Mathematics of MP3 players

Summary: Mathematics and mathematical data compression algorithms make MP3 players possible.

MP3 players have revolutionized the way people listen to music. MPEG Audio Layer III (MP3) is an audio compression standard that reduces music files with little perceptible loss of quality. It is one of the Motion Pictures Expert Group standards for lossy compression. The inventors of MP3, according to the United States MP3 patent, are engineers Bernhard Grill, Karl-Heinz Brandenburg, and Bernd Kurten; computer scientist Thomas Sporer; and mathematician Ernst Eberlein. The development was mathematically and technically challenging according to Brandenburg, who is sometimes called a specialist in mathematics and electronics. He stated, “in 1991, the project almost died. During modification tests, the encoding simply did not want to work properly. Two days before submission of the first version… we found the compiler error.” Scientists at Fraunhofer-Gesellshaft developed an MP3 player in the early 1990s.

In 1997, engineer Tomislav Uzelac invented the AMP MP3 Playback Engine, which is regarded as the first successful MP3 player. Computer science student Justin Frankel, who also helped develop the peer-to-peer Gnutella network, and fellow student Dmitry Boldyrev created the free MP3 player Winamp in 1998. Inventor Briton Kramer contributed to the first mass-produced player MPMan. The ability to share files over the Internet, legally and illegally, for free or for purchase, was a significant factor in the rapid spread of the MP3 format. By the twenty-first century, iPods became one of the most popular MP3 players, in part because of the availability of music and video via the iTunes store. The ability to hold thousands of songs, videos, and other types of files is one of the benefits of MP3 players, all of which would not be possible without mathematics and mathematical data compression algorithms.

Compression and Encoding

Data compression is either “lossy” or “lossless,” referring to whether any data is discarded in the process of creating a smaller file. Huffman coding, developed by mathematician David Huffman, is used for MP3 compression. It employs a mathematical idea called a “frequency-sorted binary tree” to look for recurring strings of binary information in the digital file. These strings are replaced by shorter binary codes. The most frequently occurring strings are assigned the shortest replacement codes, optimizing compression. In lossless compression, all original information is preserved in some way. In lossy compression, some information is discarded to decrease file size. MP3 compression relies, in part, on perceptual coding.

In a human ear, certain waveforms are indistinguishable. Psychoacoustic models prioritize data according to the ear’s ability to distinguish the sounds the data produce. Mathematical models of auditory processing yield encoding information and algorithms, such as frequency threshold curves, masking functions, and critical bandwidths. Signal processing typically relies on Fourier transforms, named for mathematician Joseph Fourier, to enable coding and decoding. Ultimately, an MP3 music file consists of a series of short, dependent frames, like a filmstrip. Each frame has a header with information about the data in the frame. Inside the frame is audio information in frequencies and amplitudes. Sometimes, at the beginning or end, there is an ID3 data block, which stores the artist name, track title, album name, recording year, or other information.

Sound

Optimization of audio playback depends not just on the human ear but on the equipment used. Speakers are common on computers, while most MP3 players use some form of over-ear headphones or earbuds that fit into the ear canal. Empirical studies suggest that noise from internal earbuds may be damaging to hearing because the decibel level experienced by listeners is higher on average than with external earphones, and long-life batteries players allow people to listen longer. Some researchers have reported average listening levels of about 110–120 decibels, equivalent to a rock concert. Based on findings of such studies, many audiologists recommend using noise-canceling headphones rather than turning up the volume. Engineer Lawrence Fogel first explored noise-canceling headphones for aviation in the 1950s. Noise cancellation uses the mathematical properties of waves to create a signal with the same amplitude but with an inverted phase to unwanted noise, creating a combined wave inaudible to the human ear.

Shuffle

One other interesting mathematical problem related to MP3 players is the shuffle function. Various mathematical algorithms are used to permute the play order of songs in an MP3 player’s library. In the early twenty-first century, the iPod’s default shuffle system reorders songs much like someone shuffling a deck of cards, giving each song an equal chance to end up in any position in the shuffle and resulting in no repeats. However, many factors can affect perceived randomness and equal likelihood of orderings. For example, users can request higher chances of play for songs with high user ratings. Songs can also be marked “Skip When Shuffling” so that they are completely excluded. Most people frequently reshuffle, generating new random orderings before completing the library, and so some tracks appear to repeat or group in nonrandom ways.

Bibliography

Kallen, Stuart. iPods and MP3 Players. Florence, KY: Gale Cengage, 2010.

Salomon, David. A Guide to Data Compression Methods. New York: Springer, 2002.