Idle-Channel Noise Characteristic of a Delta Modulator with Step Imbalance and Quantizer Hysteresis

Abstract

In 1969 Iwersen [4] analyzed the quantizing noise (assuming no slope overload) associated with a delta modulator ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaM</tex> ) with step-size imbalance. In this concise paper we consider the effects of both step-size imbalance and quantizer threshold hysteresis on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\DeltaM</tex> quantizing noise. The analysis is restricted to the important case of zero input signal. For this case we extend Iwersen's work to show that quantizer threshold hysteresis can significantly change the spectral shape of the idle channel noise. Our approach is to first construct two locus functions, one for the samples (of the steady-state noise waveform) taken at arbitrarily defined even sampling times and one for those taken at the odd sampling times. The noise waveform is then obtained as a superposition of interleaved samples of the even and odd loci. It is shown that the noise spectrum has a discrete tonal struture whose frequencies are independent of hysteresis and are determined solely by step imbalance as in Iwersen's analysis. Hysteresis changes the magnitude and phase of the discrete tones. We characterize the tone magnitude changes due to hysteresis and show that hysteresis can amplify, attenuate, or completely eliminate a tone. We present curves which specify the tone magnitude attenuation as a function of step imbalance ε, average step size σ, and hysteresis <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> . We also calculate P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> , the total noise power in the baseband up to half the sampling frequency. It is shown that for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 \leq h \leq \epsilon</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{T} = \sigma^{2}/3</tex> independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</tex> and ε ;for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h &gt; \epsilon</tex> , P <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</inf> is an increasing function of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon/\sigma</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h/\epsilon</tex> .