P. Elias, "Coding of Noisy Channels," IRE Conv. Record, vol. 4, pp. 37-47, 1955.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the block code construction, so practical trellis based soft decision decoding algorithms have recently been introduced for some linear block codes [29] The drawbacks of block codes can be avoided by the second main type of error correcting codes, convolutional codes, introduced by Elias in 1955 [30]. Instead of treating each code block independently, convolutional codes add redundancy to a continuous stream of input data by using a linear shift register and add dependence between successive symbols. Thus the current output data depends not only on the information bits in the current input ....

P. Elias, \Coding for noisy channels," IRE Conv. Record, vol. 4, pp. 37-47, 1955.

.... we know that for large N , xed rate R C, and for the best possible codes of rate R, the decoding error probability P e (C) takes the form P e (C) 2 E(R;p)N o(N) where E(R; p) called the error exponent, is a positive quantity that depends only on p and the rate R and can be computed exactly [7]. Unfortunately, the only known way to achieve this best possible error exponent is to use a random code C together with decoding algorithms that nd the closest codeword by essentially performing exhaustive search over C. Turning to manageable decoding algorithms, until fairly recently only one ....

.... and of the code vector x and equals the average error probability P e (C) 2 x2C P e (x) Therefore, for the purposes of analysis we always assume that the transmitted vector always is the all zero one: x = 0 and write P e (C) instead of P e (0) By the classical results of coding theory [7], 10] there exist sequences of binary linear codes such that the probability P e (C) under maximum likelihood decoding falls exponentially with the code length N . Therefore, de ne the error exponent E(C) N log P e (C) We de ne the best attainable error exponent for the rate R as E(R; p) ....

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P. Elias, Coding for noisy channels, IRE Conv. Rec., Mar. 1955.

.... decoding for Reed Solomon codes [19] and trellis based soft decision decoding algorithms for other classes of block codes [30] The drawbacks of block codes can be avoided by taking a different approach to coding, that of convolutional coding which was first introduced in 1955 by Elias [31]. Rather than segmenting data into distinct blocks, convolutional encoders add redundancy to a continuous stream of input data by using a linear shift register. Each set of n output bits is a linear combination of the current set of k input bits and the m bits stored in the shift register. The ....

P. Elias, "Coding for noisy channels," IRE Cony. Record, vol. 4, pp. 37 47, 1955.

.... decoding for Reed Solomon codes [15] and trellis based soft decision decoding algorithms for other classes of block codes [26] The drawbacks of block codes can be avoided by taking a different approach to coding, that of convolutional coding which was first introduced in 1955 by Elias [27]. Rather than segmenting data into distinct blocks, convolutional encoders add redundancy to a continuous stream of input data by using a linear shift register. Each set of n output bits is a linear combination of the current set of k input bits and the ra bits stored in the shift register. The ....

P. Elias, "Coding for noisy channels," IRE Cony. Record, vol. 4, pp. 37-47, 1955.

....channels, the performance of the algebraic linear block codes can sometimes be rather poor, because of its associated hard decision decoding. However these drawbacks can be easily overcome by considering a convolutional code. Convolutional code is a non block code invented by Peter Elias in 1955 [34]. A convolutional code is characterized by three parameters: The alphabet over which the code is defined. Generally a convolutional code is defined on a finite field. We will restrict our discussion to convolutional codes defined on binary alphabets 0, 1 . The rate of the convolutional ....

P. Elias, "Coding for Noisy Channels," IRE Convention Record, pp. 37--47, 1955.

....can be divided into codewords, then these codewords have variable lengths, from a minimum length (depending on the code structure) to infinity. If the encoder output symbols are taken from a finite field, then the code is known as a convolutional code (CC) These were originally proposed by Elias [7] in 1955. Otherwise, if the output symbols are taking from a real field, as of a modulator, it is best known as trellis coded modulation (TCM) as proposed by Ungerboeck [8] in 1982. Also it is possibly a CPM scheme [9] or a system unintentionally coded by a channel having memory, that is, an ISI ....

P. Elias, "Coding for noisy channels," IRE Conv. Rec., vol. Part 4, pp. 37--47, 1955.

....m(z) P t i=0 m i z i 2 F k [z] and c(z) P t i=0 c i z i 2 F n [z] it is possible to describe the encoding procedure through the module homomorphism: 1 : F k [z] F n [z] m(z) 7 c(z) Gm(z) 2. 1) The original idea of a convolutional code goes back to the paper of Elias [2], where it was suggested to use a polynomial matrix G(z) in the encoding procedure (2.1) Polynomial encoders G(z) are physically easily implemented through a feedforward linear sequential circuit. Massey and Sain [34, 45] showed that there is a close connection between linear systems and ....

P. Elias. Coding for noisy channels. IRE Conv. Rec., 4:37-46, 1955.

....u 0 ; u 1 ; u 2 ; u r u 0 u 1 s u 2 s 2 u r s r If we denote this polynomial by u(s) then the encoding of the entire sequence can be written simply as u(s) 7 u(s)G, where the matrix multiplication is done on each coe cient of the polynomial. It was Elias in 1955 [22] who suggested that the matrix G need not be scalar. In doing so, the notion of a convolutional code was born. The classical de nition of convolutional codes can now be stated. De nition 1.3.1 [23, 52] A rate k=n convolutional code is de ned as a k dimensional F linear subspace of F n , where ....

P. Elias. Coding for noisy channels. IRE Conv. Rec., 4:37-46, 1955.

.... i=0 m i z i 2 F k [z] and v(z) fl X i=0 v i z i 2 F n [z] then the total encoding scheme m i 7 Gamma v i = Gm i ; i = 0; fl can be compactly described through the module homomorphism : F k [z] Gamma F n [z] m(z) 7 Gamma v(z) Gm(z) It was the idea of Elias [2] to replace in above encoding scheme the generator matrix G with a generator matrix G(z) and to allow in this way general module homomorphisms as encoding schemes. Using this point of view we define a convolutional code C as a F[z] submodule of F n [z] If V (z) is a k Theta k unimodular matrix ....

P. Elias (1955). Coding for Noisy Channels. IRE Conv. Rec. 4, 37--46.

....theorem: All codes are good, except those that we know of [3] Rather, it was the problem of decoding complexity. By the early sixties at M.I.T. the decoding problem was considered to be essentially solved by the combination of: ffl Long, randomly chosen ( probabilistic ) convolutional codes [4]; ffl Sequential decoding [3] Long convolutional codes have a dynamical structure which was at that time pictured as a branching tree structure. Sequential decoding refers to a class of exhaustive tree search techniques that are characterized by a probabilistic distribution of decoding ....

P. Elias, "Coding for noisy channels," IRE Conv. Rec, pt. 4, pp. 37-46, 1955.

....from a traditional (non systems) viewpoint see [Dho94, For70, LC83, Pir88] 3.1 Definitions and Basic Properties Definition 3.1.1 A subset C W T is called a convolutional code if C is linear, right shift invariant and has compact support. Convolutional codes were first defined by P. Elias in [Eli55]. We would like to comment on some of the requirements we impose. First note that the condition of right shift invariance is a condition of time invariance: any message that is sent at time zero can be sent at any time t 0: The fact that we impose compact support is new. Note that many authors ....

P. Elias. Coding for noisy channels. IRE Conv. Rec., 4:37--46, 1955.

....the bit error rate achievable by systematic codes for the binary symmetric channel and we show that when the source entropy is higher than the channel capacity, the bit error rate is strictly higher than that achievable by non systematic codes. This is in contrast to the well known fact [1], 2] that for any rate below capacity, systematic (linear or nonlinear) codes achieve the random coding error exponent of the binary symmetric channel. Another motivation for the model of systematic transmission discussed in this paper arises when a source is transmitted over a channel with ....

P. Elias, "Coding for Noisy Channels," IRE Convention Record, Vol. 4, pp. 37--46, March 1955.

.... but arbitrary long message sequences with the polynomial vector m(z) 2 F k [z] If G is an encoder of a linear block code then extends to a module homomorphism : F k [z] 7 F n [z] m(z) 7 c(z) Gm(z) 1) 2 Convolutional codes and the relation to linear systems In 1955 Elias [2] proposed to replace in the encoding scheme (1) the encoding matrix G by a polynomial matrix G(z) introducing in this way the concept of a convolutional code. Formally we can define a convolutional code through: C =fc 2 F n [z] j 9m(z) 2 F k [z] c(z) G(z)m(z)g: From a mathematical point ....

P. Elias. Coding for noisy channels. IRE Conv. Rec., 4:37--46, 1955.

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P. Elias, "Coding of Noisy Channels," IRE Conv. Record, vol. 4, pp. 37-47, 1955.

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Peter Elias. Coding for noisy channels. IRE Conv. Rec., 3(4):37--46, 1955.

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