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Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. Informally, it is the amount of wasted "space" used to transmit certain data. Data compression is a way to reduce or eliminate unwanted redundancy, while checksums are a way of adding desired redundancy for purposes of error detection when communicating over a noisy channel of limited capacity. Contents 1 Quantitative definition 2 Other notions of redundancy 3 See also 4 References // Quantitative definition In describing the redundancy of raw data, recall that the rate of a source of information is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the most general case of a stochastic process, it is the limit, as n goes to infinity, of the joint entropy of the first n symbols divided by n. It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose. The rate of a memoryless source is simply H(M), since by definition there is no interdependence of the successive messages of a memoryless source. The absolute rate of a language or source is simply the logarithm of the cardinality of the message space, or alphabet. (This formula is sometimes called the Hartley function.) This is the maximum possible rate of information that can be transmitted with that alphabet. (The logarithm should be taken to a base appropriate for the unit of measurement in use.) The absolute rate is equal to the actual rate if the source is memoryless and has a uniform distribution. The absolute redundancy can then be defined as the difference between the absolute rate and the rate. The quantity is called the relative redundancy and gives the maximum possible data compression ratio, when expressed as the percentage by which a file size can be decreased. (When expressed as a ratio of original file size to compressed file size, the quantity R:r gives the maximum compression ratio that can be achieved.) Complementary to the concept of relative redundancy is efficiency, defined as so that . A memoryless source with a uniform distribution has zero redundancy (and thus 100% efficiency), and cannot be compressed. Other notions of redundancy A measure of redundancy between two variables is the mutual information or a normalized variant. A measure of redundancy among many variables is given by the total correlation. Redundancy of compressed data refers to the difference between the expected compressed data length of n messages (or expected data rate ) and the entropy (or entropy rate ). (Here we assume the data is ergodic and stationary, e.g., a memoryless source.) Although the rate difference can be arbitrarily small as increased, the actual difference , cannot, although it can be theoretically upper-bounded by 1 in the case of finite-entropy memoryless sources. See also Data compression Hartley function Negentropy Source coding theorem References Fazlollah M. Reza. An Introduction to Information Theory. New York: McGraw-Hill 1961. New York: Dover 1994. ISBN 0-486-68210-2 B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C. New York: John Wiley & Sons, Inc. 1996. ISBN 0-471-12845-7 . Bank. "Redundancy versus Video and Audio Human Perception", INTERNATIONAL JOURNAL of COMMUNICATIONS Issue 1, Volume 2, 2008, (1 – 16) v • d • e Data compression methods Lossless Theory Entropy · Complexity · Redundancy · Lossy Entropy encoding Shannon–Fano · Shannon–Fano–Elias · Huffman · Adaptive Huffman · Arithmetic · Range · Golomb · Universal (Gamma · Exp-Golomb · Fibonacci · Levenshtein) Dictionary RLE · Byte pair encoding · DEFLATE · Lempel–Ziv (LZ77/78 · LZSS · LZW · LZWL · LZO · LZMA · LZX · LZRW · LZJB · LZS · LZT · ROLZ) Others CTW · BWT · PPM · DMC · Delta Audio Theory Companding · Convolution · Dynamic range · Latency · Sampling · Nyquist–Shannon theorem · Sound quality Audio codec parts LPC (LAR · LSP) · WLPC · CELP · ACELP · A-law · μ-law · ADPCM · DPCM · MDCT · Fourier transform · Psychoacoustic model Others Bit rate (CBR · ABR · VBR) · Speech compression · Sub-band coding Image Terms Color space · Pixel · Chroma subsampling · Compression artifact · Image resolution Methods RLE · Fractal · Wavelet · EZW · SPIHT · LP · DCT · Chain code · KLT Others Test images · PSNR quality measure · Quantization Video Terms Video characteristics · Frame · Frame rate · Interlace · Frame types · Video quality · Video resolution Video codec parts Motion compensation · DCT · Quantization Others Video codecs · Rate distortion theory · Bit rate (CBR · ABR · VBR) Timeline of information theory, data compression, and error-correcting codes See Compression formats for formats and Compression software implementations for codecs