Analytical Derivation of Quantization Error in Threshold Level Quantizers Using Bipolar PFM

TL;DR

The paper addresses the analytic description of quantization noise, challenging the common reliance on statistical spectral descriptions by introducing a bipolar Pulse Frequency Modulation (PFM) equivalence to a uniform quantizer. It derives a Fourier-domain representation of the quantized signal using PFM series, showing that the output spectrum $Y_p(ω)$ comprises the input spectrum plus a quantization-error term $E(ω)$, with modulation sidebands described by Bessel-function weights. A bipolar-PFM model is developed (including $|P_k|=Δ$) to handle nonzero-mean inputs, and the analysis predicts that quantization error forms a delta-series of sidebands centered around DC. The theory is validated via simulations of level-crossing ADCs with zero-order-hold interpolation, demonstrating good agreement with the analytical predictions and highlighting practical applicability for accurate performance estimation in low-power time-sparse-signal processing systems.

Abstract

Uniform quantization is a topic that has been extensively studied. However and although an analytical description of quantization noise has been proposed, most descriptions of the spectral properties of quantization error resort to statistical descriptions. In this paper, we show how the spectrum of a quantized signal can be expressed using pulse frequency modulation. We first establish the equivalence of a uniform quantizer with a system based on the bipolar pulse frequency modulation and we define afterwards the Fourier transform of the quantized signal using pulse frequency modulation properties. This model brings a more intuitive understanding of the spectral structure of quantization noise and complements prior research in the topic. The results of the paper can be directly applied to level crossing ADCs with zero-order-hold interpolators, giving an accurate estimation of their performance.

Figures (6)

• Figure 1: a) Integrator followed by uniform quantizer. b) PFM equivalent of a)
• Figure 2: a) Uniform quantizer. b) Differential-Integral model.
• Figure 3: a) Input sinusoid. b) Input derivative. c) bipolar PFM. d) Signal $y_p(t)$. e) Signal $y_q(t)$
• Figure 4: a) Single polarity PFM b) Bipolar PFM
• Figure 5: a) simulated PFM. b) Analytical PFM. c) Analytical quantized signal. d) Simulated quantized signal.