Principle

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The principle of the sigma-delta architecture is to make rough evaluations of the signal, to measure the error, integrate it and then compensate for that error. The mean output value is then equal to the mean input value if the integral of the error is finite. A nice applet simulating the whole architecture can be found here.
The number of integrators, and consequently, the numbers of feedback loops, indicates the order of a ΔΣ-modulator; a 2^nd order ΔΣ modulator is shown in Fig. 2. First order modulators are stable, but for higher order ones stability must be taken into great account.

Quantization theory formulas

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When a signal is quantized, the resulting signal approximately has the second-order statistics of a signal with independent additive white noise. Assuming that the signal value is in the range of one step of the quantized value with an equal distribution, the mean square value of this quantization noise is

$e\_\backslash mathrm\{rms\}^2\backslash ,\; =\backslash ,\; \backslash frac\{1\}\{\backslash Delta\}\backslash int\_\{-\backslash Delta/2\}^\{+\backslash Delta/2\}\; e^2\backslash ,\; de\backslash ,\; =\backslash ,\; \backslash frac\{\backslash Delta^2\}\{12\}$

In reality, the quantization noise is of course not independent of the signal; this dependence is the source of idle tones and pattern noise in Sigma-Delta converters.

Oversampling ratio, where $f\_\backslash mathrm\{s\}$ is the sampling frequency and $2f\_0$ is Nyquist rate

$\backslash mathrm\{OSR\}\backslash ,=\backslash ,\backslash frac\{f\_s\}\{2f\_0\}\backslash ,=\backslash ,\backslash frac\{1\}\{2f\_0\backslash tau\}$

The noise power within the band of interest can be expressed in term of OSR

$\backslash mathrm\{n\_0\}\backslash ,=\backslash ,\; \backslash frac\{e\_\{rms\}\}\{\backslash sqrt\{OSR\}\}$

Oversampling

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Let's consider a signal at frequency $f\_0$ and a sampling frequency of $f\_\backslash mathrm\{s\}$ much higher than Nyquist rate (see Fig. 3). ΔΣ modulation is based on oversampling technique to reduce the noise in the band of interest (green), which also avoid the using of high-precision analog circuits for the anti-aliasing filter. The quantization noise is the same both in a Nyquist converter (in yellow) and in an oversampling one (in blue), but it is distributed in a larger spectrum; in ΔΣ-converters, noise is furtherly reduced at low frequencies, that is the band of interest where signal is, and it is increased at the highest one, where it can be filtered. This property is known as noise shaping.

From a mathematical point of view, the previous noise power formula can be re-written for a $\backslash mathrm\{N\}$-order ΔΣ-modulator

$\backslash mathrm\{n\_0\}\backslash ,=\backslash ,\; \backslash frac\{e\_\{rms\}\; \backslash pi^N\}\{\backslash sqrt\{2N\; +\; 1\}\}\backslash ,\; (2f\_0\backslash tau)^\{(N+\backslash frac\{1\}\{2\})\}$

That means that the higher is the oversampling ratio, the higher is the Signal-to-noise ratio and the higher is the resolution in bit.

Another key aspect given by oversampling is the exchange speed-resolution; in fact 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 frequency of the signal increasing its resolution; this is obtained by a sort of averaging of the higher data rate bitstream.

Example of decimation

Let's have, for instance, an 8:1 decimation filter and a 1-bit stream; if we have an input stream like 10010110, counting the number of ones, the decimation result is 4/8 = 0.5 = 100 in binary; in other words, we

• reduce by eight the frequency of the stream and
• the serial (1-bit) input bus become a parallel (3-bits) output bus.

Changes from Δ-modulation

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Δ-modulation requires an integrator to reconstruct the analog signal; moving this integrator (Σ) in front of the Δ-modulator simplify the design of the last stage filter. This is due to the different spectrum shaping of the two types of modulation: ΔΣ-modulator shapes the noise, leaving the signal as it is, while Δ-modulator leaves the noise as it is and shapes the spectrum of the signal, which has to been reconstructed by the previous cited integrator.

Naming

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As can be easily recognized from the previous section, the name Delta-Sigma comes directly from the presence of a Delta modulator and an integrator, as firstly introduced by Inose et al. from Japan in 1962 in their patent application. Very often, the name Sigma-Delta is used as a synonym, but nowadays IEEE publications mostly use Delta-Sigma.

See also

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• Pulse-code modulation
• Pulse-density modulation

References

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• "Sigma-delta techniques extend DAC resolution" article by Tim Wescott 2004-06-23
• "Tutorial on Designing Delta-Sigma Modulators: Part I" (2004-03-30) and "Part II" (2004-04-01) a tutorial by Mingliang Liu
• "Gabor Temes' Publications" contributed by Southcaltree 2005-11-22
• "Bruce Wooley's Delta-Sigma Converter Projects" contributed by Southcaltree 2005-11-22
• "An Introduction to Delta Sigma Converters" (which covers both ADC's and DAC's sigma-delta)
• "Demystifying Sigma-Delta ADCs". This in-depth article covers the theory behind a Delta-Sigma analog-to-digital converter.
• "Motorola digital signal processors: Principles of sigma-delta modulation for analog-to-digital converters"

Relevant publications

• J. Candy, G. Temes, Oversampling Delta-sigma Data Converters, ISBN 0-879-42285-8
• S. Norsworthy, R. Schreier, G. Temes, Delta-Sigma Data Converters, ISBN 0-7803-1045-4
• Mingliang Liu, Demystifying Switched-Capacitor Circuits, ISBN 0-750-67907-7
• R. Schreier, G. Temes, Understanding Delta-Sigma Data Converters, ISBN 0-471-46585-2

Digital signal processing

Delta-Sigma-Modulation | ΔΣ変調 | Modulacja Delta-Sigma