AN2407 查看數據表（PDF） - Freescale Semiconductor

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AN2407

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples

Freescale Semiconductor 

Theory

The Reed-Solomon codes are block codes. Unlike convolutional codes, Reed-Solomon codes operate on multi-bit

symbols rather than on individual bits. The question of whether to choose convolutional codes or block codes

depends on several variables. In low-speed, low-integrity applications, convolutional codes are the better choice,

and block codes are suitable for high-speed, high-integrity applications. An example of an application suited to

convolutional codes is a digitized voice communication in which a relatively high bit-error rate (about 10–3) is

acceptable. For blocks of machine-oriented data in which the desired bit-error rate ranges from 10–10 to 10–14,

block codes are the natural choice. Some applications use both convolutional and block codes. In such applications,

concatenated codes result in strong performance by operating in two steps. The inner decoder, usually

convolutional, reduces the bit-error rate to a medium-low level, and the outer decoder, usually a block type,

reduces the bit-error rate further, to a very low level.

The errors introduced by the communications channel are classified into two main categories:

• Random errors. The bit-error probabilities are independent of each other. For example, thermal noise

in communication channels typically causes random errors.

• Burst errors. The bit errors occur sequentially in time. Burst errors can be caused by such conditions as

a fading communications channel or mechanical defects in a storage system.

When an FEC system is designed, the statistical nature of the noise environment must be considered, as well as the

acceptable output bit-error rate. When the environment consists predominately of random errors, convolutional

codes provide a low bit-error rate solution. However, when the environment has lower bit-error rates, long-length

block codes often perform even better. In burst-error channels, Reed-Solomon codes are among the best codes

because errors composed of many consecutive corrupted bits translate into only a few erroneous symbols.

2 Theory

The Reed-Solomon code was developed in 1960 by I. Reed and G. Solomon [4]. This code is an error detection and

correction scheme based on the use of Galois field arithmetic. This section provides background information on

binary and extended Galois fields and summarizes the essence of the Reed-Solomon codes. For details on Reed-

Solomon codes, consult the literature, for example, [5] and [6].

2.1 Galois Fields

A number field has the following properties:

• Both an addition and a multiplication operation that satisfy the commutative, associative, and

distributive laws.

• Closure, so that adding or multiplying elements always yields field elements as results.

• Both zero and unity elements. The zero element leaves an element unchanged under addition. The

unity element leaves an element unchanged under multiplication.

• An additive/multiplicative inverse for each field element. The sole exception is the zero element,

which has no multiplicative inverse.

Division is defined as the inverse of multiplication such that if a × b = c, it follows that c divided by a yields b. An

example of a number field is the set of real numbers together with the addition and multiplication operations.

Galois fields differ from real number fields in that they have only a finite number of elements. Otherwise, they

share all the properties common to number fields.

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples, Rev. 1

Freescale Semiconductor

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