A Brief Introduction to Sigma Delta Conversion
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16-02-2011, 01:20 PM

The sigma delta conversion technique has been in existencefor many years, but recent technological advances now makethe devices practical and their use is becoming widespread.The converters have found homes in such applications ascommunications systems, consumer and professional audio,industrial weight scales, and precision measurement devices.The key feature of these converters is that they are the onlylow cost conversion method which provides both highdynamic range and flexibility in converting low bandwidth inputsignals. This application note is intended to give an engineerwith little or no sigma delta background an overview of how asigma delta converter works.The following are brief definitions of terms that will be usedin this application note:
Noise Shaping Filter or Integrator: The noise shaping filteror integrator of a sigma delta converter distributes theconverter quantization error or noise such that it is very lowin the band of interest.
Oversampling. Oversampling is simply the act of samplingthe input signal at a frequency much greater than theNyquist frequency (two times the input signal bandwidth).Oversampling decreases the quantization noise in the bandof interest.
Digital Filter. An on-chip digital filter is used to attenuatesignals and noise that are outside the band of interest.
Decimation: Decimation is the act of reducing the data ratedown from the oversampling rate without losing information
Figure 1 shows a simple block diagram of a first order sigmadelta Analog-to-Digital Converter (ADC). The input signal Xcomes into the modulator via a summing junction. It thenpasses through the integrator which feeds a comparator thatacts as a one-bit quantizer. The comparator output is fedback to the input summing junction via a one-bit digital-toanalogconverter (DAC), and it also passes through the digitalfilter and emerges at the output of the converter. Thefeedback loop forces the average of the signal W to be equalto the input signal X. A quick review of quantization noisetheory and signal sampling theory will be useful before divingdeeper into the sigma delta converter.
Signal Sampling
The sampling theorem states that the sampling frequency of asignal must be at least twice the signal frequency in order torecover the sampled signal without distortion. When a signal issampled its input spectrum is copied and mirrored at multiplesof the sampling frequency fS. Figure 2A shows the spectrum ofa sampled signal when the sampling frequency fS is less thantwice the input signal frequency 2f0. The shaded area on theplot shows what is commonly referred to as aliasing whichresults when the sampling theorem is violated. Recovering asignal contaminated with aliasing results in a distorted outputsignal. Figure 2B shows the spectrum of an oversampled signal.The oversampling process puts the entire input bandwidthat less than fS/2 and avoids the aliasing trap.[1]
Quantization Noise
Quantization noise (or quantization error) is one limiting factorfor the dynamic range of an ADC. This error is actuallythe “round-off” error that occurs when an analog signal isquantized. For example, Figure 3 shows the output codesand corresponding input voltages for a 2-bit A/D converterwith a 3V full scale value. The figure shows that input valuesof 0V, 1V, 2V, and 3V correspond to digital output codes of00, 01, 10, and 11 respectively. If an input of 1.75V is appliedto this converter, the resulting output code would be 10which corresponds to a 2V input. The 0.25V error (2V -1.75V) that occurs during the quantization process is calledthe quantization error. Assuming the quantization error israndom, which is normally true, the quantization error can betreated as random or white noise.
Sigma Delta Modulator Quantization Noise
The results of the above sampling and noise theory can nowbe used to show how a sigma delta modulator shapesquantization noise. Figure 4 shows the sampled dataequivalent block diagram of a first order sigma delta modulator.

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