Delta-Sigma Modulator
Classification
- Continuous-Time (CT) Analog Modulator
- Sampled Quantizer: Synchronous Modulator
- Unsampled Quantizer: Asynchronous Modulator
- Discrete-Time (DT) Analog Modulator
- The main application is in \(\Delta\Sigma\) ADCs
- Discrete-Time Digital Modulators (DDSM)
- Single Quantizer DDSMs - output feedback
- Error feedback modulators (EFM) - error feedback
- Multi stAge noise SHaping (MASH)
"Quantizers" and "truncators", and "integrators" and "accumulators" are used in delta-sigma ADCs and DACs, respectively
P. Kiss, J. Arias and Dandan Li, "Stable high-order delta-sigma DACS," 2003 IEEE International Symposium on Circuits and Systems (ISCAS), Bangkok, 2003 [https://www.ele.uva.es/~jesus/analog/tcasi2003.pdf]
Oversampling
David Johns and Ken Martin. Oversampling Converters [https://www.eecg.toronto.edu/~johns/ece1371/slides/14_oversampling.pdf]
Nyquist sampling theorem @signal: no aliasing, signal remain the same
noise folding @noise: same total noise power spread over a wider frequency
Noise Shaping
1 | [h1, w1] = freqz([1 -1], 1); |
SQNR improvement
In general, for an \(l\)th order modulator with \(\text{NTF}(z) = (1 − z^{−1})^l\), the SQNR increases by \((6l + 3)\) dB for every doubling of the OSR, which provides \(l+0.5\) extra bits resolution
without the delta-sigma loop
\(10\log (2) \approx 3\)dB
first order delta-sigma modulator
\(30\log (2) \approx 9\)dB
second order delta-sigma modulator
\(50\log (2) \approx 15\)dB
1 | OSR= linspace(1,16,16); |
TI. ADC12EU050 Continuous-Time Sigma-Delta ADCs [https://www.ti.com/lit/an/snaa098/snaa098.pdf]
where \(N\) is the number of bits in the output, \(M\) is known as the over-sampling ratio, \(L\) is loop orders
quantizer levels
The greater the number of quantizer levels, the smaller quantization error
quantizer overload
\(\Delta \Sigma\) vs. \(\Delta\) modulation
\(\Delta \Sigma\) modulators, and other noise-shaping modulators, change the spectrum of the noise but leave the signal unchanged
\(\Delta\) modulators and other signal-predicting modulators shape the spectrum of the modulated signal but leave the quantization noise unchanged at the receiver
output vs. error-feedback
The error-feedback architecture is problematic for analog implementation, since it is sensitive to variations of its parameters (subtractor realization)
- The error-feedback structure is thus of limited utility in \(\Delta \Sigma\) ADCs
- The error-feedback structure is very useful and applied in digital loops required in \(\Delta \Sigma\) DACs
ADC
DAC
P. Kiss, J. Arias and Dandan Li, "Stable high-order delta-sigma DACS," 2003 IEEE International Symposium on Circuits and Systems (ISCAS), Bangkok, 2003 [https://www.ele.uva.es/~jesus/analog/tcasi2003.pdf]
output-feedback
1 | // https://github.com/hamsternz/second_order_sigma_delta_DAC |
Time and Frequency Domain
\(M \gt N\)
Chun-Hsien Su ( 蘇純賢). Fundamentals of Sigma-Delta Data Converters,July, 2006 [pdf]
ADC
hackaday. Tearing Into Delta Sigma ADC’s [https://hackaday.com/2016/07/07/tearing-into-delta-sigma-adcs-part-1/]
DAC
an interpolation filter effectively up-samples its low-rate input and lowpass-filters the resulting high-rate data to produce a high-rate output devoid of images
P.E. Allen -CMOS Analog Circuit Design: Lecture 39 – Oversampling ADCs – Part I (6/26/14) [https://aicdesign.org/wp-content/uploads/2018/08/lecture39-140626.pdf]
P.E. Allen -CMOS Analog Circuit Design: Lecture 40 – Oversampling ADCs – Part II (7/17/15) [https://aicdesign.org/wp-content/uploads/2018/08/lecture40-150717.pdf]
David Johns and Ken Martin. Oversampling Converters [https://www.eecg.toronto.edu/~johns/ece1371/slides/14_oversampling.pdf]
No delay-free loops
Any such physically feasible device will take a finite time to operate – in other words, the quantized output will only be available a small time after the quantizer has "looked" at the input - insert a one-sample delay
there cannot be a "delay free loop" is a common idea in sequential digital state machine design
Both integrator and quantizer are delay free
NTF realizability criterion: No delay-free loops in the modulator
linear settling & GBW of amplifier
TODO 📅
Switched capacitor has been the common realization technique of discrete-time (DT) modulators, and in order to achieve a linear settling, the sampling frequency used in these converters needs to be significantly lower than the gain bandwidth product (GBW) of the amplifiers.
MOD1 & MOD2
MOD1: first-order noise-shaped converter (\(\Delta\Sigma\) modulator)
MOD2: second-order noise-shaped converter (\(\Delta\Sigma\) modulator)
MOD1
\[
V(z) = U(z) +(1-z^{-1})E(z)
\]
- A binary DAC (and hence a binary modulator) is inherently linear
- With a CT loop filter, MOD1 has inherent anti-alising
\[\begin{align}
v[1] &= u - (0) + e[1] \\
v[2] &= 2u - (v[1]) + e[2] \\
v[3] &= 3u - (v[1]+v[2]) + e[3] \\
v[4] &= 4u - (v[1]+v[2]+v[3]) + e[4]
\end{align}\]
That is \[ v[n] = nu - \sum_{k=1}^{n-1}v[k] + e[n] \] Therefore, we have \(v[n-1] = (n-1)u - \sum_{k=1}^{n-2}v[k] + e[n-1]\), then \[\begin{align} v[n] &= nu - \sum_{k=1}^{n-1}v[k] + e[n] \\ &= u + \left((n-1)u - \sum_{k=1}^{n-2}v[k]\right) - v[n-1] + e[n] \\ &= u + v[n-1] - e[n-1] -v[n-1] + e[n] \\ &= u + e[n] - e[n-1] \end{align}\]
Dout, the low frequency component of ADC out is same with Vin
MOD2
[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/2nd_Higher_Order.pdf]
MOD1 with DC Excitation
TODO 📅
Mismatch Shaping
Data-Weighted Averaging (DWA)
\[\begin{align}
\sum_{i=0}^{n}v[i] + e_\text{DAC}[n] &= y[n] \\
\sum_{i=0}^{n-1}v[i] + e_\text{DAC}[n-1] &= y[n-1]
\end{align}\]
and we have \(w[n] = y[n] - y[n-1]\), then \[ w[n] = v[n] + e_\text{DAC}[n] - e_\text{DAC}[n-1] \] i.e. \[ W = V + (1-z^{-1})e_\text{DAC} \]
Element Rotation:
[http://individual.utoronto.ca/schreier/lectures/12-2.pdf], [http://individual.utoronto.ca/trevorcaldwell/course/Mismatch.pdf]
integrator leakage
When the integrator includes leakage (\(\alpha\))
\[ x[n-1] + \alpha y[n-1] = y[n] \]
then, \[ \frac{Y}{X} = \frac{z^{_1}}{1-\alpha z^{-1}} \]
reference
Pavan, Shanthi, Richard Schreier, and Gabor Temes. (2016). Understanding Delta-Sigma Data Converters. 2nd ed. Wiley.
Norsworthy, Steven R., Richard Schreier, Gábor C. Temes and Ieee Circuits. “Delta-sigma data converters : theory, design, and simulation.” (1997).
Horowitz, P., & Hill, W. (2015). The art of electronics (3rd ed.). Cambridge University Press. [pdf]
R. Schreier, ISSCC2006 tutorial: Understanding Delta-Sigma Data Converters
Shanthi Pavan, ISSCC2013 T5: Simulation Techniques in Data Converter Design [https://www.nishanchettri.com/isscc-slides/2013%20ISSCC/TUTORIALS/ISSCC2013Visuals-T5.pdf]
Bruce A. Wooley , 2012, "The Evolution of Oversampling Analog-to-Digital Converters" [https://r6.ieee.org/scv-sscs/wp-content/uploads/sites/80/2012/06/Oversampling-Wooley_SCV-ver2.pdf]
Venkatesh Srinivasan, ISSCC 2019 T5: Noise Shaping in Data Converters
B. Razavi, "The Delta-Sigma Modulator [A Circuit for All Seasons]," IEEE Solid-State Circuits Magazine, Volume. 8, Issue. 20, pp. 10-15, Spring 2016. [http://www.seas.ucla.edu/brweb/papers/Journals/BRSpring16DeltaSigma.pdf]
P. M. Aziz, H. V. Sorensen and J. vn der Spiegel, "An overview of sigma-delta converters," in IEEE Signal Processing Magazine, vol. 13, no. 1, pp. 61-84, Jan. 1996 [https://sci-hub.st/10.1109/79.482138]
Richard E. Schreier, ECE 1371 Advanced Analog Circuits - 2015 [http://individual.utoronto.ca/schreier/ece1371-2015.html]
Gabor C. Temes. ECE 627-Oversampled Delta-Sigma Data Converters [https://classes.engr.oregonstate.edu/eecs/spring2017/ece627/lecturenotes.html]
Joshua Reiss. Understanding sigma delta modulation: the solved and unsolved issues
V. Medina, P. Rombouts and L. Hernandez-Corporales, "A Different View of Sigma-Delta Modulators Under the Lens of Pulse Frequency Modulation [Feature]," in IEEE Circuits and Systems Magazine, vol. 24, no. 2, pp. 80-97, Secondquarter 2024