## Brief Study of Noise-Shaping SAR ADC – Part A

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.

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).

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:

[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.

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### 3 Responses to Brief Study of Noise-Shaping SAR ADC – Part A

1. Zeinab says:

Hey, I’m really fortunate to find your blog. It seems you’re an expert in SAR ADC design! 🙂
I did research on Sigma-delta and Incremental ADCs, and now I want to use some special SAR structures in my design. But I’m new in this field and I even don’t know how I can simulate a SAR or a noise-shaping SAR ADC in system level! 😦
Should I write Matlab codes? Or better to use Simulink? Or maybe there are some other simulators which work better with this special type of ADC?!!

I’ll be very grateful if you could help me in SAR ADC design and simulation.