Digital Sine Oscillators Based on Higher Order Sigma–Delta Modulation

Abstract

In this paper, a new class of digital sine oscillators based on higher-order multibit sigma-delta modulation is presented. The samples of a given pure sine signal are quantized, resulting in a pulse-code modulated (PCM) sequence. A numeric condition for oscillation (NCO) based on the PCM signal is disclosed. If the NCO is fulfilled, a system constant <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> with a finite bit width can be derived and a hardware-efficient practical realization of a digital oscillator based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> -order sigma-delta modulation is possible. The oscillator output sequence is an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> -times differentiated version of the PCM signal and thus the PCM signal can be reconstructed by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> -subsequent accumulations. All signals are absolutely periodic and are represented by binary numbers with well-defined finite bit widths. Initial conditions for the registers of the oscillator are determined. An example for the generation of a high-purity PCM signal with 2048 samples per period is demonstrated. The noise spectrum of this signal is white and the ratio between the mean noise amplitude and the signal amplitude is about one over one billion. The resulting oscillator requires a third order sigma-delta modulator. The chip size is estimated for the oscillator being integrated in a commonly used CMOS technology.