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NextAudio storage and retrieval
We have seen how audio signals can be represented as sequences of numbers. Digital audio storage and retrieval is all about finding a physical and persistent representation for these numbers. These numbers can be converted into a single sequence of bits (binary digits) and then "printed" onto some physical medium from which they can later be read back.
In the case of compact discs (CDs), the physical medium is a layer of aluminum on a platter into which tiny pits are etched. A laser aimed at the platter uses an interference pattern to determine whether or not a pit exists at a particular point in the platter. These pits, thus, naturally represent binary digits, since they can have two states (present or not present).
While a modem transmitter converts bit sequences into sounds, a musical recording studio does the reverse, creating a representation of the sound that is a bit sequence:
The CD itself, in raw form, is just a ring-shaped aluminum platter, a region of two-dimensional space at each point of which there may be a pit or not. Let the set Pits = { pit, no_pit } represent the presence or absence of pits, and let Ring represent the writable ring-shaped area of the CD. We can use predicates to define the set Ring as follows:
(This assumes the units of measure are centimeters.) Then a CD is a function:
Let CDs represent the set all such functions. The CD manufacturer then completes the job of the recording studio by implementing the function
There is a great deal of engineering in the details, however. For instance, CDs are vulnerable to surface defects, which may arise in manufacturing or in the hands of the user. These defects may obscure some of the pits, or fool the reading laser into detecting a pit where there is none. To guard against this, a very clever error-correcting code called a Reed-Solomon code is used. The coding process can be viewed as a function
where RedundantBitSequences ⊂ BitSequences is the set of all possible encoded bit sequences. These bit sequences are redundant, in that they contain more bits than are necessary to represent the original bit sequence. The extra bits are used to first detect errors, then (sometimes) correct them. Of course, if the surface of the CD is too badly damaged, even this clever scheme fails, and the audio data will not be recoverable.
CDs also contain meta data, which is extra information about the audio signal. This information allows the CD player to identify the start of a musical number and its length, and sometimes the title and the artist.
The CD format can also be used to contain purely digital data. Such a CD is called a CD ROM (read-only memory). It is called this because, like a computer memory, it contains digital information. But unlike a computer memory, that information cannot be modified.
DVD (digital video discs) take this concept much further, including much more meta data. They may eventually replace CDs. They are entire compatible, in that they can contain exactly the same audio data that a CD can. DVD players can play CDs, but not the reverse, however. DVDs can also contain digital video information and, in fact, any other digital data. DAT (digital audio tape) is also a competitor to CDs (a corner of such a tape is visible in the image above), but has failed to capture much of a market.