DSP Tricks: Reducing A/D Converter Quantization NoiseTo reduce or eliminate the ill effects of quantization noise in analog-to-digital (A/D) converters, . DSP practitioners can use two tricks to reduce converter quantization noise. Those schemes are called oversampling and dithering.
Oversampling. The process of oversampling to reduce A/D converter quantization noise is straightforward. We merely sample an analog signal at an fs sample rate higher than the minimum rate needed to satisfy the Nyquist criterion (twice the analog signal's bandwidth), and then lowpass filter. What could be simpler?
The theory behind oversampling is based on the assumption that an A/D converter's total quantization noise power (variance) is the converter's least significant bit (lsb) value squared over 12, or
The next assumption is: the quantization noise values are truly random, and in the frequency domain the quantization noise has a flat spectrum. (These assumptions are valid if the A/D converter is being driven by an analog signal that covers most of the converter's analog input voltage range, and is not highly periodic.)
Next we consider the notion of quantization noise power spectral density (PSD), a frequency-domain characterization of quantization noise measured in noise power per hertz as shown in Figure 13"17 below. Thus we can consider the idea that quantization noise can be represented as a certain amount of power (watts, if we wish) per unit bandwidth.
In our world of discrete systems, the flat noise spectrum assumption results in the total quantization noise (a fixed value based on the converter's lsb voltage) being distributed equally in the frequency domain, from "fs/2 to + fs/2 as indicated in Figure 13"17 below. The amplitude of this quantization noise PSD is the rectangle area (total quantization noise power) divided by the rectangle width (fs), or
measured in watts/Hz.
|Figure 13"17. Frequency-domain power spectral density of an ideal A/D converter.|
The next question is: "How can we reduce the PSDnoise level defined by Eq. (13"65)?"
We could reduce the lsb value (volts) in the numerator by using an A/D converter with additional bits. That would make the lsb value smaller and certainly reduce PSDnoise, but that's an expensive solution. Extra converter bits cost money. Better yet, let's increase the denominator of Eq. (13"65) by increasing the sample rate fs.
Consider a low-level discrete signal of interest whose spectrum is depicted in Figure 13"18(a) below. By increasing the fs,old sample rate to some larger value fs,new (oversampling), we spread the total noise power (a fixed value) over a wider frequency range as shown in Figure 13"18(b).
The area under the shaded curves in Figure 13"18(a) and 13"18(b) are equal. Next we lowpass filter the converter's output samples. At the output of the filter, the quantization noise level contaminating our signal will be reduced from that at the input of the filter.
The improvement in signal to quantization noise ratio, measured in dB, achieved by oversampling is:
|Figure 13"18. Oversampling example: (a) noise PSD at an fs,old samples rate; (b) noise PSD at the higher fs,new samples rate; (c) processing steps.|
For example: if fs,old = 100 kHz, and fs,new = 400 kHz, the SNRA/D-gain = 10log10(4) = 6.02 dB. Thus oversampling by a factor of 4 (and filtering), we gain a single bit's worth of quantization noise reduction.
Consequently we can achieve N+1-bit performance from an N-bit A/D converter, because we gain signal amplitude resolution at the expense of higher sampling speed.
After digital filtering, we can decimate to the lower fs,old without degrading the improved SNR. Of course, the number of bits used for the lowpass filter's coefficients and registers must exceed the original number of A/D converter bits, or this oversampling scheme doesn't work.
With the use of a digital lowpass filter, depending on the interfering analog noise in x(t), it's possible to use a lower performance (simpler) analog anti-aliasing filter relative to the analog filter necessary at the lower sampling rate.
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