Arbitrary Waveform DDFS Utilizing Chebyshev Polynomials Interpolation

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 m sections, and each section is approximated by a series of Chebyshev polynomials up to degree d. By expanding the resultant Chebyshev polynomials, a power series of degree d 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.