Delta-sigma modulation - Knowledge (XXG)

869:: These are called sinc filters, equivalent to cascading the above sinc filter N times and rearranging the order of operations for computational efficiency. Lower N filters are simpler, settle faster, and have less attenuation in the baseband, while higher N filters are slightly more complex and settle slower and have more droop in the passband, but better attenuate undesired high frequency noise. Compensation filters can however be applied to counteract undesired passband attenuation. Sinc filters are appropriate for decimating sigma delta modulation down to four times the Nyquist rate. The height of the first sideload is -13·N dB and the height of successive lobes fall off gradually, but only the areas around the nulls will alias into the low frequency band of interest; for instance when downsampling by 8, the largest 1356: 708: 4746: 756: 4760: 29: 2353: 87: 65: 5334: 1208: 673:(PDM). A sequence of positive and negative pulses, representing bits at a known fixed rate, is very easy to generate, transmit, and accurately regenerate at the receiver, given only that the timing and sign of the pulses can be recovered. Given such a sequence of pulses from a delta-sigma modulator, the original waveform can be reconstructed with adequate precision. 935: 4384: 4788:(Figure 8b voltages). It is functionally the same Analog-to-Digital ΔΣ modulation loop in Figure 1 (note: the 2-input inverting integrator combines the summing junction and integrator and produces a negative feedback result, and the flip-flop combines the sampled quantizer and conveniently naturally functions as a 1-bit DAC too). 3270: 1956: 2345:

implicitly treats the continuous analog input as a discrete-time signal. This may be a valid approximation provided that the input signal is already bandlimited and can be assumed to be not changing on time scales higher than the sampling rate. It is particularly appropriate when the modulator is implemented as a

2267: 1679: 3000: 863:. The filter can be implemented by simply counting how many samples during a larger sampling interval are high. The 1974 paper from another Bell Labs researcher, J. C. Candy, "A Use of Limit Cycle Oscillations to Obtain Robust Analog-to-Digital Converters" was one of the early examples of this. 5212:

articles explain, a periodically-sampled signal inherently contains multiple higher frequency copies or "images" of the signal. It is often desirable to remove these higher-frequency images prior to the performing the actual delta-sigma modulation stage, in order to ease requirements on the eventual

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In other words, doing delta-sigma modulation instead of delta modulation has effectively swapped the ordering of the integrator and quantizer operations. The net effect is a simpler implementation that has the profound added benefit of shaping the quantization noise to be mostly in frequencies above

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to frequencies higher than the signal. Since the accumulated error signal is lowpass filtered by the delta-sigma modulator's integrator before being quantized, the subsequent negative feedback of its quantized result effectively subtracts the low frequency components of the quantization noise while

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The delta-sigma configuration as described by Inose et al. in 1962 was devised to solve problems in the accurate transmission of analog signals. In that application it was the pulse stream that was transmitted and the original analog signal recovered with a lowpass filter after the received pulses

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Figure 9: 1-bit asynchronous ΔΣ modulation produces a PWM output (blue in bottom plot) which is subtracted from the input signal (green in top plot) to form an error signal (blue in top plot). This error is integrated (magenta in middle plot). When the integral of the error exceeds the limits (the

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of 1 μs, so every minor division corresponds to a sampling event. Since the flip-flop is assumed to be ideal, it treats any input voltage greater than 0 V as logical high and any input voltage smaller than 0 V as logical low, no matter how close it is to 0 V (ignoring issues of

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As illustrated in Figure 4, the total amount of quantization noise is the same both in a Nyquist converter (yellow + green areas) and in an oversampling converter (blue + green areas). But oversampling converters distribute that noise over a much wider frequency range. The benefit is that the

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and is often the first step prior to performing delta-sigma modulation in DACs. Upsampling is strongly associated with delta-sigma DACs but not strictly part of the actual delta-sigma modulation stage (similar to how decimation is strongly associated with delta-sigma ADCs but not strictly part of

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Another key aspect given by oversampling is the frequency/resolution tradeoff. The decimation filter put after the modulator not only filters the whole sampled signal in the band of interest (cutting the noise at higher frequencies), but also reduces the sampling rate, and hence the representable

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The synchronous ΔΣ DAC's modulation loop (Figure 6) meanwhile is in discrete-time and so its analysis is in the z-domain. It is very similar to the above analysis in Laplace domain and produces similar curves. Note: many sources also analyze a ΔΣ ADC's modulation loop in the z-domain, which

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had been reformed. This low pass filter performed the summation function associated with Σ. The highly mathematical treatment of transmission errors was introduced by them and is appropriate when applied to the pulse stream but these errors are lost in the accumulation process associated with Σ.

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at a high-bitdepth into a low-bitdepth (often 1-bit) signal, usually at a much higher sampling rate. That delta-modulated signal can then be accurately converted into analog (since lower bitdepth DACs are easier to be highly-linear), which then goes through inexpensive low-pass filtering in the

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Amplitude information is converted, without quantization noise, into time information of the output PWM. To convert this continuous time PWM to discrete time, the PWM may be sampled by a time-to-digital converter, whose limited resolution adds noise which can be shaped by feeding it back.

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property, so is commonly used in digital audio. The carry outputs are combined through summations and delays to produce a binary output, the width of which depends on the number of stages (order) of the MASH. Besides its noise shaping function, it has two more attractive properties:

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Figure 3: Top: a sine wave input overlaid with its synchronous ΔΣ representation made using a high oversampling ratio. Middle: filtering the ΔΣ representation produces an approximation of the original sine wave. Bottom: residual error of the ΔΣ ADC, with and without adding

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high frequency component may be -16 dB below the peak of the band of interest with a sinc filter but -40 dB below for a sinc filter, and if only interested in a narrower bandwidth, even fewer high frequency components will alias into it (see Figures 7–9 of Lyons

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Wooley's "The Evolution of Oversampling Analog-to-Digital Converters" gives more history and references to relevant patents. Some avenues of variation (which may be applied in different combinations) are the modulator's order, the quantizer's bitdepth, the manner of

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Noise of the quantizer can be further shaped by replacing the quantizer itself with another ΔΣ modulator. This creates a 2-order modulator, which can be rearranged in a cascaded fashion (Figure 2). This process can be repeated to increase the order even more.

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to further reduce the amount of quantization noise in the baseband in exchange for increasing noise at higher frequencies (where it can be easily filtered out). The curves of higher-order ΔΣ modulators achieve even greater reduction of noise in the baseband.

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to demodulate it to a high-bit digital output at a lower sampling-frequency. A delta-sigma DAC (e.g. Figure 1 bottom) encodes a high-resolution digital input signal into a lower-resolution but higher sample-frequency signal that may then be mapped to

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Decimation is strongly associated with delta-sigma modulation, but is distinct and outside the scope of this article. The original 1962 paper didn't describe decimation. Oversampled data in the early days was sent as is. The proposal to

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and smoothed with an analog filter for demodulation. In both cases, the temporary use of a low bit depth signal at a higher sampling frequency simplifies circuit design and takes advantage of the efficiency and high accuracy in time of

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This simple 1-order modulation can be improved by cascading two or more overflowing accumulators, each of which is equivalent to a 1-order delta-sigma modulator. The resulting multi-stage noise shaping (MASH) structure has a steeper

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it will have a steeper noise shaping curve, for improved properties of greater attenuation in the baseband, so a dramatically larger portion of the noise is above the baseband and can be easily filtered by an ideal low-pass filter.

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While 1-order modulators are unconditionally stable, stability analysis must be performed for higher-order noise-feedback modulators. Alternatively, noise-feedforward configurations are always stable and have simpler analysis.

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From the first block diagram in Figure 7, the integrator in the feedback path can be removed if the feedback is taken directly from the input of the low-pass filter. Hence, for delta modulation of input signal

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bit quantizer. For example, a simple comparator has 2 levels and so is 1 bit quantizer; a 3-level quantizer is called a "1.5" bit quantizer; a 4-level quantizer is a 2-bit quantizer; a 5-level quantizer is called a

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and quantizer of a delta modulator, so that the output carries information corresponding to the amplitude of the input signal instead of just its derivative. This also has the benefit of incorporating desirable

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to reconstruct the signal's amplitude. Delta modulation has several drawbacks. The differentiation alters the signal's spectrum by amplifying high-frequency noise, attenuating low-frequencies, and dropping the

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by spreading it out and shaping it so it is mostly in higher frequencies. It can then be easily filtered out with inexpensive digital filters, without high-precision analog circuits needed by Nyquist ADCs.

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total amount of noise in the frequency band of interest is dramatically smaller for oversampling converters (just the small green area), than for a Nyquist converter (yellow + green total area).

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Higher linearity afforded by low-bit ADCs and DACs (for instance, a 1-bit DAC that only outputs two values of a precise high voltage and a precise low voltage is perfectly linear, in principle).

2661:(DDC), because its input is already digital and quantized but is simply reducing from a higher bitdepth to a lower bitdepth digital signal. This is represented in the z-domain by another 3265:{\displaystyle \Delta \Sigma {\text{M}}_{\Theta }({\text{z}})={\text{in}}({\text{z}})\cdot {\text{z}}^{\text{-1}}+{\text{noise}}({\text{z}})\cdot (1-{\text{z}}^{\text{-1}})^{\Theta }\,.} 1297: 723:

The principle of improving the resolution of a coarse quantizer by use of feedback, which is the basic principle of delta-sigma conversion, was first described in a 1954-filed patent by

3041: 1951:{\displaystyle {\frac {\Delta \Sigma {\text{M}}_{\text{in}}({\text{s}})}{{\text{in}}({\text{s}})}}={\frac {\tfrac {1}{\text{s}}}{1+{\tfrac {1}{\text{s}}}}}={\frac {1}{{\text{s}}+1}}\,.} 5288:

comes directly from the presence of a delta modulator and an integrator, as firstly introduced by Inose et al. in their patent application. That is, the name comes from integrating or

1693: 4698:) used in delta modulation has a small output representing a small step up and down the quantized approximation of the input while the quantizer used in delta-sigma must take values 2330: 2717: 2688: 2519: 1992: 1549: 1386: 1338: 4457: 3496: 3339: 3131: 1558:

The process of quantization is approximated as addition with a quantization error noise source. The noise is often assumed to be white and independent of the signal, though as

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In the specific case of a single-bit synchronous ΔΣ ADC, an analog voltage signal is effectively converted into a pulse frequency, or pulse density, which can be understood as

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with approximately equal intensity across the entire spectrum. In reality, the quantization noise is, of course, not independent of the signal and this dependence results in

3551: 3075: 2490: 2262:{\displaystyle {\frac {\Delta \Sigma {\text{M}}_{\text{noise}}({\text{s}})}{{\text{noise}}({\text{s}})}}={\frac {1}{1+{\tfrac {1}{\text{s}}}}}={\frac {s}{{\text{s}}+1}}\,.} 2297: 1674:{\displaystyle \Delta \Sigma {\text{M}}_{\text{total}}({\text{s}})=\Delta \Sigma {\text{M}}_{\text{in}}({\text{s}})+\Delta \Sigma {\text{M}}_{\text{noise}}({\text{s}})\,.} 1565:

Since the system and Laplace transform are linear, the total behavior of this system can be analyzed by separating how it affects the input from how it affects the noise:

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These datapoints are theoretical. In practice, circuits inevitably experience other noise sources that limit resolution, making the higher-resolution cells impractical.

3294: 3103: 2995:{\displaystyle \Delta \Sigma {\text{M}}({\text{z}})={\text{in}}({\text{z}})\cdot {\text{z}}^{\text{-1}}+{\text{noise}}({\text{z}})\cdot (1-{\text{z}}^{\text{-1}})\,.} 2459: 1151: 1011: 484:

Primarily because of its cost efficiency and reduced circuit complexity, this technique has found increasing use in modern electronic components such as DACs, ADCs,

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Figure 8b: Simulated scope view of key voltage signals over time. Each minor vertical division is 1 μs, which corresponds to a sampling event of the 1 MHz clock.

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Oversampling converters instead produce a lower-bitdepth result at a much higher sampling frequency. This can achieve comparable quality by taking advantage of:

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Delta-sigma ADCs vary in complexity. The below circuit focuses on a simple 1st-order, 2-level quantization synchronous delta-sigma ADC without decimation.

2525: 2872:{\displaystyle \cdot {\frac {1}{1-{\text{z}}^{\text{-1}}}}\cdot {\text{z}}^{\text{-1}}+{\text{noise}}({\text{z}})=\Delta \Sigma {\text{M}}({\text{z}})\,,} 4883: 4462:

Again, the linearity property of integration allows the two integrators to be combined and a delta-sigma modulator/demodulator block diagram is obtained.

496:. The coarsely-quantized output of a delta-sigma ADC is occasionally used directly in signal processing or as a representation for signal storage (e.g., 333: 5929: 6321: 5477: 859:

or sinc filter): This is the easiest digital filter and retains a sharp step response, but is mediocre at separating frequency bands and suffers from

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frequency range, of the signal, while increasing the sample amplitude resolution. This improvement in amplitude resolution is obtained by a sort of

4676:{\displaystyle \operatorname {Quantize} \left(\int \left({\text{v}}_{\text{in}}-{\text{v}}_{{\text{feedback}}_{\Delta \Sigma }}\right)dt\right).\,} 739:

in 1961 filed a patent "Feedback integrating system" with a feedback loop containing an integrator with multi-bit quantization shown in its Fig 1.

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engineers used the terms "sigma-delta" because the precedent was to name variations on delta modulation with adjectives preceding "delta", and an

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and noise. Analog is green. Digital is blue. The DDC (Digital-to-Digital Converter) requantizes its input from a high-bitdepth to a low-bitdepth.

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Kirkkert and Miller published a continuous-time variant called "Asynchronous Delta Sigma Modulation" (ADSM or ASDM) in 1975 which uses either a

546:. Such higher-bit methods seek accuracy in amplitude directly, but require extremely precise components and so may suffer from poor linearity. 5811: 6634: 6615: 6596: 6577: 6489: 6188: 6013: 5146:

can be low-pass filtered to isolate the original signal. As the sampling rate is increased relative to the input signal's maximum frequency,

4691:, which allows for quantization noise to be somewhat programmable. On the other hand, delta modulation shapes both noise and signal equally. 4459:, makes it possible to move the integrator, which reconstructs the analog signal in the demodulator section, in front of the delta modulator. 6694: 829:. Decimation may be done in a separate chip on the receiving end of the delta-sigma bit stream, sometimes by a dedicated module inside of a 5867: 866: 825:

audio was made by D. J. Goodman at Bell Labs in 1969, to reduce the ΔΣ signal from its high sampling rate while increasing its

362: 278: 6217: 5695: 6788: 1355: 1019: 711:"Feedback Integrating System" by Charles B Brahm: The entire top half of its Fig 1 is a delta-sigma modulator. Box #10 is a two-input 409: 5689: 5550: 5314:

magazine editor justified in 1990 that the functional hierarchy is "sigma-delta", because it computes the integral of a difference.

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It is worth noting that if no decimation ever took place, the digital representation from a 1-bit delta-sigma modulator is simply a

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bits when incrementing the order. For comparison, oversampling a Nyquist ADC (without any noise shaping) only improves its ENOB by

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F. de Jager, "Delta modulation, a method of PCM transmission using the 1-unit code," Philips Res. Rep., vol. 7, pp. 442–466, 1952.

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for the quantization, which carries over the most-significant bit(s) from the integrator to be the feedback for the next cycle.

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Continuous-Time Sigma-Delta Modulation for A/D Conversion in Radio Receivers: Chapter 4: Continuous-time sigma-delta modulation

5209: 4575:{\displaystyle \int \operatorname {Quantize} \left({\text{v}}_{\text{in}}-{\text{v}}_{{\text{feedback}}_{\Delta }}\right)dt.\,} 338: 252: 224: 191: 166: 118: 6081: 5716: 6287: 6151: 5973: 2131:{\displaystyle \cdot {\frac {1}{\text{s}}}+{\text{noise}}({\text{s}})=\Delta \Sigma {\text{M}}_{\text{noise}}({\text{s}})\,,} 814: 355: 2607: 917:

time is inversely proportional to their cutoff transition steepness), and higher computational requirements. They are the

715:. The 4-bit analog-to-digital quantizer uses designations "S" (sign), "1", "2", and "4" for each bit. Each "F" stands for 701: 636:

component. This makes its dynamic range and SNR inversely proportional to signal frequency. Delta modulation suffers from

441: 5769: 735:, which came up with the idea of adding a filter in the forward path of the delta modulator. However, Charles B Brahm of 1349: 531: 489: 6685: 676:

The use of PDM as a signal representation is an alternative to PCM. Alternatively, the high frequency PDM can later be

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In the example in Figure 9, when the integral of the error exceeds its limits, the output changes state, producing a

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The modulator can also be classified by the bit depth of its quantizer. A quantizer that distinguishes between

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analog domain to remove the high-frequency quantization noise inherent to the delta-sigma modulation process.

4478:), the above would be a sufficient derivation of their hypothetical equivalence. But because the quantizer is 1509:{\displaystyle \cdot {\frac {1}{\text{s}}}+{\text{noise}}({\text{s}})=\Delta \Sigma {\text{M}}({\text{s}})\,.} 6721: 3452:{\displaystyle {\text{SNR}}_{\text{dB}}\approx 3.01\cdot (2\cdot \Theta +1)\cdot d-9.36\cdot \Theta -2.76\,.} 3370:

OSR (and followed by an ideal low-pass decimation filter) can be mathematically derived to be approximately:

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The technique was first presented in the early 1960s by professor Yasuhiko Yasuda while he was a student at

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Both ADCs and DACs can employ delta-sigma modulation. A delta-sigma ADC (e.g. Figure 1 top) encodes an

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delay stage in series with adding quantization noise. (Note: some sources may have swapped ordering of the

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cannot be used in the feedback loop, so distortion can be heard under some conditions (more discussion at

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The input simply comes out of the system delayed by one clock cycle. The noise term's multiplication by

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To understand how the system affects the noise only, the input instead is temporarily imagined to be 0:

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in Laplace notation. The integrator is assumed to be an ideal integrator to keep the math simple, but a

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Decimation filters most commonly used for ΔΣ ADCs, in order of increasing complexity and quality, are:

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To understand how the system affect the input signal only, the noise is temporarily imagined to be 0:

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Figure 5 represents the 1-order ΔΣ ADC modulation loop (from Figure 1) as a continuous-time

1097: 571:: moving noise to higher frequencies above the signal of interest, so they can be easily removed with 64: 5599:, Cutler, Cassius C., "Transmission systems employing quantization", issued March 8, 1960 4471: 3556: 1245: 579: 527:

it enables noise-free transmission, storage, and processing. There are many methods of digitization.

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the demodulator can be a simple linear filter (e.g., RC or LC filter) to reconstruct the signal, and

3050: 1248:, e.g. in a DAC's modulation loop). These transforms are useful for converting harder math from the 6548: 5818: 5673: 5261: 5235: 5134:, which reconstructs the original sine wave input as piece-wise linear diagonal segments. Although 3521: 2464: 838: 732: 520: 504: 478: 196: 6715: 2274: 1961: 684:

and requantized to convert it into a multi-bit PCM code at lower sampling frequency closer to the

6762: 5899: 5892:"AN4990: Getting started with sigma-delta digital interface on applicable STM32 microcontrollers" 5643: 4821:

with inexpensive discrete components (note some variations use different biasing and use simpler

4785: 2346: 1809:{\displaystyle \cdot {\frac {1}{\text{s}}}+0=\Delta \Sigma {\text{M}}_{\text{in}}({\text{s}})\,,} 657: 572: 559: 445: 437: 433: 413: 229: 206: 186: 6210: 176: 6710: 5114:

is sent out as the resulting PDM output and also fed back to the 2-input inverting integrator.

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and is the source of idle tones and pattern noise in delta-sigma converters. However, adding

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include decimation. Some microcontrollers even incorporate both the modulator and decimator.

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if signals move too fast. And it is susceptible to transmission disturbances that result in

637: 604: 493: 303: 298: 123: 113: 6741:. This in-depth article covers the theory behind a Delta-Sigma analog-to-digital converter. 6732: 6727: 3346: 1160: 86: 6689: 5343: 5230:

can easily be laid out with basic digital elements of a subtractor for the difference, an

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Figure 4: Noise shaping curves and noise spectrum in 1, 2, and 3-order ΔΣ modulators.

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undesired high-frequency noise. The downside is poor performance in the time domain (e.g.

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discusses architectural trade-offs for continuous-time sigma-delta noise-shaping filters

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explains that is not always a valid assumption (particularly for low-bit quantization).

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However, delta-sigma modulation of the same input signal places at the low-pass filter

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delta-sigma modulation either), and the details are out of the scope of this article.

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so the system effectively acts as a high-pass filter on the noise that starts at 0 at

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Figure 5: ΔΣ modulation loop in Laplace domain. Integration is multiplication by

6782: 6446: 5248: 4811: 2595:{\displaystyle y({\text{z}})=x({\text{z}})+y({\text{z}})\cdot {\text{z}}^{\text{-1}}} 2349:

circuit, which work by transferring charge between capacitors in clocked time steps.

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could be adjusted as desired by including multiplication by a constant in the loop).

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Higher accuracy in time (afforded by high-speed digital circuits and highly accurate

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Sigma-delta class-D amplifier and control method for a sigma-delta class-D amplifier

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When a signal is quantized, the resulting signal can be approximated by addition of

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in the system and transmission is added to the analog signal, reducing its quality.

6481: 6068: 4688: 2336:, then gradually rises until it reaches the cutoff frequency, and then levels off. 1154: 952: 685: 535: 453: 449: 398: 343: 283: 262: 6274: 6143: 5960: 5740: 5631: 5596: 6031:"A Use of Limit Cycle Oscillations to Obtain Robust Analog-to-Digital Converters" 5743:, Brahm, Charles B., "Feedback integrating system", issued 1965-06-29 5651: 5532: 5265: 5049: 4833: 1249: 1241: 948: 914: 890: 878: 856: 677: 20: 6645: 6469: 6106: 6030: 5672:. The International Series in Engineering and Computer Science. Vol. 634. 5619: 5502: 4383: 6349: 6122: 6046: 5837:"Why 1-Bit Sigma-Delta Conversion is Unsuitable for High-Quality Applications" 5797: 5778: 5473: 5347: 5333: 5295:, which, in mathematics, are operations usually associated with Greek letters 5214: 5199: 5175: 4989:

that is repeatedly corrected by the feedback mechanism. While both its inputs

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which can be rearranged to express the output in terms of the input and noise:

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unconditionally stable (there are no feedback loops outside the accumulators)

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quantizer. Higher bit quantizers inherently produce less quantization noise.

6682: 6275:"Application Report: General Oversampling of MSP ADCs for Higher Resolution" 5681: 5307: 4792: 2461:

This is represented in the z-domain by feeding back a summing node's output

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Theoretical SNR & ENOB versus ΔΣM order & oversampling ratio (OSR)

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loop during quantization to the lower bit depth that continuously corrects

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Continuous time sigma-delta ADC noise shaping filter circuit architectures

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the signals of interest. This effect becomes more dramatic with increased

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Reduction of baseband noise by increasing oversampling ratio and ΔΣM order

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upper and lower grey lines in middle plot), the PWM output changes state.

4810:. 20 kHz is used as an example because that is considered the upper 4403: 3310: 967: 628: 456:

to achieve high accuracy in amplitude which can be ultimately encoded as

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Digital accelerometer with feedback control using sigma-delta modulation

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Example Simulink model & scripts for continuous-time sigma-delta ADC

6507:"Asynchronous Sigma Delta Modulators for Data Conversion - Ph.D. Thesis" 5717:"Charles Brahm Obituary (1926 - 2021) - Hartford, CT - Hartford Courant" 5350:) or (as the paper argues is equivalent) a comparator with fixed delay. 934: 6354: 4976: 4826: 2333: 1013:) may be reduced by increasing the oversampling ratio (OSR) defined by 980: 473: 402: 6474:

2015 25th International Conference Radioelektronika (RADIOELEKTRONIKA)

6243:"UNDERSTANDING SIGMA–DELTA MODULATION: The Solved and Unsolved Issues" 6076: 5771:

Principles of sigma-delta modulation for analog-to-digital converters

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Principles of Sigma-Delta Modulation for Analog-to-Digital Converters

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In general, delta-sigma has some advantages versus delta modulation:

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result. The full conversion process for each typically includes post-

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result on each clock cycle, which is fed back for computing the next

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simple to implement in hardware; only common digital blocks such as

5062:. The scope view (Figure 8b) has a minor division equal to the 4723:

The quantized value is the integral of the difference signal, which

6662: 4968:{\displaystyle \varepsilon (t)=-{\frac {1}{RC}}\int (s(t)+Q(t))dt.} 3047:

filter (which has a single pole at the origin and a single zero at

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attenuation of the lower-frequency band of interest, still removes

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The rightmost integrator performs digital-to-analog conversion on

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which can be rearranged to yield the following transfer function:

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The seminal paper combining feedback with oversampling to achieve

293: 6312:"Activity: Delta - Sigma Modulator [Analog Devices Wiki]" 6181:

The Scientist & Engineer's Guide to Digital Signal Processing

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The modulator's z-domain equation arranged like Figure 6 is:

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is an earlier related low-bit oversampling method that also uses

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article by Randy Yates presented at the 2004 comp.dsp conference

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makes it less sensitive to the rate of change of the signal, and

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One criticism of 1-bit quantization is that adequate amounts of

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representing up or down of the signal's movement, which must be

6248:. J. Audio Eng. Soc., Vol. 56, No. 1/2, 2008 January/February. 6006:

The Scientist and Engineer's Guide to Digital Signal Processing

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using high-frequency delta-sigma modulation and then applies a

6144:"A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters" 5059: 4387:

Figure 7: Derivation of delta-sigma from delta modulation

1522:

of a function of time corresponds to simply multiplication by

822: 5812:"Why Professional 1-Bit Sigma-Delta Conversion is a Bad Idea" 4718:

the quantizer (e.g., comparator) can have full-scale outputs.

1232:

These curves are derived using mathematical tools called the

719:

and each "G" is a gate, controlled by the 110 kHz oscillator.

507:

modulation, which requires a precise clock for quantization,

6765:– intuitive motivation for why a delta-sigma modulator works 6183:(2nd ed.). San Diego, Calif: California Technical Pub. 6008:(2nd ed.). San Diego, Calif: California Technical Pub. 5551:"The Evolution of Oversampling Analog-to-Digital Converters" 4486:

by delta modulation, but the two are distinct in operation.

4399:

Start with a block diagram of a delta modulator/demodulator.

1560:

quantization (signal processing) § Additive noise model

1555:(or similar filter) may have a more complicated expression. 901:

filters. This approximated filter design, while maintaining

68:

1-bit synchronous ΔΣ modulation (blue) of a sine wave (red).

6753:

with both theory and a block-level implementation of a MASH

5620:"A Telemetering System by Code Modulation - Δ- ΣModulation" 1184:). Since oversampling is typically done in powers of two, 6470:"Asynchronous sigma-delta modulator and fast demodulator" 3085:

Without getting into the mathematical details, cascading

979:

Quantization noise in the baseband frequency range (from

500:

stores the raw output of a 1-bit delta-sigma modulator).

665:

1-bit delta-sigma modulation is pulse-density modulation

32:

Figure 1: Full process of a 1-order synchronous ΔΣ

16:

Method for converting signals between digital and analog

6647:

Asynchronous sigma delta modulators for data conversion

759:

Figure 2: Noise-feedback 2-order ΔΣ modulator ADC.

440:

to higher frequencies well above the original signal's

6718:

Contains Block diagrams, code, and simple explanations

6567: 5810:

Lipschitz, Stanley P.; Vanderkooy, John (2000-09-22).

5455:"A Circuit for all Seasons: The Delta-Sigma Modulator" 4412: 3561: 3532: 3524: 3477: 3469: 2650:

is often used to label integrators in block diagrams.

2643:{\displaystyle {\tfrac {1}{1-{\text{z}}^{\text{-1}}}}} 2612: 2340:

Analysis of synchronous ΔΣ modulation loop in z-domain

2218: 1907: 1889: 1530: 1367: 661:

leaving the higher frequency components of the noise.

578:

Reduced steepness requirement for the analog low-pass

534:

at a relatively low sampling frequency just above its

6774:

Analog Devices Sigma-Delta ADC tutorial (interactive)

5835:

Lipshitz, Stanley P.; Vanderkooy, John (2001-05-12).

4886: 4751:

Figure 8a: Schematic of simple delta-sigma converter.

4594: 4506: 3559: 3504: 3378: 3349: 3320: 3282: 3141: 3112: 3091: 3053: 3008: 2885: 2728: 2696: 2667: 2610: 2528: 2498: 2467: 2399: 2370: 2305: 2277: 2148: 2018: 1970: 1830: 1696: 1573: 1528: 1397: 1365: 1305: 1268: 1190: 1163: 1129: 1100: 1022: 989: 511:

delta-sigma modulation instead runs without a clock.

6701:

Tutorial on Designing Delta-Sigma Modulators: Part I

1344:

Analysis of ΔΣ ADC modulation loop in Laplace domain

19:"Sigma delta" redirects here. For the sorority, see 5624:

IRE Transactions on Space Electronics and Telemetry

5168:

PDM signal, which can easily be converted to analog

2521:) into another input of the summing node, yielding 6211:"Signals From Noise: Calculating Delta-Sigma SNRs" 5618:Inose, H.; Yasuda, Y.; Murakami, J. (1962-05-06). 5181:However, in general, a delta-sigma DAC converts a 4967: 4675: 4574: 4451: 3582:bits for every doubling of the OSR, which is only 3574: 3545: 3510: 3490: 3451: 3362: 3333: 3288: 3264: 3125: 3097: 3069: 3035: 2994: 2871: 2711: 2682: 2642: 2594: 2513: 2484: 2453: 2385: 2356:Figure 6: ΔΣ modulation loop in the z-domain. 2324: 2291: 2261: 2130: 1986: 1950: 1808: 1673: 1543: 1508: 1380: 1332: 1291: 1196: 1176: 1145: 1115: 1083: 1005: 656:into the conversion process, to deliberately move 542:by a multi-level quantizer to produce a multi-bit 6745:One-Bit Delta Sigma D/A Conversion Part I: Theory 6390:"発見と発明のデジタル博物館卓越研究データベース・電気・情報通信関連・研究情報(登録番号671)" 5152:will more closely approximate the original input 1388:and quantization is approximated by adding noise. 519:When transmitting an analog signal directly, all 6069:"AN-455: Understanding CIC Compensation Filters" 5503:"A unity bit coding method by negative feedback" 913:overshoot and ripple), higher delay (i.e. their 6724:Contains example matlab code and Simulink model 6549:"ADC Architectures III: Sigma-Delta ADC Basics" 3313:(dB) for a sinusoid input travelling through a 2492:though a 1-clock cycle delay stage (notated as 1818:which can be rearranged to yield the following 966:ΔΣ ADCs reduce the amount of this noise in the 5140:appears coarse at this 50x oversampling rate, 4729:helps capture low frequency and DC components. 879:sinc-in-time (brick-wall in frequency) filters 6735:(which covers both ADCs and DACs sigma-delta) 6728:Bruce Wooley's Delta-Sigma Converter Projects 6606:S. Norsworthy; R. Schreier; G. Temes (1997). 5626:. SET-8 (3) (published 1962-09-01): 204–209. 5200:Upsampling § Interpolation filter design 975:Oversampling to spread out quantization noise 452:easily removes this high frequency noise and 424:(DACs). Delta-sigma modulation achieves high 363: 56:for demodulation and pre-filtering to remove 8: 6695:Sigma-delta techniques extend DAC resolution 6445:Kikkert, C. J.; Miller, D. J. (1975-04-01). 5376:Continuously variable slope delta modulation 5213:analog low-pass filter. This can be done by 2271:This transfer function has a single zero at 2653:In a ΔΣ DAC, the quantizer may be called a 1292:{\displaystyle {\text{s}}=\sigma +j\omega } 1240:, e.g. in an ADC's modulation loop) or the 833:, which is useful for interfacing with PDM 538:(twice the signal's highest frequency) and 6751:MASH (Multi-stAge noise SHaping) structure 3592:of the ENOB growth rate of a 1-order ΔΣM. 3036:{\displaystyle (1-{\text{z}}^{\text{-1}})} 2393:with the previous result of its summation 1204:represents how many times OSR is doubled. 963:by making quantization noise more random. 530:In Nyquist-rate ADCs, an analog signal is 370: 356: 71: 6733:An Introduction to Delta Sigma Converters 6589:Understanding Delta-Sigma Data Converters 5868:"Data Converter Architectures: Chapter 3" 5222:Digital-to-digital delta-sigma modulation 4905: 4885: 4784:(Figure 8a) using ideal elements is 4672: 4642: 4637: 4635: 4630: 4620: 4615: 4593: 4571: 4549: 4544: 4542: 4537: 4527: 4522: 4505: 4411: 3560: 3558: 3531: 3523: 3503: 3498:bits when doubling the OSR (incrementing 3476: 3468: 3445: 3385: 3380: 3377: 3354: 3348: 3325: 3319: 3281: 3258: 3252: 3242: 3237: 3216: 3208: 3199: 3194: 3182: 3174: 3163: 3154: 3149: 3140: 3117: 3111: 3090: 3059: 3054: 3052: 3024: 3019: 3007: 2988: 2979: 2974: 2953: 2945: 2936: 2931: 2919: 2911: 2900: 2892: 2884: 2865: 2857: 2849: 2832: 2824: 2815: 2810: 2797: 2792: 2779: 2765: 2757: 2740: 2732: 2727: 2703: 2698: 2695: 2674: 2669: 2666: 2630: 2625: 2611: 2609: 2586: 2581: 2569: 2552: 2535: 2527: 2505: 2500: 2497: 2474: 2466: 2398: 2369: 2314: 2306: 2304: 2278: 2276: 2255: 2241: 2235: 2217: 2205: 2191: 2183: 2173: 2164: 2159: 2149: 2147: 2124: 2116: 2107: 2102: 2084: 2076: 2063: 2049: 2040: 2035: 2017: 1979: 1971: 1969: 1944: 1930: 1924: 1906: 1887: 1873: 1865: 1855: 1846: 1841: 1831: 1829: 1802: 1794: 1785: 1780: 1754: 1740: 1731: 1726: 1708: 1700: 1695: 1667: 1659: 1650: 1645: 1627: 1618: 1613: 1595: 1586: 1581: 1572: 1529: 1527: 1502: 1494: 1486: 1469: 1461: 1448: 1434: 1426: 1409: 1401: 1396: 1366: 1364: 1352:in the Laplace domain with the equation: 1321: 1306: 1304: 1269: 1267: 1189: 1168: 1162: 1137: 1128: 1106: 1105: 1099: 1075: 1059: 1045: 1039: 1038: 1034: 1023: 1021: 997: 988: 6627:Oversampling Delta-sigma Data Converters 5666:"Continuous-time sigma-delta modulation" 5332: 5226:The modulation loop in Figure 6 in 5215:upsampling using an interpolation filter 5040:feedback from the previous clock cycle. 4395:by the following steps (Figure 7): 3594: 3077:) and thus high-pass filters the noise. 2351: 1354: 933: 802:Direct Stream Digital § DSD vs. PCM 754: 63: 44:(the curly bracket) which outputs a new 40:(bottom). Each contains a ΔΣ modulation 27: 6107:"Decimation for Sigma Delta Modulation" 5613: 5611: 5414: 5393: 5128:to produce a demodulated analog output 4836:is powered by dual supply voltages of V 2360:Integration in discrete-time can be an 2325:{\displaystyle {\text{s}}={\text{-1}},} 270: 237: 214: 136: 93: 74: 5383:(sometimes use delta-sigma modulation) 5099:is below the 0 V threshold, then 5084:is above the 0 V threshold, then 4497:, the low-pass filter sees the signal 3463:(ENOB) resolution is thus improved by 2712:{\displaystyle {\text{z}}^{\text{-1}}} 2683:{\displaystyle {\text{z}}^{\text{-1}}} 2514:{\displaystyle {\text{z}}^{\text{-1}}} 1987:{\displaystyle {\text{s}}={\text{-1}}} 1544:{\displaystyle {\tfrac {1}{\text{s}}}} 1381:{\displaystyle {\tfrac {1}{\text{s}}}} 1333:{\displaystyle {\text{z}}=Ae^{j\phi }} 1224:Figure 4 shows how ΔΣ modulation 647:Delta-sigma modulation rearranges the 416:as part of the process of delta-sigma 6716:Sigma-Delta Modulation Primer Part II 6447:"Asynchronous Delta Sigma Modulation" 6340: 6338: 6204: 6202: 6200: 5793: 5791: 5763: 5761: 5759: 5757: 5755: 5036:in this integral actually represents 4452:{\textstyle \int a+\int b=\int (a+b)} 4391:Delta-sigma modulation is related to 7: 6376:"15-25 MHZ Fractional-N Synthesizer" 5999:"Chapter 15: Moving Average Filters" 5544: 5542: 5501:Inose, H.; Yasuda, Y. (1963-11-01). 5448: 5446: 5444: 5234:for the integrator, and a lower-bit 5067:sample-and-hold time violations and 4736:Analog-to-digital conversion example 3491:{\textstyle \Theta +{\tfrac {1}{2}}} 3301:Theoretical effective number of bits 1960:This transfer function has a single 6273:Brown, Ryan; Singh, Sameer (2016). 6174:"Chapter 16: Windowed-Sinc Filters" 6111:IEEE Transactions on Communications 6035:IEEE Transactions on Communications 5800:by Jwin-Yen Guo and Teng-Hung Chang 5329:Asynchronous delta-sigma modulation 4694:Additionally, the quantizer (e.g., 3334:{\displaystyle \Theta ^{\text{th}}} 3126:{\displaystyle \Theta ^{\text{th}}} 959:noise (Figure 3) reduces such 893:, the sinc function can instead be 817:oversampled delta-sigma data using 688:of the frequency band of interest. 596:of the higher-data-rate bitstream. 6763:Delta Sigma Converters: Modulation 5462:IEEE Solid-State Circuits Magazine 5227: 5074:Whenever a sampling event occurs: 4850:is either +1 V or -1 V. 4844:= -1 V, so its binary output 4817:This circuit can be laid out on a 4702:of the range of the input signal. 4646: 4643: 4550: 3470: 3436: 3409: 3322: 3283: 3276:filter is now raised to the power 3253: 3155: 3145: 3142: 3114: 3092: 2889: 2886: 2846: 2843: 2754: 2751: 2155: 2152: 2098: 2095: 2031: 2028: 1837: 1834: 1776: 1773: 1722: 1719: 1641: 1638: 1609: 1606: 1577: 1574: 1483: 1480: 1423: 1420: 1107: 1030: 1027: 1024: 14: 6697:article by Tim Wescott 2004-06-23 6451:Proceedings of the IREE Australia 5930:"Understanding PDM Digital Audio" 4404:linearity property of integration 600:Improvement over delta modulation 6569:CMOS Mixed-Signal Circuit Design 5007:instead only varies by a couple 4780:To ease understanding, a simple 4758: 4744: 4379:Relationship to delta modulation 2364:which repeatedly sums its input 1520:Laplace transform of integration 1116:{\displaystyle f_{\mathrm {s} }} 867:Cascaded integrator–comb filters 700:was by F. de Jager of 85: 6525:from the original on 2020-07-10 6427:from the original on 2023-06-29 6324:from the original on 2023-04-01 6293:from the original on 2022-09-01 6255:from the original on 2023-09-01 6223:from the original on 2023-08-06 6154:from the original on 2023-10-22 6087:from the original on 2023-04-05 5979:from the original on 2022-10-15 5942:from the original on 2023-06-30 5910:from the original on 2022-01-20 5849:from the original on 2023-04-30 5698:from the original on 2023-06-29 5569:from the original on 2023-06-28 5549:Wooley, Bruce A. (2012-03-22). 5483:from the original on 2024-02-09 5210:discrete-time Fourier transform 5044:Quantizer and sampler flip-flop 4709:The structure is simplified as 3575:{\displaystyle {\tfrac {1}{2}}} 6587:R. Schreier; G. Temes (2005). 6482:10.1109/RADIOELEK.2015.7129003 6350:"One Bit ADC - Short Circuits" 4953: 4950: 4944: 4935: 4929: 4923: 4896: 4890: 4712:only one integrator is needed, 4446: 4434: 3546:{\textstyle d-{\tfrac {3}{2}}} 3418: 3400: 3249: 3227: 3221: 3213: 3187: 3179: 3168: 3160: 3070:{\displaystyle {\text{z}}{=}1} 3030: 3009: 2985: 2964: 2958: 2950: 2924: 2916: 2905: 2897: 2862: 2854: 2837: 2829: 2773: 2770: 2762: 2745: 2737: 2729: 2574: 2566: 2557: 2549: 2540: 2532: 2479: 2471: 2445: 2433: 2424: 2418: 2409: 2403: 2380: 2374: 2196: 2188: 2178: 2170: 2121: 2113: 2089: 2081: 2057: 2054: 2046: 2019: 1878: 1870: 1860: 1852: 1799: 1791: 1748: 1745: 1737: 1713: 1705: 1697: 1664: 1656: 1632: 1624: 1600: 1592: 1499: 1491: 1474: 1466: 1442: 1439: 1431: 1414: 1406: 1398: 1123:is the sampling frequency and 747:, and the oversampling ratio. 503:While this article focuses on 1: 6739:Demystifying Sigma-Delta ADCs 6409:"Editor's Notes: Σ-∆ or ∆-Σ?" 5632:10.1109/IRET-SET.1962.5008839 5453:Razavi, Behzad (2016-06-21). 5024:next gets sampled to produce 5001:vary between -1 and 1 volts, 2485:{\displaystyle y({\text{z}})} 702:Philips Research Laboratories 587:Frequency/resolution tradeoff 6683:1-bit A/D and D/A Converters 6572:(2nd ed.). Wiley-IEEE. 6547:Walt Kester (October 2008). 5162:Digital-to-analog conversion 5090:will go high (+1 V), or 5058:at the clock rate of 1 5014:Because of the integrator's 4854:2-input inverting integrator 4482:homogeneous, delta-sigma is 2719:and additive noise stages.) 2659:digital-to-digital converter 2292:{\displaystyle {\text{s}}=0} 1350:linear time-invariant system 849:Boxcar moving average filter 623:. The result is a stream of 490:switched-mode power supplies 422:digital-to-analog converters 418:analog-to-digital converters 6625:J. Candy; G. Temes (1992). 6608:Delta-Sigma Data Converters 4825:for integration instead of 1299:(in the Laplace domain) or 680:through a processed called 6805: 6707:by Mingliang (Michael) Liu 6515:Imperial College of London 6142:Lyons, Rick (2020-03-26). 5777:, Motorola, archived from 5206:discrete Fourier transform 5197: 4776:Simplified circuit example 3274:first difference backwards 861:intermodulation distortion 821:before converting it into 550:Advantages of oversampling 279:Capacity-approaching codes 18: 6789:Digital signal processing 6711:Gabor Temes' Publications 6241:Reiss, Joshua D. (2008). 6172:Smith, Steven W. (1999). 6123:10.1109/TCOM.1986.1096432 6047:10.1109/TCOM.1974.1092194 5997:Smith, Steven W. (1999). 5474:10.1109/MSSC.2016.2543061 5242:Multi-stage noise shaping 4985:contains the accumulated 4254: 4235: 4229: 4210: 4204: 4161: 4155: 4136: 4130: 4111: 4068: 4049: 4030: 4011: 3992: 3949: 3930: 3911: 3892: 3873: 3830: 3811: 3792: 3773: 3754: 3711: 3692: 3673: 3654: 3635: 3605: 3602: 3600: 3105:integrators to create an 3045:first difference backward 2006:High-pass filter on noise 1252:into simpler math in the 925:digital audio converters. 891:realizable in finite time 5676:. 2001. pp. 29–71. 5346:(i.e. a comparator with 5105:will go low (-1 V). 4800:is converted to a 1-bit 3461:effective number of bits 1684:Low-pass filter on input 671:pulse-density modulation 6655:Imperial College London 6566:R. Jacob Baker (2009). 6407:Sheingold, Dan (1990). 5682:10.1007/0-306-48004-2_3 5508:Proceedings of the IEEE 5282:the University of Tokyo 3289:{\displaystyle \Theta } 3098:{\displaystyle \Theta } 3081:Higher order modulators 1238:continuous-time signals 899:finite impulse response 889:prevents it from being 611:, but only encodes the 216:Hierarchical modulation 5521:10.1109/PROC.1963.2622 5371:Pulse-width modulation 5339: 5176:resistor and capacitor 4969: 4858:The 2-input inverting 4812:limit of human hearing 4791:The 20 kHz input 4677: 4576: 4453: 4388: 4281:each additional order: 3576: 3547: 3512: 3492: 3453: 3364: 3335: 3290: 3266: 3127: 3099: 3071: 3037: 2996: 2873: 2713: 2684: 2644: 2604:Its transfer function 2596: 2515: 2486: 2455: 2454:{\displaystyle y=x+y.} 2387: 2357: 2326: 2293: 2263: 2132: 1988: 1952: 1810: 1675: 1545: 1510: 1389: 1382: 1334: 1293: 1212: 1198: 1178: 1147: 1146:{\displaystyle 2f_{0}} 1117: 1085: 1007: 1006:{\displaystyle 2f_{0}} 944: 760: 751:Higher-order modulator 720: 692:History and variations 486:frequency synthesizers 69: 61: 42:negative feedback loop 6468:Stork, Milan (2015). 5928:Kite, Thomas (2012). 5597:US patent 2967962 5355:pulse-width modulated 5336: 5325:are frequently used. 4970: 4678: 4577: 4454: 4386: 3577: 3548: 3513: 3493: 3454: 3365: 3363:{\displaystyle 2^{d}} 3336: 3307:signal-to-noise ratio 3291: 3267: 3135:modulator results in: 3128: 3100: 3072: 3038: 2997: 2874: 2714: 2685: 2645: 2597: 2516: 2487: 2456: 2388: 2355: 2327: 2299:and a single pole at 2294: 2264: 2133: 1989: 1953: 1811: 1676: 1546: 1511: 1383: 1358: 1335: 1294: 1246:discrete-time signals 1210: 1199: 1179: 1177:{\displaystyle f_{0}} 1148: 1118: 1086: 1008: 937: 853:simple moving average 837:, though many ΔΣ ADC 808:Subsequent decimation 758: 710: 580:anti-aliasing filters 458:pulse-code modulation 67: 31: 6476:. pp. 180–183. 6105:Candy, J.C. (1986). 5228:§ Noise shaping 5183:discrete time series 4884: 4832:For simplicity, the 4592: 4504: 4410: 3557: 3522: 3502: 3467: 3376: 3347: 3318: 3280: 3139: 3110: 3089: 3051: 3006: 2883: 2726: 2694: 2665: 2608: 2526: 2496: 2465: 2397: 2368: 2303: 2275: 2146: 2016: 1968: 1828: 1694: 1571: 1526: 1395: 1363: 1303: 1266: 1188: 1161: 1127: 1098: 1020: 987: 737:United Aircraft Corp 401:method for encoding 5961:"MSP430i2xx Family" 5674:Springer Publishing 5357:(PWM) output wave. 4823:RC low-pass filters 3603:Oversampling ratio 3597: 1340:(in the z-domain). 839:integrated circuits 772:Multi-bit quantizer 733:University of Tokyo 615:of the signal (its 479:digital electronics 434:quantization errors 397:) modulation is an 6688:2021-02-25 at the 6644:Chen, Wei (2013). 6505:Wei, Chen (2014). 6029:Candy, J. (1974). 5900:STMicroelectronics 5340: 4965: 4673: 4572: 4474:(e.g., if it were 4449: 4389: 3595: 3572: 3570: 3543: 3541: 3508: 3488: 3486: 3449: 3360: 3331: 3286: 3262: 3123: 3095: 3067: 3033: 2992: 2869: 2709: 2680: 2640: 2638: 2592: 2511: 2482: 2451: 2383: 2358: 2347:switched capacitor 2322: 2289: 2259: 2227: 2128: 1984: 1948: 1916: 1898: 1806: 1671: 1541: 1539: 1506: 1390: 1378: 1376: 1330: 1289: 1213: 1194: 1174: 1143: 1113: 1081: 1003: 945: 761: 721: 658:quantization noise 619:) rather than its 573:low-pass filtering 446:low-pass filtering 438:quantization noise 138:Digital modulation 70: 62: 6636:978-0-87942-285-1 6617:978-0-7803-1045-2 6598:978-0-471-46585-0 6579:978-0-470-29026-2 6491:978-1-4799-8117-5 6283:Texas Instruments 6190:978-0-9660176-4-9 6015:978-0-9660176-4-9 5969:Texas Instruments 5515:(11): 1524–1535. 5381:Class-D amplifier 5174:, as simple as a 4975:The Greek letter 4918: 4860:op amp integrator 4840:= +1 V and V 4782:circuit schematic 4640: 4633: 4623: 4618: 4547: 4540: 4530: 4525: 4373: 4372: 3569: 3540: 3511:{\displaystyle d} 3485: 3388: 3383: 3343:modulator with a 3328: 3245: 3240: 3219: 3211: 3202: 3197: 3185: 3177: 3166: 3152: 3120: 3057: 3027: 3022: 2982: 2977: 2956: 2948: 2939: 2934: 2922: 2914: 2903: 2895: 2860: 2852: 2835: 2827: 2818: 2813: 2804: 2800: 2795: 2768: 2760: 2743: 2735: 2706: 2701: 2677: 2672: 2637: 2633: 2628: 2589: 2584: 2572: 2555: 2538: 2508: 2503: 2477: 2386:{\displaystyle x} 2317: 2309: 2281: 2253: 2244: 2230: 2226: 2225: 2200: 2194: 2186: 2176: 2167: 2162: 2119: 2110: 2105: 2087: 2079: 2071: 2070: 2052: 2043: 2038: 1982: 1974: 1942: 1933: 1919: 1915: 1914: 1897: 1896: 1882: 1876: 1868: 1858: 1849: 1844: 1820:transfer function 1797: 1788: 1783: 1762: 1761: 1743: 1734: 1729: 1711: 1703: 1662: 1653: 1648: 1630: 1621: 1616: 1598: 1589: 1584: 1538: 1537: 1497: 1489: 1472: 1464: 1456: 1455: 1437: 1429: 1412: 1404: 1375: 1374: 1309: 1272: 1234:Laplace transform 1197:{\displaystyle d} 1066: 919:de facto standard 857:sinc-in-frequency 819:digital filtering 609:negative feedback 494:motor controllers 430:negative feedback 380: 379: 95:Analog modulation 6796: 6672: 6670: 6669: 6652: 6640: 6621: 6602: 6583: 6562: 6560: 6559: 6554:. Analog Devices 6553: 6534: 6533: 6531: 6530: 6524: 6511: 6502: 6496: 6495: 6465: 6459: 6458: 6442: 6436: 6435: 6433: 6432: 6426: 6413: 6404: 6398: 6397: 6392:. Archived from 6386: 6380: 6379: 6372: 6366: 6365: 6363: 6362: 6342: 6333: 6332: 6330: 6329: 6308: 6302: 6301: 6299: 6298: 6292: 6279: 6270: 6264: 6263: 6261: 6260: 6254: 6247: 6238: 6232: 6231: 6229: 6228: 6222: 6215: 6206: 6195: 6194: 6178: 6169: 6163: 6162: 6160: 6159: 6139: 6133: 6132: 6130: 6129: 6102: 6096: 6095: 6093: 6092: 6086: 6073: 6065: 6059: 6058: 6026: 6020: 6019: 6003: 5994: 5988: 5987: 5985: 5984: 5978: 5965: 5957: 5951: 5950: 5948: 5947: 5941: 5934: 5925: 5919: 5918: 5916: 5915: 5909: 5896: 5888: 5882: 5881: 5879: 5877: 5872: 5864: 5858: 5857: 5855: 5854: 5848: 5841: 5832: 5826: 5825: 5823: 5817:. Archived from 5816: 5807: 5801: 5795: 5786: 5785: 5783: 5776: 5765: 5750: 5749: 5748: 5744: 5737: 5731: 5730: 5728: 5727: 5713: 5707: 5706: 5704: 5703: 5662: 5656: 5655: 5615: 5606: 5605: 5604: 5600: 5593: 5587: 5584: 5578: 5577: 5575: 5574: 5568: 5555: 5546: 5537: 5536: 5498: 5492: 5491: 5489: 5488: 5482: 5459: 5450: 5439: 5438: 5437: 5436: 5430: 5419: 5402: 5398: 5156: 5150: 5144: 5138: 5132: 5126: 5112: 5103: 5097: 5088: 5082: 5056: 5034: 5028: 5022: 5011:about 0 V. 5005: 4999: 4993: 4983: 4979:is used because 4974: 4972: 4971: 4966: 4919: 4917: 4906: 4878: 4872: 4866: 4848: 4808: 4798: 4762: 4748: 4682: 4680: 4679: 4674: 4668: 4664: 4657: 4653: 4652: 4651: 4650: 4649: 4641: 4638: 4634: 4631: 4625: 4624: 4621: 4619: 4616: 4581: 4579: 4578: 4573: 4561: 4557: 4556: 4555: 4554: 4553: 4548: 4545: 4541: 4538: 4532: 4531: 4528: 4526: 4523: 4496: 4458: 4456: 4455: 4450: 4393:delta modulation 4366: 4365: 4361: 4358: 4349: 4348: 4344: 4341: 4332: 4331: 4327: 4324: 4315: 4314: 4310: 4307: 4298: 4297: 4293: 4290: 4274: 4273: 4269: 4266: 4250: 4249: 4245: 4242: 4225: 4224: 4220: 4217: 4194: 4193: 4189: 4186: 4176: 4175: 4171: 4168: 4151: 4150: 4146: 4143: 4126: 4125: 4121: 4118: 4101: 4100: 4096: 4093: 4083: 4082: 4078: 4075: 4064: 4063: 4059: 4056: 4045: 4044: 4040: 4037: 4026: 4025: 4021: 4018: 4007: 4006: 4002: 3999: 3982: 3981: 3977: 3974: 3964: 3963: 3959: 3956: 3945: 3944: 3940: 3937: 3926: 3925: 3921: 3918: 3907: 3906: 3902: 3899: 3888: 3887: 3883: 3880: 3863: 3862: 3858: 3855: 3845: 3844: 3840: 3837: 3826: 3825: 3821: 3818: 3807: 3806: 3802: 3799: 3788: 3787: 3783: 3780: 3769: 3768: 3764: 3761: 3744: 3743: 3739: 3736: 3726: 3725: 3721: 3718: 3707: 3706: 3702: 3699: 3688: 3687: 3683: 3680: 3669: 3668: 3664: 3661: 3650: 3649: 3645: 3642: 3598: 3591: 3590: 3586: 3581: 3579: 3578: 3573: 3571: 3562: 3552: 3550: 3549: 3544: 3542: 3533: 3517: 3515: 3514: 3509: 3497: 3495: 3494: 3489: 3487: 3478: 3459:The theoretical 3458: 3456: 3455: 3450: 3390: 3389: 3386: 3384: 3381: 3369: 3367: 3366: 3361: 3359: 3358: 3342: 3340: 3338: 3337: 3332: 3330: 3329: 3326: 3305:The theoretical 3295: 3293: 3292: 3287: 3271: 3269: 3268: 3263: 3257: 3256: 3247: 3246: 3243: 3241: 3238: 3220: 3217: 3212: 3209: 3204: 3203: 3200: 3198: 3195: 3186: 3183: 3178: 3175: 3167: 3164: 3159: 3158: 3153: 3150: 3134: 3132: 3130: 3129: 3124: 3122: 3121: 3118: 3104: 3102: 3101: 3096: 3076: 3074: 3073: 3068: 3063: 3058: 3055: 3042: 3040: 3039: 3034: 3029: 3028: 3025: 3023: 3020: 3001: 2999: 2998: 2993: 2984: 2983: 2980: 2978: 2975: 2957: 2954: 2949: 2946: 2941: 2940: 2937: 2935: 2932: 2923: 2920: 2915: 2912: 2904: 2901: 2896: 2893: 2878: 2876: 2875: 2870: 2861: 2858: 2853: 2850: 2836: 2833: 2828: 2825: 2820: 2819: 2816: 2814: 2811: 2805: 2803: 2802: 2801: 2798: 2796: 2793: 2780: 2769: 2766: 2761: 2758: 2744: 2741: 2736: 2733: 2718: 2716: 2715: 2710: 2708: 2707: 2704: 2702: 2699: 2689: 2687: 2686: 2681: 2679: 2678: 2675: 2673: 2670: 2649: 2647: 2646: 2641: 2639: 2636: 2635: 2634: 2631: 2629: 2626: 2613: 2603: 2601: 2599: 2598: 2593: 2591: 2590: 2587: 2585: 2582: 2573: 2570: 2556: 2553: 2539: 2536: 2520: 2518: 2517: 2512: 2510: 2509: 2506: 2504: 2501: 2491: 2489: 2488: 2483: 2478: 2475: 2460: 2458: 2457: 2452: 2392: 2390: 2389: 2384: 2331: 2329: 2328: 2323: 2318: 2315: 2310: 2307: 2298: 2296: 2295: 2290: 2282: 2279: 2268: 2266: 2265: 2260: 2254: 2252: 2245: 2242: 2236: 2231: 2229: 2228: 2223: 2219: 2206: 2201: 2199: 2195: 2192: 2187: 2184: 2181: 2177: 2174: 2169: 2168: 2165: 2163: 2160: 2150: 2137: 2135: 2134: 2129: 2120: 2117: 2112: 2111: 2108: 2106: 2103: 2088: 2085: 2080: 2077: 2072: 2068: 2064: 2053: 2050: 2045: 2044: 2041: 2039: 2036: 2000:cutoff frequency 1993: 1991: 1990: 1985: 1983: 1980: 1975: 1972: 1957: 1955: 1954: 1949: 1943: 1941: 1934: 1931: 1925: 1920: 1918: 1917: 1912: 1908: 1894: 1890: 1888: 1883: 1881: 1877: 1874: 1869: 1866: 1863: 1859: 1856: 1851: 1850: 1847: 1845: 1842: 1832: 1815: 1813: 1812: 1807: 1798: 1795: 1790: 1789: 1786: 1784: 1781: 1763: 1759: 1755: 1744: 1741: 1736: 1735: 1732: 1730: 1727: 1712: 1709: 1704: 1701: 1680: 1678: 1677: 1672: 1663: 1660: 1655: 1654: 1651: 1649: 1646: 1631: 1628: 1623: 1622: 1619: 1617: 1614: 1599: 1596: 1591: 1590: 1587: 1585: 1582: 1550: 1548: 1547: 1542: 1540: 1535: 1531: 1515: 1513: 1512: 1507: 1498: 1495: 1490: 1487: 1473: 1470: 1465: 1462: 1457: 1453: 1449: 1438: 1435: 1430: 1427: 1413: 1410: 1405: 1402: 1387: 1385: 1384: 1379: 1377: 1372: 1368: 1339: 1337: 1336: 1331: 1329: 1328: 1310: 1307: 1298: 1296: 1295: 1290: 1273: 1270: 1261:complex variable 1257:frequency domain 1203: 1201: 1200: 1195: 1183: 1181: 1180: 1175: 1173: 1172: 1152: 1150: 1149: 1144: 1142: 1141: 1122: 1120: 1119: 1114: 1112: 1111: 1110: 1090: 1088: 1087: 1082: 1080: 1079: 1067: 1065: 1064: 1063: 1050: 1049: 1040: 1033: 1012: 1010: 1009: 1004: 1002: 1001: 887:infinite support 835:MEMS microphones 725:C. Chapin Cutler 698:delta modulation 642:cumulative error 625:marks and spaces 605:Delta modulation 414:sample-frequency 372: 365: 358: 89: 72: 51: 47: 6804: 6803: 6799: 6798: 6797: 6795: 6794: 6793: 6779: 6778: 6690:Wayback Machine 6679: 6667: 6665: 6650: 6643: 6637: 6624: 6618: 6605: 6599: 6586: 6580: 6565: 6557: 6555: 6551: 6546: 6543: 6541:Further reading 6538: 6537: 6528: 6526: 6522: 6509: 6504: 6503: 6499: 6492: 6467: 6466: 6462: 6444: 6443: 6439: 6430: 6428: 6424: 6411: 6406: 6405: 6401: 6388: 6387: 6383: 6374: 6373: 6369: 6360: 6358: 6346:Ellsworth, Jeri 6344: 6343: 6336: 6327: 6325: 6310: 6309: 6305: 6296: 6294: 6290: 6277: 6272: 6271: 6267: 6258: 6256: 6252: 6245: 6240: 6239: 6235: 6226: 6224: 6220: 6213: 6209:Van Ess, Dave. 6208: 6207: 6198: 6191: 6176: 6171: 6170: 6166: 6157: 6155: 6141: 6140: 6136: 6127: 6125: 6104: 6103: 6099: 6090: 6088: 6084: 6071: 6067: 6066: 6062: 6028: 6027: 6023: 6016: 6001: 5996: 5995: 5991: 5982: 5980: 5976: 5963: 5959: 5958: 5954: 5945: 5943: 5939: 5932: 5927: 5926: 5922: 5913: 5911: 5907: 5894: 5890: 5889: 5885: 5875: 5873: 5870: 5866: 5865: 5861: 5852: 5850: 5846: 5839: 5834: 5833: 5829: 5821: 5814: 5809: 5808: 5804: 5796: 5789: 5781: 5774: 5767: 5766: 5753: 5746: 5739: 5738: 5734: 5725: 5723: 5715: 5714: 5710: 5701: 5699: 5692: 5664: 5663: 5659: 5617: 5616: 5609: 5602: 5595: 5594: 5590: 5585: 5581: 5572: 5570: 5566: 5553: 5548: 5547: 5540: 5500: 5499: 5495: 5486: 5484: 5480: 5457: 5452: 5451: 5442: 5434: 5432: 5428: 5421: 5420: 5416: 5411: 5406: 5405: 5399: 5395: 5390: 5367: 5344:Schmitt trigger 5331: 5278: 5244: 5224: 5202: 5196: 5187:digital samples 5172:low-pass filter 5164: 5154: 5148: 5142: 5136: 5130: 5124: 5120: 5110: 5101: 5095: 5086: 5080: 5064:sampling period 5054: 5046: 5032: 5026: 5020: 5003: 4997: 4991: 4981: 4910: 4882: 4881: 4876: 4870: 4864: 4856: 4846: 4843: 4839: 4806: 4804:digital result 4796: 4778: 4770: 4769: 4768: 4767: 4766: 4763: 4754: 4753: 4752: 4749: 4738: 4636: 4629: 4614: 4613: 4609: 4605: 4601: 4590: 4589: 4543: 4536: 4521: 4520: 4516: 4502: 4501: 4495: 4491: 4408: 4407: 4381: 4363: 4359: 4356: 4354: 4346: 4342: 4339: 4337: 4329: 4325: 4322: 4320: 4312: 4308: 4305: 4303: 4295: 4291: 4288: 4286: 4282: 4271: 4267: 4264: 4262: 4247: 4243: 4240: 4238: 4222: 4218: 4215: 4213: 4202: 4191: 4187: 4184: 4182: 4173: 4169: 4166: 4164: 4148: 4144: 4141: 4139: 4123: 4119: 4116: 4114: 4109: 4098: 4094: 4091: 4089: 4080: 4076: 4073: 4071: 4061: 4057: 4054: 4052: 4042: 4038: 4035: 4033: 4023: 4019: 4016: 4014: 4004: 4000: 3997: 3995: 3990: 3979: 3975: 3972: 3970: 3961: 3957: 3954: 3952: 3942: 3938: 3935: 3933: 3923: 3919: 3916: 3914: 3904: 3900: 3897: 3895: 3885: 3881: 3878: 3876: 3871: 3860: 3856: 3853: 3851: 3842: 3838: 3835: 3833: 3823: 3819: 3816: 3814: 3804: 3800: 3797: 3795: 3785: 3781: 3778: 3776: 3766: 3762: 3759: 3757: 3752: 3741: 3737: 3734: 3732: 3723: 3719: 3716: 3714: 3704: 3700: 3697: 3695: 3685: 3681: 3678: 3676: 3666: 3662: 3659: 3657: 3647: 3643: 3640: 3638: 3633: 3588: 3584: 3583: 3555: 3554: 3520: 3519: 3500: 3499: 3465: 3464: 3379: 3374: 3373: 3350: 3345: 3344: 3321: 3316: 3315: 3314: 3303: 3278: 3277: 3248: 3236: 3193: 3148: 3137: 3136: 3113: 3108: 3107: 3106: 3087: 3086: 3083: 3049: 3048: 3018: 3004: 3003: 2973: 2930: 2881: 2880: 2809: 2791: 2784: 2724: 2723: 2697: 2692: 2691: 2668: 2663: 2662: 2624: 2617: 2606: 2605: 2580: 2524: 2523: 2522: 2499: 2494: 2493: 2463: 2462: 2395: 2394: 2366: 2365: 2342: 2301: 2300: 2273: 2272: 2240: 2210: 2182: 2158: 2151: 2144: 2143: 2101: 2034: 2014: 2013: 2008: 1966: 1965: 1929: 1899: 1864: 1840: 1833: 1826: 1825: 1779: 1725: 1692: 1691: 1686: 1644: 1612: 1580: 1569: 1568: 1553:real integrator 1524: 1523: 1393: 1392: 1361: 1360: 1346: 1317: 1301: 1300: 1264: 1263: 1222: 1186: 1185: 1164: 1159: 1158: 1133: 1125: 1124: 1101: 1096: 1095: 1071: 1055: 1051: 1041: 1018: 1017: 993: 985: 984: 977: 932: 881:: Although the 831:microcontroller 810: 785: 774: 753: 694: 667: 602: 589: 552: 517: 428:by utilizing a 412:at a very high 410:digital signals 376: 239:Spread spectrum 49: 45: 24: 17: 12: 11: 5: 6802: 6800: 6792: 6791: 6781: 6780: 6777: 6776: 6771: 6766: 6760: 6754: 6748: 6742: 6736: 6730: 6725: 6719: 6713: 6708: 6698: 6692: 6678: 6677:External links 6675: 6674: 6673: 6663:10.25560/23651 6653:(PhD thesis). 6641: 6635: 6622: 6616: 6603: 6597: 6584: 6578: 6563: 6542: 6539: 6536: 6535: 6518:. p. 88. 6497: 6490: 6460: 6437: 6417:Analog Devices 6399: 6396:on 2022-04-08. 6381: 6367: 6348:(2012-11-05). 6334: 6320:. 2021-01-09. 6317:Analog Devices 6303: 6265: 6233: 6196: 6189: 6164: 6148:dsprelated.com 6134: 6097: 6060: 6041:(3): 298–305. 6021: 6014: 5989: 5952: 5920: 5903:. March 2018. 5883: 5859: 5827: 5824:on 2022-11-02. 5802: 5787: 5751: 5732: 5708: 5690: 5657: 5607: 5588: 5579: 5538: 5493: 5440: 5413: 5412: 5410: 5407: 5404: 5403: 5392: 5391: 5389: 5386: 5385: 5384: 5378: 5373: 5366: 5363: 5330: 5327: 5312:Analog Devices 5306:In the 1970s, 5303:respectively. 5277: 5274: 5273: 5272: 5269: 5243: 5240: 5223: 5220: 5198:Main article: 5195: 5192: 5163: 5160: 5119: 5116: 5107: 5106: 5091: 5045: 5042: 4964: 4961: 4958: 4955: 4952: 4949: 4946: 4943: 4940: 4937: 4934: 4931: 4928: 4925: 4922: 4916: 4913: 4909: 4904: 4901: 4898: 4895: 4892: 4889: 4855: 4852: 4841: 4837: 4777: 4774: 4764: 4757: 4756: 4755: 4750: 4743: 4742: 4741: 4740: 4739: 4737: 4734: 4733: 4732: 4731: 4730: 4727: 4721: 4720: 4719: 4716: 4713: 4684: 4683: 4671: 4667: 4663: 4660: 4656: 4648: 4645: 4628: 4612: 4608: 4604: 4600: 4597: 4583: 4582: 4570: 4567: 4564: 4560: 4552: 4535: 4519: 4515: 4512: 4509: 4493: 4464: 4463: 4460: 4448: 4445: 4442: 4439: 4436: 4433: 4430: 4427: 4424: 4421: 4418: 4415: 4400: 4380: 4377: 4371: 4370: 4368: 4351: 4334: 4317: 4300: 4283: 4280: 4277: 4276: 4259: 4253: 4234: 4228: 4209: 4203: 4200: 4197: 4196: 4179: 4160: 4154: 4135: 4129: 4110: 4107: 4104: 4103: 4086: 4067: 4048: 4029: 4010: 3991: 3988: 3985: 3984: 3967: 3948: 3929: 3910: 3891: 3872: 3869: 3866: 3865: 3848: 3829: 3810: 3791: 3772: 3753: 3750: 3747: 3746: 3729: 3710: 3691: 3672: 3653: 3634: 3631: 3628: 3627: 3624: 3621: 3618: 3615: 3611: 3610: 3604: 3601: 3568: 3565: 3539: 3536: 3530: 3527: 3507: 3484: 3481: 3475: 3472: 3448: 3444: 3441: 3438: 3435: 3432: 3429: 3426: 3423: 3420: 3417: 3414: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3357: 3353: 3324: 3302: 3299: 3285: 3261: 3255: 3251: 3235: 3232: 3229: 3226: 3223: 3215: 3207: 3192: 3189: 3181: 3173: 3170: 3162: 3157: 3147: 3144: 3116: 3094: 3082: 3079: 3066: 3062: 3032: 3017: 3014: 3011: 2991: 2987: 2972: 2969: 2966: 2963: 2960: 2952: 2944: 2929: 2926: 2918: 2910: 2907: 2899: 2891: 2888: 2868: 2864: 2856: 2848: 2845: 2842: 2839: 2831: 2823: 2808: 2790: 2787: 2783: 2778: 2775: 2772: 2764: 2756: 2753: 2750: 2747: 2739: 2731: 2623: 2620: 2616: 2579: 2576: 2568: 2565: 2562: 2559: 2551: 2548: 2545: 2542: 2534: 2531: 2481: 2473: 2470: 2450: 2447: 2444: 2441: 2438: 2435: 2432: 2429: 2426: 2423: 2420: 2417: 2414: 2411: 2408: 2405: 2402: 2382: 2379: 2376: 2373: 2341: 2338: 2321: 2313: 2288: 2285: 2258: 2251: 2248: 2239: 2234: 2222: 2216: 2213: 2209: 2204: 2198: 2190: 2180: 2172: 2157: 2154: 2127: 2123: 2115: 2100: 2097: 2094: 2091: 2083: 2075: 2067: 2062: 2059: 2056: 2048: 2033: 2030: 2027: 2024: 2021: 2007: 2004: 1978: 1947: 1940: 1937: 1928: 1923: 1911: 1905: 1902: 1893: 1886: 1880: 1872: 1862: 1854: 1839: 1836: 1805: 1801: 1793: 1778: 1775: 1772: 1769: 1766: 1758: 1753: 1750: 1747: 1739: 1724: 1721: 1718: 1715: 1707: 1699: 1685: 1682: 1670: 1666: 1658: 1643: 1640: 1637: 1634: 1626: 1611: 1608: 1605: 1602: 1594: 1579: 1576: 1534: 1505: 1501: 1493: 1485: 1482: 1479: 1476: 1468: 1460: 1452: 1447: 1444: 1441: 1433: 1425: 1422: 1419: 1416: 1408: 1400: 1371: 1345: 1342: 1327: 1324: 1320: 1316: 1313: 1288: 1285: 1282: 1279: 1276: 1221: 1218: 1193: 1171: 1167: 1140: 1136: 1132: 1109: 1104: 1092: 1091: 1078: 1074: 1070: 1062: 1058: 1054: 1048: 1044: 1037: 1032: 1029: 1026: 1000: 996: 992: 976: 973: 931: 928: 927: 926: 875: 864: 809: 806: 783: 773: 770: 752: 749: 693: 690: 666: 663: 638:slope overload 601: 598: 588: 585: 584: 583: 576: 566: 563: 551: 548: 544:digital signal 516: 513: 498:Super Audio CD 469:digital filter 378: 377: 375: 374: 367: 360: 352: 349: 348: 347: 346: 341: 336: 331: 326: 321: 316: 311: 306: 301: 296: 291: 286: 281: 273: 272: 268: 267: 266: 265: 260: 255: 250: 242: 241: 235: 234: 233: 232: 227: 219: 218: 212: 211: 210: 209: 204: 199: 194: 189: 184: 179: 174: 169: 164: 159: 154: 149: 141: 140: 134: 133: 132: 131: 126: 121: 116: 111: 106: 98: 97: 91: 90: 82: 81: 15: 13: 10: 9: 6: 4: 3: 2: 6801: 6790: 6787: 6786: 6784: 6775: 6772: 6770: 6767: 6764: 6761: 6758: 6755: 6752: 6749: 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5887: 5884: 5869: 5863: 5860: 5845: 5838: 5831: 5828: 5820: 5813: 5806: 5803: 5799: 5794: 5792: 5788: 5784:on 2006-06-21 5780: 5773: 5772: 5768:Sangil Park, 5764: 5762: 5760: 5758: 5756: 5752: 5742: 5736: 5733: 5722: 5718: 5712: 5709: 5697: 5693: 5691:9780306480041 5687: 5683: 5679: 5675: 5671: 5667: 5661: 5658: 5653: 5649: 5645: 5641: 5637: 5633: 5629: 5625: 5621: 5614: 5612: 5608: 5598: 5592: 5589: 5583: 5580: 5565: 5561: 5560: 5552: 5545: 5543: 5539: 5534: 5530: 5526: 5522: 5518: 5514: 5510: 5509: 5504: 5497: 5494: 5479: 5475: 5471: 5467: 5463: 5456: 5449: 5447: 5445: 5441: 5427: 5426: 5422:Sangil Park, 5418: 5415: 5408: 5397: 5394: 5387: 5382: 5379: 5377: 5374: 5372: 5369: 5368: 5364: 5362: 5358: 5356: 5351: 5349: 5345: 5335: 5328: 5326: 5324: 5320: 5315: 5313: 5309: 5304: 5302: 5298: 5294: 5291: 5287: 5283: 5275: 5270: 5267: 5263: 5259: 5255: 5254: 5253: 5250: 5249:noise shaping 5241: 5239: 5237: 5233: 5229: 5221: 5219: 5216: 5211: 5207: 5201: 5193: 5191: 5188: 5184: 5179: 5177: 5173: 5169: 5161: 5159: 5157: 5151: 5145: 5139: 5133: 5127: 5117: 5115: 5113: 5104: 5098: 5092: 5089: 5083: 5077: 5076: 5075: 5072: 5070: 5069:metastability 5065: 5061: 5057: 5051: 5043: 5041: 5039: 5035: 5029: 5023: 5017: 5012: 5010: 5006: 5000: 4994: 4988: 4984: 4978: 4962: 4959: 4956: 4947: 4941: 4938: 4932: 4926: 4920: 4914: 4911: 4907: 4902: 4899: 4893: 4887: 4879: 4873: 4867: 4861: 4853: 4851: 4849: 4835: 4830: 4828: 4824: 4820: 4815: 4813: 4809: 4803: 4799: 4794: 4789: 4787: 4783: 4775: 4773: 4761: 4747: 4735: 4728: 4725: 4724: 4722: 4717: 4714: 4711: 4710: 4708: 4707: 4706: 4703: 4701: 4697: 4692: 4690: 4669: 4665: 4661: 4658: 4654: 4626: 4610: 4606: 4602: 4598: 4595: 4588: 4587: 4586: 4568: 4565: 4562: 4558: 4533: 4517: 4513: 4510: 4507: 4500: 4499: 4498: 4487: 4485: 4481: 4477: 4473: 4469: 4461: 4443: 4440: 4437: 4431: 4428: 4425: 4422: 4419: 4416: 4413: 4405: 4401: 4398: 4397: 4396: 4394: 4385: 4378: 4376: 4369: 4352: 4335: 4318: 4301: 4284: 4279: 4278: 4260: 4258: 4252: 4233: 4227: 4208: 4199: 4198: 4180: 4178: 4159: 4153: 4134: 4128: 4106: 4105: 4087: 4085: 4066: 4047: 4028: 4009: 3987: 3986: 3968: 3966: 3947: 3928: 3909: 3890: 3868: 3867: 3849: 3847: 3828: 3809: 3790: 3771: 3749: 3748: 3730: 3728: 3709: 3690: 3671: 3652: 3630: 3629: 3625: 3622: 3619: 3616: 3613: 3612: 3609: 3599: 3593: 3566: 3563: 3537: 3534: 3528: 3525: 3505: 3482: 3479: 3473: 3462: 3446: 3442: 3439: 3433: 3430: 3427: 3424: 3421: 3415: 3412: 3406: 3403: 3397: 3394: 3391: 3371: 3355: 3351: 3312: 3308: 3300: 3298: 3275: 3259: 3233: 3230: 3224: 3205: 3190: 3171: 3080: 3078: 3064: 3060: 3046: 3043:represents a 3015: 3012: 2989: 2970: 2967: 2961: 2942: 2927: 2908: 2866: 2840: 2821: 2806: 2788: 2785: 2781: 2776: 2748: 2720: 2660: 2656: 2651: 2621: 2618: 2614: 2577: 2563: 2560: 2546: 2543: 2529: 2468: 2448: 2442: 2439: 2436: 2430: 2427: 2421: 2415: 2412: 2406: 2400: 2377: 2371: 2363: 2354: 2350: 2348: 2339: 2337: 2335: 2319: 2311: 2286: 2283: 2269: 2256: 2249: 2246: 2237: 2232: 2220: 2214: 2211: 2207: 2202: 2141: 2138: 2125: 2092: 2073: 2065: 2060: 2025: 2022: 2011: 2005: 2003: 2001: 1997: 1996:complex plane 1976: 1963: 1958: 1945: 1938: 1935: 1926: 1921: 1909: 1903: 1900: 1891: 1884: 1823: 1821: 1816: 1803: 1770: 1767: 1764: 1756: 1751: 1716: 1689: 1683: 1681: 1668: 1635: 1603: 1566: 1563: 1561: 1556: 1554: 1532: 1521: 1516: 1503: 1477: 1458: 1450: 1445: 1417: 1369: 1357: 1353: 1351: 1343: 1341: 1325: 1322: 1318: 1314: 1311: 1286: 1283: 1280: 1277: 1274: 1262: 1258: 1255: 1251: 1247: 1243: 1239: 1235: 1230: 1227: 1220:Noise shaping 1219: 1217: 1209: 1205: 1191: 1169: 1165: 1156: 1138: 1134: 1130: 1102: 1076: 1072: 1068: 1060: 1056: 1052: 1046: 1042: 1035: 1016: 1015: 1014: 998: 994: 990: 982: 974: 972: 969: 964: 962: 958: 954: 950: 942: 936: 929: 924: 923:high fidelity 920: 916: 912: 911:step response 908: 904: 900: 896: 892: 888: 884: 883:sinc function 880: 876: 872: 868: 865: 862: 858: 854: 850: 847: 846: 845: 842: 840: 836: 832: 828: 824: 820: 816: 807: 805: 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Subsequent 443: 439: 435: 431: 427: 423: 419: 415: 411: 408: 404: 400: 396: 392: 388: 384: 373: 368: 366: 361: 359: 354: 353: 351: 350: 345: 342: 340: 337: 335: 332: 330: 327: 325: 322: 320: 317: 315: 312: 310: 307: 305: 302: 300: 297: 295: 292: 290: 287: 285: 282: 280: 277: 276: 275: 274: 269: 264: 261: 259: 256: 254: 251: 249: 246: 245: 244: 243: 240: 236: 231: 228: 226: 223: 222: 221: 220: 217: 213: 208: 205: 203: 200: 198: 195: 193: 190: 188: 185: 183: 180: 178: 175: 173: 170: 168: 165: 163: 160: 158: 155: 153: 150: 148: 145: 144: 143: 142: 139: 135: 130: 127: 125: 122: 120: 117: 115: 112: 110: 107: 105: 102: 101: 100: 99: 96: 92: 88: 84: 83: 80: 77: 73: 66: 59: 55: 43: 39: 36:(top) and ΔΣ 35: 30: 26: 22: 6666:. Retrieved 6646: 6626: 6607: 6588: 6568: 6556:. Retrieved 6527:. Retrieved 6513: 6500: 6473: 6463: 6454: 6450: 6440: 6429:. Retrieved 6415: 6402: 6394:the original 6384: 6370: 6359:. Retrieved 6353: 6326:. Retrieved 6315: 6306: 6295:. Retrieved 6281: 6268: 6257:. 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Retrieved 5468:(2): 10–15. 5465: 5461: 5433:, retrieved 5424: 5417: 5396: 5359: 5352: 5341: 5322: 5318: 5316: 5305: 5292: 5289: 5285: 5279: 5268:are required 5266:D flip-flops 5258:accumulators 5245: 5225: 5203: 5180: 5165: 5153: 5147: 5141: 5135: 5129: 5123: 5121: 5118:Demodulation 5109: 5108: 5100: 5094: 5085: 5079: 5073: 5053: 5047: 5037: 5031: 5025: 5019: 5015: 5013: 5008: 5002: 4996: 4990: 4980: 4875: 4869: 4863: 4857: 4845: 4831: 4816: 4805: 4795: 4790: 4779: 4771: 4704: 4699: 4693: 4689:oversampling 4685: 4584: 4488: 4483: 4479: 4468:quantization 4465: 4390: 4374: 4256: 4237: 4231: 4212: 4206: 4163: 4157: 4138: 4132: 4113: 4070: 4051: 4032: 4013: 3994: 3951: 3932: 3913: 3894: 3875: 3832: 3813: 3794: 3775: 3756: 3713: 3694: 3675: 3656: 3637: 3607: 3372: 3304: 3273: 3084: 3044: 2721: 2658: 2654: 2652: 2359: 2343: 2270: 2142: 2139: 2012: 2009: 1959: 1824: 1817: 1690: 1687: 1567: 1564: 1557: 1517: 1391: 1347: 1231: 1226:shapes noise 1223: 1214: 1155:Nyquist rate 1093: 978: 965: 953:limit cycles 946: 906: 902: 870: 843: 811: 795: 790: 781: 780:is called a 777: 775: 766: 762: 741: 722: 695: 686:Nyquist rate 675: 668: 646: 616: 603: 593: 590: 553: 536:Nyquist rate 529: 518: 509:asynchronous 502: 483: 462: 450:demodulation 399:oversampling 394: 390: 386: 382: 381: 344:Multiplexing 328: 284:Demodulation 25: 6457:(4): 83–88. 5876:October 27, 5652:IEEE Xplore 5533:IEEE Xplore 5323:delta-sigma 5319:sigma-delta 5317:Both names 5293:differences 5286:delta-sigma 5284:. The name 5232:accumulator 5050:D flip-flop 5018:sign, when 4874:to produce 4834:D flip-flop 4472:homogeneous 3272:Since this 2655:requantizer 2362:accumulator 1250:time domain 1242:Z-transform 949:white noise 915:convolution 897:to realize 678:downsampled 505:synchronous 420:(ADCs) and 391:sigma-delta 383:Delta-sigma 289:Line coding 21:Sigma Delta 6668:2024-01-19 6558:2010-11-02 6529:2024-03-16 6431:2023-06-28 6361:2023-06-29 6328:2023-07-01 6297:2023-09-01 6259:2023-09-01 6227:2023-08-22 6158:2024-01-03 6128:2024-01-03 6091:2024-01-03 5983:2023-09-03 5946:2023-08-24 5914:2023-09-03 5853:2023-08-28 5741:US3192371A 5726:2024-03-16 5721:Legacy.com 5702:2023-06-28 5573:2023-06-28 5487:2024-03-16 5435:2017-09-01 5431:, Motorola 5409:References 5348:hysteresis 5194:Upsampling 5185:signal of 5009:millivolts 4819:breadboard 4696:comparator 3518:), and by 961:distortion 907:almost all 745:decimation 713:integrator 682:decimation 649:integrator 629:integrated 613:derivative 525:Digitizing 515:Motivation 436:and moves 79:modulation 6610:. Wiley. 6591:. Wiley. 6117:: 72–76. 6055:1558-0857 5640:2331-1657 5529:1558-2256 5308:Bell Labs 5048:An ideal 4921:∫ 4903:− 4888:ε 4862:combines 4793:sine wave 4786:simulated 4647:Σ 4644:Δ 4627:− 4607:∫ 4599:⁡ 4551:Δ 4534:− 4514:⁡ 4508:∫ 4432:∫ 4423:∫ 4414:∫ 3608:doubling 3606:each OSR 3529:− 3471:Θ 3440:− 3437:Θ 3434:⋅ 3428:− 3422:⋅ 3410:Θ 3407:⋅ 3398:⋅ 3392:≈ 3323:Θ 3309:(SNR) in 3284:Θ 3254:Θ 3234:− 3225:⋅ 3191:⋅ 3156:Θ 3146:Σ 3143:Δ 3115:Θ 3093:Θ 3016:− 2971:− 2962:⋅ 2928:⋅ 2890:Σ 2887:Δ 2847:Σ 2844:Δ 2807:⋅ 2789:− 2777:⋅ 2755:Σ 2752:Δ 2749:− 2622:− 2578:⋅ 2440:− 2156:Σ 2153:Δ 2099:Σ 2096:Δ 2061:⋅ 2032:Σ 2029:Δ 2026:− 1838:Σ 1835:Δ 1777:Σ 1774:Δ 1752:⋅ 1723:Σ 1720:Δ 1717:− 1642:Σ 1639:Δ 1610:Σ 1607:Δ 1578:Σ 1575:Δ 1484:Σ 1481:Δ 1446:⋅ 1424:Σ 1421:Δ 1418:− 1326:ϕ 1287:ω 1278:σ 957:dithering 903:almost no 877:Windowed 874:article). 827:bit depth 729:Bell Labs 717:flip-flop 704:in 1952. 621:amplitude 594:averaging 540:quantized 442:bandwidth 407:bit depth 405:into low 54:filtering 6783:Category 6686:Archived 6520:Archived 6422:Archived 6322:Archived 6288:Archived 6250:Archived 6218:Archived 6152:Archived 6082:Archived 6080:. 2007. 5974:Archived 5972:. 2014. 5937:Archived 5905:Archived 5844:Archived 5696:Archived 5648:51647729 5564:Archived 5478:Archived 5365:See also 5236:register 5170:using a 5052:samples 5038:negative 5016:negative 4639:feedback 4596:Quantize 4546:feedback 4511:Quantize 4484:inspired 4257:42 bits 4232:29 bits 4207:16 bits 4201:6-order: 4158:30 bits 4133:19 bits 4108:5-order: 3989:4-order: 3870:3-order: 3751:2-order: 3632:1-order: 3311:decibels 968:baseband 895:windowed 815:decimate 778:N-levels 474:voltages 271:See also 76:Passband 6705:Part II 6355:YouTube 5290:summing 5204:As the 5030:, the + 4977:epsilon 4827:op amps 4700:outside 4362:⁄ 4345:⁄ 4328:⁄ 4311:⁄ 4294:⁄ 4270:⁄ 4255:254 dB 4246:⁄ 4236:215 dB 4230:176 dB 4221:⁄ 4211:137 dB 4190:⁄ 4172:⁄ 4162:215 dB 4156:182 dB 4147:⁄ 4137:149 dB 4131:116 dB 4122:⁄ 4097:⁄ 4079:⁄ 4069:177 dB 4060:⁄ 4050:149 dB 4041:⁄ 4031:112 dB 4022:⁄ 4003:⁄ 3978:⁄ 3960:⁄ 3950:138 dB 3941:⁄ 3931:117 dB 3922:⁄ 3903:⁄ 3884:⁄ 3859:⁄ 3841:⁄ 3822:⁄ 3803:⁄ 3784:⁄ 3765:⁄ 3740:⁄ 3722:⁄ 3703:⁄ 3684:⁄ 3665:⁄ 3646:⁄ 3587:⁄ 1994:in the 1259:of the 1254:complex 1153:is the 871:aliased 791:2.5-bit 532:sampled 460:(PCM). 426:quality 403:signals 58:aliases 6633: 6614: 6595: 6576: 6488: 6187: 6077:Altera 6053: 6012: 5747: 5688: 5646: 5638: 5603: 5527: 5276:Naming 5264:, and 5262:adders 4476:linear 4205:99 dB 4112:83 dB 4012:95 dB 3993:68 dB 3912:96 dB 3893:75 dB 3874:53 dB 3831:99 dB 3812:84 dB 3793:69 dB 3774:54 dB 3755:39 dB 3712:60 dB 3693:51 dB 3674:42 dB 3655:33 dB 3636:24 dB 3626:2 OSR 3623:2 OSR 3620:2 OSR 3617:2 OSR 3614:2 OSR 3341:-order 3133:-order 1094:where 943:noise. 941:dither 798:dither 560:clocks 197:SC-FDE 6651:(PDF) 6552:(PDF) 6523:(PDF) 6510:(PDF) 6425:(PDF) 6412:(PDF) 6291:(PDF) 6278:(PDF) 6253:(PDF) 6246:(PDF) 6221:(PDF) 6214:(PDF) 6177:(PDF) 6085:(PDF) 6072:(PDF) 6002:(PDF) 5977:(PDF) 5964:(PDF) 5940:(PDF) 5933:(PDF) 5908:(PDF) 5895:(PDF) 5871:(PDF) 5847:(PDF) 5840:(PDF) 5822:(PDF) 5815:(PDF) 5782:(PDF) 5775:(PDF) 5644:S2CID 5567:(PDF) 5554:(PDF) 5481:(PDF) 5458:(PDF) 5429:(PDF) 5388:Notes 5301:delta 5297:sigma 4987:error 4868:with 4470:were 4367:bits 4350:bits 4333:bits 4316:bits 4299:bits 4275:bits 4251:bits 4226:bits 4195:bits 4177:bits 4152:bits 4127:bits 4102:bits 4084:bits 4065:bits 4046:bits 4027:bits 4008:bits 3983:bits 3965:bits 3946:bits 3927:bits 3908:bits 3889:bits 3864:bits 3846:bits 3827:bits 3808:bits 3789:bits 3770:bits 3745:bits 3727:bits 3708:bits 3689:bits 3670:bits 3651:bits 3210:noise 2947:noise 2826:noise 2657:or a 2185:noise 2166:noise 2109:noise 2078:noise 2042:noise 1652:noise 1588:total 1463:noise 1244:(for 1236:(for 617:delta 521:noise 389:; or 294:Modem 6703:and 6631:ISBN 6612:ISBN 6593:ISBN 6574:ISBN 6486:ISBN 6185:ISBN 6051:ISSN 6010:ISBN 5878:2018 5686:ISBN 5636:ISSN 5559:IEEE 5525:ISSN 5321:and 5299:and 5208:and 5155:s(t) 5149:r(t) 5143:r(t) 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