Download Digital Circuits for Binary Arithmetic by R. M. M. Oberman (auth.) PDF

By R. M. M. Oberman (auth.)

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Parity checks By expanding the code words of a set with one or more redundant elements, it can be given error-detecting features. The most simple such method is to extend the code words with one element in such a way that the number of 1 bits in the code words becomes odd or even. These checks are called the odd parity check or the even parity check. With this type of parity check the number of words in the code is doubled. In an odd parity check system the code words with an odd number of 1 bits are the correct code words, and the code words with an even number of 1 bits are the erroneous code words which originate from the correct code words by a single error (in fact, an odd number of errors).

2-2. This operating time can be greatly reduced by introducing fast carry-generating circuits. The multiplication, to be treated in chapter 4, can be considered as a more or less sophisticated application of the addition. In the multiplying operation the digits of the multiplier and multiplicand generate groups of partial product digits with successively increasing weights, however, these words have to be added to obtain the desired product. In a parallel multiplication several rows of partial products can be formed which can be added with two different viewpoints in mind.

The solution of the reconstruction of the erroneous pair of information bits has already been given in the second example corresponding to the explanation of Table 1-15. Other types of errors can now be readily corrected in these types of codes, for example, all the errors in the check bits, or errors in the information bits and in the check bits. 12 THE NUMBER SYSTEM OF RESIDUAL CLASSES When the algebra of residual classes is used for the representation of numbers, it is possible to perform the addition, subtraction and multiplication without any carry transfer between the digits of the numbers used.

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