A Brief Introduction to Sigma Delta Conversion

Introduction

The sigma delta conversion technique has been in existence

for many years, but recent technological advances now make

the devices practical and their use is becoming widespread.

The converters have found homes in such applications as

communications systems, consumer and professional audio,

industrial weight scales, and precision measurement devices.

The key feature of these converters is that they are the only

low cost conversion method which provides both high

dynamic range and flexibility in converting low bandwidth input

signals. This application note is intended to give an engineer

with little or no sigma delta background an overview of how a

sigma delta converter works.

The following are brief definitions of terms that will be used

in this application note:

Noise Shaping Filter or Integrator: The noise shaping fil-

ter or integrator of a sigma delta converter distributes the

converter quantization error or noise such that it is very low

in the band of interest.

Oversampling. Oversampling is simply the act of sampling

the input signal at a frequency much greater than the

Nyquist frequency (two times the input signal bandwidth).

Oversampling decreases the quantization noise in the band

of interest.

Digital Filter. An on-chip digital filter is used to attenuate

signals and noise that are outside the band of interest.

Decimation: Decimation is the act of reducing the data rate

down from the oversampling rate without losing information.

Discussion

Figure 1 shows a simple block diagram of a first order sigma

delta Analog-to-Digital Converter (ADC). The input signal X

comes into the modulator via a summing junction. It then

passes through the integrator which feeds a comparator that

acts as a one-bit quantizer. The comparator output is fed

back to the input summing junction via a one-bit digital-to-

analog converter (DAC), and it also passes through the digi-

tal filter and emerges at the output of the converter. The

feedback loop forces the average of the signal W to be equal

to the input signal X. A quick review of quantization noise

theory and signal sampling theory will be useful before div-

ing deeper into the sigma delta converter.

FIGURE 1. FIRST ORDER SIGMA DELTA ADC BLOCK DIAGRAM

Signal Sampling

The sampling theorem states that the sampling frequency of a

signal must be at least twice the signal frequency in order to

recover the sampled signal without distortion. When a signal is

sampled its input spectrum is copied and mirrored at multiples

of the sampling frequency fS. Figure 2A shows the spectrum of

a sampled signal when the sampling frequency fSis less than

twice the input signal frequency 2f0. The shaded area on the

plot shows what is commonly referred to as aliasing which

results when the sampling theorem is violated. Recovering a

signal contaminated with aliasing results in a distorted output

signal. Figure 2B shows the spectrum of an oversampled sig-

nal. The oversampling process puts the entire input bandwidth

at less than fS/2 and avoids the aliasing trap.[1]

FIGURE 2A. UNDERSAMPLED SIGNAL SPECTRUM

FIGURE 2B. OVERSAMPLED SIGNAL SPECTRUM

Quantization Noise

Quantization noise (or quantization error) is one limiting fac-

tor for the dynamic range of an ADC. This error is actually

the “round-off” error that occurs when an analog signal is

quantized. For example, Figure 3 shows the output codes

and corresponding input voltages for a 2-bit A/D converter

with a 3V full scale value. The figure shows that input values

of 0V, 1V, 2V, and 3V correspond to digital output codes of

00, 01, 10, and 11 respectively. If an input of 1.75V is applied

to this converter, the resulting output code would be 10

which corresponds to a 2V input. The 0.25V error (2V -

1.75V) that occurs during the quantization process is called

the quantization error. Assuming the quantization error is

random, which is normally true, the quantization error can be

treated as random or white noise. Therefore, the quantiza-

tion noise power and RMS quantization voltage for an A/D

converter are given by the following equations:

∑∫

INTEGRATOR QUANTIZER

(COMPARATOR)

MODULATOR

DIGITAL Y

FILTER

1-BIT

DAC

BCD

AMPLITUDE

INPUT

BANDWIDTH

FREQUENCY fS/2 fS

f02f0