Introduction
This in-depth article covers the theory behind a
Delta-Sigma analog-to-digital converter converter (ADC). It specifically focuses on the
difficult to understand key digital concepts of over-sampling, noise shaping,
and decimation filtering. A description of new converter, the MAX1402, and
several applications for Delta-Sigma converters are included.
Sigma-delta converters offer high resolution,
high integration, and low cost, making them a good ADC choice for applications
such as process control and weighing scales. Designers often choose a classic
SAR ADC instead, because they don't understand the sigma-delta types.
The analog side of a sigma-delta converter (a
1-bit ADC) is very simple. The digital side, which is what makes the sigma-delta
ADC inexpensive to produce, is more complex. It performs filtering and
decimation. To understand how it works, you must become familiar with the
concepts of oversampling, noise shaping, digital filtering, and decimation.
This application note covers these topics.
Over- sampling
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 ħLSB.
Figure 1. FFT diagram of a
multi-bit ADC with a sampling frequency FS
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 frequency increased by the over-sampling ratio k, to kFs
(Figure 2). An FFT analysis shows that the noise floor has dropped. SNR is the
same as before, but the noise energy has been spread over a wider frequency
range. Sigma-delta converters exploit this effect by following the 1-bit ADC
with a digital filter (Figure 3). The RMS noise is less, because most of the
noise passes through the digital filter. This action enables sigma-delta
converters to achieve wide dynamic range from a low-resolution ADC.
Figure 2. FFT diagram of a
multi-bit ADC with a sampling frequency kFS
Figure 3. Effect of the digital
filter on the noise bandwidth
Does the SNR improvement come simply from
over-sampling and filtering? Note that the SNR for a 1-bit ADC is 7.78dB (6.02 +
1.76). Each factor-of-4 over-sampling increases the SNR by 6dB, and each 6dB
increase is equivalent to gaining one bit. A 1-bit ADC with 24x oversampling
achieves a resolution of four bits, and to achieve 16-bit resolution you must
oversample be a factor of 415, which is not realizable. But,
sigma-delta converters overcome this limitation with the technique of noise
shaping, which enables a gain of more than 6dB for each factor of 4x
oversampling.