# How are radio waves mathematically expressed?

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That article describes radio waves as frequency and wavelength. The [1977 Wow! Signal](https://en.wikipedia.org/wiki/Wow!_signal) is expressed in frequency and bandwidth.

Can someone help elucidate the difference for me?

And how is this mathematically expressed? Would a radio wave be expressed as a “1420 Mhz wavelength signalin the 10 kHz band?” Something else?

Thanks!

In: Mathematics

The frequency of the transmission is the center frequency, and the bandwidth tells you how much up and down the frequency wavers. So if you had a 1420 Mhz (1,420,000,000 Hz) signal with a 10 Khz (10,000 Hz) bandwidth, the frequency would wobble between 1,419,995,000 Hz and 1,420,005,000 Hz.

It depends on who you’re communicating to and what radio waves you’re talking about.

Any waveform can be represented as a sum of sine waves of varying frequencies. Frequency and wavelength are directly related as the inverse of each other. If you record some radio transmission and apply a Fourier transform to it, what you’ll see is a “spectrum” showing how much of each sine wave exists in the signal. You can measure this through an instrument called a [spectrum analyzer](https://en.wikipedia.org/wiki/Spectrum_analyzer?wprov=sfti1). A sine wave is expressed as A x sin(2 x pi x f + phi), where “A” is its amplitude (how big it is), “f” is its frequency (how fast the sin cycles), and “phi” is its phase (sin(0) is 0. A phase would offset that).

A signals bandwidth typically refers to how “wide” the signal is on the spectrum. Frequency may generally refer to the center of the band. Something like an FM transmission would be described as an FM transmission on channel 103.3. “Channel” represents a center frequency of 103.3 MHz and FM channels have standardized bandwidths of 0.2 MHz, which means a particular FM transmission on that channel can have frequency content between 103.2 and 103.4 MHz.

When a signal is transmitted, a “carrier wave” is first generated. In the case above, that would be 103.3 MHz, or A x sin(2 x pi x (103.3MHz) + phi) It would then be “modulated” with the data to be transmitted, which in the case of an FM radio station is music or a host’s voice. This mathematically is represented by multiplying the data with the carrier wave. Practically, what this does is moves the 0.2 MHz-wide data into the 103.3 MHz channel from “baseband”.

>And how is this mathematically expressed? Would a radio wave be expressed as a “1420 Mhz wavelength signalin the 10 kHz band?” Something else?

If someone talks about it in a similar way, it would be the opposite. That is, they would say it’s a 10 kilohertz bandwidth signal in the 1420 MHz band (except that isn’t quite a band, see below).

The way humans use the electromagnetic spectrum is to divide it up into particular frequency bands (or, equivalently, wavelength bands, but talking about them in terms of wavelength is no longer very common). This is because it turns out that the total amount of information you can convey over a periodic signal like an electromagnetic wave is directly related to the bandwidth, meaning the range of frequencies over which the signal can vary. The way that we transmit information over radio is by using a baseline or fundamental frequency of a particular value, and then changing or *modulating* it in some way.

The reason we have to modulate it is that a single constant signal doesn’t convey any information, unless you’re using an extremely basic communication system where the communication is just binary and based on whether you can receive the signal or not. Imagine somebody whistling at a constant pitch. You have no idea if they are trying to communicate anything, other than perhaps the fact that they are currently whistling. But if they change the pitch and start whistling a tune, then you might recognize it, and that just conveyed to you information about what that person was trying to communicate, namely which song they were whistling.

The fact that they have to change the pitch of their whistling in order for you to get any information at all is exactly what *bandwidth* is talking about. Imagine they could only whistle two notes. Then it would still be pretty hard to figure out what song they were attempting to whistle. You could develop a coding scheme, where they could whistle the name of the song in Morse code or something. But that would be much slower than them being able to whistle the first few notes of the song. They can convey more information in the same amount of time if they can whistle more than two notes.

Now imagine a bunch of people are trying to whistle tunes at the same time. It would be difficult to impossible to figure out what they were whistling. They would all interfere with each other. But imagine that you could design a system where one person whistled within a particular range of notes, and somebody else used a different one, that didn’t overlap. If you could selectively listen to only a particular range of notes, then you wouldn’t have any trouble figuring out what any given person was trying to whistle.

We can design radios that can listen to almost any given range of frequencies. That means we can split up the entire range of electromagnetic frequencies, which are equivalent to pitches of notes, into huge numbers of communications bands. But the amount of information that can be sent across any given band is still limited by how much the frequency is allowed to vary. That is the bandwidth. The electromagnetic spectrum is chopped up into many different bands, which are reserved by law for particular uses. Those bands are themselves typically divided into channels, so that many users can stay within the band but still make use of the spectrum. And those channels have what is somewhat confusingly called a bandwidth, which is how much the frequency of the signal can vary around whatever the main frequency is.

Hence if you were talking about a signal which varied by 10 kHz, with a center frequency of 1,420 MHz, at least in the US it would be a 10 kHz signal within the 1,400 – 1,427 MHz band, with a center frequency of 1,420 MHz.

One of the interesting things about the Wow! signal is that it wasn’t modulated at all, as far as we could tell based on the reception we got. That is, it was more like somebody whistling a constant pitch than somebody trying to actually communicate info. That means it didn’t have any bandwidth at all, at least not one we could measure.

E: the reason that it is described as having a 10 kHz or less bandwidth is that the radio telescope itself could only measure signal amplitude within ranges of frequencies, ie channels. Those channels were 10 kHz wide, and the signal was only detected on a single channel. The nature of the telescope means that all of the signal intensity across that entire channel gets added together, so technically if the frequency was changing within that channel, the telescope could not have seen it. That is why people say it was less than 10 kilohertz. And that 10 kilohertz would have had to be centered right at the middle of the channel. But there is no evidence that the signal was modulated at all, whether in frequency or in amplitude.