Historical Perspective

The sigma-delta (Σ-Δ) ADC architecture had its origins in the early development phases of pulse code modulation (PCM) systems—specifically, those related to transmission techniques called delta modulation and differential PCM. (An excellent discussion of both the history and concepts of the sigma-delta ADC can be found by Max Hauser in Reference 1).

The driving force behind delta modulation and differential PCM was to achieve higher transmission efficiency by transmitting the changes (delta) in value between consecutive samples rather than the actual samples themselves.

In delta modulation, the analog signal is quantized by a one-bit ADC (a comparator) as shown in Figure 6.85A. The comparator output is converted back to an analog signal with a 1-bit DAC, and subtracted from the input after passing through an integrator. The shape of the analog signal is transmitted as follows: a 1 indicates that a positive excursion has occurred since the last sample, and a 0 indicates that a negative excursion has occurred since the last sample.

Figure 6.85: Delta Modulation and Differential PCM

If the analog signal remains at a fixed dc level for a period of time, a pattern alternating of 0s and 1s is obtained. It should be noted that differential PCM (see Figure 6.85B) uses

∑

exactly the same concept except a multibit ADC is used rather than a comparator to derived the transmitted information.

Since there is no limit to the number of pulses of the same sign that may occur, delta modulation systems are capable of tracking signals of any amplitude. In theory, there is no peak clipping. However, the theoretical limitation of delta modulation is that the analog signal must not change too rapidly. The problem of slope clipping is shown in Figure 6.86. Here, although each sampling instant indicates a positive excursion, the analog signal is rising too quickly, and the quantizer is unable to keep pace.

SLOPE OVERLOAD

Figure 6.86: Quantization Using Delta Modulation

Slope clipping can be reduced by increasing the quantum step size or increasing the sampling rate. Differential PCM uses a multibit quantizer to effectively increase the quantum step sizes at the increase of complexity. Tests have shown that in order to obtain the same quality as classical PCM, delta modulation requires very high sampling rates, typically 20× the highest frequency of interest, as opposed to Nyquist rate of 2×.

For these reasons, delta modulation and differential PCM have never achieved any significant degree of popularity, however a slight modification of the delta modulator leads to the basic sigma-delta architecture, one of the most popular high resolution ADC architectures in use today.

The basic single and multibit first-order sigma-delta ADC architecture is shown in

Figures 6.87A and 6.87B, respectively. Note that the integrator operates on the error signal, whereas in a delta modulator, the integrator is in the feedback loop. The basic oversampling sigma-delta modulator increases the overall signal-to-noise ratio at low frequencies by shaping the quantization noise such that most of it occurs outside the bandwidth of interest. The digital filter then removes the noise outside the bandwidth of

Figure 6.87: Single and Multibit Sigma-Delta ADCs

The IC sigma-delta ADC offers several advantages over the other architectures, especially for high resolution, low frequency applications. First and foremost, the single-bit delta ADC is inherently monotonic and requires no laser trimming. The sigma-delta ADC also lends itself to low cost foundry CMOS processes because of the digitally intensive nature of the architecture. Examples of early monolithic sigma-delta ADCs are given in References 13-21. Since that time there has been a constant stream of process and design improvements in the fundamental architecture proposed in the early works cited above.

Sigma-Delta (Σ-Δ) or Delta-Sigma (Δ-Σ)? Editor's Notes from Analog

This is not the most earth-shaking of controversies, and many readers may wonder what the fuss is all about—if they wonder at all. The issue is important to both editor and readers because of the need for consistency; we’d like to use the same name for the same thing whenever it appears. But which name? In the case of the modulation technique that led to a new oversampling A/D conversion mechanism, we chose sigma-delta. Here’s why.

Ordinarily, when a new concept is named by its creators, the name sticks; it should not be changed unless it is erroneous or flies in the face of precedent. The seminal paper on this subject was published in 1962 (References 9, 10), and its authors chose the name “delta-sigma modulation,” since it was based on delta modulation but included an integration (summation, hence Σ).

Delta-sigma was apparently unchallenged until the 1970s, when engineers at AT&T were publishing papers using the term sigma-delta. Why? According to Hauser (Reference 1), the precedent had been to name variants of delta modulation with adjectives preceding the word “delta.” Since the form of modulation in question is a variant of delta modulation, the sigma, used as an adjective—so the argument went—should precede the delta.

Many engineers who came upon the scene subsequently used whatever term caught their fancy, often without knowing why. It was even possible to find both terms used interchangeably in the same paper. As matters stand today, sigma-delta is in widespread use, probably for the majority of citations. Would its adoption be an injustice to the inventors of the technique?

We think not. Like others, we believe that the name delta-sigma is a departure from precedent. Not just in the sense of grammar, but also in relation to the hierarchy of operations. Consider a block diagram for embodying an analog root-mean-square (finding the square root of the mean of a squared signal) computer. First the signal is squared, then it is integrated, and finally it is rooted (see Figure 6.88).

If we were to name the overall function after the causal order of operations, it would have to be called a “square mean root” function. But naming in order of the hierarchy of its mathematical operations gives us the familiar—and undisputed—name, root mean-square. Consider now a block diagram for taking a difference (delta), and then integrating it (sigma).

Its causal order would give delta-sigma, but in functional hierarchy it is sigma-delta, since it computes the integral of a difference. We believe that the latter term is correct and follows precedent; and we have adopted it as our standard.

Dan Sheingold, 1990.

Figure 6.88: Sigma-Delta (Σ-Δ) or Delta-Sigma (Δ-Σ)?

Root-Mean-Square:

SQUARE MEAN ROOT

Hierarchy of Mathematical Operations: ROOT MEAN (SQUARE) 1 2

3

∑ ∫∫

+

Sigma-Delta:

Hierarchy of Mathematical Operations: SIGMA (DELTA) 1 2

Delta Sigma