# Continuous-Time Sigma-Delta A D Conversion: Fundamentals, by Friedel Gerfers, Maurits Ortmanns

By Friedel Gerfers, Maurits Ortmanns

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**Extra resources for Continuous-Time Sigma-Delta A D Conversion: Fundamentals, Performance Limits and Robust Implementations (Advanced Microelectronics, Volume 21)**

**Sample text**

15) the performance of an ideal N th order Σ∆ modulator could be derived. But actually, the agreement of calculation and simulation results would be poor and not transferable to real implementations. 24 2 Basic Understanding of Σ∆ A/D Conversion Fig. 13. NTF(z) for an ideal N th order Σ∆ modulator The reason for this is twofold: First, due to the required modeling of the quantizer as gain and additive white noise source as in Sect. 2, where the gain was not yet speciﬁed in detail for single-bit quantizers.

5a the total quantization noise power can be calculated as: ∞ σe2 e2 pdf e de = = −∞ ∆2 . 4) It should be noted at this point that the total quantization error power is independent of the sampling frequency and is only determined by the quantizer resolution. Since the signals at the quantizer are sampled (discrete-time) signals, all the quantization noise power σe2 is folded into the frequency range [−fS /2, fS /2]. Thus, with the white noise approximation in Fig. 5b the power spectral density of the quantization noise is: Se (f ) = ∆2 1 .

As a consequence, the IBN and the DR will be improved. A detailed analysis concerning the performance gain as well as the optimal zero placements for modulators up to the eighth order is given in [4]. The optimal ai H(z) ai+1 H(z) H(z) Fig. 17. 1 f /fS f /fS (a) (b) Fig. 18. Power spectrum, IBN and calculated NTF of the feed-forward structure with and without optimized zeros. Psig = −15 dB, fsig = fB and OSR = 48. IBN ≈ −87 dB, IBNOpt ≈ −93 dB. (a) Without optimized zero; (b) with optimized zero zero placement can be found by minimizing the integrated IBN with respect to the feedback coeﬃcient γ: ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ fB 2 |NTF(f, γ)| df .