Sometimes it is much easier to become a fan of something when you only know something about it. Just like Sigma-Delta ADC, it is so complicated that even though I have learned it for several times I still can’t fully understand it!

Nevertheless, I am still a fan of it ;-).

**Sigma-Delta ADCs dominate in the high-resolution domain (though they are not extremely fast, actually kind of slow…).**

Fig. 1 Signal-to-noise-and-distortion ratio (SNDR) versus sampling frequency of Sigma-Delta ADCs and other Nyquist ADCs (SAR, Pipeline, and Flash). The data were reported in ISSCC, and collected by Murmann’s ADC survey [1].

I am currently doing successive-approximation-register (SAR) ADC.
**SAR ADCs are quite energy-efficient, but less accurate than Sigma-Delta ADCs. **

Fig. 2 Energy (P/fs) versus SNDR of SAR ADCs, Sigma-Delta ADCs, and other Nyquist ADCs (Pipeline and Flash). The data were again extracted from Murmann’s ADC survey [1].

**In order to achieve high resolution, can SARs shape the noise just as Sigma-Delta ADCs do? **
People have tried to imploy noise-shaping technique into the SAR architecture [2, 3], but so far the reported performance (with chip measurement) is not very compelling (SNDR = 62dB , Power = 806uW, Bandwidth = 11MHz, FoM = 35.8fJ/conv) [3].

**Nevertheless, the idea of noise-shaping SAR is so intriguing.**

Before entering into this topic, I would like to do some warm-ups – some basics of Sigma-Delta ADCs (yes, that’s all I know about it).

**Some basics of Sigma-Delta ADCs:**

**1. Oversampling **

Fig.3 Brief illustration of oversampling (OSR is the abbreviation of oversampling ratio)

**Doubling the sampling frequency gives 3 dB increase of SNR. **However, oversampling is seldom used alone, and it is commonly used together with the noise-shaping technique.

**2. Noise-shaping**

Fig.4 Brief illustration of noise-shaping and the sigma-delta modulator

Filtering is introduced into the ADC to further suppress the in-band quantization noise power. At the same time, the filtering does not affect the input signal. By applying a loop filter before the quantizer and introducing the feedback, a sigma delta modulator is built.

**3. Linear model of a sigma-delta modulator **

Fig.5 Linear model of a sigma-delta modulator, STF and NTF are abbreviations of signal transfer function and noise transfer funcion, respectively. (More information can be referred to Schreier’s book[4])

According to STF and NTF, if the transfer function of the loop filter H(z) is designed to have a large gain inside the band of interest and small gain outside the band of interest, the signal can pass the modulator and the noise can be greatly reduced.

**4. If an integrator is chosen to be the loop filter**

Fig. 6 Modulator with an integrator as the loop filter and its STF and NTF

We do a plot of H(f), STF(f), and NTF(f) (Matlab ‘*fvtool’* is used):

Fig. 7 Magnitude response of H(f), STF(f), NTF(f)

Bingo! The signal is passed to the output with a delay of a clock cycle, while the quantization noise is passed through a high-pass filter.

**Doubling the sampling frequency gives 9 dB increase of SNR for 1st order noise shaping.**

**5. Get more aggressive on the order**

Fig. 8 Magnitude response of NTF from 0th – 3rd order

This post tells the basic story of noise-shaping. In the next post, I will try to learn how noise-shaping can be used in SAR ADCs.

References:

[1] B. Murmann, “ADC Performance Survey 1997-2014,” [Online]. Available: http://www.stanford.edu/~murmann/adcsurvey.html.

[2] K. S. Kim, J. Kim, and S. H. Cho, “nth-order multi-bit \Sigma-\Delta ADC using SAR quantiser”, *Electronics Letters*, vol. 46, 2010.

[3] J. A. Fredenburg and M. P. Flynn, “A 90-MS/s 11-MHz-Bandwidth 62-dB SNDR Noise-shaping SAR ADC”, *JSSC*, vol.47, 2012.

[4] R. Schreier and G. C. Temes, *Understanding Delta-Sigma Data Converters*, 2005.