No matter how many bits an analog-to-digital converter (ADC) provides, the digital output can only approximate the original signal. This approximation gives rise to quantization errors, or quantization "noise." The error values fall between ±1/2 the voltage represented by the ADC's least-significant bit (LSB) and they have a fairly uniform distribution within this range. Although ADCs with higher resolution reduce the quantization errors, they always remain within ±1/2 LSB. You can think of the quantization error as adding white noise to the digitized information. By definition, white noise has a "flat" power spectrum over the f_{sample}/2 bandwidth.

So how can a data-acquisition system reduce quantization errors? Because these errors depend only on an ADCs resolution, sampling at a much higher rate than you would normally spreads the quantization noise over a larger bandwidth. And thus the power density for a fixed bandwidth decreases as f_{sample} increases. In practice, a higher sample rate decreases the quantization noise superimposed on the digital data for the signal you want to measure. But the reduction of the noise comes at a price -- more data to process and the need to digitally filter the data. Doubling the sample rate increases the ADC resolution by 1/2 LSB.

This sigma-delta modulator converts an analog-input signal to a stream of logic-1 bits proportional to the signal voltage.

This type of oversampling becomes impractical, though, unless you also use a technique called noise shaping. This technique decreases noise in the bandwidth of interest by "shifting" it to higher frequencies where it has less effect on your signals of interest. Sigma-delta, also called delta-sigma, ADCs provide this function and produce high resolutions for relatively low-frequency signals. A sound card, for example, relies on a sigma-delta converter that oversamples at up to 192 ksamples/sec. Some converters operate at 256 times the Nyquist frequency and provide resolutions of 20 or more bits.

The simplest sigma-delta ADC uses a difference amplifier, an integrator, a comparator, and a 1-bit digital-to-analog converter (DAC), as shown in the figure, in what engineers call a sigma-delta modulator. The DAC provides either a positive or a negative reference voltage and the loop acts to hold the integrator’s output at the reference input (ground) at the comparator.

The comparator's output produces logic-1 pulses at a "density" commensurate with the voltage at the signal input. As the voltage increases, the comparator generates more 1s and a decreasing voltage generates more 0s. By summing the error voltage, the integrator acts as a low-pass filter for the input signal and as a high-pass filter for the quantization noise. This in effect "shifts" quantization into higher frequencies. Oversampling does not decrease the total noise power, it simply distributes it at higher frequencies.

Although this type of ADC includes only a 1-bit DAC, it can achieve high resolutions because the modulator output goes through several stages of processing, including decimation, reducing the amount of data to a small fraction of the number of raw samples processed.

I really appreciated your explanation of quantization errors and possible solutions, as well as the trade-offs that are involved. It seems to me from reading your blog that an important first step of any project would be to have a very good understanding of the precision required so that one knows what effect quantization errors would have and how far one should go in attempting to reduce or eliminate them. Thanks for the great information, Jon!

Thanks, Nancy. Yes, before you think about digitizing analog signals you must know much about them. Unfortunately, some engineers jump in and specify data-acquisition equipment they later find doesn't give them the results they expect. Early in my career I made similar mistakes.

>A sound card, for example, relies on a sigma-delta converter that oversamples at up to 192 ksamples/sec.

Maybe just a typo, but of course, that isn't oversampling. Audio HW that samples at 192k does so to get 90kHz of BW. The actual sampling rate will likely be 128*192k = 24.576MHz.

Great article! Sometimes you have a fast ADC and a lot of time for a precision measurement. In 1989 we scrapped a whole board of high precision analog components on a Shuttle experiment which took 5 seconds to produce a single 12 bit digitization by supeimposing a precision sine wave on the DC raw data and summing 2048 samples from the 12bit ADC. We achieved 18 bit precision in one second. The accuracy was improved by intermingling precision references and board temperature measurements, and applying post processing corrections. (US Patent 4973914).

Tthe study quantization error is usually modelled as a uniform random variable and so while carrying on this you actually need to be very precised and accurate!!

However, oversampling can become impractical but, it can be helped by using a technique called noise shaping. This technique decreases noise in the bandwidth and has less effect on the signal.

Just wanted to let you know that you are doing a great job. Your articles are very interesting and informative and I just couldn't leave your website before telling you that. Have a great day.

In practice, oversampling is implemented in order to achieve cheaper higher-resolution A/D and D/A conversion. For instance, to implement a 24-bit converter, it is sufficient to use a 20-bit converter that can run at 256 times the target sampling rate.

Suppose you wanted to create a FIR filter with your own requirements. How would you find the necessary coefficients, and how many of them would you need?

Switched-capacitor filters have a few disadvantages. They exhibit greater sensitivity to noise than their op-amp-based filter siblings, and they have low-amplitude clock-signal artifacts -- clock feedthrough -- on their outputs.

The Machinist Calc Pro computes speeds and feed rates for milling, turning, and drilling: cutting speed, spindle speed, feed rate (inches/minute), cutting feed, etc.

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