No abstract

No abstract

A b s t r a c t -BCH codes are powerful errorcorrecting codes. Algorithms used for decoding must be able to find the error locations, and for nonbinary codes, the error magnitudes. One of the most efficient algorithms for decoding BCH codes is Berlekamp's algorithm. To find the error locations the algorithm must solve a set of t equations in t unknowns.This paper explores, for binary BCH codes, a new method that uses half of the unknowns to determine the other unknowns, thus solving tl2 equations in tl2 unknowns.However, because of the reduced number of equations, the algorithm only iterates half the number of times, The performance of the new algorithm is shown to be superior to both Berlekamp's algorithm and a simplified algorithm in terms of execution times, which includes the field multiplications and additions and required memory.

No abstract

A new technique of arbitrary waveform direct digital frequency synthesis (DDFS) is introduced. In this method, one period of the desired periodic waveform is divided into sections, and each section is approximated by a series of Chebyshev polynomials up to degree. By expanding the resultant Chebyshev polynomials, a power series of degree is produced. The coefficients of this power series are obtained by a closed-form direct formula. To reconstruct the desired signal, the coefficients of the approximated power series are placed in a small ROM, which delivers the coefficients to the inputs of a digital system. This digital system contains digital multipliers and adders to simulate the desired polynomial, as well as a phase accumulator for generating the digital time base. The output of this system is a reconstructed signal that is a good approximation of the desired waveform. The accuracy of the output signal depends on the degree of the reconstructing polynomial, the number of subsections, the wordlength of the truncated phase accumulator output, as well as the word length of the DDFS system output. The coefficients are not dependent on the sampling frequency; therefore, the proposed system is ideal for frequency sweeping. The proposed method is adopted to build a traditional DDFS to generate a sinusoidal signal. The tradeoff between the ROM capacity, number of sections, and spectral purity for an infinite output wordlength is also investigated.