Delta-Sigma Modulator

Posted on Edited on In

image-20250611074830238

"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]

image-20250616223003455

  • 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

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Oversampling Advantage

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David Johns and Ken Martin. Oversampling Converters [https://www.eecg.toronto.edu/~johns/ece1371/slides/14_oversampling.pdf]

Over Sampling

Nyquist sampling theorem @signal: no aliasing, signal remain the same

noise folding @noise: same total noise power spread over a wider frequency

[https://dsp.stackexchange.com/a/40261/59253]

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Noise Shaping

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[h1, w1] = freqz([1 -1], 1);
[h2, w2] = freqz([1 -2 1], 1);

plot(w1/2/pi, abs(h1), LineWidth=3)
hold on
plot(w2/2/pi, abs(h2), LineWidth=3)
grid on
legend('MOD1', 'MOD2')
xlabel('fs')
ylabel('mag')
title('NTF of MOD1 & MOD2')

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

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DAC

image-20250617223537672

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

img

[https://www.linkedin.com/posts/danboschen_signalprocessing-dsp-pythonforengineers-activity-7345777588746788866-SprG?utm_source=share&utm_medium=member_desktop&rcm=ACoAAD-cuiIBDJ62eh9q3qTSSdslYXr-XMd8TGw]

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// https://github.com/hamsternz/second_order_sigma_delta_DAC

`timescale 1ns / 1ps
module second_order_dac(
  input wire i_clk,
  input wire i_res,
  input wire i_ce,
  input wire [15:0] i_func, 
  output wire o_DAC
);

  reg this_bit;
 
  reg [19:0] DAC_acc_1st;
  reg [19:0] DAC_acc_2nd;
  reg [19:0] i_func_extended;
   
  assign o_DAC = this_bit;

  always @(*)
     i_func_extended = {i_func[15],i_func[15],i_func[15],i_func[15],i_func};
    
  always @(posedge i_clk or negedge i_res)
    begin
      if (i_res==0)
        begin
          DAC_acc_1st<=16'd0;
          DAC_acc_2nd<=16'd0;
          this_bit = 1'b0;
        end
      else if(i_ce == 1'b1) 
        begin
          if(this_bit == 1'b1)
            begin
              DAC_acc_1st = DAC_acc_1st + i_func_extended - (2**15);
              DAC_acc_2nd = DAC_acc_2nd + DAC_acc_1st     - (2**15);
            end
          else
            begin
              DAC_acc_1st = DAC_acc_1st + i_func_extended + (2**15);
              DAC_acc_2nd = DAC_acc_2nd + DAC_acc_1st + (2**15);
            end
          // When the high bit is set (a negative value) we need to output a 0 and when it is clear we need to output a 1.
          this_bit = ~DAC_acc_2nd[19];
        end
    end
endmodule

Time and Frequency Domain

image-20250627193435726

\(M \gt N\)

[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/Intro_to_Delta_Sigma_Data_Converters.pdf]

ADC

image-20250611234653738

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hackaday. Tearing Into Delta Sigma ADC’s [https://hackaday.com/2016/07/07/tearing-into-delta-sigma-adcs-part-1/]

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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

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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]

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David Johns and Ken Martin. Oversampling Converters [https://www.eecg.toronto.edu/~johns/ece1371/slides/14_oversampling.pdf]

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[https://web.engr.oregonstate.edu/~temes/ece627/Lecture_Notes/Intro_to_Delta_Sigma_Data_Converters.pdf]

interpolation filter

Notice that the requirements of the first stage are very demanding

image-20250617001439043

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

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there cannot be a "delay free loop" is a common idea in sequential digital state machine design

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Both integrator and quantizer are delay free

NTF realizability criterion: No delay-free loops in the modulator

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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

image-20241005120659945 \[ V(z) = U(z) +(1-z^{-1})E(z) \]

  • A binary DAC (and hence a binary modulator) is inherently linear
  • With a CT loop filter, MOD1 has inherent anti-alising

image-20241005202024498 \[\begin{align} v[1] &= u - (0) + e[1] \\ v[2] &= 2u - (v[1]) + e[2] \\ v[3] &= 3u - (v[1]+v[2]) + e[3] \\ v[4] &= 4u - (v[1]+v[2]+v[3]) + e[4] \end{align}\]

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 \[\begin{align} v[n] &= nu - \sum_{k=1}^{n-1}v[k] + e[n] \\ &= u + \left((n-1)u - \sum_{k=1}^{n-2}v[k]\right) - v[n-1] + e[n] \\ &= u + v[n-1] - e[n-1] -v[n-1] + e[n] \\ &= u + e[n] - e[n-1] \end{align}\]

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Dout, the low frequency component of ADC out is same with Vin

MOD2

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MOD1 with DC Excitation

TODO 📅

decimation filter

The combination of the the digital post-filter and downsampler is called the decimation filter or decimator

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\(\text{sinc}\) filter

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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 \(f=\frac{n}{N}, n\in \mathbb{Z}\)

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\(\text{sinc}^2\) filter

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[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

truncator_1bit.drawio

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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 \[ V = \frac{1}{2^{m_2}}U + (1-z^{-1})Q \]

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m1 = 4  # MSBs
m2 = 2  # LSBs
Vmax = 2**(m1 + m2) - 1

u = 1

ylist = [0]
vlist = [0]
elist = []

Niter = 2**10
for _ in range(Niter):
    ecur = vlist[-1] - ylist[-1]
    elist.append(ecur)
    ycur = (u - ecur) % Vmax	# overflow
    ylist.append(ycur)
    ycur_bin = format(ycur, '06b')
    vcur = int(ycur_bin[:-2]+'00', 2)
    vlist.append(vcur)

print(ylist)
print(vlist)
print(sum(vlist)/len(vlist))

image-20250607161739820

u v_avg
0 image-20250609233713939 0000_00
1 image-20250609233741985 0000_01
60 image-20250609233808262 1111_00
61 image-20250609233837090 1111_01
62 image-20250609233903431 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]

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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

image-20241022233749243

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

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Mismatch Shaping

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Data-Weighted Averaging (DWA)

image-20241113000942025 \[\begin{align} \sum_{i=0}^{n}v[i] + e_\text{DAC}[n] &= y[n] \\ \sum_{i=0}^{n-1}v[i] + e_\text{DAC}[n-1] &= y[n-1] \end{align}\]

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:

image-20241112233059745

[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

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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]

    image-20250602100424369

  • 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

image-20250530190858386 \[ \tau[n-1] + (N+y[n])T_{PLL} - (N+\alpha)T_{PLL} = \tau[n] \]

i.e. \[ \tau[n] = \tau[n-1] + (y[n] - \alpha)T_{PLL} \]

where \(\tau[n] = t_{v_{DIV}} - t_{v_{DIV}, desired}\)

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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 \[\begin{align} y[n] &= \alpha[n-2] + d[n-2] + q[n]-2q[n-1]+q[n-2] \\ &= \alpha + d[n-2] + q[n]-2q[n-1]+q[n-2] \end{align}\]

image-20250601201952868

quantizer overload

TODO 📅

CIC filter

Cascaded Integrator-Comb (CIC) Filters

Let’s focus on decimation: if we decimate by a factor 4, we simply retain one output sample out of every 4 input samples.

In the example below, the downsampler at the right drops those 3 samples out of 4, and the output rate, \(y^\prime(n)\), is one fourth of the input rate \(x(n)\):

moving_average_filters-decimation_trivial \[\begin{align} Y(z) &= X(z)\frac{1-z^{-4}}{1-z^{-1}} \\ Y^\prime(\xi) &= \frac{1}{4}Y(\xi^{1/4}) = \frac{1}{4}X(\xi^{1/4})\frac{1-\xi^{-1}}{1-\xi^{-1/4}} \end{align}\]

with \(z=e^{j\Omega/f_s}\) and \(\xi =z^4\), we have \[ Y^\prime(z) = \frac{1}{4}X(z)\frac{1-z^{-4}}{1-z^{-1}} \]

But if we're going to be throwing away 75% of the calculated values, can't we just move the downsampler from the end of the pipeline to somewhere in the middle? Right between the integrator stage and the comb stage? That answer is yes, but to keep the math working, we also need to divide the number of delay elements in the comb stage by the decimation rate:

moving_average_filters-decimation_smart

\[\begin{align} A(z) &= X(z)\frac{1}{1-z^{-1}} \\ A^\prime(\xi) &= \frac{1}{4}A(\xi^{1/4}) = \frac{1}{4}X(\xi^{1/4})\frac{1}{1-\xi^{-1/4}} \\ Y^\prime(\xi) &= A^\prime(\xi) (1-\xi^{-1}) = \frac{1}{4}X(\xi^{1/4})\frac{1-\xi^{-1}}{1-\xi^{-1/4}} \end{align}\]

with \(z=e^{j\Omega/f_s}\) and \(\xi =z^4\), we have \[ Y^\prime(z) = \frac{1}{4}X(z)\frac{1-z^{-4}}{1-z^{-1}} \]

And we can do this just the same with cascaded sections (without downsampler or updampler) where integrators and combs have been grouped

  • for decimation, the integrators come first and the combs second with the downsampler in between
  • For interpolation, the reverse is true
    • the incoming sample rate is fraction of the outgoing sample rate, the combs must come first and the interpolators second

moving_average_filters-integrator_comb_decimated

moving_average_filters-comb_integrator_interpolated

Tom Verbeure. An Intuitive Look at Moving Average and CIC Filters [https://tomverbeure.github.io/2020/09/30/Moving-Average-and-CIC-Filters.html]

—. Half-Band Filters, a Workhorse of Decimation Filters [https://tomverbeure.github.io/2020/12/15/Half-Band-Filters-A-Workhorse-of-Decimation-Filters.html]

—. Design of a Multi-Stage PDM to PCM Decimation Pipeline [https://tomverbeure.github.io/2020/12/20/Design-of-a-Multi-Stage-PDM-to-PCM-Decimation-Pipeline.html]

Arash Loloee, Ph.D. Exploring Decimation Filters [https://www.highfrequencyelectronics.com/Archives/Nov13/1311_HFE_decimationFilters.pdf]

Rick Lyons. A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters [https://www.dsprelated.com/showarticle/1337.php]

DC Gain in Interpolation Filtering

[https://raytroop.github.io/2025/06/21/data-converter-in-action/#dac-zoh]

DC gain is used to compensate the ratio of sampling rate before and after upsample

image-20250701070539064

Given \[ X_e = X = \propto \frac{1}{T} = \frac{1}{L\cdot T_i} \] Then, the lowpass filter (ZOH, FOH .etc) gain shall be \(L\)

Employ definition of DTFT, \(X(e^{j\hat{\omega}}) =\sum_{n=-\infty}^{+\infty}x[n]e^{-j\hat{\omega} n}\), and set \(\hat{\omega} = 0\) \[ X(e^{j0}) = \sum_{n=-\infty}^{+\infty}x[n] \] That is, \(\sum_{n=-\infty}^{+\infty}x[n] = \sum_{n=-\infty}^{+\infty}x_e[n]\), so \[ \overline{x_e[n]} = \frac{1}{L} \overline{x[n]} \] It also indicate that dc gain of upsampling is \(1/L\)

ZOH

Zero-Order Hold (ZOH)

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dc gain = \(N\)

FOH

First-Order Hold (FOH)

image-20250630235714996

dc gain = \(N\)

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

[https://www.eecs.qmul.ac.uk/~josh/documents/2008/Reiss-JAES-UnderstandingSigmaDeltaModulation-SolvedandUnsolvedIssues.pdf]

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

Venkatesh Srinivasan, ISSCC 2019 T5: Noise Shaping in Data Converters

Arash Loloee, Ph.D. Exploring Decimation Filters [https://www.highfrequencyelectronics.com/Archives/Nov13/1311_HFE_decimationFilters.pdf]