What are the differences between AM and FM radio?

AM is amplitude Modulation, and FM is Frequency Modulation. With both AM and FM, a carrier wave must be broadcasted. And it is just that... A carrier wave for the information. A greyhound bus for the information you are trying to tune in. (music, etc.). The carrier wave is at a particular frequency, which is singled out as you tune in your radio station. All other frequencies are filtered out.

Simplified explanation:

The difference is how the "information" rides on that carrier frequency. And the names give it away. In AM or Amplitude Modulation, the information is added to the amplitude of the carrier signal. When your receiver receives the signal, it does an analog sample of the wave to determine it's average strength, then demodulates the differences in the amplitude to pull out the original information. This is the worst and most inaccurate way of demodulation, for the signal strength is continuously changing, with the analog demodulator always working "behind the times". This is why it took so many years for the occurance of AM Stereo. It's hard enough to accurately pull one stream of info out of the carrier wave. Try pulling two distinct signals out of one carrier wave that is Amplitude modulated. How in the world do you do that??? (But they do have AM stereo now, yet not very popular) The reason it came first, is it is very easy to mix amplitudes, and no complicated circuitry was required in the old days to pull in the correct carrier wave. They couldn't do frequency modulation at first cause uncle sam wouldn't let commercial businesses transmit on high enough frequencies to pull out the original information. Also since AM carrier waves have relatively low frequencies, they can travel quite a distance.

In Frequency Modulation, the information is added to the frequency of the carrier wave. The amplitude of the incoming signal is a mute point as long as it is strong enough where your receiver can lock onto the carrier wave. Carrier frequencies in high ranges were released for commercial use, so they could transmit in the MegaHertz range. Since all information needed in the hearing range is commonly in the 20 to 20KiloHertz range, it was easy to pull the signal in. The information didn't cause enough of a difference in the signal to cause the tuner to filter it out. And since the original carrier frequency is known, and locked in, it was extremely easy to pull out the original information. In fact, you could add 6 or 8 different information signals if you wanted to. Complex, but achievable. Since the demodulation is so accurate, this gives it better clarity, and the ability to modulate 2 separate signals for stereo. However, since the carrier frequency is so high, it bounces off obstacles instead of going through them, thus reducing range.

Let's see if I can make the difference more clear:

For AM radio, imagine a sine wave swinging back and forth at a single frequency. That is the 'carrier wave'. Now imagine the peaks of those waves being different heights. Then draw a continuous curve between those wave peaks.

That curve represents the audio signal. The tuner converts that curve into a voltage signal that drives a speaker.

http://www.electronics-radio.com/articles/radio/modulation/amplitude_modulation/am.php

For FM radio, imagine that you are graphing the instantaneous *frequency* of the carrier signal, which is centered around a single 'station' but is varying above and below that value. That continuous frequency change represents the audio signal. The FM tuner outputs a voltage proportional to that frequency that drives a speaker.

http://www.electronics-radio.com/articles/radio/modulation/frequency_modulation/fm.php

AM radio has the advantage of transmitting further than FM, especially at night when AM signals will bounce off of the upper atmosphere.

FM radio has the advantage of being inherently more resistant to noise and interference and can produce a higher-fidelity audio signal.
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The difference between AM and FM radio is that FM has shorter range, within the line of sight of the radio transmitter.

the differences is that both of the radio stations signals carries an output at a signal frequency FM is usually much higher than AM

All radio stations are made out of electromagnetic waves. The audio signal is transferred from a microphone, or a CD player, Tape Deck or Record Player, through wires to the transmitter, a large apparatus which generates the bulk of the radio wave, the carrier wave. If your listening to 830 on the AM dial, your stations carrier wave is at 830 kilocycles (KC or kHz). If your listening to 101.3 on the FM dial, your stations carrier wave frequency is 101.3 Megacycles (MC or MHz), and so on.

The difference between say, the same 830 on the AM dial and the 101.3 on the FM dial is how the carrier. To add music, talking, or morse code to a radio wave, you have to MODULATE the carrier wave.

In the case of the AM 830 kHz signal, the transmitter will generate the carrier wave and add the talk programming into the carrier through amplitude modulation.
This varies the strength of the Radio wave. Its akin in the visible light world to dimming and brightening a lightbulb to send information.
There are several flaws with this system, say next to the brightening and dimming bulb, someone decided to take a photograph of this bulb. The momentary bright flash of the bulb outshines the bulb, making the information hard to see. In the radio world, this is what happens to AM radio stations in a Thunderstorm. THe bright lightning makes electrical noise which blots out the information that was encoded in the carrier by the strength of the electrical noise "outshining" the encoded information causing the momentary fading of the Radio station and a buzz of electrical noise.
This also happens with steady sources of electric noise, like appliances with motors, fluorescent lighting and Television screens and Computer Monitors. The noise will compete with the information encoded in the signal, causing buzzing, clicks, pops and, in the worst cases, distortion.
This is why most programming on AM Radio is talk, sports and religious programming, as high-quality is not generally demanded, and not music programs.

The 101.3 FM signal is different, the carrier wave is generated in the transmitter, but this time the information is encoded in frequency modulation. This varies the frequency of the radio wave from 101.300-101.400, once every second, altering the frequency of the carrier wave. In the lightbulb analogy, its like varying the color of the lightbulb rather than the brightness, so if anyone decides to photograph the bulb, no information will be lost, which is why FM is less susceptible to electrical interference. This also has drawbacks, as the FM method of transmission also is a line-of-sight transmission medium, which means that FM signals can only travel as far as you can see the transmitters antenna (broadcast tower) and about 25% farther than that, which travel about 30 miles. The main benefit to AM Radio is that it can follow the curvature of the earth for a distance of usually 90 miles. And better yet, at night, changes in the ionosphere cause AM signals to bounce of the sky, causing, for instance, Chicago stations such as WBBM 780 AM, will reach the city of Minneapolis, at a listenable quality, around 400 miles away! FM is also susceptible to multi-path interference, where the signals will bounce off buildings and trees to your receiver, causing hissing and chattering on weaker stations. Most Music and Public radio stations are on the FM band because of the bands ability to provide crystal-clear, stereo broadcasts.

AM and FM aren't the only ways to modulate Radio signals, airport and military radio use Pulse-code modulation, which pulses a burst of radio waves that reflect off metal objects to track them.

As counter intuitive as it may seem. AM stations use solid-state transmitters, while FM and TV transmitters usually use a giant vacuum-tube to make the carrier wave.
(Just a random, interesting note :)