Delta-sigma_modulation

# Delta-sigma modulation **Delta-sigma modulation** (ΔΣ, also written sigma-delta, ΣΔ) is a method of encoding a signal as a rapid stream of very coarse codes, often a single bit, in which a [[Negative_feedback|negative-feedback]] loop around the quantizer arranges the unavoidable rounding error so that almost all of it falls at frequencies far above the signal. The loop subtracts the last output from the input, accumulates the difference in an integrator, and lets a comparator decide the next bit; the average of the bits tracks the input, and the error, viewed in the [[Frequency_domain|frequency domain]], is multiplied by a high-pass noise transfer function that is nearly zero across the signal band. Because the loop runs at many times the rate the signal needs, a practice called [[Oversampling|oversampling]], a [[Low-pass_filter|low-pass]] [[Digital_filter|digital filter]] can then remove the shaped noise and reduce the rate, delivering a word of 16 to 24 bits from a stream of ones and zeros. The method is the core of most precision [[Analog-to-digital_converter|analog-to-digital converters]] and of nearly every audio [[Digital-to-analog_converter|digital-to-analog converter]]. Its worth lies in what it does not need. A conventional converter of `N` bits must hold `2^N` levels to an accuracy of a fraction of one step, a matching problem that grows harder with every bit; a one-bit quantizer has two levels and is linear by construction, and the modulator buys resolution with speed and arithmetic instead of with matched components. The price is bandwidth: plain oversampling gains only 3 dB of in-band [[Signal-to-noise_ratio|signal-to-noise ratio]] for each doubling of the rate, noise shaping raises that to `3·(2L + 1)` dB for a loop of order `L`, and the final rate reduction, called decimation, is where the word grows to its full width. The technique grew out of [[Delta_modulation|delta modulation]] in the early 1960s, was named for the order of its operations, difference then sum, and now exists in higher-order, multi-bit, multi-stage and clockless forms. The page's primary microsim, *Delta-sigma modulation*, lets the reader set the input level, the tone frequency and the oversampling ratio of a first- or second-order loop and watch the one-bit stream thicken and thin with the wave while a measured spectrum shows the noise floor tilting up and away from the signal band and a diagnostics panel reports the in-band signal-to-noise ratio and the effective number of bits. ## Motivation ### Advantages of oversampling A quantizer with step `Q` produces an error that is bounded by half a step and, for a busy input, behaves like noise with root-mean-square value `Q/√12`,[^dyer38] about 0.29 of a step.[^stiber59] Its total power does not depend on how often the input is sampled, but its distribution in frequency does: sampled at `fs`, the noise spreads evenly from zero to `fs/2`. If the signal occupies only the band up to `fB`, and `fs = 2·OSR·fB` for an oversampling ratio `OSR`, then only the fraction `1/OSR` of the noise power lies in the band the signal needs, and a filter that keeps the band and discards the rest removes the other `1 − 1/OSR`. Every doubling of the rate halves the in-band noise power, a gain of 3 dB, which by the `6.02` dB per bit of the quantization formula[^dyer39] is half a bit of resolution. This is the frequency/resolution trade-off: a converter may spend its speed on bandwidth, or on bits, and the same one-bit quantizer run 64 times faster than the signal requires is worth three more bits before any shaping is done. Oversampling has a second advantage. With the sample rate far above the signal band, the [[Anti-aliasing_filter|anti-aliasing filter]] ahead of the converter has a wide transition region between the top of the signal band and the [[Nyquist_frequency|Nyquist frequency]] where [[Aliasing|aliasing]] begins, so a gentle analog filter suffices and the steep filtering moves into the digital domain, where it is exact and repeatable. ### Improvement over delta modulation Delta modulation encodes the change of a signal rather than the signal itself: a one-bit stream says at each sample whether the input has risen or fallen relative to a running estimate, and the receiver integrates the bits to rebuild the waveform. The scheme fails on two fronts. A signal whose slope exceeds one step per sample outruns the estimate, an error called slope overload, and any bit lost in transmission shifts the receiver's integrator permanently. Moving the integrator from the receiver to the transmitter, in front of the quantizer, cures both: the loop now encodes the amplitude, so there is no slope limit within the range, the receiver needs only a low-pass filter, and a lost bit is a momentary blip rather than a lasting offset. That rearrangement is the delta-sigma modulator, and the section on the relationship to delta modulation below gives the reasoning. ### 1-bit delta–sigma modulation is pulse-density modulation When the quantizer has one bit, the output is a stream of pulses whose local density carries the signal: a high input produces mostly ones, a low input mostly zeros, and an input at the middle of the range alternates. In the microsim the bit density readout shows this at the coarsest scale, reporting about 50 percent ones for the default zero-mean sine, while the time panel shows the stream crowding toward the top when the wave is high and toward the bottom when it is low. The same stream can be treated as a signal in its own right: a one-bit stream fed to a switch and a low-pass filter reproduces the input, which is why the one-bit format is used directly for storage and for driving output stages. ## History and variations The modulator descends from delta modulation, which de Jager described in 1952 as a method of transmitting pulse-code signals with a one-unit code.[^dejager] Cutler's patent, filed in 1954 and granted in 1960, described a quantizing transmission system in which the error of the quantizer is fed back and subtracted from later samples, the first statement of noise-shaping feedback.[^cutler] The delta-sigma form itself, with the integrator moved ahead of the quantizer, was published by Inose, Yasuda and Murakami in 1962 as a code-modulation telemetering system and developed by Inose and Yasuda in 1963 as a unity-bit coding method by negative feedback.[^inose62][^inose63] The method waited for [[Integrated_circuit|integrated circuits]] fast and dense enough to run the loop and the decimation filter cheaply; from the 1970s onward work at [[Bell_Labs]] and elsewhere turned it into the standard architecture for audio and instrumentation converters, Candy's 1985 analysis of the two-integrator loop being the reference for the stable second-order structure the microsim implements.[^candy85] The variations answer the first-order loop's limits. A higher-order modulator puts `L` integrators in the loop and raises the noise-shaping slope, but stability is no longer guaranteed: a second-order loop needs a limited input range, which is why the microsim keeps its input level below 0.8 of full scale, and a loop of order three or more also needs scaled coefficients. A multi-bit quantizer, of two to five bits, lowers the noise before shaping and eases stability, at the cost of reintroducing a matching problem that is then solved by dynamic element matching, shuffling the elements so that their errors average out. Subsequent decimation, the digital filter that follows every modulator, has its own design lineage, from cascaded moving-average filters to multi-stage [[Finite_impulse_response|finite-impulse-response]] designs. Other loop filters replace the pure integrators with resonators or band-pass sections, so that the noise null sits at a carrier frequency rather than at zero, which allows a radio's intermediate frequency to be digitised directly. ## Reduction of baseband noise by increasing oversampling ratio and ΔΣM order ### Oversampling to spread out quantization noise The arithmetic of plain oversampling is set out above: the in-band noise power falls as `1/OSR`, so the signal-to-noise ratio gains `10·log10(OSR)` dB over the same quantizer run at the Nyquist rate. For a one-bit quantizer this is not much. Stiber, Stiber and Larson budget 59 dB for an 8-bit converter on their span-to-noise convention, and 107 dB for 16 bits,[^stiber60] and reaching the latter from one bit by oversampling alone would need a ratio of about `2^30`. Oversampling on its own therefore serves the multi-bit converter as a way of relaxing its analog filter; the one-bit converter needs the loop. ### Noise shaping The loop is analysed by treating the quantizer as a source of additive error `E` and the rest as a [[Linear_time-invariant_system|linear time-invariant system]]. In the continuous-time picture, with an integrator `1/s` in the forward path and the output fed back to the input, the [[Laplace_transform|Laplace-domain]] output is `Y = X/(1 + s) + s·E/(1 + s)`: the signal passes through a low-pass [[Transfer_function|transfer function]] while the error passes through a high-pass one, small at low frequency and rising toward unity. The synchronous, sampled loop is analysed in the [[Z-transform|z-domain]]. With a delaying integrator `z⁻¹/(1 − z⁻¹)` the output is `Y = z⁻¹·X + (1 − z⁻¹)·E`; the signal transfer function is a single delay and the noise transfer function is `NTF(z) = 1 − z⁻¹`, whose magnitude on the unit circle is `|NTF(f)| = 2·sin(πf/fs)`. It is zero at zero frequency, `√2` at `fs/4` and 2 at the Nyquist frequency: the error is not removed but moved. A loop of `L` integrators, arranged as Candy's cascade with the output fed back to every stage, has `NTF(z) = (1 − z⁻¹)^L` and `|NTF(f)| = (2·sin(πf/fs))^L`, the curve the microsim draws as the ideal NTF tilt over its measured spectrum.[^candy85] Integrating the shaped noise over the signal band gives the closed form on which converters are designed. For a quantizer of `b` bits, an oversampling ratio `OSR` and a loop of order `L`, the in-band signal-to-noise ratio for a full-scale sine is `SNR = 6.02·b + 1.76 + (20L + 10)·log10(OSR) − 10·log10(π^(2L)/(2L + 1))` dB. Each doubling of the ratio now buys `3·(2L + 1)` dB, 9 dB or 1.5 bits for a first-order loop and 15 dB or 2.5 bits for a second-order one, against 3 dB for oversampling alone. The microsim prints the formula's value as ideal SNR (theory), omitting the `6.02·b` term for its one-bit quantizer: 50.8 dB at the default ratio of 64 for the first-order loop and 79.2 dB for the second, rising to 68.8 and 109.3 dB at a ratio of 256. The theoretical effective number of bits follows by inverting the quantization formula, `ENOB = (SNR − 1.76)/6.02`,[^dyer39] so that the two default figures correspond to about 8.1 and 12.9 bits from a one-bit stream. The sketch's measured values fall below the ideal ones, 41.0 dB and 6.5 bits for the first-order loop at the defaults and 67.1 dB and 10.9 bits for the second-order, because its input level of 0.40 is 8 dB below full scale and because the plain two-integrator loop with clamped integrators does not quite meet the idealised noise model; at an input level of 0.80 the first-order measurement, 50.7 dB, meets the theory row within a tenth of a decibel. ## Relationship to delta modulation A delta modulator quantizes the difference between the input and an integrated copy of its own output, and the receiver integrates the bits: in block form, quantizer at the transmitter, integrator at the receiver, with a second integrator inside the transmitter's feedback path. Because integration and the linear parts of the loop commute, the receiver's integrator can be moved to the transmitter's input, where it merges with the integrator already in the loop; the result is a single integrator ahead of the quantizer, fed by the difference between input and output. That is the delta-sigma modulator, and the change of position is the whole difference between the two. The delta modulator's stream encodes slope and its receiver must integrate, accumulating any error; the delta-sigma stream encodes amplitude and its receiver merely averages. [[Differential_pulse-code_modulation|Differential pulse-code modulation]] is the multi-bit cousin of delta modulation in the same family, and [[Pulse-code_modulation|pulse-code modulation]] itself is what the decimation filter at the end of a delta-sigma converter produces. ## Analog-to-digital conversion example The simplest analog delta-sigma converter has three parts, and each corresponds to a line of the microsim's loop. The first is a two-input inverting integrator: an operational amplifier with a [[Capacitor|capacitor]] in feedback, one input [[Resistor|resistor]] carrying the signal [[Voltage|voltage]] and a second carrying a reference current switched positive or negative by the last output bit, so that the capacitor accumulates the input-minus-feedback difference, the `i1 += x − y` of the sketch. The second is the quantizer and sampler, a comparator whose output is captured by a D-type [[Flip-flop_(electronics)|flip-flop]] on every edge of the oversampling clock, the `y = sign(i1)` step made synchronous; the flip-flop's output is the bitstream and also the signal that steers the reference current. The third is demodulation. In its crudest form a counter tallies the ones over a frame of `OSR` clocks and the count is the output word, a moving-average filter that is exactly the boxcar average ≈ x trace the sketch draws in green; a practical converter replaces the counter with a cascade of averaging stages followed by a [[Finite_impulse_response|finite-impulse-response]] low-pass filter, which removes the shaped noise far more completely before the rate is reduced. Two integrators in series, each fed back from the output, make the second-order loop that the sketch's loop order button selects, with a steeper noise tilt and a lower in-band floor. ## Digital-to-analog conversion The same loop runs entirely in arithmetic to convert the other way. An `N`-bit input stream at the signal rate is first upsampled: zeros are inserted between samples and an interpolation filter, usually a [[Finite_impulse_response|finite-impulse-response]] design, fills them with the values the band-limited signal would have taken, so that the stream now runs at `OSR` times the original rate with its spectral images removed. A digital-to-digital delta-sigma modulator then requantizes the wide words to one bit, or to a few bits, with the same integrate-and-feedback loop, pushing the requantization noise out of band. The analog part is now trivial: a one-bit switch between two reference voltages, which is linear by construction, followed by an analog low-pass filter that removes the shaped noise. This is the architecture of the audio [[Digital-to-analog_converter|digital-to-analog converter]] and of the one-bit output stages of digital amplifiers. Multi-stage noise shaping, known as MASH, reaches high order without the stability limits of a single high-order loop by cascading first- or second-order stages, each digitising the quantization error of the one before, and combining their outputs through digital differentiators so that the errors of all but the last stage cancel.[^mash] ## Naming Inose and Yasuda named the method for the order of its two operations: a difference, delta, is formed first, and its accumulation, sigma, follows.[^inose63] The reversed name, sigma-delta, entered use later and is at least as common in the converter industry, and the two are used interchangeably; the pair article discusses the disagreement at length. Wikitube uses the originators' order. ## Asynchronous delta–sigma modulation An asynchronous modulator has no clock. The comparator is given hysteresis, so that the loop oscillates on its own, and the output is a continuous-time square wave whose duty cycle follows the input and whose switching frequency falls as the input moves away from mid-scale. Because no sampling occurs, there is no quantization noise in the usual sense; the information is carried in the timing of the transitions, and the signal is recovered by a low-pass filter exactly as in the clocked case. The form is used in switching audio amplifiers and in time-encoding circuits, and it can be converted to a synchronous stream by sampling the square wave with a fast clock, at which point the timing error of the sampling reappears as shaped noise. ## Microsims The primary microsim, *Delta-sigma modulation*, runs a real first- or second-order loop over 1,024 samples and measures the result. The TIME panel plots the input `x[n]`, the one-bit output `y[n]` at ±1 and the boxcar average ≈ x, a 24-sample moving average of the bitstream. The SPECTRUM panel shows a Hann-[[Window_function|windowed]] [[Fast_Fourier_transform|fast Fourier transform]] of the bitstream in [[Decibel|decibels]], the shaded signal band up to `fB = fs/(2·OSR)`, the signal spike inside it and the measured noise floor climbing away from it, with the ideal NTF tilt `(2·sin(πf/fs))^L` drawn over the measurement. The controls are input level, from 0.05 to 0.80 of full scale, default 0.40; freq, the tone's position in the signal band as a fraction of `fB`, from 0.10 to 0.90, snapped to an FFT bin; OSR, the oversampling ratio, at powers of two from 8 to 256, default 64; and the loop order button, which switches between 1st and 2nd. The DIAGNOSTICS panel reports the loop order, the OSR, the signal band edge as a fraction of `fs`, the bit density, the measured in-band SNR, the effective bits ENOB and the ideal SNR (theory), and states the gain per doubling of OSR for the current order. The model is ILLUSTRATIVE: the loop coefficients are all unity, the integrators are clamped, the ideal row omits the one-bit quantizer's `6.02` dB term, and at OSR 256 the signal band holds too few of the 1,024 FFT bins for the measured SNR to mean anything. A three.js companion renders this concept in depth. *Try:* Leave the input level at 0.40 and step OSR from 8 to 64, reading the measured in-band SNR rise from 18.2 to 41.0 dB (ENOB 6.5) as the noise floor sinks beneath the shaded band; then press loop order to 2nd and watch the tilt steepen and the readout jump to 67.1 dB (10.9 bits), and finally drag input level to 0.80 with the first-order loop to see the measured 50.7 dB meet the 50.8 dB theory row.  **Microsim — three.js (Wikitube framework), pending deploy:** *Delta-sigma modulation: one crude bit, made precise by speed* will play here once `https://wikitube-3d-microsims.netlify.app/signal/Delta-sigma_modulation.html` is live.  *Built from `MICROSIM_GUIDE/specs/sims/Delta-sigma_modulation.json`; part of the [[Signal_processing]] set ([[PORTAL_Signal_Processing]]).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Delta-sigma_modulation) : [Wikitube](https://en.wikitube.io/wiki/Delta-sigma_modulation) Skeleton mirrored at revision 1375131374. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Oversampling]] - [[Digital-to-analog_converter]] - [[Quantization_(signal_processing)]] - [[Analog-to-digital_converter]] - [[Delta_modulation]] - [[Pulse-code_modulation]] - [[Negative_feedback]] - [[Digital_filter]] - [[Noise_(electronics)]] - [[PORTAL_Signal_Processing]] ## Notes The microsim is ILLUSTRATIVE: its loops use unity coefficients with clamped integrators, its ideal SNR row omits the 6.02 dB one-bit term of the textbook formula, and its measured SNR is a 1,024-point estimate that becomes meaningless at an oversampling ratio of 256, where only two FFT bins fall inside the signal band. Page numbers in the references are PDF pages of the open editions. ## References [^dyer38]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 38-39 (PDF pages): the rms quantization noise Q/√12. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer39]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 39-40 (PDF pages): SNR = 6.02 N + 1.76 dB for a full-scale sine. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^stiber59]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 59 (PDF page): the uniform error model, σ = 1/√12 LSB ≈ 0.29 LSB. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^stiber60]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 60-61 (PDF pages): quantization budgets of 59, 83 and 107 dB at 8, 12 and 16 bits on the span/σ convention. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^dejager]: de Jager, F. "Deltamodulation, a method of P.C.M. transmission using the 1-unit code". *Philips Research Reports*, vol. 7, 1952, pp. 442-466. [^cutler]: Cutler, C. C. "Transmission systems employing quantization". U.S. Patent 2,927,962, filed 1954, granted March 8, 1960. https://patents.google.com/patent/US2927962 [^inose62]: Inose, H.; Yasuda, Y.; Murakami, J. "A telemetering system by code modulation: Δ-Σ modulation". *IRE Transactions on Space Electronics and Telemetry*, vol. SET-8, 1962, pp. 204-209. [^inose63]: Inose, H.; Yasuda, Y. "A unity bit coding method by negative feedback". *Proceedings of the IEEE*, vol. 51, no. 11, 1963, pp. 1524-1535. [^candy85]: Candy, J. C. "A use of double integration in sigma delta modulation". *IEEE Transactions on Communications*, vol. COM-33, no. 3, 1985, pp. 249-258. [^mash]: Citation needed: the multi-stage noise-shaping (MASH) architecture; the 1987 *IEEE Journal of Solid-State Circuits* paper by Matsuya and colleagues on 16-bit oversampling conversion with triple-integration noise shaping, or the 1986 conference paper that introduced the cascade, would settle the attribution and date. ## Further reading - Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain* (2020). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain - Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*, 1st ed. (2020). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods - Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory* (2026). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory ## External links - *Delta-sigma modulation*, the primary sketch: https://editor.p5js.org/sciencenibber/full/GA5IIXbtZ (fork: https://editor.p5js.org/sciencenibber/sketches/GA5IIXbtZ)