# How Does Signal Modulation Work?

Blog Post created by mike1305 on Sep 1, 2016

To understand how wireless data transfer happens, we need to understand:

•    What is frequency?
•    Message / Data Signals
•    time representation
•    frequency representation, and why is it important?
•    How do filters work?
•    FCC Communication bands
•    Modulation and demodulation

You can spend years at University learning these subjects in depth (or on Wikipedia, if that’s your style!). This is designed to be a flash flood of knowledge. This was originally put together as a PowerPoint for non-EE students in my senior project group who were curious about our lingo when we talk about “900 MHz” or “2.4 GHz” or “Frequency Hopping”. As such, it is not complete, thorough, and skips many details that one would include in a professional analysis of a system. This is only to provide a concept of wireless transmission.

What is Frequency?

Frequency is the unit describing how often something oscillates, or goes back and forth. Units are Hertz (Hz), or the inverse of a second. Something oscillating 60 times per second has frequency 60 Hz. For our purposes, we are going to focus on audio waves (oscillation of air pressure) and how it gets broadcasted from a radio station to your car in the range of hundreds of kilohertz (or any AM radio station). Any wave has a frequency – light, for example. Generally light and other higher frequency waves (e.g. x-rays, gamma rays, microwaves) are represented by their wavelength, not frequency. For example, green light is around 400 nanometers. Here is a picture showing the relationship of units on a traveling wave:

Basic units of a sine wave.

Assuming constant speed of the signal, wavelength and frequency are interchangeable. That is outside the scope of this article, though.

Message Signals of Varying Complexity

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. Here is an image of a sine wave, with time on the X axis and voltage on the Y axis. This is 150 Hz for reference.

Single tone signal (time domain)

Okay, so why am I showing you this? Let’s take a look at increasingly complex signals in the time domain. Here is a two tone signal (two tones, added together). It is the same sine wave above, added together with another sine wave with twice the frequency, 300 Hz.

Dual tone signal (time domain)

How about a signal composed of many tones of varying frequencies:

Multi-tone signal (time domain)

It’s starting to get a bit hairy. The only real information you can gather from that is voltage level at a specified time. That’s the essence of a message, and extremely important – but makes for difficult analysis, and even more difficult for understanding the way modulation works. This is why you may want to use a different way of graphing a signal: the frequency domain. It is a representation of how strong the signal is over a range of frequencies. Let’s look.

Why is the Frequency Spectrum of a Signal Important?

There is a precise mathematical operation to convert a chunk of a signal into the frequency domain. It is dense, difficult, and takes practice to master. I even struggle with convolution of non-trivial signals regularly. Regardless, 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.

Single tone signal (frequency domain)

Dual tone signal (frequency domain)

Multi-tone signal (frequency domain)

Notice the clear spikes? That is the mathematical representation of a sine wave at that particular frequency (X-axis). Ideally, these spikes would be infinitely narrow (width) and infinitely tall, but due the techniques used by my Spice software, it is imperfect. This is called an impulse signal. Read more on this here! For the tone, we see one spike at 150 Hz. The dual tone has two spikes, 150 Hz and 300 Hz. 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. In the below picture, I have taken a 15 second sample of the song “White Room” by Cream. Don’t worry, no microphones were damaged during Eric Clapton’s guitar solo!

Audio Signal

This is how 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 (even though some of that content is almost inaudible). This range is taken from 3 Hz to 20 kHz, the approximate range of the human ear. Bass notes are lower in the range, while treble is higher. The Y-scale is represented in dB, which is a unitless representation of proportion. In essence, the higher the dB value, the more of that frequency is in the signal.

In theory, we can represent this analog signal as the sum of an infinite number of tones added together.

Filters!

Hopefully the graphical representation of frequency domains will shed some light on filter design. There are four types of filters:

•    Low Pass filter: all frequencies over the “cutoff” are removed.
•    High Pass filter: all frequencies under the “cutoff” are removed.
•    Band Pass filter: All frequencies outside a distance from the “center” are removed.
•    Band Stop filter: All frequencies within a distance from the “center” are removed.

Clockwise: Band Pass, High Pass, Low Pass filters

The “3dB” point is where signal output is reduced by ~30%. It has to do with how “log” magnitude is calculated (dB is a log scale):

x [dB] = 10 * log(x[linear])
x [linear] = 10^(x[dB]/10)

Based on this, a gain of 0.7 [linear] is approximately -3.0dB (and change). It’s what is referred to as the cutoff frequency of a filter. A practical example of this is your car stereo, which may include a “crossover”. This is a special filter design that routes low frequencies to your woofer, high ones to the tweeters, etc. This is very important in radio receivers.

FCC Communication Bands

The FCC and other organizations worldwide have agreed that it would be absolute chaos to allow anyone to use any frequency for their own use. Thus, there are special allocations of frequency ranges for different uses. Examples include FM radio, AM radio, WiFi, cell phones, maritime communications, air traffic control, HAM radios, walkie talkies, military communications, police radios, and the list goes on. We haven’t even talked about satellites or space communication, either! It’s a crazy world out there and thankfully the FCC helps organize it all. A quick Google search will provide you a more detailed image and tables if you’re curious.

The FCC Spectral Allocation Table

The FCC has left a few bands open for low range personal use, hobbyists, and other general use in the “ISM bands” (Industrial, Scientific, Medical). This is where WiFi, walkie talkies, wireless sensors, and other commercial devices operate. Let’s talk frequencies again! The human ear has a range of 20 Hz to 20 kHz. What if our AM talk station is 680 kHz? How does the radio tower get the sound up to that frequency? How does it not interfere with other stations? How does the receiver bring the signal frequency back to an audible range?

Modulation

Let’s step away from the frequency domain and go back into the time domain. I am again making generous use of my earlier disclaimer: this is over-simplified and skips many details! This is only to get the concept. The reason I say this is because the math works out best in the time domain, and a graphical representation is best served in the frequency domain.

Modulation is what takes a signal from low frequencies (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 680 kHz. Voila, that’s AM radio! Wait, is it really that easy? Let’s look at a few mathematical relationships. In this case, theta is the message (the audible stuff) and phi is the carrier (the AM radio frequency, for example).

Our AM solution involves multiplying signals, but that’s hard to imagine in the time or frequency domain, since we only have seen what tones look like. But the nifty relationships above show us that two signals multiplied can be represented as two signals added together! Now it’s easy to plot a multiplied signal in the frequency domain.

A single tone (150 Hz) modulated on a carrier (1000 Hz)

In this picture, we have multiplied a 150 Hz tone with a 1000 Hz carrier. The table above shows us to expect two, half-powered signals at 1000-150 and 1000+150 Hz, 850 Hz and 1150 Hz. What does our sound byte look like when it’s been modulated?

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).

Here is a crude image of an AM frequency spectrum and signal content.

Demodulation

Now let’s talk about receivers. All signals start at the antenna, which sees all signals at the same time as one big jumbled mess. It isn’t the antenna’s job to sort through the mess of data it is picking up, but that of the tuner and other hardware. The theory of demodulating a signal is identical to modulating it, conveniently enough! To bring our audio signal back to “baseband” where it can be sent to a speaker, we multiply everything by the carrier again.

That’s a bunch of math, parenthesis, and f’s all over the place. But it’s correct, and we see that there are four signals that result from it:

•    1/4 power signal, (2*carrier + message)
•    1/4 power signal, (message)
•    1/4 power signal, (2*carrier – message)
•    1/4 power signal, (-message)

Let’s immediately disregard the term with a negative frequency. It is a mathematical artifact which occurs quite often when talking about modulation and the math involved. 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, which will block all higher frequency content of a signal. That just leaves us with the original message, which can be boosted with an amplifier and then sent to a speaker. Cool! Here’s a picture of it, but backwards.

Conclusion

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 data streams over the same channel 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! Please leave a comment below if you want to join the conversation!