# Principles of Sigma Delta Modulator

**Topics:**Digital signal processing, Signal processing, Analog-to-digital converter

**Pages:**4 (1273 words)

**Published:**December 25, 2012

Abstract: This article covers the theory behind a Delta-Sigma analog-to-digital converter (ADC). It specifically focuses on key digital concepts of oversampling, noise shaping, and decimation filtering. Introduction Sigma-delta converters offer high resolution, high integration, and low cost, making them a good ADC choice for applications such as process control and monitoring. The analog side of a sigma-delta converter (a 1-bit ADC) is very simple. The digital side, performing filtering and decimation, makes the sigma-delta ADC inexpensive to produce, but is more complex. To understand how it works, you must become familiar with the concepts of oversampling, noise shaping, digital filtering, and decimation. Oversampling First, consider the frequency-domain transfer function of a traditional multi-bit ADC with a sine-wave input signal. This input is sampled at a frequency Fs. According to Nyquist theory, Fs must be at least twice the bandwidth of the input signal. When observing the result of an FFT analysis on the digital output, we see a single tone and lots of random noise extending from DC to Fs/2 (Figure 1). Known as quantization noise, this effect results from the following consideration: the ADC input is a continuous signal with an infinite number of possible states, but the digital output is a discrete function whose number of different states is determined by the converter's resolution. So, the conversion from analog to digital loses some information and introduces some distortion into the signal. The magnitude of this error is random, with values up to ±LSB. If we divide the fundamental amplitude by the RMS sum of all the frequencies representing noise, we obtain the signal to noise ratio (SNR). For an N-bit ADC, SNR = 6.02N + 1.76dB. To improve the SNR in a conventional ADC (and consequently the accuracy of signal reproduction) you must increase the number of bits. Consider again the above example, but with a sampling...

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