radio

Radio is pervasive in our lives; we get much of our news, entertainment, and cultural awareness from listening to the “airwaves.” Did you ever wonder how all this information gets sent? This section uses Mathcad to demonstrate the mechanics of radio transmission and reception.

To understand radio, it’s important to understand the wave nature of sound. Sound is a series of pressure waves that make air vibrate at different rates (frequencies). These frequencies are interpreted by the ear as different tones. Slow, or low frequencies correspond to low-pitched sounds, and fast or high frequencies to high-pitched sounds.

If sound is to be stored or transmitted electronically, these pressure wave frequencies must be represented as electronic waves. The human ear can hear between about 10 Hz and 20 kilohertz (kHz). Although it’s uncommon to find an electronic system which can produce this full range, or bandwidth, with fidelity, let’s use this range in this discussion.

Examine what an (imaginary) sound wave might look like. It will be a sum of sinusoidal waves of different frequencies between 10 Hz and 20 kHz. Let’s pick a relatively high-pitched set.

Notice that all of the frequencies chosen fall between 10 Hz and 20 kHz. This sound wave is represented in time (t). If we plot it over time, we’ll see the oscillation as the wave travels.

Another useful way to represent a sound wave, or any other kind of wave, is in the frequency domain: the wave is represented as a spectrum of frequency components. These frequency components have been written specifically above, so we know them. But, if we didn’t, it would be possible to recover them using the Fourier transform. The Fourier transform coefficients are given by a built-in Mathcad function, the fast Fourier transform, or FFT.

There is a single spike for each frequency in the sound wave above. If you look at the y-axis, you’ll see that the height of each spike is the specified height of that corresponding frequency in the expression for sound(t). If you change the frequencies used in sound(t), or the relative magnitude of each of the terms, you’ll see these changes reflected in the spectrum, as well as in the time plot.

Radio broadcasting is essentially an exercise in attaching sound waves, such as the one above, to a suitable carrier (broadcasting) frequency, transmitting the carrier with the attached sound, and then removing the carrier at the receiver. Since the sound is already stored electronically using some set of frequencies, why bother with the carrier? In order to transmit or receive sound through the air, an antenna is required. The physics of antenna design require that, for a low frequency signal, on the order of 10 Hz, the required antenna is several miles long! This would certainly be difficult to install in your home or car. Additionally, if your radio had an antenna designed for 60 Hz, you’d probably spend most of your time listening to the hum of power transmission lines. These, and other considerations, such as system bandwidth, make a carrier frequency a necessity.

So, a manageable antenna length requires a high frequency carrier, and one that is much higher in frequency than the sound wave components. This last requirement will insure that the sound frequencies form a little bunch around each carrier frequency, so that multiple carrier frequencies can be used. This allows you to have multiple radio stations. Appropriate frequencies would start around 10 times the bandwidth:

You can clearly see what happens when there is a large (>10X) difference between the carrier and sound frequencies. The carrier frequency is amplitude modulated by the sound. That is, the carrier continues to oscillate in a recognizable sinusoidal pattern, but its amplitude, or height, moves up and down inside an envelope defined by the sound wave. The electronic multiplication is performed by a device called a mixer. The sound can then be transmitted on this high frequency wave by an appropriate antenna, received, and unmixed, which will be demonstrated below.

You might be able to anticipate what has happened in frequency. The same components observable in the original sound wave are still there, but they have been shifted up around the carrier frequency.

This should give you an inkling about how many radio stations can be transmitted at once. If every broadcast frequency is at least 10 times as large as the highest sound frequency, then the broadcast frequencies could be placed apart by just a little more than the sound bandwidth, and each could carry a full complement of sound without interference. This is why station numbering isn’t exactly sequential. The station number corresponds to a frequency, and there has to be a little "space" between them.

The alternate, and somewhat more robust, method of combining the carrier and the sound is to find the difference in phase between the carrier and the sound wave at any particular moment in time. This is done with an electric circuit known as a phase-locked loop. The effect is to modulate the frequency of the carrier wave, rather than the amplitude, and hence is known as frequency modulation (FM), as demonstrated below.

An unreasonably small carrier frequency and an excessively large modulation index are used as an example, so that you can see the frequency modulation easily. The waves get very close together in some places, and far apart in others. With realistic values of carrier frequency, between 88 MHz and 108 MHz, the phase shift introduced by the sound is minimal, typically less than ±5 degrees.

You may have noticed that FM stations are much higher in frequency than AM stations. The demonstration above shows you the reason. If sound(t) distorted the signal as much as it does in the graph above, the sound would be unrecoverable, because the carrier frequency would be indistinguishable. Sufficiently high frequencies must be picked, along with the modulation index, to insure that the phase shifts are always small, but still detectable by your radio.

This decomposes, as you can see, into the original sound and a cosine term at twice the carrier frequency. If you lowpass filter this signal, i.e., allow the low frequencies to pass through, all that remains is the original sound(t).

Examine the received signal in the frequency domain to see if the analytical expression is confirmed. What you should see is a signal at twice the frequency, with the sound wave frequencies bunched around it, and the original sound frequencies in their proper locations.

It is possible to improve the filter significantly with more electronics (more mathematical terms), but this is typical of what will result. There will be some small residual signal at twice the carrier frequency. To see if we’ve recovered the original signal adequately, find the inverse Fourier transform through the use of another built-in function, IFFT:

Pretty good! The red line is the received output, and the blue, dotted line is the original sound input. The “fluffiness” on the output signal is an example of noise. This is why radios (particularly AM radios) sometimes have static on the signal. In practical systems, noise is partly the result of attenuation and distortion in the electronics, and partly the limitations on filtering. The output is also ever so slightly phase shifted (occurs later in time). This is also the result of filtering.

As a final test of your understanding, let’s look at how a radio tuner would function. Start with a few different sound waves, or “songs,” and put each one on a different carrier frequency.

Select a radio channel by changing the value of station below. The signal carried on that station will be displayed below. The available AM stations on your “dial” are 550 kHz, 780 kHz, and 1010 kHz. Try values between these stations too, to see the result. It will be visually equivalent to static. What happens when you get close to a station value, but not on it?

These look a lot sloppier than the single station example. The reason is interference from the high frequency carriers of the other stations which are not completely filtered. As mentioned earlier, you can improve the filter in lots of ways, electronically, and so most radios are not this noisy.

AM radio carries lots of information these days; for example, you can listen to a baseball game on a summer day at the beach. To understand the player stats that get broadcast, see the section on baseball statistics.