Delta-Sigma Modulator
"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]
- a delta–sigma ADC consists of an analog modulator followed by a digital filter
- a delta–sigma DAC consists of a digital modulator followed by an analog filter
Analog Delta Sigma Modulators (ADSM) are used in the context of analog-to-digital conversion
- In a CT delta-sigma ADC, there is no need for an anti-aliasing filter or a front-end sampler
Digital Delta Sigma Modulators (DDSM) are commonly used in digital to-analog conversion and fractional-N frequency synthesis
- In a DDSM, the input is digital and the filters are implemented digitally
- the input to the DDSM is often a constant digital word, this covers delta-sigma fractional-N synthesizers in the frequency generation application
Oversampling Advantage
David Johns and Ken Martin. Oversampling Converters [https://www.eecg.toronto.edu/~johns/ece1371/slides/14_oversampling.pdf]
[https://dsp.stackexchange.com/a/40261/59253]
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]
Time and Frequency Domain
\(M \gt N\)
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]
interpolation filter
Notice that the requirements of the first stage are very demanding
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
MOD1 with DC Excitation
TODO 📅
decimation filter
The combination of the the digital post-filter and downsampler is called the decimation filter or decimator
\(\text{sinc}\) filter
Provided that \(T=1\) \[ H_1(e^{j2\pi f}) = \frac{\text{sinc}(Nf)}{\text{sinc}(f)} = \frac{1}{N}\frac{\sin(\pi Nf)}{\sin(\pi f)} \] that is \(\lim_{f\to 0^+}H_1(e^{j2\pi f}) = 1\) and \(H_1 = 0\) when \(f=\frac{n}{N}, n\in \mathbb{Z}\)
\(\text{sinc}^2\) filter
[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/First_Order_DS_ADC_scan1.pdf]
[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/First_Order_DS_ADC_scan2.pdf]
Truncation DAC
The noise-shaping loop output must contain a faithful reproduction of the input signal \(u_0[n]\) in the baseband,
but it will also include the filtered truncation noise caused by the reduction of the word length in the loop.
Idealy, the DAC will reproduce its input digital signal in an analog form without any distortion
with \(\frac{y}{2^{m_2}} + q= v\), where \(v = \lfloor\frac{y}{2^{m_2}}\rfloor\)
\[ \left\{ \begin{array}{cl} Y + 2^{m_2} Q &= 2^{m_2}V \\ U - z^{-1}2^{m_2}Q &= Y \end{array} \right. \]
The STF & NTF is shown as below \[ V = \frac{1}{2^{m_2}}U + (1-z^{-1})Q \]
1 | m1 = 4 # MSBs |
u | v_avg | |
---|---|---|
0 | ![]() |
0000_00 |
1 | ![]() |
0000_01 |
60 | ![]() |
1111_00 |
61 | ![]() |
1111_01 |
62 | ![]() |
1111_10 |
!!! The \(u\) is limited between 0 and 60 (MSBs_LSBs - LSBs)
Tuan Minh Vo, S. Levantino and C. Samori, "Analysis of fractional-n bang-bang digital PLLs using phase switching technique," 2016 12th Conference on Ph.D. Research in Microelectronics and Electronics (PRIME), Lisbon, Portugal, [https://sci-hub.se/10.1109/PRIME.2016.7519545]
An implementation of a high-resolution integral path using a digital delta-sigma modulator, low-resolution Nyquist DAC, and a lowpass filter
- \(\Delta \Sigma\) truncates \(n\)-bit accumulator output to \(m\)-bits with \(m\le n\)
- A \(m\)-bit Nyquist DAC outputs current, which is fed into a low pass filter that suppresses \(\Delta \Sigma\)'s quantization noise
The remaining 11 bits are truncated to 3-levels using a second-order delta-sigma modulator (DSM), thus, obviating the need for a high resolution DAC
Hanumolu, Pavan Kumar. "Design techniques for clocking high performance signaling systems" [https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/1v53k219r]
1st order DDSM
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]
LSB Dither
dithering break periodicity and convert them to noise while input is constant
drawback of Integer-N PLL
integer-N PLL frequency synthesizers
the frequency resolution, is equal to the reference frequency, meaning that only integer multiples of the reference frequency can be synthesized
if fine tuning is required, only choice in an integer-N PLL is to decrease the reference frequency
Stability requirements limit the loop bandwidth to about one tenth of the reference frequency; therefore, decreasing the reference frequency increases the settling time as the loop bandwidth also has to be decreased
Another drawback of the integer-N PLL is the trade-off between phase noise and settling time when the divider ratio becomes large (The contributions to the output phase noise of almost all PLL building blocks, except the VCO, are multiplied by the division ratio)
[https://people.engr.tamu.edu/spalermo/ecen620/lecture03_ee620_pll_system.pdf]
if a small reference frequency is chosen, the reference spur in the output phase noise is located at a smaller offset frequency
Fractional-N PLL
\[
\tau[n-1] + (N+y[n])T_{PLL} - (N+\alpha)T_{PLL} = \tau[n]
\]
i.e. \[ \tau[n] = \tau[n-1] + (y[n] - \alpha)T_{PLL} \]
where \(\tau[n] = t_{v_{DIV}} - t_{v_{DIV}, desired}\)
In \(z\)-domain \[ \left\{(A + D - Y)\frac{z^{-1}}{1-z^{-1}} - 2Y \right\}\frac{z^{-1}}{1-z^{-1}} + Q = Y \] That is \[ Y = A z^{-2} + Dz^{-2} + Q(1-z^{-1})^2 \] In time domain \[\begin{align} y[n] &= \alpha[n-2] + d[n-2] + q[n]-2q[n-1]+q[n-2] \\ &= \alpha + d[n-2] + q[n]-2q[n-1]+q[n-2] \end{align}\]
quantizer overload
TODO 📅
reference
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]
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]
Pavan, Shanthi, Richard Schreier, and Gabor Temes. (2016) 2016. Understanding Delta-Sigma Data Converters. 2nd ed. Wiley.
Horowitz, P., & Hill, W. (2015). The art of electronics (3rd ed.). Cambridge University Press. [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]
Boris Murmann, ISSCC2022 SC1: Introduction to ADCs/DACs: Metrics, Topologies, Trade Space, and Applications [link]
Ian Galton. Delta-Sigma Fractional-N Phase-Locked Loops [https://ispg.ucsd.edu/wordpress/wp-content/uploads/2022/10/fnpll_ieee_tutorial_2003_corrected.pdf]
Joshua Reiss. Understanding sigma delta modulation: the solved and unsolved issues
Ian Galton ISSCC 2010 SC3: Fractional-N PLLs [https://www.nishanchettri.com/isscc-slides/2010%20ISSCC/Short%20Course/SC3.pdf]
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
Sudhakar Pamarti. CICC 2020 ES2-2: Basics of Closed- and Open-Loop Fractional Frequency Synthesis [https://youtu.be/t1TY-D95CY8?si=tbav3J2yag38HyZx]
S. Pamarti, J. Welz and I. Galton, "Statistics of the Quantization Noise in 1-Bit Dithered Single-Quantizer Digital Delta–Sigma Modulators," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 3, pp. 492-503, March 2007 [https://ispg.ucsd.edu/wordpress/wp-content/uploads/2017/05/2007-TCASI-S.-Pamarti-Statistics-of-the-Quantization-Noise-in-1-Bit-Dithered-Single-Quantizer-Digital-Delta-Sigma-Modulators.pdf]
S. Pamarti and I. Galton, "LSB Dithering in MASH Delta–Sigma D/A Converters," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 4, pp. 779-790, April 2007 [https://sci-hub.se/10.1109/TCSI.2006.888780]
Michael Peter Kennedy. An Introduction to Digital Delta-Sigma Modulators [https://site.ieee.org/scv-cas/files/2014/07/2014Kennedy.pdf]
Kaveh Hosseini , Michael Peter Kennedy. Springer 2011. Minimizing Spurious Tones in Digital Delta-Sigma Modulators