A Different View of Sigma-Delta Modulators Under the Lens of Pulse Frequency Modulation

TL;DR

This work reinterprets Sigma-Delta modulation through a sampling-invariance connection to Pulse Frequency Modulation (PFM), showing that CTSD and VCO-based implementations can be viewed as PFM encoders with sampling either inside or outside the loop. By rigorously establishing first-order equivalence and extending it to second and third order (and CIFB structures), it decomposes quantization noise into PFM sidebands and alias components, enabling prediction of spurious tones beyond the white-noise model. The paper also analyzes overload constraints, deriving conditions under which the PFM equivalence remains valid and highlighting the dynamic-range limits of higher-order and multi-bit modulators. Overall, the PFM perspective provides a complementary, physically intuitive framework to understand nonlinear phenomena and spurious content in Sigma-Delta modulators, with practical implications for CTSD design and overload handling.

Abstract

The fact that VCO-ADCs produce noise-shaped quantization noise suggests that a link between frequency modulation and Sigma-Delta modulation should exist. The connection between a VCO-ADC and a first-order Sigma-Delta modulator has been already explained using Pulse Frequency Modulation. In this paper, we attempt to extend the theory based on Pulse Frequency Modulation to a generic Sigma-Delta modulator. We show that this link between Sigma-Delta modulation and Pulse Frequency Modulation relies in a sampling invariance property that defines the equivalence between both entities. This novel point of view, allows to go beyond the white quantization noise model of a Sigma-Delta modulator, revealing the origin of some nonlinear phenomena. We first predict spurious tones which cannot be explained by circuit non linearity. Multi-bit and single bit modulators are shown to belong to a same generic class of systems. Finally, quantizer overload is analyzed using our model. The results are applied to Continuous-Time Sigma-Delta modulators of orders one, two and three and then extended to a generic case.

Figures (28)

• Figure 1: Model of a PFM modulator.
• Figure 2: Input signal, PFM and pulse shaping filter output.
• Figure 3: Calculated spectra of $d(t)$ with the first six side bands for the case of an input signal with A=1/8 and $f_0=128f_x$, (a) linear frequency axis, (b) logarithmic frequency axis, x(t) can be observed at $f_x=1/128$ with an amplitude of -19 dB corresponding to A=1/8.
• Figure 4: Equivalence between a first-order single-bit CTSD and a PFM based model.
• Figure 5: Time domain waveforms of $v(t)$, $p(t)$, $y_{SD}(t)$ and $e(t)$ of Fig. \ref{['fig:fig1_VM']}