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bits, the number of information bits and the minimum Hamming distance, respectively The number of parity bits is equal to - k and the coding rate is: The BCH encoder accepts n k information bits and by using a specific polynomial code generator [157], the parity bits are added in order to produce n coded bits Thus, according to the encoding rules, only certain encoded sequences are legitimateIt is this distinction that enables the decoder to recognize and correct corrupted or illegitimate codewords We will refrain from discussing the code generation and decoding mechanism, referring the reader to references [13,157-1591 By referring to Figure 51, the generated codewords are punctured and multiplexed,in order to produce the turbo code However, puncturing of the parity bits is not applied to turbo BCH codes as proposed by Hagenauer [l601 and Pyndiah [161].

Unique solution. The argument will show that a unique solution KOexists if, and only if, the covariance matrix R, is positive-definite. Indeed, if 4 > 0 then all its eigenvalues are positive and, consequently, R, is nonsingular. In this case, equation (4.7) will have a unique solution KOgiven by KO = R,,R;I. Conversely, assume a unique solution KO exists to the normal equations (4.7) and let us establish that it must hold that R, > 0. Assume, to the contrary, that R, is singular. Then there should exist a nonzero row vector c* such that c* R, = 0 or, equivalently, R,c = 0. The vector c belongs to the nullspace of R,, written as c E N ( 4 ) .It is now easy to see that by adding to the rows of KOany such vector c*, we obtain a matrix KL that satisfies the same equations (4.7),i.e., KLR, = Rzy. This contradicts the fact that KOis unique so that R, must be positive-definite. Infinitely many solutions. We now show that the normal equations (4.7) will have infinitely many solutions KOif, and only if, is singular. One direction of the proof is obvious. Assume KOis one solution and that R, is singular. Then by adding to the rows of KOany vector c* (or any combination of vectors) from the nullspace of R,, we obtain another solution KL (as explained above). Hence, infinitely many solutions exist in this case. Conversely assume that many solutions exist. Let KOand KL denote any two of these solutions. Then subtracting the equations KoR, = R,, and KLR, = R,, we obtain (KO- KL)R, = 0, which means that R, is singular, since there is at least one nonzero row in KO- KL and this row annihilates R,.

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MIN() MAX() QUARTILE() PERCENTILE()

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Consequently, for example, a of component codeof BCH (31,26, 3) will yield a turbo block code BCH (36,26), where the additional five parity bits of the second encoder are included the output turbo block code, in while the systematic information bits produced the second encoder are discarded by The other family constituent turbo encoders that can be utilized is Recursive Systematic of Convolutional (RSC) codes, which is shown in Figure 52 Here, the constraint length is set in to K = 3, and the generator polynomials are set octal terms, to7 and 5 [ 1571 Referring to Figure 52, a stream systematic bits, which represents the original information sequence is of generated along with the corresponding parity sequenceIn forming the convolutional-based turbo code, the systematic bits the second convolutional encoder are discarded andtwo of the sets of parity sequences are punctured accordingly.

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