I highly recommend reading the full length article for more details, fun facts about the FCC, as well as more in depth examples of signals being modulated for better understanding.