Delta-sigma modulation

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The Delta-Sigma (ΔΣ) modulation is a kind of analog-to-digital signal or digital-to-analog conversion derived from delta modulation. An analog to digital converter (ADC) or DAC circuit which implements this technique can be easily realized using low-cost CMOS processes, such as the processes used to produce digital integrated circuits; for this reason, even though it was first presented in the early 1960s, it is only in recent years that it has come into widespread use with improvements in silicon technology. Almost all analog integrated circuit vendors offer sigma delta converters.

Contents 1 Principle 2 Quantization theory formulas 3 Structures 4 Oversampling 4.1 Example of decimation 5 Changes from Δ-modulation 6 Naming 7 See also 8 References 8.1 Relevant publications

[edit] Principle

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.

[edit] Quantization theory formulas

Main article: Quantization

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

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_s is the sampling frequency and 2f₀ is Nyquist rate

The rms noise voltage within the band of interest can be expressed in terms of OSR

[edit] Structures

The MASH structure is a type of delta sigma modulator having a noise shaping property. It is commonly used in Fractional-N frequency synthesizers. Besides its noise shaping function, it has two more attractive properties:

• simple to implement in hardware; only common digital blocks such as accumulators, adders, and D flip-flops are required
• unconditionally stable

[edit] Oversampling

Main article: Oversampling

Let's consider a signal at frequency f₀ and a sampling frequency of f_s much higher than Nyquist rate (see Fig. 3). ΔΣ modulation is based on the technique of oversampling to reduce the noise in the band of interest (green), which also avoids the use 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 convertor (in blue), but it is distributed over a larger spectrum. In ΔΣ-converters, noise is further reduced at low frequencies, which is the band where the signal of interest is, and it is increased at the higher frequencies, 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 N-order ΔΣ-modulator

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

Another key aspect given by oversampling is the speed/resolution tradeoff. 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.

[edit] 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,

• the sample frequency is reduced by a factor of eight
• the serial (1-bit) input bus becomes a parallel (3-bits) output bus.

[edit] Changes from Δ-modulation

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

[edit] Naming

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.

[edit] See also

• Pulse-code modulation
• Pulse-density modulation

[edit] References

• "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"
• "Bruce Wooley's Delta-Sigma Converter Projects"
• "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"
• "MASH (Multi-stAge noise SHaping) structure" Simulation and Z-transform analysis
• "One-Bit Delta Sigma D/A Conversion Part I: Theory" article by Randy Yates presented at the 2004 comp.dsp conference

[edit] Relevant publications

• J. Candy, G. Temes, Oversampling Delta-sigma Data Converters, ISBN 0-87942-285-8
• S. Norsworthy, R. Schreier, G. Temes, Delta-Sigma Data Converters, ISBN 0-7803-1045-4
• Mingliang Liu, Demystifying Switched-Capacitor Circuits, ISBN 0-7506-7907-7
• R. Schreier, G. Temes, Understanding Delta-Sigma Data Converters, ISBN 0-471-46585-2