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E-book Error-Correction Coding and Decoding : Bounds, Codes, Decoders, Analysis and Applications

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Penilaian

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The sphere packing bound by Shannon [18] provides a lower bound to the frameerror rate (FER) achievable by an(n,k,d)code but is not directly applicable tobinary codes. Gallager [4] presented his coding theorem for the average FER forthe ensemble of all random binary(n,k,d)codes. There are 2npossible binarycombinations for each codeword which in terms of then-dimensional signal spacehypercube corresponds to one vertex taken from 2npossible vertices. There are2kcodewords, and therefore 2nkdifferent possible random codes. The receiver isconsidered to be composed of 2kmatched filters, one for each codeword and adecoder error occurs if any of the matched filter receivers has a larger output thanthe matched filter receiver corresponding to the transmitted codeword. Consider thismatched filter receiver and another different matched filter receiver, and assume thatthe two codewords differ indbit positions. The Hamming distance between the twocodewords isd. The energy per transmitted bit isEs=knEb, whereEbis the energyper information bit. The noise variance per matched filtered received bit,?2=N02,whereN0is the single sided noise spectral density. In the absence of noise, the outputof the matched filter receiver for the transmitted codeword isn?Esand the outputof the other codeword matched filter receiver is(n?2d)?Es. The noise voltage atthe output of the matched filter receiver for the transmitted codeword is denoted asnc?n1, and the noise voltage at the output of the other matched filter receiver willbenc+n1. This result may be interpreted as providing the number of information bits of thecode is less than the length of the code times the cut-off rate, then the probabilityof decoder error will approach zero as the length of the code approaches infinity.Alternatively, provided the rate of the code,kn, is less than the cut-off rate,R0, then theprobability of decoder error will approach zero as the length of the code approachesinfinity. The cut-off rateR0, particularly in the period from the late 1950s to the 1970swas used as a practical measure of the code rate of an achievable error-correctionsystem [11,20–22]. However, plotting the exact expression for probability of decodererror, Eq. (1.10), in comparison to the cut-off rate approximation Eq. (1.17), shows asignificant difference in performance, as shown in Fig.1.1. The codes shown are the(128,264),(256,2128)and(512,2256)code ensembles of nonlinear, random binarycodes. It is recommended that the exact expression, Eq. (1.10) be evaluated unlessthe code in question is a long code. As a consequence, in the following sections weshall only use the exact Gallager bound.Shown in Fig.1.2is the sphere packing lower bound, offset by the loss attributableto binary transmission and the Gallager upper bound for the(128,264),(256,2128)and(512,2256)nonlinear binary codes. For each code, the exact Gallager upperbound given by (1.10), is shown. One reason why Gallager’s bound is some way from the sphere packing lower bound as shown in Fig.1.2is that the bound is basedon the union bound and counts all error events as if these are independent. Except fororthogonal codes, this produces increasing inaccuracy as theEbN0is reduced. Equiva-lently expressed, double counting is taking place since some codewords include thesupport of other codewords. It is shown in the next section that for linear codes theGallager bound may be improved by considering the erasure correcting capabilityof codes, viz. no(n,k)code can correct more thann?k erasures.

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Ketersediaan

Informasi Detail

Judul Seri

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No. Panggil

621.382 TOM e

Penerbit

Switzerland : Springer Nature., 2017

Deskripsi Fisik

527 hlm

Bahasa

English

ISBN/ISSN

978-3319511030

Klasifikasi

621.382

Tipe Isi

text

Tipe Media

computer

Tipe Pembawa

online resource

Edisi

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Subjek

Communication Engineering

Info Detail Spesifik

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Pernyataan Tanggungjawab

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Versi lain/terkait

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